Survey_Zadrozny_2016

Abstract

"The present paper makes two contributions. First, the paper extends the original XYW method to an extended XYW method that determines all ARMA parameters of a VARMA model with available covariances of its variables observed with SFD or MFD. Second, the paper proves that if the parameters of the model satisfy conditions I-VI of stationarity, regularity, miniphaseness, controllability, observability, and diagonalizability, then, the extended XYW method produces unique ARMA parameter values and the VARMA model is identified (not just ""generically"") with population covariances of its variables. Although the paper is not directly concerned with parameter estimation, the extended XYW method becomes a consistent method for estimating VARMA parameters simply by replacing population covariances with consistent sample covariances. However, experience with the XYW method (Chen and Zadrozny, 1998) suggests that such a consistent estimation method is unlikely to be accurate in small samples but that a generalized method of moments (GMM) extension of the method could be accurate in small samples. However, such an extension is beyond the scope of this paper and is left for the future. The extended XYW method solves one linear system to determine the AR parameters and solves two linear systems and does one matrix spectral factorization to determine the MA parameters. Spectral factorization is a linear operation except for an initial step of computing eigenvalues, which can be done reliably, accurately, and quickly using the QR algorithm (Golub and Van Loan, 1996, Zadrozny, 1998). The key to the proof in the paper is exploiting the block-Vandermonde structure of eigenvectors of a blockcompanion-form state-transition matrix of a state-space representation of a VARMA model."