I am working on the following problem:
In a problem of simple linear regression, $$Y = \hat\beta_0 + \hat\beta_1 x(bar),$$ show that the random variables $\hat\beta_1$ and $Y$ are un-correlated (All the betas have "hats").
I have come up with a solution, but I am not convinced whether it is correct logically. Any alternative ways/suggestions/corrections would be helpful.
My solution goes through the following steps:
- Express $\hat\beta_0 = Y - \hat\beta_1 x$
- Find the variance of the expression: $Var(\hat\beta_0) = Var(Y - \hat\beta_1 x)$
- Use the variance formula for $β_0$ on the left side, and split up the right side, so that we have a term that is $Cov(Y,\hat\beta_1)$
- Simplify and show that the covariance is $0$. Therefore the correlation will be $0$.