1. EC220 - Introduction to econometrics (chapter 8)
Christopher Dougherty
EC220 - Introduction to econometrics (chapter 8)
Slideshow: instrumental variables
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 8). [Teaching Resource]
© 2012 The Author
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Available in LSE Learning Resources Online: May 2012
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2. INSTRUMENTAL VARIABLES
Suppose that you have a model in which Y is determined by X but you have reason to believe that Assumption B.7 is invalid and u is not distributed independently of X. An OLS regression would then yield inconsistent estimates. 1

3. INSTRUMENTAL VARIABLES
However, suppose that you have reason to believe that another variable Z is related to X but is unrelated to u. We will see that we can use it to obtain consistent estimates of the parameters. As a first step, suppose that we use it as a proxy for X. 2

4. INSTRUMENTAL VARIABLES
We will demonstrate that the resulting estimates will be biased. However, we will be able to do something about the bias. To investigate the properties of b2?, we first substitute for Y from the true model. 3

5. INSTRUMENTAL VARIABLES
We decompose the expression. 4

6. INSTRUMENTAL VARIABLES
Under the assumption that u is distributed independently of Z, the second term disappears when we take expectations. We see that b2? is nevertheless a biased estimator of b2. 5

7. INSTRUMENTAL VARIABLES
However, we can neutralize the bias by multiplying b2? By the reciprocal of the bias factor. We will call the new estimator b2IV, for reasons that will be explained later. 6

8. INSTRUMENTAL VARIABLES
The new estimator simplifies as shown. 7

9. INSTRUMENTAL VARIABLES
If you compare this with the OLS estimator of the slope coefficient, you see that Z replaces X in the numerator, but in the denominator it replaces only one of the two X arguments. 8

10. INSTRUMENTAL VARIABLES
The new estimator is described as an instrumental variables (IV) estimator, with Z being described as an instrument. We will check its properties. 9

11. INSTRUMENTAL VARIABLES
To do this, we start as usual by substituting for Y from the true model. 10

12. INSTRUMENTAL VARIABLES
We rearrange the numerator. 11

13. INSTRUMENTAL VARIABLES
Thus the new estimator simplifies as shown. 12

14. INSTRUMENTAL VARIABLES
We would like to check whether the estimator is unbiased. Unfortunately we are unable to do this. 13

15. INSTRUMENTAL VARIABLES
We could demonstrate that the expected value of the error term is 0 if u were distributed independently of X and Z. But because Assumption B.7 is invalid, u is not distributed independently of X. Otherwise we would use OLS. 14

16. INSTRUMENTAL VARIABLES
Instead, as a second-best measure, we will demonstrate that b2IV is a consistent estimator of b2. 15

17. INSTRUMENTAL VARIABLES
We focus on the error term. We would like to use the plim quotient rule. The plim of a quotient is the plim of the numerator divided by the plim of the denominator, provided tha both of these limits exist. 16

18. INSTRUMENTAL VARIABLES
However, as the expression stands, the numerator nor the denominator do not have limits. Both increase indefinitely as the sample size increases. 17

19. INSTRUMENTAL VARIABLES
To deal with this problem, we divide both of them by n. 18

20. INSTRUMENTAL VARIABLES
Now they do have limits and we can apply the plim quotient rule. 19

21. INSTRUMENTAL VARIABLES
It can be shown that the limit of the numerator is the covariance of Z with u and the limit of the denominator is the covariance of Z with X. 20

22. INSTRUMENTAL VARIABLES
Cov(Z, u) = 0 by assumption if Z is a valid instrument. Thus the IV estimator is consistent. 21

23. INSTRUMENTAL VARIABLES
The variance of the IV estimator is given by the expression shown. It is the expression for the variance of the OLS estimator, multiplied by the square of the reciprocal of the correlation between X and Z. 22

24. INSTRUMENTAL VARIABLES
Obviously, we would like the population variance to be as small as possible. This means that we want the correlation between X and Z to be as large (positive or negative) as possible. 23

25. INSTRUMENTAL VARIABLES
A famous early example of the use of instrumental variables to overcome the problem of measurement error is provided by Liviatan's use of it to fit a consumption function within the framework of Friedman's Permanent Income Hypothesis. 24

26. INSTRUMENTAL VARIABLES
Liviatan had data on the consumption and income of the same set of households for two years, several years apart. 25

27. INSTRUMENTAL VARIABLES
Measured consumption and income in Year 1 will be denoted C1 and Y1, respectively. According to the Permanent Income Hypothesis, each of these has its permanent and transitory components. 26

28. INSTRUMENTAL VARIABLES
The same will be true for Year 2. Measured consumption and income in Year 2 will be denoted C2 and Y2, respectively, and each of these has its permanent and transitory components. 27

29. INSTRUMENTAL VARIABLES
Suppose that we now use the Year 1 data to fit the consumption function. Permanent consumption and income are unobservable so we first derive a regression model in terms of the observable variables C1 and Y1. 28

30. INSTRUMENTAL VARIABLES
An OLS regression would yield inconsistent estimates because Y1T is a component of both Y1 and the compound disturbance term. 29

31. INSTRUMENTAL VARIABLES
However, consistent estimates would be obtained if one could find a suitable instrument. 30

32. INSTRUMENTAL VARIABLES
Liviatan used income from Year 2 for this purpose, arguing that both permanent and transitory income in Year 2 should be distributed independently of both transitory income and transitory consumption in Year 1. 31

33. INSTRUMENTAL VARIABLES
He argued that Y2 should be highly correlated with Y1, the other desirable attribute of an instrument, because families that were rich in Year 1 would tend to be rich in Year 2, and families that were poor would tend to remain poor. 32

34. INSTRUMENTAL VARIABLES
Thus he could obtain consistent estimates. He could also obtain (alternative) consistent estimates by fitting the consumption function using Year 2 data, this time using Y1 as an instrument. 33

35. Copyright Christopher Dougherty 2011.
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The content of this slideshow comes from Section 8.5 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
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
or the University of London International Programmes distance learning course
20 Elements of Econometrics