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Fixed-smoothing asymptotics in the generalized empirical likelihood estimation framework. (English) Zbl 1420.62405

Summary: 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 [Y. Sun, “Fixed-smoothing asymptotics in a two-step generalized method of moments framework”, Econometrica 82, No. 6, 2327–2370 (2014; doi:10.3982/ECTA11684)] continue to hold under suitable assumptions. For continuous updating estimator obtained through solving a saddle point problem [W. K. Newey and R. J. Smith, Econometrica 72, No. 1, 219–255 (2004; Zbl 1151.62313)] and the maximum blockwise empirical likelihood estimator [Y. Kitamura, Ann. Stat. 25, No. 5, 2084–2102 (1997; Zbl 0881.62095)], 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.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics
62G20 Asymptotic properties of nonparametric inference
62G10 Nonparametric hypothesis testing

Software:

Stata; ivregress
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Full Text: DOI

References:

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