Realized skewness/kurosis of returns How can we precisely measure the realized skewness/kurtosis of returns over various horizons? 
Suppose that I would like to measure the realized skewness/kurtosis of stock returns last month. There are two main estimators that I found in the literature, the moment-based skewness/kurtosis and the quantile-based (robust) skewness/kurtosis. 
However, the values of these measurments would depends heavily on the frequency of returns that I have in hand. For instance, the value would be very different depending on whether I use daily returns or 15-mins returns or 5-min returns over the month. 
How could I precisely measure the ex-post skewness/kurtosis? Which frequency should be used?
Regards to my objectives, what I'm trying to do is to compare the skewness/kurtosis of several models proposed in the literature to see which model could give the best forecasts. I am interested with several time-horizons from daily, weekly, biweekly and monthly. As the true skewness and kurtosis is unobservable, I try to use measures of the ex-post realized skewness and kurtosis of stock returns as proxies. For the realized volatility, its is of popular that we could utilize the high-frequency of returns to capture both the long-intergrated part and jumps part in volatility. But for the higher moments like skewness and kurtosis, much less proxies can be found in literature.  
 A: One possible response to your first question about measuring "skewness/kurtosis" would be to estimate a tail index. While there are several rigorous methods for this, e.g., Pickands or Hills methods, Gabaix proposes an easily implemented estimator based on log ranks and OLS here ... http://www.eco.uc3m.es/temp/jbes.2009.06157.pdf . Once you have this metric, then this wiki page on Tweedie distributions (the Examples section)  provides a "lookup" table of extreme value distributions that would classify the fatness of the tails (based on the magnitude of the index) here ... https://en.wikipedia.org/wiki/Tweedie_distribution . This should help with deeper insight into the behavior of your returns.
However, a single month of returns, even high frequency returns, is not very much information, particularly wrt the stock market. There are many regularities, e.g., seasonalities over hourly, daily, weekly, monthly, etc., time periods, that simply won't be estimable, not to mention enabling analysis of important long range dependence structures. It would be much better to push for a significantly longer time series. 
What you haven't described for us is the objectives or purpose behind your analysis. This information should guide a decision as to the frequency of the returns. In its absence and in the abstract, not much can be said. 
