4 Instrumental Variables Single endogenous variable One continuous instrument. 2

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1 Econ Econometric Review 1 Contents 4 Instrumental Variables Single endogenous variable One continuous instrument Single endogenous variable more than one continuous instrument Testing for Endogeneity and Overidentifying Restrictions. 27

2 Econ Econometric Review 2 4 Instrumental Variables 4.1 Single endogenous variable One continuous instrument Instrumental Variables (IV) estimation is used when a model Y = β 0 + β 1 X + u (1) has an endogenous X, that is, whenever Cov(X, u) 0 In other words, IV can be used to address the problem of omitted variable bias

3 Econ Econometric Review 3 For example, we are concerned that educ in a wage equation may be an endogeneous variable and the OLS coefficients over-estimate the returns to education. use c:\data\card; /* sample of men in 1976 as in Wooldridge example 15.4 */. reg lwage educ exper expersq black south smsa smsa66 reg661-reg668 ; Source SS df MS Number of obs = F( 15, 2994) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lwage Coef. Std. Err. t P> t [95% Conf. Interval] educ exper

4 Econ Econometric Review 4 expersq black south smsa smsa reg reg reg reg reg reg reg reg _cons Additionally, IV can be used to solve the classic errors-in-variables problem

5 Econ Econometric Review 5 But what is an instrumental variable? In order for a variable, Z, to serve as a valid instrument for X, the following must be true Assumption 1: Exclusion Restriction The instrument must be exogenous, that is, uncorrelated with the error term, Cov(Z, u) = 0 Assumption 2: Instrument Relevance The instrument must be correlated with the endogenous variable X that is, Cov(Z, X) 0 How do we know that Z is a valid instrument?

6 Econ Econometric Review 6 The main problem is that we have to use common sense and economic theory to decide if it makes sense to assume Cov(Z, u) = 0 In the case of multiple instruments, we can use the overid test below However, we can test whether Cov(Z, X) 0 We simply test H 0 : π 1 = 0 in the regression X = π 0 + π 1 Z + v (2) This regression is called the first-stage regression

7 Econ Econometric Review 7 Card (1995) has used proximity to a four-year college nearc4 as instrument for education. reg educ nearc4 exper expersq black south smsa smsa66 reg661-reg668 ; Source SS df MS Number of obs = F( 15, 2994) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = educ Coef. Std. Err. t P> t [95% Conf. Interval] nearc exper expersq black south smsa

8 Econ Econometric Review 8 smsa reg reg reg reg reg reg reg reg _cons test nearc4; ( 1) nearc4 = 0 F( 1, 2994) = Prob > F =

9 Econ Econometric Review 9 Rule-of-thumb: you need to worry about weak instruments if the firststage F-statistic is less than 10 Given equation (1) and our assumptions 1 and 2 Cov(Z, Y ) = Cov[Z, (β 0 + β 1 X + u)] = β 1 Cov(Z, X) + Cov(Z, u), so β IV 1 = Cov(Z, Y ) Cov(Z, X)

10 Econ Econometric Review 10 Equivalently, this yields the conditions for a method-of-moments estimator E(u) = E(Y β 0 β 1 X) = m 1 = 0 E(Zu) = E[Z(Y β 0 β 1 X)] = m 2 = 0 Either way, the IV estimator for β 1 is β IV 1 = ni=1 (Z i Z)(Y i Ȳ ) ni=1 (Z i Z)(X i X) (3)

11 Econ Econometric Review 11. ivreg lwage (educ=nearc4) exper expersq black south smsa smsa66 reg661-reg668 ; Instrumental variables (2SLS) regression Source SS df MS Number of obs = F( 15, 2994) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lwage Coef. Std. Err. t P> t [95% Conf. Interval] educ exper expersq black south smsa smsa

12 Econ Econometric Review 12 reg reg reg reg reg reg reg reg _cons Instrumented: educ Instruments: exper expersq black south smsa smsa66 reg661 reg662 reg663 reg664 reg665 reg666 reg667 reg668 nearc4 Notice that β IV educ = > β OLS educ = (see LATE effect below) Which estimator should we prefer IV or OLS?

13 Econ Econometric Review 13 When the regressor is endogenous, Cov(X, u) 0, IV estimation is consistent, while OLS is inconsistent, plim β OLS 1 = β 1 + Corr(X, u) σu σ X (4) But when R 2 < 1 in the first stage, IV standard errors are larger than the OLS The stronger the correlation between Z and X (strong instrument), the smaller the IV standard errors On the other hand, if the instrument is weak, the IV standard errors will be large

14 Econ Econometric Review 14 When the instrument is not really exogenous, i.e. if our assumption that Cov(Z, u) = 0 is false, Then the IV estimator will be inconsistent, too plim β IV 1 = β 1 + Corr(Z, u) Corr(Z, X) σu σ X (5) We will prefer IV if Corr(Z, u)/corr(z, X) < Corr(X, u).

15 Econ Econometric Review 15 Potential problems with IV estimation: IV can be very biased (much more than OLS), when the instrument is not truly exogenous Even instruments that are randomly assigned can be invalid if they affect the outcome is some way (not double-blind) Not always representative of the whole population, but may capture a local average treatment effect (LATE) e.g. proximity to school may affect more lower income students, compulsory schooling may affect marginal students Specification searching and publication bias lead to higher IV estimates since they have larger standard errors.

