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Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: exercise r.22 Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (review chapter). [Teaching Resource] © 2012 The Author This version available at: http://learningresources.lse.ac.uk/141/http://learningresources.lse.ac.uk/141/ Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms. http://creativecommons.org/licenses/by-sa/3.0/ http://creativecommons.org/licenses/by-sa/3.0/ http://learningresources.lse.ac.uk/

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EXERCISE R.22 R.22A random variable X has unknown population mean X and population variance X. A sample of n observations {X 1,..., X n } is generated. Show that is an unbiased estimator of X. Show that the variance of Z does not tend to zero as n tends to infinity and that therefore Z is an inconsistent estimator, despite being unbiased. 1 2

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EXERCISE R.22 2 To demonstrate unbiasedness, we start by using the first expected value rule to decompose the expression and then the second to take the factors of 2 out of each expectation.

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EXERCISE R.22 3 Each expectation is equal to X. (We are thinking about the sample at the planning stage, before the sample is actually generated.) The coefficients of X add to 1, so the estimator is unbiased.

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EXERCISE R.22 4 Now consider the variance. Using variance rule 1, we can decompose it into the sum of the variances plus twice the covariance of each observation with every other observation.

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EXERCISE R.22 5 We are assuming that the observations are generated independently and hence that the covariances are all zero.

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EXERCISE R.22 6 We use variance rule 2 to take the coefficients out of the variances. Remember that we have to square the coefficients when we do this.

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EXERCISE R.22 7 Each variance is equal to X. (Again, we are thinking about the sample at the planning stage, before the sample is actually generated.) 2

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EXERCISE R.22 8 Hence we obtain an expression for the variance. It decreases as n increases.

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EXERCISE R.22 9 But it does not tend to zero. The distribution of the estimator does not collapse to a spike and so it is inconsistent.

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EXERCISE R.22 10 The reason is that, as n increases, the additional observations are being given sharply decreasing weights, and hence the estimator is not benefiting from the increase in the sample size.

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Copyright Christopher Dougherty 1999–2006. This slideshow may be freely copied for personal use. 27.08.06

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