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Asymptotic F and t Tests in an Efficient GMM Setting

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  • Hwang, Jungbin
  • Sun, Yixiao
Abstract
This paper considers two-step efficient GMM estimation and inference where the weighting matrix and asymptotic variance matrix are based on the series long run variance estimator. We propose a simple and easy-to-implement modification to the trinity of test statistics in the two-step efficient GMM setting and show that the modified test statistics are all asymptotically F distributed under the so-called fixed-smoothing asymptotics. The modification is multiplicative and involves the J statistic for testing over-identifying restrictions. This leads to convenient asymptotic F tests that use standard F critical values. Simulation shows that, in terms of both size and power, the asymptotic F tests perform as well as the nonstandard tests proposed recently by Sun (2014b) in finite samples. But the F tests are more appealing as the critical values are readily available from standard statistical tables. Compared to the conventional chi-square tests, the F tests are as powerful, but are much more accurate in size.

Suggested Citation

  • Hwang, Jungbin & Sun, Yixiao, 2015. "Asymptotic F and t Tests in an Efficient GMM Setting," University of California at San Diego, Economics Working Paper Series qt1c62d8xf, Department of Economics, UC San Diego.
    • Handle: RePEc:cdl:ucsdec:qt1c62d8xf
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    References listed on IDEAS

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    1. Phillips, Peter C.B., 2005. "Hac Estimation By Automated Regression," Econometric Theory, Cambridge University Press, vol. 21(1), pages 116-142, February.
    2. Hwang, Jungbin & Sun, Yixiao, 2018. "Should we go one step further? An accurate comparison of one-step and two-step procedures in a generalized method of moments framework," Journal of Econometrics, Elsevier, vol. 207(2), pages 381-405.
    3. Yixiao Sun, 2013. "A heteroskedasticity and autocorrelation robust F test using an orthonormal series variance estimator," Econometrics Journal, Royal Economic Society, vol. 16(1), pages 1-26, February.
    4. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-1054, July.
    5. Sun, Yixiao, 2011. "Robust trend inference with series variance estimator and testing-optimal smoothing parameter," Journal of Econometrics, Elsevier, vol. 164(2), pages 345-366, October.
    6. Sun, Yixiao & Phillips, Peter C.B. & Jin, Sainan, 2011. "Power Maximization And Size Control In Heteroskedasticity And Autocorrelation Robust Tests With Exponentiated Kernels," Econometric Theory, Cambridge University Press, vol. 27(6), pages 1320-1368, December.
    7. Yixiao Sun & Peter C. B. Phillips & Sainan Jin, 2008. "Optimal Bandwidth Selection in Heteroskedasticity-Autocorrelation Robust Testing," Econometrica, Econometric Society, vol. 76(1), pages 175-194, January.
    8. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, vol. 59(3), pages 817-858, May.
    9. Yixiao Sun & Min Seong Kim, 2015. "Asymptotic F-Test in a GMM Framework with Cross-Sectional Dependence," The Review of Economics and Statistics, MIT Press, vol. 97(1), pages 210-233, March.
    10. Nicholas M. Kiefer & Timothy J. Vogelsang, 2002. "Heteroskedasticity-Autocorrelation Robust Standard Errors Using The Bartlett Kernel Without Truncation," Econometrica, Econometric Society, vol. 70(5), pages 2093-2095, September.
    11. Gonçalves, Sílvia & Vogelsang, Timothy J., 2011. "Block Bootstrap Hac Robust Tests: The Sophistication Of The Naive Bootstrap," Econometric Theory, Cambridge University Press, vol. 27(4), pages 745-791, August.
    12. Sun, Yixiao & Kim, Min Seong, 2012. "Simple and powerful GMM over-identification tests with accurate size," Journal of Econometrics, Elsevier, vol. 166(2), pages 267-281.
    13. Zhang, Xianyang, 2016. "Fixed-smoothing asymptotics in the generalized empirical likelihood estimation framework," Journal of Econometrics, Elsevier, vol. 193(1), pages 123-146.
    14. Sun, Yixiao, 2014. "Let’s fix it: Fixed-b asymptotics versus small-b asymptotics in heteroskedasticity and autocorrelation robust inference," Journal of Econometrics, Elsevier, vol. 178(P3), pages 659-677.
    15. Kiefer, Nicholas M. & Vogelsang, Timothy J., 2005. "A New Asymptotic Theory For Heteroskedasticity-Autocorrelation Robust Tests," Econometric Theory, Cambridge University Press, vol. 21(6), pages 1130-1164, December.
    16. Kiefer, Nicholas M. & Vogelsang, Timothy J., 2002. "Heteroskedasticity-Autocorrelation Robust Testing Using Bandwidth Equal To Sample Size," Econometric Theory, Cambridge University Press, vol. 18(6), pages 1350-1366, December.
    17. Newey, Whitney & West, Kenneth, 2014. "A simple, positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 33(1), pages 125-132.
    18. Yixiao Sun, 2014. "Fixed-smoothing Asymptotics and AsymptoticFandtTests in the Presence of Strong Autocorrelation," Advances in Econometrics, in: Essays in Honor of Peter C. B. Phillips, volume 14, pages 23-63, Emerald Group Publishing Limited.
    19. Michael Jansson, 2004. "The Error in Rejection Probability of Simple Autocorrelation Robust Tests," Econometrica, Econometric Society, vol. 72(3), pages 937-946, May.
    20. Bester, C. Alan & Conley, Timothy G. & Hansen, Christian B. & Vogelsang, Timothy J., 2016. "FIXED-b ASYMPTOTICS FOR SPATIALLY DEPENDENT ROBUST NONPARAMETRIC COVARIANCE MATRIX ESTIMATORS," Econometric Theory, Cambridge University Press, vol. 32(1), pages 154-186, February.
    21. Preinerstorfer, David & Pötscher, Benedikt M., 2016. "On Size And Power Of Heteroskedasticity And Autocorrelation Robust Tests," Econometric Theory, Cambridge University Press, vol. 32(2), pages 261-358, April.
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    Cited by:

