Arima Procedure Ebook p642v

Chapter 7

The ARIMA Procedure Chapter Table of Contents OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 GETTING STARTED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 The Three Stages of ARIMA Modeling . . . . . . . . . . . . . . . . . . . . 194 Identification Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Estimation and Diagnostic Checking Stage . . . . . . . . . . . . . . . . . . . 200 Forecasting Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Using ARIMA Procedure Statements . . . . . . . . . . . . . . . . . . . . . . 205 General Notation for ARIMA Models . . . . . . . . . . . . . . . . . . . . . 206 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Differencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Subset, Seasonal, and Factored ARMA Models . . . . . . . . . . . . . . . . 211 Input Variables and Regression with ARMA Errors . . . . . . . . . . . . . . 213 Intervention Models and Interrupted Time Series . . . . . . . . . . . . . . . 215 Rational Transfer Functions and Distributed Lag Models . . . . . . . . . . . 217 Forecasting with Input Variables . . . . . . . . . . . . . . . . . . . . . . . . 219 Data Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 SYNTAX . . . . . . . . . . Functional Summary . . . PROC ARIMA Statement . BY Statement . . . . . . . IDENTIFY Statement . . . ESTIMATE Statement . . FORECAST Statement . .

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DETAILS . . . . . . . . . . . . . . . . The Inverse Autocorrelation Function The Partial Autocorrelation Function . The Cross-Correlation Function . . . The ESACF Method . . . . . . . . . The MINIC Method . . . . . . . . . . The SCAN Method . . . . . . . . . . Stationarity Tests . . . . . . . . . . . Prewhitening . . . . . . . . . . . . . Identifying Transfer Function Models

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Part 2. General Information Missing Values and Autocorrelations . . . . . . Estimation Details . . . . . . . . . . . . . . . . Specifying Inputs and Transfer Functions . . . Initial Values . . . . . . . . . . . . . . . . . . Stationarity and Invertibility . . . . . . . . . . Naming of Model Parameters . . . . . . . . . . Missing Values and Estimation and Forecasting Forecasting Details . . . . . . . . . . . . . . . Forecasting Log Transformed Data . . . . . . . Specifying Series Periodicity . . . . . . . . . . OUT= Data Set . . . . . . . . . . . . . . . . . OUTCOV= Data Set . . . . . . . . . . . . . . OUTEST= Data Set . . . . . . . . . . . . . . . OUTMODEL= Data Set . . . . . . . . . . . . OUTSTAT= Data Set . . . . . . . . . . . . . . Printed Output . . . . . . . . . . . . . . . . . . ODS Table Names . . . . . . . . . . . . . . .

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EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Example 7.1 Simulated IMA Model . . . . . . . . . . . . . . . . . . . . . . 265 Example 7.2 Seasonal Model for the Airline Series . . . . . . . . . . . . . . 270 Example 7.3 Model for Series J Data from Box and Jenkins . . . . . . . . . . 275 Example 7.4 An Intervention Model for Ozone Data . . . . . . . . . . . . . 287 Example 7.5 Using Diagnostics to Identify ARIMA models . . . . . . . . . . 292 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

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Chapter 7

The ARIMA Procedure Overview The ARIMA procedure analyzes and forecasts equally spaced univariate time series data, transfer function data, and intervention data using the AutoRegressive Integrated Moving-Average (ARIMA) or autoregressive moving-average (ARMA) model. An ARIMA model predicts a value in a response time series as a linear combination of its own past values, past errors (also called shocks or innovations), and current and past values of other time series. The ARIMA approach was first popularized by Box and Jenkins, and ARIMA models are often referred to as Box-Jenkins models. The general transfer function model employed by the ARIMA procedure was discussed by Box and Tiao (1975). When an ARIMA model includes other time series as input variables, the model is sometimes referred to as an ARIMAX model. Pankratz (1991) refers to the ARIMAX model as dynamic regression. The ARIMA procedure provides a comprehensive set of tools for univariate time series model identification, parameter estimation, and forecasting, and it offers great flexibility in the kinds of ARIMA or ARIMAX models that can be analyzed. The ARIMA procedure s seasonal, subset, and factored ARIMA models; intervention or interrupted time series models; multiple regression analysis with ARMA errors; and rational transfer function models of any complexity. The design of PROC ARIMA closely follows the Box-Jenkins strategy for time series modeling with features for the identification, estimation and diagnostic checking, and forecasting steps of the Box-Jenkins method. Before using PROC ARIMA, you should be familiar with Box-Jenkins methods, and you should exercise care and judgment when using the ARIMA procedure. The ARIMA class of time series models is complex and powerful, and some degree of expertise is needed to use them correctly. If you are unfamiliar with the principles of ARIMA modeling, refer to textbooks on time series analysis. Also refer to SAS/ETS Software: Applications Guide 1, Version 6, First Edition. You might consider attending the SAS Training Course "Forecasting Techniques Using SAS/ETS Software." This course provides in-depth training on ARIMA modeling using PROC ARIMA, as well as training on the use of other forecasting tools available in SAS/ETS software.

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Part 2. General Information

Getting Started This section outlines the use of the ARIMA procedure and gives a cursory description of the ARIMA modeling process for readers less familiar with these methods.

The Three Stages of ARIMA Modeling The analysis performed by PROC ARIMA is divided into three stages, corresponding to the stages described by Box and Jenkins (1976). The IDENTIFY, ESTIMATE, and FORECAST statements perform these three stages, which are summarized below. 1. In the identification stage, you use the IDENTIFY statement to specify the response series and identify candidate ARIMA models for it. The IDENTIFY statement reads time series that are to be used in later statements, possibly differencing them, and computes autocorrelations, inverse autocorrelations, partial autocorrelations, and cross correlations. Stationarity tests can be performed to determine if differencing is necessary. The analysis of the IDENTIFY statement output usually suggests one or more ARIMA models that could be fit. Options allow you to test for stationarity and tentative ARMA order identification. 2. In the estimation and diagnostic checking stage, you use the ESTIMATE statement to specify the ARIMA model to fit to the variable specified in the previous IDENTIFY statement, and to estimate the parameters of that model. The ESTIMATE statement also produces diagnostic statistics to help you judge the adequacy of the model. Significance tests for parameter estimates indicate whether some in the model may be unnecessary. Goodness-of-fit statistics aid in comparing this model to others. Tests for white noise residuals indicate whether the residual series contains additional information that might be utilized by a more complex model. If the diagnostic tests indicate problems with the model, you try another model, then repeat the estimation and diagnostic checking stage. 3. In the forecasting stage you use the FORECAST statement to forecast future values of the time series and to generate confidence intervals for these forecasts from the ARIMA model produced by the preceding ESTIMATE statement. These three steps are explained further and illustrated through an extended example in the following sections.

Identification Stage Suppose you have a variable called SALES that you want to forecast. The following example illustrates ARIMA modeling and forecasting using a simulated data set TEST containing a time series SALES generated by an ARIMA(1,1,1) model. The output produced by this example is explained in the following sections. The simulated SALES series is shown in Figure 7.1. 194 SAS OnlineDoc : Version 8

Chapter 7. Getting Started

Figure 7.1.

Simulated ARIMA(1,1,1) Series SALES

Using the IDENTIFY Statement You first specify the input data set in the PROC ARIMA statement. Then, you use an IDENTIFY statement to read in the SALES series and plot its autocorrelation function. You do this using the following statements: proc arima data=test; identify var=sales nlag=8; run; Descriptive Statistics

The IDENTIFY statement first prints descriptive statistics for the SALES series. This part of the IDENTIFY statement output is shown in Figure 7.2. The ARIMA Procedure Name of Variable = sales Mean of Working Series Standard Deviation Number of Observations

Figure 7.2.

137.3662 17.36385 100

IDENTIFY Statement Descriptive Statistics Output

Autocorrelation Function Plots

The IDENTIFY statement next prints three plots of the correlations of the series with its past values at different lags. These are the sample autocorrelation function plot 195 SAS OnlineDoc : Version 8

Part 2. General Information sample partial autocorrelation function plot sample inverse autocorrelation function plot The sample autocorrelation function plot output of the IDENTIFY statement is shown in Figure 7.3. The ARIMA Procedure Autocorrelations Lag

Covariance

Correlation

0 1 2 3 4 5 6 7 8

301.503 288.454 273.437 256.787 238.518 219.033 198.617 177.150 154.914

1.00000 0.95672 0.90691 0.85169 0.79110 0.72647 0.65876 0.58755 0.51381

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | |

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|********************| |******************* | |****************** | |***************** | |**************** | |*************** | |************* | |************ | |********** . |

"." marks two standard errors

Figure 7.3.

IDENTIFY Statement Autocorrelations Plot

The autocorrelation plot shows how values of the series are correlated with past values of the series. For example, the value 0.95672 in the "Correlation" column for the Lag 1 row of the plot means that the correlation between SALES and the SALES value for the previous period is .95672. The rows of asterisks show the correlation values graphically. These plots are called autocorrelation functions because they show the degree of correlation with past values of the series as a function of the number of periods in the past (that is, the lag) at which the correlation is computed. The NLAG= option controls the number of lags for which autocorrelations are shown. By default, the autocorrelation functions are plotted to lag 24; in this example the NLAG=8 option is used, so only the first 8 lags are shown. Most books on time series analysis explain how to interpret autocorrelation plots and partial autocorrelation plots. See the section "The Inverse Autocorrelation Function" later in this chapter for a discussion of inverse autocorrelation plots. By examining these plots, you can judge whether the series is stationary or nonstationary. In this case, a visual inspection of the autocorrelation function plot indicates that the SALES series is nonstationary, since the ACF decays very slowly. For more formal stationarity tests, use the STATIONARITY= option. (See the section "Stationarity" later in this chapter.) The inverse and partial autocorrelation plots are printed after the autocorrelation plot. These plots have the same form as the autocorrelation plots, but display inverse and partial autocorrelation values instead of autocorrelations and autocovariances. The partial and inverse autocorrelation plots are not shown in this example.


Chapter 7. Getting Started White Noise Test

The last part of the default IDENTIFY statement output is the check for white noise. This is an approximate statistical test of the hypothesis that none of the autocorrelations of the series up to a given lag are significantly different from 0. If this is true for all lags, then there is no information in the series to model, and no ARIMA model is needed for the series. The autocorrelations are checked in groups of 6, and the number of lags checked depends on the NLAG= option. The check for white noise output is shown in Figure 7.4. The ARIMA Procedure Autocorrelation Check for White Noise To Lag

ChiSquare

DF

Pr > ChiSq

6

426.44

6

<.0001

Figure 7.4.

---------------Autocorrelations--------------0.957

0.907

0.852

0.791

0.726

0.659

IDENTIFY Statement Check for White Noise

In this case, the white noise hypothesis is rejected very strongly, which is expected since the series is nonstationary. The p value for the test of the first six autocorrelations is printed as <0.0001, which means the p value is less than .0001. Identification of the Differenced Series Since the series is nonstationary, the next step is to transform it to a stationary series by differencing. That is, instead of modeling the SALES series itself, you model the change in SALES from one period to the next. To difference the SALES series, use another IDENTIFY statement and specify that the first difference of SALES be analyzed, as shown in the following statements: identify var=sales(1) nlag=8; run;

The second IDENTIFY statement produces the same information as the first but for the change in SALES from one period to the next rather than for the total sales in each period. The summary statistics output from this IDENTIFY statement is shown in Figure 7.5. Note that the period of differencing is given as 1, and one observation was lost through the differencing operation. The ARIMA Procedure Name of Variable = sales Period(s) of Differencing Mean of Working Series Standard Deviation Number of Observations Observation(s) eliminated by differencing

Figure 7.5.

1 0.660589 2.011543 99 1

IDENTIFY Statement Output for Differenced Series

The autocorrelation plot for the differenced series is shown in Figure 7.6. 197 SAS OnlineDoc : Version 8

Part 2. General Information The ARIMA Procedure Autocorrelations Lag

Covariance

Correlation

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1

0 1 2 3 4 5 6 7 8

4.046306 3.351258 2.390895 1.838925 1.494253 1.135753 0.801319 0.610543 0.326495

1.00000 0.82823 0.59088 0.45447 0.36929 0.28069 0.19804 0.15089 0.08069

| | | | | | | | |

. . . . . . . .

|********************| |***************** | |************ | |********* | |*******. | |****** . | |**** . | |*** . | |** . |


Figure 7.6.

Autocorrelations Plot for Change in SALES

The autocorrelations decrease rapidly in this plot, indicating that the change in SALES is a stationary time series. The next step in the Box-Jenkins methodology is to examine the patterns in the autocorrelation plot to choose candidate ARMA models to the series. The partial and inverse autocorrelation function plots are also useful aids in identifying appropriate ARMA models for the series. The partial and inverse autocorrelation function plots are shown in Figure 7.7 and Figure 7.8. The ARIMA Procedure Inverse Autocorrelations Lag

Correlation

1 2 3 4 5 6 7 8

Figure 7.7.

-0.73867 0.36801 -0.17538 0.11431 -0.15561 0.18899 -0.15342 0.05952

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | |

***************| . . |******* ****| . . |** . .***| . . |**** .***| . . |* .

| | | | | | | |

Inverse Autocorrelation Function Plot for Change in SALES


Chapter 7. Getting Started The ARIMA Procedure Partial Autocorrelations Lag

Correlation

1 2 3 4 5 6 7 8

0.82823 -0.30275 0.23722 -0.07450 -0.02654 -0.01012 0.04189 -0.17668

Figure 7.8.

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | |

. |***************** ******| . . |***** . *| . . *| . . | . . |* . ****| .

| | | | | | | |

Partial Autocorrelation Plot for Change in SALES

In the usual Box and Jenkins approach to ARIMA modeling, the sample autocorrelation function, inverse autocorrelation function, and partial autocorrelation function are compared with the theoretical correlation functions expected from different kinds of ARMA models. This matching of theoretical autocorrelation functions of different ARMA models to the sample autocorrelation functions computed from the response series is the heart of the identification stage of Box-Jenkins modeling. Most textbooks on time series analysis discuss the theoretical autocorrelation functions for different kinds of ARMA models. Since the input data is only a limited sample of the series, the sample autocorrelation functions computed from the input series will only approximate the true autocorrelation functions of the process generating the series. This means that the sample autocorrelation functions will not exactly match the theoretical autocorrelation functions for any ARMA model and may have a pattern similar to that of several different ARMA models. If the series is white noise (a purely random process), then there is no need to fit a model. The check for white noise, shown in Figure 7.9, indicates that the change in sales is highly autocorrelated. Thus, an autocorrelation model, for example an AR(1) model, might be a good candidate model to fit to this process. The ARIMA Procedure Autocorrelation Check for White Noise To Lag

ChiSquare

DF

Pr > ChiSq

6

154.44

6

<.0001

Figure 7.9.


0.591

0.454

0.369

0.281

0.198

IDENTIFY Statement Check for White Noise



Estimation and Diagnostic Checking Stage The autocorrelation plots for this series, as shown in the previous section, suggest an AR(1) model for the change in SALES. You should check the diagnostic statistics to see if the AR(1) model is adequate. Other candidate models include an MA(1) model, and low-order mixed ARMA models. In this example, the AR(1) model is tried first. Estimating an AR(1) Model The following statements fit an AR(1) model (an autoregressive model of order 1), which predicts the change in sales as an average change, plus some fraction of the previous change, plus a random error. To estimate an AR model, you specify the order of the autoregressive model with the P= option on an ESTIMATE statement, as shown in the following statements: estimate p=1; run;

The ESTIMATE statement fits the model to the data and prints parameter estimates and various diagnostic statistics that indicate how well the model fits the data. The first part of the ESTIMATE statement output, the table of parameter estimates, is shown in Figure 7.10. The ARIMA Procedure Conditional Least Squares Estimation

Parameter MU AR1,1

Figure 7.10.

Estimate

Approx Std Error

t Value

Pr > |t|

Lag

0.90280 0.86847

0.65984 0.05485

1.37 15.83

0.1744 <.0001

0 1

Parameter Estimates for AR(1) Model

The table of parameter estimates is titled "Condidtional Least Squares Estimation," which indicates the estimation method used. You can request different estimation methods with the METHOD= option. The table of parameter estimates lists the parameters in the model; for each parameter, the table shows the estimated value and the standard error and t value for the estimate. The table also indicates the lag at which the parameter appears in the model. In this case, there are two parameters in the model. The mean term is labeled MU; its estimated value is .90280. The autoregressive parameter is labeled AR1,1; this is the coefficient of the lagged value of the change in SALES, and its estimate is .86847. The t values provide significance tests for the parameter estimates and indicate whether some in the model may be unnecessary. In this case, the t value for the autoregressive parameter is 15.83, so this term is highly significant. The t value for MU indicates that the mean term adds little to the model.


Chapter 7. Getting Started The standard error estimates are based on large sample theory. Thus, the standard errors are labeled as approximate, and the standard errors and t values may not be reliable in small samples. The next part of the ESTIMATE statement output is a table of goodness-of-fit statistics, which aid in comparing this model to other models. This output is shown in Figure 7.11. The ARIMA Procedure Constant Estimate 0.118749 Variance Estimate 1.15794 Std Error Estimate 1.076076 AIC 297.4469 SBC 302.6372 Number of Residuals 99 * AIC and SBC do not include log determinant.

Figure 7.11.

Goodness-of-Fit Statistics for AR(1) Model

The "Constant Estimate" is a function of the mean term MU and the autoregressive parameters. This estimate is computed only for AR or ARMA models, but not for strictly MA models. See the section "General Notation for ARIMA Models" later in this chapter for an explanation of the constant estimate. The "Variance Estimate" is the variance of the residual series, which estimates the innovation variance. The item labeled "Std Error Estimate" is the square root of the variance estimate. In general, when comparing candidate models, smaller AIC and SBC statistics indicate the better fitting model. The section "Estimation Details" later in this chapter explains the AIC and SBC statistics. The ESTIMATE statement next prints a table of correlations of the parameter estimates, as shown in Figure 7.12. This table can help you assess the extent to which collinearity may have influenced the results. If two parameter estimates are very highly correlated, you might consider dropping one of them from the model. The ARIMA Procedure Correlations of Parameter Estimates Parameter MU AR1,1

Figure 7.12.

MU

AR1,1

1.000 0.114

0.114 1.000

Correlations of the Estimates for AR(1) Model

The next part of the ESTIMATE statement output is a check of the autocorrelations of the residuals. This output has the same form as the autocorrelation check for white noise that the IDENTIFY statement prints for the response series. The autocorrelation check of residuals is shown in Figure 7.13.


Part 2. General Information The ARIMA Procedure Autocorrelation Check of Residuals To Lag

ChiSquare

DF

Pr > ChiSq

6 12 18 24

19.09 22.90 31.63 32.83

5 11 17 23

0.0019 0.0183 0.0167 0.0841

Figure 7.13.

---------------Autocorrelations--------------0.327 0.072 -0.233 0.009

-0.220 0.116 -0.129 -0.057

-0.128 -0.042 -0.024 -0.057

0.068 -0.066 0.056 -0.001

-0.002 0.031 -0.014 0.049

-0.096 -0.091 -0.008 -0.015

Check for White Noise Residuals for AR(1) Model

The test statistics for the residuals series indicate whether the residuals are uncorrelated (white noise) or contain additional information that might be utilized by a more complex model. In this case, the test statistics reject the no-autocorrelation hypothesis at a high level of significance. (p=0.0019 for the first six lags.) This means that the residuals are not white noise, and so the AR(1) model is not a fully adequate model for this series. The final part of the ESTIMATE statement output is a listing of the estimated model using the back shift notation. This output is shown in Figure 7.14. The ARIMA Procedure Model for variable sales Estimated Mean Period(s) of Differencing

0.902799 1

Autoregressive Factors Factor 1:

Figure 7.14.

1 - 0.86847 B**(1)

Estimated ARIMA(1,1,0) Model for SALES

This listing combines the differencing specification given in the IDENTIFY statement with the parameter estimates of the model for the change in sales. Since the AR(1) model is for the change in sales, the final model for sales is an ARIMA(1,1,0) model. Using , the back shift operator, the mathematical form of the estimated model shown in this output is as follows:

See the section "General Notation for ARIMA Model" later in this chapter for further explanation of this notation. Estimating an ARMA(1,1) Model The IDENTIFY statement plots suggest a mixed autoregressive and moving average model, and the previous ESTIMATE statement check of residuals indicates that an AR(1) model is not sufficient. You now try estimating an ARMA(1,1) model for the change in SALES. 202 SAS OnlineDoc : Version 8

Chapter 7. Getting Started An ARMA(1,1) model predicts the change in SALES as an average change, plus some fraction of the previous change, plus a random error, plus some fraction of the random error in the preceding period. An ARMA(1,1) model for the change in sales is the same as an ARIMA(1,1,1) model for the level of sales. To estimate a mixed autoregressive moving average model, you specify the order of the moving average part of the model with the Q= option on an ESTIMATE statement in addition to specifying the order of the autoregressive part with the P= option. The following statements fit an ARMA(1,1) model to the differenced SALES series: estimate p=1 q=1; run;

The parameter estimates table and goodness-of-fit statistics for this model are shown in Figure 7.15. The ARIMA Procedure Conditional Least Squares Estimation Parameter

Estimate

Approx Std Error

t Value

Pr > |t|

Lag

MU MA1,1 AR1,1

0.89288 -0.58935 0.74755

0.49391 0.08988 0.07785

1.81 -6.56 9.60

0.0738 <.0001 <.0001

0 1 1

Constant Estimate 0.225409 Variance Estimate 0.904034 Std Error Estimate 0.950807 AIC 273.9155 SBC 281.7009 Number of Residuals 99 * AIC and SBC do not include log determinant.

Figure 7.15.

Estimated ARMA(1,1) Model for Change in SALES

The moving average parameter estimate, labeled "MA1,1", is . Both the moving average and the autoregressive parameters have significant t values. Note that the variance estimate, AIC, and SBC are all smaller than they were for the AR(1) model, indicating that the ARMA(1,1) model fits the data better without overparameterizing. The check for white noise residuals is shown in Figure 7.16. The tests show that we cannot reject the hypothesis that the residuals are uncorrelated. Thus, you conclude that the ARMA(1,1) model is adequate for the change in sales series, and there is no point in trying more complex models.


