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

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Author Info
Emma Iglesias
Jean Marie Dufour

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

Find related papers by JEL classification:
C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Hypothesis Testing
C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Statistical Simulation Methods
C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions

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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. [Downloadable!] (restricted)
  2. Andrews, Donald W K & Ploberger, Werner, 1994. "Optimal Tests When a Nuisance Parameter Is Present Only under the Alternative," Econometrica, Econometric Society, vol. 62(6), pages 1383-1414, November. [Downloadable!] (restricted)
    Other versions:
  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. [Downloadable!]
    Other versions:
  4. 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. [Downloadable!] (restricted)
  5. Dufour, Jean-Marie, 2006. "Monte Carlo tests with nuisance parameters: A general approach to finite-sample inference and nonstandard asymptotics," Journal of Econometrics, Elsevier, vol. 133(2), pages 443-477, August. [Downloadable!] (restricted)
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  6. Lee, John H. H., 1991. "A Lagrange multiplier test for GARCH models," Economics Letters, Elsevier, vol. 37(3), pages 265-271, November. [Downloadable!] (restricted)
  7. Elliott, Graham & Rothenberg, Thomas J & Stock, James H, 1996. "Efficient Tests for an Autoregressive Unit Root," Econometrica, Econometric Society, vol. 64(4), pages 813-36, July. [Downloadable!] (restricted)
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  8. 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. [Downloadable!] (restricted)
    Other versions:
  9. Tim Bollerslev & Jeffrey Wooldridge, 1992. "Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances," Econometric Reviews, Taylor and Francis Journals, vol. 11(2), pages 143-172. [Downloadable!] (restricted)
  10. Tim Bollerslev & Robert F. Engle & Daniel B. Nelson, 1993. "ARCH Models," University of California at San Diego, Economics Working Paper Series 93-49, Department of Economics, UC San Diego. [Downloadable!]
    Other versions:
    • 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. [Downloadable!] (restricted)
  11. 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.
  12. Jean-Marie Dufour & Jan F. Kiviet, 1998. "Exact Inference Methods for First-Order Autoregressive Distributed Lag Models," Econometrica, Econometric Society, vol. 66(1), pages 79-104, January.
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  13. 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. [Downloadable!] (restricted)
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