I am a scientist and new to this group so apologies in advance if this has been covered before.
In my work, I use experiments to calculate the slope and intercept of experimental data from a simple regression model:
$$y=\beta_0+\beta_1x+\epsilon$$
The OLS estimators from this model: intercept ($\hat{\beta}_0^{OLS}$) and slope ($\hat{\beta}_1^{OLS}$) are then used as parameters in other functions to calculate other variables.
Because these OLS estimators are correlated, I also need to know the covariance $\mathrm{Cov} (\hat{\beta}_0^{OLS},\hat{\beta}_1^{OLS})$ between them in the corresponding error propagation calculations in order to calculate the uncertainty of these other variables.
I have discovered the discussion in this group on Correlation between OLS estimators for intercept and slope, which has been helpful, but I have some additional questions:
(1) In the posting above, @RichardHardy states that the correlation between the two estimators is:
$$\mathrm{Corr} (\hat{\beta}_0^{OLS},\hat{\beta}_1^{OLS})=\frac{-\sum^n_{i=1}x_i}{\sqrt[]{n}\sum^n_{i=1}x_i^2}$$
Is there a published reference for the derivation of this expression? I would like to know how this is derived (for myself), and as this will eventually be used in publishing the research, I would like to ensure there is a proper and valid reference for it.
(2) To determine $\mathrm{Cov} (\hat{\beta}_0^{OLS},\hat{\beta}_1^{OLS})$, am I correct in assuming that:
$$\mathrm{Corr} (\hat{\beta}_0^{OLS},\hat{\beta}_1^{OLS})=\frac{\mathrm{Cov} (\hat{\beta}_0^{OLS},\hat{\beta}_1^{OLS})}{\sigma_{\hat{\beta}_0^{OLS}}\sigma_{\hat{\beta}_1^{OLS}}}$$
If this is not correct, could someone point me to a reference which will allow me to derive $\mathrm{Cov} (\hat{\beta}_0^{OLS},\hat{\beta}_1^{OLS})$? Thanks in advance for any help or advice.
