The ARIMA Procedure
- Overview
-
Getting Started
The Three Stages of ARIMA Modeling Identification Stage Estimation and Diagnostic Checking Stage Forecasting Stage Using ARIMA Procedure Statements General Notation for ARIMA Models Stationarity Differencing Subset, Seasonal, and Factored ARMA Models Input Variables and Regression with ARMA Errors Intervention Models and Interrupted Time Series Rational Transfer Functions and Distributed Lag Models Forecasting with Input Variables Data Requirements -
Syntax
-
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 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 Detecting Outliers OUT= Data Set OUTCOV= Data Set OUTEST= Data Set OUTMODEL= SAS Data Set OUTSTAT= Data Set Printed Output ODS Table Names Statistical Graphics
-
Examples
- References
| 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 in the ESTIMATE statement also controls the way forecasts are produced by the FORECAST statement. If METHOD=CLS is used, the forecasts are infinite memory forecasts, also called conditional forecasts. If METHOD=ULS or METHOD=ML, the forecasts are finite memory forecasts, also called unconditional forecasts. A complete description of the steps to produce the series forecasts and their standard errors by using either of these methods is quite involved, and only a brief explanation of the algorithm is given in the next two sections. Additional details about the finite and infinite memory forecasts can be found in Brockwell and Davis (1991). The prediction of stationary ARMA processes is explained in Chapter 5, and the prediction of nonstationary ARMA processes is given in Chapter 9 of Brockwell and Davis (1991).
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 are set to zero and 
is obtained from the estimated parameters 
and 
.
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.
That is, the k-step forecast of , given 
, is
 |
where 
is the covariance of and 
and 
is the covariance matrix of the vector . and are derived from the estimated parameters.
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 the forecast are 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 by 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.