The Matrix-Logarithmic Covariance Model

Multivariate Stochastic Volatility: A Review

“…Motivated by the requirement for the positivity of the covariance, Chiu et al (1996) proposed a general framework for the logarithmic covariance matrix based on the matrix exponential transformation, which is well known in the mathematics literature (see, for example, Bellman, 1970). In this subsection, we denote Exp(·) as the matrix exponential operation to distinguish it from the standard exponential operation.…”

“…The properties of the matrix exponential and matrix logarithm are summarized in Chiu et al (1996). For any real symmetric matrix A, we note the singular value decomposition A = TDT , where the columns of the m × m orthonormal matrix T denote the appropriate eigenvectors of A, and D is an m × m diagonal matrix with elements equal to the eigenvalues of A.…”

A Latent Dynamic Factor Approach to Forecasting Multivariate Stock Market Volatility

“…Taking the matrix logarithm of a real, positive de nite matrix R t results in a real, symmetric matrix Y t and applying the matrix exponential function to a real symmetric matrix results in a real symmetric positive de nite matrix (see Chiu et al, 1996, Lemma 1).…”

Modelling the random effects covariance matrix in longitudinal data

“…First, we define b̃i k to be the linear least-squares predictor of b ik based on its predecessors b i, k − 1 ,…, b i1 , and e ik = b ik − b̃i k to be its prediction error with variance . Thus, we have (2) The special Cholesky decomposition of Σ i is defined as where and T i is the unit lower triangular matrix with −ϕ i, kj as its (k, j)th entry. We refer to the ϕ as the GARP and the σ 2 as IV.…”

“…In marginal models, Chiu et al, [2] model the covariance matrix using a log matrix parameterization and obtain estimates using estimating equations. In non-linear mixed models, heterogeneous covariance structures are frequently used, but typically with variance a function of the mean and constant correlations across subjects [3].…”