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Regression Analysis.tex
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\documentclass[10pt]{article} \usepackage{amssymb,amsmath} \usepackage[hmargin=1cm,vmargin=1cm]{geometry} \usepackage{cancel} \begin{document} {\large Regression Analysis} \begin{align*} \text{\bf The }&\text{\bf Basics}\\ &\mathbf{Y}=\beta_0+\beta_1 x+\epsilon, \qquad\hat{\beta_1}=\frac{S_{XY}}{S_{XX}}, \qquad\hat{\beta_0} =\overline{\mathbf{Y}}-\hat{\beta_1}\overline{\mathbf{X}} =\overline{\mathbf{Y}}-\frac{S_{XY}}{S_{XX}}\overline{\mathbf{X}},\\ &\quad\text{where } S_{XX}=\sum_i(\mathbf{X}_i-\overline{\mathbf{X}})^2 \quad\text{and}\quad S_{XY}=\sum_i(\mathbf{X}_i-\overline{\mathbf{X}})(\mathbf{Y}_i-\overline{\mathbf{Y}}).\\ % &\text{Fixed Design: If $x_i$ is fixed, $\sum_i(x_i-\bar{x})=0$ and therefore, } S_{XX}=\sum_i(\mathbf{X}_i-\overline{\mathbf{X}})^2=\sum_i(x_i-\bar{x})^2=S_{xx},\\ &\text{and } S_{XY}=\sum_i(\mathbf{X}_i-\overline{\mathbf{X}})(\mathbf{Y}_i-\overline{\mathbf{Y}}) =\sum_i(x_i-\bar{x})(\mathbf{Y}_i-\overline{\mathbf{Y}}) =\sum_i(x_i-\bar{x})\mathbf{Y}_i-\cancel{\overline{\mathbf{Y}}\sum_i(x_i-\bar{x})} =\sum_i(x_i-\bar{x})\mathbf{Y}_i.\\ \\ \text{\bf The }&\text{\bf Gradient}\\ &\mathbf{E}(\hat{\beta_1}) =\mathbf{E}\left(\frac{S_{XY}}{S_{XX}}\right) =\frac{\mathbf{E}(S_{XY})}{S_{xx}} =\frac{\mathbf{E}(\sum_i(x_i-\bar{x})\mathbf{Y}_i)}{S_{xx}} =\frac{\sum_i(x_i-\bar{x})\mathbf{E}(\mathbf{Y}_i)}{S_{xx}} =\frac{\sum_i(x_i-\bar{x})\mathbf{E}(\beta_0+\beta_1 x_i)}{S_{xx}}\\ &\quad =\cancel{\frac{\beta_0}{S_{xx}}\sum_i(x_i-\bar{x})}+\frac{\beta_1}{S_{xx}}\sum_i x_i(x_i-\bar{x}) =\frac{\beta_1}{S_{xx}}\left(\sum_i x_i(x_i-\bar{x})-\sum_i \bar{x}(x_i-\bar{x})\right) =\frac{\beta_1}{\cancel{S_{xx}}}\cancel{\sum_i (x_i-\bar{x})^2} =\beta_1.\\ % &\mathbf{Var}(\hat{\beta_1}) =\mathbf{Var}\left(\frac{S_{XY}}{S_{XX}}\right) %=\frac{\mathbf{Var}(S_{XY})}{S_{xx}^2} =\mathbf{Var}\left(\frac{\sum_i(x_i-\bar{x})\mathbf{Y}_i}{S_{xx}}\right) =\mathbf{Var}\left(\sum_i\frac{x_i-\bar{x}}{S_{xx}}\mathbf{Y}_i\right) =\sum_i\left(\frac{x_i-\bar{x}}{S_{xx}}\right)^2\mathbf{Var}(\mathbf{Y}_i)\\ &\quad =\frac{\mathbf{Var}(\mathbf{Y}_i)}{S_{xx}^2}\sum_i(x_i-\bar{x})^2 =\frac{\sigma^2}{S_{xx}^2}S_{xx} =\frac{\sigma^2}{S_{xx}}.\\ \\ &\boxed{\hat{\beta_1}\sim\mathbf{N}\left(\beta_1,\frac{\sigma}{\sqrt{S_{xx}}}\right).