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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.
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We have
β 1
Cov(y, x 1
V ar(x 1
Cov(β 0
x 1
x 2
V ar(x 1
Cov(β 0
, x 1
V ar(x 1 )
Cov(x 1
, x 1
V ar(x 1 )
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
Cov(x 2 , x 1 )
V ar(x 1
= β 1
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
β 1
) = β 1
δ 1
