Principle of Least Squares: Estimates
Statistical Model
\[ \large y_{i} = \beta_0 + \beta_1 x_{i} + \epsilon_{i} \]
Residual Sum of Squares
\[ \large RSS = \sum\limits_{i=1}^{n} \hat\epsilon_{i}^2 \]
\(\beta_1\) Estimates
The least squares approach chooses \(\large \beta_0\) and \(\large \beta_1\) to minimize the Residual Sum of Square (RSS).
\[ \large \hat \beta_1 = \frac{\sum\limits_{i=1}^{n}(x_i-\bar x)(y_i-\bar y)}{\sum\limits_{i=1}^{n}(x_i-\bar x)^2} = \frac{S_{xy}}{S_{xx}} \]
\(\beta_0\) Estimates
\[ \large \hat \beta_0 = \bar y - \hat \beta_1 \bar x \]
Here,
\[ \large \bar{x} = \frac{1}{n}\sum\limits_{i=1}^{n} x_{i} \]
\[ \large \bar{y} = \frac{1}{n}\sum\limits_{i=1}^{n} y_{i} \]