1152 The Methods of Growth Econometrics
impliesT−2 extra moment conditions of the form:
E[
logyi,t− 1 (αi+εi,t)]
=0 fori=1,...,Nandt=3, 4,...,T. (24.37)Intuitively, as is clear from the new moment conditions, the extra assumptions
ensure that the lagged first difference of the dependent variable is a valid instru-
ment for untransformed equations in levels, since it is uncorrelated with the
composite error term in the levels equation. The additional moment conditions
build in some insurance against weak identification, because if the series are per-
sistent and lagged levels are weak instruments for first differences, it may still be
the case that lagged first differences will have some explanatory power for levels.^21
Nevertheless, the analysis of Bun and Windmeijer (2007) indicates that weak instru-
ment problems can emerge even for the system GMM estimator, especially when
the variance of the country effects is high relative to the variance of the transitory
shocks, which may well be the case for growth data.
Moreover, the extra moment conditions are based on assumptions about the ini-
tial conditions that are unlikely to command universal assent. In principle, these
assumptions can be tested using the incremental Sargan statistic (or C statistic)
associated with the additional moment conditions. Yet the validity of the restric-
tion should arguably be evaluated in wider terms, based on some knowledge of
the historical forces giving rise to the observed initial conditions. This point,
that key statistical assumptions should not always be evaluated only in statisti-
cal terms, is one that we will return to later, when discussing the wider application
of instrumental variable (IV) methods.
Alternatives to GMM have been proposed. Kiviet (1995, 1999) derives an analyt-
ical approximation to the Nickell bias that can be used to construct a bias-adjusted
within-country estimator for dynamic panels. The simulation evidence reported
in Judson and Owen (1999) and Bun and Kiviet (2001) suggests that this estima-
tor performs well relative to standard alternatives whenNandTare small. More
recently, Bun and Carree (2005) have developed an alternative bias-adjusted esti-
mator. One serious limitation of the currently available bias-adjusted estimators,
relative to GMM, is that they do not address the possible correlation between the
explanatory variables and the disturbances due to simultaneity and measurement
error. Nevertheless, there is a clear case for implementing these estimators, at least
as a complement to other methods.
A further issue that arises when estimating dynamic panel data models is that
of parameter heterogeneity. If a slope parameter such asβvaries across countries,
and the relevant explanatory variable is serially correlated, this will induce serial
correlation in the error term. If we focus on a simple case where a researcher
wrongly assumes homogeneity in the coefficient on lagged output, orβi=βfor
alli=1,...,N, then the error process for a given country will contain a compo-
nent that resembles
(
βi−β)
logyi,t− 1. Hence there is serial correlation in the errors,
given the persistence of output. The estimates of a dynamic panel data model will
be inconsistent even if GMM methods are applied. This problem was analyzed in
more general terms by Robertson and Symons (1992) and Pesaran and Smith (1995)