Boubacar Mainassara, Yacouba and Francq, Christian (2009): Estimating structural VARMA models with uncorrelated but nonindependent error terms.

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Abstract
The asymptotic properties of the quasimaximum likelihood estimator (QMLE) of vector autoregressive movingaverage (VARMA) models are derived under the assumption that the errors are uncorrelated but not necessarily independent. Relaxing the independence assumption considerably extends the range of application of the VARMA models, and allows to cover linear representations of general nonlinear processes. Conditions are given for the consistency and asymptotic normality of the QMLE. A particular attention is given to the estimation of the asymptotic variance matrix, which may be very different from that obtained in the standard framework. Modified versions of the Wald, Lagrange Multiplier and Likelihood Ratio tests are proposed for testing linear restrictions on the parameters.
Item Type:  MPRA Paper 

Original Title:  Estimating structural VARMA models with uncorrelated but nonindependent error terms 
Language:  English 
Keywords:  Echelon form; Lagrange Multiplier test; Likelihood Ratio test; Nonlinear processes; QMLE; Structural representation; VARMA models; Wald test. 
Subjects:  C  Mathematical and Quantitative Methods > C3  Multiple or Simultaneous Equation Models ; Multiple Variables > C32  TimeSeries Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes ; State Space Models 
Item ID:  15141 
Depositing User:  Christian Francq 
Date Deposited:  09. May 2009 17:37 
Last Modified:  31. May 2014 21:11 
References:  Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. {\em Econometrica} 59, 817858. Berk, K.N. (1974) Consistent Autoregressive Spectral Estimates. {\it Annals of Statistics}, {\bf 2}, 489502. Brockwell, P.J. and Davis, R.A. (1991) {\em Time series: theory and methods.} Springer Verlag, New York. 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), 291341. Duchesne, P. and Roy, R. (2004) On consistent testing for serial correlation of unknown form in vector time series models. {\em Journal of Multivariate Analysis} 89, 148180. Dufour, JM., 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.} Dunsmuir, W.T.M. and Hannan, E.J. (1976) Vector linear time series models, {\em Advances in Applied Probability} 8, 339364. Fan, J. and Yao, Q. (2003) {\em Nonlinear time series: Nonparametric and parametric methods.} Springer Verlag, New York. Francq, C. and Raïssi, H. (2007) Multivariate Portmanteau Test for Autoregressive Models with Uncorrelated but Nonindependent Errors, {\em Journal of Time Series Analysis} 28, 454470. Francq, C., Roy, R. and Zakoïan, JM. (2005) Diagnostic checking in ARMA Models with Uncorrelated Errors, {\em Journal of the American Statistical Association} 100, 532544. Francq, C. and Zakoïan, JM. (1998) Estimating linear representations of nonlinear processes, {\em Journal of Statistical Planning and Inference} 68, 145165. Francq, and Zakoïan, JM. (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 Rémillard}). New York: Springer Verlag, 137161. Francq, and Zakoïan, JM. (2007) HAC estimation and strong linearity testing in weak ARMA models, {\em Journal of Multivariate Analysis} 98, 114144. Hannan, E.J. (1976) The identification and parametrization of ARMAX and state space forms, {\em Econometrica} 44, 713723. Hannan, E.J. and Deistler, M. (1988) \emph{The Statistical Theory of Linear Systems} John Wiley, New York. Hannan, E.J., Dunsmuir, W.T.M. and Deistler, M. (1980) Estimation of vector ARMAX models, {\em Journal of Multivariate Analysis} 10, 275295. Hannan, E.J. and Rissanen (1982) Recursive estimation of mixed of Autoregressive Moving Average order, {\em Biometrika} 69, 8194. Herrndorf, N. (1984) A Functional Central Limit Theorem for Weakly Dependent Sequences of Random Variables. {\em The Annals of Probability} 12, 141153. Imhof, J.P. (1961) Computing the distribution of quadratic forms in normal variables. {\em Biometrika} 48, 419426. Kascha, C. (2007) A Comparison of Estimation Methods for Vector Autoregressive MovingAverage Models. \emph{ECO Working Papers, EUI ECO 2007/12,} http://hdl.handle.net/1814/6921. Lütkepohl, H. (1993) {\em Introduction to multiple time series analysis.} Springer Verlag, Berlin. \bibitem{Lu05} Lütkepohl, H. (2005) {\em New introduction to multiple time series analysis.} Springer Verlag, Berlin. Newey, W.K., and West, K.D. (1987) A simple, positive semidefinite, heteroskedasticity and autocorrelation consistent covariance matrix. {\em Econometrica} 55, 703708. Nsiri, S., and Roy, R.] (1996) Identification of refined ARMA echelon form models for multivariate time series. {\em Journal of Multivariate Analysis} 56, 207231. Reinsel, G.C. (1997) {\em Elements of multivariate time series Analysis.} Second edition. Springer Verlag, New York. \bibitem{RiCa79} Rissanen, J. and Caines, P.E. (1979) The strong consistency of maximum likelihood estimators for ARMA processes. {\em Annals of Statistics} 7, 297315. Romano, J.L. and Thombs, L.A. (1996) Inference for autocorrelations under weak assumptions, {\em Journal of the American Statistical Association} 91, 590600. Tong, H. (1990) Nonlinear time series: {\em A Dynamical System Approach,} Clarendon Press Oxford. van der Vaart, A.W. (1998) Asymptotic statistics, {\em Cambridge University Press,} Cambridge. van der Vaart, A.W. (1998) Asymptotic statistics, {\em Cambridge University Press,} Cambridge. Wald, A. (1949) Note on the consistency of the maximum likelihood estimate. {\em Annals of Mathematical Statistics} 20, 595601. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/15141 