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Wilks’ theorem for semiparametric regressions with weakly dependent data. (English) Zbl 1486.62239

Summary: The empirical likelihood inference is extended to a class of semiparametric models for stationary, weakly dependent series. A partially linear single-index regression is used for the conditional mean of the series given its past, and the present and past values of a vector of covariates. A parametric model for the conditional variance of the series is added to capture further nonlinear effects. We propose suitable moment equations which characterize the mean and variance model. We derive an empirical log-likelihood ratio which includes nonparametric estimators of several functions, and we show that this ratio behaves asymptotically as if the functions were given.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G05 Nonparametric estimation
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference

Software:

dlnm
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References:

[1] Bossaerts, P., Hafner, C. and Härdle, W. (1996). A New Method for Volatility Estimation with Applications in Foreign Exchange Rate Series. In Finanzmarktanalyse und -prognose Mit Innovativen Quantitativen Verfahren: Ergebnisse des 5. Karlsruher Ökonometrie-Workshops 71-83. Physica-Verlag HD, Heidelberg.
[2] Bradley, R. C. (2005). Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surv. 2 107-144. · Zbl 1189.60077 · doi:10.1214/154957805100000104
[3] Bravo, F., Escanciano, J. C. and Van Keilegom, I. (2020). Two-step semiparametric empirical likelihood inference. Ann. Statist. 48 1-26. · Zbl 1439.62188 · doi:10.1214/18-AOS1788
[4] Carroll, R. J., Fan, J., Gijbels, I. and Wand, M. P. (1997). Generalized partially linear single-index models. J. Amer. Statist. Assoc. 92 477-489. · Zbl 0890.62053 · doi:10.2307/2965697
[5] Chang, J., Chen, S. X. and Chen, X. (2015). High dimensional generalized empirical likelihood for moment restrictions with dependent data. J. Econometrics 185 283-304. · Zbl 1331.62188 · doi:10.1016/j.jeconom.2014.10.011
[6] Chang, J., Chen, S. X., Tang, C. Y. and Wu, T. T. (2021). High-dimensional empirical likelihood inference. Biometrika 108 127-147. · Zbl 1462.62760 · doi:10.1093/biomet/asaa051
[7] Chang, J., Tang, C. Y. and Wu, Y. (2013). Marginal empirical likelihood and sure independence feature screening. Ann. Statist. 41 2123-2148. · Zbl 1277.62109 · doi:10.1214/13-AOS1139
[8] Chang, J., Tang, C. Y. and Wu, Y. (2016). Local independence feature screening for nonparametric and semiparametric models by marginal empirical likelihood. Ann. Statist. 44 515-539. · Zbl 1486.62082 · doi:10.1214/15-AOS1374
[9] Chen, S. X. and Van Keilegom, I. (2009). A review on empirical likelihood methods for regression. TEST 18 415-447. · Zbl 1203.62035 · doi:10.1007/s11749-009-0159-5
[10] Chen, X. and Cui, H. (2008). Empirical likelihood inference for partial linear models under martingale difference sequence. Statist. Probab. Lett. 78 2895-2901. · Zbl 1317.62069 · doi:10.1016/j.spl.2008.04.012
[11] Dong, C., Gao, J. and TjØstheim, D. (2016). Estimation for single-index and partially linear single-index integrated models. Ann. Statist. 44 425-453. · Zbl 1331.62190 · doi:10.1214/15-AOS1372
[12] du Roy de Chaumaray, M., Marbac, M. and Patilea, V. (2021). Supplement to “Wilks’ theorem for semiparametric regressions with weakly dependent data.” https://doi.org/10.1214/21-AOS2081SUPP
[13] Fan, G.-L. and Liang, H.-Y. (2010). Empirical likelihood inference for semiparametric model with linear process errors. J. Korean Statist. Soc. 39 55-65. · Zbl 1293.62105 · doi:10.1016/j.jkss.2009.04.001
[14] Fryzlewicz, P. and Subba Rao, S. (2011). Mixing properties of ARCH and time-varying ARCH processes. Bernoulli 17 320-346. · Zbl 1284.62550 · doi:10.3150/10-BEJ270
[15] Gasparrini, A. (2011). Distributed lag linear and non-linear models in R: The package dlnm. J. Stat. Softw. 43 1-20.
