Estimating structural VARMA models with uncorrelated but non-independent error terms

The asymptotic properties of the quasi-maximum likelihood estimator (QMLE) of vector autoregressive moving-average (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.

[1]  Christian Francq,et al.  HAC estimation and strong linearity testing in weak ARMA models , 2007 .

[2]  E. Hannan,et al.  Recursive estimation of mixed autoregressive-moving average order , 1982 .

[3]  Ching-Zong Wei,et al.  Modeling of time series arrays by multistep prediction or likelihood methods , 2004 .

[4]  Helmut Ltkepohl,et al.  New Introduction to Multiple Time Series Analysis , 2007 .

[5]  J. Zakoian,et al.  Estimating linear representations of nonlinear processes , 1998 .

[6]  E. Hannan,et al.  Estimation of vector ARMAX models , 1980 .

[7]  Jorma Rissanen,et al.  THE STRONG CONSISTENCY OF MAXIMUM LIKELIHOOD ESTIMATORS FOR ARMA PROCESSES , 1979 .

[8]  Pierre Duchesne,et al.  On consistent testing for serial correlation of unknown form in vector time series models , 2004 .

[9]  D. Andrews.  Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation , 1991 .

[10]  H. Tong.  Non-linear time series. A dynamical system approach , 1990 .

[11]  J. Zakoian,et al.  Recent Results for Linear Time Series Models with Non Independent Innovations , 2005 .

[12]  G. C. Tiao,et al.  An introduction to multiple time series analysis. , 1993, Medical care.

[13]  Christian Kascha,et al.  A Comparison of Estimation Methods for Vector Autoregressive Moving-Average Models , 2012 .

[14]  Christian Francq,et al.  Diagnostic Checking in ARMA Models With Uncorrelated Errors , 2005 .

[15]  Andrew T. Levin,et al.  A Practitioner's Guide to Robust Covariance Matrix Estimation , 1996 .

[16]  E. Hannan.  The Identification and Parameterization of ARMAX and State Space Forms , 1976 .

[17]  J. Imhof.  Computing the distribution of quadratic forms in normal variables , 1961 .

[18]  Y. Davydov.  Convergence of Distributions Generated by Stationary Stochastic Processes , 1968 .

[19]  Pierre Duchesne,et al.  Statistical Modeling and Analysis for Complex Data Problems , 2010 .

[20]  K. Berk.  Consistent Autoregressive Spectral Estimates , 1974 .

[21]  Robert Kohn,et al.  Asymptotic Estimation and Hypothesis Testing Results for Vector Linear Time Series Models , 1979 .

[22]  A. Wald.  Note on the Consistency of the Maximum Likelihood Estimate , 1949 .

[23]  E. Hannan,et al.  The statistical theory of linear systems , 1989 .

[24]  N. Herrndorf.  A Functional Central Limit Theorem for Weakly Dependent Sequences of Random Variables , 1984 .

[25]  D. Findley.  ASYMPTOTIC SECOND MOMENT PROPERTIES OF OUT-OF-SAMPLE FORECAST ERRORS OF MISSPECIFIED REGARIMA MODELS AND THE OPTIMALITY OF GLS , 2005 .

[26]  E. J. Hannan,et al.  Vector linear time series models , 1976, Advances in Applied Probability.

[27]  G. Reinsel.  Elements of Multivariate Time Series Analysis, 2nd Edition , 1998 .

[28]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[29]  Joseph P. Romano,et al.  Inference for Autocorrelations under Weak Assumptions , 1996 .

[30]  C. Francq,et al.  Multivariate Portmanteau Test For Autoregressive Models with Uncorrelated but Nonindependent Errors , 2007 .

[31]  R. Roy,et al.  Identification of Refined ARMA Echelon Form Models for Multivariate Time Series , 1996 .

[32]  N. Wermuth,et al.  Nonlinear Time Series: Nonparametric and Parametric Methods , 2005 .

[33]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .