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 <formula format="inline"> <file name="ecta_499_m1.gif" type="gif" /> <alternativemath type="latex2e"> <tex>$n^{1/6}$</tex> </alternativemath> <alternativemath type="mathml"> <math overflow="scroll"> <msup> <mi>n</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>6</mn> </mrow> </msup> </math> </alternativemath> </formula>, slower than <formula format="inline"> <file name="ecta_499_m2.gif" type="gif" /> <alternativemath type="latex2e"> <tex>$\sqrt{n}$</tex> </alternativemath> <alternativemath type="mathml"> <math overflow="scroll"> <msqrt> <mi>n</mi> </msqrt> </math> </alternativemath> </formula>, 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. Copyright The Econometric Society 2004.