# What is the correlation between Y and $\epsilon$ in a linear regression?

Suppose I have the multivariate linear regression:

$$Y=a_0 + a_1*X_1 + .... + a_k*X_k + \epsilon$$

What is $$Cov(Y, \epsilon)$$? Testing it empirically with random values of Y and X, I find that the correlation is close to 1. Why is that the case?

In linear model, $$\epsilon$$ always results to be orthogonal to de predictors, then $$Cov(y, \epsilon) = Var(\epsilon)$$. Their correlation depends on the proportion of y which is explained by the model, so $$Cor(y, \epsilon)^2 = 1-R^2$$.