Identification and inference in moments based analysis of linear dynamic panel data models

We show that Dif(ference), see Arellano and Bond (1991), Lev(el), see Arellano and Bover (1995) and Blundell and Bond (1998), or the N(on-)L(inear) moment conditions of Ahn and Schmidt (1995) do not identify the parameters of a first-order autoregressive panel data model when the autoregressive parameter is equal to one. Combinations of the Dif and Lev, resulting in Sys(tem), moment conditions and the Dif and NL, resulting in A(hn-)S(chmidt), moment conditions identify the parameters when there are four or more time periods. The behaviour of one step and two step GMM estimators, however, remains non-standard. We therefore use size correct GMM statistics, like, the GMM-AR, GMM-LM or KLM statistic, to conduct inference. We compare their worst case large sample distributions with the power envelope to determine the optimal statistic. The power envelope involves a quartic root convergence rate which further indicates the non-standard identification issues. The worst case large sample distribution of the KLM statistic coincides with the power envelope whilst the one of the GMM-LM statistic only does so when there are four time periods. It shows that the KLM statistic is efficient both when the autoregressive parameter is one or less than one. The power envelopes for the AS and Sys moment conditons are identical so assuming mean stationarity does not help for identification.