×

zbMATH — the first resource for mathematics

Goodness-of-fit tests for kernel regression with an application to option implied volatilities. (English) Zbl 1004.62042
Summary: This paper proposes a test of a restricted specification of regression, based on comparing residual sum of squares from kernel regression. Our main case is where both the restricted specification and the general model are nonparametric, with our test equivalently viewed as a test of dimension reduction. We discuss practical features of implementing the test, and variations applicable to testing parametric models as the null hypothesis, or semiparametric models that depend on a finite parameter vector as well as unknown functions. We apply our testing procedure to option prices; we reject a parametric version of the Black-Scholes formula but fail to reject a semiparametric version against a general nonparametric regression.

MSC:
62G10 Nonparametric hypothesis testing
62F03 Parametric hypothesis testing
62P05 Applications of statistics to actuarial sciences and financial mathematics
62G08 Nonparametric regression and quantile regression
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aı̈t-Sahalia, Y., Nonparametric pricing of interest rate derivative securities, Econometrica, 64, 527-560, (1996) · Zbl 0844.62094
[2] Aı̈t-Sahalia, Y., 1998. The delta method for nonparametric kernel functionals. Working paper, Graduate School of Business, University of Chicago.
[3] Aı̈t-Sahalia, Y.; Lo, A.W., Nonparametric estimation of state price densities implicit in financial asset prices, Journal of finance, 53, 499-547, (1998)
[4] Aı̈t-Sahalia, Y.; Lo, A.W., Nonparametric risk management and implied risk aversion, Journal of econometrics, 94, 9-51, (2000) · Zbl 0952.62091
[5] Bickel, P.J.; Rosenblatt, M., On some global measures of the deviations of density function estimates, Annals of statistics, 1, 1071-1096, (1973) · Zbl 0275.62033
[6] Bickel, P.J.; Götze, F.; van Zwet, W.R., Resampling fewer than n observations: gains, losses and remedies for losses, Statistica sinica, 1, 1-31, (1997) · Zbl 0927.62043
[7] Bickel, P.J., Ritov, Y., Stoker, T.M., 1998. Testing and the method of sieves, Working paper, Department of Statisties, University of California at Berkeley.
[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] Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of political economy, 81, 637-654, (1973) · Zbl 1092.91524
[11] Blundell, R.; Duncan, A., Kernel regression in empirical microeconomics, Journal of human resources, 33, 62-87, (1998)
[12] Chen, X.; Fan, Y., Consistent hypothesis testing in semiparametric and nonparametric models of economic time series, Journal of econometrics, 91, 373-401, (1999) · Zbl 1041.62506
[13] Christoffersen, P., Hahn, J., 1998. Nonparametric testing of ARCH for option pricing. Working paper, University of Pennsylvania.
[14] Doksum, K.; Samarov, A., Global functionals and a measure of the explanatory power of covariates in nonparametric regression, Annals of statistics, 23, 1443-1473, (1995) · Zbl 0843.62045
[15] Ellison, G.; Fisher-Ellison, S., A simple framework for nonparametric specification testing, Journal of econometrics, 96, 1-23, (2000) · Zbl 0968.62046
[16] Eubank, R.; Spiegelman, C., Testing the goodness-of-fit of linear models via nonparametric regression techniques, Journal of the American statistical association, 85, 387-392, (1990) · Zbl 0702.62037
[17] Fan, J., Testing the goodness-of-fit of a parametric density function by kernel method, Econometric theory, 10, 316-356, (1994)
[18] Fan, J.; Li, Q., Consistent model specification tests: omitted variables and semiparametric functional forms, Econometrica, 64, 865-890, (1996) · Zbl 0854.62038
[19] Fernandes, M., 1999. Nonparametric tests for Markovian dynamics. Working paper, European University Institute in Florence.
[20] Fernholz, L., Von Mises calculus for statistical functionals, Lecture notes in statistics, Vol. 19, (1983), Springer New York
[21] Filippova, A.A., Mises’ theorem on the asymptotic behavior of functionals of empirical distribution functions and its statistical applications, Theoretical probability and applications, 7, 24-57, (1962) · Zbl 0118.14501
[22] Fraga, M., 1999. Parametric and semiparametric estimation of sample selection models: an application to the female labor force. Working paper, Université Libre de Bruxelles.
[23] Gozalo, P.L., A consistent model specification test for nonparametric estimation of regression functions models, Econometric theory, 9, 451-477, (1993)
