USING SUBSPACE METHODS FOR ESTIMATING ARMA MODELS FOR MULTIVARIATE TIME SERIES WITH CONDITIONALLY HETEROSKEDASTIC INNOVATIONS

This paper deals with the estimation of linear dynamic models of the autoregressive moving average type for the conditional mean for stationary time series with conditionally heteroskedastic innovation process. Estimation is performed using a particular class of subspace methods that are known to have computational advantages as compared to estimation based on criterion minimization. These advantages are especially strong for high-dimensional time series. Conditions to ensure consistency and asymptotic normality of the subspace estimators are derived in this paper. Moreover asymptotic equivalence to quasi maximum likelihood estimators based on the Gaussian likelihood in terms of the asymptotic distribution is proved under mild assumptions on the innovations. Furthermore order estimation techniques are proposed and analyzed.

[1]  Guido M. Kuersteiner,et al.  Efficiency IV Estimation for Autoregressive Models with Conditional Heterogeneity , 2000 .

[2]  Dietmar Bauer,et al.  Comparing the CCA Subspace Method to Pseudo Maximum Likelihood Methods in the case of No Exogenous Inputs , 2005 .

[3]  George Kapetanios,et al.  Testing the rank of the Hankel covariance matrix: a statistical approach , 2001, IEEE Trans. Autom. Control..

[4]  Lennart Ljung,et al.  Some facts about the choice of the weighting matrices in Larimore type of subspace algorithms , 2002, Autom..

[5]  W. Larimore.  System Identification, Reduced-Order Filtering and Modeling via Canonical Variate Analysis , 1983, 1983 American Control Conference.

[6]  Michel Verhaegen,et al.  Identification of the deterministic part of MIMO state space models given in innovations form from input-output data , 1994, Autom..

[7]  Michael McAleer,et al.  ASYMPTOTIC THEORY FOR A VECTOR ARMA-GARCH MODEL , 2003, Econometric Theory.

[8]  Masanao Aoki,et al.  State Space Modeling of Time Series , 1987 .

[9]  Thierry Jeantheau,et al.  STRONG CONSISTENCY OF ESTIMATORS FOR MULTIVARIATE ARCH MODELS , 1998, Econometric Theory.

[10]  Dietmar Bauer,et al.  Consistency and asymptotic normality of some subspace algorithms for systems without observed inputs , 1999, Autom..

[11]  Β. L. HO,et al.  Editorial: Effective construction of linear state-variable models from input/output functions , 1966 .

[12]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .

[13]  K. Taira.  Proof of Theorem 1.3 , 2004 .

[14]  Daniel B. Nelson.  CONDITIONAL HETEROSKEDASTICITY IN ASSET RETURNS: A NEW APPROACH , 1991 .

[15]  Optimal Instrumental Variables Estimation For ARMA Models , 2001 .

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

[17]  Dietmar Bauer,et al.  Order estimation for subspace methods , 2001, Autom..

[18]  J. Davidson.  Stochastic Limit Theory , 1994 .

[19]  Michael McAleer,et al.  NECESSARY AND SUFFICIENT MOMENT CONDITIONS FOR THE GARCH(r,s) AND ASYMMETRIC POWER GARCH(r,s) MODELS , 2002, Econometric Theory.

[20]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.

[21]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[22]  R. Jong.  Consistency and asymptotic normality , 2007 .

[23]  Richard S. Varga,et al.  Proof of Theorem 5 , 1983 .

[24]  E. Hannan,et al.  The Statistical Theory of Linear Systems. , 1990 .

[25]  M. Deistler,et al.  On the Impact of Weighting Matrices in Subspace Algorithms , 2000 .

[26]  R. Varga,et al.  Proof of Theorem 4 , 1983 .

[27]  P. Bougerol,et al.  Stationarity of Garch processes and of some nonnegative time series , 1992 .

[28]  R. Engle,et al.  Multivariate Simultaneous Generalized ARCH , 1995, Econometric Theory.

[29]  D. Bauer.  Some asymptotic theory for the estimation of linear systems using maximum likelihood methods or subspace algorithms , 1998 .

[30]  Offer Lieberman,et al.  Asymptotic theory for multivariate GARCH processes , 2003 .

[31]  李幼升,et al.  Ph , 1989 .

[32]  Bart De Moor,et al.  N4SID: Subspace algorithms for the identification of combined deterministic-stochastic systems , 1994, Autom..

[33]  Bronwyn H Hall,et al.  Estimation and Inference in Nonlinear Structural Models , 1974 .

[34]  Felix Schlenk,et al.  Proof of Theorem 3 , 2005 .

[35]  C. Gouriéroux.  ARCH Models and Financial Applications , 1997 .

[36]  Manfred Deistler,et al.  Consistency and relative efficiency of subspace methods , 1994, Autom..

[37]  Dietmar Bauer,et al.  ESTIMATING LINEAR DYNAMICAL SYSTEMS USING SUBSPACE METHODS , 2005, Econometric Theory.

[38]  M. Aoki,et al.  State space modeling of multiple time series , 1991 .

[39]  Andrew A. Weiss,et al.  Asymptotic Theory for ARCH Models: Estimation and Testing , 1986, Econometric Theory.

[40]  R. Baillie,et al.  INTRA DAY AND INTER MARKET VOLATILITY IN FOREIGN EXCHANGE RATES , 1991 .

[41]  David F. Findley,et al.  Uniform convergence of sample second moments of families of time series arrays , 2001 .