Christopher Dougherty EC220 - Introduction to econometrics (chapter 2) Slideshow: exercise 2.11 Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 2). [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

2.11 An investigator correctly believes that the relationship between two variables X and Y is given by Y =  1 +  2 X + u Given a sample of n observations on Y, X, and a third variable Z, which is not a determinant of Y, the investigator estimates  2 as Discuss the properties of this estimator. It can be shown that its population variance is given by EXERCISE

2 As always in an analysis of the properties of an estimator, the first step is to substitute for Y from the true relationship.

3 EXERCISE 2.11 The  1 terms cancel. We rearrange the rest of the numerator.

4 Hence we can decompose the estimator into the true value and an error term. EXERCISE 2.11

5 We will now investigate whether the estimator is biased. We take expectations and use the first expected value rule to decompose the expression into two terms.

6 EXERCISE 2.11 E(  2 ) is just  2 since  2 is a constant. The denominator of the error term can be taken out of the expectation as a multiplicative factor since we are assuming that both Z and X are nonstochastic.

7 EXERCISE 2.11 In the numerator of the error term we have used the fact that the expectation of a sum is equal to the sum of the expectations (first expected value rule).

8 EXERCISE 2.11 We again use the second expected value rule to take the term involving Z out of the expectation, Z and its mean being nonstochastic.

9 EXERCISE 2.11 The expected values of u i and its mean are both zero by virtue of Assumption A.3. Hence the estimator is unbiased.

10 EXERCISE 2.11 So why do we prefer the OLS estimator? It is not enough to say that Z is not a determinant of Y, and therefore it should not be a component of an estimator. Later on, we will be using estimators that incorporate extraneous variables.

11 It can be shown that the variance of the estimator is equal to the variance of the OLS estimator multiplied by the reciprocal of the correlation between X and Z. EXERCISE 2.11

12 EXERCISE 2.11 If Z were an exact linear function of X, this estimator would yield the same estimates as OLS and its population variance be the same.

13 EXERCISE 2.11 In general, however, the correlation coefficient squared will be less than 1 and the variance of the estimator will be greater, in which case the estimator is inefficient.

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