Part 2. General Information The ARIMA Procedure Autocorrelation Check of Residuals To Lag

ChiSquare

DF

Pr > ChiSq

6 12 18 24

3.95 7.03 15.41 16.96

4 10 16 22

0.4127 0.7227 0.4951 0.7657

Figure 7.16.

---------------Autocorrelations--------------0.016 0.088 -0.221 0.011

-0.044 0.087 -0.033 -0.066

-0.068 -0.037 -0.092 -0.022

0.145 -0.075 0.086 -0.032

0.024 0.051 -0.074 0.062

-0.094 -0.053 -0.005 -0.047

Check for White Noise Residuals for ARMA(1,1) Model

The output showing the form of the estimated ARIMA(1,1,1) model for SALES is shown in Figure 7.17. The ARIMA Procedure Model for variable sales Estimated Mean Period(s) of Differencing

0.892875 1


1 - 0.74755 B**(1)

Moving Average Factors Factor 1:

Figure 7.17.

1 + 0.58935 B**(1)


The estimated model shown in this output is

Since the model diagnostic tests show that all the parameter estimates are significant and the residual series is white noise, the estimation and diagnostic checking stage is complete. You can now proceed to forecasting the SALES series with this ARIMA(1,1,1) model.



Forecasting Stage To produce the forecast, use a FORECAST statement after the ESTIMATE statement for the model you decide is best. If the last model fit were not the best, then repeat the ESTIMATE statement for the best model before using the FORECAST statement. Suppose that the SALES series is monthly, that you wish to forecast one year ahead from the most recently available sales figure, and that the dates for the observations are given by a variable DATE in the input data set TEST. You use the following FORECAST statement: forecast lead=12 interval=month id=date out=results; run;

The LEAD= option specifies how many periods ahead to forecast (12 months, in this case). The ID= option specifies the ID variable used to date the observations of the SALES time series. The INTERVAL= option indicates that data are monthly and enables PROC ARIMA to extrapolate DATE values for forecast periods. The OUT= option writes the forecasts to an output data set RESULTS. See the section "OUT= Data Set" later in this chapter for information on the contents of the output data set. By default, the FORECAST statement also prints the forecast values, as shown in Figure 7.18. This output shows for each forecast period the observation number, forecast value, standard error estimate for the forecast value, and lower and upper limits for a 95% confidence interval for the forecast. The ARIMA Procedure Forecasts for variable sales Obs

Forecast

Std Error

101 102 103 104 105 106 107 108 109 110 111 112

171.0320 174.7534 177.7608 180.2343 182.3088 184.0850 185.6382 187.0247 188.2866 189.4553 190.5544 191.6014

0.9508 2.4168 3.9879 5.5658 7.1033 8.5789 9.9841 11.3173 12.5807 13.7784 14.9153 15.9964

Figure 7.18.

95% Confidence Limits 169.1684 170.0165 169.9445 169.3256 168.3866 167.2707 166.0698 164.8433 163.6289 162.4501 161.3209 160.2491

172.8955 179.4903 185.5770 191.1430 196.2310 200.8993 205.2066 209.2061 212.9443 216.4605 219.7879 222.9538


Normally, you want the forecast values stored in an output data set, and you are not interested in seeing this printed list of the forecast. You can use the NOPRINT option on the FORECAST statement to suppress this output.

Using ARIMA Procedure Statements The IDENTIFY, ESTIMATE, and FORECAST statements are related in a hierarchy. An IDENTIFY statement brings in a time series to be modeled; several 205 SAS OnlineDoc : Version 8

Part 2. General Information ESTIMATE


Chapter 7. Getting Started statements can follow to estimate different ARIMA models for the series; for each model estimated, several FORECAST statements can be used. Thus, a FORECAST statement must be preceded at some point by an ESTIMATE statement, and an ESTIMATE statement must be preceded at some point by an IDENTIFY statement. Additional IDENTIFY statements can be used to switch to modeling a different response series or to change the degree of differencing used. The ARIMA procedure can be used interactively in the sense that all ARIMA procedure statements can be executed any number of times without reinvoking PROC ARIMA. You can execute ARIMA procedure statements singly or in groups by following the single statement or group of statements with a RUN statement. The output for each statement or group of statements is produced when the RUN statement is entered. A RUN statement does not terminate the PROC ARIMA step but tells the procedure to execute the statements given so far. You can end PROC ARIMA by submitting a QUIT statement, a DATA step, another PROC step, or an ENDSAS statement. The example in the preceding section illustrates the interactive use of ARIMA procedure statements. The complete PROC ARIMA program for that example is as follows: proc arima data=test; identify var=sales nlag=8; run; identify var=sales(1) nlag=8; run; estimate p=1; run; estimate p=1 q=1; run; forecast lead=12 interval=month id=date out=results; run; quit;

General Notation for ARIMA Models ARIMA is an acronym for AutoRegressive Integrated Moving-Average. The order of an ARIMA model is usually denoted by the notation ARIMA(p,d,q), where p

is the order of the autoregressive part

d

is the order of the differencing

q

is the order of the moving-average process

If no differencing is done (d = 0), the models are usually referred to as ARMA(p,q) models. The final model in the preceding example is an ARIMA(1,1,1) model since the IDENTIFY statement specified d = 1, and the final ESTIMATE statement specified p = 1 and q = 1.


Part 2. General Information Notation for Pure ARIMA Models Mathematically the pure ARIMA model is written as

where t

indexes time is the response series

or a difference of the response series

is the mean term is the backshift operator; that is, is the autoregressive operator, represented as a polynomial in the back shift operator: is the moving-average operator, represented as a polynomial in the back shift operator: is the independent disturbance, also called the random error. The series is computed by the IDENTIFY statement and is the series processed by the ESTIMATE statement. Thus, is either the response series Y or a difference of specified by the differencing operators in the IDENTIFY statement. For simple (nonseasonal) differencing, . For seasonal differencing , where d is the degree of nonseasonal differencing, D is the degree of seasonal differencing, and s is the length of the seasonal cycle. For example, the mathematical form of the ARIMA(1,1,1) model estimated in the preceding example is

Model Constant Term The ARIMA model can also be written as

or

where


Chapter 7. Getting Started Thus, when an autoregressive operator and a mean term are both included in the model, the constant term for the model can be represented as . This value is printed with the label "Constant Estimate" in the ESTIMATE statement output. Notation for Transfer Function Models The general ARIMA model with input series, also called the ARIMAX model, is written as

where is the ith input time series or a difference of the ith input series at time t is the pure time delay for the effect of the ith input series is the numerator polynomial of the transfer function for the ith input series is the denominator polynomial of the transfer function for the ith input series. The model can also be written more compactly as

where is the transfer function weights for the ith input series modeled as a ratio of the and polynomials: is the noise series: This model expresses the response series as a combination of past values of the random shocks and past values of other input series. The response series is also called the dependent series or output series. An input time series is also referred to as an independent series or a predictor series. Response variable, dependent variable, independent variable, or predictor variable are other often used. Notation for Factored Models ARIMA models are sometimes expressed in a factored form. This means that the , , , or polynomials are expressed as products of simpler polynomials. For example, we could express the pure ARIMA model as

where

and

. 209 SAS OnlineDoc : Version 8

Part 2. General Information When an ARIMA model is expressed in factored form, the order of the model is usually expressed using a factored notation also. The order of an ARIMA model expressed as the product of two factors is denoted as ARIMA(p,d,q) (P,D,Q). Notation for Seasonal Models ARIMA models for time series with regular seasonal fluctuations often use differencing operators and autoregressive and moving average parameters at lags that are multiples of the length of the seasonal cycle. When all the in an ARIMA model factor refer to lags that are a multiple of a constant s, the constant is factored out and suffixed to the ARIMA(p,d,q) notation. Thus, the general notation for the order of a seasonal ARIMA model with both seasonal and nonseasonal factors is ARIMA(p,d,q) (P,D,Q) . The term (p,d,q) gives the order of the nonseasonal part of the ARIMA model; the term (P,D,Q) gives the order of the seasonal part. The value of s is the number of observations in a seasonal cycle: 12 for monthly series, 4 for quarterly series, 7 for daily series with day-of-week effects, and so forth. For example, the notation ARIMA(0,1,2) (0,1,1) describes a seasonal ARIMA model for monthly data with the following mathematical form:

Stationarity The noise (or residual) series for an ARMA model must be stationary, which means that both the expected values of the series and its autocovariance function are independent of time. The standard way to check for nonstationarity is to plot the series and its autocorrelation function. You can visually examine a graph of the series over time to see if it has a visible trend or if its variability changes noticeably over time. If the series is nonstationary, its autocorrelation function will usually decay slowly. Another way of checking for stationarity is to use the stationarity tests described in the section “Stationarity Tests” on page 241. Most time series are nonstationary and must be transformed to a stationary series before the ARIMA modeling process can proceed. If the series has a nonstationary variance, taking the log of the series may help. You can compute the log values in a DATA step and then analyze the log values with PROC ARIMA. If the series has a trend over time, seasonality, or some other nonstationary pattern, the usual solution is to take the difference of the series from one period to the next and then analyze this differenced series. Sometimes a series may need to be differenced more than once or differenced at lags greater than one period. (If the trend or seasonal effects are very regular, the introduction of explanatory variables may be an appropriate alternative to differencing.)



Differencing Differencing of the response series is specified with the VAR= option of the IDENTIFY statement by placing a list of differencing periods in parentheses after the variable name. For example, to take a simple first difference of the series SALES, use the statement identify var=sales(1);

In this example, the change in SALES from one period to the next will be analyzed. A deterministic seasonal pattern will also cause the series to be nonstationary, since the expected value of the series will not be the same for all time periods but will be higher or lower depending on the season. When the series has a seasonal pattern, you may want to difference the series at a lag corresponding to the length of the cycle of seasons. For example, if SALES is a monthly series, the statement identify var=sales(12);

takes a seasonal difference of SALES, so that the series analyzed is the change in SALES from its value in the same month one year ago. To take a second difference, add another differencing period to the list. For example, the following statement takes the second difference of SALES: identify var=sales(1,1);

That is, SALES is differenced once at lag 1 and then differenced again, also at lag 1. The statement identify var=sales(2);

creates a 2-span difference, that is current period sales minus sales from two periods ago. The statement identify var=sales(1,12);

takes a second-order difference of SALES, so that the series analyzed is the difference between the current period-to-period change in SALES and the change 12 periods ago. You might want to do this if the series had both a trend over time and a seasonal pattern. There is no limit to the order of differencing and the degree of lagging for each difference.


Part 2. General Information Differencing not only affects the series used for the IDENTIFY statement output but also applies to any following ESTIMATE and FORECAST statements. ESTIMATE statements fit ARMA models to the differenced series. FORECAST statements forecast the differences and automatically sum these differences back to undo the differencing operation specified by the IDENTIFY statement, thus producing the final forecast result. Differencing of input series is specified by the CROSSCORR= option and works just like differencing of the response series. For example, the statement identify var=y(1) crosscorr=(x1(1) x2(1));

takes the first difference of Y, the first difference of X1, and the first difference of X2. Whenever X1 and X2 are used in INPUT= options in following ESTIMATE statements, these names refer to the differenced series.

Subset, Seasonal, and Factored ARMA Models The simplest way to specify an ARMA model is to give the order of the AR and MA parts with the P= and Q= options. When you do this, the model has parameters for the AR and MA parts for all lags through the order specified. However, you can control the form of the ARIMA model exactly as shown in the following section. Subset Models You can control which lags have parameters by specifying the P= or Q= option as a list of lags in parentheses. A model like this that includes parameters for only some lags is sometimes called a subset or additive model. For example, consider the following two ESTIMATE statements: identify var=sales; estimate p=4; estimate p=(1 4);

Both specify AR(4) models, but the first has parameters for lags 1, 2, 3, and 4, while the second has parameters for lags 1 and 4, with the coefficients for lags 2 and 3 constrained to 0. The mathematical form of the autoregressive models produced by these two specifications is shown in Table 7.1. Table 7.1.

Option P=4 P=(1 4)

Saturated versus Subset Models

Autoregressive Operator


Chapter 7. Getting Started Seasonal Models One particularly useful kind of subset model is a seasonal model. When the response series has a seasonal pattern, the values of the series at the same time of year in previous years may be important for modeling the series. For example, if the series SALES is observed monthly, the statements identify var=sales; estimate p=(12);

model SALES as an average value plus some fraction of its deviation from this average value a year ago, plus a random error. Although this is an AR(12) model, it has only one autoregressive parameter. Factored Models A factored model (also referred to as a multiplicative model) represents the ARIMA model as a product of simpler ARIMA models. For example, you might model SALES as a combination of an AR(1) process reflecting short term dependencies and an AR(12) model reflecting the seasonal pattern. It might seem that the way to do this is with the option P=(1 12), but the AR(1) process also operates in past years; you really need autoregressive parameters at lags 1, 12, and 13. You can specify a subset model with separate parameters at these lags, or you can specify a factored model that represents the model as the product of an AR(1) model and an AR(12) model. Consider the following two ESTIMATE statements: identify var=sales; estimate p=(1 12 13); estimate p=(1)(12);

The mathematical form of the autoregressive models produced by these two specifications are shown in Table 7.2. Table 7.2.

Subset versus Factored Models

Option P=(1 12 13) P=(1)(12)

Autoregressive Operator

Both models fit by these two ESTIMATE statements predict SALES from its values 1, 12, and 13 periods ago, but they use different parameterizations. The first model has three parameters, whose meanings may be hard to interpret. The factored specification P=(1)(12) represents the model as the product of two different AR models. It has only two parameters: one that corresponds to recent effects and one that represents seasonal effects. Thus the factored model is more parsimonious, and its parameter estimates are more clearly interpretable.



Input Variables and Regression with ARMA Errors In addition to past values of the response series and past errors, you can also model the response series using the current and past values of other series, called input series. Several different names are used to describe ARIMA models with input series. Transfer function model, intervention model, interrupted time series model, regression model with ARMA errors, Box-Tiao model, and ARIMAX model are all different names for ARIMA models with input series. Pankratz (1991) refers to these models as dynamic regression. Using Input Series To use input series, list the input series in a CROSSCORR= option on the IDENTIFY statement and specify how they enter the model with an INPUT= option on the ESTIMATE statement. For example, you might use a series called PRICE to help model SALES, as shown in the following statements: proc arima data=a; identify var=sales crosscorr=price; estimate input=price; run;

This example performs a simple linear regression of SALES on PRICE, producing the same results as PROC REG or another SAS regression procedure. The mathematical form of the model estimated by these statements is

The parameter estimates table for this example (using simulated data) is shown in Figure 7.19. The intercept parameter is labeled MU. The regression coefficient for PRICE is labeled NUM1. (See the section "Naming of Model Parameters" later in this chapter for information on how parameters for input series are named.) The ARIMA Procedure Conditional Least Squares Estimation

Parameter MU NUM1

Estimate

Approx Std Error

t Value

Pr > |t|

Lag

199.83602 -9.99299

2.99463 0.02885

66.73 -346.38

<.0001 <.0001

0 0

Figure 7.19.

Variable sales price

Shift 0 0

Parameter Estimates Table for Regression Model

Any number of input variables can be used in a model. For example, the following statements fit a multiple regression of SALES on PRICE and INCOME: proc arima data=a; identify var=sales crosscorr=(price income); estimate input=(price income); run;


Chapter 7. Getting Started The mathematical form of the regression model estimated by these statements is

Lagging and Differencing Input Series You can also difference and lag the input series. For example, the following statements regress the change in SALES on the change in PRICE lagged by one period. The difference of PRICE is specified with the CROSSCORR= option and the lag of the change in PRICE is specified by the 1 $ in the INPUT= option. proc arima data=a; identify var=sales(1) crosscorr=price(1); estimate input=( 1 $ price ); run;

These statements estimate the model

Regression with ARMA Errors You can combine input series with ARMA models for the errors. For example, the following statements regress SALES on INCOME and PRICE but with the error term of the regression model (called the noise series in ARIMA modeling terminology) assumed to be an ARMA(1,1) process. proc arima data=a; identify var=sales crosscorr=(price income); estimate p=1 q=1 input=(price income); run;


Stationarity and Input Series Note that the requirement of stationarity applies to the noise series. If there are no input variables, the response series (after differencing and minus the mean term) and the noise series are the same. However, if there are inputs, the noise series is the residual after the effect of the inputs is removed. There is no requirement that the input series be stationary. If the inputs are nonstationary, the response series will be nonstationary, even though the noise process may be stationary. When nonstationary input series are used, you can fit the input variables first with no ARMA model for the errors and then consider the stationarity of the residuals before identifying an ARMA model for the noise part. 215 SAS OnlineDoc : Version 8

Part 2. General Information Identifying Regression Models with ARMA Errors Previous sections described the ARIMA modeling identification process using the autocorrelation function plots produced by the IDENTIFY statement. This identification process does not apply when the response series depends on input variables. This is because it is the noise process for which we need to identify an ARIMA model, and when input series are involved the response series adjusted for the mean is no longer an estimate of the noise series. However, if the input series are independent of the noise series, you can use the residuals from the regression model as an estimate of the noise series, then apply the ARIMA modeling identification process to this residual series. This assumes that the noise process is stationary. The PLOT option on the ESTIMATE statement produces for the model residuals the same plots as the IDENTIFY statement produces for the response series. The PLOT option prints an autocorrelation function plot, an inverse autocorrelation function plot, and a partial autocorrelation function plot for the residual series. The following statements show how the PLOT option is used to identify the ARMA(1,1) model for the noise process used in the preceding example of regression with ARMA errors: proc arima data=a; identify var=sales crosscorr=(price income) noprint; estimate input=(price income) plot; run; estimate p=1 q=1 input=(price income) plot; run;

In this example, the IDENTIFY statement includes the NOPRINT option since the autocorrelation plots for the response series are not useful when you know that the response series depends on input series. The first ESTIMATE statement fits the regression model with no model for the noise process. The PLOT option produces plots of the autocorrelation function, inverse autocorrelation function, and partial autocorrelation function for the residual series of the regression on PRICE and INCOME. By examining the PLOT option output for the residual series, you that the residual series is stationary and identify an ARMA(1,1) model for the noise process. The second ESTIMATE statement fits the final model. Although this discussion addresses regression models, the same remarks apply to identifying an ARIMA model for the noise process in models that include input series with complex transfer functions.

Intervention Models and Interrupted Time Series One special kind of ARIMA model with input series is called an intervention model or interrupted time series model. In an intervention model, the input series is an indicator variable containing discrete values that flag the occurrence of an event affecting the response series. This event is an intervention in or an interruption of the normal 216 SAS OnlineDoc : Version 8

Chapter 7. Getting Started evolution of the response time series, which, in the absence of the intervention, is usually assumed to be a pure ARIMA process. Intervention models can be used both to model and forecast the response series and to analyze the impact of the intervention. When the focus is on estimating the effect of the intervention, the process is often called intervention analysis or interrupted time series analysis. Impulse Interventions The intervention can be a one-time event. For example, you might want to study the effect of a short-term advertising campaign on the sales of a product. In this case, the input variable has the value of 1 for the period during which the advertising campaign took place and the value 0 for all other periods. Intervention variables of this kind are sometimes called impulse functions or pulse functions. Suppose that SALES is a monthly series, and a special advertising effort was made during the month of March 1992. The following statements estimate the effect of this intervention assuming an ARMA(1,1) model for SALES. The model is specified just like the regression model, but the intervention variable AD is constructed in the DATA step as a zero-one indicator for the month of the advertising effort. data a; set a; ad = date = ’1mar1992’d; run; proc arima data=a; identify var=sales crosscorr=ad; estimate p=1 q=1 input=ad; run;

Continuing Interventions Other interventions can be continuing, in which case the input variable flags periods before and after the intervention. For example, you might want to study the effect of a change in tax rates on some economic measure. Another example is a study of the effect of a change in speed limits on the rate of traffic fatalities. In this case, the input variable has the value 1 after the new speed limit went into effect and the value 0 before. Intervention variables of this kind are called step functions. Another example is the effect of news on product demand. Suppose it was reported in July 1996 that consumption of the product prevents heart disease (or causes cancer), and SALES is consistently higher (or lower) thereafter. The following statements model the effect of this news intervention: data a; set a; news = date >= ’1jul1996’d; run; proc arima data=a; identify var=sales crosscorr=news; estimate p=1 q=1 input=news; run;


Part 2. General Information Interaction Effects You can include any number of intervention variables in the model. Intervention variables can have any pattern–impulse and continuing interventions are just two possible cases. You can mix discrete valued intervention variables and continuous regressor variables in the same model. You can also form interaction effects by multiplying input variables and including the product variable as another input. Indeed, as long as the dependent measure forms a regular time series, you can use PROC ARIMA to fit any general linear model in conjunction with an ARMA model for the error process by using input variables that correspond to the columns of the design matrix of the linear model.

