On random correlation matrices
SIAM Journal on Matrix Analysis and Applications
Priors for ordered conditional variance and vector partial correlation
Journal of Multivariate Analysis
Time Series Analysis: Forecasting and Control
Time Series Analysis: Forecasting and Control
Journal of Multivariate Analysis
Generating random correlation matrices based on partial correlations
Journal of Multivariate Analysis
Generating random AR(p) and MA(q) Toeplitz correlation matrices
Journal of Multivariate Analysis
Rejoinder---Estimation Issues for Copulas Applied to Marketing Data
Marketing Science
Journal of Multivariate Analysis
Journal of Multivariate Analysis
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We study the role of partial autocorrelations in the reparameterization and parsimonious modeling of a covariance matrix. The work is motivated by and tries to mimic the phenomenal success of the partial autocorrelations function (PACF) in model formulation, removing the positive-definiteness constraint on the autocorrelation function of a stationary time series and in reparameterizing the stationarity-invertibility domain of ARMA models. It turns out that once an order is fixed among the variables of a general random vector, then the above properties continue to hold and follow from establishing a one-to-one correspondence between a correlation matrix and its associated matrix of partial autocorrelations. Connections between the latter and the parameters of the modified Cholesky decomposition of a covariance matrix are discussed. Graphical tools similar to partial correlograms for model formulation and various priors based on the partial autocorrelations are proposed. We develop frequentist/Bayesian procedures for modelling correlation matrices, illustrate them using a real dataset, and explore their properties via simulations.