In a simple regression model estimated using OLS, the covariance between the estimated errors and regressors is zero by construction

Question

Is this statement true or false?

I seem to remember that this relationship does not hold when the regression has no intercept, however my teacher said that this was true regardless of whether we have an intercept or not. Doesn't having an intercept negate this statement?

Answer 1

Let $e$ be the estimated errors. I'll focus on a simple linear regression

$cov(x,e) = cov(x,y-a-xb) = cov(x,y-(\bar{y}-\bar{x}b)-x\frac{cov(x,y)}{var(x)})=cov(x,(y-\bar{y})-(x-\bar{x})\frac{cov(x,y)}{var(x)}) $

$= cov(x,y)-cov(x,(x-\bar{x})\frac{cov(x,y)}{var(x)}) = cov(x,y)-\frac{cov(x,y)}{var(x)}cov(x,(x-\bar{x})) = cov(x,y)-\frac{cov(x,y)}{var(x)}var(x) = 0$