INSTRUMENTAL VARIABLES 1 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. X not independent of u

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 INSTRUMENTAL VARIABLES

X not independent of u 3 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 b 2 ?, we first substitute for Y from the true model. INSTRUMENTAL VARIABLES

4 We decompose the expression. INSTRUMENTAL VARIABLES X not independent of u

5 Under the assumption that u is distributed independently of Z, the second term disappears when we take expectations. We see that b 2 ? is nevertheless a biased estimator of  2. INSTRUMENTAL VARIABLES X not independent of u

6 However, we can neutralize the bias by multiplying b 2 ? By the reciprocal of the bias factor. We will call the new estimator b 2 IV, for reasons that will be explained later. INSTRUMENTAL VARIABLES X not independent of u

7 The new estimator simplifies as shown. INSTRUMENTAL VARIABLES X not independent of u

8 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. INSTRUMENTAL VARIABLES X not independent of u

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

X not independent of u 10 To do this, we start as usual by substituting for Y from the true model. INSTRUMENTAL VARIABLES

X not independent of u 11 We rearrange the numerator. INSTRUMENTAL VARIABLES

12 Thus the new estimator simplifies as shown. INSTRUMENTAL VARIABLES X not independent of u

13 We would like to check whether the estimator is unbiased. Unfortunately we are unable to do this. INSTRUMENTAL VARIABLES X not independent of u

14 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. INSTRUMENTAL VARIABLES It is not possible to take expectations since X is not independent of u X not independent of u

15 Instead, as a second-best measure, we will demonstrate that b 2 IV is a consistent estimator of  2. INSTRUMENTAL VARIABLES X not independent of u

16 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. INSTRUMENTAL VARIABLES X not independent of u if A and B have probability limits and plim B is not 0.

X not independent of u if A and B have probability limits and plim B is not However, as the expression stands, the numerator and the denominator do not converge on limits. INSTRUMENTAL VARIABLES

18 To deal with this problem, we divide both of them by n. INSTRUMENTAL VARIABLES X not independent of u if A and B have probability limits and plim B is not 0.

19 Now they do have limits. It can be shown that the limit of the numerator is the covariance of Z with u. INSTRUMENTAL VARIABLES X not independent of u

20 The limit of the denominator is the covariance of Z with X. INSTRUMENTAL VARIABLES X not independent of u

21 Hence we may now apply the plim quotient rule, provided that cov(Z,X) is not zero. A requirement for Z, therefore, is that it is correlated with X, while remaining independent of u. INSTRUMENTAL VARIABLES X not independent of u

22 cov(Z, u) = 0 by assumption if Z is independent of u. Thus the IV estimator is consistent. INSTRUMENTAL VARIABLES X not independent of u

23 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. INSTRUMENTAL VARIABLES X not independent of u

24 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. INSTRUMENTAL VARIABLES X not independent of u

25 However, we would not want Z to be perfectly correlated with X, because then it could not be independent of u and the plim of the error term would not be zero. INSTRUMENTAL VARIABLES X not independent of u

Copyright Christopher Dougherty 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 8.5 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 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 EC2020 Elements of Econometrics