[16] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application: Probability and Mathematical Statistics. Academic Press, New York. · Zbl 0462.60045
[17] Hansen, B. E. (2008). Uniform convergence rates for kernel estimation with dependent data. Econometric Theory 24 726-748. · Zbl 1284.62252 · doi:10.1017/S0266466608080304
[18] Härdle, W., Lütkepohl, H., Chen, R., Hardle, W. and Lutkepohl, H. (1997). A review of nonparametric time series analysis. Int. Stat. Rev. 65 49-72. · Zbl 0887.62043 · doi:10.2307/1403432
[19] Härdle, W., Tsybakov, A. and Yang, L. (1998). Nonparametric vector autoregression. J. Statist. Plann. Inference 68 221-245. · Zbl 0937.62042 · doi:10.1016/S0378-3758(97)00143-2
[20] Hjort, N. L., McKeague, I. W. and Van Keilegom, I. (2009). Extending the scope of empirical likelihood. Ann. Statist. 37 1079-1111. · Zbl 1160.62029 · doi:10.1214/07-AOS555
[21] Kanai, H., Ogata, H. and Taniguchi, M. (2010). Estimating function approach for CHARN models. Metron 68 1-21. · Zbl 1301.62036 · doi:10.1007/bf03263521
[22] Kato, H., Taniguchi, M. and Honda, M. (2006). Statistical analysis for multiplicatively modulated nonlinear autoregressive model and its applications to electrophysiological signal analysis in humans. IEEE Trans. Signal Process. 54 3414-3425. · Zbl 1373.94626 · doi:10.1109/tsp.2006.877663
[23] Kitamura, Y. (1997). Empirical likelihood methods with weakly dependent processes. Ann. Statist. 25 2084-2102. · Zbl 0881.62095 · doi:10.1214/aos/1069362388
[24] Li, G., Zhu, L., Xue, L. and Feng, S. (2010). Empirical likelihood inference in partially linear single-index models for longitudinal data. J. Multivariate Anal. 101 718-732. · Zbl 1181.62034 · doi:10.1016/j.jmva.2009.08.006
[25] Lian, H., Liang, H. and Carroll, R. J. (2015). Variance function partially linear single-index models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 171-194. · Zbl 1414.62301 · doi:10.1111/rssb.12066
[26] Liang, H., Liu, X., Li, R. and Tsai, C.-L. (2010). Estimation and testing for partially linear single-index models. Ann. Statist. 38 3811-3836. · Zbl 1204.62068 · doi:10.1214/10-AOS835
[27] Liebscher, E. (2005). Towards a unified approach for proving geometric ergodicity and mixing properties of nonlinear autoregressive processes. J. Time Series Anal. 26 669-689. · Zbl 1092.62091 · doi:10.1111/j.1467-9892.2005.00412.x
[28] Lu, X. (2009). Empirical likelihood for heteroscedastic partially linear models. J. Multivariate Anal. 100 387-396. · Zbl 1154.62033 · doi:10.1016/j.jmva.2008.05.006
[29] Lu, Z. and Jiang, Z. (2001). \[{L_1}\] geometric ergodicity of a multivariate nonlinear AR model with an ARCH term. Statist. Probab. Lett. 51 121-130. · Zbl 1059.62585 · doi:10.1016/S0167-7152(00)00138-3
[30] Ma, Y. and Zhu, L. (2013). Doubly robust and efficient estimators for heteroscedastic partially linear single-index models allowing high dimensional covariates. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 305-322. · Zbl 07555449 · doi:10.1111/j.1467-9868.2012.01040.x
[31] Masry, E. and Tjostheim, D. (1995). Nonparametric estimation and identification of nonlinear ARCH time series strong convergence and asymptotic normality: Strong convergence and asymptotic normality. Econometric Theory 11 258-289. · Zbl 1401.62171
