High Dimensional Generalized Empirical Likelihood for Moment Restrictions with Dependent Data

Chang, Jinyuan and Chen, Song Xi and Chen, Xiaohong (2014): High Dimensional Generalized Empirical Likelihood for Moment Restrictions with Dependent Data. Forthcoming in: Journal of Econometrics

Abstract

This paper considers the maximum generalized empirical likelihood (GEL) estimation and inference on parameters identified by high dimensional moment restrictions with weakly dependent data when the dimensions of the moment restrictions and the parameters diverge along with the sample size. The consistency with rates and the asymptotic normality of the GEL estimator are obtained by properly restricting the growth rates of the dimensions of the parameters and the moment restrictions, as well as the degree of data dependence. It is shown that even in the high dimensional time series setting, the GEL ratio can still behave like a chi-square random variable asymptotically. A consistent test for the over-identification is proposed. A penalized GEL method is also provided for estimation under sparsity setting.

• Item Type: MPRA Paper
• Language: English
• Keywords: Generalized empirical likelihood; High dimensionality; Penalized likelihood; Variable selection; Over-identification test; Weak dependence.
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General

References

Ai, C. and Chen, X. (2003). Efficient estimation of models with conditional moment restrictions containing unknown functions, Econometrica, 71, 1795-1843.

Anatolyev, S. (2005). GMM, GEL, serial correlation, and asymptotic bias, Econometrica, 73, 983-1002.

Bai, J. and Ng, S. (2002). Determining the number of factors in approximating factor models, Econometrica, 70, 191-221.

Billingsley, P. (1995). Probability and Measure (3rd edition), Wiley, New York.

Carlstein, E. (1986). The use of subseries values for estimating the variance of a general statistic from a stationary sequence, The Annals of Statistics, 14, 1171-1179.

Chang, J., Tang, C. Y. and Wu, Y. (2013). Marginal empirical likelihood and sure independence feature screening, The Annals of Statistics, 41, 2123-2148.

Chen, S. X and Cui, H. (2003). An extended empirical likelihood for generalized linear models, Statistica Sinica, 13, 69-81.

Chen, S. X. and Cui, H. (2006). On Bartlett correction of empirical likelihood in the presence of nuisance parameters, Biometrika, 93, 215-220.

Chen, S. X. and Cui, H. (2007). On the second properties of empirical likelihood with moment restrictions, Journal of Econometrics, 141, 492-516.

Chen, S. X., Peng, L. and Qin, Y. L. (2009). Effects of data dimension on empirical likelihood, Biometrika, 96, 711-722.

Chen, S. X. and Van Keilegom, I. (2009). A review on empirical likelihood methods for regression (with discussion), Test, 18, 415-447.

Chen, X. (2007). Large sample sieve estimation of semi-nonparametric models in The Handbook of Econometrics, 6B, ed. by J. J. Heckman and E. Leamer. Amsterdam: North-Holland.

Chen, X. and Pouzo, D. (2012). Estimation of nonparametric conditional moment models with possibly nonsmooth generalized residuals, Econometrica, 80, 277-321.

Davis, R. A., Zhang, P. and Zheng, T. (2012). Sparse vector autoregressive modelling. Available at arXiv:1207.0520.

Davydov, Y. A. (1968). On convergence of distributions generated by stationary stochastic processes, Theory of Probability and its Applications, 13, 691-696.

Donald, S. G., Imbens, G. W. and Newey, W. K. (2003). Empirical likelihood estimation and consistent tests with conditional moment restrictions, Journal of Econometrics, 117, 55-93.

Doukhan, P. (1994). Mixing: properties and examples, Springer-Verlag, Berlin.

Durrett, R. (2010). Probability: Theory and Examples (4th edition), Cambridge University Press.

Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties, Journal of the American Statistical Association, 96, 1348-1360.

Fan, J. and Liao, Y. (2014). Endogeneity in high dimensions, The Annals of Statistics, 42, 872{917. Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods, Springer, New York.

Fan, J., Zhang, C. and Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon, The Annals of Statistics, 29, 153-193.

Francq, C. and Zakoian, J. M. (2005). A central limit theorem for mixing triangular arrays of variables whose dependence is allowed to grow with the sample size, Econometric Theory, 21, 1165-1171.

