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SIMULTANEOUS SPECIFICATION TESTING OF MEANAND VARIANCE STRUCTURES IN NONLINEAR TIME SERIESREGRESSION

Published online by Cambridge University Press:  03 March 2011

Song Xi Chen  and
Jiti Gao
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Abstract

This paper proposes a nonparametric simultaneous testfor parametric specification of the conditional meanand variance functions in a time series regressionmodel. The test is based on an empirical likelihood(EL) statistic that measures the goodness of fitbetween the parametric estimates and thenonparametric kernel estimates of the mean andvariance functions. A unique feature of the test isits ability to distribute natural weightsautomatically between the mean and the variancecomponents of the goodness-of-fit measure. To reducethe dependence of the test on a single pair ofsmoothing bandwidths, we construct an adaptive testby maximizing a standardized version of theempirical likelihood test statistic over a set ofsmoothing bandwidths. The test procedure is based ona bootstrap calibration to the distribution of theempirical likelihood test statistic. We demonstratethat the empirical likelihood test is able todistinguish local alternatives that are differentfrom the null hypothesis at an optimal rate.

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Type
Research Article
Information
Econometric Theory , Volume 27 , Issue 4 , August 2011 , pp. 792 - 843
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

We thank Peter C.B. Phillips, the editor, YuichiKitamura, the associate editor, and two refereesfor their constructive and insightful comments andsuggestions, which have improved the presentationof the paper. We also thank Ming Li and IsabelCasas Villalba for their valuable computationalassistance. Chen acknowledges the financialsupport from National Science Foundation grantsSES-0518904 and DMS-0604533, and Gao acknowledgesthe financial support by Australian ResearchCouncil Discovery Grants under grant numbersDP0558602 and DP0879088.

