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# Example we have shown that bias y1 0 however var y1

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Unformatted text preview: estimators are not necessarily consistent. EXAMPLE: we have shown that Bias [Y1 ] = 0; however Var [Y1 ] = σ2 and so Var [Y1 ] 6= 0 and ? n. This estimator uses only the …rst observation. Y1 is an unbiased but inconsistent estimator of µ. Unbiased estimators whose variance ! 0 as n ! ∞ are consistent. Unbiased estimators that use the entire sample will usually satisfy this requirement. Melissa Tartari (Yale) Econometrics 8 / 27 Consistency: A Review IV Unbiased estimators are not necessarily consistent. EXAMPLE: we have shown that Bias [Y1 ] = 0; however Var [Y1 ] = σ2 and so Var [Y1 ] 6= 0 and ? n. This estimator uses only the …rst observation. Y1 is an unbiased but inconsistent estimator of µ. Unbiased estimators whose variance ! 0 as n ! ∞ are consistent. Unbiased estimators that use the entire sample will usually satisfy this requirement. ¯ EXAMPLE: we have shown that Bias [Yn ] = 0; since 2 ¯ ¯ Var [Yn ] = σ ! 0 as n ! ∞ then we conclude that Yn is n consistent for µ E [Yi ]. Melissa Tartari (Yale) Econometrics 8 / 27 Consistency of the OLS Estimator Can we gain an intuitive understanding behind consistency of OLS estimators? Melissa Tartari (Yale) Econometrics 9 / 27 Consistency of the OLS Estimator Can we gain an intuitive understanding behind consistency of OLS estimators? ˆ Let βj be the OLS estimator for βj . For each sample size n, we saw ˆ that β has a certain pdf, denote this distribution f n for clarity ˆ βj j (notice the indexing to the sample size ). Melissa Tartari (Yale) Econometrics 9 / 27 Consistency of the OLS Estimator Can we gain an intuitive understanding behind consistency of OLS estimators? ˆ Let βj be the OLS estimator for βj . For each sample size n, we saw ˆ that β has a certain pdf, denote this distribution f n for clarity ˆ βj j (notice the indexing to the sample size ). ˆ Under LR.1 to LR.4 βj is unbiased, then it must be that the mean of n is the true β . fβ ˆ j j Melissa Tartari (Yale) Econometrics 9 / 27 Consistency of the OLS Estimator Can we gain an intuitive understanding behind consistency of OLS estimators? ˆ Let βj be the OLS estimator for βj . For each sample size n, we saw ˆ that β has a certain pdf, denote this distribution f n for clarity ˆ βj j (notice the indexing to the sample size ). ˆ Under LR.1 to LR.4 βj is unbiased, then it must be that the mean of n is the true β . fβ ˆ j j n ˆ If βj is also consistent, then fβ becomes more and more tightly ˆ j n distributed around βj as n grows; as n ! ∞, fβ collapses to the ˆ j single point βj . Melissa Tartari (Yale) Econometrics 9 / 27 Consistency of the OLS Estimator Can we gain an intuitive understanding behind consistency of OLS estimators? ˆ Let βj be the OLS estimator for βj . For each sample size n, we saw ˆ that β has a certain pdf, denote this distribution f n for clarity ˆ βj j (notice the indexing to the sample size ). ˆ Under LR.1 to LR.4 βj is unbiased, then it must be that the mean...
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