第5講 回帰分析

$$\begin{align} V[\hat{\beta}_1] &= V\left[\sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{S_{xx}}\right)y_i\right] \\ &= \sum_{i=1}^{n}V\left[\left(\frac{x_i - \bar{x}}{S_{xx}}\right)y_i\right] \\ &= \sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{S_{xx}}\right)^2 V\left[y_i\right] \\ &= \sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{S_{xx}}\right)^2 V\left[\beta_0 + \beta_1 x_i + \epsilon_i\right] \\ &= \sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{S_{xx}}\right)^2 V\left[\epsilon_i\right] \\ &= \sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{S_{xx}}\right)^2 \sigma^2 \\ &= \frac{\sigma^2}{S_{xx}^2}\sum_{i=1}^{n}(x_i - \bar{x})^2 \\ &= \frac{\sigma^2}{S_{xx}^2}S_{xx} \\ &= \frac{\sigma^2}{S_{xx}} \end{align}$$

$$\begin{align} V[\hat{\beta}_0] &= V\left[\sum_{i=1}^{n}\left(\frac{1}{n} - \bar{x}\frac{x_i - \bar{x}}{S_{xx}}\right)y_i\right] \\ &= \sum_{i=1}^{n}V\left[\left(\frac{1}{n} - \bar{x}\frac{x_i - \bar{x}}{S_{xx}}\right)y_i\right] \\ &= \sum_{i=1}^{n}\left(\frac{1}{n} - \bar{x}\frac{x_i - \bar{x}}{S_{xx}}\right)^2 V\left[y_i\right] \\ &= \sigma^2\sum_{i=1}^{n}\left(\frac{1}{n} - \bar{x}\frac{x_i - \bar{x}}{S_{xx}}\right)^2 \\ &= \sigma^2\sum_{i=1}^{n}\left(\frac{1}{n^2} - 2 \frac{1}{n}\bar{x}\frac{x_i - \bar{x}}{S_{xx}} + \bar{x}^2\frac{(x_i - \bar{x})^2}{{S_{xx}}^2}\right) \\ &= \sigma^2\left(\sum_{i=1}^{n}\frac{1}{n^2} - \frac{2 \bar{x}}{n S_{xx}}\sum_{i=1}^{n}(x_i - \bar{x}) + \frac{\bar{x}^2}{{S_{xx}}^2}\sum_{i=1}^{n}(x_i - \bar{x})^2\right) \\ &= \sigma^2\left(\frac{1}{n} + \frac{\bar{x}^2}{S_{xx}}\right) \end{align}$$

$$\begin{align} Cov[\hat{\beta}_0,\hat{\beta}_1] &= Cov\left[\sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{S_{xx}}\right)y_i,~\sum_{i=1}^{n}\left(\frac{1}{n} - \bar{x}\frac{x_i - \bar{x}}{S_{xx}}\right)y_i\right] \\ &= \sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{S_{xx}}\right)\left(\frac{1}{n} - \bar{x}\frac{x_i - \bar{x}}{S_{xx}}\right)V[y_i]~~~~(\because Cov[y_i,y_i] = V[y_i],~Cov[y_i,y_j] = 0) \\ &= \sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{nS_{xx}} - \frac{\bar{x}(x_i - \bar{x})^2}{{S_{xx}}^2}\right)\sigma^2 \\ &= \sum_{i=1}^{n}\left(- \frac{\bar{x}(x_i - \bar{x})^2}{{S_{xx}}^2}\right)\sigma^2 \\ &= - \frac{\bar{x}\sigma^2}{{S_{xx}}^2}\sum_{i=1}^{n}(x_i - \bar{x})^2 \\ &= - \frac{\bar{x}\sigma^2}{{S_{xx}}^2}S_{xx} \\ &= - \frac{\bar{x}\sigma^2}{S_{xx}} \end{align}$$