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.