Where does this formula for the OLS estimate as a function of the true parameters come from? Suppose that in the population:
$$ y = \alpha + \beta_1 x_1 + \epsilon $$
We now estimate the model:
$$ \hat{y} = \hat{\alpha} + \hat{\beta_1} x_1 $$
I have seen the following formula which writes the OLS estimate as a function of the true parameter:
$$ \hat{\beta}_\text{OLS} = \beta + \text{corr}(x_1, \epsilon) \dfrac{\sigma_\epsilon}
{\sigma_{x_1}}$$
Where does this formula comes from?
Do you now nay bibliographic references where I can find a proof?
 A: Starting from the standard result for a bivariate regression model, we have:
$$
\hat \beta_{OLS} = \frac{\sum_{i=1}^n(x_{i1}-\bar x_1)(y_{i}-\bar y)}{\sum_{i=1}^n(x_{i1}-\bar x_1)^2}.
$$
Plugging in $y_i = \alpha + \beta x_{i1} + \varepsilon_i$ and $\bar y = \alpha + \beta \bar x_{1} + \bar \varepsilon$, we get:
$$
\hat \beta_{OLS} = \frac{\beta \sum_{i=1}^n(x_{i1}-\bar x_1)^2 + \sum_{i=1}^n(x_{i1}-\bar x_1)(\varepsilon_{i}-\bar \varepsilon)}{\sum_{i=1}^n(x_{i1}-\bar x_1)^2} = \beta + \frac{\sum_{i=1}^n(x_{i1}-\bar x_1)(\varepsilon_{i}-\bar \varepsilon)}{\sum_{i=1}^n(x_{i1}-\bar x_1)^2} 
$$
Imagine multiplying both numerator and denominator with $1/(n-1)$, then:
$$
\hat \beta_{OLS} = \beta + \frac{\widehat{cov}(x_1, \varepsilon)}{\hat \sigma_{x_1}^2} = \beta + \widehat{corr}(x_1, \varepsilon)\frac{\hat\sigma_{\varepsilon}}{\hat\sigma_{x_1}},
$$
where $\widehat{cov}$ represents the sample covariance, $\hat \sigma^2$ sample variance, $\hat \sigma$ sample standard deviation and $\widehat{corr}$ sample correlation.
