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Re: st: Constant terms in AR1 error regressions


From   Michael Hanson <mshanson@mac.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Constant terms in AR1 error regressions
Date   Thu, 18 Dec 2008 17:06:57 -0500

On Dec 18, 2008, at 9:43 AM, Clive Nicholas wrote:

Michael Hanson replied:

An "error regression equation" is a little ambiguous: after all, the errors are unobservable (and thus cannot be "put" into a regression), while the residuals are by construction mean zero, so a constant term is unnecessary.
 Although you could run a regression on the residuals of a previously
estimated model (and many tests of serial dependence have that form),
typically what one does is model the (assumed) auto-regressive properties of
the error term as part of the specification to be estimated -- in a
univariate or single-equation context, this can be accomplished in Stata
with the -arima- command.

I was pressed was for time when posting this query, so apologies for
using the wrong terminology: I did, of course, mean 'residuals'.
Although you say a constant term in such residual-on-residual
regressions are unnecesary, a constant term nevertheless appears, and
my task is to do this in -reg-, not -arima-. Essentially, what I'm
asking is is it best to leave it there or to apply the -nocons-
option?

You've defined your "task" very narrowly as using -regress- to estimate an AR(1) equation on residuals from (what I presume to be) a prior regression. If you could give a more general idea of what you are trying to accomplish, I and others on the list might be able to make better suggestions. For example, one might give different advice if you were concerned that the residuals were I(1) than if you were fairly confident they were stationary.

That said, Wooldridge (2006, p. 418) discusses testing for AR(1) serial correlation with strictly exogenous regressors, and advises "this regression may or may not contain an intercept; the t statistic for \hat{\rho} will be slightly affected, but it is asymptotically valid either way." Later, he notes that strictly exogenous regressors are not very common with time series data, and that such simple tests are not robust to higher order autocorrelation. (You did test for higher order terms before settling on an AR(1) specification, right?) Wooldridge recommends a Breusch-Godfrey test, but there are others: see -help regress postestimationts- (yes, that is a "ts" at the end) for discussion of what is implemented in Stata.

Two final thoughts: First, if you include the intercept in a regression of a residual series on its first lag, and the estimated intercept is significantly different from zero, then you probably should revisit your prior estimation: your residuals should be mean- zero by definition. Second, if your results are very different when the intercept is excluded -- if that one extra degree of freedom is enough to change your results -- then I would caution you to be very skeptical of them to begin with, as you are working with large-T asymptotics by using -reg-.

Hope this helps,
Mike

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