The Simple Regression Model
simple OLS Regression
We would like to estimate \(\beta_0\) and \(\beta_1\) from a random sample of \(y\) and \(x\)
\[\begin{equation}\label{E1}
y = \beta_0 + \beta_1 x +\nu
\end{equation}\]
According to Wooldridge(2016, Section 2.2), the OLS estimators are
\[\begin{equation}\label{E2}
\hat{\beta_0} = \bar{y} - \hat{\beta_1} \bar{x}
\end{equation}\]
where \(\bar{x}\) and \(\bar{y}\) are the sample average of the \(x\) and \(y\), respectively
\[\begin{equation}\label{E3}
\hat{\beta_1} = \frac{Cov(x,y)}{Var(x)}
\end{equation}\]
Based on these estimated parameters, the OLS regression line is
\[\begin{equation}\label{E4}
\hat{y} = \hat{\beta_0} + \hat{\beta_1}x
\end{equation}\]
Coefficients, Fitted Values and Residuals
- Coefficients: \(\hat{\beta_0}\), \(\hat{\beta_1}\)
- Intercept: \(\hat{\beta_0}\)
- Slop parameter: \(\hat{\beta_1}\)
- Predicted values \(\hat{y}\): \(\hat{y}= \hat{\beta_0} + \hat{\beta_1}x\)
- Residuals \(\hat{u}\): \(\hat{u} = y - \hat{y}\) {} \[\sum_{i=1}^n \hat{u_i} = 0, \Leftarrow \bar{hat{u_i}} = 0\]