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Unformatted text preview: lationship between y and x is Ushaped while the
relationship between y and x 2 is linear: in the do …le we con…rm this
graphicaly. Melissa Tartari (Yale) Econometrics 24 / 27 Part I: Normal Disturbances II
Observe that the relationship between y and x is Ushaped while the
relationship between y and x 2 is linear: in the do …le we con…rm this
graphicaly.
ˆ
Observe that the histogram for β1 is centered about 1 (as it should by
unbiasedness) and has a bellshaped symmetrical form, also 95% of
the probability mass is within 2 standard deviations of the mean: all
of these are typical features of a normal distribution (the red line
represents the best …tting N and helps us make comparisons). Melissa Tartari (Yale) Econometrics 24 / 27 Part I: Normal Disturbances II
Observe that the relationship between y and x is Ushaped while the
relationship between y and x 2 is linear: in the do …le we con…rm this
graphicaly.
ˆ
Observe that the histogram for β1 is centered about 1 (as it should by
unbiasedness) and has a bellshaped symmetrical form, also 95% of
the probability mass is within 2 standard deviations of the mean: all
of these are typical features of a normal distribution (the red line
represents the best …tting N and helps us make comparisons).
This result does not surprise us since under LR.1 through LR.6 the
OLS estimator has an exact normal distribution and the histogram is
an estimate of that distribution. Melissa Tartari (Yale) Econometrics 24 / 27 Part I: Normal Disturbances II
Observe that the relationship between y and x is Ushaped while the
relationship between y and x 2 is linear: in the do …le we con…rm this
graphicaly.
ˆ
Observe that the histogram for β1 is centered about 1 (as it should by
unbiasedness) and has a bellshaped symmetrical form, also 95% of
the probability mass is within 2 standard deviations of the mean: all
of these are typical features of a normal distribution (the red line
represents the best …tting N and helps us make comparisons).
This result does not surprise us since under LR.1 through LR.6 the
OLS estimator has an exact normal distribution and the histogram is
an estimate of that distribution.
By increasing the size of each sample (namely M ) you see that the
distribution becomes more and more concentrated (as it should, by
consistency of the OLS estimator).
Melissa Tartari (Yale) Econometrics 24 / 27 Part II: NonNormal Disturbances I Now we consider a di¤erent distributional assumption for u , namely
we assume
u Uniform [ 1, 1]
where the choice of the support is meant to preserve comparability
with the previous example in terms of the …rst two moments (indeed
you can verify that E [u ] = 0 and Var [u ] ' 0.66). Melissa Tartari (Yale) Econometrics 25 / 27 Part II: NonNormal Disturbances I Now we consider a di¤erent distributional assumption for u , namely
we assume
u Uniform [ 1, 1]
where the choice of the support is meant to preserve comparability
with the previous example in terms of the …rst two moments (indeed
you can verify that E [u ] = 0 and Var [u ] ' 0.66). Once again, In STATA I draw N samples from the population, for
each of them I compute the OLS estimates of the "unknown&quo...
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 Fall '10
 DonaldBrown
 Econometrics

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