Multivariate portmanteau test for structural VARMA models with uncorrelated but non-independent error terms

Boubacar Mainassara, Yacouba (2009): Multivariate portmanteau test for structural VARMA models with uncorrelated but non-independent error terms.

Abstract

We consider portmanteau tests for testing the adequacy of structural vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. The structural forms are mainly used in econometrics to introduce instantaneous relationships between economic variables. We first study the joint distribution of the quasi-maximum likelihood estimator (QMLE) and the noise empirical autocovariances. We then derive the asymptotic distribution of residual empirical autocovariances and autocorrelations under weak assumptions on the noise. We deduce the asymptotic distribution of the Ljung-Box (or Box-Pierce) portmanteau statistics in this framework. It is shown that the asymptotic distribution of the portmanteau tests is that of a weighted sum of independent chi-squared random variables, which can be quite different from the usual chi-squared approximation used under independent and identically distributed (iid) assumptions on the noise. Hence we propose a method to adjust the critical values of the portmanteau tests. Monte carlo experiments illustrate the finite sample performance of the modified portmanteau test.

• Item Type: MPRA Paper
• Language: English
• Keywords: Goodness-of-fit test, QMLE/LSE, Box-Pierce and Ljung-Box portmanteau tests, residual autocorrelation, Structural representation, weak VARMA models
• Subjects: C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C32 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes ; State Space Models; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables; C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics

References

Ahn, S. K. (1988) Distribution for residual autocovariances in multivariate autoregressive models with structured parameterization. {\em Biometrika} 75, 590-93.

Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. {\em Econometrica} 59, 817-858.

Arbués, I. (2008) An extended portmanteau test for VARMA models with mixing nonlinear constraints, {\em Journal of Time Series Analysis} 29, 741-761.

Bauwens, L., Laurent, S. and Rombouts, J. V. K. (2006) Multivariate GARCH models: a survey. {\it Journal of Applied Econometrics} 21, 79--109.

Boubacar Mainassara, Y. (2010) Selection of weak VARMA models by modified Akaike's information criteria. \emph{Working Papers,} http://mpra.ub.uni-muenchen.de/24981/.

Boubacar Mainassara, Y. and Francq, C. (2009) Estimating structural VARMA models with uncorrelated but non-independent error terms. \emph{Working Papers,} http://perso.univ-lille3.fr/~cfrancq/pub.html.

Box, G. E. P. and Pierce, D. A. (1970) Distribution of residual autocorrelations in autoregressive integrated moving average time series models. {\em Journal of the American Statistical Association} 65, 1509-26.

Brockwell, P. J. and Davis, R. A. (1991) {\em Time series: theory and methods.} Springer Verlag, New York.

Chabot-Hallé, D. and Duchesne, P. (2008) Diagnostic checking of multivariate nonlinear time series models with martingale difference errors, {\em Statistics and Probability Letters} 78, 997-1005.

Chitturi, R. V. (1974) Distribution of residual autocorrelations in multiple autoregressive schemes. {\em Journal of the American Statistical Association} 69, 928-934.

Chitturi, R. V. (1976) Distribution of multivariate white noise autocorrelations. {\em Journal of the American Statistical Association} 71, 223--226.

Davydov, Y. A. (1968) Convergence of Distributions Generated by Stationary Stochastic Processes. {\em Theory of Probability and Applications} 13, 691-696.

den Hann, W.J. and Levin, A. (1997) A Practitioner's Guide to Robust Covariance Matrix Estimation. {\em In Handbook of Statistics} 15, Rao, C.R. and G.S. Maddala (eds), 291-341.

Dufour, J-M., and Pelletier, D. (2005) Practical methods for modelling weak VARMA processes: identification, estimation and specification with a macroeconomic application. \emph{ Technical report, Département de sciences économiques and CIREQ, Université de Montréal, Montréal, Canada.}

Francq, C. and Raïssi, H. (2007) Multivariate Portmanteau Test for Autoregressive Models with Uncorrelated but Nonindependent Errors, {\em Journal of Time Series Analysis} 28, 454--470.

Francq, C., Roy, R. and Zakoïan, J-M. (2003) Goodness-of-fit tests for ARMA models with uncorrelated errors, \emph{Discussion Paper CRM-2925, Centre de recherches mathématiques, Université de Montréal.} http://www.crm.umontreal.ca/pub/Rapports/2900-2999/2925.pdf

Francq, C., Roy, R. and Zakoïan, J-M. (2005) Diagnostic checking in ARMA Models with Uncorrelated Errors, {\em Journal of the American Statistical Association} 100, 532--544.

Francq, C. and Zakoïan, J-M. (1998) Estimating linear representations of nonlinear processes, {\em Journal of Statistical Planning and Inference} 68, 145--165.

Francq, C. and Zakoïan, J-M. (2005) Recent results for linear time series models with non independent innovations. In {\em Statistical Modeling and Analysis for Complex Data Problems,} Chap. 12 (eds P. {\sc Duchesne} and B. {\sc R\'{e}millard}). New York: Springer Verlag, 241--265.

Hannan, E. J. (1976) The identification and parametrization of ARMAX and state space forms, {\em Econometrica} 44, 713--723.

Herrndorf, N. (1984) A Functional Central Limit Theorem for Weakly Dependent Sequences of Random Variables. {\em The Annals of Probability} 12, 141--153.

Hong, Y. (1996) Consistent testing for serial correlation of unknown form, {\em Econometrica} 64, 837--864.

Hosking, J. R. M. (1980) The multivariate portmanteau statistic, {\em Journal of the American Statistical Association} 75, 602--608.

Hosking, J. R. M. (1981a) Equivalent forms of the multivariate portmanteau statistic, {\em Journal of the Royal Statistical Society} B 43, 261--262.

Hosking, J. R. M. (1981b) Lagrange-tests of multivariate time series models, {\em Journal of the Royal Statistical Society} B 43, 219--230.

Imhof, J. P.] (1961) Computing the distribution of quadratic forms in normal variables. {\em Biometrika} 48, 419--426.

Jeantheau, T. (1998) Strong consistency of estimators for multivariate ARCH models, {\em Econometric Theory} 14, 70--86.

Li, W. K. and McLeod, A. I. (1981) Distribution of the residual autocorrelations in multivariate ARMA time series models, {\em Journal of the Royal Statistical Society} B 43, 231--239.

Ljung, G. M. and Box, G. E. P. (1978) On measure of lack of fit in time series models. {\em Biometrika} 65, 297--303.

L\"utkepohl, H.] (2005) {\em New introduction to multiple time series analysis.} Springer Verlag, Berlin.

Newey, W. K. and West, K. D. (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. {\em Econometrica} 55, 703--708.

Reinsel, G. C. (1997) {\em Elements of multivariate time series Analysis.} Second edition. Springer Verlag, New York.

Romano, J. L. and Thombs, L. A. (1996) Inference for autocorrelations under weak assumptions, {\em Journal of the American Statistical Association} 91, 590--600.