Dovonon, Prosper and Renault, Eric (2011): Testing for Common GARCH Factors.
Download (467kB) | Preview
This paper proposes a test for common GARCH factors in asset returns. Following Engle and Kozicki (1993), the common GARCH factors 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 chi-squares with H-1 and H degrees of freedom, where H is the number of moment conditions, as opposed to a chi-square with H-1 degree of freedom. With more than two assets, this distribution lies between the chi-square with H-p and the chi-square with H degrees of freedom (p, the number of parameters) and both bounds are conditionally sharp. 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.
|Item Type:||MPRA Paper|
|Original Title:||Testing for Common GARCH Factors|
|Keywords:||GARCH factors, Nonstandard asymptotics, GMM, GMM overidenti fication test, identifi cation, first order identification|
|Subjects:||C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics|
|Depositing User:||Prosper Dovonon|
|Date Deposited:||23. Jul 2012 14:16|
|Last Modified:||21. Sep 2015 02:05|
Billingsley, P., 1961. "The Lindeberg-Levy Theorem for Martingales," Proc. Amer. Math. Soc., 12, 788-792.
Chamberlain, G., 1986. "Asymptotic Efficiency in Semi-Parametric Models with Censoring," Journal of Econometrics, 32, 189-218.
Cragg, J.G. and S. G. Donald, 1993. "Testing Identifiability and Specification in Instrumental Variable Models," Econometric Theory, 9, 222-240.
Cragg, J.C. and S.G. Donald, 1996. "Testing Overidentifying Restrictions in Unidentified Models," Unpublished UBC discussion paper, 96/20.
Diebold, F. and M. Nerlove, 1989. "The Dynamics of Exchange Rate Volatility: A Multivariate Latent Factor Arch Model," Journal of Applied Econometrics, 4, 1-21.
Dovonon, P. and S. Goncalves, 2011. "Bootstrapping Gmm Tests Under First Order Underidentification," work in progress, Concordia University.
Dovonon, P. and E. Renault, 2009. "Gmm Overidentification Test with First Order Underidentification," working paper, UNC, http://www.unc.edu/depts/econ/profs/renault/ J testDR20090824.pdf.
Doz, C. and E. Renault, 2006. "Factor Volatility in Mean Models: a Gmm Approach," Econometric Reviews, 25, 275-309.
Engle, R. F., V. K. Ng and M. Rothschild, 1990. "Asset Pricing with a Factor-Arch Covariance Structure: Empirical Estimates for Treasury Bills," Journal of Econometrics, 45, 213-237.
Engle, R. F. and S. Kozicki, 1993. "Testing For Common Features," Journal of Business and Economic Statistics, 11(4), 369-395.
Fiorentini, G., E. Sentana and N, Shephard, 2004. "Likelihood-based Estimation of Generalised arch Structures," Econometrica, 72, 1481-1517.
Hansen, L. P., 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, 50, 1029-1054.
Horn, A. R. and C. R. Johnson 1985. "Matrix Analysis," Cambridge University Press.
Koul, H. L., 2002. "Weighted Empirical Processes in Dynamic Nonlinear Models," Springer-Verlag, New York.
Lee, L. F. and A. Chesher, 1986. "Specification Testing when Score Test Statistics are Identically Zero," Journal of Econometrics, 31, 121-149.
Melino, A., 1982. "Testing for Sample Selection Bias," Review of Economic Studies, 49, 151-153.
Phillips, P. C. B., 1989. "Partially Identified Econometric Models," Econometric Theory, 5, 151-240.
Rotnitzky, A., D. R. Cox, M. Bottai and J. Robins, 2000. "Likelihood-based Inference with Singular Information Matrix," Bernoulli, 6(2), 243-284.
Sargan, J. D., 1983. "Identification and lack of Identification," Econometrica, 51, 1605-1633.
Schott, J. R., 1984. "Optimal Bounds for the Distribution of some Test Criteria for Dimensionality," Biometrica, 71, 561-567.
Staiger, D. and J. H. Stock, 1997. "Instrumental Variables Regression with Weak Instruments," Econometrica, 65, 557-586.
Stock, J. H. and J. H. Wright, 2000. "Gmm with Weak Identification," Econometrica, 68, 1055-1096.
van der Vaart, A. W., 1998. "Asymptotic Statistics," Cambridge University Press.
van der Vaart, A. W. and J. A.Wellner, 1996. "Stochastic Convergence and Empirical Processes," Springer-Verlag, New York.