A robust goodness-of-fit test for generalized autoregressive conditional heteroscedastic models

Yao ZHENG, Wai Keung LI, Guodong LI

Research output: Contribution to journalArticle

1 Citation (Scopus)


The estimation of time series models with heavy-tailed innovations has been widely discussed, but corresponding goodness-of-fit tests have attracted less attention, primarily because the autocorrelation function commonly used in constructing goodness-of-fit tests necessarily imposes certain moment conditions on the innovations. As a bounded random variable has finite moments of all orders, we address the problem by first transforming the residuals with a bounded function. More specifically, we consider the sample autocorrelation function of the transformed absolute residuals of a fitted generalized autoregressive conditional heteroscedastic model. With the corresponding residual empirical distribution function naturally employed as the transformation, a robust goodness-of-fit test is then constructed. The asymptotic distributions of the test statistic under the null hypothesis and local alternatives are derived, and Monte Carlo experiments are conducted to examine finite-sample properties. The proposed test is shown to be more powerful than existing tests when the innovations are heavy-tailed. Copyright © 2017 Biometrika Trust.
Original languageEnglish
Pages (from-to)73-89
Issue number1
Early online dateNov 2017
Publication statusPublished - Mar 2018


Heteroscedastic Model
Robust Tests
Conditional Model
Goodness of Fit Test
Autocorrelation Function
Local Alternatives
Empirical Distribution Function
Moment Conditions
Monte Carlo Experiment
Time Series Models
Random variables
Null hypothesis
Asymptotic distribution
Test Statistic
Distribution functions
Time series


Zheng, Y., Li, W. K., & Li, G. (2018). A robust goodness-of-fit test for generalized autoregressive conditional heteroscedastic models. Biometrika, 105(1), 73-89. doi: 10.1093/biomet/asx063


  • Conditional heteroscedastic model
  • Goodness-of-fit test
  • Heavy tail
  • Residual empirical process
  • Robustness