Multivariate stochastic volatility

Hi. I’m trying to run stochastic volatility model on panel data (n_times x m_assets), and while I scaled the documented example (Example: Stochastic Volatility — NumPyro documentation) to multiple assets (thanks to some help from @fehiepsi), now I also would like to extend the model to fit asset correlations. I understand that multivariate SV models, or time-varying covariance models may be non-trivial, but I wonder if there are simple things I could try (perhaps assume that correlation is constant through time) and get a better fit out of my model while preserving correlations in the posterior predictive samples.

Right now my model has the following structure:

def model(data):

  n_times, n_assets = data.shape

  # hyperpriors
  sigma_lam = ...
  nu_lam = ...
  with numpyro.plate('asset', n_assets, dim=-1):
      # volatility priors
      sigma = numpyro.sample('sigma', dist.Exponential(sigma_lam))
      h = numpyro.sample(
         dist.GaussianRandomWalk(scale=sigma, num_steps=n_times)
      v = numpyro.deterministic('volatility', jnp.exp(-2 * h))
      # returns' normality prior
      nu = numpyro.sample('nu', dist.Exponential(nu_lam)).reshape(-1, 1)
      # likelihood distribution
      dist_kwargs = {
          'df': nu,
          'loc': jnp.zeros((n_assets, 1)),
          'scale': v
      like_dist = dist.StudentT(**dist_kwargs).to_event(1)
      # likelihood
      y = numpyro.sample('y_like', like_dist, obs=data)
  return y

Naive attempts to use multivariate likelihood and construct a covariance prior (with Cholesky decomposition) fail because volatility matrix v is not square (as it is time-varying stochastic volatility). I also tried introducing a second .plate statement along the time dimension and updating a covariance prior at each time point, but I couldn’t quite figure out how to make it work, and even if I could, I suspect it would be computationally inefficient. I was also thinking of introducing a shared factor or component into the model to induce correlations (but haven’t tried anything specific yet).

If you have any thoughts or suggestions on this, please share. Thank you!