INTRODUCTION TO ECONOMETRICS II
ECO 306
NOUN
64
(
) (
̅ ) ̅
…[2.39- Thus b
1
is an unbiased estimator of provided that the Gauss–Markov conditions 1 and 4 are satisfied. Of course in any given sample the random factor will cause b
1
to differ from
2.2.3.5 Precision of the Regression Coefficients
Now we shall consider and
, the population variances of b
1
and b
2 about their population means. The following expressions give these
,
̅
( )
] and
( )
…[2.40] Equation [2.40] has three obvious implications. First, the variances of both b
1
and b
2
are directly inversely proportional to the number of observations in the sample. This makes good sense. The more information you have, the more accurate your estimates are likely to be. Second, the variances are proportional to the variance of the disturbance term. The bigger the the variance of the random factor in the relationship, the worse the estimates of the parameters are likely to be. Third, the variance of the regression coefficients is inversely related to the variance of
X. What is the reason for this Remember that (1) the regression coefficients are calculated on the assumption that the observed variations in Y are due to variations in
X, but (2) they are in reality partly due to variations in X and partly to variations in u. The smaller the variance of X, the greater is likely to be the relative influence of the random factor in determining the variations in Yand the more likely is regression analysis give inaccurate estimates.
2.2.3.6 Testing Hypotheses Relating to the Regression Coefficients
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