The AUTOREG procedure estimates and forecasts linear regression models for time series data when the errors are autocorrelated or heteroscedastic. The autoregressive error model is used to correct for autocorrelation, and the generalized autoregressive conditional heteroscedasticity (GARCH) model and its variants are used to model and correct for heteroscedasticity.
When time series data are used in regression analysis, often the error term is not independent through time. Instead, the errors are serially correlated (autocorrelated). If the error term is autocorrelated, the efficiency of ordinary least squares (OLS) parameter estimates is adversely affected and standard error estimates are biased.
The autoregressive error model corrects for serial correlation. The AUTOREG procedure can fit autoregressive error models of any order and can fit subset autoregressive models. You can also specify stepwise autoregression to select the autoregressive error model automatically.
To diagnose autocorrelation, the AUTOREG procedure produces generalized Durbin-Watson (DW) statistics and their marginal probabilities. Exact p-values are reported for generalized DW tests to any specified order. For models with lagged dependent regressors, PROC AUTOREG performs the Durbin t test and the Durbin h test for first-order autocorrelation and reports their marginal significance levels.
Ordinary regression analysis assumes that the error variance is the same for all observations. When the error variance is not constant, the data are said to be heteroscedastic, and ordinary least squares estimates are inefficient. Heteroscedasticity also affects the accuracy of forecast confidence limits. More efficient use of the data and more accurate prediction error estimates can be made by models that take the heteroscedasticity into account.
To test for heteroscedasticity, the AUTOREG procedure uses the portmanteau Q test statistics (McLeod and Li, 1983), Engle’s Lagrange multiplier tests (Engle, 1982), tests from Lee and King (1993), and tests from Wong and Li (1995). Test statistics and significance p-values are reported for conditional heteroscedasticity at lags 1 through 12. The Bera-Jarque normality test statistic and its significance level are also reported to test for conditional nonnormality of residuals. The following tests for independence are also supported by the AUTOREG procedure for residual analysis and diagnostic checking: Brock-Dechert-Scheinkman (BDS) test, runs test, turning point test, and the rank version of the von Neumann ratio test.
The family of GARCH models provides a means of estimating and correcting for the changing variability of the data. The GARCH process assumes that the errors, although uncorrelated, are not independent, and it models the conditional error variance as a function of the past realizations of the series.
The AUTOREG procedure supports the following variations of the GARCH models:
generalized ARCH (GARCH)
integrated GARCH (IGARCH)
exponential GARCH (EGARCH)
quadratic GARCH (QGARCH)
threshold GARCH (TGARCH)
power GARCH (PGARCH)
GARCH-in-mean (GARCH-M)
For GARCH-type models, the AUTOREG procedure produces the conditional prediction error variances in addition to parameter and covariance estimates.
The AUTOREG procedure can also analyze models that combine autoregressive errors and GARCH-type heteroscedasticity. PROC AUTOREG can output predictions of the conditional mean and variance for models with autocorrelated disturbances and changing conditional error variances over time.
Four estimation methods are supported for the autoregressive error model:
Yule-Walker
iterated Yule-Walker
unconditional least squares
exact maximum likelihood
The maximum likelihood method is used for GARCH models and for mixed AR-GARCH models.
The AUTOREG procedure produces forecasts and forecast confidence limits when future values of the independent variables are included in the input data set. PROC AUTOREG is a useful tool for forecasting because it uses the time series part of the model in addition to the systematic part in generating predicted values. The autoregressive error model takes into account recent departures from the trend in producing forecasts.
The AUTOREG procedure permits embedded missing values for the independent or dependent variables. The procedure should be used only for ordered and equally spaced time series data.