×

zbMATH — the first resource for mathematics

A consistent model specification test with mixed discrete and continuous data. (English) Zbl 1247.62126
Summary: In this paper we propose a nonparametric kernel-based model specification test that can be used when the regression model contains both discrete and continuous regressors. We employ discrete variable kernel functions and we smooth both the discrete and continuous regressors using least squares cross-validation (CV) methods. The test statistic is shown to have an asymptotic normal null distribution. We also prove the validity of using the wild bootstrap method to approximate the null distribution of the test statistic, the bootstrap being our preferred method for obtaining the null distribution in practice. Simulations show that the proposed test has significant power advantages over conventional kernel tests which rely upon frequency-based nonparametric estimators that require sample splitting to handle the presence of discrete regressors.

MSC:
62G10 Nonparametric hypothesis testing
62G08 Nonparametric regression and quantile regression
62E20 Asymptotic distribution theory in statistics
62G09 Nonparametric statistical resampling methods
65C60 Computational problems in statistics (MSC2010)
PDF BibTeX Cite
Full Text: DOI
References:
[1] Ahmad, I.A.; Cerrito, P.B., Nonparametric estimation of joint discrete-continuous probability densities with applications, Journal of statistical planning and inference, 41, 349-364, (1994) · Zbl 0803.62033
[2] Ait-Sahalia, Y.; Bickel, P.; Stoker, T.M., Goodness-of-fit tests for kernel regression with an application to option implied volatilities, Journal of econometrics, 105, 363-412, (2001) · Zbl 1004.62042
[3] Aitchison, J.; Aitken, C.G.G., Multivariate binary discrimination by the kernel method, Biometrika, 63, 413-420, (1976) · Zbl 0344.62035
[4] Andrews, D.W.K., A conditional Kolmogorov test, Econometrica, 65, 1097-1128, (1997) · Zbl 0928.62019
[5] Bhattacharya, R.N.; Rao, R.R., Normal approximations and asymptotic expansions, (1986), Krieger Malabar, FL · Zbl 0657.41001
[6] Bierens, H.J., Uniform consistency of kernel estimators of a regression function under generalized conditions, Journal of American statistical association, 77, 699-707, (1983) · Zbl 0565.62027
[7] Bierens, H.J., Kernel estimators of regression functions, (), 99-144 · Zbl 0850.62348
[8] Bierens, H.J.; Ploberger, W., Asymptotic theory of integrated conditional moment tests, Econometrica, 65, 1129-1152, (1997) · Zbl 0927.62085
[9] Billingsley, P., Convergence of probability measure, (1968), Wiley New York · Zbl 0172.21201
[10] Bowman, A.W., A note on consistency of the kernel method for the analysis of categorical data, Biometrika, 67, 682-684, (1980)
[11] de Jong, P., A central limit theorem for generalized quadratic forms, Probability and related fields, 75, 261-277, (1987) · Zbl 0596.60022
[12] de Jong, R.M., The bierens test under data dependence, Journal of econometrics, 72, 1-32, (1996) · Zbl 0855.62073
[13] Eubank, R.; Hart, J., Testing goodness-of-fit in regression via order selection criteria, The annals of statistics, 20, 1412-1425, (1992) · Zbl 0776.62045
[14] Eubank, R.; Spiegelman, S., Testing the goodness-of-fit of a linear model via nonparametric regression techniques, Journal of the American statistical association, 85, 387-392, (1990) · Zbl 0702.62037
[15] Fan, Y.; Li, Q., Consistent model specification tests: omitted variables and semiparametric functional forms, Econometrica, 64, 865-890, (1996) · Zbl 0854.62038
[16] Fan, Y.; Li, Q., Consistent model specification tests: kernel-based test versus bierens’ ICM tests, Econometric theory, 16, 1016-1041, (2000) · Zbl 1180.62071
[17] Fahrmeir, L.; Tutz, G., Multivariate statistical modeling based on generalized models, (1994), Springer New York
[18] Freeman, R.B.; Medoff, J.L., What do unions do?, (1984), Basic Books New York
