Introduction to econometrics II eco 356 faculty of social sciences course guide course Developers: Dr. Adesina-Uthman

INTRODUCTION TO ECONOMETRICS II

ECO 306

NOUN
62 In addition to the Gauss–Markov conditions, one usually assumes that the disturbance term is normally distributed. The reason is that if u is normally distributed, so will be the regression coefficients, and this is useful when performing tests of hypotheses and constructing confidence intervals for and using the regression results. The justification for the assumption depends on the Central Limit Theorem that, if a random variable is the composite result of the effects of a large number of other random variables, it will have an approximately normal distribution even if its components do not, provided that none of them is dominant. The disturbance term u is composed of a number of factors not appearing explicitly in the regression equation so, even if we know nothing about the distribution of these factors (or even their identity, we are entitled to assume that they are normally distributed.
2.2.3.4 Unbiasedness of the Regression Coefficients
We can show that b
2
must bean unbiased estimator of if the fourth Gauss–Markov condition is satisfied
(
) 0
( )
( )
1 0
( )
( )
1
…[2.31] since is a constant. If we adopt the strong version of the fourth Gauss–Markov condition and assume that X is nonrandom, we may also take Var(X) as a given constant, and so
(
)
( )
, ( )-
…[2.32] To demonstrate that
, ( )- :
, ( )- 0
∑
(
̅)(
̅)
1
∑
,(
̅)(
̅)-
∑
(
̅) ,(
̅)
-
…[2.33]

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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