# Multiplying regressors in linear regression model and multicollinearity

I am trying to train a linear regression model on $$n$$ data samples. My training set contains two features sets, $$X_i$$ and $$Y_i$$ for $$i \in \{1, 2, \dots, n\}$$. I want to create a transformed third feature $$Z_i = X_i Y_i$$ (i.e. the element-wise product of features $$X_i$$ and $$Y_i$$).

For example, $$X = [1, 2, 3]^T$$, $$Y = [4, 5, 6]^T$$, $$Z = X \cdot Y = [4, 10, 18]^T$$.

Linear regressions should avoid multicollinearity in their regressors, so I was wondering:

1. What do we know about the correlation between $$Z_i$$ and $$X_i$$ or $$Y_i$$? We can assume that $$X_i$$ and $$Y_i$$ are standardized to have $$0$$ mean and unit variance.

2. Can we simply do a correlation test between $$Z_i$$ and $$X_i$$ or $$Y_i$$ to see if our regressors are multicollinear or not (and therefore appropriate to use in our linear regression model)? Or is the fact that $$Z_i$$ depends on $$X_i$$ and $$Y_i$$ automatically make it a troubling regressor to use in our model along with $$X_i$$ and $$Y_i$$?

• Are $X$ and $Y$ both predictors of a third variable? It would be conventional to reserve $Y$ for the response variable. – Dave Nov 25 '20 at 3:47
• Yes, $X$ and $Y$ are predictors of a third variable which is I believe irrelevant to this post. I wasn't sure what variables to use and hope it's not too confusing to use $Y$ as a predictor. – user3667125 Nov 25 '20 at 5:13