# Financial time series data: Imputing before or after calculating returns?

I've got several time series of daily prices ($(p_t^j)_{t=1,\dots,n}$ ) of different tradable cards $j=1,\dots,k$. I'd like to calculate the time series of the (log)returns $r_t= \log(\frac{p_t}{p_{t-1}})$ (omitting the index $j$ here in favor of readability. For some $t$, the value of $p_t$ is missing.

To make the different time series of returns comparable for different cards, I'd like to use some kind of imputation technique (haven't decided on what to use yet - probably hot deck for a start).

If $p_t$ is missing, should I rather use an imputed value $p'_t$ to calculate $r_{t}= \log(\frac{p'_t}{p_{t-1}})$ and $r_{t+1}= \log(\frac{p_{t+1}}{p'_{t}})$, or should I consider $r_{t}$ and $r_{t+1}$ as missing and impute them directly as $r'_t$ and $r'_{t+1}$?

Apart from trying both variants and validating which will create the better time series (by whatever measure), are there more general aspects to consider when doing this kind of imputation?

• Why do you have missing data in price returns? Those only occur on holidays for a given market, and if a market is closed, there are no data! – user32398 Jan 16 '16 at 4:28
• @LEP: There are several reasons: One is that there is missing liquidity and there is simply noone selling the item in question, thus at this given day, there is no existing price. Another reason are technical problems on days where I wasn't able to retrieve prices (e.g. missing internet connection). – Roland Jan 16 '16 at 11:05
• Probably explore imputation in financial time series. ARMA, ARCH, and GARCH autocorrelated time series models come to mind. – user32398 Jan 17 '16 at 14:01
• @LEP: These were indeed the kind of models I was planning to look into. Are there model-specific imputation approaches, or should I rather go looking for model-agnostic approaches. – Roland Jan 17 '16 at 20:05
• Time series are autocorrelated, so any imputation must take into account day-day correlation. Maybe search Prof Kit Baum, Boston Univ, who provides online power points for quite detailed ARMA, ARCH, GARCH fitting. Stack exchange has a Quantitative Finance forum which is where your request should be made. – user32398 Jan 18 '16 at 22:52