Using correlation coefficients as x in a regression?

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SUMMARY

The discussion centers on the use of rolling correlation coefficients as predictor variables in linear regression models. Specifically, the author references a univariate linear regression model where the predictor variable, X, is the Pearson correlation coefficient calculated over a 30-day window for two random variables, C and D. While some participants express skepticism about the appropriateness of this approach, it is established that there are no fundamental mathematical violations of regression assumptions. The model can effectively illustrate relationships, particularly in scenarios where the dependent variable, Y, is influenced by the balance between C and D, such as in predator-prey dynamics.

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Using correlation coefficients as x in a regression??

I was reading an article in the Wall street journal and the author was using a rolling correlation coefficient, on a set of variables, as his predictor variable in a linear regression.

Basically it was a uni-variate linear regression , y= mx+b and x was the Pearson correlation coefficient calculated using a 30 day window on two random variables.

This seems "wrong" but I am not sure that it is. I don't know of any regression assumption this violates but at the same time it just doesn't seem like you can do this sort of thing.

What do you think ?
 
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I think this isn't a mathematical question until we know what is assumed and what is asserted. For example, what are the assumptions of "regression"? What consequences follow from those assumptions? There are several types of regression.
 
There is nothing mathematically wrong with this model, per say. It's easy to define an example . Suppose we have two random variables, C and D and X is the rolling correlation between C and D. We can define a random variable Y as Y = ax+b for real constants a,b. Then the linear regression would be a perfect model for Y.

I thought of a type of situation that would naturally lead to a model like you have. When the subject of interest, Y, is related to how well two other variables, C and D, are in or out of balance, this type of model might easily occur. Example: Suppose you are studying predator/pray (C and D) and trying to predict the rate, Y, that predators are killing that prey. When they are out of balance, too many predators or not enough predators, the rate of kills, Y, will be smaller than when they are in balance. The rolling correlation, X, between predator and prey numbers indicates how well they stay in balance. So a natural model would be a linear regression between X and Y.

Other examples are where the rate of something, Y, increases when two other things, C and D, get out of balance.
 
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