References

REFERENCES

Ahn, D.H. & Gao, B. (1999) A parametric nonlinear model of term structure dynamics. Review of Financial Studies 12, 721–762.10.1093/rfs/12.4.721CrossRefGoogle Scholar
Aït-Sahalia, Y. (1996) Testing continuous–time models of the spot interest rate. Review of Financial Studies 9, 385–426.Google Scholar
Aït-Sahalia, Y. (1999) Transition densities for interest rate and other nonlinear diffusions. Journal of Finance 54, 1361–1395.Google Scholar
Aït-Sahalia, Y., Fan, J., & Peng, H. (2009) Nonparametric transition–based tests for jump diffusions. Journal of the American Statistical Association 487, 1102–1116.CrossRefGoogle Scholar
Benjamini, Y. & Hochberg, Y. (1995) Controlling the false discovery rate: A practical and powerful approach to multiple testing. Journal of Royal Statistical Society Series B 57, 289–300.Google Scholar
Bosq, D. (1998) Nonparametric Statistics for Stochastic Processes (Lecture Note in Statistics, 110). Springer.CrossRefGoogle Scholar
Brown, B. & Chen, S.X. (1998) Combined and least squares empirical likelihood. Annals of the Institute of Statistical Mathematics 50, 697–714.CrossRefGoogle Scholar
Chan, K.C., Karolyi, D.A., Longstaff, F.A., & Sanders, A.B. (1992) An empirical comparison of alternative models of the short–term interest rate. Journal of Finance 47, 1209–1227.10.1111/j.1540-6261.1992.tb04011.xGoogle Scholar
Chen, S.X. & Cui, H. (2006) On Bartlett correction of empirical likelihood in the presence of nuisance parameters. Biometrika 93, 215–220.10.1093/biomet/93.1.215CrossRefGoogle Scholar
Chen, S.X. & Cui, H. (2007) On the second order properties of empirical likelihood with moment restrictions. Journal of Econometrics 141, 492–516.Google Scholar
Chen, S.X. & Gao, J. (2007) An adaptive empirical likelihood test for parametric time series models. Journal of Econometrics 141, 950–972.10.1016/j.jeconom.2006.12.002CrossRefGoogle Scholar
Chen, S.X., Gao, J., & Tang, C. (2008) A test for model specification of diffusion processes. Annals of Statistics 36, 167–198.CrossRefGoogle Scholar
Chen, S.X., Härdle, W., & Li, M. (2003) An empirical likelihood goodness-of-fit test for time series. Journal of the Royal Statistical Society, Series B 65, 663–678.CrossRefGoogle Scholar
Chen, X. & Fan, Y. (1999) Consistent hypothesis testing in semiparametric and nonparametric models for econometric time series. Journal of Econometrics 91, 373–401.Google Scholar
Cox, J.C., Ingersoll, J.E., & Ross, S.A. (1985) A theory of term structure of interest rates. Econometrica 53, 385–407.CrossRefGoogle Scholar
Dette, H. & Hetzler, B. (2007) Specification tests indexed by bandwidths. Sankhy Series A 69, 28–54.Google Scholar
Donald, S., Imbens, G., & Newey, W.K. (2003) Empirical likelihood estimation and tests with conditional moment restrictions. Journal of Econometrics 117, 55–93.CrossRefGoogle Scholar
Einmahl, J.H.J. & McKeague, I.W. (2003) Empirical likelihood based hypothesis testing. Bernoulli 9, 267–290.CrossRefGoogle Scholar
Escanciano, J.C. (2008) Joint diagnostic tests for conditional mean and variance specifications. Journal of Econometrics 143, 74–87.Google Scholar
Escanciano, J.C. & Velasco, C. (2008) Specification tests of parametric dynamic conditional quantiles. CAEPR Working paper #2008–021. Indiana University Bloomington.10.2139/ssrn.1228528Google Scholar
Eubank, R.L. & Spiegelman, C.H. (1990) Testing the goodness of fit of a linear model via nonparametric regression techniques. Journal of the American Statistical Association 85, 387–392.Google Scholar
Fan, J. & Gijbels, I. (1996) Local Polynomial Modeling and Its Applications. Chapman & Hall.Google Scholar
Fan, J. & Yao, Q. (1998) Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85, 645–660.10.1093/biomet/85.3.645CrossRefGoogle Scholar
Fan, J. & Yao, Q. (2003) Nonlinear Time Series: Nonparametric and Parametric Methods. Springer Series in Statistics. Springer.Google Scholar
Fan, J. & Zhang, C. (2003) A re-examination of diffusion estimators with applications to financial model validation. Journal of the American Statistical Association 98, 118–134.10.1198/016214503388619157CrossRefGoogle Scholar
Fan, J., Zhang, C., & Zhang, J. (2001) Generalized likelihood ratio statistics and Wilks phenomenon. Annals of Statistics 29, 153–193.CrossRefGoogle Scholar
Fan, J. & Zhang, J. (2004) Sieve empirical likelihood ratio tests for nonparametric functions. Annals of Statistics 32, 1858–1907.Google Scholar
Fan, Y. & Li, Q. (1996) Consistent model specification tests: Omitted variables and semiparametric functional forms. Econometrica 64, 865–890.Google Scholar
Francke, J., Kreiss, J.P., & Mammen, E. (2002) Bootstrap of kernel smoothing in nonlinear time series. Bernoulli 8, 1–38.Google Scholar
Gao, J. (2007) Nonlinear Time Series: Semiparametric and Nonparametric Methods. Chapman & Hall/CRC.10.1201/9781420011210Google Scholar
Gao, J. & Gijbels, I. (2008) Bandwidth selection in nonparametric kernel testing. Journal of the American Statistical Association 484, 1584–1594.Google Scholar
