简单线性回归模型
OLS的基本思想
Ordinary Least Squares:
[hat{Y}_{i}=hat{eta}_{1}+hat{eta}_{2} X_{i}
]
[min sum e_{i}^{2}=min sumleft(Y_{i}-hat{eta}_{1}-hat{eta}_{2} X_{i}
ight)^{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{partialleft(sum e_{i}^{2}
ight)}{partial hat{eta}_{1}}=-2 sumleft(Y_{i}-hat{eta}_{1}-hat{eta}_{2} X_{i}
ight)=0)
(frac{partialleft(sum e_{i}^{2}
ight)}{partial hat{eta}_{2}}=-2 sumleft(Y_{i}-hat{eta}_{1}-hat{eta}_{2} X_{i}
ight) X_{i}=0)
So that:
(sum Y_{i}=n hat{eta}_{1}+hat{eta}_{2} sum X_{i})
(sum X_{i} Y_{i}=hat{eta}_{1} sum X_{i}+hat{eta}_{2} sum X_{i}^{2})