Consider the simple linear regression: Yi = β0 + β1Xi + ui whereYi and Xi...

Consider the simple linear regression: Yi = β0 + β1Xi + ui where Yi and Xi are random variables, β0 and β1 are population intercept and slope parameters, respectively, ui is the error term. Suppose the estimated regression equation is given by: Yˆ i = βˆ 0 + βˆ 1Xi where βˆ 0 and βˆ 1 are OLS estimates for β0 and β1. Define residuals ˆui as: uˆi = Yi − Yˆ i Show that: (a) (2 pts.) Pn i=1 uˆ 2 i = 0 (b) (2 pts.) Pn i=1 uˆiXi = 0

Expert Solution

FIRST WE WROTE THE GIVEN EQUATIONS

THEN WROTE RESIDUAL AS GIVEN

AND THEN SQUARED IT AND TOOK A SUMMATION ON BOTH SIDES

AFTER THAT WE DIFFERENTIATED IT WITH RESPECT TO 0 AND 1

AND HENCE PROVED THE REQUIRED

Rahul Sunny answered 2 years ago

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