Dummy Variables and Autocorrelation

488 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

As the preceding discussion points out, the critical observation is the first observation in the second period. If this is taken care of in the manner just suggested, there should be no problem in estimating regressions like (12.13.1) subject to AR(1) autocorrelation. In exercise 12.37, the reader is asked to carry such a transformation for the data on U.S. savings and income given in Chapter 9.

Just as the error term u at time t can be correlated with the error term at time (t - 1) in an AR(1) scheme or with various lagged error terms in a general AR(p) scheme, can there be autocorrelation in the variance o2 at time t with its values lagged one or more periods? Such an autocorrelation has been observed by researchers engaged in forecasting financial time series, such as stock prices, inflation rates, and foreign exchange rates. Such autocorrelation is given the rather daunting names autoregressive conditional heteroscedasticity (ARCH) if the error variance is related to the squared error term in the previous term and generalized autoregressive conditional heteroscedasticity (GARH) if the error variance is related to squared error terms several periods in the past. Since this topic belongs in the general area of time series econometrics, we will discuss it in some depth in the chapters on time series econometrics. Our objective here is to point out that autocorrelation is not confined to relationships between current and past error terms but also with current and past error variances.

What happens if a regression model suffers from both heteroscedasticity and autocorrelation? Can we solve the problem sequentially, that is, take care of heteroscedasticity first and then autocorrelation? As a matter of fact, one author contends that "Autoregression can only be detected after the het-eroscedasticity is controlled for."49 But can we develop an omnipotent test that can solve these and other problems (e.g., model specification) simultaneously? Yes, such tests exist, but their discussion will take us far afield. It is better to leave them for references.50

1. If the assumption of the classical linear regression model—that the errors or disturbances ut entering into the population regression function (PRF) are random or uncorrelated—is violated, the problem of serial or autocorrelation arises.

49Lois W. Sayrs, Pooled Time Series Analysis, Sage Publications, California, 1989, p. 19.

50See Jeffrey M. Wooldridge, op. cit., pp. 402-403, and A. K. Bera and C. M. Jarque, "Efficient Tests for Normality, Homoscedasticity and Serial Independence of Regression Residuals: Monte Carlo Evidence," Economic Letters, vol. 7, 1981, pp. 313-318.

ARCH and GARCH Models

Coexistence of Autocorrelation and Heteroscedasticity 