    1. Hwang, Jungbin & Sun, Yixiao, 2018. "SIMPLE, ROBUST, AND ACCURATE F AND t TESTS IN COINTEGRATED SYSTEMS," Econometric Theory, Cambridge University Press, vol. 34(5), pages 949-984, October.
    2. Casini, Alessandro, 2023. "Theory of evolutionary spectra for heteroskedasticity and autocorrelation robust inference in possibly misspecified and nonstationary models," Journal of Econometrics, Elsevier, vol. 235(2), pages 372-392.
    3. Yixiao Sun & Peter C. B. Phillips & Igor L. Kheifets, 2024. "Estimation and Inference in a Possibly Multi-cointegrated System with a Fixed Number of Instruments," Cowles Foundation Discussion Papers 2410, Cowles Foundation for Research in Economics, Yale University.
    4. Martínez-Iriarte, Julián & Sun, Yixiao & Wang, Xuexin, 2020. "Asymptotic F tests under possibly weak identification," Journal of Econometrics, Elsevier, vol. 218(1), pages 140-177.
    5. Casini, Alessandro, 2024. "The fixed-b limiting distribution and the ERP of HAR tests under nonstationarity," Journal of Econometrics, Elsevier, vol. 238(2).
    6. Ulrich K. Müller & Mark W. Watson, 2020. "Low-Frequency Analysis of Economic Time Series," Working Papers 2020-13, Princeton University. Economics Department..
    7. Jungbin Hwang, 2017. "Simple and Trustworthy Cluster-Robust GMM Inference," Working papers 2017-19, University of Connecticut, Department of Economics, revised Aug 2020.
    8. Hwang, Jungbin, 2021. "Simple and trustworthy cluster-robust GMM inference," Journal of Econometrics, Elsevier, vol. 222(2), pages 993-1023.
    9. Liu, Cheng & Sun, Yixiao, 2019. "A simple and trustworthy asymptotic t test in difference-in-differences regressions," Journal of Econometrics, Elsevier, vol. 210(2), pages 327-362.
    10. Xiaoqing Ye & Yixiao Sun, 2018. "Heteroskedasticity- and autocorrelation-robust F and t tests in Stata," Stata Journal, StataCorp LP, vol. 18(4), pages 951-980, December.
    11. Hwang, Jungbin & Valdés, Gonzalo, 2023. "Finite-sample corrected inference for two-step GMM in time series," Journal of Econometrics, Elsevier, vol. 234(1), pages 327-352.

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    More about this item

    Keywords

    Social and Behavioral Sciences; Efficient GMM; F distribution; F test; Fixed-smoothing Asymptotics; Heteroskedasticity and Autocorrelation Robust; Two-step GMM;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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