1. Christopher Dougherty EC220 - Introduction to econometrics (chapter 8) Slideshow: measurement error Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 8). [Teaching Resource] © 2012 The Author This version available at: http://learningresources.lse.ac.uk/134/http://learningresources.lse.ac.uk/134/ 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/

2. MEASUREMENT ERROR 1 In this sequence we will investigate the consequences of measurement errors in the variables in a regression model. To keep the analysis simple, we will confine it to the simple regression model.

3. MEASUREMENT ERROR 2 We will start with measurement errors in the explanatory variable. Suppose that Y is determined by a variable Z, but Z is subject to measurement error, w. We will denote the measured explanatory variable X.

4. MEASUREMENT ERROR 3 Substituting for Z from the second equation, we can rewrite the model as shown.

5. MEASUREMENT ERROR 4 We are thus able to express Y as a linear function of the observable variable X, with the disturbance term being a compound of the disturbance term in the original model and the measurement error.

6. MEASUREMENT ERROR 5 However if we fit this model using OLS, Assumption B.7 will be violated. X has a random component, the measurement error w.

7. MEASUREMENT ERROR 6 And w is also one of the components of the compound disturbance term. Hence u is not distributed independently of X.

8. MEASUREMENT ERROR 7 We will demonstrate that the OLS estimator of the slope coefficient is inconsistent and that in large samples it is biased downwards if  2 is positive, and upwards if  2 is negative.

9. MEASUREMENT ERROR 8 We begin by writing down the OLS estimator and substituting for Y from the true model. In this case there are alternative versions of the true model. The analysis is simpler if you use the equation relating Y to X.

10. MEASUREMENT ERROR 9 Simplifying, we decompose the slope coefficient into the true value and an error term as usual.

11. MEASUREMENT ERROR 10 We have reached this point many times before. We would like to investigate whether b 2 is biased. This means taking the expectation of the error term.

12. MEASUREMENT ERROR 11 However, it is not possible to obtain a closed-form expression for the expectation of the error term. Both its numerator and its denominator are functions of w and there are no expected value rules that can allow us to simplify.

13. MEASUREMENT ERROR 12 As a second-best measure, we take plims and investigate what would happen in large samples. The plim quotient rule allows us to write the plim of the error term as the plim of the numerator divided by the plim of the denominator, provided that these plims exist.

14. MEASUREMENT ERROR 13 For the plims to exist, the numerator and denominator must both be divided by n. Otherwise they would increase indefinitely with the sample size.

15. MEASUREMENT ERROR 14 Having divided by n, it can be shown that the plim of the numerator is cov(X, u) and the plim of the denominator is var(X).

16. MEASUREMENT ERROR 15 We can decompose both the numerator and the denominator of the error term. We will start by substituting for X and u in the numerator.

17. MEASUREMENT ERROR 16 We expand the expression using the first covariance rule.

18. MEASUREMENT ERROR 17 If we assume that Z, v, and w are distributed indepndently of each other, the first 3 terms are 0. The last term gives us –  2  w 2.

19. MEASUREMENT ERROR 18 We next expand the denominator of the error term. The first two terms are variances. The covariance is 0 if we assume w is distributed independently of Z.

20. MEASUREMENT ERROR 19 Thus in large samples, b 2 is biased towards 0 and the size of the bias depends on the relative sizes of the variances of w and Z.

21. MEASUREMENT ERROR 20 Since b 2 is an inconsistent estimator, it is safe to assume that it is biased in finite samples as well.

22. MEASUREMENT ERROR 21 If our assumptions concerning Z, v, and w are incorrect, b 2 would almost certainly still be a biased estimator, but the expression for the bias would be more complicated.

23. MEASUREMENT ERROR 22 A further consequence of the violation of Assumption B.7 is that the standard errors, t tests and F test are invalid.

24. MEASUREMENT ERROR 23 Measurement error in the dependent variable has less serious consequences. Suppose that the true dependent variable is Q, that the measured variable is Y, and that the measurement error is r.

25. MEASUREMENT ERROR 24 We can rewrite the model in terms of the observable variables by substituting for Q from the second equation.

26. MEASUREMENT ERROR 25 In this case the presence of the measurement error does not lead to a violation of Assumption B.7. If v satisfies that assumption in the original model, u will satisfy it in the revised one, unless for some strange reason r is not distributed independently of X.

27. MEASUREMENT ERROR 26 The standard errors and tests will remain valid. However the standard errors will tend to be larger than they would have been if there had been no measurement error, reflecting the fact that the variances of the coefficients are larger.

28. Copyright Christopher Dougherty 2011. These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 8.4 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/. http://www.oup.com/uk/orc/bin/9780199567089/ Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course 20 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 11.07.24