Hausman tests for the error distribution in conditionally heteroskedastic models

Zhu, Ke (2015): Hausman tests for the error distribution in conditionally heteroskedastic models.

Abstract

This paper proposes some novel Hausman tests to examine the error distribution in conditionally heteroskedastic models. Unlike the existing tests, all Hausman tests are easy-to-implement with the limiting null distribution of $\chi^{2}$, and moreover, they are consistent and able to detect the local alternative of order n−1=2. The scope of the Hausman test covers all Generalized error distributions and Student’s t distributions. The performance of each Hausman test is assessed by simulated and real data sets.

• Item Type: MPRA Paper
• Language: English
• Keywords: Conditionally heteroskedastic model; Consistent test; GARCH model; Goodness-of-fit test; Hausman test; Nonlinear time series.
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General

References

BERKES, I. & HORVATH, L. (2001). Strong approximation of the empirical process of GARCH sequences. Annals of Applied Probability 11, 789-809.

BERKES, I. & HORVATH, L. (2004). The efficiency of the estimators of the parameters in GARCH processes. Annals of Statistics 32, 633-655.

BOLDIN, M.V. (1982). Estimation of the distribution of noise in an autoregressive scheme. Theory of Probability and Its Applications 27, 866-871.

BOLDIN, M.V. (1989). On hypothesis testing in a moving average scheme by the Kolmogorov-Smirnov and the $\omega^{2}$ tests. Theory of Probability and Its Applications 34, 699-704.

BOLLERSLEV, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.

BOLLERSLEV, T., CHOU, R.Y. & KRONER, K.F. (1992). ARCH modelling in finance: A review of the theory and empirical evidence. Journal of Econometrics 52, 5–59.

CHEN, M. & ZHU, K. (2015). Sign-based portmanteau test for ARCH-type models with heavy-tailed innovations. Forthcoming in Journal of Econometrics.

CHRISTOFFERSEN, P.F. & DIEBOLD, F.X. (1997). Optimal prediction under asymmetric loss. Econometric Theory 13, 808-817,

D’AGOSTINO, R.B. & STEPHENS, M.A. (1986). Goodness-of-Fit Techniques. Dekker, New York.

DING, Z., GRANGER, C.W.J. & ENGLE, R.F. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83-106.

ENGLE, R.F. (1982). Autoregressive conditional heteroskedasticity with estimates of variance of U.K. inflation. Econometrica 50, 987-1008.

ENGLE, R.F. (2004). Risk and volatility: Econometric models and financial practice. American Economic Review 94, 405-420.

FAN, J., QI, L. & XIU, D. (2014). Quasi maximum likelihood estimation of GARCH models with heavy-tailed likelihoods (with discussion). Journal of Business and Economic Statistics 32, 178-205.

FAN, J. & YAO, Q. (2003). Nonlinear time series: Nonparametric and parametric methods. Springer, New York.

FRANCQ, C., WINTENBERGER, O. & ZAKOIAN, J.M. (2013). Garch models without positivity constraints: exponential or log garch? Journal of Econometrics 177, 34-46.

FRANCQ, C. & ZAKOIAN, J.M. (2010). GARCH Models: Structure, Statistical Inference and Financial Applications. Chichester, UK: John Wiley.

FRANCQ, C. & ZAKOIAN, J.M. (2012). Strict stationarity testing and estimation of explosive and stationary GARCH models. Econometrica 80, 821-861.

FRANCQ, C. & ZAKOIAN, J.M. (2013). Optimal predictions of powers of conditionally heteroscedastic processes. Journal of the Royal Statistical Society: Series B 75, 345-367.

GEWEKE, J. (1986). Modeling the persistence of conditional variances: A comment. Econometric Review 5, 57-61.

HAFNER, C.M. & PREMINGER, A. (2015). An ARCH model without intercept. Economics Letters 129, 13-17.

HAMADEH, T. & ZAKOIAN, J.M. (2011). Asymptotic properties of LS and QML estimators for a class of nonlinear GARCH processes. Journal of Statistical Planning and Inference 141, 488-507.

HAUSMAN, J.A. (1978). Specification tests in econometrics. Econometrica 46, 1251-1271.

HORVATH, L., KOKOSZKA, P. & TEYSSIERE, G. (2001). Empirical process of the squared residuals of an ARCH sequence. Annals of Statistics 29, 445-469.

HORVATH, L., KOKOSZKA, P. & TEYSSIERE, G. (2004). Bootstrap misspecification tests for ARCH based on the empirical process of squared residuals. Journal of Statistical Computation and Simulation 74, 469-485.

HORVATH, L. & ZITIKIS, R. (2006). Testing goodness of fit based on densities of GARCH innovations. Econometric theory 22, 457-482.

KLAR, B., LINDNER, F., & MEINTANIS, S.G. (2012). Specification tests for the error distribution in GARCH models. Computational Statistics and Data Analysis 56, 3587-3598.

KOUL, H.L. (1996). Asymptotics of some estimators and sequential residual empiricals in nonlinear time series. Annals of Statistics 24 380-404.

KOUL, H.L. (2002). Weighted Empirical Processes in Dynamic Nonlinear Models, second ed., in: Lecture Notes in Statistics, vol. 166, Springer-Verlag, New York.

KOUL, H.L. & LING, S. (2006). Fitting an error distribution in some heteroscedastic time series models. Annals of Statistics 34, 994-1012.

LI, W.K. & MAK, T.K. (1994). On the squared residual autocorrelations in non-linear time series with conditional heteroscedasticity. Journal of Time Series Analysis 15, 627-636.

LING, S. (2007). Self-weighted and local quasi-maximum likelihood estimators for ARMA-GARCH/IGARCH models. Journal of Econometrics 140, 849-873.

LINTON, O., PAN, J. & WANG, H. (2010). Estimation for a non-stationary semi-strong GARCH(1, 1) model with heavy-tailed errors. Econometric Theory 26, 1-28.

LJUNG, G.M. & BOX, G.E.P. (1978). On a measure of lack of fit in time series models. Biometrika 65, 297-303.

PENG, L. & YAO, Q. (2003). Least absolute deviations estimation for ARCH and GARCH models. Biometrika 90, 967-975.

ZAKOIAN, J.M. (1994). Threshold heteroskedastic models. Journal of Economic Dynamics and Control 18, 931-955.

ZHU, K. & LI, W.K. (2015). A new Pearson-type QMLE for conditionally heteroskedastic models. Forthcoming in Journal of Business and Economic Statistics.

ZHU, K. & LING, S. (2015). Model-based pricing for financial derivatives. Journal of Econometrics 187, 447-457.