I am trying to construct a daily time series of prices and returns for some large universe of securities. However, all I have available are a monthly time series of the prices/returns (as well as other characteristics) of the individual securities, a daily time series of a market-value-weighted index of all securities, and weekly time series of various sub-indices.
The constructed time series will ultimately be used to estimate parameters of a more general model, such as the probability of a security's issuer taking some action (e.g. refinancing their debt) as a function of the security's price. Therefore I feel it is not important to maintain causality. The issuer presumably knows the true price when taking the action, even though I do not, and I need to construct a best guess as to what the price was given everything I know today.
Note: it is not possible to obtain higher frequency data at the individual security level, either because the securities themselves do not trade that often, or because (AFAIK) nobody collects the data. The goal is to interpolate a reasonable-looking set of daily prices and returns based on all available information. Any advice on how to carry out this estimation would be appreciated.
I have some of my own ideas, which I may share after a while, but right now I'm still in the exploratory phase and I'm looking for some additional inspiration.
Just to make it clear what I mean by way of example, suppose I wanted to find the daily prices of all 1500 stocks in the S&P 1500, but all I had were monthly prices for the stocks, weekly prices for the 10 GICS sector indices and for the large cap 500, mid cap 400, and small cap 600, and daily prices for the S&P 1500 as a whole.
The purpose, in that example, would be to fit a model of announcements of share buybacks and secondary offerings based on interpolated valuation metrics.
UPDATE: One answer suggested applying the Expectation-Maximization algorithm. As far as I can tell, EM is not applicable to this problem. Applying EM to price, one gets a sawtooth-pattern where the filled values are on a different plane from the known values. I can't figure out a way to apply EM to returns, since I'm not missing any monthly returns, and I'm missing all daily/weekly returns for the individual securities.