In a simple linear regression model, B1 has important relationships with a samplecorrelation coefficient r. Prove the following two identities.Tyn – 2SyySIT(2)se(3)(1) B

Answered: In a simple linear regression model, B1…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

In a simple linear regression model, B1 has important relationships with a sample
correlation coefficient r. Prove the following two identities.
Tyn – 2
Syy
SIT
(2)
se(3)
(1) B

Expert Solution

Step 1

Consider, the regression equation is,

$Y = β_{0} + β_{1} X + ε$

By minimizing the least squares equation,

${\hat{β}}_{1} = \frac{\sum (X - \bar{X}) (Y - \bar{Y})}{\sum {(X - \bar{X})}^{2}}$

Formula for correlation between X and Y is,

$r = \frac{\sum (X - \bar{X}) (Y - \bar{Y})}{\sqrt{\sum {(X - \bar{X})}^{2} \sum {(Y - \bar{Y})}^{2}}}$

Variance of X and variance of Y are,

$S_{x x} = \sum {(X - \bar{X})}^{2} S_{y y} = \sum {(Y - \bar{Y})}^{2}$

Consider,

$r \sqrt{\frac{S_{y y}}{S_{x x}}} = \frac{\sum (X - \bar{X}) (Y - \bar{Y})}{\sqrt{\sum {(X - \bar{X})}^{2} \sum {(Y - \bar{Y})}^{2}}} * \sqrt{\frac{\sum {(Y - \bar{Y})}^{2}}{\sum {(X - \bar{X})}^{2}}} = \frac{\sum (X - \bar{X}) (Y - \bar{Y})}{\sqrt{\sum {(X - \bar{X})}^{2}} \sqrt{\sum {(Y - \bar{Y})}^{2}}} * \frac{\sqrt{\sum {(Y - \bar{Y})}^{2}}}{\sqrt{\sum {(X - \bar{X})}^{2}}} = \frac{\sum (X - \bar{X}) (Y - \bar{Y})}{\sum {(X - \bar{X})}^{2}} = {\hat{β}}_{1}$