Centred simple linear regression ($\hat{\alpha }=\overline{Y},\hat{\beta }=S_{xy}/S_{xx}$)

In simple linear regression written as $Y_i=\alpha +\beta (x_i-\bar{x})+{\epsilon }_i$, the constant and centred predictor columns are orthogonal. Differentiating the sum of squared residuals gives the displayed ordinary least squares estimators, provided $S_{xx}={\sum }_i(x_i-\bar{x}{)}^2>0$. For independent mean-zero errors of common variance ${\sigma }^2$, both estimators are unbiased, their variances are ${\sigma }^2/n$ and ${\sigma }^2/S_{xx}$, and their covariance is zero. The uncentred intercept is $\hat{\alpha }-\hat{\beta }\bar{x}$.