Show that, for the simple linear regression model, the

Chapter , Problem 96E

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Show that, for the simple linear regression model, the following statements are true:

a. $\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)=0$

b. $\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right) x_{i}=0$\

c. $\frac{1}{n} \sum_{i=1}^{n} \hat{y}_{i}=\bar{y}$

Equation Transcription:

$\sum\limits_{i=1}^{n}({y}_{i}-{\widehat{y}}_{i})=0$

$\sum\limits_{i=1}^{n}({y}_{i}-{\widehat{y}}_{i}){x}_{i}=0$

$\frac{1}{n}\sum\limits_{i=1}^{n}{\widehat{y}}_{i}=\overline{y}$

Text Transcription:

sum_i=1^n (y_i-\hat y_i)=0

sum_i=1^n (y_i-\haty_i) x_i=0

\frac1 n \sum_i=1^n \hat y_i=\bary

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