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:
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.
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 $Y_i$?
