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ASYMPTOTIC DISTRIBUTION-FREE DIAGNOSTIC TESTS FOR HETEROSKEDASTIC TIME SERIES MODELS

Published online by Cambridge University Press:  26 October 2009

J. Carlos Escanciano
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Abstract

This article investigates model checks for a class of possibly nonlinear heteroskedastic time series models, including but not restricted to ARMA-GARCH models. We propose omnibus tests based on functionals of certain weighted standardized residual empirical processes. The new tests are asymptotically distribution-free, suitable when the conditioning set is infinite-dimensional, and consistent against a class of Pitman’s local alternatives converging at the parametric rate n−1/2, with n the sample size. A Monte Carlo study shows that the simulated level of the proposed tests is close to the asymptotic level already for moderate sample sizes and that tests have a satisfactory power performance. Finally, we illustrate our methodology with an application to the well-known S&P 500 daily stock index. The paper also contains an asymptotic uniform expansion for weighted residual empirical processes when initial conditions are considered, a result of independent interest.

Type
Research Article
Information
Econometric Theory , Volume 26 , Issue 3 , June 2010 , pp. 744 - 773
Copyright
Copyright © Cambridge University Press 2009

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Footnotes

Research was funded by the Spanish Plan Nacional de I+D+I, reference number SEJ2007-62908, and by the Spanish Ministerio de Educación y Ciencia, reference number SEJ2005-07657/ECON. I would like to thank Miguel A. Delgado, Oliver Linton, Carlos Velasco, and two anonymous referees for helpful comments. I also thank Wenceslao González-Manteiga for pointing out an important reference. All errors are mine.

