We describe and estimate for the first time a natural multivariate extension of the
univariate stochastic volatility model with leverage. The model, which we call the multivariate
stochastic volatility with cross leverage, is fit by a tuned Bayesian MCMC method.
Of particular general interest is our approach for sampling the state variables from the
posterior distribution conditioned on the parameters. The state variables are sampled
in blocks by the Metropolis-Hastings algorithm in which the proposal density is derived
from an approximating linear Gaussian state space model. The conditional modes of the
latent volatility variables are computed using a method of scoring where the covariance
matrix of the proposal density is guaranteed to be positive definite. The auxiliary particle
filter to compute the likelihood function is also shown and the model and the techniques
are illustrated with daily stock returns data from the Tokyo Stock Exchange.
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