简单线性回归模型
OLS的基本思想
Ordinary Least Squares:
\[ \hat{Y}_{i}=\hat{\beta}_{1}+\hat{\beta}_{2} X_{i} \]
\[ \min \sum e_{i}^{2}=\min \sum\left(Y_{i}-\hat{\beta}_{1}-\hat{\beta}_{2} X_{i}\right)^{2} \]
正规方程和估计量
取偏导数并令其为0，可得正规方程：
\(\frac{\partial\left(\sum e_{i}^{2}\right)}{\partial \hat{\beta}_{1}}=-2 \sum\left(Y_{i}-\hat{\beta}_{1}-\hat{\beta}_{2} X_{i}\right)=0\)
\(\frac{\partial\left(\sum e_{i}^{2}\right)}{\partial \hat{\beta}_{2}}=-2 \sum\left(Y_{i}-\hat{\beta}_{1}-\hat{\beta}_{2} X_{i}\right) X_{i}=0\)
So that:
\(\sum Y_{i}=n \hat{\beta}_{1}+\hat{\beta}_{2} \sum X_{i}\)
\(\sum X_{i} Y_{i}=\hat{\beta}_{1} \sum X_{i}+\hat{\beta}_{2} \sum X_{i}^{2}\)