Gao, J. & King, M.L. (2004) Adaptive testing in continuous-time diffusion models. Econometric Theory 20, 844–882.10.1017/S0266466604205023CrossRefGoogle Scholar
Gao, J., Tong, H. & Wolff, R.C.L. (2002) Model specification tests in nonparametric stochastic regression models. Journal of Multivariate Analysis 83, 324–359.CrossRefGoogle Scholar
Genon-Caralot, V., Jeantheau, T., & Laredo, C. (2000) Stochastic volatility models as hidden Markov models and statistical applications. Bernoulli 6, 1051–1079.CrossRefGoogle Scholar
Härdle, W. & Mammen, E. (1993) Comparing nonparametric versus parametric regression fits. Annals of Statistics 21, 1926–1947.CrossRefGoogle Scholar
Hart, J. (1997) Nonparametric Smoothing and Lack-of-Fit Tests. Springer.Google Scholar
Hjellvik, V. & Tjøstheim, D. (1995) Nonparametric tests of linearity for time series. Biometrika 82, 351–368.Google Scholar
Hjellvik, V., Yao, Q., & Tjøstheim, D. (1998) Linearity testing using local polynomial approximation. Journal of Statistical Planning and Inference 68, 295–321.Google Scholar
Hong, Y. & Li, H. (2005) Nonparametric specification testing for continuous-time models with application to spot interest rates. Review of Financial Studies 18, 37–84.10.1093/rfs/hhh006Google Scholar
Horowitz, J.L. & Spokoiny, V.G. (2001) An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69, 599–632.Google Scholar
Ingster, Y.I. (1993a) Asymptotically minimax hypothesis testing for nonparametric alternatives. I. Mathematical Methods of Statistics 2, 85–114.Google Scholar
Ingster, Y.I. (1993b) Asymptotically minimax hypothesis testing for nonparametric alternatives. II. Mathematical Methods of Statistics 2, 171–189.Google Scholar
Ingster, Y.I. (1993c) Asymptotically minimax hypothesis testing for nonparametric alternatives. III. Mathematical Methods of Statistics 2, 249–268.Google Scholar
Kitamura, Y. (1997) Empirical likelihood methods with weakly dependent processes. Annals of Statistics 25, 2084–2102.10.1214/aos/1069362388CrossRefGoogle Scholar
Kitamura, Y. (2001) Asymptotic optimality of empirical likelihood for testing moment restrictions. Econometrica 69, 1661–1672.10.1111/1468-0262.00261CrossRefGoogle Scholar
Kitamura, Y., Tripathi, G., & Ahn, H., (2004) Empirical likelihood–based inference in conditional moment restriction models. Econometrica 72, 1667–1714.Google Scholar
Kreiss, J.P., Neumann, M., & Yao, Q. (1999) Bootstrap tests for simple structures in nonparametric time series regression. Private communication.Google Scholar
Li, G. (2003) Nonparametric likelihood ratio goodness-of-fit tests for survival data. Journal of Multivariate Analysis 86, 166–182.CrossRefGoogle Scholar
Li, G., Hollander, M., McKeague, I. W., & Yang, J. (1996) Nonparametric likelihood ratio confidence bands for quantile functions from incomplete survival data. Annals of Statistics 24, 628–640.10.1214/aos/1032894455Google Scholar
Li, Q. (1999) Consistent model specification tests for time series econometric models. Journal of Econometrics 92, 101–147.Google Scholar
Li, Q. & Racine, J. (2007) Nonparametric Econometrics: Theory and Practice. Princeton University Press.Google Scholar
Li, Q. & Wang, S. (1998) A simple consistent bootstrap test for a parametric regression functional form. Journal of Econometrics 87, 145–165.10.1016/S0304-4076(98)00011-6Google Scholar
Masry, E. & Tjøstheim, D. (1995) Nonparametric estimation and identification of nonlinear ARCH time series. Econometric Theory 11, 258–289.CrossRefGoogle Scholar
McKeague, I.W. & Zhang, M.J. (1994) Identification of nonlinear time series from first order cumulative characteristics. Annals of Statistics 22, 495–514.Google Scholar
Owen, A. (1988) Empirical likelihood confidence intervals for a single functional. Biometrika 75, 237–249.10.1093/biomet/75.2.237Google Scholar
Owen, A. (1990) Empirical likelihood ratio confidence regions. Annals of Statistics 18, 90–120.CrossRefGoogle Scholar
Owen, A. (2001) Empirical Likelihood. Chapman & Hall.Google Scholar
Qin, J. & Lawless, J. (1994) Empirical likelihood and general estimating functions. Annals of Statistics 22, 300–325.CrossRefGoogle Scholar
Simes, R.J. (1986) An improved Bonferroni procedure for multiple tests of significance. Biometrika 73, 751–754.10.1093/biomet/73.3.751Google Scholar
Spokoiny, V.G. (1996) Adaptive hypothesis testing using wavelets. Annals of Statistics 24, 2477–2498.10.1214/aos/1032181163CrossRefGoogle Scholar
Tripathi, G. & Kitamura, Y. (2003) Testing conditional moment restrictions. Annals of Statistics 31, 2059–2095.Google Scholar
Vasicek, O. (1977) An equilibrium characterization of the term structure. Journal of Financial Economics 5, 177–188.Google Scholar
Xu, K.L. & Phillips, P.C.B. (2006) Empirical likelihood re–weighted functional estimation of diffusion models. Mimeo, Yale University.CrossRefGoogle Scholar
Zhang, C. & Dette, H. (2004) A power comparison between nonparametric regression tests. Statistics and Probability Letters 66, 289–301.Google Scholar