GMM Estimation of Autoregressive Roots near Unity with Panel Data

This paper investigates a generalized method of moments (GMM) approach to the
estimation of autoregressive roots near unity with panel data and incidental deterministic
trends. Such models arise in empirical econometric studies of firm size and in dynamic
panel data modeling with weak instruments. The two moment conditions in the
GMM approach are obtained by constructing bias corrections to the score functions
under OLS and GLS detrending, respectively. It is shown that the moment condition
under GLS detrending corresponds to taking the projected score on the Bhattacharya
basis, linking the approach to recent work on projected score methods for models with
infinite numbers of nuisance parameters (Waterman and Lindsay (1998)). Assuming
that the localizing parameter takes a nonpositive value. we establish consistency of the
GMM estimator and find its limiting distribution. A notable new finding is that the
GMM estimator has convergence rate n"', slower than A.when the true localizing parameter
is zero (i.e., when there is a panel unit root) and the deterministic trends in the
panel are linear. These results. which rely on boundary point asymptotics, point to the
continued difficulty of distinguishing unit roots from local alternatives, even when there
is an infinity of additional data.
KEYWORDS:Bias, boundary point asymptotics, GMM estimation, local to unity, moment
conditions, nuisance parameters, panel data, pooled regression, projected score.

1. INTRODUCTION
RECENTYEARS HAVE SEEN the introduction of several important panel data
sets where the cross sectional dimension (n) and the time series dimension (T)
are comparable in magnitude. Some of these panel data sets, like the Penn
World Tables, involve time series that are manifestly nonstationary and have
persistent or slowly decaying serial correlations. These features distinguish the
new data from the characteristics that are conventionally assumed in the analysis
of panel data where T is very small and n is very large.