Unbiasedness of the Regression Coefficients

INTRODUCTION TO ECONOMETRICS II

ECO 306

NOUN
63 In the second line, the second expected value rule has been used to bring
( ) out of the expression as a common factor, and the first rule has been used to breakup the expectation of the sum into the sum of the expectations. In the third line, the term involving has been brought out because X is non-stochastic. By virtue of the first
Gauss–Markov condition,
( is , and hence ( ) is also 0. Therefore
, ( )- is 0 and
(
)



…[2.34] In other words, b
2
is an unbiased estimator of

. We can obtain the same result with the weak version of the fourth Gauss–Markov condition (allowing X to have a random component but assuming that it is distributed independently of u), unless the random factor in the nobservations happens to cancel out exactly, which can happen only by coincidence. b
2
will be different from
for any given sample, but in view of unbiased regression coefficient, there will be no systematic tendency for it to be either higher or lower. The same is true for the regression coefficient b
1
Using [2.22]

̅
̅


…[2.35] Hence
(
) ( ̅) ̅ (
)

…[2.36] Since
is determined by




We have
(
)


(
)



…[2.37] because ( is 0 if the first Gauss–Markov condition is satisfied. Hence
( ̅)

̅

…[2.38] Substituting this into [2.36], and using the result that
(
)

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