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Re: st: Constant terms in AR1 error regressions

From   "Clive Nicholas" <>
Subject   Re: st: Constant terms in AR1 error regressions
Date   Thu, 18 Dec 2008 14:43:27 +0000

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-

> Also, note that u_{0} is the initial observation in time of the
> (hypothetical) u time series, u_{t} for t = 0, \dots, T.  It is not a
> parameter to be estimated (like a constant).

When I wrote u_{0} in this equation, I meant this to represent the
constant term, since it appears in all of the residual-on-residual
equations I have run in Stata.

> I'm not certain what literature you are referring to, but I know from
> teaching time series that textbooks often do not clearly distinguish
> hypothetical concepts from specifications that can be estimated on "real"
> data.
> By the way, if you are estimating an AR(1) model on "real" data (not
> residuals), you will certainly want to include a constant term.  Whether it
> is of interest or not depends in part on your application.  But its
> exclusion is likely to yield biased estimates of the other parameters (such
> as \rho), just as in the OLS case.

Thanks for this.

Clive Nicholas

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