Is it valid to get a correlation between moving averages?

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I have a 10 day moving average of one set of return data for one stock and a ten day moving ave of another stock. 400 data points and correlating 390. Can I now get a correlation between the two or does it have to be by day? Thanks

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The moving average is a smoothing (low-pass) filter, and if you then compute the correlation what you get is the correlation of filtered time series. If that is interesting to you, then yes, you can do that.

This correlation can however be both be higher or larger than the correlation of unfiltered time series, because the part that is attenuated by the filter (the high-frequency components) may contribute positively or negatively to the overall correlation.