$$$$

$$\Huge\mathbf{y} = \mathbf{X}\beta+\epsilon$$

$$\Huge\mathbf{y} = \mathbf{X}\beta+\epsilon$$

$$\Huge E[\hat\beta]=\beta$$

$$\Huge\textrm{var}(\hat\beta)=\hat\sigma^2(\mathbf{X}^T\mathbf{X})^{-1}$$

$$\Huge E[\hat\beta-\beta] = 0$$

$$\Huge\textrm{var}(\hat\beta)=\hat\sigma^2(\mathbf{X}^T\mathbf{X})^{-1}$$

$$\Large\textrm{var}(\hat\beta)=\hat\sigma^2(\mathbf{X}^T\mathbf{X})^{-1}\leq \textrm{var}(\tilde\beta)$$

$$\Huge E[\tilde\beta-\beta]=0$$

$$\Large\textrm{var}(\hat\beta)=\hat\sigma^2(\mathbf{X}^T\mathbf{X})^{-1}\leq \textrm{var}(\tilde\beta)$$

$$\Huge\textrm{RSS}=\sum e^2$$

$$\Huge\textrm{RSS}=\sum e^2$$

$$\Huge E[\mathbf{e}^T\mathbf{e}] = \sigma^2(n-(p+1))$$

$$\Large E[\mathbf{e}^T\mathbf{e}]=E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]$$

$$\Huge\hat\sigma^2=\frac{\textrm{RSS}}{(n-(p+1))}$$

$$\Huge\hat\sigma^2=\frac{\textrm{RSS}}{(n-(p+1))}$$

$$\Huge\textrm{var}(\hat\beta) = \hat\sigma^2(\mathbf{X}^T\mathbf{X})^{-1}$$

$$F = \frac{RSS_{small} - RSS_{larger} / (df_{small}- df_{larger})}{RSS_{larger}/df_{larger}}\sim F_{df_{small}- df_{larger}, df_{larger}}$$

$$F = \frac{RSS_{small} - RSS_{larger} / (df_{small}- df_{larger})}{RSS_{larger}/df_{larger}}\sim F_{df_{small}- df_{larger}, df_{larger}}$$

$$F = \frac{RSS_{small} - RSS_{larger} / (df_{small}- df_{larger})}{RSS_{larger}/df_{larger}}\sim F_{df_{small}- df_{larger}, df_{larger}}$$

$$F = \frac{RSS_{small} - RSS_{larger} / (df_{small}- df_{larger})}{RSS_{larger}/df_{larger}}\sim F_{df_{small}- df_{larger}, df_{larger}}$$

$$\Huge\hat\beta\pm t^*SE_{\hat\beta}$$