SSR is defined by SSR = E ( - 9). Prove(1) SSR = 3 S.(2) E(SSR) = o + B} S.(3) ShowY under Ho : B1 = 0.

Given, SSR=∑i=1ny^i-y¯2. The fitted equation of yi=β0+β1xi+εi is y^i=β^0+β^1xi, for i=1,2,...,n. yi…

Answered: SSR is defined by SSR = E9 - 9). Prove…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.3: The Gram-schmidt Process And The Qr Factorization

Problem 11AEXP

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Question

SSR is defined by SSR = E ( - 9). Prove
(1) SSR = 3 S.
(2) E(SSR) = o + B} S.
(3) Show
Y under Ho : B1 = 0.

Expert Solution

Step 1

Given, $S S R = \sum_{i = 1}^{n} {({\hat{y}}_{i} - \bar{y})}^{2}$ .

The fitted equation of $y_{i} = β_{0} + β_{1} x_{i} + ε_{i}$ is ${\hat{y}}_{i} = {\hat{β}}_{0} + {\hat{β}}_{1} x_{i}$ , for i=1,2,...,n.

$y_{i} i s t h e r e s p o n s e v a r i a b l e β_{0} i s t h e i n t e r c e p t β_{1} i s t h e s l o p e x_{i} i s t h e r e g r e s s o r v a r i a b l e ε_{i} i s t h e r a n d o m e r r o r$

${\hat{y}}_{i} i s t h e f i t t e d o r p r e d i c t e d r e s p o n s e v a r i a b l e {\hat{β}}_{0} i s t h e l e a s t s q u a r e e s t i m a t e o f β_{0} {\hat{β}}_{1} i s t h e l e a s t s q u a r e e s t i m a t e o f β_{1}$

The normal equations are:

${\hat{β}}_{0} = \bar{y} - {\hat{β}}_{1} \bar{x} {\hat{β}}_{1} = \frac{\sum_{i = 1}^{n} (y_{i} - \bar{y}) (x_{i} - \bar{x})}{\sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}} = \frac{S_{x y}}{S_{x x}}$

$\bar{y}$ is the mean or average, $\bar{y} = \frac{\sum_{i = 1}^{n} (y_{i})}{n}$ , which implies:

$\begin{array}{rcl} \bar{y} & = & \frac{\sum_{i = 1}^{n} (β_{0} + β_{1} x_{i} + ε_{i})}{n} \\ = & \frac{\sum_{i = 1}^{n} (β_{0})}{n} + \frac{\sum_{i = 1}^{n} (β_{1} x_{i})}{n} + \frac{\sum_{i = 1}^{n} (ε_{i})}{n} \\ = & \frac{n β_{0}}{n} + β_{1} \frac{\sum_{i = 1}^{n} (x_{i})}{n} + \frac{\sum_{i = 1}^{n} (ε_{i})}{n} \\ = & β_{0} + β_{1} \bar{x} + \bar{ε} \end{array}$