Rational Transfer Functions and Distributed Lag Models How an input series enters the model is called its transfer function. Thus, ARIMA models with input series are sometimes referred to as transfer function models. In the preceding regression and intervention model examples, the transfer function is a single scale parameter. However, you can also specify complex transfer functions composed of numerator and denominator polynomials in the backshift operator. These transfer functions operate on the input series in the same way that the ARMA specification operates on the error term. Numerator Factors For example, suppose you want to model the effect of PRICE on SALES as taking place gradually with the impact distributed over several past lags of PRICE. This is illustrated by the following statements: proc arima data=a; identify var=sales crosscorr=price; estimate input=( (1 2 3) price ); run;


This example models the effect of PRICE on SALES as a linear function of the current and three most recent values of PRICE. It is equivalent to a multiple linear regression of SALES on PRICE, LAG(PRICE), LAG2(PRICE), and LAG3(PRICE). This is an example of a transfer function with one numerator factor. The numerator factors for a transfer function for an input series are like the MA part of the ARMA model for the noise series. Denominator Factors You can also use transfer functions with denominator factors. The denominator factors for a transfer function for an input series are like the AR part of the ARMA model for the noise series. Denominator factors introduce exponentially weighted, infinite distributed lags into the transfer function. 218 SAS OnlineDoc : Version 8

Chapter 7. Getting Started To specify transfer functions with denominator factors, place the denominator factors after a slash (/) in the INPUT= option. For example, the following statements estimate the PRICE effect as an infinite distributed lag model with exponentially declining weights: proc arima data=a; identify var=sales crosscorr=price; estimate input=( / (1) price ); run;

The transfer function specified by these statements is as follows:

This transfer function also can be written in the following equivalent form:

This transfer function can be used with intervention inputs. When it is used with a pulse function input, the result is an intervention effect that dies out gradually over time. When it is used with a step function input, the result is an intervention effect that increases gradually to a limiting value. Rational Transfer Functions By combining various numerator and denominator factors in the INPUT= option, you can specify rational transfer functions of any complexity. To specify an input with a general rational transfer function of the form

use an INPUT= option in the ESTIMATE statement of the form input=( k $ (

-lags ) / ( -lags) x)

See the section "Specifying Inputs and Transfer Functions" later in this chapter for more information. Identifying Transfer Function Models The CROSSCORR= option of the IDENTIFY statement prints sample crosscorrelation functions showing the correlations between the response series and the input series at different lags. The sample cross-correlation function can be used to help identify the form of the transfer function appropriate for an input series. See textbooks on time series analysis for information on using cross-correlation functions to identify transfer function models. For the cross-correlation function to be meaningful, the input and response series must be filtered with a prewhitening model for the input series. See the section "Prewhitening" later in this chapter for more information on this issue. 219 SAS OnlineDoc : Version 8


Forecasting with Input Variables To forecast a response series using an ARIMA model with inputs, you need values of the input series for the forecast periods. You can supply values for the input variables for the forecast periods in the DATA= data set, or you can have PROC ARIMA forecast the input variables. If you do not have future values of the input variables in the input data set used by the FORECAST statement, the input series must be forecast before the ARIMA procedure can forecast the response series. If you fit an ARIMA model to each of the input series for which you need forecasts before fitting the model for the response series, the FORECAST statement automatically uses the ARIMA models for the input series to generate the needed forecasts of the inputs. For example, suppose you want to forecast SALES for the next 12 months. In this example, we predict the change in SALES as a function of the lagged change in PRICE, plus an ARMA(1,1) noise process. To forecast SALES using PRICE as an input, you also need to fit an ARIMA model for PRICE. The following statements fit an AR(2) model to the change in PRICE before fitting and forecasting the model for SALES. The FORECAST statement automatically forecasts PRICE using this AR(2) model to get the future inputs needed to produce the forecast of SALES. proc arima data=a; identify var=price(1); estimate p=2; identify var=sales(1) crosscorr=price(1); estimate p=1 q=1 input=price; forecast lead=12 interval=month id=date out=results; run;

Fitting a model to the input series is also important for identifying transfer functions. (See the section "Prewhitening" later in this chapter for more information.) Input values from the DATA= data set and input values forecast by PROC ARIMA can be combined. For example, a model for SALES might have three input series: PRICE, INCOME, and TAXRATE. For the forecast, you assume that the tax rate will be unchanged. You have a forecast for INCOME from another source but only for the first few periods of the SALES forecast you want to make. You have no future values for PRICE, which needs to be forecast as in the preceding example. In this situation, you include observations in the input data set for all forecast periods, with SALES and PRICE set to a missing value, with TAXRATE set to its last actual value, and with INCOME set to forecast values for the periods you have forecasts for and set to missing values for later periods. In the PROC ARIMA step, you estimate ARIMA models for PRICE and INCOME before estimating the model for SALES, as shown in the following statements: proc arima data=a; identify var=price(1);


Chapter 7. Getting Started estimate identify estimate identify estimate forecast run;

p=2; var=income(1); p=2; var=sales(1) crosscorr=( price(1) income(1) taxrate ); p=1 q=1 input=( price income taxrate ); lead=12 interval=month id=date out=results;

In forecasting SALES, the ARIMA procedure uses as inputs the value of PRICE forecast by its ARIMA model, the value of TAXRATE found in the DATA= data set, and the value of INCOME found in the DATA= data set, or, when the INCOME variable is missing, the value of INCOME forecast by its ARIMA model. (Because SALES is missing for future time periods, the estimation of model parameters is not affected by the forecast values for PRICE, INCOME, or TAXRATE.)

Data Requirements PROC ARIMA can handle time series of moderate size; there should be at least 30 observations. With 30 or fewer observations, the parameter estimates may be poor. With thousands of observations, the method requires considerable computer time and memory.



Syntax The ARIMA procedure uses the following statements: PROC ARIMA options; BY variables; IDENTIFY VAR=variable options; ESTIMATE options; FORECAST options;

Functional Summary The statements and options controlling the ARIMA procedure are summarized in the following table.

Description

Statement

Data Set Options specify the input data set

PROC ARIMA IDENTIFY specify the output data set PROC ARIMA FORECAST include only forecasts in the output data set FORECAST write autocovariances to output data set IDENTIFY write parameter estimates to an output data set ESTIMATE write correlation of parameter estimates ESTIMATE write covariance of parameter estimates ESTIMATE write estimated model to an output data set ESTIMATE write statistics of fit to an output data set ESTIMATE

Options for Identifying the Series difference time series and plot autocorrelations specify response series and differencing specify and cross correlate input series center data by subtracting the mean exclude missing values delete previous models and start fresh specify the significance level for tests perform tentative ARMA order identification using the ESACF Method perform tentative ARMA order identification using the MINIC Method perform tentative ARMA order identification using the SCAN Method 222 SAS OnlineDoc : Version 8

Option

DATA= DATA= OUT= OUT= NOOUTALL OUTCOV= OUTEST= OUTCORR OUTCOV OUTMODEL= OUTSTAT=

IDENTIFY IDENTIFY IDENTIFY IDENTIFY IDENTIFY IDENTIFY IDENTIFY IDENTIFY

VAR= CROSSCORR= CENTER NOMISS CLEAR ALPHA= ESACF

IDENTIFY

MINIC

IDENTIFY

SCAN

Chapter 7. Syntax

Description

Statement

Option

specify the range of autoregressive model orders for estimating the error series for the MINIC Method determines the AR dimension of the SCAN, ESACF, and MINIC tables determines the MA dimension of the SCAN, ESACF, and MINIC tables perform stationarity tests

IDENTIFY

PERROR=

IDENTIFY

P=

IDENTIFY

Q=

IDENTIFY

STATIONARITY=

Options for Defining and Estimating the Model specify and estimate ARIMA models ESTIMATE specify autoregressive part of model ESTIMATE specify moving average part of model ESTIMATE specify input variables and transfer functions ESTIMATE drop mean term from the model ESTIMATE specify the estimation method ESTIMATE use alternative form for transfer functions ESTIMATE suppress degrees-of-freedom correction in ESTIMATE variance estimates

Printing Control Options limit number of lags shown in correlation plots suppress printed output for identification plot autocorrelation functions of the residuals print log likelihood around the estimates control spacing for GRID option print details of the iterative estimation process suppress printed output for estimation suppress printing of the forecast values print the one-step forecasts and residuals

IDENTIFY IDENTIFY ESTIMATE ESTIMATE ESTIMATE ESTIMATE ESTIMATE FORECAST FORECAST

P= Q= INPUT= NOINT METHOD= ALTPARM NODF

NLAG= NOPRINT PLOT GRID GRIDVAL= PRINTALL NOPRINT NOPRINT PRINTALL

Options to Specify Parameter Values specify autoregressive starting values ESTIMATE specify moving average starting values ESTIMATE specify a starting value for the mean parameter ESTIMATE specify starting values for transfer functions ESTIMATE

AR= MA= MU= INITVAL=

Options to Control the Iterative Estimation Process specify convergence criterion ESTIMATE specify the maximum number of iterations ESTIMATE

CONVERGE= MAXITER=



Description

Statement

specify criterion for checking for singularity ESTIMATE suppress the iterative estimation process ESTIMATE omit initial observations from objective ESTIMATE specify perturbation for numerical derivatives ESTIMATE omit stationarity and invertibility checks ESTIMATE use preliminary estimates as starting values for ESTIMATE ML and ULS

Options for Forecasting forecast the response series specify how many periods to forecast specify the ID variable specify the periodicity of the series specify size of forecast confidence limits start forecasting before end of the input data specify the variance term used to compute forecast standard errors and confidence limits control the alignment of SAS Date values BY Groups specify BY group processing

Option SINGULAR= NOEST BACKLIM= DELTA= NOSTABLE NOLS

FORECAST FORECAST FORECAST FORECAST FORECAST FORECAST FORECAST

LEAD= ID= INTERVAL= ALPHA= BACK= SIGSQ=

FORECAST

ALIGN=

BY

PROC ARIMA Statement PROC ARIMA options; The following options can be used in the PROC ARIMA statement: DATA= SAS-data-set

specifies the name of the SAS data set containing the time series. If different DATA= specifications appear in the PROC ARIMA and IDENTIFY statements, the one in the IDENTIFY statement is used. If the DATA= option is not specified in either the PROC ARIMA or IDENTIFY statement, the most recently created SAS data set is used. OUT= SAS-data-set

specifies a SAS data set to which the forecasts are output. If different OUT= specifications appear in the PROC ARIMA and FORECAST statement, the one in the FORECAST statement is used.


Chapter 7. Syntax

BY Statement BY variables; A BY statement can be used in the ARIMA procedure to process a data set in groups of observations defined by the BY variables. Note that all IDENTIFY, ESTIMATE, and FORECAST statements specified are applied to all BY groups. Because of the need to make data-based model selections, BY-group processing is not usually done with PROC ARIMA. You usually want different models for the different series contained in different BY-groups, and the PROC ARIMA BY statement does not let you do this. Using a BY statement imposes certain restrictions. The BY statement must appear before the first RUN statement. If a BY statement is used, the input data must come from the data set specified in the PROC statement; that is, no input data sets can be specified in IDENTIFY statements. When a BY statement is used with PROC ARIMA, interactive processing only applies to the first BY group. Once the end of the PROC ARIMA step is reached, all ARIMA statements specified are executed again for each of the remaining BY groups in the input data set.

IDENTIFY Statement IDENTIFY VAR=variable options; The IDENTIFY statement specifies the time series to be modeled, differences the series if desired, and computes statistics to help identify models to fit. Use an IDENTIFY statement for each time series that you want to model. If other time series are to be used as inputs in a subsequent ESTIMATE statement, they must be listed in a CROSSCORR= list in the IDENTIFY statement. The following options are used in the IDENTIFY statement. The VAR= option is required. ALPHA= significance-level

The ALPHA= option specifies the significance level for tests in the IDENTIFY statement. The default is 0.05. CENTER

centers each time series by subtracting its sample mean. The analysis is done on the centered data. Later, when forecasts are generated, the mean is added back. Note that centering is done after differencing. The CENTER option is normally used in conjunction with the NOCONSTANT option of the ESTIMATE statement. CLEAR

deletes all old models. This option is useful when you want to delete old models so 225 SAS OnlineDoc : Version 8

Part 2. General Information that the input variables are not prewhitened. (See the section "Prewhitening" later in this chapter for more information.) CROSSCORR= variable (d11, d12, ..., d1k) CROSSCORR= (variable (d11, d12, ..., d1k) ... variable (d21, d22, ..., d2k))

names the variables cross correlated with the response variable given by the VAR= specification. Each variable name can be followed by a list of differencing lags in parentheses, the same as for the VAR= specification. If differencing is specified for a variable in the CROSSCORR= list, the differenced series is cross correlated with the VAR= option series, and the differenced series is used when the ESTIMATE statement INPUT= option refers to the variable. DATA= SAS-data-set

specifies the input SAS data set containing the time series. If the DATA= option is omitted, the DATA= data set specified in the PROC ARIMA statement is used; if the DATA= option is omitted from the PROC ARIMA statement as well, the most recently created data set is used. ESACF

computes the extended sample autocorrelation function and uses these estimates to tentatively identify the autoregressive and moving average orders of mixed models. The ESACF option generates two tables. The first table displays extended sample autocorrelation estimates, and the second table displays probability values that can be used to test the significance of these estimates. The P= and Q= options determine the size of the table. The autoregressive and moving average orders are tentatively identified by finding a triangular pattern in which all values are insignificant. The ARIMA procedure finds these patterns based on the IDENTIFY statement ALPHA= option and displays possible recommendations for the orders. The following code generates an ESACF table with dimensions of p=(0:7) and q=(0:8). proc arima data=test; identify var=x esacf p=(0:7) q=(0:8); run;

See the “The ESACF Method” section on page 236 for more information. MINIC

uses information criteria or penalty functions to provide tentative ARMA order identification. The MINIC option generates a table containing the computed information criterion associated with various ARMA model orders. The PERROR= option determines the range of the autoregressive model orders used to estimate the error series. The P= and Q= options determine the size of the table. The ARMA orders are tentatively identified by those orders that minimize the information criterion.


Chapter 7. Syntax The following code generates a MINIC table with default dimensions of p=(0:5) and q=(0:5) and with the error series estimated by an autoregressive model with an order, , that minimizes the AIC in the range from 8 to 11. proc arima data=test; identify var=x minic perror=(8:11); run;

See the “The MINIC Method” section on page 238 for more information. NLAG= number

indicates the number of lags to consider in computing the autocorrelations and cross correlations. To obtain preliminary estimates of an ARIMA( p,d,q) model, the NLAG= value must be at least p+q+d. The number of observations must be greater than or equal to the NLAG= value. The default value for NLAG= is 24 or one-fourth the number of observations, whichever is less. Even though the NLAG= value is specified, the NLAG= value can be changed according to the data set. NOMISS

uses only the first continuous sequence of data with no missing values. By default, all observations are used. NOPRINT

suppresses the normal printout (including the correlation plots) generated by the IDENTIFY statement. OUTCOV= SAS-data-set

writes the autocovariances, autocorrelations, inverse autocorrelations, partial autocorrelations, and cross covariances to an output SAS data set. If the OUTCOV= option is not specified, no covariance output data set is created. See the section "OUTCOV= Data Set" later in this chapter for more information. P= (

)

see the ESCAF, MINIC, and SCAN options for details. PERROR= (

)

see the ESCAF, MINIC, and SCAN options for details. Q= (

)

see the ESACF, MINIC, and SCAN options for details. SCAN

computes estimates of the squared canonical correlations and uses these estimates to tentatively identify the autoregressive and moving average orders of mixed models. The SCAN option generates two tables. The first table displays squared canonical correlation estimates, and the second table displays probability values that can be used to test the significance of these estimates. The P= and Q= options determine the size of each table. The autoregressive and moving average orders are tentatively identified by finding a rectangular pattern in which all values are insignificant. The ARIMA procedure finds these patterns based on the IDENTIFY statement ALPHA= option and displays possible recommendations for the orders. 227 SAS OnlineDoc : Version 8

Part 2. General Information The following code generates a SCAN table with default dimensions of p=(0:5) and q=(0:5). The recommended orders are based on a significance level of 0.1. proc arima data=test; identify var=x scan alpha=0.1; run;

See the “The SCAN Method” section on page 239 for more information. STATIONARITY=

performs stationarity tests. Stationarity tests can be used to determine whether differencing should be included in the model specification. In each stationarity test, the autoregressive orders can be specified by a range, test= , or as a list of values, test= , where test is ADF, PP, or RW. The default is (0,1,2). See the “Stationarity Tests” section on page 241 for more information. STATIONARITY=(ADF= AR orders DLAG= s) STATIONARITY=(DICKEY= AR orders DLAG= s)

performs augmented Dickey-Fuller tests. If the DLAG= option specified with is greater than one, seasonal Dickey-Fuller tests are performed. The maximum allowable value of is 12. The default value of is one. The following code performs augmented Dickey-Fuller tests with autoregressive orders 2 and 5. proc arima data=test; identify var=x stationarity=(adf=(2,5)); run; STATIONARITY=(PP= AR orders) STATIONARITY=(PHILLIPS= AR orders)

performs Phillips-Perron tests. The following code performs Augmented PhillipsPerron tests with autoregressive orders ranging from 0 to 6. proc arima data=test; identify var=x stationarity=(pp=6); run; STATIONARITY=(RW= AR orders) STATIONARITY=(RANDOMWALK= AR orders)

performs random-walk with drift tests. The following code performs random-walk with drift tests with autoregressive orders ranging from 0 to 2. proc arima data=test; identify var=x stationarity=(rw); run;


Chapter 7. Syntax VAR= variable VAR= variable ( d1, d2, ..., dk )

names the variable containing the time series to analyze. The VAR= option is required. A list of differencing lags can be placed in parentheses after the variable name to request that the series be differenced at these lags. For example, VAR=X(1) takes the first differences of X. VAR=X(1,1) requests that X be differenced twice, both times with lag 1, producing a second difference series, which is . VAR=X(2) differences X once at lag two

.

If differencing is specified, it is the differenced series that is processed by any subsequent ESTIMATE statement.

ESTIMATE Statement ESTIMATE options; The ESTIMATE statement specifies an ARMA model or transfer function model for the response variable specified in the previous IDENTIFY statement, and produces estimates of its parameters. The ESTIMATE statement also prints diagnostic information by which to check the model. Include an ESTIMATE statement for each model that you want to estimate. Options used in the ESTIMATE statement are described in the following sections. Options for Defining the Model and Controlling Diagnostic Statistics The following options are used to define the model to be estimated and to control the output that is printed. ALTPARM

specifies the alternative parameterization of the overall scale of transfer functions in the model. See the section "Alternative Model Parameterization" later in this chapter for details. INPUT= variable INPUT= ( transfer-function variable ... )

specifies input variables and their transfer functions. The variables used on the INPUT= option must be included in the CROSSCORR= list in the previous IDENTIFY statement. If any differencing is specified in the CROSSCORR= list, then the differenced series is used as the input to the transfer function. The transfer function specification for an input variable is optional. If no transfer function is specified, the input variable enters the model as a simple regressor. If specified, the transfer function specification has the following syntax:


Part 2. General Information Here, S is a shift or lag of the input variable, the before the slash (/) are numerator factors, and the after the slash (/) are denominator factors of the transfer function. All three parts are optional. See the section "Specifying Inputs and Transfer Functions" later in this chapter for details. METHOD=ML METHOD=ULS METHOD=CLS

specifies the estimation method to use. METHOD=ML specifies the maximum likelihood method. METHOD=ULS specifies the unconditional least-squares method. METHOD=CLS specifies the conditional least-squares method. METHOD=CLS is the default. See the section "Estimation Details" later in this chapter for more information. NOCONSTANT NOINT

suppresses the fitting of a constant (or intercept) parameter in the model. (That is, the parameter is omitted.) NODF

estimates the variance by dividing the error sum of squares (SSE) by the number of residuals. The default is to divide the SSE by the number of residuals minus the number of free parameters in the model. NOPRINT

suppresses the normal printout generated by the ESTIMATE statement. If the NOPRINT option is specified for the ESTIMATE statement, then any error and warning messages are printed to the SAS log. P= order P= (lag, ..., lag) ... (lag, ..., lag)

specifies the autoregressive part of the model. By default, no autoregressive parameters are fit. P=(l , l , ..., l ) defines a model with autoregressive parameters at the specified lags. P= order is equivalent to P=(1, 2, ..., order). A concatenation of parenthesized lists specifies a factored model. P=(1,2,5)(6,12) specifies the autoregressive model

For example,

PLOT

plots the residual autocorrelation functions. The sample autocorrelation, the sample inverse autocorrelation, and the sample partial autocorrelation functions of the model residuals are plotted.


Chapter 7. Syntax Q= order Q= (lag, ..., lag) ... (lag, ..., lag)

specifies the moving-average part of the model. By default, no moving-average part is included in the model. Q=(l , l , ..., l ) defines a model with moving-average parameters at the specified lags. Q= order is equivalent to Q=(1, 2, ..., order). A concatenation of parenthesized lists specifies a factored model. The interpretation of factors and lags is the same as for the P= option. Options for Output Data Sets The following options are used to store results in SAS data sets: OUTEST= SAS-data-set

writes the parameter estimates to an output data set. If the OUTCORR or OUTCOV option is used, the correlations or covariances of the estimates are also written to the OUTEST= data set. See the section "OUTEST= Data Set" later in this chapter for a description of the OUTEST= output data set. OUTCORR

writes the correlations of the parameter estimates to the OUTEST= data set. OUTCOV

writes the covariances of the parameter estimates to the OUTEST= data set. OUTMODEL= SAS-data-set

writes the model and parameter estimates to an output data set. If OUTMODEL= is not specified, no model output data set is created. See the section "OUTMODEL= Data Set" for a description of the OUTMODEL= output data set. OUTSTAT= SAS-data-set

writes the model diagnostic statistics to an output data set. If OUTSTAT= is not specified, no statistics output data set is created. See the section "OUTSTAT= Data Set" later in this chapter for a description of the OUTSTAT= output data set. Options to Specify Parameter Values The following options enable you to specify values for the model parameters. These options can provide starting values for the estimation process, or you can specify fixed parameters for use in the FORECAST stage and suppress the estimation process with the NOEST option. By default, the ARIMA procedure finds initial parameter estimates and uses these estimates as starting values in the iterative estimation process. If values for any parameters are specified, values for all parameters should be given. The number of values given must agree with the model specifications. AR= value ...

lists starting values for the autoregressive parameters. See "Initial Values" later in this chapter for more information. INITVAL= (initializer-spec variable ... )

specifies starting values for the parameters in the transfer function parts of the model. See "Initial Values" later in this chapter for more information. 231 SAS OnlineDoc : Version 8

Part 2. General Information MA= value ...

lists starting values for the moving-average parameters. See "Initial Values" later in this chapter for more information. MU= value

specifies the MU parameter. NOEST

uses the values specified with the AR=, MA=, INITVAL=, and MU= options as final parameter values. The estimation process is suppressed except for estimation of the residual variance. The specified parameter values are used directly by the next FORECAST statement. When NOEST is specified, standard errors, t values, and the correlations between estimates are displayed as 0 or missing. (The NOEST option is useful, for example, when you wish to generate forecasts corresponding to a published model.) Options to Control the Iterative Estimation Process The following options can be used to control the iterative process of minimizing the error sum of squares or maximizing the log likelihood function. These tuning options are not usually needed but may be useful if convergence problems arise. BACKLIM=

omits the specified number of initial residuals from the sum of squares or likelihood function. Omitting values can be useful for suppressing transients in transfer function models that are sensitive to start-up values. CONVERGE= value

specifies the convergence criterion. Convergence is assumed when the largest change in the estimate for any parameter is less that the CONVERGE= option value. If the absolute value of the parameter estimate is greater than 0.01, the relative change is used; otherwise, the absolute change in the estimate is used. The default is CONVERGE=.001. DELTA= value

specifies the perturbation value for computing numerical derivatives. The default is DELTA=.001. GRID

prints the error sum of squares (SSE) or concentrated log likelihood surface in a small grid of the parameter space around the final estimates. For each pair of parameters, the SSE is printed for the nine parameter-value combinations formed by the grid, with a center at the final estimates and with spacing given by the GRIDVAL= specification. The GRID option may help you judge whether the estimates are truly at the optimum, since the estimation process does not always converge. For models with a large number of parameters, the GRID option produces voluminous output. GRIDVAL= number

controls the spacing in the grid printed by the GRID option. The default is GRIDVAL=0.005.


