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On Diagnostic Checking of Vector ARMA-GARCH Models with Gaussian and Student-t Innovations

Author

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  • Yongning Wang

    (Booth School of Business, University of Chicago, 5807 South Woodlawn Avenue, Chicago,IL 60637, USA)

  • Ruey S. Tsay

    (Booth School of Business, University of Chicago, 5807 South Woodlawn Avenue, Chicago,IL 60637, USA)

Abstract

This paper focuses on the diagnostic checking of vector ARMA (VARMA) models with multivariate GARCH errors. For a fitted VARMA-GARCH model with Gaussian or Student-t innovations, we derive the asymptotic distributions of autocorrelation matrices of the cross-product vector of standardized residuals. This is different from the traditional approach that employs only the squared series of standardized residuals. We then study two portmanteau statistics, called Q1(M) and Q2(M), for model checking. A residual-based bootstrap method is provided and demonstrated as an effective way to approximate the diagnostic checking statistics. Simulations are used to compare the performance of the proposed statistics with other methods available in the literature. In addition, we also investigate the effect of GARCH shocks on checking a fitted VARMA model. Empirical sizes and powers of the proposed statistics are investigated and the results suggest a procedure of using jointly Q1(M) and Q2(M) in diagnostic checking. The bivariate time series of FTSE 100 and DAX index returns is used to illustrate the performance of the proposed portmanteau statistics. The results show that it is important to consider the cross-product series of standardized residuals and GARCH effects in model checking.

Suggested Citation

  • Yongning Wang & Ruey S. Tsay, 2013. "On Diagnostic Checking of Vector ARMA-GARCH Models with Gaussian and Student-t Innovations," Econometrics, MDPI, vol. 1(1), pages 1-31, April.
    • Handle: RePEc:gam:jecnmx:v:1:y:2013:i:1:p:1-31:d:24773
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    References listed on IDEAS

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    1. Engle, Robert F. & Kroner, Kenneth F., 1995. "Multivariate Simultaneous Generalized ARCH," Econometric Theory, Cambridge University Press, vol. 11(1), pages 122-150, February.
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    3. Christian M. Hafner, 2003. "Fourth Moment Structure of Multivariate GARCH Models," Journal of Financial Econometrics, Oxford University Press, vol. 1(1), pages 26-54.
    4. Helmut Lütkepohl, 2005. "New Introduction to Multiple Time Series Analysis," Springer Books, Springer, number 978-3-540-27752-1, December.
    5. Bénédicte Vidaillet & V. d'Estaintot & P. Abécassis, 2005. "Introduction," Post-Print hal-00287137, HAL.
    6. Shiqing Ling & W. K. Li, 1997. "Diagnostic checking of nonlinear multivariate time series with multivariate arch errors," Journal of Time Series Analysis, Wiley Blackwell, vol. 18(5), pages 447-464, September.
    7. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    8. Ruey S. Tsay, 1992. "Model Checking Via Parametric Bootstraps in Time Series Analysis," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 41(1), pages 1-15, March.
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    Cited by:

    1. Ke, Rui & Jia, Jing & Tan, Changchun, 2021. "A residual-based test for multivariate GARCH models using transformed quadratic residuals," Economics Letters, Elsevier, vol. 206(C).
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    3. Julia Adamska & Łukasz Bielak & Joanna Janczura & Agnieszka Wyłomańska, 2022. "From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t -Distribution Case," Mathematics, MDPI, vol. 10(18), pages 1-29, September.

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