[19] Grund, B.; Hall, P., On the performance of kernel estimators for high-dimensional sparse binary data, Journal of multivariate analysis, 44, 321-344, (1993) · Zbl 0766.62019
[20] Hall, P., On nonparametric multivariate binary discrimination, Biometrika, 68, 287-294, (1981) · Zbl 0463.62059
[21] Hall, P., Central limit theorem for integrated square error of multivariate nonparametric density estimators, Journal of multivariate analysis, 14, 1-16, (1984) · Zbl 0528.62028
[22] Hall, P.; Wand, M., On nonparametric discrimination using density differences, Biometrika, 75, 541-547, (1988) · Zbl 0651.62029
[23] Hall, P.; Racine, J.; Li, Q., Cross-validation and the estimation of conditional probability densities, Journal of the American statistical association, 99, 1015-1026, (2004) · Zbl 1055.62035
[24] Hall, P., Li, Q., Racine, J., 2005. Nonparametric estimation of regression functions in the presence of irrelevant regressors. Manuscript, Texas A&M University.
[25] Härdle, W.; Mammen, E., Comparing nonparametric versus parametric regression fits, The annals of statistics, 21, 1926-1947, (1993) · Zbl 0795.62036
[26] Hart, J.D., Nonparametric smoothing and lack-of-fit tests, (1997), Springer New York · Zbl 0886.62043
[27] Hong, Y.; White, H., Consistent specification testing via nonparametric series regression, Econometrica, 63, 1133-1159, (1995) · Zbl 0941.62125
[28] Horowitz, J.T.; Spokoiny, V.G., An adaptive, rate-optimal test of a parametric model against a nonparametric alternative, Econometrica, 69, 599-631, (2001) · Zbl 1017.62012
[29] Ichimura, H., 2000. Asymptotic distribution of non-parametric and semiparametric estimators with data dependent smoothing parameters. Manuscript.
[30] Lanot, G.; Walker, I., The union/non-union wage differential: an application of semi-parametric methods, Journal of econometrics, 84, 327-349, (1998) · Zbl 1041.62528
[31] Li, Q.; Racine, J., Cross-validated local linear nonparametric regression, Statistica sinica, 14, 485-512, (2004) · Zbl 1045.62033
[32] Li, Q.; Racine, J., Nonparametric econometrics: theory and practice, (2006), Princeton University Press Princeton, NJ
[33] Li, Q.; Wang, S., A simple consistent bootstrap test for a parametric regression function, Journal of econometrics, 87, 145-165, (1998) · Zbl 0943.62031
[34] Li, Q.; Hsiao, C.; Zinn, J., Consistent specification tests for semiparametric/nonparametric models based on series estimation methods, Journal of econometrics, 112, 295-325, (2003) · Zbl 1027.62027
[35] Mammen, E., When does bootstrap work? asymptotic results and simulations, (1992), Springer New York · Zbl 0760.62038
[36] Murphy, K.M.; Welch, F., Empirical age-earnings profiles, Journal of labor economics, 8, 202-229, (1990)
[37] Ossiander, M., A central limit theorem under metric entropy with \(L_2\) bracketing, Annals of probability, 15, 897-919, (1987) · Zbl 0665.60036
[38] Racine, J.; Li, Q., Nonparametric estimation of regression functions with both categorical and continuous data, Journal of econometrics, 119, 99-130, (2004) · Zbl 1337.62062
[39] 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
[40] Robinson, P.M., Consistent nonparametric entropy-based testing, Review of economic studies, 58, 437-453, (1991) · Zbl 0719.62055
[41] Scott, D., Multivariate in density estimation: theory, practice, and visualization, (1992), Wiley New York · Zbl 0850.62006
[42] Simonoff, J.S., Smoothing methods in statistics, (1996), Springer New York · Zbl 0859.62035
[43] Wooldridge, J., A test for functional form against nonparametric alternatives, Econometric theory, 8, 452-475, (1992)
[44] Yatchew, A.J., Nonparametric regression tests based on least squares, Econometric theory, 8, 435-451, (1992)
[45] Zheng, J.X., A consistent test of functional form via nonparametric estimation technique, Journal of econometrics, 75, 263-289, (1996) · Zbl 0865.62030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.