Chapter 7. Syntax MAXITER= MAXIT=

specifies the maximum number of iterations allowed. The default is MAXITER=50. (The default was 15 in previous releases of SAS/ETS software.) NOLS

begins the maximum likelihood or unconditional least-squares iterations from the preliminary estimates rather than from the conditional least-squares estimates that are produced after four iterations. See the section "Estimation Details" later in this chapter for details. NOSTABLE

specifies that the autoregressive and moving-average parameter estimates for the noise part of the model not be restricted to the stationary and invertible regions, respectively. See the section "Stationarity and Invertibility" later in this chapter for more information. PRINTALL

prints preliminary estimation results and the iterations in the final estimation process. SINGULAR= value

specifies the criterion for checking singularity. If a pivot of a sweep operation is less than the SINGULAR= value, the matrix is deemed singular. Sweep operations are performed on the Jacobian matrix during final estimation and on the covariance matrix when preliminary estimates are obtained. The default is SINGULAR=1E-7.

FORECAST Statement FORECAST options; The FORECAST statement generates forecast values for a time series using the parameter estimates produced by the previous ESTIMATE statement. See the section "Forecasting Details" later in this chapter for more information on calculating forecasts. The following options can be used in the FORECAST statement: ALIGN= option

controls the alignment of SAS dates used to identify output observations. The ALIGN= option allows the following values: BEGINNING|BEG|B, MIDDLE|MID|M, and ENDING|END|E. BEGINNING is the default. ALPHA= n

sets the size of the forecast confidence limits. The ALPHA= value must be between 0 and 1. When you specify ALPHA= , the upper and lower confidence limits will have a confidence level. The default is ALPHA=.05, which produces 95% confidence intervals. ALPHA values are rounded to the nearest hundredth.


Part 2. General Information BACK= n

specifies the number of observations before the end of the data that the multistep forecasts are to begin. The BACK= option value must be less than or equal to the number of observations minus the number of parameters. The default is BACK=0, which means that the forecast starts at the end of the available data. The end of the data is the last observation for which a noise value can be calculated. If there are no input series, the end of the data is the last nonmissing value of the response time series. If there are input series, this observation can precede the last nonmissing value of the response variable, since there may be missing values for some of the input series. ID= variable

names a variable in the input data set that identifies the time periods associated with the observations. The ID= variable is used in conjunction with the INTERVAL= option to extrapolate ID values from the end of the input data to identify forecast periods in the OUT= data set. If the INTERVAL= option specifies an interval type, the ID variable must be a SAS date or datetime variable with the spacing between observations indicated by the INTERVAL= value. If the INTERVAL= option is not used, the last input value of the ID= variable is incremented by one for each forecast period to extrapolate the ID values for forecast observations. INTERVAL= interval INTERVAL= n

specifies the time interval between observations. See Chapter 3, “Date Intervals, Formats, and Functions,” for information on valid INTERVAL= values. The value of the INTERVAL= option is used by PROC ARIMA to extrapolate the ID values for forecast observations and to check that the input data are in order with no missing periods. See the section "Specifying Series Periodicity" later in this chapter for more details. LEAD= n

specifies the number of multistep forecast values to compute. For example, if LEAD=10, PROC ARIMA forecasts for ten periods beginning with the end of the input series (or earlier if BACK= is specified). It is possible to obtain fewer than the requested number of forecasts if a transfer function model is specified and insufficient data are available to compute the forecast. The default is LEAD=24. NOOUTALL

includes only the final forecast observations in the output data set, not the one-step forecasts for the data before the forecast period. NOPRINT

suppresses the normal printout of the forecast and associated values.


Chapter 7. Syntax OUT= SAS-data-set

writes the forecast (and other values) to an output data set. If OUT= is not specified, the OUT= data set specified in the PROC ARIMA statement is used. If OUT= is also not specified in the PROC ARIMA statement, no output data set is created. See the section "OUT= Data Set" later in this chapter for more information. PRINTALL

prints the FORECAST computation throughout the whole data set. The forecast values for the data before the forecast period (specified by the BACK= option) are onestep forecasts. SIGSQ=

specifies the variance term used in the formula for computing forecast standard errors and confidence limits. The default value is the Variance Estimate computed by the preceding ESTIMATE statement. This option is useful when you wish to generate forecast standard errors and confidence limits based on a published model. It would often be used in conjunction with the NOEST option in the preceding ESTIMATE statement.

Details The Inverse Autocorrelation Function The sample inverse autocorrelation function (SIACF) plays much the same role in ARIMA modeling as the sample partial autocorrelation function (SPACF) but generally indicates subset and seasonal autoregressive models better than the SPACF. Additionally, the SIACF may be useful for detecting over-differencing. If the data come from a nonstationary or nearly nonstationary model, the SIACF has the characteristics of a noninvertible moving average. Likewise, if the data come from a model with a noninvertible moving average, then the SIACF has nonstationary characteristics and therefore decays slowly. In particular, if the data have been over-differenced, the SIACF looks like a SACF from a nonstationary process. The inverse autocorrelation function is not often discussed in textbooks, so a brief description is given here. More complete discussions can be found in Cleveland (1972), Chatfield (1980), and Priestly (1981). Let

be generated by the ARMA(p,q) process

where is a white noise sequence. If ( ) is invertible (that is, if considered as a polynomial in has no roots less than or equal to 1 in magnitude), then the model

is also a valid ARMA(q,p) model. This model is sometimes referred to as the dual model. The autocorrelation function (ACF) of this dual model is called the inverse autocorrelation function (IACF) of the original model. 235 SAS OnlineDoc : Version 8

Part 2. General Information Notice that if the original model is a pure autoregressive model, then the IACF is an ACF corresponding to a pure moving-average model. Thus, it cuts off sharply when the lag is greater than p; this behavior is similar to the behavior of the partial autocorrelation function (PACF). The sample inverse autocorrelation function (SIACF) is estimated in the ARIMA procedure by the following steps. A high-order autoregressive model is fit to the data by means of the Yule-Walker equations. The order of the autoregressive model used to calculate the SIACF is the minimum of the NLAG= value and one-half the number of observations after differencing. The SIACF is then calculated as the autocorrelation function that corresponds to this autoregressive operator when treated as a moving-average operator. That is, the autoregressive coefficients are convolved with themselves and treated as autocovariances. Under certain conditions, the sampling distribution of the SIACF can be approximated by the sampling distribution of the SACF of the dual model (Bhansali 1980). In the plots generated by ARIMA, the confidence limit marks (.) are located at . These limits bound an approximate 95% confidence interval for the hypothesis that the data are from a white noise process.

The Partial Autocorrelation Function The approximation for a standard error for the estimated partial autocorrelation function at lag k is based on a null hypothesis that a pure autoregressive Gaussian process of order k-1 generated the time series. This standard error is and is used to produce the approximate 95% confidence intervals depicted by the dots in the plot.

The Cross-Correlation Function The autocorrelation, partial and inverse autocorrelation functions described in the preceding sections help when you want to model a series as a function of its past values and past random errors. When you want to include the effects of past and current values of other series in the model, the correlations of the response series and the other series must be considered. The CROSSCORR= option on the IDENTIFY statement computes cross correlations of the VAR= series with other series and makes these series available for use as inputs in models specified by later ESTIMATE statements. When the CROSSCORR= option is used, PROC ARIMA prints a plot of the crosscorrelation function for each variable in the CROSSCORR= list. This plot is similar in format to the other correlation plots, but shows the correlation between the two series at both lags and leads. For example identify var=y crosscorr=x ...;

plots the cross-correlation function of Y and X, , for to , where is the value of the NLAG= option. Study of the cross-correlation functions can indicate the transfer functions through which the input series should enter the model for the response series. 236 SAS OnlineDoc : Version 8

Chapter 7. Details The cross-correlation function is computed after any specified differencing has been done. If differencing is specified for the VAR= variable or for a variable in the CROSSCORR= list, it is the differenced series that is cross correlated (and the differenced series is processed by any following ESTIMATE statement). For example, identify var=y(1) crosscorr=x(1);

computes the cross correlations of the changes in Y with the changes in X. Any following ESTIMATE statement models changes in the variables rather than the variables themselves.

The ESACF Method The Extended Sample Autocorrelation Function (ESACF) method can tentatively identify the orders of a stationary or nonstationary ARMA process based on iterated least squares estimates of the autoregressive parameters. Tsay and Tiao (1984) proposed the technique, and Choi (1990) provides useful descriptions of the algorithm. Given a stationary or nonstationary time series with mean corrected form , with a true autoregressive order of , and with a true movingaverage order of , you can use the ESACF method to estimate the unknown orders and by analyzing the autocorrelation functions associated with filtered series of the form

where represents the backshift operator, where are the autoregressive test orders, where are the moving average test orders, and where are the autoregressive parameter estimates under the assumption that the series is an ARMA( ) process. For purely autoregressive models ( ), ordinary least squares (OLS) is used to consistently estimate . For ARMA models, consistent estimates are obtained by the iterated least squares recursion formula, which is initiated by the pure autoregressive estimates:

The th lag of the sample autocorrelation function of the filtered series, extended sample autocorrelation function, and it is denoted as

, is the .

The standard errors of are computed in the usual way using Bartlett’s approximation of the variance of the sample autocorrelation function, . 237 SAS OnlineDoc : Version 8

Part 2. General Information If the true model is an ARMA ( an MA( ) model for so that

) process, the filtered series,

, follows

Additionally, Tsay and Tiao (1984) show that the extended sample autocorrelation satisfies

where bounded by -1 and 1.

is a nonzero constant or a continuous random variable

An ESACF table is then constructed using the for and to identify the ARMA orders (see Table 7.3). The orders are tentatively identified by finding a right (maximal) triangular pattern with vertices located at and and in which all elements are insignificant (based on asymptotic normality of the autocorrelation function). The vertex identifies the order. Table 7.4 depicts the theoretical pattern associated with an ARMA(1,2) series. Table 7.3.

ESACF Table

AR 0 1 2 3

Table 7.4.

0

1

MA 2

3

Theoretical ESACF Table for an ARMA(1,2) Series

AR 0 1 2 3 4

MA 0 1 2 3 4 5 * X X X X X * X 0 0 0 0 * X X 0 0 0 * X X X 0 0 * X X X X 0 X = significant 0 = insignificant * = no pattern 238

SAS OnlineDoc : Version 8

6 X 0 0 0 0

7 X 0 0 0 0

Chapter 7. Details

The MINIC Method The MINimum Information Criterion (MINIC) method can tentatively identify the order of a stationary and invertible ARMA process. Note that Hannan and Rissannen (1982) proposed this method, and Box et al (1994) and Choi (1990) provide useful descriptions of the algorithm. Given a stationary and invertible time series with mean corrected form , with a true autoregressive order of , and with a true movingaverage order of , you can use the MINIC method to compute information criteria (or penalty functions) for various autoregressive and moving average orders. The following paragraphs provide a brief description of the algorithm. If the series is a stationary and invertible ARMA(p,q) process of the form

the error series can be approximated by a high-order AR process

where the parameter estimates are obtained from the Yule-Walker estimates. The choice of the autoregressive order, , is determined by the order that minimizes the Akaike information criterion (AIC) in the range

where

Note that Hannan and Rissannen (1982) use the Bayesian information criterion (BIC) to determine the autoregressive order used to estimate the error series. Box et al (1994) and Choi (1990) recommend the AIC. Once the error series has been estimated for autoregressive test order and for moving-average test order , the OLS estimates and are computed from the regression model

From the preceding parameter estimates, the BIC is then computed


Part 2. General Information where

where

.

A MINIC table is then constructed using (see Table 7.5). If , the preceding regression may fail due to linear dependence on the estimated error series and the mean-corrected series. Values of that cannot be computed are set to missing. For large autoregressive and moving average test orders with relatively few observations, a nearly perfect fit can result. This condition can be identified by a large negative value. Table 7.5.

MINIC Table

MA AR 0 1 2 3

0

1

2

3

The SCAN Method The Smallest CANonical (SCAN) correlation method can tentatively identify the orders of a stationary or nonstationary ARMA process. Tsay and Tiao (1985) proposed the technique, and Box et al (1994) and Choi (1990) provide useful descriptions of the algorithm. Given a stationary or nonstationary time series with mean corrected form , with a true autoregressive order of , and with a true movingaverage order of , you can use the SCAN method to analyze eigenvalues of the correlation matrix of the ARMA process. The following paragraphs provide a brief description of the algorithm. For autoregressive test order and for moving-average test order , perform the following steps. 1. Let matrix

. Compute the following


Chapter 7. Details

where ranges from

to .

2. Find the smallest eigenvalue, malized eigenvector, canonical correlation estimate is

, of

and its corresponding nor. The squared .

3. Using the as AR( ) coefficients, obtain the residuals for to , by following the formula: . 4. From the sample autocorrelations of the residuals, , approximate the standard error of the squared canonical correlation estimate by

where

.

The test statistic to be used as an identification criterion is

which is asymptotically if and or if and . For and , there is more than one theoretical zero canonical correlation between and . Since the are the smallest canonical correlations for each , the percentiles of are less than those of a ; therefore, Tsay and Tiao (1985) state that it is safe to assume a . For and , no conclusions about the distribution of are made. A SCAN table is then constructed using to determine which of the are significantly different from zero (see Table 7.6). The ARMA orders are tentatively identified by finding a (maximal) rectangular pattern in which the are insignificant for all test orders and . There may be more than one pair of values ( ) that permit such a rectangular pattern. In this case, parsimony and the number of insignificant items in the rectangular pattern should help determine the model order. Table 7.7 depicts the theoretical pattern associated with an ARMA(2,2) series. Table 7.6.

SCAN Table

AR 0 1 2 3

0

1

MA 2

3


Part 2. General Information Table 7.7.

Theoretical SCAN Table for an ARMA(2,2) Series

AR 0 1 2 3 4

MA 0 1 2 3 4 5 * X X X X X * X X X X X * X 0 0 0 0 * X 0 0 0 0 * X 0 0 0 0 X = significant 0 = insignificant * = no pattern

6 X X 0 0 0

7 X X 0 0 0

Stationarity Tests When a time series has a unit root, the series is nonstationary and the ordinary least squares (OLS) estimator is not normally distributed. Dickey (1976) and Dickey and Fuller (1979) studied the limiting distribution of the OLS estimator of autoregressive models for time series with a simple unit root. Dickey, Hasza and Fuller (1984) obtained the limiting distribution for time series with seasonal unit roots. Hamilton (1994) discusses the various types of unit root testing. For a description of Dickey-Fuller tests, refer to , "PROBDF Function for Dickey- Fuller Tests" in Chapter 4. Refer to Chapter 4, “SAS Macros and Functions,” for a description of the augmented Dickey-Fuller tests. Refer to Chapter 8, “The AUTOREG Procedure,” for a description of Phillips-Perron tests. The random walk with drift test recommends whether or not an integrated times series has a drift term. Hamilton (1994) discusses this test.

Prewhitening If, as is usually the case, an input series is autocorrelated, the direct cross-correlation function between the input and response series gives a misleading indication of the relation between the input and response series. One solution to this problem is called prewhitening. You first fit an ARIMA model for the input series sufficient to reduce the residuals to white noise; then, filter the input series with this model to get the white noise residual series. You then filter the response series with the same model and cross correlate the filtered response with the filtered input series. The ARIMA procedure performs this prewhitening process automatically when you precede the IDENTIFY statement for the response series with IDENTIFY and ESTIMATE statements to fit a model for the input series. If a model with no inputs was previously fit to a variable specified by the CROSSCORR= option, then that model is used to prewhiten both the input series and the response series before the cross correlations are computed for the input series. 242 SAS OnlineDoc : Version 8

Chapter 7. Details For example, proc arima data=in; identify var=x; estimate p=1 q=1; identify var=y crosscorr=x;

Both X and Y are filtered by the ARMA(1,1) model fit to X before the cross correlations are computed. Note that prewhitening is done to estimate the cross-correlation function; the unfiltered series are used in any subsequent ESTIMATE or FORECAST statements, and the correlation functions of Y with its own lags are computed from the unfiltered Y series. But initial values in the ESTIMATE statement are obtained with prewhitened data; therefore, the result with prewhitening can be different from the result without prewhitening. To suppress prewhitening for all input variables, use the CLEAR option on the IDENTIFY statement to make PROC ARIMA forget all previous models. Prewhitening and Differencing If the VAR= and CROSSCORR= options specify differencing, the series are differenced before the prewhitening filter is applied. When the differencing lists specified on the VAR= option for an input and on the CROSSCORR= option for that input are not the same, PROC ARIMA combines the two lists so that the differencing operators used for prewhitening include all differences in either list (in the least common multiple sense).

Identifying Transfer Function Models When identifying a transfer function model with multiple input variables, the crosscorrelation functions may be misleading if the input series are correlated with each other. Any dependencies among two or more input series will confound their cross correlations with the response series. The prewhitening technique assumes that the input variables do not depend on past values of the response variable. If there is from the response variable to an input variable, as evidenced by significant cross-correlation at negative lags, both the input and the response variables need to be prewhitened before meaningful cross correlations can be computed. PROC ARIMA cannot handle models. The STATESPACE procedure is more appropriate for models with .

Missing Values and Autocorrelations To compute the sample autocorrelation function when missing values are present, PROC ARIMA uses only cross products that do not involve missing values and employs divisors that reflect the number of cross products used rather than the total length of the series. Sample partial autocorrelations and inverse autocorrelations are then computed using the sample autocorrelation function. If necessary, a taper is 243 SAS OnlineDoc : Version 8

Part 2. General Information employed to transform the sample autocorrelations into a positive definite sequence before calculating the partial autocorrelation and inverse correlation functions. The confidence intervals produced for these functions may not be valid when there are missing values. The distributional properties for sample correlation functions are not clear for finite samples. See Dunsmuir (1984) for some asymptotic properties of the sample correlation functions.

Estimation Details The ARIMA procedure primarily uses the computational methods outlined by Box and Jenkins. Marquardt’s method is used for the nonlinear least-squares iterations. Numerical approximations of the derivatives of the sum-of-squares function are taken using a fixed delta (controlled by the DELTA= option). The methods do not always converge successfully for a given set of data, particularly if the starting values for the parameters are not close to the least-squares estimates. Back-forecasting The unconditional sum of squares is computed exactly; thus, back-forecasting is not performed. Early versions of SAS/ETS software used the back-forecasting approximation and allowed a positive value of the BACKLIM= option to control the extent of the back-forecasting. In the current version, requesting a positive number of backforecasting steps with the BACKLIM= option has no effect. Preliminary Estimation If an autoregressive or moving-average operator is specified with no missing lags, preliminary estimates of the parameters are computed using the autocorrelations computed in the IDENTIFY stage. Otherwise, the preliminary estimates are arbitrarily set to values that produce stable polynomials. When preliminary estimation is not performed by PROC ARIMA, then initial values of the coefficients for any given autoregressive or moving average factor are set to 0.1 if the degree of the polynomial associated with the factor is 9 or less. Otherwise, the coefficients are determined by expanding the polynomial ( ) to an appropriate power using a recursive algorithm. These preliminary estimates are the starting values in an iterative algorithm to compute estimates of the parameters. Estimation Methods Maximum Likelihood

The METHOD= ML option produces maximum likelihood estimates. The likelihood function is maximized via nonlinear least squares using Marquardt’s method. Maximum likelihood estimates are more expensive to compute than the conditional least-squares estimates, however, they may be preferable in some cases (Ansley and Newbold 1980; Davidson 1981). The maximum likelihood estimates are computed as follows. Let the univariate ARMA model be


Chapter 7. Details where is an independent sequence of normally distributed innovations with mean 0 and variance . Here is the mean parameter plus the transfer function inputs. The log likelihood function can be written as follows:

In this equation, n is the number of observations, is the variance of x as a function of the and parameters, and denotes the determinant. The vector x is the time series minus the structural part of the model , written as a column vector, as follows:

.

.

