Consider a standard multiple linear regression model with time series data:
yt = b0 + b1xt1 + . . . + bkxtk + ut. Assume that Assumptions TS.1, TS.2, TS.3, and TS.4 all hold.
(i) Suppose we think that the errors {ut} follow an AR(1) model with parameter r and so we apply the Prais-Winsten method. If the errors do not follow an AR(1) model- for example, suppose they follow an AR(2) model, or an MA(1) model-why will the usual Prais-Winsten standard errors be incorrect?
(ii) Can you think of a way to use the Newey-West procedure, in conjunction with Prais- Winsten estimation, to obtain valid standard errors? Be very specific about the steps you would follow. [Hint: It may help to study equation (12.32) and note that, if {ut} does not follow an AR(1) process, et generally should be replaced by ut - rut-1, where r is the probability limit of the estimator rˆ. Now, is the error {ut - rut-1} serially uncorrelated in general? What can you do if it is not?]
(iii) Explain why your answer to part (ii) should not change if we drop Assumption TS.4.