National open university of nigeria introduction to econometrics II eco 356

INTRODUCTION TO ECONOMETRICS II

ECO 306

NOUN
113 where
(
)
(X )
(
)
i
i
X
X
f
Var X


. Now, if X and uare independently distributed,
[ (
)(
)]
i
i
E f X
u
u

may be decomposed as the product of
[ (x )]
i
E f
and
[(
)]
i
E u
u

. Hence
[ (X )(
)]
i
i
E f
u
u

=
[ XX 0
i
i
i
E f
E u
u
E f




…[4.04] since by assumption )
i
E u
is 0 in each observation. This implies,of course, that )
E u
is also 0. Hence, when we take the expectation of

∑
(


)(
̅), each term within the summation has expected value 0. Thus the error term as a whole has expected value 0 and b
2 is an unbiased estimator of

4.1.3.3 Consistency
Generally stated,
( ) is equal to ( ) ( ) where A and B are any two stochastic quantities, on condition that both
( ) and ( ) exist and that
( ) is nonzero ( is the limiting value as the sample size becomes large. As also stated, sample expressions tend to their population counterparts as the sample size becomes large, so
( ) is the population covariance of X and u and
( ) is
, the population variance of X. If X and u are independent, the population covariance of X and u is 0 and we can write that


( )
( )






…[4.05]

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