Econometrics Midterm Formula Sheet

\[ y = \beta_0 + \beta_1 x + u \]

\[ \hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}, \quad \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x} \]

\[ SST = SSE + SSR \] \[ R^2 = 1 - \frac{SSR}{SST} \qquad \overline{R}^2=1-\frac{SSR/(n-k-1)}{SST/(n-1)} \]

\[ Var(\hat{\beta}|X) = \frac{\sigma^2}{\sum (x-\overline{x})^2} \] \[ \hat{\sigma}^2 = \frac{SSR}{n-k-1} \]

\[ t = \frac{\hat{\beta}_j - \beta_{j}}{se(\hat{\beta}_j)} \]

\[ F = \frac{(SSR_{r} - SSR_{ur})/q}{SSR_{ur}/(n-k-1)} \qquad F = \frac{(R^2_{ur} - R^2_{r})/q}{(1 - R^2_{ur})/(n-k-1)} \]

\[ \hat{\beta}_1^{IV} = \frac{\sum z_i y_i}{\sum z_i x_i} \]