The ARIMA Procedure

1 Chapter 7 The ARIMA ProcedureChapter Table of ..194 IdentificationStage ..194 Estimation and Diagnostic Checking Stage ..200 forecasting ..206 Stationarity .. ,Seasonal,andFactoredARMAM odels ..211 Input Variables and Regression with ARMA Errors ..213 InterventionModelsandInterruptedTimeSeri es ..215 Rational Transfer Functions and Distributed Lag with Input Variables ..219 DataRequirements .. Statement ..234 The Partial Autocorrelation ..235 TheESACFM ethod ..239 Stationarity ..241 Identifying Transfer Function 2. General Inputs and Transfer Functions ..248 Initial Values ..249 Stationarity and Values and Estimation and Log Transformed Data.

2 254 OUTCOV=DataSet ..255 OUTEST= Data ..259 OUTSTAT=DataSet ..265 Example Seasonal Model for the Airline series ..287 Example Using Diagnostics to Identify ARIMA OnlineDoc : Version 8192 Chapter 7 The ARIMA ProcedureOverviewThe ARIMA Procedure analyzes and forecasts equally spaced univariate time se-ries data, transfer function data, and intervention data using theAutoRegressiveIntegratedMoving-Averag e ( ARIMA ) or autoregressive moving-average (ARMA)model. An ARIMA model predicts a value in a response time series as a linear com-bination of its own past values, past errors (also called shocks or innovations), andcurrent and past values of other time ARIMA approach was first popularized by Box and Jenkins, and ARIMA modelsare often referred to as Box-Jenkins models.

3 The general transfer function modelemployed by the ARIMA Procedure was discussed by Box and Tiao (1975). When anARIMA model includes other time series as input variables, the model is sometimesreferred to as an ARIMAX model. Pankratz (1991) refers to the ARIMAX model asdynamic ARIMA Procedure provides a comprehensive set of tools for univariate time se-ries model identification, parameter estimation, and forecasting , and it offers greatflexibility in the kinds of ARIMA or ARIMAX models that can be analyzed. TheARIMA Procedure supports seasonal, subset, and factored ARIMA models; inter-vention or interrupted time series models; multiple regression analysis with ARMA errors; and rational transfer function models of any design of PROC ARIMA closely follows the Box-Jenkins strategy for time seriesmodeling with features for the identification, estimation and diagnostic checking, andforecasting steps of the Box-Jenkins using PROC ARIMA , you should be familiar with Box-Jenkins methods, andyou should exercise care and judgment when using the ARIMA Procedure .

4 TheARIMA class of time series models is complex and powerful, and some degree ofexpertise is needed to use them you are unfamiliar with the principles of ARIMA modeling, refer to textbooks ontime series analysis. Also refer toSAS/ETS Software: Applications Guide 1, Version6, First Edition. You might consider attending the SAS Training Course "Forecast-ing Techniques Using SAS/ETS Software." This course provides in-depth trainingon ARIMA modeling using PROC ARIMA , as well as training on the use of otherforecasting tools available in SAS/ETS 2. General InformationGetting StartedThis section outlines the use of the ARIMA Procedure and gives a cursory descriptionof the ARIMA modeling process for readers less familiar with these Three Stages of ARIMA ModelingThe analysis performed by PROC ARIMA is divided into three stages, correspondingto the stages described by Box and Jenkins (1976).

5 The IDENTIFY, ESTIMATE, andFORECAST statements perform these three stages, which are summarized In theidentificationstage, you use the IDENTIFY statement to specify the re-sponse series and identify candidate ARIMA models for it. The IDENTIFY statement reads time series that are to be used in later statements, possibly dif-ferencing them, and computes autocorrelations, inverse autocorrelations, par-tial autocorrelations, and cross correlations. Stationarity tests can be performedto determine if differencing is necessary. The analysis of the IDENTIFY state-ment output usually suggests one or more ARIMA models that could be allow you to test for stationarity and tentative ARMA order In theestimation and diagnostic checkingstage, you use the ESTIMATE state-ment to specify the ARIMA model to fit to the variable specified in the previousIDENTIFY statement, and to estimate the parameters of that model.

6 The ES-TIMATE statement also produces diagnostic statistics to help you judge theadequacy of the tests for parameter estimates indicate whether some terms in themodel may be unnecessary. Goodness-of-fit statistics aid in comparing thismodel to others. Tests for white noise residuals indicate whether the residualseries contains additional information that might be utilized by a more complexmodel. If the diagnostic tests indicate problems with the model, you try anothermodel, then repeat the estimation and diagnostic checking In theforecastingstage you use the FORECAST statement to forecast futurevalues of the time series and to generate confidence intervals for these forecastsfrom the ARIMA model produced by the preceding ESTIMATE three steps are explained further and illustrated through an extended examplein the following StageSuppose you have a variable called SALES that you want to forecast.

7 The follow-ing example illustrates ARIMA modeling and forecasting using a simulated data setTEST containing a time series SALES generated by an ARIMA (1,1,1) model. Theoutput produced by this example is explained in the following sections. The simu-lated SALES series is shown in Figure OnlineDoc : Version 8194 Chapter 7. Getting StartedFigure ARIMA (1,1,1) series SALESU sing the IDENTIFY StatementYou first specify the input data set in the PROC ARIMA statement. Then, you usean IDENTIFY statement to read in the SALES series and plot its autocorrelationfunction. You do this using the following statements:proc ARIMA data=test;identify var=sales nlag=8;run;Descriptive StatisticsThe IDENTIFY statement first prints descriptive statistics for the SALES series .

8 Thispart of the IDENTIFY statement output is shown in Figure ARIMA ProcedureName of Variable = salesMean of Working series Deviation of Observations 100 Figure Statement Descriptive Statistics OutputAutocorrelation Function PlotsThe IDENTIFY statement next prints three plots of the correlations of the series withits past values at different lags. These are the sample autocorrelation function plot195 SAS OnlineDoc : Version 8 Part 2. General Information sample partial autocorrelation function plot sample inverse autocorrelation function plotThe sample autocorrelation function plot output of the IDENTIFY statement is shownin Figure ARIMA ProcedureAutocorrelationsLag Covariance Correlation -1 9 87654321012345678910 | |**|1 |.

9 |** |2 | . |** |3 | . |** |4 | . |** |5 | . |** |6 | . |** |7 | . |** |8 | . |** . |"." marks two standard errorsFigure Statement Autocorrelations PlotThe autocorrelation plot shows how values of the series are correlated with past valuesof the series . For example, the value in the "Correlation" column for the Lag1 row of the plot means that the correlation between SALES and the SALES valuefor the previous period is.

10 95672. The rows of asterisks show the correlation plots are called autocorrelation functions because they show the degree of cor-relation with past values of the series as a function of the number of periods in thepast (that is, the lag) at which the correlation is NLAG= option controls the number of lags for which autocorrelations are default, the autocorrelation functions are plotted to lag 24; in this example theNLAG=8 option is used, so only the first 8 lags are books on time series analysis explain how to interpret autocorrelation plots andpartial autocorrelation plots. See the section "The Inverse Autocorrelation Function"later in this chapter for a discussion of inverse autocorrelation examining these plots, you can judge whether the series isstationaryornonsta-tionary.