The maximum likelihood estimate (MLE) of

is

Note that the default estimator of the variance divides by , where r is the number of parameters in the model, instead of by n. Specifying the NODF option causes a divisor of n to be used. The log likelihood concentrated with respect to stants as

can be taken up to additive con-

Let H be the lower triangular matrix with positive elements on the diagonal such that . Let e be the vector . The concentrated log likelihood with respect to can now be written as

or

The MLE is produced by using a Marquardt algorithm to minimize the following sum of squares: 245 SAS OnlineDoc : Version 8

Part 2. General Information The subsequent analysis of the residuals is done using e as the vector of residuals. Unconditional Least Squares

The METHOD=ULS option produces unconditional least-squares estimates. The ULS method is also referred to as the exact least-squares (ELS) method. For METHOD=ULS, the estimates minimize

where is the covariance matrix of and , and is the variance matrix of . In fact, is the same as and, hence, . Therefore, the unconditional least-squares estimates are obtained by minimizing the sum of squared residuals rather than using the log likelihood as the criterion function. Conditional Least Squares

The METHOD=CLS option produces conditional least-squares estimates. The CLS estimates are conditional on the assumption that the past unobserved errors are equal to 0. The series can be represented in of the previous observations, as follows:

The

weights are computed from the ratio of the and polynomials, as follows:

The CLS method produces estimates minimizing

where the unobserved past values of estimates of and at each iteration.

are set to 0 and

are computed from the

For METHOD=ULS and METHOD=ML, initial estimates are computed using the METHOD=CLS algorithm. Start-up for Transfer Functions When computing the noise series for transfer function and intervention models, the start-up for the transferred variable is done assuming that past values of the input series are equal to the first value of the series. The estimates are then obtained by applying least squares or maximum likelihood to the noise series. Thus, for transfer 246 SAS OnlineDoc : Version 8

Chapter 7. Details function models, the ML option does not generate the full (multivariate ARMA) maximum likelihood estimates, but it uses only the univariate likelihood function applied to the noise series. Because PROC ARIMA uses all of the available data for the input series to generate the noise series, other start-up options for the transferred series can be implemented by prefixing an observation to the beginning of the real data. For example, if you fit a transfer function model to the variable Y with the single input X, then you can employ a start-up using 0 for the past values by prefixing to the actual data an observation with a missing value for Y and a value of 0 for X. Information Criteria PROC ARIMA computes and prints two information criteria, Akaike’s information criterion (AIC) (Akaike 1974; Harvey 1981) and Schwarz’s Bayesian criterion (SBC) (Schwarz 1978). The AIC and SBC are used to compare competing models fit to the same series. The model with the smaller information criteria is said to fit the data better. The AIC is computed as

where L is the likelihood function and k is the number of free parameters. The SBC is computed as

where n is the number of residuals that can be computed for the time series. Sometimes Schwarz’s Bayesian criterion is called the Bayesian Information criterion (BIC). If METHOD=CLS is used to do the estimation, an approximation value of L is used, where L is based on the conditional sum of squares instead of the exact sum of squares, and a Jacobian factor is left out. Tests of Residuals A table of test statistics for the hypothesis that the model residuals are white noise is printed as part of the ESTIMATE statement output. The chi-square statistics used in the test for lack of fit are computed using the Ljung-Box formula

where

and

is the residual series. 247 SAS OnlineDoc : Version 8

Part 2. General Information This formula has been suggested by Ljung and Box (1978) as yielding a better fit to the asymptotic chi-square distribution than the Box-Pierce Q statistic. Some simulation studies of the finite sample properties of this statistic are given by Davies, Triggs, and Newbold (1977) and by Ljung and Box (1978). Each chi-square statistic is computed for all lags up to the indicated lag value and is not independent of the preceding chi-square values. The null hypotheses tested is that the current set of autocorrelations is white noise. t-values The t values reported in the table of parameter estimates are approximations whose accuracy depends on the validity of the model, the nature of the model, and the length of the observed series. When the length of the observed series is short and the number of estimated parameters is large with respect to the series length, the t approximation is usually poor. Probability values corresponding to a t distribution should be interpreted carefully as they may be misleading. Cautions During Estimation The ARIMA procedure uses a general nonlinear least-squares estimation method that can yield problematic results if your data do not fit the model. Output should be examined carefully. The GRID option can be used to ensure the validity and quality of the results. Problems you may encounter include the following: Preliminary moving-average estimates may not converge. Should this occur, preliminary estimates are derived as described previously in "Preliminary Estimation." You can supply your own preliminary estimates with the ESTIMATE statement options. The estimates can lead to an unstable time series process, which can cause extreme forecast values or overflows in the forecast. The Jacobian matrix of partial derivatives may be singular; usually, this happens because not all the parameters are identifiable. Removing some of the parameters or using a longer time series may help. The iterative process may not converge. PROC ARIMA’s estimation method stops after n iterations, where n is the value of the MAXITER= option. If an iteration does not improve the SSE, the Marquardt parameter is increased by a factor of ten until parameters that have a smaller SSE are obtained or until the limit value of the Marquardt parameter is exceeded. For METHOD=CLS, the estimates may converge but not to least-squares estimates. The estimates may converge to a local minimum, the numerical calculations may be distorted by data whose sum-of-squares surface is not smooth, or the minimum may lie outside the region of invertibility or stationarity. If the data are differenced and a moving-average model is fit, the parameter estimates may try to converge exactly on the invertibility boundary. In this case, the standard error estimates that are based on derivatives may be inaccurate.


Chapter 7. Details

Specifying Inputs and Transfer Functions Input variables and transfer functions for them may be specified using the INPUT= option on the ESTIMATE statement. The variables used on the INPUT= option must be included in the CROSSCORR= list in the previous IDENTIFY statement. If any differencing is specified in the CROSSCORR= list, then the differenced variable is used as the input to the transfer function. General Syntax of the INPUT= Option The general syntax of the INPUT= option is ESTIMATE

INPUT=( transfer-function variable ... )

The transfer function for an input variable is optional. The name of a variable by itself can be used to specify a pure regression term for the variable. If specified, the syntax of the transfer function is

S is the number of periods of time delay (lag) for this input series. Each term in parentheses specifies a polynomial factor with parameters at the lags specified by the values. The before the slash (/) are numerator factors. The after the slash (/) are denominator factors. All three parts are optional. Commas can optionally be used between input specifications to make the INPUT= option more readable. The $ sign after the shift is also optional. Except for the first numerator factor, each of the factor of the form

indicates a

The form of the first numerator factor depends on the ALTPARM option. By default, the constant 1 in the first numerator factor is replaced with a free parameter . Alternative Model Parameterization When the ALTPARM option is specified, the parameter is factored out so it multiplies the entire transfer function, and the first numerator factor has the same form as the other factors. The ALTPARM option does not materially affect the results; it just presents the results differently. Some people prefer to see the model written one way, while others prefer the alternative representation. Table 7.8 illustrates the effect of the ALTPARM option. Table 7.8.

The ALTPARM Option

INPUT= Option INPUT=((1 2)(12)/(1)X);

ALTPARM No Yes

Model


Part 2. General Information Differencing and Input Variables If you difference the response series and use input variables, take care that the differencing operations do not change the meaning of the model. For example, if you want to fit the model

then the IDENTIFY statement must read identify var=y(1,12) crosscorr=x(1,12); estimate q=1 input=(/(1)x) noconstant;

If instead you specify the differencing as identify var=y(1,12) crosscorr=x; estimate q=1 input=(/(1)x) noconstant;

then the model being requested is

which is a very different model. The point to is that a differencing operation requested for the response variable specified by the VAR= option is applied only to that variable and not to the noise term of the model.

Initial Values The syntax for giving initial values to transfer function parameters in the the INITVAL= option parallels the syntax of the INPUT= option. For each transfer function in the INPUT= option, the INITVAL= option should give an initialization specification followed by the input series name. The initialization specification for each transfer function has the form

where C is the lag 0 term in the first numerator factor of the transfer function (or the overall scale factor if the ALTPARM option is specified), and is the coefficient of the element in the transfer function. To illustrate, suppose you want to fit the model


Chapter 7. Details and start the estimation process with the initial values =10, =1, =.5, =.03, =.8, =-.1, =.002, =.1, =.01. (These are arbitrary values for illustration only.) You would use the following statements: identify var=y crosscorr=x; estimate p=(1,3) input=(3$(1,2)/(1,2,3)x) mu=10 ar=.1 .01 initval=(1$(.5,.03)/(.8,-.1,.002)x);

Note that the lags specified for a particular factor will be sorted, so initial values should be given in sorted order. For example, if the P= option had been entered as P=(3,1) instead of P=(1,3), the model would be the same and so would the AR= option. Sorting is done within all factors, including transfer function factors, so initial values should always be given in order of increasing lags. Here is another illustration, showing initialization for a factored model with multiple inputs. The model is

and the initial values are =10, =5, =.8, =.01. You would use the following statements:

=1,

=.5,

=.1,

=.05, and

identify var=y crosscorr=(w x); estimate p=(1)(6,12) input=(/(1)w, 3$(1)x) mu=10 ar=.1 .05 .01 initval=(5$/(.8)w 1$(.5)x);

Stationarity and Invertibility By default PROC ARIMA requires that the parameter estimates for the AR and MA parts of the model always remain in the stationary and invertible regions, respectively. The NOSTABLE option removes this restriction and for high-order models may save some computer time. Note that using the NOSTABLE option does not necessarily result in an unstable model being fit, since the estimates may leave the stable region for some iterations, but still ultimately converge to stable values.

Naming of Model Parameters In the table of parameter estimates produced by the ESTIMATE statement, model parameters are referred to using the naming convention described in this section. The parameters in the noise part of the model are named as ARi,j or MAi,j, where AR refers to autoregressive parameters and MA to moving-average parameters. The subscript i refers to the particular polynomial factor, and the subscript j refers to the 251 SAS OnlineDoc : Version 8

Part 2. General Information jth term within the ith factor. These are sorted in order of increasing lag within factors, so the subscript j refers to the jth term after sorting. When inputs are used in the model, the parameters of each transfer function are named NUMi,j and DENi,j. The jth term in the ith factor of a numerator polynomial is named NUMi,j. The jth term in the ith factor of a denominator polynomial is named DENi,j. This naming process is repeated for each input variable, so if there are multiple inputs, parameters in transfer functions for different input series have the same name. The table of parameter estimates shows in the "Variable" column the input with which each parameter is associated. The parameter name shown in the "Parameter" column and the input variable name shown in the "Variable" column must be combined to fully identify transfer function parameters. The lag 0 parameter in the first numerator factor for the first input variable is named NUM1. For subsequent input variables, the lag 0 parameter in the first numerator factor is named NUMk, where k is the position of the input variable in the INPUT= option list. If the ALTPARM option is specified, the NUMk parameter is replaced by an overall scale parameter named SCALEk. For the mean and noise process parameters, the response series name is shown in the "Variable" column. The "Lag" and "Shift" for each parameter are also shown in the table of parameter estimates when inputs are used.

Missing Values and Estimation and Forecasting Estimation and forecasting are carried out in the presence of missing values by forecasting the missing values with the current set of parameter estimates. The maximum likelihood algorithm employed was suggested by Jones (1980) and is used for both unconditional least-squares (ULS) and conditional least-squares (CLS) estimation. The CLS algorithm simply fills in missing values with infinite memory forecast values, computed by forecasting ahead from the nonmissing past values as far as required by the structure of the missing values. These artificial values are then employed in the nonmissing value CLS algorithm. Artificial values are updated at each iteration along with parameter estimates. For models with input variables, embedded missing values (that is, missing values other than at the beginning or end of the series) are not generally ed. Embedded missing values in input variables are ed for the special case of a multiple regression model having ARIMA errors. A multiple regression model is specified by an INPUT= option that simply lists the input variables (possibly with lag shifts) without any numerator or denominator transfer function factors. One-step-ahead forecasts are not available for the response variable when one or more of the input variables have missing values. When embedded missing values are present for a model with complex transfer functions, PROC ARIMA uses the first continuous nonmissing piece of each series to do the analysis. That is, PROC ARIMA skips observations at the beginning of each series until it encounters a nonmissing value and then uses the data from there until it 252 SAS OnlineDoc : Version 8

Chapter 7. Details encounters another missing value or until the end of the data is reached. This makes the current version of PROC ARIMA compatible with earlier releases that did not allow embedded missing values.

Forecasting Details If the model has input variables, a forecast beyond the end of the data for the input variables is possible only if univariate ARIMA models have previously been fit to the input variables or future values for the input variables are included in the DATA= data set. If input variables are used, the forecast standard errors and confidence limits of the response depend on the estimated forecast error variance of the predicted inputs. If several input series are used, the forecast errors for the inputs should be independent; otherwise, the standard errors and confidence limits for the response series will not be accurate. If future values for the input variables are included in the DATA= data set, the standard errors of the forecasts will be underestimated since these values are assumed to be known with certainty. The forecasts are generated using forecasting equations consistent with the method used to estimate the model parameters. Thus, the estimation method specified on the ESTIMATE statement also controls the way forecasts are produced by the FORECAST statement. Infinite Memory Forecasts If METHOD=CLS is used, the forecasts are infinite memory forecasts, also called conditional forecasts. The term conditional is used because the forecasts are computed by assuming that the unknown values of the response series before the start of the data are equal to the mean of the series. Thus, the forecasts are conditional on this assumption. The series

can be represented as

where The k-step forecast of

. is computed as

where unobserved past values of mated parameters and .

are set to zero, and

is obtained from the esti-

Finite Memory Forecasts For METHOD=ULS or METHOD=ML, the forecasts are finite memory forecasts, also called unconditional forecasts. For finite memory forecasts, the covariance function of the ARMA model is used to derive the best linear prediction equation. 253 SAS OnlineDoc : Version 8

Part 2. General Information That is, the k-step forecast of

where is the covariance of matrix of the vector parameters.

, given

, is

and .

and

, and is the covariance are derived from the estimated

Finite memory forecasts minimize the mean-squared error of prediction if the parameters of the ARMA model are known exactly. (In most cases, the parameters of the ARMA model are estimated, so the predictors are not true best linear forecasts.) If the response series is differenced, the final forecast is produced by summing the forecast of the differenced series. This summation, and thus the forecast, is conditional on the initial values of the series. Thus, when the response series is differenced, the final forecasts are not true finite memory forecasts because they are derived assuming that the differenced series begins in a steady-state condition. Thus, they fall somewhere between finite memory and infinite memory forecasts. In practice, there is seldom any practical difference between these forecasts and true finite memory forecasts.

Forecasting Log Transformed Data The log transformation is often used to convert time series that are nonstationary with respect to the innovation variance into stationary time series. The usual approach is to take the log of the series in a DATA step and then apply PROC ARIMA to the transformed data. A DATA step is then used to transform the forecasts of the logs back to the original units of measurement. The confidence limits are also transformed using the exponential function. As one alternative, you can simply exponentiate the forecast series. This procedure gives a forecast for the median of the series, but the antilog of the forecast log series underpredicts the mean of the original series. If you want to predict the expected value of the series, you need to take into the standard error of the forecast, as shown in the following example, which uses an AR(2) model to forecast the log of a series Y: data in; set in; ylog = log( y ); run; proc arima data=in; identify var=ylog; estimate p=2; forecast lead=10 out=out; run; data out; set out; y = exp( ylog );


Chapter 7. Details l95 = exp( l95 ); u95 = exp( u95 ); forecast = exp( forecast + std*std/2 ); run;

Specifying Series Periodicity The INTERVAL= option is used together with the ID= variable to describe the observations that make up the time series. For example, INTERVAL=MONTH specifies a monthly time series in which each observation represents one month. See Chap- ter 3, “Date Intervals, Formats, and Functions,” for details on the interval values ed. The variable specified by the ID= option in the PROC ARIMA statement identifies the time periods associated with the observations. Usually, SAS date or datetime values are used for this variable. PROC ARIMA uses the ID= variable in the following ways: to validate the data periodicity. When the INTERVAL= option is specified, PROC ARIMA uses the ID variable to check the data and that successive observations have valid ID values corresponding to successive time intervals. When the INTERVAL= option is not used, PROC ARIMA verifies that the ID values are nonmissing and in ascending order. to check for gaps in the input observations. For example, if INTERVAL=MONTH and an input observation for April 1970 follows an observation for January 1970, there is a gap in the input data with two omitted observations (namely February and March 1970). A warning message is printed when a gap in the input data is found. to label the forecast observations in the output data set. PROC ARIMA extrapolates the values of the ID variable for the forecast observations from the ID value at the end of the input data according to the frequency specifications of the INTERVAL= option. If the INTERVAL= option is not specified, PROC ARIMA extrapolates the ID variable by incrementing the ID variable value for the last observation in the input data by 1 for each forecast period. Values of the ID variable over the range of the input data are copied to the output data set. The ALIGN= option is used to align the ID variable to the beginning, middle or end of the time ID interval specified by the INTERVAL= option.

OUT= Data Set The output data set produced by the OUT= option of the PROC ARIMA or FORECAST statements contains the following: the BY variables the ID variable 255 SAS OnlineDoc : Version 8

Part 2. General Information the variable specified by the VAR= option in the IDENTIFY statement, which contains the actual values of the response series FORECAST, a numeric variable containing the one-step-ahead predicted values and the multistep forecasts STD, a numeric variable containing the standard errors of the forecasts a numeric variable containing the lower confidence limits of the forecast. This variable is named L95 by default but has a different name if the ALPHA= option specifies a different size for the confidence limits. RESIDUAL, a numeric variable containing the differences between actual and forecast values a numeric variable containing the upper confidence limits of the forecast. This variable is named U95 by default but has a different name if the ALPHA= option specifies a different size for the confidence limits. The ID variable, the BY variables, and the time series variable are the only ones copied from the input to the output data set. Unless the NOOUTALL option is specified, the data set contains the whole time series. The FORECAST variable has the one-step forecasts (predicted values) for the input periods, followed by n forecast values, where n is the LEAD= value. The actual and RESIDUAL values are missing beyond the end of the series. If you specify the same OUT= data set on different FORECAST statements, the latter FORECAST statements overwrite the output from the previous FORECAST statements. If you want to combine the forecasts from different FORECAST statements in the same output data set, specify the OUT= option once on the PROC ARIMA statement and omit the OUT= option on the FORECAST statements. When a global output data set is created by the OUT= option in the PROC ARIMA statement, the variables in the OUT= data set are defined by the first FORECAST statement that is executed. The results of subsequent FORECAST statements are vertically concatenated onto the OUT= data set. Thus, if no ID variable is specified in the first FORECAST statement that is executed, no ID variable appears in the output data set, even if one is specified in a later FORECAST statement. If an ID variable is specified in the first FORECAST statement that is executed but not in a later FORECAST statement, the value of the ID variable is the same as the last value processed for the ID variable for all observations created by the later FORECAST statement. Furthermore, even if the response variable changes in subsequent FORECAST statements, the response variable name in the output data set will be that of the first response variable analyzed.

OUTCOV= Data Set The output data set produced by the OUTCOV= option of the IDENTIFY statement contains the following variables: LAG, a numeric variable containing the lags corresponding to the values of the covariance variables. The values of LAG range from 0 to N for covariance 256 SAS OnlineDoc : Version 8

Chapter 7. Details functions and from -N to N for cross-covariance functions, where N is the value of the NLAG= option. VAR, a character variable containing the name of the variable specified by the VAR= option. CROSSVAR, a character variable containing the name of the variable specified in the CROSSCORR= option, which labels the different cross-covariance functions. The CROSSVAR variable is blank for the autocovariance observations. When there is no CROSSCORR= option, this variable is not created. N, a numeric variable containing the number of observations used to calculate the current value of the covariance or cross-covariance function. COV, a numeric variable containing the autocovariance or cross-covariance function values. COV contains the autocovariances of the VAR= variable when the value of the CROSSVAR variable is blank. Otherwise COV contains the cross covariances between the VAR= variable and the variable named by the CROSSVAR variable. CORR, a numeric variable containing the autocorrelation or cross-correlation function values. CORR contains the autocorrelations of the VAR= variable when the value of the CROSSVAR variable is blank. Otherwise CORR contains the cross correlations between the VAR= variable and the variable named by the CROSSVAR variable. STDERR, a numeric variable containing the standard errors of the autocorrelations. The standard error estimate is based on the hypothesis that the process generating the time series is a pure moving-average process of order LAG-1. For the cross correlations, STDERR contains the value , which approximates the standard error under the hypothesis that the two series are uncorrelated. INVCORR, a numeric variable containing the inverse autocorrelation function values of the VAR= variable. For cross-correlation observations, (that is, when the value of the CROSSVAR variable is not blank), INVCORR contains missing values. PARTCORR, a numeric variable containing the partial autocorrelation function values of the VAR= variable. For cross-correlation observations (that is, when the value of the CROSSVAR variable is not blank), PARTCORR contains missing values.

OUTEST= Data Set PROC ARIMA writes the parameter estimates for a model to an output data set when the OUTEST= option is specified in the ESTIMATE statement. The OUTEST= data set contains the following: the BY variables – NAME– , a character variable containing the name of the parameter for the covariance or correlation observations, or blank for the observations containing 257 SAS OnlineDoc : Version 8

Part 2. General Information the parameter estimates. (This variable is not created if neither OUTCOV nor OUTCORR is specified.) – TYPE– , a character variable that identifies the type of observation. A description of the – TYPE– variable values is given below. variables for model parameters The variables for the model parameters are named as follows: ERRORVAR

This numeric variable contains the variance estimate. The TYPE =EST observation for this variable contains the estimated – – error variance, and the remaining observations are missing.

MU

This numeric variable contains values for the mean parameter for the model. (This variable is not created if NOCONSTANT is specified.)

MAj– k

These numeric variables contain values for the moving average parameters. The variables for moving average parameters are named MAj– k, where j is the factor number, and k is the index of the parameter within a factor.

ARj– k

These numeric variables contain values for the autoregressive parameters. The variables for autoregressive parameters are named ARj– k, where j is the factor number, and k is the index of the parameter within a factor.

Ij– k

These variables contain values for the transfer function parameters. Variables for transfer function parameters are named Ij– k, where j is the number of the INPUT variable associated with the transfer function component, and k is the number of the parameter for the particular INPUT variable. INPUT variables are numbered according to the order in which they appear in the INPUT= list.

– STATUS–

This variable describes the convergence status of the model. A value of 0– CONVERGED indicates that the model converged.

The value of the – TYPE– variable for each observation indicates the kind of value contained in the variables for model parameters for the observation. The OUTEST= data set contains observations with the following – TYPE– values: EST

the observation contains parameter estimates

STD

the observation contains approximate standard errors of the estimates

CORR

the observation contains correlations of the estimates. OUTCORR must be specified to get these observations.

