Fixed-smoothing asymptotics in the generalized empirical likelihood estimation framework

This paper concerns the fixed-smoothing asymptotics for two commonly used estimators in the generalized empirical likelihood estimation framework for time series data, namely the continuous updating estimator and the maximum blockwise empirical likelihood estimator. For continuously updating generalized method of moments (GMM) estimator, we show that the results for the two-step GMM estimator in Sun (2014a) continue to hold under suitable assumptions. For continuous updating estimator obtained through solving a saddle point problem (Newey and Smith, 2004) and the maximum blockwise empirical likelihood estimator (Kitamura, 1997), we show that their fixed-smoothing asymptotic distributions (up to an unknown linear transformation) are mixed normal. Based on these results, we derive the asymptotic distributions of the specification tests (including the over-identification testing and testing on parameters) under the fixed-smoothing asymptotics, where the corresponding limiting distributions are nonstandard yet pivotal. Simulation studies show that (i) the fixed-smoothing asymptotics provides better approximation to the sampling distributions of the continuous updating estimator and the maximum blockwise empirical likelihood estimator as compared to the standard normal approximation. The testing procedures based on the fixed-smoothing critical values are more accurate in size than the conventional chi-square based tests; (ii) the continuously updating GMM estimator is asymptotically more efficient and the corresponding specification tests are generally more powerful than the other two competitors. Finite sample results from an empirical data analysis are also reported.