Imputation (consistent with a given correlation matrix) of missing values in a time series

Question

There is a model which is based on N factors which are correlated through a correlation matrix of size NxN. For a subset of M factors the corresponding values are explicitly given over a specific time period [A,B].

Which methods are recommended to obtain the most appropriate "missing" values for remaining N-M factors over that time period [A,B]?

The problem is a little different compared to "imputing missing values for a time series", since these missing values have to be consistent with a given correlation matrix.

I am looking at various alternatives, and any relevant reference would be appreciated.

On a side note, a promising avenue (with some enhancements needed) is 2013 the article "How to Combine Long and Short Return Histories Efficiently" by S. Page

Answer 1

I hope you have some information on the mean of the missing variables, otherwise there is no hope of recovering the missing data with just a covariance matrix, let alone a correlation matrix.

I assume that you have some estimates on variances and means of all series. Then assuming normality and stationarity, we can exploit the fact that $$X_t|Y_t \sim \mathcal{N}(\mu_{X}+\Sigma_{XY}\Sigma_{Y}^{-1}(Y_t-\mu_{Y}),\Sigma_{X}-\Sigma_{XY}\Sigma_{Y}^{-1}\Sigma_{YX})$$ where $X_t$'s are missing values and $Y_t$'s are observed values. Given $Y_t$ at any $t$, the best predictor of $X_t$ is therefore $$\mu_{X}+\Sigma_{XY}\Sigma_{Y}^{-1}(Y_t-\mu_{Y})$$

If you have more information,such as auto-covariance matrices, then you can extend the above approach to have more conditioning variables, and thus more 'accurate' imputation.