COV

the observation contains covariances of the estimates. OUTCOV must be specified to get these observations.


Chapter 7. Details FACTOR

the observation contains values that identify for each parameter the factor that contains it. Negative values indicate denominator factors in transfer function models.

LAG

the observation contains values that identify the lag associated with each parameter

SHIFT

the observation contains values that identify the shift associated with the input series for the parameter

The values given for – TYPE– =FACTOR, – TYPE– =LAG, or – TYPE– =SHIFT observations enable you to reconstruct the model employed when provided with only the OUTEST= data set. OUTEST= Examples This section clarifies how model parameters are stored in the OUTEST= data set with two examples. Consider the following example: proc arima data=input; identify var=y cross=(x1 x2); estimate p=(1)(6) q=(1,3)(12) input=(x1 x2) outest=est; quit; proc print data=est; run;

The model specified by these statements is

The OUTEST= data set contains the values shown in Table 7.9. Table 7.9. Obs 1 2 3 4 5

OUTEST= Data Set for First Example

– TYPE– EST STD FACTOR LAG SHIFT

Y

MU

MA1– 1

MA1– 2

MA2– 1

AR1– 1

AR2– 1

. . . .

se 0 0 0

se

se

se

se

se

1 1 0

1 3 0

2 12 0

1 1 0

I1– 1 se

2 6 0

I2– 1 se

1 0 0

1 0 0

Note that the symbols in the rows for – TYPE– =EST and – TYPE– =STD in Table 7.9 would be numeric values in a real data set. Next, consider the following example: proc arima data=input; identify var=y cross=(x1(2) x2(1)); estimate p=1 q=1 input=(2 $ (1)/(1,2)x1 1 $ /(1)x2) outest=est; quit; proc print data=est; run;


Part 2. General Information The model specified by these statements is

The OUTEST= data set contains the values shown in Table 7.10. Table 7.10. Obs 1 2 3 4 5

OUTEST= Data Set for Second Example

– TYPE– EST STD FACTOR LAG SHIFT

Y

MU

. . . .

se 0 0 0

MA1– 1 se

AR1– 1 se

1 1 0

I1– 1 se

1 1 0

I1– 2 se

1 0 2

I1– 3 se

1 1 2

I1– 4 se

-1 1 2

I2– 1 se

-1 2 2

I2– 2 se

1 0 1

-1 1 1

OUTMODEL= Data Set The OUTMODEL= option in the ESTIMATE statement writes an output data set that enables you to reconstruct the model. The OUTMODEL= data set contains much the same information as the OUTEST= data set but in a transposed form that may be more useful for some purposes. In addition, the OUTMODEL= data set includes the differencing operators. The OUTMODEL data set contains the following: the BY variables – NAME– , a character variable containing the name of the response or input variable for the observation. – TYPE– , a character variable that contains the estimation method that was employed. The value of – TYPE– can be CLS, ULS, or ML. – STATUS– This variable describes the convergence status of the model. A value of 0– CONVERGED indicates that the model converged. – PARM– , a character variable containing the name of the parameter given by the observation. – PARM– takes on the values ERRORVAR, MU, AR, MA, NUM, DEN, and DIF. – VALUE– , a numeric variable containing the value of the estimate defined by the – PARM– variable. – STD– , a numeric variable containing the standard error of the estimate. – FACTOR– , a numeric variable indicating the number of the factor to which the parameter belongs. – LAG– , a numeric variable containing the number of the term within the factor containing the parameter. – SHIFT– , a numeric variable containing the shift value for the input variable associated with the current parameter. The values of – FACTOR– and – LAG– identify which particular MA, AR, NUM, or DEN parameter estimate is given by the – VALUE– variable. The – NAME– variable 260 SAS OnlineDoc : Version 8

Chapter 7. Details contains the response variable name for the MU, AR, or MA parameters. Otherwise, – NAME– contains the input variable name associated with NUM or DEN parameter estimates. The – NAME– variable contains the appropriate variable name associated with the current DIF observation as well. The – VALUE– variable is 1 for all DIF observations, and the – LAG– variable indicates the degree of differencing employed. The observations contained in the OUTMODEL= data set are identified by the – PARM– variable. A description of the values of the – PARM– variable follows: NUMRESID NPARMS

– VALUE– contains the number of residuals. – VALUE– contains the number of parameters in the model.

NDIFS

– VALUE– contains the sum of the differencing lags employed for the response variable.

ERRORVAR

– VALUE– contains the estimate of the innovation variance. – VALUE– contains the estimate of the mean term.

MU AR

– VALUE– contains the estimate of the autoregressive parameter indexed by the – FACTOR– and – LAG– variable values.

MA

– VALUE– contains the estimate of a moving average parameter indexed by the – FACTOR– and – LAG– variable values.

NUM

– VALUE– contains the estimate of the parameter in the numerator factor of the transfer function of the input variable indexed by the – FACTOR– , – LAG– , and – SHIFT– variable values. – VALUE– contains the estimate of the parameter in the denominator factor of the transfer function of the input variable indexed by the – FACTOR– , – LAG– , and – SHIFT– variable values.

DEN

DIF

– VALUE– contains the difference operator defined by the difference lag given by the value in the – LAG– variable.

OUTSTAT= Data Set PROC ARIMA writes the diagnostic statistics for a model to an output data set when the OUTSTAT= option is specified in the ESTIMATE statement. The OUTSTAT data set contains the following: the BY variables. – TYPE– , a character variable that contains the estimation method used. – TYPE– can have the value CLS, ULS, or ML. – STAT– , a character variable containing the name of the statistic given by the – VALUE– variable in this observation. – STAT– takes on the values AIC, SBC, LOGLIK, SSE, NUMRESID, NPARMS, NDIFS, ERRORVAR, MU, CONV, and NITER. – VALUE– , a numeric variable containing the value of the statistic named by the – STAT– variable. 261 SAS OnlineDoc : Version 8

Part 2. General Information The observations contained in the OUTSTAT= data set are identified by the – STAT– variable. A description of the values of the – STAT– variable follows: AIC

Akaike’s information criterion

SBC

Schwarz’s Bayesian criterion

LOGLIK

the log likelihood, if METHOD=ML or METHOD=ULS is specified

SSE

the sum of the squared residuals

NUMRESID

the number of residuals

NPARMS

the number of parameters in the model

NDIFS

the sum of the differencing lags employed for the response variable

ERRORVAR

the estimate of the innovation variance

MU

the estimate of the mean term

CONV

tells if the estimation converged

NITER

the number of iterations

Printed Output The ARIMA procedure produces printed output for each of the IDENTIFY, ESTIMATE, and FORECAST statements. The output produced by each ARIMA statement is described in the following sections. IDENTIFY Statement Printed Output The printed output of the IDENTIFY statement consists of the following: 1. a table of summary statistics, including the name of the response variable, any specified periods of differencing, the mean and standard deviation of the response series after differencing, and the number of observations after differencing 2. a plot of the sample autocorrelation function for lags up to and including the NLAG= option value. Standard errors of the autocorrelations also appear to the right of the autocorrelation plot if the value of LINESIZE= option is sufficiently large. The standard errors are derived using Bartlett’s approximation (Box and Jenkins 1976, p. 177). The approximation for a standard error for the estimated autocorrelation function at lag k is based on a null hypothesis that a pure moving-average Gaussian process of order k-1 generated the time series. The relative position of an approximate 95% confidence interval under this null hypothesis is indicated by the dots in the plot, while the asterisks represent the relative magnitude of the autocorrelation value. 3. a plot of the sample inverse autocorrelation function. See the section "The Inverse Autocorrelation Function" for more information on the inverse autocorrelation function.


Chapter 7. Details 4. a plot of the sample partial autocorrelation function 5. a table of test statistics for the hypothesis that the series is white noise. These test statistics are the same as the tests for white noise residuals produced by the ESTIMATE statement and are described in the section "Estimation Details" earlier in this chapter. 6. if the CROSSCORR= option is used, a plot of the sample cross-correlation function for each series specified in the CROSSCORR= option. If a model was previously estimated for a variable in the CROSSCORR= list, the cross correlations for that series are computed for the prewhitened input and response series. For each input variable with a prewhitening filter, the cross-correlation report for the input series includes (a) a table of test statistics for the hypothesis of no cross correlation between the input and response series (b) the prewhitening filter used for the prewhitening transformation of the predictor and response variables 7. if the ESACF option is used, ESACF tables are printed 8. if the MINIC option is used, a MINIC table is printed 9. if the SCAN option is used, SCAN table is printed 10. if the STATIONARITY option is used, STATIONARITY tests results are printed ESTIMATE Statement Printed Output The printed output of the ESTIMATE statement consists of the following: 1. when the PRINTALL option is specified, the preliminary parameter estimates and an iteration history showing the sequence of parameter estimates tried during the fitting process 2. a table of parameter estimates showing the following for each parameter: the parameter name, the parameter estimate, the approximate standard error, t value, approximate probability ( ), the lag for the parameter, the input variable name for the parameter, and the lag or "Shift" for the input variable 3. the estimates of the constant term, the innovation variance (Variance Estimate), the innovation standard deviation (Std Error Estimate), Akaike’s information criterion (AIC), Schwarz’s Bayesian criterion (SBC), and the number of residuals 4. the correlation matrix of the parameter estimates 5. a table of test statistics for hypothesis that the residuals of the model are white noise titled "Autocorrelation Check of Residuals" 6. if the PLOT option is specified, autocorrelation, inverse autocorrelation, and partial autocorrelation function plots of the residuals 263 SAS OnlineDoc : Version 8

Part 2. General Information 7. if an INPUT variable has been modeled in such a way that prewhitening is performed in the IDENTIFY step, a table of test statistics titled "Crosscorrelation Check of Residuals." The test statistic is based on the chi-square approximation suggested by Box and Jenkins (1976, pp. 395–396). The cross-correlation function is computed using the residuals from the model as one series and the prewhitened input variable as the other series. 8. if the GRID option is specified, the sum-of-squares or likelihood surface over a grid of parameter values near the final estimates 9. a summary of the estimated model showing the autoregressive factors, moving average factors, and transfer function factors in back shift notation with the estimated parameter values. FORECAST Statement Printed Output The printed output of the FORECAST statement consists of the following: 1. a summary of the estimated model 2. a table of forecasts, with columns for the observation numbers (Obs), the forecast values (Forecast), the forecast standard errors (Std Error), lower and upper limits of the approximate 95% confidence interval (95% confidence limits). The ALPHA= option can be used to change the confidence interval for forecasts. If the PRINTALL option is specified, the forecast table also includes columns for the actual values of the response series (Actual) and the residual values (Residual), and the table includes the input observations used to estimate the model.

ODS Table Names PROC ARIMA assigns a name to each table it creates. You can use these names to reference the table when using the Output Delivery System (ODS) to select tables and create output data sets. These names are listed in the following table. For more information on ODS, see Chapter 6, “Using the Output Delivery System.” Table 7.11.

ODS Table Name

ODS Tables Produced in PROC ARIMA

Description

Option

ODS Tables Created by the IDENTIFY Statement DescStats InputDescStats CorrGraph StationarityTests

Descriptive Statistics Input Descriptive Statistics Correlations graph Stationarity tests

TentativeOrders

Tenative Order Selection

PACFGraph

Partial autocorrelations graph 264

SAS OnlineDoc : Version 8

STATIONARITY option MINIC, ESACF, or SCAN option

Chapter 7. Details Table 7.11.

(continued)

ODS Table Name IACFGraph ChiSqAuto ChiSqCross MINIC ESACF ESACFPValues SCAN SCANPValues

Description Inverse autocorrelations graph Chi-Square statistics table for autocorrelation Chi-Square statistics table for crosscorrelations Minimum Information Criterion Extended Sample Autocorrelation Function ESACF Probability Values Squared Canonical Correlation Estimates SCAN Chi-Square[1] Probability Values

Option

CROSSCORR= MINIC option ESACF option ESACF option SCAN option SCAN option

ODS Tables Created by the ESTIMATE Statement FitStatistics ARPolynomial MAPolynomial NumPolynomial DenPolynomial ParameterEstimates ChiSqAuto ChiSqCross InitialAREstimates InitialMAEstimates PrelimEstimates IterHistory OptSummary ModelDescription InputDescription ObjectiveGrid CorrB

Fit Statistics Filter Equations Filter Equations Filter Equations Filter Equations ParameterEstimates Chi-Square statistics table for autocorrelation Chi-Square statistics table for crosscorrelations Initial autoregressive parameter estimates Initial moving average parameter estimates Preliminary Estimation Conditional Least Squares Estimation ARIMA Estimation Optimization Model description Input description Objective function grid matrix Correlations of the Estimates

METHOD=CLS PRINTALL option

GRID option

ODS Tables Created by the FORECAST Statement Forecasts

Forecast



Examples Example 7.1. Simulated IMA Model This example illustrates the ARIMA procedure results for a case where the true model is known. An integrated moving average model is used for this illustration. The following DATA step generates a pseudo-random sample of 100 periods from the ARIMA(0,1,1) process , . title1 ’Simulated IMA(1,1) Series’; data a; u1 = 0.9; a1 = 0; do i = -50 to 100; a = rannor( 32565 ); u = u1 + a - .8 * a1; if i > 0 then output; a1 = a; u1 = u; end; run;

The following ARIMA procedure statements identify and estimate the model. proc arima identify run; identify run; estimate run; quit;

data=a; var=u nlag=15; var=u(1) nlag=15; q=1 ;

The results of the first IDENTIFY statement are shown in Output 7.1.1. The output shows the behavior of the sample autocorrelation function when the process is nonstationary. Note that in this case the estimated autocorrelations are not very high, even at small lags. Nonstationarity is reflected in a pattern of significant autocorrelations that do not decline quickly with increasing lag, not in the size of the autocorrelations.


Chapter 7. Examples Output 7.1.1.

Output from the First IDENTIFY Statement Simulated IMA(1,1) Series The ARIMA Procedure Name of Variable = u Mean of Working Series Standard Deviation Number of Observations

0.099637 1.115604 100

Autocorrelations Lag

Covariance

Correlation

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1.244572 0.547457 0.534787 0.569849 0.384428 0.405137 0.253617 0.321830 0.363871 0.271180 0.419208 0.298127 0.186460 0.313270 0.314594 0.156329

1.00000 0.43988 0.42970 0.45787 0.30888 0.32552 0.20378 0.25859 0.29237 0.21789 0.33683 0.23954 0.14982 0.25171 0.25277 0.12561

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | | | | | | | |

. . . . . . . . . . . . . . .

|********************| |********* | |********* | |********* | |****** | |******* | |**** . | |***** . | |******. | |**** . | |******* | |***** . | |*** . | |***** . | |***** . | |*** . |



Part 2. General Information The ARIMA Procedure Inverse Autocorrelations Lag

Correlation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-0.12382 -0.17396 -0.19966 -0.01476 -0.02895 0.20612 0.01258 -0.09616 0.00025 -0.16879 0.05680 0.14306 -0.02466 -0.15549 0.08247

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | | | | | | |

. **| . .***| . ****| . . | . . *| . . |**** . | . . **| . . | . .***| . . |* . . |***. . | . .***| . . |** .

| | | | | | | | | | | | | | |

Partial Autocorrelations Lag

Correlation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.43988 0.29287 0.26499 -0.00728 0.06473 -0.09926 0.10048 0.12872 0.03286 0.16034 -0.03794 -0.14469 0.06415 0.15482 -0.10989

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | | | | | | |

. |********* . |****** . |***** . | . . |* . . **| . . |** . . |***. . |* . . |***. . *| . .***| . . |* . . |***. . **| .

| | | | | | | | | | | | | | |

The ARIMA Procedure Autocorrelation Check for White Noise To Lag

ChiSquare

DF

Pr > ChiSq

6 12

87.22 131.39

6 12

<.0001 <.0001

---------------Autocorrelations--------------0.440 0.259

0.430 0.292

0.458 0.218

0.309 0.337

0.326 0.240

0.204 0.150

The second IDENTIFY statement differences the series. The results of the second IDENTIFY statement are shown in Output 7.1.2. This output shows autocorrelation, inverse autocorrelation, and partial autocorrelation functions typical of MA(1) processes.



Output from the Second IDENTIFY Statement The ARIMA Procedure Name of Variable = u

Period(s) of Differencing Mean of Working Series Standard Deviation Number of Observations Observation(s) eliminated by differencing

1 0.019752 1.160921 99 1


Covariance

Correlation

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1.347737 -0.699404 -0.036142 0.245093 -0.234167 0.181778 -0.184601 0.0088659 0.146372 -0.241579 0.240512 0.031005 -0.250954 0.095295 0.194110 -0.219688

1.00000 -.51895 -.02682 0.18186 -.17375 0.13488 -.13697 0.00658 0.10861 -.17925 0.17846 0.02301 -.18620 0.07071 0.14403 -.16300

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | | | | | | | |

|********************| **********| . | . *| . | . |****. | . ***| . | . |*** . | . ***| . | . | . | . |** . | .****| . | . |****. | . | . | . ****| . | . |* . | . |*** . | . ***| . |




Correlation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.72538 0.48987 0.35415 0.34169 0.33466 0.34003 0.24192 0.12899 0.06597 0.01654 0.06434 0.08659 0.02485 -0.03545 -0.00113

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | | | | | | |

. . . . . . . . . . . . . . .

|*************** |********** |******* |******* |******* |******* |***** |***. |* . | . |* . |** . | . *| . | .

| | | | | | | | | | | | | | |


Correlation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-0.51895 -0.40526 -0.07862 -0.14588 0.02735 -0.13782 -0.16741 -0.06041 -0.18372 -0.01478 0.14277 -0.04345 -0.19959 0.08302 0.00278

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | | | | | | |

**********| . ********| . . **| . .***| . . |* . .***| . .***| . . *| . ****| . . | . . |***. . *| . ****| . . |** . . | .

| | | | | | | | | | | | | | |


ChiSquare

DF

Pr > ChiSq

6 12

38.13 50.62

6 12

<.0001 <.0001

---------------Autocorrelations---------------0.519 0.007

-0.027 0.182 0.109 -0.179

-0.174 0.178

0.135 0.023

-0.137 -0.186

The ESTIMATE statement fits an ARIMA(0,1,1) model to the simulated data. Note that in this case the parameter estimates are reasonably close to the values used to generate the simulated data. ( ) The ESTIMATE statement results are shown in Output 7.1.3.



Output from Fitting ARIMA(0,1,1) Model The ARIMA Procedure Conditional Least Squares Estimation

Parameter MU MA1,1

Estimate

Approx Std Error

t Value

Pr > |t|

Lag

0.02056 0.79142

0.01972 0.06474

1.04 12.22

0.2997 <.0001

0 1

Constant Estimate 0.020558 Variance Estimate 0.819807 Std Error Estimate 0.905432 AIC 263.2594 SBC 268.4497 Number of Residuals 99 * AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter MU MA1,1

MU

MA1,1

1.000 -0.124

-0.124 1.000

Autocorrelation Check of Residuals To Lag

ChiSquare

DF

Pr > ChiSq

6 12 18 24

6.48 13.11 20.12 24.73

5 11 17 23

0.2623 0.2862 0.2680 0.3645

---------------Autocorrelations---------------0.033 -0.048 0.069 0.064

0.030 0.046 0.130 0.032

0.153 -0.086 -0.099 0.076

-0.096 0.159 0.006 -0.077

0.013 0.027 0.164 -0.075

-0.163 -0.145 -0.013 0.114

Model for variable u Estimated Mean Period(s) of Differencing

0.020558 1

Moving Average Factors Factor 1:

1 - 0.79142 B**(1)

Example 7.2. Seasonal Model for the Airline Series The airline enger data, given as Series G in Box and Jenkins (1976), has been used in time series analysis literature as an example of a nonstationary seasonal time series. This example uses PROC ARIMA to fit the "airline model," ARIMA(0,1,1) (0,1,1) , to Box and Jenkins’ Series G. The following statements read the data and log transform the series. The PROC GPLOT step plots the series, as shown in Output 7.2.1.


Part 2. General Information title1 ’International Airline engers’; title2 ’(Box and Jenkins Series-G)’; data seriesg; input x @@; xlog = log( x ); date = intnx( ’month’, ’31dec1948’d, _n_ format date monyy.; datalines; 112 118 132 129 121 135 148 148 136 119 104 115 126 141 135 125 149 170 170 158 133 114 145 150 178 163 172 178 199 199 184 162 146 171 180 193 181 183 218 230 242 209 191 172 196 196 236 235 229 243 264 272 237 211 180 204 188 235 227 234 264 302 293 259 229 203 242 233 267 269 270 315 364 347 312 274 237 284 277 317 313 318 374 413 405 355 306 271 315 301 356 348 355 422 465 467 404 347 305 340 318 362 348 363 435 491 505 404 359 310 360 342 406 396 420 472 548 559 463 407 362 417 391 419 461 472 535 622 606 508 461 390 ;

);

118 140 166 194 201 229 278 306 336 337 405 432

symbol1 i= v=dot; proc gplot data=seriesg; plot x * date = 1 / haxis= ’1jan49’d to ’1jan61’d by year; run;

Output 7.2.1.

Plot of Data

The following PROC ARIMA step fits an ARIMA(0,1,1) (0,1,1) model without a mean term to the logarithms of the airline engers series. The model is forecast, and the results stored in the data set B.


Chapter 7. Examples proc arima data=seriesg; identify var=xlog(1,12) nlag=15; run; estimate q=(1)(12) noconstant method=uls; run; forecast out=b lead=24 id=date interval=month noprint; quit;

The printed output from the IDENTIFY statement is shown in Output 7.2.2. The autocorrelation plots shown are for the twice differenced series . Note that the autocorrelation functions have the pattern characteristic of a first-order moving average process combined with a seasonal moving average process with lag 12. Output 7.2.2.

