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Semiparametric quantile regression estimation in dynamic models with partially varying coefficients. (English) Zbl 1441.62623
Summary: We study quantile regression estimation for dynamic models with partially varying coefficients so that the values of some coefficients may be functions of informative covariates. Estimation of both parametric and nonparametric functional coefficients are proposed. In particular, we propose a three stage semiparametric procedure. Both consistency and asymptotic normality of the proposed estimators are derived. We demonstrate that the parametric estimators are root-$$n$$ consistent and the estimation of the functional coefficients is oracle. In addition, efficiency of parameter estimation is discussed and a simple efficient estimator is proposed. A simple and easily implemented test for the hypothesis of a varying-coefficient is proposed. A Monte Carlo experiment is conducted to evaluate the performance of the proposed estimators.

##### MSC:
 62P20 Applications of statistics to economics 62G08 Nonparametric regression and quantile regression 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62G20 Asymptotic properties of nonparametric inference
quantreg
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##### References:
 [1] Ahmad, I.; Leelahanon, S.; Li, Q., Efficient estimation of a semiparametric partially linear varying coefficient mode, The annals of statistics, 33, 258-283, (2005) · Zbl 1064.62043 [2] Cai, Z., Regression quantile for time series, Econometric theory, 18, 169-192, (2002) · Zbl 1181.62124 [3] Cai, Z., Two-step likelihood estimation procedure for varying-coefficient models, Journal of multivariate analysis, 82, 189-209, (2002) · Zbl 0995.62039 [4] Cai, Z.; Fan, J., Average regression surface for dependent data, Journal of multivariate analysis, 75, 112-142, (2000) · Zbl 0960.62096 [5] Cai, Z.; Fan, J.; Yao, Q., Functional-coefficient regression models for nonlinear time series, Journal of the American statistical association, 95, 941-956, (2000) · Zbl 0996.62078 [6] Cai, Z.; Gu, J.; Li, Q., Recent developments in nonparametric econometrics, Advances in econometrics, 25, 495-549, (2009) · Zbl 1190.62195 [7] Cai, Z.; Masry, E., Nonparametric estimation of additive nonlinear ARX time series: local linear Fitting and projection, Econometric theory, 16, 465-501, (2000) · Zbl 0997.62065 [8] Cai, Z.; Xu, X., Nonparametric quantile estimations for dynamic smooth coefficient models, Journal of the American statistical association, 103, 1596-1608, (2008) · Zbl 1286.62029 [9] Chaudhuri, P., Nonparametric estimates of regression quantiles and their local bahadur representation, The annals of statistics, 19, 760-777, (1991) · Zbl 0728.62042 [10] Chaudhuri, P.; Doksum, K.; Samarov, A., On average derivative quantile regression, The annuals of statistics, 25, 715-744, (1997) · Zbl 0898.62082 [11] De Gooijer, J.; Zerom, D., On additive conditional quantiles with high dimensional covariates, Journal of the American statistical association, 98, 135-146, (2003) · Zbl 1047.62027 [12] Dette, H.; Spreckelsen, I., Some comments on specification tests in nonparametric absolutely regular processes, Journal of time series analysis, 25, 159-172, (2004) · Zbl 1051.62043 [13] Doukhan, P., Mixing, Lecture notes in statistics, vol. 85, (1994), Springer-Verlag Heielberg [14] Engle, R.; Granger, C.W.J.; Rice, R.; Weiss, A., Nonparametric estimates of the relation between weather and electricity sales, Journal of the American statistical association, 81, 310-320, (1986) [15] Fan, J.; Gijbels, I., Local polynomial modeling and its applications, (1996), Chapman and Hall London · Zbl 0873.62037 [16] Fan, J.; Huang, T., Profile likelihood inferences on semiparametric varying-coefficient partially