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Re: st: Constant terms in AR1 error regressions

From   Michael Hanson <>
Subject   Re: st: Constant terms in AR1 error regressions
Date   Wed, 17 Dec 2008 14:30:37 -0500

On Dec 17, 2008, at 2:00 PM, Clive Nicholas wrote:

When the AR1 error-regression equation

u_{t} = \rhou{t-1} + e_{t}

is displayed without the constant - presumably u_{0} - is this because
it is suppressed (-nocons-) or simply because it isn't shown as it's
not of interest? The literature doesn't make this clear, but I'm
guessing it's the former.


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.

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).

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.

Hope this helps,

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