Abstract
We prove asymptotic normality of a suitably standardized integrated square difference between a kernel type error density estimator based on residuals and the expected value of the error density estimator based on innovations in GARCH models. This result is similar to that of Bickel–Rosenblatt under i.i.d. set up. Consequently the goodness-of-fit test for the innovation density of GARCH processes based on this statistic is asymptotically distribution free, unlike the tests based on the residual empirical process. A simulation study comparing the finite sample behavior of this test with Kolmogorov–Smirnov test and the test based on integrated square difference between the kernel density estimate and null density shows some superiority of the proposed test.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bachmann D, Dette H (2005) A note on the Bickel-Rosenblatt test in autoregressive time series. Stat Prob Lett 74: 221–234
Berkes I, Horváth L, Kokoszka P (2003) GARCH process: structure and estimation. Bernoulli 9: 201–227
Bickel P, Rosenblatt M (1973) On some global measures of the deviations of density function estimates. Ann Stat 1: 1071–1095
Bougerol P, Picard N (1992a) Strict stationarity of generalized autoregressive process. Ann Prob 20: 1714–1730
Bougerol P, Picard N (1992b) Stationarity of GARCH processes and of some nonnegative time series. J Econometr 52: 115–117
Cheng F (2008) Asymptotic properties in ARCH(p)-time series. J Nonparametr Stat 20: 47–60
Horváth L, Zitikis R (2006) Testing goodness of fit based on densities of GARCH innovations. Econometr Theory 22: 457–482
Koul HL (2002) Weighted empirical processes in dynamic nonlinear models. Second edition of Weighted empiricals and linear models [Inst. Math. Statist., Hayward, CA, 1992]. Lecture Notes in Statistics, 166. Springer, New York
Lee S, Na S (2002) On the BickelRosenblatt test for first-order autoregressive models. Stat Prob Lett 56: 23–35
Lee SW, Hansen BE (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometr Theory 10: 29–52
Lumsdaine RL (1996) Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica 6: 575–596
Mimoto N (2008) Convergence in distribution for the sup-norm of a kernel density estimator for GARCH innovations. Stat Prob Lett 78: 915–923
Nelson DB (1990) Stationarity and persistence in the GARCH(1,1) model. Econometr Theory 6: 318–334
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported in part by the NSF Grant DMS-07-04130.
Rights and permissions
About this article
Cite this article
Koul, H.L., Mimoto, N. A goodness-of-fit test for GARCH innovation density. Metrika 75, 127–149 (2012). https://doi.org/10.1007/s00184-010-0318-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-010-0318-4