# Multivariate time series model evaluation with conditional moments

Consider multivariate time series models that estimate potentially time-varying conditional means, variances, and correlations (one type of model might be a VAR(p)+Garch(1,1)+DCC Gaussian Copula model). I will simulate from one "true" distribution and compare other potential models to it. I don't particularly care about RMSE because I care just as much about the conditional covariance matrix as the conditional means. For the sake of simplicity, assume that the conditional mean and conditional covariance are sufficient to explain "true" distribution.

I can calculate the conditional mean vector and conditional covariance matrix of each model and the true model at any given point in time. Alternately, I can calculate the forecasted conditional means and covariances on a rolling basis.

1. Is there any reason to prefer the Kullback-Leibler divergence or Bhattacharyya distance as comparison methodologies or should I just do both? Alternately, is there a different method that is even more appropriate.
2. In the constant multivariate Gaussian case, is it possible to make statistical inferences based on these comparison methodologies, such as that the divergence is not statistically bigger than zero or that the divergence of one is bigger than another?
3. If I calculate the conditional means and variances at any given point in time, I could calculate the comparison statistic in each period. This would produce a time series of divergences/distances. Is it reasonable to make statistical inferences in this case by just assuming it is a typical time series?
• I don't get why you are throwing away essentially all the reputation you have accumulated so far to hope for an answer to this. The reason it is not getting attention is probably because it is so specicalized that no one who has seen it thus far has the expertise to answer it. If you get no answer for the next six days you lose 50 reputation points and get no benefit. Commented Jun 1, 2012 at 19:26
• I got +100 reputation for joining since I have more reputation on another stackexchange site. Since the question is important, figured it couldn't hurt.
– John
Commented Jun 4, 2012 at 16:23
• Why would you compare means and covariances between models, where one is such that you know it's true? Or are you trying to say you're knowingly fitting 'wrong' models on (real or simulated) data, and trying to come up with a routine that flags when something is wrong or when the model is dangerous or unsuitable or fits poorly? Is this for a trading strategy? Commented Nov 2, 2016 at 3:43
• @Taylor The idea was that I cared about the density as much as the mean. This question was from a few years ago. Nowadays I would probably use WAIC or Leave one out cross validation.
– John
Commented Nov 2, 2016 at 12:40
• @John my fault I didn't see the datestamp Commented Nov 2, 2016 at 15:32