IDENTIFY Statement Output The ARIMA Procedure Name of Variable = xlog

Period(s) of Differencing Mean of Working Series Standard Deviation Number of Observations Observation(s) eliminated by differencing

1,12 0.000291 0.045673 131 13


Covariance

Correlation

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.0020860 -0.0007116 0.00021913 -0.0004217 0.00004456 0.00011610 0.00006426 -0.0001159 -1.5867E-6 0.00036791 -0.0001593 0.00013431 -0.0008065 0.00031624 -0.0001202 0.00031200

1.00000 -.34112 0.10505 -.20214 0.02136 0.05565 0.03080 -.05558 -.00076 0.17637 -.07636 0.06438 -.38661 0.15160 -.05761 0.14957

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | | | | | | | |

|********************| *******| . | . |** . | ****| . | . | . | . |* . | . |* . | . *| . | . | . | . |**** | . **| . | . |* . | ********| . | . |*** . | . *| . | . |*** . |




Correlation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.41027 0.12711 0.10189 0.01978 -0.10310 -0.11886 -0.04088 -0.05086 -0.06022 0.06460 0.19907 0.31709 0.12434 0.06583 0.01515

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | | | | | | |

. |******** . |*** . |**. . | . .**| . .**| . . *| . . *| . . *| . . |* . . |**** . |****** . |**. . |* . . | .

| | | | | | | | | | | | | | |


Correlation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-0.34112 -0.01281 -0.19266 -0.12503 0.03309 0.03468 -0.06019 -0.02022 0.22558 0.04307 0.04659 -0.33869 -0.10918 -0.07684 -0.02175

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | | | | | | |

*******| . . | . ****| . ***| . . |* . . |* . . *| . . | . . |***** . |* . . |* . *******| . .**| . .**| . . | .

| | | | | | | | | | | | | | |


ChiSquare

DF

Pr > ChiSq

6 12

23.27 51.47

6 12

0.0007 <.0001

---------------Autocorrelations---------------0.341 -0.056

0.105 -0.001

-0.202 0.176

0.021 -0.076

0.056 0.064

The results of the ESTIMATE statement are shown in Output 7.2.3.


0.031 -0.387


ESTIMATE Statement Output The ARIMA Procedure Unconditional Least Squares Estimation

Parameter MA1,1 MA2,1

Estimate

Approx Std Error

t Value

Pr > |t|

Lag

0.39594 0.61331

0.08149 0.07961

4.86 7.70

<.0001 <.0001

1 12

Variance Estimate Std Error Estimate AIC SBC Number of Residuals

0.001363 0.036921 -484.755 -479.005 131

Correlations of Parameter Estimates Parameter MA1,1 MA2,1

MA1,1

MA2,1

1.000 -0.055

-0.055 1.000


ChiSquare

DF

Pr > ChiSq

6 12 18 24

5.56 8.49 13.23 24.99

4 10 16 22

0.2349 0.5816 0.6560 0.2978

---------------Autocorrelations--------------0.022 -0.065 0.022 -0.106

0.024 -0.042 0.039 -0.104

-0.125 0.102 0.045 -0.037

-0.129 -0.060 -0.162 -0.027

0.057 0.023 0.035 0.219

0.065 0.007 0.001 0.040

Model for variable xlog Period(s) of Differencing

1,12

Moving Average Factors Factor 1: Factor 2:

1 - 0.39594 B**(1) 1 - 0.61331 B**(12)

The following statements retransform the forecast values to get forecasts in the original scales. See the section "Forecasting Log Transformed Data" earlier in this chapter for more information. data c; set b; x forecast l95 u95 run;

= = = =

exp( exp( exp( exp(

xlog ); forecast + std*std/2 ); l95 ); u95 );


Part 2. General Information The forecasts and their confidence limits are plotted using the following PROC GPLOT step. The plot is shown in Output 7.2.4. symbol1 i=none v=star; symbol2 i= v=circle; symbol3 i= v=none l=3; proc gplot data=c; where date >= ’1jan58’d; plot x * date = 1 forecast * date = 2 l95 * date = 3 u95 * date = 3 / overlay haxis= ’1jan58’d to ’1jan62’d by year; run; Output 7.2.4.

Plot of the Forecast for the Original Series

Example 7.3. Model for Series J Data from Box and Jenkins This example uses the Series J data from Box and Jenkins (1976). First the input series, X, is modeled with a univariate ARMA model. Next, the dependent series, Y, is cross correlated with the input series. Since a model has been fit to X, both Y and X are prewhitened by this model before the sample cross correlations are computed. Next, a transfer function model is fit with no structure on the noise term. The residuals from this model are identified by means of the PLOT option; then, the full model, transfer function and noise is fit to the data. The following statements read Input Gas Rate and Output CO from a gas furnace. (Data values are not shown. See "Series J" in Box and Jenkins (1976) for the values.)


Chapter 7. Examples title1 ’Gas Furnace Data’; title2 ’(Box and Jenkins, Series J)’; data seriesj; input x y @@; label x = ’Input Gas Rate’ y = ’Output CO2’; datalines; ;

The following statements produce Output 7.3.1 through Output 7.3.5. proc arima data=seriesj; /*--- Look at the input process -------------------*/ identify var=x nlag=10; run; /*--- Fit a model for the input -------------------*/ estimate p=3; run; /*--- Crosscorrelation of prewhitened series ------*/ identify var=y crosscorr=(x) nlag=10; run; /*--- Fit transfer function - look at residuals ---*/ estimate input=( 3 $ (1,2)/(1,2) x ) plot; run; /*--- Estimate full model -------------------------*/ estimate p=2 input=( 3 $ (1,2)/(1) x ); run; quit;

The results of the first IDENTIFY statement for the input series X are shown in Output 7.3.1.


Part 2. General Information Output 7.3.1.

IDENTIFY Statement Results for X Gas Furnace Data (Box and Jenkins, Series J) The ARIMA Procedure Name of Variable = x Mean of Working Series Standard Deviation Number of Observations

-0.05683 1.070952 296


Covariance

Correlation

0 1 2 3 4 5 6 7 8 9 10

1.146938 1.092430 0.956652 0.782051 0.609291 0.467380 0.364957 0.298427 0.260943 0.244378 0.238942

1.00000 0.95247 0.83409 0.68186 0.53123 0.40750 0.31820 0.26019 0.22751 0.21307 0.20833

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | | |

|********************| . |******************* | . |***************** | . |************** | . |*********** | . |******** | . |****** | . |*****. | . |*****. | . |**** . | . |**** . |

"." marks two standard errors Inverse Autocorrelations Lag 1 2 3 4 5 6 7 8 9 10

Correlation -0.71090 0.26217 -0.13005 0.14777 -0.06803 -0.01147 -0.01649 0.06108 -0.04490 0.01100

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | |

**************| . . |***** ***| . . |*** .*| . . | . . | . . |*. .*| . . | .


| | | | | | | | | |

Chapter 7. Examples The ARIMA Procedure Partial Autocorrelations Lag

Correlation

1 2 3 4 5 6 7 8 9 10

0.95247 -0.78796 0.33897 0.12121 0.05896 -0.11147 0.04862 0.09945 0.01587 -0.06973

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | |

. |******************* ****************| . . |******* . |** . |*. **| . . |*. . |** . | . .*| .

| | | | | | | | | |

Autocorrelation Check for White Noise To Lag

ChiSquare

DF

Pr > ChiSq

6

786.35

6

<.0001


0.834

0.682

0.531

0.408

0.318

The ESTIMATE statement results for the AR(3) model for the input series X are shown in Output 7.3.2.



Estimates of the AR(3) Model for X The ARIMA Procedure Conditional Least Squares Estimation

Parameter

Estimate

Approx Std Error

t Value

Pr > |t|

Lag

MU AR1,1 AR1,2 AR1,3

-0.12280 1.97607 -1.37499 0.34336

0.10902 0.05499 0.09967 0.05502

-1.13 35.94 -13.80 6.24

0.2609 <.0001 <.0001 <.0001

0 1 2 3

Constant Estimate -0.00682 Variance Estimate 0.035797 Std Error Estimate 0.1892 AIC -141.667 SBC -126.906 Number of Residuals 296 * AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter MU AR1,1 AR1,2 AR1,3

MU

AR1,1

AR1,2

AR1,3

1.000 -0.017 0.014 -0.016

-0.017 1.000 -0.941 0.790

0.014 -0.941 1.000 -0.941

-0.016 0.790 -0.941 1.000


ChiSquare

DF

Pr > ChiSq

6 12 18 24 30 36 42 48

10.30 19.89 27.92 31.05 34.58 38.84 41.18 42.73

3 9 15 21 27 33 39 45

0.0162 0.0186 0.0221 0.0729 0.1499 0.2231 0.3753 0.5687

---------------Autocorrelations---------------0.042 0.014 0.099 -0.078 -0.007 0.010 0.002 0.018

0.068 0.002 0.043 0.024 -0.004 0.002 0.033 0.051

0.056 -0.055 -0.082 0.015 0.073 0.082 -0.061 -0.012

-0.145 0.035 0.017 0.030 -0.038 0.045 -0.003 0.015

-0.009 0.143 0.066 0.045 -0.062 0.056 -0.006 -0.027

0.059 -0.079 -0.052 0.004 0.003 -0.023 -0.043 0.020

The ARIMA Procedure Model for variable x Estimated Mean

-0.1228


1 - 1.97607 B**(1) + 1.37499 B**(2) - 0.34336 B**(3)

The IDENTIFY statement results for the dependent series Y cross correlated with the input series X is shown in Output 7.3.3. Since a model has been fit to X, both Y and X are prewhitened by this model before the sample cross correlations are computed. 280 SAS OnlineDoc : Version 8


IDENTIFY Statement for Y Cross Correlated with X The ARIMA Procedure Partial Autocorrelations

Lag

Correlation

1 2 3 4 5 6 7 8 9 10

0.97076 -0.80388 0.18833 0.25999 0.05949 -0.06258 -0.01435 0.05490 0.00545 0.03141

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | |

. |******************* ****************| . . |**** . |***** . |*. .*| . . | . . |*. . | . . |*.

| | | | | | | | | |

Autocorrelation Check for White Noise To Lag

ChiSquare

DF

Pr > ChiSq

6

1023.15

6

<.0001


0.896

0.793

0.680

0.574

0.485


Part 2. General Information The ARIMA Procedure Correlation of y and x Number of Observations Variance of transformed series y Variance of transformed series x

296 0.131438 0.035357

Both series have been prewhitened. Crosscorrelations Lag -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Covariance

Correlation

0.0015683 0.00013502 -0.0060480 -0.0017624 -0.0080539 -0.0000944 -0.0012802 -0.0031078 0.00065212 -0.0019166 -0.0003673 0.0038939 -0.0016971 -0.019231 -0.022479 -0.030909 -0.018122 -0.011426 -0.0017355 0.0022590 -0.0035152

0.02301 0.00198 -.08872 -.02585 -.11814 -.00138 -.01878 -.04559 0.00957 -.02811 -.00539 0.05712 -.02489 -.28210 -.32974 -.45341 -.26583 -.16761 -.02546 0.03314 -.05156

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | | | | | | | | | | | | |

. | . . | . **| . .*| . **| . . | . . | . .*| . . | . .*| . . | . . |*. . | . ******| . *******| . *********| . *****| . ***| . .*| . . |*. .*| .

| | | | | | | | | | | | | | | | | | | | |

"." marks two standard errors Crosscorrelation Check Between Series To Lag

ChiSquare

DF

5

117.75

6

Pr > ChiSq <.0001

---------------Crosscorrelations--------------0.005

0.057

-0.025

-0.282

-0.330

-0.453

The ARIMA Procedure Both variables have been prewhitened by the following filter: Prewhitening Filter


1 - 1.97607 B**(1) + 1.37499 B**(2) - 0.34336 B**(3)

The ESTIMATE statement results for the transfer function model with no structure on the noise term is shown in Output 7.3.4. The PLOT option prints the residual autocorrelation functions from this model. 282 SAS OnlineDoc : Version 8


Estimates of the Transfer Function Model The ARIMA Procedure Conditional Least Squares Estimation

Parameter

Estimate

Approx Std Error

t Value

Pr > |t|

Lag

MU NUM1 NUM1,1 NUM1,2 DEN1,1 DEN1,2

53.32237 -0.62868 0.47258 0.73660 0.15411 0.27774

0.04932 0.25385 0.62253 0.81006 0.90483 0.57345

1081.24 -2.48 0.76 0.91 0.17 0.48

<.0001 0.0138 0.4484 0.3640 0.8649 0.6285

0 0 1 2 1 2

Variable

Shift

y x x x x x

0 3 3 3 3 3


Correlations of Parameter Estimates Variable Parameter y x x x x x

MU NUM1 NUM1,1 NUM1,2 DEN1,1 DEN1,2

y MU

x NUM1

x NUM1,1

x NUM1,2

x DEN1,1

x DEN1,2

1.000 0.013 0.002 -0.002 0.004 -0.006

0.013 1.000 0.755 -0.447 0.089 -0.065

0.002 0.755 1.000 0.121 -0.538 0.565

-0.002 -0.447 0.121 1.000 -0.892 0.870

0.004 0.089 -0.538 -0.892 1.000 -0.998

-0.006 -0.065 0.565 0.870 -0.998 1.000

The ARIMA Procedure Autocorrelation Check of Residuals To Lag

ChiSquare

DF

Pr > ChiSq

6 12 18 24 30 36 42 48

496.45 498.58 539.38 561.87 585.90 592.42 593.44 601.94

6 12 18 24 30 36 42 48

<.0001 <.0001 <.0001 <.0001 <.0001 <.0001 <.0001 <.0001

---------------Autocorrelations--------------0.893 -0.003 -0.045 -0.163 0.158 -0.048 0.042 0.043

0.711 -0.040 -0.083 -0.102 0.156 -0.018 0.025 0.068

0.502 -0.054 -0.131 -0.026 0.131 0.038 0.013 0.084

0.312 -0.040 -0.170 0.047 0.081 0.070 0.004 0.082

0.167 -0.022 -0.196 0.106 0.013 0.079 0.006 0.061

0.064 -0.021 -0.195 0.142 -0.037 0.067 0.019 0.023


Part 2. General Information The ARIMA Procedure Autocorrelation Plot of Residuals Lag

Covariance

Correlation

0 1 2 3 4 5 6 7 8 9 10

0.704241 0.628846 0.500490 0.353404 0.219895 0.117330 0.044967 -0.0023551 -0.028030 -0.037891 -0.028378

1.00000 0.89294 0.71068 0.50182 0.31224 0.16660 0.06385 -.00334 -.03980 -.05380 -.04030

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | | |

|********************| . |****************** | . |************** | . |********** | . |****** | . |*** . | . |* . | . | . | . *| . | . *| . | . *| . |

"." marks two standard errors Inverse Autocorrelations Lag 1 2 3 4 5 6 7 8 9 10

Correlation -0.57346 0.02264 0.03631 0.03941 -0.01256 -0.01618 0.02680 -0.05895 0.07043 -0.02987

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | |

***********| . . | . . |*. . |*. . | . . | . . |*. .*| . . |*. .*| .


| | | | | | | | | |

Chapter 7. Examples The ARIMA Procedure Partial Autocorrelations Lag

Correlation

1 2 3 4 5 6 7 8 9 10

0.89294 -0.42765 -0.13463 0.02199 0.03891 -0.02219 -0.02249 0.01538 0.00634 0.07737

-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | | | | |

. |****************** *********| . ***| . . | . . |*. . | . . | . . | . . | . . |**

| | | | | | | | | |

Crosscorrelation Check of Residuals with Input x To Lag

ChiSquare

DF

Pr > ChiSq

5 11 17 23 29 35 41 47

0.48 0.93 2.63 19.19 20.12 24.22 30.66 31.65

2 8 14 20 26 32 38 44

0.7855 0.9986 0.9996 0.5092 0.7857 0.8363 0.7953 0.9180

---------------Crosscorrelations--------------0.009 -0.006 0.012 -0.076 -0.039 -0.022 0.108 0.008

-0.005 0.008 0.035 -0.108 -0.013 -0.031 0.091 -0.011

0.026 0.022 0.037 -0.122 0.010 -0.074 0.046 -0.040

0.013 0.023 0.039 -0.122 -0.020 -0.036 0.018 -0.030

-0.017 -0.017 -0.005 -0.094 -0.031 0.014 0.003 -0.002

-0.022 -0.013 -0.040 -0.041 -0.005 0.076 0.009 0.028

Model for variable y Estimated Intercept

53.32237

Input Number 1 Input Variable Shift

x 3

Numerator Factors Factor 1:

-0.6287 - 0.47258 B**(1) - 0.7366 B**(2) Denominator Factors

Factor 1:

1 - 0.15411 B**(1) - 0.27774 B**(2)

The ESTIMATE statement results for the final transfer function model with AR(2) noise are shown in Output 7.3.5.



Estimates of the Final Model The ARIMA Procedure Conditional Least Squares Estimation

Parameter

Estimate

Approx Std Error

t Value

Pr > |t|

Lag

MU AR1,1 AR1,2 NUM1 NUM1,1 NUM1,2 DEN1,1

53.26307 1.53292 -0.63297 -0.53522 0.37602 0.51894 0.54842

0.11926 0.04754 0.05006 0.07482 0.10287 0.10783 0.03822

446.63 32.25 -12.64 -7.15 3.66 4.81 14.35

<.0001 <.0001 <.0001 <.0001 0.0003 <.0001 <.0001

0 1 2 0 1 2 1

Variable y y y x x x x


Correlations of Parameter Estimates Variable Parameter y y y x x x x


y MU

y AR1,1

y AR1,2

x NUM1

1.000 -0.063 0.047 -0.008 -0.016 0.017 -0.049

-0.063 1.000 -0.927 -0.003 0.007 -0.002 0.015

0.047 -0.927 1.000 0.023 -0.005 0.005 -0.022

-0.008 -0.003 0.023 1.000 0.713 -0.178 -0.013

Correlations of Parameter Estimates Variable Parameter y y y x x x x


x NUM1,1

x NUM1,2

x DEN1,1

-0.016 0.007 -0.005 0.713 1.000 -0.467 -0.039

0.017 -0.002 0.005 -0.178 -0.467 1.000 -0.720

-0.049 0.015 -0.022 -0.013 -0.039 -0.720 1.000


Shift 0 0 0 3 3 3 3

Chapter 7. Examples The ARIMA Procedure Autocorrelation Check of Residuals To Lag

ChiSquare

DF

Pr > ChiSq

6 12 18 24 30 36 42 48

8.61 15.43 21.13 27.52 36.94 44.26 45.62 48.60

4 10 16 22 28 34 40 46

0.0717 0.1172 0.1734 0.1922 0.1202 0.1119 0.2500 0.3689

---------------Autocorrelations--------------0.024 0.032 -0.038 -0.118 0.034 -0.025 -0.017 0.024

0.055 0.028 0.052 -0.002 -0.021 -0.057 -0.036 0.069

-0.073 -0.081 -0.093 -0.007 0.020 0.113 -0.029 0.024

-0.054 0.047 -0.013 0.076 0.094 0.022 -0.013 0.017

-0.054 0.022 -0.073 0.024 -0.118 0.030 -0.033 0.022

0.119 0.107 -0.005 -0.004 0.065 0.065 0.017 -0.044

Crosscorrelation Check of Residuals with Input x To Lag

ChiSquare

DF

Pr > ChiSq

5 11 17 23 29 35 41 47

0.93 6.60 13.86 18.55 27.99 35.18 37.15 42.42

3 9 15 21 27 33 39 45

0.8191 0.6784 0.5365 0.6142 0.4113 0.3654 0.5544 0.5818

---------------Crosscorrelations-------------0.008 0.075 0.050 -0.074 -0.071 -0.014 0.031 0.036

0.004 -0.024 0.043 -0.078 -0.001 0.015 -0.029 -0.038

0.010 -0.019 0.014 0.023 0.038 -0.039 -0.070 -0.053

0.008 -0.026 0.014 -0.016 -0.156 0.028 -0.006 0.107

-0.045 -0.111 -0.141 0.021 0.031 0.046 0.012 0.029

0.030 0.013 -0.028 0.060 0.035 0.142 -0.004 0.021

The ARIMA Procedure Model for variable y Estimated Intercept

53.26307


1 - 1.53292 B**(1) + 0.63297 B**(2) Input Number 1 Input Variable Shift

x 3

Numerator Factors Factor 1:

-0.5352 - 0.37602 B**(1) - 0.51894 B**(2) Denominator Factors Factor 1:

1 - 0.54842 B**(1)



Example 7.4. An Intervention Model for Ozone Data This example fits an intervention model to ozone data as suggested by Box and Tiao (1975). Notice that since the response variable, OZONE, is differenced, the innovation, X1, must also be differenced to generate a step function change in the response. If X1 had not been differenced, the change in the response caused by X1 would be a (seasonal) ramp and not a step function. Notice that the final model for the differenced data is a multiple regression model with a moving-average structure assumed for the residuals. The model is fit by maximum likelihood. The seasonal moving-average parameter and its standard error are fairly sensitive to which method is chosen to fit the model, in agreement with the observations of Davidson (1981) and Ansley and Newbold (1980); thus, fitting the model by the unconditional or conditional least squares methods produce somewhat different estimates for these parameters. Some missing values are appended to the end of the input data to generate additional values for the independent variables. Since the independent variables are not modeled, values for them must be available for any times at which predicted values are desired. In this case, predicted values are requested for 12 periods beyond the end of the data. Thus, values for X1, WINTER, and SUMMER must be given for 12 periods ahead. The following statements read in the data and compute dummy variables for use as intervention inputs: title1 ’Intervention Data for Ozone Concentration’; title2 ’(Box and Tiao, JASA 1975 P.70)’; data air; input ozone @@; label ozone = ’Ozone Concentration’ x1 = ’Intervention for post 1960 period’ summer = ’Summer Months Intervention’ winter = ’Winter Months Intervention’; date = intnx( ’month’, ’31dec1954’d, _n_ ); format date monyy.; month = month( date ); year = year( date ); x1 = year >= 1960; summer = ( 5 < month < 11 ) * ( year > 1965 ); winter = ( year > 1965 ) - summer; datalines; 2.7 2.0 3.6 5.0 6.5 6.1 5.9 5.0 6.4 7.4 8.2 4.1 4.5 5.5 3.8 4.8 5.6 6.3 5.9 8.7 5.3 5.7 3.0 3.4 4.9 4.5 4.0 5.7 6.3 7.1 8.0 5.2 5.0 3.7 3.1 2.5 4.0 4.1 4.6 4.4 4.2 5.1 4.6 4.4 2.9 2.4 4.7 5.1 4.0 7.5 7.7 6.3 5.3 5.7 4.8 1.7 2.0 3.4 4.0 4.3 5.0 5.5 5.0 5.4 3.8 2.4 2.2 2.5 2.6 3.3 2.9 4.3 4.2 4.2 3.9 3.9 2.5 2.4 1.9 2.1 4.5 3.3 3.4 4.1 5.7 4.8 5.0 2.8 1.7 3.2 2.7 3.0 3.4 3.8 5.0 4.8 4.9 3.5 2.5


3.9 5.7 4.7 4.0 2.7 2.0 2.2 2.9 2.4

Chapter 7. Examples 1.6 2.1 1.8 1.8 2.1 2.0 1.3 1.8 1.5 ;

2.3 2.9 1.9 2.5 2.6 1.7 2.3 2.0 2.0

2.5 2.7 3.7 2.6 2.8 2.8 2.7 2.2 3.1

3.1 4.2 4.4 1.8 3.2 3.2 3.3 3.0 3.0

3.5 3.9 3.8 3.7 3.5 4.4 3.7 2.4 3.5

4.5 4.1 5.6 3.7 3.5 3.4 3.0 3.5 3.4

5.7 4.6 5.7 4.9 4.9 3.9 3.8 3.5 4.0

5.0 5.8 5.1 5.1 4.2 5.5 4.7 3.3 3.8

4.6 4.4 5.6 3.7 4.7 3.8 4.6 2.7 3.1

4.8 6.1 4.8 5.4 3.7 3.2 2.9 2.5 2.1

2.1 3.5 2.5 3.0 3.2 2.3 1.7 1.6 1.6

1.4 1.9 1.5 1.8 1.8 2.2 1.3 1.2 1.3

The following statements produce Output 7.4.1 and Output 7.4.2: proc arima data=air; /*--- Identify and seasonally difference ozone series ---*/ identify var=ozone(12) crosscorr=( x1(12) summer winter ) noprint; /*--- Fit a multiple regression with a seasonal MA model ---*/ /*--by the maximum likelihood method ---*/ estimate q=(1)(12) input=( x1 summer winter ) noconstant method=ml itprint; /*--- Forecast ---*/ forecast lead=12 id=date interval=month; run;

The ESTIMATE statement results are shown in Output 7.4.1. Output 7.4.1.