References

REFERENCES

Bai, J. & Ng, S. (2001) A test for conditional symmetry in time series models. Journal of Econometrics 103, 225–258.CrossRefGoogle Scholar
Bera, A.K. & Higgins, M.L. (1997) Arch and bilinearity as competing models for nonlinear dependence. Journal of Business and Economic Statistics 15, 43–50.Google Scholar
Billingsley, P. (1999) Convergence of Probability Measures, 2nd ed. Wiley.CrossRefGoogle Scholar
Bollerslev, T., Chou, R.Y., & Kroner, K.F. (1992) ARCH modelling in finance. Journal of Econometrics 52, 5–59.CrossRefGoogle Scholar
Box, G. & Pierce, D. (1970) Distribution of residual autocorrelations in autoregressive integrated moving average time series models. Journal of the American Statistical Association 65, 1509–1527.CrossRefGoogle Scholar
Chang, N.M. (1990) Weak convergence of a self-consistent estimator of a survival function with doubly censored data. Annals of Statistics 18, 391–404.CrossRefGoogle Scholar
Chen, M. & An, H.Z. (1998) A note on the stationarity and the existence of moments of the GARCH model. Statistica Sinica 8, 505–510.Google Scholar
Chen, X., Linton, O., & van Keilegom, I. (2003) Estimation of semiparametric models when the criterion function is not smooth. Econometrica 71, 1591–1608.CrossRefGoogle Scholar
Delgado, M.A. & Escanciano, J.C. (2007) Nonparametric tests for conditional symmetry in dynamic models. Journal of Econometrics 141, 652–682.CrossRefGoogle Scholar
Escanciano, J.C. (2006). Goodness-of-fit tests for linear and non-linear time series models. Journal of the American Statistical Association 101, 531–541.CrossRefGoogle Scholar
Escanciano, J.C. (2008) Joint and marginal diagnostic tests for conditional mean and variance specifications. Journal of Econometrics 143, 74–87.CrossRefGoogle Scholar
Escanciano, J.C. (2009) On the lack of power of omnibus specification tests. Econometric Theory 25, 1–33.CrossRefGoogle Scholar
Escanciano, J.C. & Velasco, C. (2006) Generalized spectral tests for the martingale difference hypothesis. Journal of Econometrics 134, 151–185.10.1016/j.jeconom.2005.06.019CrossRefGoogle Scholar
Fan, J. & Yao, Q. (2003) Nonlinear Time Series: Nonparametric and Parametric Methods. Springer-Verlag.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.M. (2004). Maximum likelihood estimation of pure GARCH and ARMA-GARCH. Bernoulli 10, 605–637.CrossRefGoogle Scholar
Gao, J. & King, M. (2004) Adaptive testing in continuous-time diffusion models. Econometric Theory 20, 844–882.CrossRefGoogle Scholar
Hall, P. & Heyde, C. (1980) Martingale Limit Theory and Its Application. Academic Press.Google Scholar
Härdle, W. & Mammen, E. (1993) Comparing nonparametric versus parametric regression fits. Annals of Statistics 21, 1926–1974.CrossRefGoogle Scholar
Hidalgo, J. & Zaffaroni, P. (2007) A goodness of fit test for ARCH(∞). Journal of Econometrics 141, 835–875.10.1016/j.jeconom.2006.11.005CrossRefGoogle Scholar
Hong, Y. & Lee, T.H. (2003) Diagnostic checking for adequacy of nonlinear time series models. Econometric Theory 19, 1065–1121.CrossRefGoogle Scholar
Jennrich, R.I. (1969) Asymptotic properties of nonlinear least squares estimators. Annals of Mathematical Statistics 40, 633–643.10.1214/aoms/1177697731CrossRefGoogle Scholar
Koenker, R. & Zhao, Q. (1996) Conditional quantile estimation and inference for ARCH models. Econometric Theory 12, 793–814.CrossRefGoogle Scholar
Koul, H.L. (2002). Weighted Empirical Processes in Dynamic Nonlinear Models, 2nd ed. Lecture Notes in Statistics, vol. 166. Springer.CrossRefGoogle Scholar
Koul, H.L. & Ling, S. (2006). Fitting an error distribution in some heteroscedastic time series models. Annals of Statistics 34, 994–1012.CrossRefGoogle Scholar
Li, W.K., Ling, S., & McAleer, M. (2002) A survey of recent theoretical results for time series models with GARCH errors. Journal of Economic Survey 16, 245–269.10.1111/1467-6419.00169CrossRefGoogle Scholar
Li, W.K. & Mak, T.K. (1994) On the squared residual autocorrelation in nonlinear time series with conditional heteroskedasticity. Journal of Time Series Analysis 15, 627–636.CrossRefGoogle Scholar
Ljung, G.M. & Box, G.E.P. (1978) A measure of lack of fit in time series models. Biometrika 65, 297–303.CrossRefGoogle Scholar
Loudon, G.F., Watt, W.H., & Yadav, P.K. (2000) An empirical analysis of alternative parametric ARCH models. Journal of Applied Econometrics, 15, 117–136.3.0.CO;2-4>CrossRefGoogle Scholar
Lundbergh, S.T. & Teräsvirta, T. (2002). Evaluating GARCH models. Journal of Econometrics 110, 417–435.CrossRefGoogle Scholar
Ngatchou-Wandji, J., (2005) Checking nonlinear heteroscedastic time series models. Journal of Statistical Planning and Inference 133, 33–68.10.1016/j.jspi.2004.03.013CrossRefGoogle Scholar
Robinson, P.M. & Zaffaroni, P. (2006) Pseudo-maximum likelihood estimation of ARCH(∞) models. Annals of Statistics 34, 1049–1074.CrossRefGoogle Scholar
Shorack, G. & Wellner, J. (1986) Empirical Processes with Applications to Statistics. Wiley.Google Scholar
Straumann, D. (2005) Estimation in Conditionally Heteroscedastic Time Series Models. Lecture Notes in Statistics 181. Berlin-Heidelberg.Google Scholar
Stute, W., Xu, W.L., & Zhu, L.X. (2008) Model diagnosis for parametric regression in high dimensional spaces. Biometrika 95, 451–467.CrossRefGoogle Scholar
Tsay, R.S. (2005). Analysis of Financial Time Series, 2nd ed. Wiley-Intersci.CrossRefGoogle Scholar
Van der Vaart, A.W. & Wellner, J.A. (1996). Weak Convergence and Empirical Processes. Springer.CrossRefGoogle Scholar
Wooldridge (1990) A unified approach to robust, regression-based specification tests. Econometric Theory 6, 17–43.CrossRefGoogle Scholar