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An empirical process central limit theorem for dependent non-identically distributed random variables

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  • Andrews, Donald W. K.

Abstract

This paper establishes a central limit theorem (CLT) for empirical processes indexed by smooth functions. The underlying random variables may be temporally dependent and non-identically distributed. In particular, the CLT holds for near epoch dependent (i.e., functions of mixing processes) triangular arrays, which include strong mixing arrays, among others. The results apply to classes of functions that have series expansions. The proof of the CLT is particularly simple; no chaining argument is required. The results can be used to establish the asymptotic normality of semiparametric estimators in time series contexts. An example is provided.

Suggested Citation

  • Andrews, Donald W. K., 1991. "An empirical process central limit theorem for dependent non-identically distributed random variables," Journal of Multivariate Analysis, Elsevier, vol. 38(2), pages 187-203, August.
    • Handle: RePEc:eee:jmvana:v:38:y:1991:i:2:p:187-203
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    References listed on IDEAS

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    1. Robinson, P M, 1987. "Asymptotically Efficient Estimation in the Presence of Heteroskedasticity of Unknown Form," Econometrica, Econometric Society, vol. 55(4), pages 875-891, July.
    2. Pollard, David, 1985. "New Ways to Prove Central Limit Theorems," Econometric Theory, Cambridge University Press, vol. 1(3), pages 295-313, December.
    3. H.J. Bierens, 1981. "Robust Methods and Asymptotic Theory in Nonlinear Econometrics," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 35(3), pages 173-173, September.
    4. Andrews, Donald W. K., 1988. "Chi-square diagnostic tests for econometric models : Introduction and applications," Journal of Econometrics, Elsevier, vol. 37(1), pages 135-156, January.
    5. Andrews, Donald W K, 1988. "Chi-Square Diagnostic Tests for Econometric Models: Theory," Econometrica, Econometric Society, vol. 56(6), pages 1419-1453, November.
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    Cited by:

    1. de Castro, Luciano & Galvao, Antonio F. & Kaplan, David M. & Liu, Xin, 2019. "Smoothed GMM for quantile models," Journal of Econometrics, Elsevier, vol. 213(1), pages 121-144.
    2. de Castro, Luciano & Galvao, Antonio F. & Kaplan, David M. & Liu, Xin, 2019. "Smoothed GMM for quantile models," Journal of Econometrics, Elsevier, vol. 213(1), pages 121-144.
    3. Corradi, Valentina & Fernandez, Andres & Swanson, Norman R., 2009. "Information in the Revision Process of Real-Time Datasets," Journal of Business & Economic Statistics, American Statistical Association, vol. 27(4), pages 455-467.
    4. Sakata, Shinichi & White, Halbert, 2001. "S-estimation of nonlinear regression models with dependent and heterogeneous observations," Journal of Econometrics, Elsevier, vol. 103(1-2), pages 5-72, July.
    5. Odendahl, Florens & Rossi, Barbara & Sekhposyan, Tatevik, 2023. "Evaluating forecast performance with state dependence," Journal of Econometrics, Elsevier, vol. 237(2).
    6. Kuersteiner, Guido M., 2019. "Invariance principles for dependent processes indexed by Besov classes with an application to a Hausman test for linearity," Journal of Econometrics, Elsevier, vol. 211(1), pages 243-261.
    7. Donald W.K. Andrews, 1992. "An Introduction to Econometric Applications of Functional Limit Theory for Dependent Random Variables," Cowles Foundation Discussion Papers 1020, Cowles Foundation for Research in Economics, Yale University.
    8. Chung-Ming Kuan, 2013. "Markov switching model (in Russian)," Quantile, Quantile, issue 11, pages 13-40, December.
    9. Blasques, F. & Gorgi, P. & Koopman, S.J., 2021. "Missing observations in observation-driven time series models," Journal of Econometrics, Elsevier, vol. 221(2), pages 542-568.
    10. Hajivassiliou, 1993. "Macroeconomic Shocks in an Aggregative Disequilibrium Model," Cowles Foundation Discussion Papers 1063, Cowles Foundation for Research in Economics, Yale University.
    11. Hansen, Bruce E., 1996. "Stochastic Equicontinuity for Unbounded Dependent Heterogeneous Arrays," Econometric Theory, Cambridge University Press, vol. 12(2), pages 347-359, June.
    12. Donald W.K. Andrews & David Pollard, 1990. "A Functional Central Limit Theorem for Strong Mixing Stochastic Processes," Cowles Foundation Discussion Papers 951, Cowles Foundation for Research in Economics, Yale University.
    13. Arcones, Miguel A., 1996. "Weak convergence of stochastic processes indexed by smooth functions," Stochastic Processes and their Applications, Elsevier, vol. 62(1), pages 115-138, March.

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