Christopher Dougherty EC220 - Introduction to econometrics (chapter 7) Slideshow: exercise 7.5 Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 7). [Teaching Resource] © 2012 The Author This version available at: 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

You ask two research assistants to obtain estimates X 1 and X 2 of the population mean  of a random variable X. You know from past experience that both of them will obtain unbiased estimates, but one of the assistants is less careful than the other, and the variance of X 2 about the mean will be three times the variance of X 1. You have to come up with a single figure yourself. Should you take the average of X 1 and X 2, or discard X 2 and take X 1 as your estimate, or do something else? 1 RESEARCH ASSISTANTS

2 We will suppose that our estimate, which we will denote Z, should be a linear combination of the estimates of the research assistants. We will derive the optimal values of 1 and 2. RESEARCH ASSISTANTS

3 We will first derive the condition that 1 and 2 must satisfy for Z to be an unbiased estimator of . We use Expected Value Rule 1 to decompose the expected value expression. RESEARCH ASSISTANTS

4 Next we use Expected Value Rule 2 to bring 1 and 2 out of their terms. We are told that E(X 1 ) and E(X 2 ) are both equal to . RESEARCH ASSISTANTS

5 Thus the condition for unbiasedness is that 1 and 2 should sum to 1. The sample mean, with 1 = 2 = 0.5, satisfies this condition, but that does not mean that it is optimal. RESEARCH ASSISTANTS

6 We want our estimator to be as efficient as possible, so we will choose 1 and 2 so that its population variance is minimized. RESEARCH ASSISTANTS

7 We have used Variance Rule 1 to decompose the variance expression. On the assumption that the research assistants are working independently, the population covariance will be 0. RESEARCH ASSISTANTS

8 We have used Variance Rule 2 to bring 1 and 2 out of their terms, squaring them as we do. RESEARCH ASSISTANTS

9 We now make use of the information about the variances of X 1 and X 2. RESEARCH ASSISTANTS

10 We substitute for 2, making use of the unbiased condition = 1. RESEARCH ASSISTANTS

11 Hence we obtain the variance of Z as a function of 1. RESEARCH ASSISTANTS

12 The optimal value of 1 will be given by the first order condition. RESEARCH ASSISTANTS

13 Hence the optimal value of 1 is The unbiasedness condition then implies that 2 should be RESEARCH ASSISTANTS

14 Note that the second derivative of the variance is positive, confirming that we have found a minimum and not a maximum. RESEARCH ASSISTANTS

15 Substituting for 1 in the variance expression, we find that the variance of Z is 0.75  2. RESEARCH ASSISTANTS

16 We will check that this is smaller than the variance of the sample mean. RESEARCH ASSISTANTS

17 The covariance term is 0 on the assumption that the research assistants work independently. We take the 0.5 factors out of the variance terms, squaring them as we do. RESEARCH ASSISTANTS

18 We now make use of the information about the variances of X 1 and X 2. RESEARCH ASSISTANTS

19 Hence we obtain the population variance of the sample mean. RESEARCH ASSISTANTS

20 The variance of the optimal estimator is 25% lower. This is because it gives relatively large weight to the more accurate observation. RESEARCH ASSISTANTS

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