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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   Fri, 19 Dec 2008 21:27:58 +0000

Michael Hanson replied:

> 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.

This example probably best illustrates what I was originally getting at:

webuse union
xtset idcode year
reg grade south union black if year==70
predict r70, r
forval i=71(1)73 {
local j = `i'-1
reg grade south union black if year==`i'
predict r`i', r
reg r`i' r`j' if year==`i'
reg r`i' r`j' if year==`i', nocons

Notice that the coefficients on \rho and their attendant t-ratios are
not always the same. Unit-root tests via -pperron- and -dfuller-,
within unit, suggests my pooled time-series is 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.

Because, like -union-, my dataset is pooled, I ran -corrgram- with
selected units (i.e., those possessing a long enough time-series to be
worth performing Q tests on) and found one AR1 process and one AR3
process; the rest had none at all. That said, a pooled regression
using BSS's -xtivreg2, bw(2) small robust i() t() fe- showed my lagged
dependent variable to be significant; running (mean-centered) pooled
regressions using -xtpcse-, however, showed my LDV failed to reach

I did look carefully at -help regress postestimationts-, as it would
be much easier to run any of those AC tests automatically than do it
'by hand', as I had to. However,

g lagrade=l.grade
quietly reg grade lagrade south union black if year==73
estat bgodfrey
sample may not include multiple panels

estat archlm
sample may not include multiple panels

estat durbinalt
sample may not include multiple panels

estat dwatson
sample may not include multiple panels

and so on.

> 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-.

Indeed, my T has a maximum of 13 in this pooled dataset, and I'm
running -reg- on single cross-sections in order to obtain \rho by

Clive Nicholas

[Please DO NOT mail me personally here, but at
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"My colleagues in the social sciences talk a great deal about
methodology. I prefer to call it style." -- Freeman J. Dyson.
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