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Derivation of Equation (1) for Beta Estimate, Study Guides, Projects, Research of Introduction to Econometrics

The derivation of equation (1) for the beta estimate in a multiple linear regression model. The equation is derived from the definition of beta as the covariance of the dependent variable with the independent variable, divided by the variance of the independent variable. The document also explains the expected value of the beta estimate, given the values of the independent variables.

Typology: Study Guides, Projects, Research

2017/2018

Uploaded on 10/30/2018

michaellamlam
michaellamlam 🇬🇧

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Derivation of Equation on page 24 of Lecure 2
We have
˜
β1=Cov(y, x1)
V ar(x1)=Cov(β0+β1x1+β2x2+u, x1)
V ar(x1)
=Cov(β0, x1)
V ar(x1)+β1
Cov(x1, x1)
V ar(x1)+β2
Cov(x2, x1)
V ar(x1)+Cov(u, x1)
V ar(x1)
= 0 + β1
V ar(x1)
V ar(x1)+β2
Cov(x2, x1)
V ar(x1)+ 0,
=β1+β2˜
δ1,(1)
where V ar(·) and Cov(·) denote variance and covariance, respectively.
Taking the expectation conditional on the value of the independent variables (so that ˜
δ1
is not random here), we have
E(˜
β1) = β1+β2˜
δ1.(2)
1

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Derivation of Equation on page 24 of Lecure 2

We have

β 1

Cov(y, x 1

V ar(x 1

Cov(β 0

  • β 1

x 1

  • β 2

x 2

  • u, x 1

V ar(x 1

Cov(β 0

, x 1

V ar(x 1 )

  • β 1

Cov(x 1

, x 1

V ar(x 1 )

  • β 2

Cov(x 2

, x 1

V ar(x 1 )

Cov(u, x 1

V ar(x 1 )

= 0 + β 1

V ar(x 1 )

V ar(x 1

  • β 2

Cov(x 2 , x 1 )

V ar(x 1

= β 1

  • β 2

δ 1

where V ar(·) and Cov(·) denote variance and covariance, respectively.

Taking the expectation conditional on the value of the independent variables (so that

δ 1

is not random here), we have

E(

β 1

) = β 1

  • β 2

δ 1