[32] Meitz, M. and Saikkonen, P. (2010). A note on the geometric ergodicity of a nonlinear AR-ARCH model. Statist. Probab. Lett. 80 631-638. · Zbl 1185.62160 · doi:10.1016/j.spl.2009.12.020
[33] Mokkadem, A. (1988). Mixing properties of ARMA processes. Stochastic Process. Appl. 29 309-315. · Zbl 0647.60042 · doi:10.1016/0304-4149(88)90045-2
[34] Mokkadem, A. (1990). Propriétés de mélange des processus autorégressifs polynomiaux. Ann. Inst. Henri Poincaré Probab. Stat. 26 219-260. · Zbl 0706.60040
[35] Nordman, D. J. and Lahiri, S. N. (2014). A review of empirical likelihood methods for time series. J. Statist. Plann. Inference 155 1-18. · Zbl 1307.62120 · doi:10.1016/j.jspi.2013.10.001
[36] Owen, A. B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75 237-249. · Zbl 0641.62032 · doi:10.1093/biomet/75.2.237
[37] Owen, A. B. (2001). Empirical Likelihood. Chapman and Hall/CRC, Boca Raton, FL. · Zbl 0989.62019
[38] Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating equations. Ann. Statist. 22 300-325. · Zbl 0799.62049 · doi:10.1214/aos/1176325370
[39] Rio, E. (2000). Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants. Mathématiques & Applications (Berlin) [Mathematics & Applications] 31. Springer, Berlin.
[40] Severini, T. A. and Wong, W. H. (1992). Profile likelihood and conditionally parametric models. Ann. Statist. 20 1768-1802. · Zbl 0768.62015 · doi:10.1214/aos/1176348889
[41] TjØstheim, D. (1990). Nonlinear time series and Markov chains. Adv. in Appl. Probab. 22 587-611. · Zbl 0712.62080 · doi:10.2307/1427459
[42] Tong, H. (1990). Nonlinear Time Series: A Dynamical System Approach. Oxford Statistical Science Series 6. The Clarendon Press, Oxford University Press, New York. · Zbl 0716.62085
[43] Wang, Q.-H. and Jing, B.-Y. (1999). Empirical likelihood for partial linear models with fixed designs. Statist. Probab. Lett. 41 425-433. · Zbl 1054.62550 · doi:10.1016/S0167-7152(98)00230-2
[44] Wang, Q.-H. and Jing, B.-Y. (2003). Empirical likelihood for partial linear models. Ann. Inst. Statist. Math. 55 585-595. · Zbl 1047.62026 · doi:10.1007/BF02517809
[45] Xia, Y. and Härdle, W. (2006). Semi-parametric estimation of partially linear single-index models. J. Multivariate Anal. 97 1162-1184. · Zbl 1089.62050 · doi:10.1016/j.jmva.2005.11.005
[46] Xia, Y., Tong, H. and Li, W. K. (1999). On extended partially linear single-index models. Biometrika 86 831-842. · Zbl 0942.62109 · doi:10.1093/biomet/86.4.831
[47] Xue, L. and Zhu, L. (2007). Empirical likelihood semiparametric regression analysis for longitudinal data. Biometrika 94 921-937. · Zbl 1156.62324 · doi:10.1093/biomet/asm066
[48] Xue, L.-G. and Zhu, L. (2006). Empirical likelihood for single-index models. J. Multivariate Anal. 97 1295-1312. · Zbl 1099.62045 · doi:10.1016/j.jmva.2005.09.004
[49] Zhu, L., Lin, L., Cui, X. and Li, G. (2010). Bias-corrected empirical likelihood in a multi-link semiparametric model. J. Multivariate Anal. 101 850-868. · Zbl 1181.62039 · doi:10.1016/j.jmva.2009.08.009
[50] Zhu, L. and Xue, L. (2006). Empirical likelihood confidence regions in a partially linear single-index model. J. R. Stat. Soc. Ser. B. Stat. Methodol. 68 549-570 · Zbl 1110.62055 · doi:10.1111/j.1467-9868.2006.00556.x
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