Econometric Theory/SLR

$$\begin{align} & Y_{i}=\beta _{1}+\beta _{2}X_{i}+U_{i} \\ & E(U|X)=0 \\ & Var(U|X)=Var(Y|X)=\sigma ^{2} \\ & Y_{i}=\beta _{1}+\beta _{2}X_{i}+U_{i} \\ & \widehat{Y_{i}}=\widehat{\beta _{1}}+\widehat{\beta _{2}}X_{i} \\ & \widehat{U_{i}}=Y_{i}-\widehat{Y_{i}} \\ \end{align}$$

Use of $$\beta_{1}$$ or $$\beta_{0}$$ to denote the Y-intercept is solely discretionary.

$$\begin{align} & \widehat{\beta _{2}}=\frac{\sum{(Y_{i}-\bar{Y}})(X_{i}-\bar{X})}{\sum{(X_{i}-\bar{X})^{2}}} \\ & \widehat{\beta _{1}}=\bar{Y}-\widehat{\beta _{2}}\bar{X} \\ & Var(\widehat{\beta _{2}})=\frac{\sigma ^{2}}{\sum{(X_{i}-\bar{X})^{2}}} \\ & Var(\widehat{\beta _{1}})=\frac{\sum{X_{i}^{2}}*\sigma ^{2}}{n*\sum{(X_{i}-\bar{X})^{2}}} \\ & \widehat{\sigma ^{2}}=\frac{\sum{\widehat{U_{i}^{2}}}}{n-2} \\ \end{align}$$

U and $$\epsilon$$ have both been used to denote the error term.

$$\begin{align} & S_{2}^{2}=\widehat{Var(\widehat{\beta _{2}})}=\frac{\widehat{\sigma ^{2}}}{\sum{(X_{i}-\bar{X})^{2}}} \\ & S_{1}^{2}=\widehat{Var(\widehat{\beta _{1}})}=\frac{\sum{X_{i}^{2}}*\widehat{\sigma ^{2}}}{n*\sum{(X_{i}-\bar{X})^{2}}} \\ & S.E.(\widehat{\beta _{2}})=\sqrt{\widehat{Var(\widehat{\beta _{2}})}} \\ & S.E.(\widehat{\beta _{1}})=\sqrt{\widehat{Var(\widehat{\beta _{1}})}} \\ \end{align}$$

$$S^2$$ is used to denote a sample variance, S.E. standard error.

$$\begin{align} & TSS=\sum{(Y_{i}-\bar{Y})^{2}} \\ & ESS=\sum{(\widehat{Y_{i}}-\bar{Y})^{2}} \\ & RSS=\sum{\widehat{U_{i}^{2}}} \\ \end{align}$$

TSS may also be presented as SST for the Total Sum of Squares, ESS as SSE (error) and RSS as SSR (residuals). Depending on the text, ESS and RSS may become very confusing, as there is great variety in the terminology used.

$$R^{2}=\frac{ESS}{TSS}=1-\frac{RSS}{TSS}$$

$$\widehat{\log (Y)}=\widehat{\beta _{1}}+\widehat{\beta _{2}}\log (X)$$