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April 2006 Fitting an error distribution in some heteroscedastic time series models
Hira L. Koul, Shiqing Ling
Ann. Statist. 34(2): 994-1012 (April 2006). DOI: 10.1214/009053606000000191


This paper addresses the problem of fitting a known distribution to the innovation distribution in a class of stationary and ergodic time series models. The asymptotic null distribution of the usual Kolmogorov–Smirnov test based on the residuals generally depends on the underlying model parameters and the error distribution. To overcome the dependence on the underlying model parameters, we propose that tests be based on a vector of certain weighted residual empirical processes. Under the null hypothesis and under minimal moment conditions, this vector of processes is shown to converge weakly to a vector of independent copies of a Gaussian process whose covariance function depends only on the fitted distribution and not on the model. Under certain local alternatives, the proposed test is shown to have nontrivial asymptotic power. The Monte Carlo critical values of this test are tabulated when fitting standard normal and double exponential distributions. The results obtained are shown to be applicable to GARCH and ARMA–GARCH models, the often used models in econometrics and finance. A simulation study shows that the test has satisfactory size and power for finite samples at these models. The paper also contains an asymptotic uniform expansion result for a general weighted residual empirical process useful in heteroscedastic models under minimal moment conditions, a result of independent interest.


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Hira L. Koul. Shiqing Ling. "Fitting an error distribution in some heteroscedastic time series models." Ann. Statist. 34 (2) 994 - 1012, April 2006.


Published: April 2006
First available in Project Euclid: 27 June 2006

zbMATH: 1095.62110
MathSciNet: MR2283401
Digital Object Identifier: 10.1214/009053606000000191

Primary: 62F05 , 62M10
Secondary: 60G10

Keywords: Goodness-of-fit test , Nonlinear time series models , weighted empirical process

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.34 • No. 2 • April 2006
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