Evaluating exact VARMA likelihood and its gradient when data are incomplete

Abstract

A detailed description of an algorithm for the evaluation and differentiation of the likelihood function for VARMA processes in the general case of missing values is presented. The method is based on combining the Cholesky decomposition method for complete data VARMA evaluation and the Sherman-Morrison-Woodbury formula. Potential saving for pure VAR processes is discussed and formulae for the estimation of missing values and shocks are provided. A theorem on the determinant of a low rank update is proved. Matlab implementation of the algorithm is in a companion article.

The exact Gaussian likelihood estimation of time-dependent VARMA models

“…Shea (1984Shea ( , 1989, Penzer and Shea (1997) via a Kalman filter and Mauricio (2002), Jónasson and Ferrando (2008) via the Cholesky decomposition method. We present an algorithm based on a Cholesky factorization of a block band matrix instead of the block matrix S, like in Jónasson and Ferrando (2008). For other references, see Jónasson and Ferrando (2008).…”

“…This is done by modifying an algorithm for standard VARMA models, i.e. with constant coefficients, developed by Jónasson and Ferrando (2008). The submatrices in (3.2) and (3.3) were used in that article without the first subscript.…”

On multiplicative seasonal modelling for vector time series

“…All the computer code has been written using the MATLAB software. For estimating VARMA models, we implemented the algorithms based on the E 4 package developed by Terceiro (1990) and Casals et al (1999), and we also used the efficient estimation procedures described in Jonasson (2008) and Jonasson and Ferrando (2008).…”

Asymptotic Distributions for Some Quasi-Efficient Estimators in Echelon VARMA Models

“…But the most widely studied estimation method is Gaussian maximum likelihood (ML) for independent and identically distributed (i.i.d.) Gaussian innovations; see Hannan (1969a), Newbold (1974), Box and Jenkins (1976), Hillmer and Tiao (1979), Hall (1979, 1980), Hannan, Kavalieris and Mackisack (1986), Kohn (1981), Tiao and Box (1981), Solo (1984), Shea (1989), Mélard, Roy and Saidi (2002), Mauricio (2002Mauricio ( , 2006, Metaxoglou and Smith (2007), Jonasson and Ferrando (2008), and Gallego (2009). However, maximizing the exact likelihood in stationary invertible VARMA models is computationally burdensome since for each autoregressive and moving average order (say p and q) a non-quadratic optimization with respect to inequality constraints must be performed using iterative algorithms.…”