GEL statistics under weak identification

Abstract

The central concern of this paper is the provision in a time series moment condition framework of practical recommendations of confidence regions for parameters whose coverage probabilities are robust to the strength or weakness of identification. To this end we develop Pearson-type test statistics based on GEL implied probabilities formed from general kernel smoothed versions of the moment indicators. We also modify the statistics suggested in Guggenberger and Smith (2008) for a general kernel smoothing function. Importantly for our conclusions, we provide GEL time series counterparts to GMM and GEL conditional likelihood ratio statistics given in Kleibergen (2005) and Smith (2007). Our analysis not only demonstrates that these statistics are asymptotically (conditionally) pivotal under both classical asymptotic theory and weak instrument asymptotics of Stock and Wright (2000) but also provides asymptotic power results in the weakly identified time series context. Consequently, the empirical null rejection probabilities of the associated tests and, thereby, the coverage probabilities of the corresponding confidence regions, should not be affected greatly by the strength or otherwise of identification. A comprehensive Monte Carlo study indicates that a number of the tests proposed here represent very competitive choices in comparison with those suggested elsewhere in the literature.