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Observe that the histogram for 1 is centered about 1

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Unformatted text preview: lationship between y and x is U-shaped 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 U-shaped 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 bell-shaped 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 U-shaped 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 bell-shaped 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 U-shaped 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 bell-shaped 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: Non-Normal 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: Non-Normal 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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