×

zbMATH — the first resource for mathematics

Consistent specification tests for semiparametric/nonparametric models based on series estimation methods. (English) Zbl 1027.62027
Summary: This paper considers the problem of consistent model specification tests using series estimation methods. The null models we consider all contain some nonparametric components. A leading case we consider is to test for an additive partially linear model. The null distribution of the test statistic is derived using a central limit theorem for Hilbert-valued random arrays. The test statistic is shown to be able to detect local alternatives that approach the null models at the order of O\(_p(n^{-1/2})\).
We show that the wild bootstrap method can be used to approximate the null distribution of the test statistic. A small Monte Carlo simulation is reported to examine the finite sample performance of the proposed test. We also show that the proposed test can be easily modified to obtain series-based consistent tests for other semiparametric/nonparametric models.

MSC:
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
62P20 Applications of statistics to economics
62G09 Nonparametric statistical resampling methods
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 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
[2] Anderson, N.T.; Gine, E.; Ossiander, M.; Zinn, J., The central limit theorems and the law of iterative logarithm for empirical processes under local conditions, Probability theory and related fields, 77, 271-303, (1988)
[3] Andrews, D.W.K., Asymptotic normality of series estimators for nonparametric and semiparametric regression models, Econometrica, 59, 307-345, (1991) · Zbl 0727.62047
[4] Andrews, D.W.K., A conditional Kolmogorov test, Econometrica, 65, 1097-1128, (1997) · Zbl 0928.62019
[5] Andrews, D.W.K.; Whang, Y.J., Additive interactive regression modelscircumvention of the curse of dimensionality, Econometric theory, 6, 466-479, (1990)
[6] Araujo, A., Gine, E., 1980. The central Limit Theorem and Banach Valued Random Variables. Wiley, New York. · Zbl 0457.60001
[7] Bierens, H.J., Consistent model specification tests, Journal of econometrics, 20, 105-134, (1982) · Zbl 0549.62076
[8] Bierens, H.J., A consistent conditional moment test of functional form, Econometrica, 58, 1443-1458, (1990) · Zbl 0737.62058
[9] Bierens, H.J.; Ploberger, W., Asymptotic theory of integrated conditional moment tests, Econometrica, 65, 1129-1151, (1997) · Zbl 0927.62085
[10] Chen, X.; Fan, Y., Consistent hypothesis testing in semiparametric and nonparametric models for econometric time series, Journal of econometrics, 91, 373-401, (1999) · Zbl 1041.62506
[11] Chen, X.; Shen, X., Sieve extremum estimates for weakly dependent data, Econometrica, 66, 289-314, (1998) · Zbl 1055.62544
[12] Chen, X.; White, H., Central limit theorems and functional central limit theorems for Hilbert-valued dependent heterogeneous arrays with applications, Econometric theory, 14, 289-314, (1997)
[13] Dechevsky, L.; Penez, S., On shape-preserving probabilistic wavelet approximators, Stochastic analysis and applications, 15, 2, 187-215, (1997) · Zbl 0902.60018
[14] De John, R.M., The bierens test under data dependence, Journal of econometrics, 72, 1-32, (1996) · Zbl 0855.62073
[15] Delgado, M.A., Testing the equality of nonparametric curves, Probability and statistics letters, 17, 199-204, (1993) · Zbl 0771.62034
[16] Delgado, M.A.; Manteiga, W.G., Significance testing in nonparametric regression based on the bootstrap, Annals of statistics, 29, 1469-1507, (2001) · Zbl 1043.62032
[17] Delgado, M.A.; Stengos, T., Semiparametric testing of non-nested econometric models, Review of economic studies, 75, 345-367, (1994)
[18] Donald, S.G., Inference concerning the number of factors in a multivariate nonparametric relationship, Econometrica, 65, 103-131, (1997) · Zbl 0873.62127
[19] Donald, S.G.; Newey, W.K., Series estimation of semilinear regression, Journal of multivariate analysis, 50, 30-40, (1994) · Zbl 0798.62074
[20] Ellison, G.; Ellison, S.F., A simple framework for nonparametric specification testing, Journal of econometrics, 96, 1-23, (2000) · Zbl 0968.62046
[21] 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
[22] Fan, Y., Li, Q., 1996a. On estimating additive partially linear models, unpublished manuscript.
[23] Fan, Y.; Li, Q., Consistent model specification testsomitted variables, parametric and semiparametric functional forms, Econometrica, 64, 865-890, (1996) · Zbl 0854.62038
[24] Fan, Y.; Li, Q., Central limit theorem for degenerate U-statistics of absolutely regular processes with applications to model specification tests, Journal of nonparametric statistics, 10, 245-271, (1999) · Zbl 0974.62044
[25] Fan, J.; Härdle, W.; Mammen, E., Direct estimation of low dimensional components in additive models, Annals of statistics, 26, 943-971, (1998) · Zbl 1073.62527
[26] Gine, E.; Zinn, J., Bootstraping general empirical measures, Annals of probability, 18, 851-869, (1990) · Zbl 0706.62017
