Consider the following 2-variable linear regression where error $e_i$'s are independently and identically distributed with mean 0 and variance 1;
$$ y_i=\alpha + \beta (x_i - \bar {x}) + e_i$$ where i ranges from 1 to n.
Let $\hat {\alpha}$ and $\hat {\beta}$ be ordinary least square estimates of $\alpha$ and $\beta$ respectively. Then what's the correlation coefficient between $\hat {\alpha}$ and $\hat {\beta}$?
I'm really confused about whether the answer is 0 or -1.
Because on one hand, these two are independent variables of the regression model and if there was a discernible correlation between the two- it would jeopardize the model and should have been compensated for. On the other hand- I have $$\hat {\alpha} = \bar {y} - \hat {\beta} \bar {x}$$and the correlation from here between the two terms seem to be -1. I feel like i'm missing something really basic, why is the correlation not equal to -1?