1 Research Method Lecture 11-1 (Ch15) Instrumental Variables Estimation and Two Stage Least Square ©

Motivation One explanatory variable case 4Consider the following regression. 2 4Since ability is not observed, we can only run the following regression. 4Since ability is correlated with educ, educ is endogenous (i.e, correlated with u). Thus, will be biased.

4We learned two methods to eliminate the bias. (1)Plug in the proxy variable for ability, such as IQ. (2)Use panel data method (either the fixed effect or the first differenced model). 4Instrumental variable method is another method to eliminate the bias. 3

Instrumental variable method: One explanatory variable case. 4Consider the following model. 4Suppose that x is endogenous, that is cov(x, u)≠0. 4Further, suppose that you have another variable, z, which satisfies the following conditions. Cov(z,u)=0 (instrument exogeneity) …....(1) Cov(z,x)≠0 (instrument relevance)………(2) 4If the above conditions are satisfied, we call z an instrumental variable. 4

4There are two ways to intuitively understand these conditions. 1.Instrumental variable is a variable that is not correlated with the omitted variable, but is correlated with the endogenous explanatory variable. 2.Instrumental variable is a variable that affects y only through x. 5

4The condition Cov(z,u)=0 involves unobserved u. Therefore, we cannot test this condition. (When you have extra instrumental variables, you can test this. This will be discussed later). 4The condition Cov(z,x)≠0 is easy to test. Just runt the following OLS, x=π 0 +π 1 z+v then test H 0 :π 1 =0 6

Instrumental variable estimation: One explanatory variable-one instrument case 4Now, consider 4Then we have Cov(z,y)= Cov(z,β 0 +β 1 x+u) So we have, Cov(z,y)= β 1 Cov(z,x)+Cov(z,u) Since Cov(z,u)=0, we have 7

 By replacing Cov(z,y) and Cov(z,x) with their sample covariances, we have the instrumental variable estimator of β 1 which is given by  You can easily show that is a consistent estimator of β 1. 8

Statistical inference with IV: Homoskedasticity case 4Homoskedasticity assumption in the case of IV regression is stated in terms of z. E(u 2 |z)=σ 2  It can be shown that the asymptotic variance of is given by: where is the variance of x, and is the correlation between x and z. 9

 Now, the estimator of var( ) is obtained by replacing σ 2,, and with their sample estimates.  Sample estimator of σ 2 is obtained in the following way. First, obtain the IV estimates for β 0 and β 1, then compute The estimator for σ 2 is then computed as 10

4The sample estimator for is given as: 4Finally, sample estimator for can be most easily obtained in the following way. First, regress x on z. Then the R-squared from this regression equals the square of the sample correlation. Let call this R 2 x,z. (Off course, you can compute the sample correlation and raise it by power 2. You will get the same result). 11

 Then, the estimator for the variance of is given by: 4You can show that this is a consistent estimator of the asymptotic variance given by (5). 12

Note: R-squared in IV regression 4The R-squared for IV regression is computed as R 2 =1-SSR/SST Where SSR is the sum of the squared IV residuals. (The IV residual is given by (6)). Unlike in the case of OLS, SSR can be greater than SST. Thus, R 2 can be negative. In IV regression, R 2 does not have a natural interpretation. 13

Finding the instrumental variable 4The most difficult part of the instrumental variable estimation is to find suitable instrumental variables. 4Consider the following regression 4Then, you have to find z that is correlated with educ, but not correlated with abil. What can be z? 14

4Consider the father’s education. Perhaps a person whose father is highly educated tends to take more education as well. So the father’s education is likely correlated with educ. 4But, for father’s education to be an instrument, this should not be correlated with the unobserved ability. A highly educated father may nurture his child better, so father’s education may be correlated with the unobserved ability. If this is the case, father’s education is not a good instrument. 4Nonetheless, many studies have used father’s and mother’s education as instruments. 15

Exercises 1. Run the following regression using OLS, using MROZ.dat 2. Using the father’s education as an instrument for edu, estimate the same model using IV regression. Also check if father’s education is correlated with educ. 16

17 OLS IV regression

18 Check if father’s education is correlated with educ.

An application Angrist and Krueger (1991), “Does Compulsory School Attendance Affect Schooling and Earning?” They used the quarter of the birth dummy as an instrument for education to estimate the effect of education on wage. 19

4In the US, the compulsory schooling law requires students to remain in school until their 16th birthday. 4At the same time, schools usually requires Children to be 6 years old on January 1 st to be admitted to school. Therefore, children who were born in the first quarter were older than children who were born in the last quarter when they were first admitted to schools (6.45 v.s years). 20

4This also means that children who were born in the first quarter of the year has shorter schooling when they reach the legal drop out age. So, children who were born in the first quarter can legally drop out of school with less education than children who were born in other quarters. 4If some people want to take as little education as possible but are constrained by the compulsory schooling law, the quarter of birth should affect the education attainment. 21

4At the same time, the quarter of birth is unlikely to be correlated with the unobserved ability. 4Therefore, the dummy variable indicating if a person was born in the first quarter of the year is a good instrument for education. 22

23 Those born in the first quarter of the year tend to have lower education attainment

24 This is the IV regression