# A regressor performs better under a certain regime — how to condition that regressor to make regression better?

I ran a regression $$Y \sim X_1 + X_2 + .. X_n$$. I find out what one regressor , $$X_1$$'s performance depends on another variable $$t$$ (not in the regression). So basically if I bucket by $$t$$, within each bucket of $$t$$, I plot the correlation of $$X_1$$ and $$Y$$. I can see that when $$t$$ is higher, the correlation between $$X_1$$ and $$Y$$ is very obviously higher.

One example is $$t$$ is time of the day. $$X_1$$ has higher correlation with $$Y$$ as time goes by during the day, for may reason. But $$t$$ itself doesn't predict $$Y$$ at all. My $$t$$ and $$X_1$$ can be positive or negative.

So I want to take advantage of this phenomenon. I scale $$X_1' = X_1 * (1+t)$$. Basically when $$t>0$$, I increase the magnitude of $$X_1$$, and vice versa. I rescale $$t$$ to be within $$(0,1)$$ or even smaller intervals. Then I run regression $$Y \sim X_1' + X_2 + .. X_n$$

Much to my surprise, this adjustment doesn't improve my regression at all, based on all the regression stats.

How do I improve my regression based on this observation?