linear models, Bernoulli, 11, 1031-1057, (2005) · Zbl 1098.62077 [17] Gao, J., Nonlinear time series: semiparametric and nonparametric methods, (2007), Chapman and Hall London [18] He, X.; Liang, H., Quantile regression estimates for a class of linear and partially linear errors-in-variables models, Statistica sinica, 10, 129-140, (2000) · Zbl 0970.62043 [19] He, X.; Ng, P., Quantile splines with several covariates, Journal of statistical planning and inference, 75, 343-352, (1999) · Zbl 0931.62031 [20] He, X.; Ng, P.; Portnoy, S., Bivariate quantile smoothing splines, Journal of the royal statistical society, series B, 60, 537-550, (1998) · Zbl 0909.62038 [21] He, X.; Portnoy, S., Some asymptotic results on bivariate quantile splines, Journal of statistical planning and inference, 91, 341-349, (2000) · Zbl 1091.62515 [22] He, X.; Shi, P., Bivariate tensor-product $$B$$-splines in a partly linear model, Journal of multivariate analysis, 58, 162-181, (1996) · Zbl 0865.62027 [23] Honda, T., Nonparametric estimation of a conditional quantile for $$\alpha$$-mixing processes, Annals of the institute of statistical mathematics, 52, 459-470, (2000) · Zbl 0960.62033 [24] Honda, T., Quantile regression in varying coefficient models, Journal of statistical planning and inferences, 121, 113-125, (2004) · Zbl 1038.62041 [25] Horowitz, J.L.; Lee, S., Nonparametric estimation of an additive quantile regression model, Journal of the American statistical association, 100, 1238-1249, (2005) · Zbl 1117.62355 [26] Khindanova, I.N.; Rachev, S.T., Value at risk: recent advances, () · Zbl 0978.91050 [27] Kim, M.-O., Quantile regression with varying coefficients, The annals of statistics, 35, 92-108, (2007) · Zbl 1114.62051 [28] Koenker, R., 2004. Quantreg: An R Package for Quantile Regression and Related Methods http://cran.r-project.org. [29] Koenker, R., Quantile regression, Econometric society monograph series, (2005), Cambridge University Press New York · Zbl 1111.62037 [30] Koenker, R.; Bassett, G.W., Regression quantiles, Econometrica, 46, 33-50, (1978) · Zbl 0373.62038 [31] Koenker, R.; Ng, P.; Portnoy, S., Quantile smoothing splines, Biometrika, 81, 673-680, (1994) · Zbl 0810.62040 [32] Koenker, R.; Xiao, Z., Quantile autoregression, Journal of the American statistical association, 101, 980-990, (2006) · Zbl 1120.62326 [33] Lee, S., Efficient semiparametric estimation of partially linear quantile regression model, Econometric theory, 19, 1-31, (2003) · Zbl 1031.62034 [34] Lee, A.J., $$U$$-statistics: theory and practice, (1990), Marcel Dekker New York · Zbl 0771.62001 [35] Li, Q.; Huang, C.J.; Li, D.; Fu, T., Semiparametric smooth coefficient model, Journal of business & economics statistics, 20, 412-422, (2002) [36] Robinson, P.M., Root-$$N$$-consistent semiparametric regression, Econometrica, 56, 931-954, (1988) · Zbl 0647.62100 [37] Robinson, P.M., Hypothesis testing in semiparametric and nonparametric models for econometric time series, Review of economic studies, 56, 511-534, (1989) · Zbl 0681.62101 [38] Ruppert, D.; Wand, M., Multivariate locally least squares regression, The annals of statistics, 22, 1346-1370, (1994) · Zbl 0821.62020 [39] Speckman, P., Kernel smoothing partial linear models, The journal of royal statistical society, series B, 50, 413-426, (1988) · Zbl 0671.62045 [40] Yu, K.; Jones, M.C., Local linear quantile regression, Journal of the American statistical association, 93, 228-237, (1998) · Zbl 0906.62038 [41] Yu, K.; Lu, Z., Local linear additive quantile regression, Scandinavian journal of statistics, 31, 333-346, (2004) · Zbl 1063.62060 [42] Zhang, W.; Lee, S.Y.; Song, X., Local polynomial Fitting in semivarying coefficient model, Journal of multivariate analysis, 82, 166-188, (2002) · Zbl 0995.62038
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