[27] Gozalo, P.; Linton, O., Testing additivity in generalized nonparametric regression models with estimated parameters, Journal of econometrics, 104, 1-48, (2001) · Zbl 0978.62032
[28] Härdle, W.; Mammen, E., Comparing nonparametric versus parametric regression fits, Annals of statistics, 21, 1926-1947, (1993) · Zbl 0795.62036
[29] Hong, Y.; White, H., Consistent specification testing via nonparametric series regression, Econometrica, 63, 1133-1159, (1995) · Zbl 0941.62125
[30] Horowitz, J.L.; Härdle, W., Testing a parametric model against a semiparametric alternative, Econometric theory, 10, 821-848, (1994)
[31] Lavergne, P., An equality test across nonparametric regressions, Journal of econometrics, 103, 307-344, (2001) · Zbl 0969.62029
[32] Lavergne, P.; Vuong, Q., Nonparametric selection of regressorsthe nonnested case, Econometrica, 64, 207-219, (1996) · Zbl 0860.62039
[33] Ledoux, M.; Talagrand, M., Probability in Banach space, (1991), Springer New York
[34] Lewbel, A., Consistent nonparametric testing with an application to testing slusky symmetry, Journal of econometrics, 67, 379-401, (1995) · Zbl 0820.62042
[35] Li, Q., Efficient estimation of additive partially linear models, International economic reviews, 41, 1073-1092, (2000)
[36] Li, Q.; Wang, S., A simple consistent bootstrap test for a parametric regression functional form, Journal of econometrics, 87, 145-165, (1998) · Zbl 0943.62031
[37] Linton, O.B., Efficient estimation of generalized additive nonparametric regression models, Econometric theory, 16, 502-523, (2000) · Zbl 0963.62037
[38] Linton, O.B.; Nielsen, J.P., A kernel method of estimating structured nonparametric regression based on marginal integration, Biometrika, 82, 91-100, (1995) · Zbl 0823.62036
[39] Lorentz, G.G., Approximation of functions, (1966), Chelsea New York · Zbl 0153.38901
[40] Mammen, E.; Linton, O.; Nielsen, J.P., The existence and asymptotic properties of a backfitting projection algorithm under weak conditions, Annals of statistics, 27, 1443-1490, (1999) · Zbl 0986.62028
[41] Newey, W.K., Kernel estimation of partial means in a general variance estimator, Econometric theory, 10, 233-253, (1994)
[42] Newey, W.K., Convergence rates for series estimators, (), 254-275
[43] Newey, W.K., Convergence rates and asymptotic normality for series estimators, Journal of econometrics, 79, 147-168, (1997) · Zbl 0873.62049
[44] Nielsen, J.P.; Linton, O.B., An optimal interpretation of integration and backfitting estimators for separable nonparametric models, Journal of the royal statistical society, series B, 60, 217-222, (1998) · Zbl 0909.62040
[45] Opsomer, J.D.; Ruppert, D., Fitting a bivariate additive model by local polynomial regression, Annals of statistics, 25, 186-211, (1997) · Zbl 0869.62026
[46] Ossiander, M., A central limit theorem under metric entropy with L2 bracketing, Annals of probability, 15, 897-919, (1987) · Zbl 0665.60036
[47] Politis, D.N.; Romano, J.P., Limit theorems for weakly dependent Hilbert space valued random variables with applications to stationary bootstrap, Statistica sinica, 4, 461-476, (1994) · Zbl 0824.60006
[48] Robinson, P., Root-N consistent semiparametric regression, Econometrica, 56, 931-954, (1988) · Zbl 0647.62100
[49] 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
[50] Robinson, P.M., Consistent nonparametric entropy-based testing, Review of economic studies, 58, 437-453, (1991) · Zbl 0719.62055
[51] Schumakers, L.L., 1980. Spline Functions: Basic Theories. Wiley, New York.
[52] Sperlich, S.; Tjostheim, D.; Yang, L., Nonparametric estimation and testing of interaction in additive models, Econometric theory, 18, 197-251, (2002) · Zbl 1109.62310
[53] Stinchcombe, M.B.; White, H., Consistent specification testing with nuisance parameters present only under the alternative, Econometric theory, 14, 295-324, (1998)
[54] Stock, J.H., Nonparametric policy analysis, Journal of the American statistical association, 84, 567-575, (1989)
[55] Stone, C.J., Additive regression and other nonparametric models, Annals of statistics, 13, 685-705, (1985) · Zbl 0605.62065
[56] Stone, C.J., The dimensionality reduction principle for generalized additive models, Annals of statistics, 14, 592-606, (1986) · Zbl 0603.62050
[57] Stute, W., Nonparametric model checks for regression, Annals of statistics, 25, 613-641, (1997) · Zbl 0926.62035
[58] Stute, W.; Gonzalez, W.G.; Presedo, M., Bootstrap approximation in model checks for regression, Journal of American statistical association, 93, 141-149, (1998) · Zbl 0902.62027
[59] Tjostheim, D.; Auestad, B.H., Nonparametric identification of nonlinear time seriesprojections, Journal of American statistical association, 89, 1398-1409, (1994) · Zbl 0813.62036
[60] Van der Vaart, A.W.; Wellner, J.A., Weak convergence and empirical processes: with applications to statistics, (1996), Springer New York · Zbl 0862.60002
[61] Wooldridge, J., A test for functional form against nonparametric alternatives, Econometric theory, 8, 452-475, (1992)
[62] Yatchew, A.J., Nonparametric regression tests based on least squares, Econometric theory, 8, 435-451, (1992)
[63] 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.