Re: st: Constant terms in AR1 error regressions

[Michael Hanson <mshanson@mac.com>]

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.


Clive:

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,
Mike

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