} \qquad\sqrt{S_{xx}}\cdot\frac{\hat{\beta_1}-\beta_1}{\sigma}\sim\mathbf{N}\left(0,1\right), \quad\sqrt{S_{xx}}\cdot\frac{\hat{\beta_1}-\beta_1}{S}\sim t_{n-2}.\\ \\ \text{\bf The }&\text{\bf Intercept}\\ &\mathbf{E}(\hat{\beta_0}) =\mathbf{E}\left(\overline{\mathbf{Y}}-\hat{\beta_1}\overline{\mathbf{X}}\right) =\mathbf{E}\left(\overline{\mathbf{Y}}\right)-\mathbf{E}\left(\hat{\beta_1}\overline{\mathbf{X}}\right) =\mathbf{E}\left(\frac{\sum_i{\mathbf{Y_i}}}{n}\right)-\mathbf{E}\left(\hat{\beta_1}\bar{x}\right) =\mathbf{E}\left(\sum_i\frac{\beta_0+\beta_1 x_i}{n}\right)-\bar{x}\mathbf{E}\left(\hat{\beta_1}\right)\\ &\quad =\frac{n\beta_0}{n}+\beta_1\sum_i\frac{x_i}{n}-\bar{x}\beta_1 =\beta_0+\beta_1\bar{x}-\beta_1\bar{x} =\beta_0.\\ % &\mathbf{Var}(\hat{\beta_0}) =\mathbf{Var}\left(\overline{\mathbf{Y}}-\hat{\beta_1}\overline{\mathbf{X}}\right) =\mathbf{Var}\left(\frac{\sum_i{\mathbf{Y_i}}}{n}\right)-\bar{x}^2\mathbf{Var}\left(\hat{\beta_1}\right) =\frac{\sum_i{\mathbf{\mathbf{Var}(Y_i})}}{n^2}-\bar{x}^2\frac{\sigma^2}{S_{xx}}\\ &\quad =\frac{n\sigma^2}{n^2}-\bar{x}^2\frac{\sigma^2}{S_{xx}} =\sigma^2\left(\frac{1}{n}-\frac{\bar{x}^2}{S_{xx}}\right).\\ \\ &\boxed{\hat{\beta_0}\sim\mathbf{N}\left(\beta_0,\sigma\sqrt{\frac{1}{n}-\frac{\bar{x}^2}{S_{xx}}}\right).} \qquad\frac{\hat{\beta_0}-\beta_0}{\sigma\sqrt{\frac{1}{n}-\frac{\bar{x}^2}{S_{xx}}}}\sim\mathbf{N}\left(0,1\right), \quad\frac{\hat{\beta_0}-\beta_0}{S\sqrt{\frac{1}{n}-\frac{\bar{x}^2}{S_{xx}}}}\sim t_{n-2}.\\ \\ \text{\bf Test }&\text{\bf Statistic}\\ &t_0=\sqrt{S_{xx}}\cdot\frac{\hat{b_1}}{S},\quad\text{where $\hat{b_1}$ is an estimate of $\hat{\beta_1}$.}\\ &\text{To test the null hypotheses }H_0:\beta_2=0\text{ against the alternative hypothese }H_a:\beta_2\le 0,\\ &\text{$H_0$ is to be rejected if }\hat{b_1}\notin\left[-t\frac{S}{\sqrt{S_{xx}}},~~t\frac{S}{\sqrt{S_{xx}}}\right], \quad\text{where }t=t_{n-1,1-\tfrac{\alpha}{2}}. \quad\text{i.e. }\left|\hat{b_1}\right|>t\frac{S}{\sqrt{S_{xx}}}=\left|\frac{\hat{b_1}}{t_0}\right|.\\ &\boxed{\text{Reject $H_0$ if $|t_0|>t_{n-1,1-\tfrac{\alpha}{2}}$.}} \quad\text{i.e. }t_{n-1,1-\tfrac{\alpha}{2}}\in[-|t_0|,|t_0|]. \quad\text{Let }T\sim t_{n-1,1-\tfrac{\alpha}{2}}.\\ &\text{The $p$-value (probability of being wrong) }p=1-\mathbf{P}\left(T\in[-|t_0|,|t_0|]\right)=2\times\mathbf{P}(T>|t_0|).\\ \end{align*} \end{document}