Properties of Least Square Estimators in SLR

{\hat{β}}_{1} = \frac{S S x y}{S S x x} $ $ w h e r e $ S S_{x y} = \sum_{i = 1}^{n} X_{i} Y_{i} - n \bar{X} \bar{Y} = \sum_{i = 1}^{n} Y_{i} (X_{i} - \bar{X}) = \sum_{i = 1}^{n} X_{i} (Y_{i} - \bar{Y}) = \sum_{i = 1}^{n} (X_{i} - \bar{X}) (Y_{i} - \bar{Y}) $ a n d $ S S_{x x} = \sum_{i = 1}^{n} X_{i}^{2} - n {\bar{X}}^{2} = \sum_{i = 1}^{n} (X_{i} - \bar{X})^{2} $

\hat\beta_0 = \bar Y-\hat\beta_1 \bar X

e r g o t h e f i t t e d l i n e (e s t i m a t e) i s :

\hat Y = \hat \beta_0 + \hat \beta_1 X_i

\sigma^2 = \sum_{i=1} \frac{(y_i-\hat y_i)^2}{n-2} = \sum_{i=1} \frac{\varepsilon_i^2}

A l l t h e s e e s t i m a t o r s a r e u n b i a s e d . R e f e r e n c e s