GMM-based inference in the AR(1) panel data model for parameter values where local identication fails

We are interested in making inference on the AR parameter in an AR(1) panel data model when the time-series dimension is …xed and the cross-section dimension is large. We consider a GMM estimator based on the second order moments of the observed variables. It turns out that when the AR parameter equals unity and certain restrictions apply to the other parameters in the model then local identi…cation fails. We show that the parameters and in particular the AR parameter can still be estimated consistently but at a slower rate than usual. Also we derive the asymptotic distribution of the parameter estimators which turns out to be non-standard. The properties of this estimator of the AR parameter are very di¤erent from those of the widely used Arellano-Bond GMM estimator which is obtained by taking …rst-di¤erences of the variables and then use lagged levels as instruments. It is well-known that this estimator performs poorly when the AR parameter is close to unity. The reason for this di¤erence between the two estimators is that with the Arellano-Bond moment conditions global identi…cation fails for the speci…c values of the parameters considered here whereas with the non-linear GMM estimator that uses all information from the second order moments only local identi…cation fails. We illustrate the results in a simulation study. Keywords: AR(1) panel data model, GMM estimation, local identi…cation failure, rate of convergence, non-standard limiting distribution