class: center, middle, inverse, title-slide # Putting it all together ### Dr. D’Agostino McGowan --- layout: true <div class="my-footer"> <span> Dr. Lucy D'Agostino McGowan </span> </div> --- class: middle $$$$  --- class: middle `$$\Huge\mathbf{y} = \mathbf{X}\beta+\epsilon$$`  --- class: middle `$$\Huge\mathbf{y} = \mathbf{X}\beta+\epsilon$$`  --- class: middle `$$\Huge E[\hat\beta]=\beta$$`  --- class: middle `$$\Huge\textrm{var}(\hat\beta)=\hat\sigma^2(\mathbf{X}^T\mathbf{X})^{-1}$$`  --- class: middle `$$\Huge E[\hat\beta-\beta] = 0$$`  --- class: middle `$$\Huge\textrm{var}(\hat\beta)=\hat\sigma^2(\mathbf{X}^T\mathbf{X})^{-1}$$`  --- class: middle `$$\Large\textrm{var}(\hat\beta)=\hat\sigma^2(\mathbf{X}^T\mathbf{X})^{-1}\leq \textrm{var}(\tilde\beta)$$`  --- class: middle `$$\Huge E[\tilde\beta-\beta]=0$$`  --- class: middle `$$\Large\textrm{var}(\hat\beta)=\hat\sigma^2(\mathbf{X}^T\mathbf{X})^{-1}\leq \textrm{var}(\tilde\beta)$$`  --- class: middle `$$\Huge\textrm{RSS}=\sum e^2$$`  --- class: middle `$$\Huge\textrm{RSS}=\sum e^2$$`  --- class: middle `$$\Huge E[\mathbf{e}^T\mathbf{e}] = \sigma^2(n-(p+1))$$`  --- class: middle `$$\Large E[\mathbf{e}^T\mathbf{e}]=E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]$$`  --- class: middle `$$\Huge\hat\sigma^2=\frac{\textrm{RSS}}{(n-(p+1))}$$`  --- class: middle `$$\Huge\hat\sigma^2=\frac{\textrm{RSS}}{(n-(p+1))}$$`  --- class: middle `$$\Huge\textrm{var}(\hat\beta) = \hat\sigma^2(\mathbf{X}^T\mathbf{X})^{-1}$$`  --- class: middle `$$F = \frac{RSS_{small} - RSS_{larger} / (df_{small}- df_{larger})}{RSS_{larger}/df_{larger}}\sim F_{df_{small}- df_{larger}, df_{larger}}$$`  --- class: middle `$$F = \frac{RSS_{small} - RSS_{larger} / (df_{small}- df_{larger})}{RSS_{larger}/df_{larger}}\sim F_{df_{small}- df_{larger}, df_{larger}}$$`  --- class: middle `$$F = \frac{RSS_{small} - RSS_{larger} / (df_{small}- df_{larger})}{RSS_{larger}/df_{larger}}\sim F_{df_{small}- df_{larger}, df_{larger}}$$`  --- class: middle `$$F = \frac{RSS_{small} - RSS_{larger} / (df_{small}- df_{larger})}{RSS_{larger}/df_{larger}}\sim F_{df_{small}- df_{larger}, df_{larger}}$$`  --- class: middle `$$\Huge\hat\beta\pm t^*SE_{\hat\beta}$$`