Parameter Estimates

Intervention Data for Ozone Concentration (Box and Tiao, JASA 1975 P.70) The ARIMA Procedure Initial Moving Average Estimates Estimate 1

-0.29241

Initial Moving Average Estimates Estimate 12

0.40740

White Noise Variance Est

0.944969


Part 2. General Information The ARIMA Procedure Conditional Least Squares Estimation Iteration

SSE

MA1,1

MA2,1

NUM1

NUM2

NUM3

Lambda

0 1 2 3 4

154.53 146.20 145.88 145.88 145.88

-0.29241 -0.29256 -0.30071 -0.29976 -0.29983

0.40740 0.59844 0.59239 0.59242 0.59234

-1.13490 -1.20292 -1.26173 -1.26246 -1.26243

-0.11731 -0.29784 -0.26252 -0.26150 -0.26154

0.05581 -0.11572 -0.08247 -0.08197 -0.08196

0.00001 1E-6 1E-7 1E-8 1E-9

Conditional Least Squares Estimation Iteration

R Crit

0 1 2 3 4

1 0.230552 0.046601 0.001345 0.000125

Maximum Likelihood Estimation Iter

Loglike

MA1,1

MA2,1

NUM1

NUM2

NUM3

Lambda

R Crit

0 1 2 3 4

-249.07778 -245.89135 -245.88484 -245.88482 -245.88481

-0.29983 -0.26830 -0.26653 -0.26689 -0.26684

0.59234 0.76634 0.76623 0.76661 0.76665

-1.26243 -1.34490 -1.33046 -1.33070 -1.33062

-0.26154 -0.23984 -0.23939 -0.23936 -0.23936

-0.08196 -0.07578 -0.08025 -0.08020 -0.08021

0.00001 1E-6 1E-7 1E-8 1E-9

1 0.169445 0.008044 0.000603 0.000073

ARIMA Estimation Optimization Summary Estimation Method Parameters Estimated Termination Criteria Iteration Stopping Value Criteria Value Alternate Criteria Alternate Criteria Value Maximum Absolute Value of Gradient R-Square Change from Last Iteration Objective Function Objective Function Value Marquardt’s Lambda Coefficient Numerical Derivative Perturbation Delta Iterations


Maximum Likelihood 5 Maximum Relative Change in Estimates 0.001 0.000195 Relative Change in Objective Function 1.247E-8 0.00712 0.000073 Log Gaussian Likelihood -245.885 1E-9 0.001 4

Chapter 7. Examples The ARIMA Procedure Maximum Likelihood Estimation Parameter

Estimate

Approx Std Error

t Value

Pr > |t|

Lag

MA1,1 MA2,1 NUM1 NUM2 NUM3

-0.26684 0.76665 -1.33062 -0.23936 -0.08021

0.06710 0.05973 0.19236 0.05952 0.04978

-3.98 12.83 -6.92 -4.02 -1.61

<.0001 <.0001 <.0001 <.0001 0.1071

1 12 0 0 0

Variance Estimate Std Error Estimate AIC SBC Number of Residuals

Variable

Shift

ozone ozone x1 summer winter

0 0 0 0 0

0.634506 0.796559 501.7696 518.3602 204

Correlations of Parameter Estimates Variable Parameter ozone ozone x1 summer winter

MA1,1 MA2,1 NUM1 NUM2 NUM3

ozone MA1,1

ozone MA2,1

x1 NUM1

summer NUM2

winter NUM3

1.000 0.090 -0.039 0.062 -0.034

0.090 1.000 -0.169 0.211 0.022

-0.039 -0.169 1.000 -0.124 -0.107

0.062 0.211 -0.124 1.000 0.097

-0.034 0.022 -0.107 0.097 1.000


ChiSquare

DF

Pr > ChiSq

6 12 18 24 30 36

7.47 10.21 14.53 19.99 27.00 32.65

4 10 16 22 28 34

0.1132 0.4220 0.5593 0.5834 0.5180 0.5336

---------------Autocorrelations--------------0.017 -0.024 0.054 0.003 -0.072 0.022

0.054 -0.059 0.006 -0.074 -0.035 -0.099

0.043 -0.047 -0.110 -0.074 0.023 -0.006

0.101 0.014 0.028 0.098 -0.028 0.087

-0.022 0.032 -0.042 -0.038 -0.107 -0.046

0.140 0.072 0.043 0.043 0.100 0.053


Part 2. General Information The ARIMA Procedure Model for variable ozone Period(s) of Differencing

12

Moving Average Factors Factor 1: Factor 2:

1 + 0.26684 B**(1) 1 - 0.76665 B**(12)

Input Number 1 Input Variable Period(s) of Differencing Overall Regression Factor

x1 12 -1.33062

Input Number 2 Input Variable Overall Regression Factor

summer -0.23936

Input Number 3 Input Variable Overall Regression Factor

winter -0.08021

The FORECAST statement results are shown in Output 7.4.2. Output 7.4.2.

Forecasts The ARIMA Procedure Forecasts for variable ozone

Obs

Forecast

Std Error

217 218 219 220 221 222 223 224 225 226 227 228

1.4205 1.8446 2.4567 2.8590 3.1501 2.7211 3.3147 3.4787 2.9405 2.3587 1.8588 1.2898

0.7966 0.8244 0.8244 0.8244 0.8244 0.8244 0.8244 0.8244 0.8244 0.8244 0.8244 0.8244


95% Confidence Limits -0.1407 0.2287 0.8408 1.2431 1.5342 1.1053 1.6989 1.8629 1.3247 0.7429 0.2429 -0.3260

2.9817 3.4604 4.0725 4.4748 4.7659 4.3370 4.9306 5.0946 4.5564 3.9746 3.4746 2.9057

Chapter 7. Examples

Example 7.5. Using Diagnostics to Identify ARIMA models Fitting ARIMA models is as much an art as it is a science. The ARIMA procedure has diagnostic options to help tentatively identify the orders of both stationary and nonstationary ARIMA processes. Consider the Series A in Box et al (1994), which consists of 197 concentration readings taken every two hours from a chemical process. Let SeriesA be a data set containing these readings in a variable named X. The following SAS statements use the SCAN option of the IDENTIFY statement to generate Output 7.5.1 and Output 7.5.2. See “The SCAN Method” for details of the SCAN method. proc arima data=SeriesA; identify var=x scan; run; Output 7.5.1.

Example of SCAN Tables

SERIES A: Chemical Process Concentration Readings The ARIMA Procedure Squared Canonical Correlation Estimates Lags AR AR AR AR AR AR

0 1 2 3 4 5

MA 0

MA 1

MA 2

MA 3

MA 4

MA 5

0.3263 0.0643 0.0061 0.0072 0.0049 0.0202

0.2479 0.0012 0.0027 <.0001 0.0010 0.0009

0.1654 0.0028 0.0021 0.0007 0.0014 0.0016

0.1387 <.0001 0.0011 0.0005 0.0014 <.0001

0.1183 0.0051 0.0017 0.0019 0.0039 0.0126

0.1417 0.0002 0.0079 0.0021 0.0145 0.0001

SCAN Chi-Square[1] Probability Values Lags AR AR AR AR AR AR

0 1 2 3 4 5

MA 0

MA 1

MA 2

MA 3

MA 4

MA 5

<.0001 0.0003 0.2754 0.2349 0.3297 0.0477

<.0001 0.6649 0.5106 0.9812 0.7154 0.7254

<.0001 0.5194 0.5860 0.7667 0.7113 0.6652

0.0007 0.9235 0.7346 0.7861 0.6995 0.9576

0.0037 0.3993 0.6782 0.6810 0.5807 0.2660

0.0024 0.8528 0.2766 0.6546 0.2205 0.9168

In Output 7.5.1, there is one (maximal) rectangular region in which all the elements are insignificant with 95% confidence. This region has a vertex at (1,1). Output 7.5.2 gives recommendations based on the significance level specified by the ALPHA=siglevel option.



Example of SCAN Option Tentative Order Selection The ARIMA Procedure ARMA(p+d,q) Tentative Order Selection Tests ----SCAN--p+d q 1

1

(5% Significance Level)

Another order identification diagnostic is the extended sample autocorrelation function or ESACF method. See “The ESACF Method” for details of the ESACF method. The following statements generate Output 7.5.3 and Output 7.5.4. proc arima data=SeriesA; identify var=x esacf; run; Output 7.5.3.

Example of ESACF Tables The ARIMA Procedure Extended Sample Autocorrelation Function

Lags AR AR AR AR AR AR

0 1 2 3 4 5

MA 0

MA 1

MA 2

MA 3

MA 4

MA 5

0.5702 -0.3907 -0.2859 -0.5030 -0.4785 -0.3878

0.4951 0.0425 -0.2699 -0.0106 -0.0176 -0.4101

0.3980 -0.0605 -0.0449 0.0946 0.0827 -0.1651

0.3557 -0.0083 0.0089 -0.0137 -0.0244 0.0103

0.3269 -0.0651 -0.0509 -0.0148 -0.0149 -0.1741

0.3498 -0.0127 -0.0140 -0.0302 -0.0421 -0.0231

ESACF Probability Values Lags AR AR AR AR AR AR

0 1 2 3 4 5

MA 0

MA 1

MA 2

MA 3

MA 4

MA 5

<.0001 <.0001 <.0001 <.0001 <.0001 <.0001

<.0001 0.5974 0.0002 0.9022 0.8380 <.0001

0.0001 0.4622 0.6106 0.2400 0.3180 0.0765

0.0014 0.9198 0.9182 0.8713 0.7737 0.9142

0.0053 0.4292 0.5683 0.8930 0.8913 0.1038

0.0041 0.8768 0.8592 0.7372 0.6213 0.8103

In Output 7.5.3, there are three right-triangular regions in which all elements are insignificant at the 5% level. The triangles have vertices (1,1), (3,1), and (4,1). Since the triangle at (1,1) covers more insignificant , it is recommended first. Similarly, the remaining recommendations are ordered by the number of insignificant contained in the triangle. Output 7.5.4 gives recommendations based on the significance level specified by the ALPHA=siglevel option. 294 SAS OnlineDoc : Version 8


Example of ESACF Option Tentative Order Selection The ARIMA Procedure ARMA(p+d,q) Tentative Order Selection Tests ---ESACF--p+d q 1 3 4

1 1 1


If you also specify the SCAN option in the same IDENTIFY statement, the two recommendations are printed side by side. proc arima data=SeriesA; identify var=x scan esacf; run; Output 7.5.5.

Example of SCAN and ESACF Option Combined The ARIMA Procedure ARMA(p+d,q) Tentative Order Selection Tests ---SCAN-p+d q 1

1

--ESACF-p+d q 1 3 4

1 1 1


From above, the autoregressive and moving average orders are tentatively identified by both SCAN and ESACF tables to be ( )=(1,1). Because both the SCAN and ESACF indicate a term of 1, a unit root test should be used to determine whether this autoregressive term is a unit root. Since a moving average term appears to be present, a large autoregressive term is appropriate for the Augmented DickeyFuller test for a unit root. Submitting the following code generates Output 7.5.6. proc arima data=SeriesA; identify var=x stationarity=(adf=(5,6,7,8)); run;



Example of STATIONARITY Option Output The ARIMA Procedure Augmented Dickey-Fuller Unit Root Tests

Type Zero Mean

Single Mean

Trend

Lags

Rho

Pr < Rho

Tau

Pr < Tau

F

Pr > F

5 6 7 8 5 6 7 8 5 6 7 8

0.0403 0.0479 0.0376 0.0354 -18.4550 -10.8939 -10.9224 -10.2992 -18.4360 -10.8436 -10.7427 -10.0370

0.6913 0.6931 0.6907 0.6901 0.0150 0.1043 0.1035 0.1208 0.0871 0.3710 0.3773 0.4236

0.42 0.63 0.49 0.48 -2.67 -2.02 -1.93 -1.83 -2.66 -2.01 -1.90 -1.79

0.8024 0.8508 0.8200 0.8175 0.0821 0.2767 0.3172 0.3650 0.2561 0.5939 0.6519 0.7081

3.67 2.27 2.00 1.81 3.54 2.04 1.91 1.74

0.1367 0.4931 0.5605 0.6108 0.4703 0.7694 0.7956 0.8293

The preceding test results show that a unit root is very likely and that the series should be differenced. Based on this test and the previous results, an ARIMA(0,1,1) would be a good choice for a tentative model for Series A. Using the recommendation that the series be differenced, the following statements generate Output 7.5.7. proc arima data=SeriesA; identify var=x(1) minic; run; Output 7.5.7.

Example of MINIC Table The ARIMA Procedure Minimum Information Criterion

Lags AR AR AR AR AR AR

0 1 2 3 4 5

MA 0

MA 1

MA 2

MA 3

MA 4

MA 5

-2.05761 -2.23291 -2.23947 -2.25092 -2.25934 -2.2751

-2.3497 -2.32345 -2.30313 -2.28088 -2.2778 -2.26805

-2.32358 -2.29665 -2.28084 -2.25567 -2.25363 -2.24249

-2.31298 -2.28644 -2.26065 -2.23455 -2.22983 -2.21789

-2.30967 -2.28356 -2.25685 -2.22997 -2.20312 -2.19667

-2.28528 -2.26011 -2.23458 -2.20769 -2.19531 -2.17426

The error series is estimated using an AR(7) model, and the minimum of this MINIC table is . This diagnostic confirms the previous result indicating that an ARIMA(0,1,1) is a tentative model for Series A. If you also specify the SCAN or MINIC option in the same IDENTIFY statement, the BIC associated with the SCAN table and ESACF table recommendations are listed. proc arima data=SeriesA; identify var=x(1) minic scan esacf; run;



Example of SCAN, ESACF, MINIC Options Combined The ARIMA Procedure

ARMA(p+d,q) Tentative Order Selection Tests ---------SCAN-------p+d q BIC 0

1

-2.3497

--------ESACF-------p+d q BIC 0 1

1 1

-2.3497 -2.32345




References Akaike, H. (1974), "A New Look at the Statistical Model Identification," IEEE Transaction on Automatic Control, AC-19, 716-723. Anderson, T.W. (1971), The Statistical Analysis of Time Series, New York: John Wiley & Sons, Inc. Andrews and Herzberg (1985), A collection of problems from many fields for the student and research worker, New York: Springer Verlag. Ansley, C. (1979), "An Algorithm for the Exact Likelihood of a Mixed Autoregressive-Moving Average Process," Biometrika, 66, 59. Ansley, C. and Newbold, P. (1980), "Finite Sample Properties of Estimators for Autoregressive Moving Average Models," Journal of Econometrics, 13, 159. Bhansali, R.J. (1980), "Autoregressive and Window Estimates of the Inverse Correlation Function," Biometrika, 67, 551-566. Box, G.E.P. and Jenkins, G.M. (1976), Time Series Analysis: Forecasting and Control, San Francisco: Holden-Day. Box, G.E.P., Jenkins, G.M., and Reinsel, G.C. (1994), Time Series Analysis: Forecasting and Control, Third Edition, Englewood Cliffs, NJ: Prentice Hall, 197199. Box, G.E.P. and Tiao, G.C. (1975), "Intervention Analysis with Applications to Economic and Environmental Problems," JASA, 70, 70-79. Brocklebank, J.C. and Dickey, D.A. (1986), SAS System for Forecasting Time Series, 1986 Edition, Cary, North Carolina: SAS Institute Inc. Chatfield, C. (1980), "Inverse Autocorrelations," Journal of the Royal Statistical So- ciety, A142, 363-377. Choi, ByoungSeon (1992), ARMA Model Identification, New York: Springer-Verlag, 129-132. Cleveland, W.S. (1972), "The Inverse Autocorrelations of a Time Series and Their Applications," Technometrics, 14, 277. Davidson, J. (1981), "Problems with the Estimation of Moving Average Models," Journal of Econometrics, 16, 295. Davies, N., Triggs, C.M., and Newbold, P. (1977), "Significance Levels of the BoxPierce Portmanteau Statistic in Finite Samples," Biometrika, 64, 517-522. Dickey, D. A., and Fuller, W.A. (1979), "Distribution of the Estimators for Autoregressive Time Series With a Unit Root," Journal of the American Statistical Association, 74 (366), 427-431. Dickey, D. A., Hasza, D. P., and Fuller, W.A. (1984), "Testing for Unit Roots in Seasonal Time Series," Journal of the American Statistical Association, 79 (386), 355-367.


Chapter 7. References Dunsmuir, William (1984), "Large sample properties of estimation in time series observed at unequally spaced times," in Time Series Analysis of Irregularly Observed Data, Emanuel Parzen, ed., New York: Springer-Verlag. Fuller, W.A. (1976), Introduction to Statistical Time Series, New York: John Wiley & Sons, Inc. Hamilton, J. D. (1994), Time Series Analysis, Princeton: Princeton University Press. Hannan, E.J. and Rissanen, J. (1982), "Recursive Estimation of Mixed Autoregressive Moving Average Order," Biometrika, 69 (1), 81-94. Harvey, A.C. (1981), Time Series Models, New York: John Wiley & Sons, Inc. Jones, Richard H. (1980), "Maximum Likelihood Fitting of ARMA Models to Time Series with Missing Observations," Technometrics, 22, 389-396. Ljung, G.M. and Box, G.E.P. (1978), "On a Measure of Lack of Fit in Time Series Models," Biometrika, 65, 297-303. Montgomery, D.C. and Johnson, L.A. (1976), Forecasting and Time Series Analysis, New York: McGraw-Hill Book Co. Morf, M., Sidhu, G.S., and Kailath, T. (1974), "Some New Algorithms for Recursive Estimation on Constant Linear Discrete Time Systems," I.E.E.E. Transactions on Automatic Control, AC-19, 315-323. Nelson, C.R. (1973), Applied Time Series for Managerial Forecasting, San Francisco: Holden-Day. Newbold, P. (1981), "Some Recent Developments in Time Series Analysis," Interna- tional Statistical Review, 49, 53-66. Newton, H. Joseph and Pagano, Marcello (1983), "The Finite Memory Prediction of Covariance Stationary Time Series," SIAM Journal of Scientific and Statistical Computing, 4, 330-339. Pankratz, Alan (1983), Forecasting with Univariate Box-Jenkins Models: Concepts and Cases, New York: John Wiley & Sons, Inc. Pankratz, Alan (1991), Forecasting with Dynamic Regression Models, New York: John Wiley & Sons, Inc. Pearlman, J.G. (1980), "An Algorithm for the Exact Likelihood of a High-order Autoregressive-Moving Average Process," Biometrika, 67, 232-233. Priestly, M.B. (1981), Spectra Analysis and Time Series, Volume 1: Univariate Series, New York: Academic Press, Inc. Schwarz, G. (1978), "Estimating the Dimension of a Model," Annals of Statistics, 6, 461-464. Tsay, R.S. and Tiao, G.C. (1984), "Consistent Estimates of Autoregressive Parame- ters and Extended Sample Autocorrelation Function for Stationary and Nonsta- tionary ARMA Models," JASA, 79 (385), 84-96. Tsay, R.S. and Tiao, G.C. (1985), "Use of Canonical Analysis in Time Series Model Identification," Biometrika, 72 (2), 299-315. 299 SAS OnlineDoc : Version 8

Part 2. General Information Woodfield, T.J. (1987), "Time Series Intervention Analysis Using SAS Software," Proceedings of the Twelfth Annual SAS s Group International Conference, 331-339. Cary, NC: SAS Institute Inc.


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