Paul Johnson, Steven Durlauf and Jonathan Temple 1151within-group estimates. Since the OLS and within-group estimates ofβare biased in
opposing directions then, leaving aside sampling variability and small-sample con-
siderations, a consistent parameter estimate should lie between these two extremes,
as discussed in Nerlove (1999, 2000). Formally, when the explanatory variables
other than lagged output are strictly exogenous, we have:
plimβˆWG<plimβ<ˆ plimβˆOLS, (24.33)whereβˆis a consistent parameter estimate,βˆWGis the within-groups estimate
andβˆOLSis the estimate from a pooled OLS regression. For the dataset and model
used by Caselliet al.(1996), this large-sample prediction is not valid, which raises
a question mark over the reliability of the first-differenced GMM estimates. The
problem may be one of weak instruments, and unless this can be resolved, it is
not difficult to imagine circumstances in which the within-groups estimator, or
bias-corrected versions of it, may be preferable to the GMM approach.
One device that can be informative in short panels is to make more restrictive
assumptions about the initial conditions. If the observations at the start of the
sample are distributed in a way that is representative of steady-state behavior, in a
sense that will be made precise below, efficiency gains are possible. Assumptions
about the initial conditions can be used to derive a “system” GMM estimator, of
the form developed and studied by Arellano and Bover (1995) and Blundell and
Bond (1998), and also discussed in Ahn and Schmidt (1995) and Hahn (1999). In
this estimator, not only are lagged levels used as instruments for first differences,
but lagged first differences are used as instruments for levels, which corresponds
to an extra set of moment conditions. Blundell and Bond (1998) provide Monte
Carlo evidence that this estimator is more robust than the Arellano–Bond method
in the presence of highly persistent series. As also shown by Blundell and Bond, the
necessary assumptions can be seen in terms of an extra restriction, namely that the
deviations of the initial values of logyi,tfrom their long-run (steady-state) values
are not systematically related to the individual effects.^20 For simplicity, we focus
on the case where there are no explanatory variables other than lagged output. The
required assumption on the initial conditions is that, for alli=1,...,N, we have:
E[(
logyi,1−y ̄i)
αi]
=0, (24.34)where the ̄yiare the long-run values of the logyi,tseries and are therefore functions
of the individual effectsαiand the autoregressive parameterβ. This assumption
on the initial conditions ensures that:
E[
logyi,2αi]
=0, (24.35)and this, together with the mild assumption that the changes in the errors are
uncorrelated with the individual effects:
E[
εi,tαi]
=0, (24.36)