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Testing for vector autoregressive dynamics under heteroskedasticity


  • Hafner, Christian M.
  • Herwartz, Helmut


In this paper we introduce a bootstrap procedure to test parameter restrictions in vector autoregressive models which is robust in cases of conditionally heteroskedastic error terms. The adopted wild bootstrap method does not require any parametric specification of the volatility process and takes contemporaneous error correlation implicitly into account. Via a Monte Carlo investigation empirical size and power properties of the new method are illustrated. We compare the bootstrap approach with standard procedures either ignoring heteroskedasticity or adopting a spirit of the White correction. In terms of empirical size the proposed method clearly outperforms competing approaches without paying any price in terms of size adjusted power. We apply the alternative tests to investigate the potential of causal relationships linking daily prices of natural gas and crude oil. Unlike standard inference ignoring time varying error variances, heteroskedasticity consistent test procedures do not deliver any evidence in favor of short run causality between the two series.

Suggested Citation

  • Hafner, Christian M. & Herwartz, Helmut, 2002. "Testing for vector autoregressive dynamics under heteroskedasticity," SFB 373 Discussion Papers 2003,4, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  • Handle: RePEc:zbw:sfb373:20034

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    References listed on IDEAS

    1. Engle, Robert F. & Kroner, Kenneth F., 1995. "Multivariate Simultaneous Generalized ARCH," Econometric Theory, Cambridge University Press, vol. 11(01), pages 122-150, February.
    2. Jeantheau, Thierry, 1998. "Strong Consistency Of Estimators For Multivariate Arch Models," Econometric Theory, Cambridge University Press, vol. 14(01), pages 70-86, February.
    3. Christian M. Hafner & Helmut Herwartz, 2000. "Testing for linear autoregressive dynamics under heteroskedasticity," Econometrics Journal, Royal Economic Society, vol. 3(2), pages 177-197.
    4. Andrew Harvey & Esther Ruiz & Neil Shephard, 1994. "Multivariate Stochastic Variance Models," Review of Economic Studies, Oxford University Press, vol. 61(2), pages 247-264.
    5. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    6. Bollerslev, Tim & Engle, Robert F & Wooldridge, Jeffrey M, 1988. "A Capital Asset Pricing Model with Time-Varying Covariances," Journal of Political Economy, University of Chicago Press, vol. 96(1), pages 116-131, February.
    7. Danielsson, Jon, 1998. "Multivariate stochastic volatility models: Estimation and a comparison with VGARCH models," Journal of Empirical Finance, Elsevier, vol. 5(2), pages 155-173, June.
    8. Fengler, Matthias R. & Herwartz, Helmut, 2001. "Multivariate volatility models," SFB 373 Discussion Papers 2001,74, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    9. Braun, Phillip A & Nelson, Daniel B & Sunier, Alain M, 1995. " Good News, Bad News, Volatility, and Betas," Journal of Finance, American Finance Association, vol. 50(5), pages 1575-1603, December.
    10. White, Halbert, 1980. "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, Econometric Society, vol. 48(4), pages 817-838, May.
    11. Bollerslev, Tim, 1990. "Modelling the Coherence in Short-run Nominal Exchange Rates: A Multivariate Generalized ARCH Model," The Review of Economics and Statistics, MIT Press, vol. 72(3), pages 498-505, August.
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    Cited by:

    1. Christian M. Hafner & Helmut Herwartz, 2009. "Testing for linear vector autoregressive dynamics under multivariate generalized autoregressive heteroskedasticity," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 63(3), pages 294-323.

    More about this item


    heteroskedasticity; bootstrap; vector autoregression; hypothesis testing; causality; energy markets;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models


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