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Testing for Common GARCH Factors

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  • Prosper Dovonon

    ()

  • Éric Renault

    ()

Abstract

This paper proposes a test for common conditionally heteroskedastic (CH) features in asset returns. Following Engle and Kozicki (1993), the common CH features property is expressed in terms of testable overidentifying moment restrictions. However, as we show, these moment conditions have a degenerate Jacobian matrix at the true parameter value and therefore the standard asymptotic results of Hansen (1982) do not apply. We show in this context that the Hansen’s (1982) J-test statistic is asymptotically distributed as the minimum of the limit of a certain empirical process with a markedly nonstandard distribution. If two assets are considered, this asymptotic distribution is a half-half mixture of x_(H-1)^2and x_H^2, where H is the number of moment conditions, as opposed to a x_(H-1)^2. With more than two assets, this distribution lies between the x_(H-p)^2 and x_H^2 (p, the number of parameters). These results show that ignoring the lack of first order identification of the moment condition model leads to oversized tests with possibly increasing over-rejection rate with the number of assets. A Monte Carlo study illustrates these findings. Cet article propose un test pour la détection de caractéristiques communes d’hétéroscédasticité conditionnelle (HC) dans des rendements d’actifs financiers. Conformément à Engle et Kozicki (1993), l’existence de caractéristiques communes HC est exprimée en termes de conditions de moment sur-identifiantes testables. Cependant, nous montrons que ces conditions de moment ne sont pas localement linéairement indépendantes; la matrice Jacobienne est nulle à la vraie valeur des paramètres et, par conséquent, la théorie asymptotique de Hansen (1982) ne s’applique pas. Nous montrons dans ce contexte que la statistique de J-test de Hansen (1982) est distribuée asymptotiquement comme le minimum de la limite d’un processus empirique avec une distribution non standard. Quand on considère deux actifs, cette distribution asymptotique est un mélange à parts égales de x_(H-1)^2 et x_H^2, où H est le nombre de conditions de moment, par opposition à x_(H-1)^2. Avec plus de deux actifs, cette distribution est comprise entre x_(H-p)^2 et x_H^2 (p, le nombre de paramètres). Ces résultats montrent que l’ignorance du défaut d’identification au premier ordre dans ce modèle de conditions de moments conduit à des tests qui rejettent trop souvent l’hypothèse nulle, le degré de sur-rejet étant croissant avec le nombre d’actifs. Une étude de Monte-Carlo illustre ces résultats.

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Bibliographic Info

Paper provided by CIRANO in its series CIRANO Working Papers with number 2012s-34.

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Date of creation: 01 Dec 2012
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Handle: RePEc:cir:cirwor:2012s-34

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Keywords: Common features; GARCH factors; Nonstandard asymptotics; GMM; GMM overidentification test; identification; first order identification;

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  1. Donald W.K. Andrews & Xu Cheng, 2010. "Estimation and Inference with Weak, Semi-strong, and Strong Identification," Cowles Foundation Discussion Papers 1773R, Cowles Foundation for Research in Economics, Yale University, revised Jul 2011.
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  5. Melino, Angelo, 1982. "Testing for Sample Selection Bias," Review of Economic Studies, Wiley Blackwell, vol. 49(1), pages 151-53, January.
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  7. Robert F. Engle & Victor Ng & Michael Rothschild, 1988. "Asset Pricing with a Factor Arch Covariance Structure: Empirical Estimates for Treasury Bills," NBER Technical Working Papers 0065, National Bureau of Economic Research, Inc.
  8. Sargan, J D, 1983. "Identification and Lack of Identification," Econometrica, Econometric Society, vol. 51(6), pages 1605-33, November.
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  13. Hecq Alain & Laurent Sébastien & Palm Franz C., 2012. "On the Univariate Representation of BEKK Models with Common Factors," Research Memorandum 018, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
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  15. Douglas Staiger & James H. Stock, 1997. "Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 65(3), pages 557-586, May.
  16. Cragg, John G. & Donald, Stephen G., 1993. "Testing Identifiability and Specification in Instrumental Variable Models," Econometric Theory, Cambridge University Press, vol. 9(02), pages 222-240, April.
  17. Engle, Robert F. & Marcucci, Juri, 2006. "A long-run Pure Variance Common Features model for the common volatilities of the Dow Jones," Journal of Econometrics, Elsevier, vol. 132(1), pages 7-42, May.
  18. Francis X. Diebold & Marc Nerlove, 1986. "The dynamics of exchange rate volatility: a multivariate latent factor ARCH model," Special Studies Papers 205, Board of Governors of the Federal Reserve System (U.S.).
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  20. Chamberlain, Gary, 1986. "Asymptotic efficiency in semi-parametric models with censoring," Journal of Econometrics, Elsevier, vol. 32(2), pages 189-218, July.
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