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Finite Sample and Optimal Inference in Possibly Nonstationary ARCH Models with Gaussian and Heavy-Tailed Errors

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  • Emma Iglesias
  • Jean Marie Dufour

Abstract

Most of the literature on testing ARCH models focuses on the null hypothesis of no-ARCH effects. In this paper, we consider the general problem of testing any possible set of coefficient values in ARCH models, which may be non-stationary, with Gaussian and non-Gaussian errors, as well as with any number exogenous regressors in the mean equation. Both Engle-type and point-optimal tests are studied. Special problems considered include the hypothesis of no-ARCH effects and IARCH structure. We propose exact inference based on pivotal Monte Carlo tests [as in Dufour and Kiviet (1996, 1998) and Dufour, Khalaf, Bernard and Genest (2004)] and maximised Monte Carlo tests [Dufour (2004))], depending on whether nuisance parameters are present. This will allow the introduction of dynamics in the mean equation as well. We show that the method suggested provides provably valid tests in both finite and large samples, in cases where standard asymptotic and bootstrap methods may fail in the presence of heavy-tailed errors [as shown by Hall and Yao (2003)]. The performance of the proposed procedures with both Gaussian and non-Gaussian errors is analyzed in a simulation experiment. Our results show that the proposed procedures work well from the viewpoints of size and power. The powers gains provided by the point optimal procedures are in many cases spectacular. The tests also exhibit good behaviour outside the stationarity region [following the work of Jensen and Rahbek (2004)]. Finally, the technique is applied to the US inflation

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Bibliographic Info

Paper provided by Econometric Society in its series Econometric Society 2004 North American Summer Meetings with number 161.

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Date of creation: 11 Aug 2004
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Handle: RePEc:ecm:nasm04:161

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Keywords: Point Optimal Test; ARCH; Non-stationarity; Fat-tails;

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References

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  1. Dufour, Jean-Marie & Kiviet, Jan F., 1996. "Exact tests for structural change in first-order dynamic models," Journal of Econometrics, Elsevier, vol. 70(1), pages 39-68, January.
  2. Ullah, Aman, 2004. "Finite Sample Econometrics," OUP Catalogue, Oxford University Press, edition 1, number 9780198774488.
  3. DUFOUR, Jean-Marie & KHALAF, Lynda & BERNARD, Jean-Thomas, 2001. "Simulation-Based Finite-Sample Tests for Heteroskedasticity and ARCH Effects," Cahiers de recherche 2001-08, Universite de Montreal, Departement de sciences economiques.
  4. 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.
  5. Dufour, J.M. & Kiviet, J.F., 1995. "Exact Inference Methods for First-Order Autoregressive Distributed Lag Models," Cahiers de recherche 9547, Universite de Montreal, Departement de sciences economiques.
  6. Bollerslev, Tim & Engle, Robert F. & Nelson, Daniel B., 1986. "Arch models," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 49, pages 2959-3038 Elsevier.
  7. Donald W.K. Andrews & Werner Ploberger, 1992. "Optimal Tests When a Nuisance Parameter Is Present Only Under the Alternative," Cowles Foundation Discussion Papers 1015, Cowles Foundation for Research in Economics, Yale University.
  8. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
  9. DUFOUR, Jean-Marie, 2005. "Monte Carlo Tests with Nuisance Parameters: A General Approach to Finite-Sample Inference and Nonstandard Asymptotics," Cahiers de recherche 2005-03, Universite de Montreal, Departement de sciences economiques.
  10. Whitney K. Newey & Kenneth D. West, 1986. "A Simple, Positive Semi-Definite, Heteroskedasticity and AutocorrelationConsistent Covariance Matrix," NBER Technical Working Papers 0055, National Bureau of Economic Research, Inc.
  11. Lee, John H. H., 1991. "A Lagrange multiplier test for GARCH models," Economics Letters, Elsevier, vol. 37(3), pages 265-271, November.
  12. Dufour, Jean-Marie & King, Maxwell L., 1991. "Optimal invariant tests for the autocorrelation coefficient in linear regressions with stationary or nonstationary AR(1) errors," Journal of Econometrics, Elsevier, vol. 47(1), pages 115-143, January.
  13. Tim Bollerslev & Jeffrey M. Wooldridge, 1988. "Quasi-Maximum Likelihood Estimation of Dynamic Models with Time-Varying Covariances," Working papers 505, Massachusetts Institute of Technology (MIT), Department of Economics.
  14. Graham Elliott & Thomas J. Rothenberg & James H. Stock, 1992. "Efficient Tests for an Autoregressive Unit Root," NBER Technical Working Papers 0130, National Bureau of Economic Research, Inc.
  15. Søren Tolver Jensen & Anders Rahbek, 2004. "Asymptotic Normality of the QMLE Estimator of ARCH in the Nonstationary Case," Econometrica, Econometric Society, vol. 72(2), pages 641-646, 03.
  16. Peter Hall & Qiwei Yao, 2003. "Inference in Arch and Garch Models with Heavy--Tailed Errors," Econometrica, Econometric Society, vol. 71(1), pages 285-317, January.
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Cited by:
  1. Dufour, Jean-Marie & Taamouti, Abderrahim, 2010. "Exact optimal inference in regression models under heteroskedasticity and non-normality of unknown form," Computational Statistics & Data Analysis, Elsevier, vol. 54(11), pages 2532-2553, November.

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