Francq, C. and Zakoian, J. M. (2007). HAC estimation and strong linearity testing in weak ARMA models, Journal of Multivariate Analysis, 98, 114-144.

Hall, P. (1985). Resampling a coverage pattern, stochastical Processes Application, 20, 231-246.

Hall, P., Horowitz, J. and Jing, B.-Y. (1995). On blocking rules for the bootstrap and dependent data, Biometrika, 82, 561-574.

Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators, Econometrica, 50, 1029-1054.

Hansen, L. P., Heaton, J. and Yaron, A. (1996). Finite-sample properties of some alternative GMM estimators, Journal of Business and Economic Statistics, 14, 262-280.

Hansen, L. P. and Singleton, K. (1982). Generalized instrumental variables estimation of nonlinear rational expectations models, Econometrica, 50, 1269-1286.

Hjort, N. L., McKeague, I. and Van Keilegom, I. (2009). Extending the scope of empirical likelihood, The Annals of Statistics, 37, 1079-1111.

Imbens, G. W., Spady, R. H. and Johnson, P. (1998). Information theoretic approaches to inference in moment condition models, Econometrica, 66, 333-357.

Kitamura, Y. (1997). Empirical likelihood methods with weakly dependent processes, The Annals of Statistics, 25, 2084-2102.

Kitamura, Y. (2007). Empirical likelihood methods in econometrics: theory and practice" in Advances in Economics and Econometrics: Ninth World Congress of the Econometric Society , R. Blundell, W. K. Newey and T. Personn (eds.), Cambridge University Press.

Kitamura, Y. and Stutzer, M. (1997). An information-theoretic alternative to generalized method of moments estimation, Econometrica, 65, 861-874.

Kunsch, H. R. (1989). The jackkinife and the bootstrap for general stationary observations, The Annals of Statistics, 17, 1217-1241.

Lahiri, S. N. (2003). Resampling Methods for Dependent Data, Springer, New York.

Lahiri, S. N. and Mukhopadhyay, S. (2012). A penalized empirical likelihood method in high dimensions, The Annals of Statistics, 40, 2511-2540.

Leng, C. and Tang, C. Y. (2012). Penalized empirical likelihood and growing dimensional general estimating equations, Biometrika, 99, 703-716.

Lutkepohl, H. (2006). New Introduction to Multiple Time Series Analysis, Springer, Berlin.

Newey, W. K. (1991). Uniform convergence in probability and stochastic equicontinuity, Econometrica, 59, 1161-1167.

Newey, W. K. and Smith, R. J. (2004). Higher order properties of GMM and generalized empirical likelihood estimators, Econometrica, 72, 219-255.

Nordman, D. J. and Lahiri, S. N. (2013). A review of empirical likelihood methods for time series, Journal of Statistical Planning and Inference, to appear.

Owen, A. (1988). Empirical likelihood ratio confidence intervals for a single functional, Biometrika, 75, 237-249.

Owen, A. (1990). Empirical likelihood ratio confidence regions, The Annals of Statistics, 18, 90-120.

Owen, A. (2001). Empirical Likelihood, Chapman and Hall-CRC, New York.

Peligrad, M. and Utev, S. (1997). Central limit theorem for linear processes, The Annals of Probability, 25, 443-456.

Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating equations, The Annals of Statistics, 22, 300-325.

Rio, E. (1993). Covariance inequalities for strongly mixing processes, Annales de l'Institut Henri Poincare, 29, 589-597.

Rothenberg, T. J. (1973). Efficient Estimation with a priori Information, Yale University Press, New Haven, USA.

Rudin, W. (1976). Principles of Mathematical Analysis, McGraw-Hill, New York.

Smith, R. J. (1997). Alternative semi-parametric likelihood approaches to generalized method of moments estimation, Economic Journal, 107, 503-519.

Stock, J. H. and Watson, M. W. (2010). Dynamic factor models", Oxford Handbook of Economic Forecasting, M. Clements and D. Hendry (eds), Chapter 2.

Tang, C. Y. and Leng, C. (2010). Penalized high dimensional empirical likelihood, Biometrika, 97, 905-920.

Zhang, C. H. (2010). Nearly unbiased variable selection under minimax concave penalty, The Annals of Statistics, 38, 894-942.