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The AUTOREG Procedure |

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* or *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 test statistics and the Engle Lagrange multiplier tests.
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 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 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)
- GARCH-in-mean (GARCH-M)

For GARCH-type models, the AUTOREG procedure produces the conditional prediction error variances as well as 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 as well as 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.

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