We have two vectors of random variables: ${\bf X}_t\in R^p$ and ${\bf Y}_t\in R^q$. Let ${\bf Z}_t=({\bf X}_t,{\bf Y}_t)$. Here $t$ represent the time in months. In my specific application, these variables represent end of month stock prices.
We define $${\bf R}_t={{\bf Z}_t \over {\bf Z}_{t-1}}-1$$ where the division is done coordinate-wise. In my specific application, this variable represents stock returns. We assume that this variable is normally distributed with mean ${\bf \mu}$ and covariance $\bf\Sigma$.
We want to estimate the covariance matrix $\bf \Sigma$.
The problem is that I have an incomplete data set for the estimation : ${\bf Y}_t$ has missing values because it is sampled quaterly. The variable ${\bf X}_t$ is sampled monthly.
How can this be done? I have no problem in assuming a distribution for any of the variables. I would like to be able to use all of the data points of ${\bf X}_t$ in my estimation (i.e. not throw away 2/3 of the data to make the frequencies match).
Thanks.