[24] Gyorfi, L.; Härdle, W.; Sarda, P.; Vieu, P., Nonparametric curve estimation from time series, Lecture notes in statistics, Vol. 60, (1989), Springer New York
[25] Hall, P., Central limit theorem for integrated squared error of multivariate nonparametric density estimators, Journal of multivariate analysis, 14, 1-16, (1984) · Zbl 0528.62028
[26] Härdle, W., Applied nonparametric regression, (1990), Cambridge University Press Cambridge, UK · Zbl 0714.62030
[27] Härdle, W.; Mammen, E., Comparing nonparametric vs. parametric regression fits, Annals of statistics, 21, 1926-1947, (1993) · Zbl 0795.62036
[28] Heckman, J.; Ichimura, I.; Smith, J.; Todd, P., Characterization of selection bias using experimental data, Econometrica, 66, 1017-1098, (1998) · Zbl 1055.62573
[29] Hidalgo, J., 1992. A general non-parametric misspecification test. Working paper, Department of Economics, London School of Economics.
[30] Hong, Y.; White, H., Consistent specification testing via nonparametric series regression, Econometrica, 63, 1133-1160, (1995) · Zbl 0941.62125
[31] Horowitz, J.L.; Härdle, W., Testing a parametric model against a semiparametric alternative, Econometric theory, 10, 821-848, (1994)
[32] de Jong, R.M.; Bierens, H.J., On the limit behavior of a chi-squared type test if the number of conditional moments tested approaches infinity, Econometric theory, 10, 70-90, (1994)
[33] Khashimov, Sh.A., Limiting behavior of generalized U-statistics of weakly dependent stationary processes, Theory of probability and its applications, 37, 148-150, (1992) · Zbl 0794.60028
[34] Lavergne, P.; Vuong, Q.H., Nonparametric selection of regressors: the nonnested case, Econometrica, 64, 207-219, (1996) · Zbl 0860.62039
[35] Lavergne, P.; Vuong, Q.H., Nonparametric significance testing, Econometric theory, 16, 576-601, (2000) · Zbl 0968.62047
[36] Lee, B.J., 1988. Nonparametric tests using a kernel estimation method. Ph.D. Dissertation. Department of Economics, University of Wisconsin at Madison.
[37] Lewbel, A., 1991. Applied consistent tests of nonparametric regression and density restrictions. Working paper, Department of Economics, Brandeis University.
[38] Li, Q., Consistent model specification tests for time series econometric models, Journal of econometrics, 92, 101-147, (1999) · Zbl 0929.62054
[39] von Mises, R., On the asymptotic distribution of differentiable statistical functions, Annals of mathematical statistics, 18, 309-348, (1947) · Zbl 0037.08401
[40] Reeds, J.A., On the definition of von Mises functionals. ph.D. dissertation., (1976), Harvard University Cambridge
[41] Robinson, P.M., Root n semiparametric regression, Econometrica, 56, 931-954, (1988) · Zbl 0647.62100
[42] 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
[43] Rodriguez, D., Stoker, T.M., 1992. A regression test of semiparametric index model specification. Working paper, Sloan School of Management, MIT.
[44] Sakov, A., 1998. Using the m out of n bootstrap in hypothesis testing. Ph.D. Dissertation, Department of Statistics, University of California at Berkeley.
[45] Staniswalis, J.G.; Severini, T.A., Diagnostics for assessing regression models, Journal of the American statistical association, 86, 684-691, (1991) · Zbl 0736.62063
[46] Stoker, T.M., 1992. Lectures on Semiparametric Econometrics. CORE Foundation, Louvain-la-Neuve.
[47] Stone, C.J., Optimal uniform rate of convergence for nonparametric estimators of a density function or its derivatives., Recent advances in statistics, Vol. 393-406, (1983), Academic Press New York
[48] Vieu, P., Choice of regressors in nonparametric estimation, Computational statistics and data analysis, 17, 575-594, (1994) · Zbl 0937.62583
[49] Whang, Y.-J.; Andrews, D.W.K., Tests of specification for parametric and semiparametric models, Journal of econometrics, 57, 277-318, (1991)
[50] White, H., Hong, Y., 1993. M-testing using finite and infinite dimensional parameter estimators. Working paper, University of California at San Diego.
[51] Wooldridge, J., A test for functional form against nonparametric alternatives, Econometric theory, 8, 452-475, (1992)
[52] Yatchew, A.J., Nonparametric regression tests based on an infinite dimensional least squares procedure, Econometric theory, 8, 435-451, (1992)
[53] Zhang, P., Variable selection in nonparametric regression with continuous covariates, Annals of statistics, 19, 1869-1882, (1991) · Zbl 0738.62051
[54] Zheng, J.X., A consistent test of functional form via nonparametric estimation function, Journal of econometrics, 75, 263-290, (1996)
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.