1
$\begingroup$

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

Improve this question
$\endgroup$
1
$\begingroup$

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.

Improve this answer
$\endgroup$
2
  • $\begingroup$ Thank you, this is an approach which would indeed work in a first stage. However, it relies on the assumptions of normality and stationarity, which is a simplification I would like to avoid. My preference is to use the distributions which correspond to actual data, and that is why that paper by Page was appealing. I can also get the historical data that was used to construct the correlation matrix, which would allow me to find the most appropriate distribution(s). $\endgroup$
    – Alex
    Feb 6 '17 at 15:55
  • 1
    $\begingroup$ If your series are non-stationary, then your correlation matrix is not meaningful anyway. $\endgroup$
    – Julius
    Feb 6 '17 at 23:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.