16 Econ Econometric Review Single endogenous variable more than one continuous instrument Consider the following structural model Y = β 0 + β 1 X 1 + β 2 X 2 + u 1 (6) where X 1 is an endogeneous variable and X 2 is an exogenous variable Suppose now that we have two exogenous variables excluded from equation (6) X 1 = π 0 + π 1 Z 1 + π 2 Z 2 + v (7) where Z 1 and Z 2 are valid instruments in that they do not appear in the structural model and are uncorrelated with the structural error term u 1, but are correlated with X 1

17 Econ Econometric Review 17 With more than one instrument, the IV estimator is also called the two-stage least squares (2SLS) estimator In our returns to education example, we can add proximity to a twoyear college nearc2. reg educ nearc4 nearc2 exper expersq black south smsa smsa66 reg661-reg668 ; Source SS df MS Number of obs = F( 16, 2993) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = 1.94 educ Coef. Std. Err. t P> t [95% Conf. Interval]

18 Econ Econometric Review 18 nearc nearc exper expersq black south smsa smsa reg reg reg reg reg reg reg reg _cons predict peduc; (option xb assumed; fitted values)

19 Econ Econometric Review 19. predict reseduc, res;. test nearc4=nearc2=0; ( 1) nearc4 - nearc2 = 0 ( 2) nearc4 = 0 F( 2, 2993) = 7.89 Prob > F = Here nearc2 is a weak instrument, so we no longer pass the rule-ofthumb test, so we would prefer the IV using only nearc4 In a more general case, we could use either Z 1 or Z 2 as an instrument

20 Econ Econometric Review 20 But the best instrument is a linear combination of all of the exogenous variables, including X 2 X 1 = π 0 + π 1 Z 1 + π 2 Z 2 + π 3 X 2 + v 2 We can estimate X 1 by regressing X 1 on Z 1, Z 2 and X 2, a regression that is an example of a reduced form equation If we use X 1 as an instrument for X 1 in the structural model, we will get same coefficient as 2SLS (see ivreg2 below) The 2 SLS expression comes from the fact that this estimation strategy is done in two steps

21 Econ Econometric Review 21. ivreg lwage (educ=peduc) exper expersq black south smsa smsa66 reg661-reg668; Instrumental variables (2SLS) regression Source SS df MS Number of obs = F( 15, 2994) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lwage Coef. Std. Err. t P> t [95% Conf. Interval] educ exper expersq black south smsa smsa

22 Econ Econometric Review 22 reg reg reg reg reg reg reg reg _cons Instrumented: educ Instruments: exper expersq black south smsa smsa66 reg661 reg662 reg663 reg664 reg665 reg666 reg667 reg668 peduc While the coefficients are the same, the standard errors from doing 2SLS by hand are incorrect, so let STATA do it for you

23 Econ Econometric Review 23 Some economists like to interpret the first stage of 2SLS as a way to purge X 1 from its correlation with u 1 before doing the second stage regression The method extends to multiple endogenous variables (say k), but we need at least as many excluded exogenous variables (instruments) as there are endogenous variables in the structural equation The standard IV(one instrument) and 2SLS(one linear combination of many instruments) estimators are special cases of the GMM estimator, where l instruments will give us a set of l moments E[Z j u] = m j = 0, j = 1,... l.

24 Econ Econometric Review 24 In a just-identified model (l=k), GMM will be identical to IV, but when there are more instruments than endogeneous variables (l > k), we will need a weighing matrix that accounts for the correlations among the moments conditions when errors are not i.i.d. When we use ivreg2 with the option gmm, STATA will compute 2SLS residuals in a first step and use these residuals to compute a weighing matrix that will give the most efficient feasible estimate. Thus for overidentified models, the GMM approach makes more efficient use of the information in the l moment conditions than the standard 2SLS approach which reduces them to k instrument, and it is heteroskedascity-efficient.

25 Econ Econometric Review 25. ivreg2 lwage (educ=nearc2 nearc4) exper expersq black south smsa smsa66 reg661-reg668, gmm ; GMM estimation Number of obs = 3010 F( 15, 2994) = Prob > F = Total (centered) SS = Centered R2 = Total (uncentered) SS = Uncentered R2 = Residual SS = Root MSE =.4028 Robust lwage Coef. Std. Err. z P> z [95% Conf. Interval] educ exper expersq black

26 Econ Econometric Review 26 south smsa smsa reg reg reg reg reg reg reg reg _cons Anderson canon. corr. LR statistic (identification/iv relevance test): Chi-sq(2) P-val = Hansen J statistic (overidentification test of all instruments): Chi-sq(1) P-val = Instrumented: educ Included instruments: exper expersq black south smsa smsa66 reg661 reg662 reg663

27 Econ Econometric Review 27 reg664 reg665 reg666 reg667 reg668 Excluded instruments: nearc2 nearc4 4.3 Testing for Endogeneity and Overidentifying Restrictions Since OLS is preferred to IV if we do not have an endogeneity problem, then we would like to be able to test for endogeneity If we do not have endogeneity, both OLS and IV are consistent

28 Econ Econometric Review 28 The idea of the Hausman test is to see if the estimates from OLS and IV are different If X 1 is endogenous, then v 2 (from the first stage equation) and u 1 from the structural model will be correlated This test is easily done by including the residual from the first stage in the OLS regression Y = β 0 + β 1 X 1 + β 2 X 2 + δˆv 2 + u 1 and testing H 0 : δ = 0 that there is no correlation between the residuals and the X s using a t statistic

29 Econ Econometric Review 29. reg lwage educ reseduc exper expersq black south smsa smsa66 reg661-reg668 ; Source SS df MS Number of obs = F( 16, 2993) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lwage Coef. Std. Err. t P> t [95% Conf. Interval] educ reseduc exper expersq black south smsa smsa reg

30 Econ Econometric Review 30 reg reg reg reg reg reg reg _cons With a t = 1.71, there is moderate evidence of positive correlation between u 1 and v 2, we conclude that educ is moderately endogenous Alternatively, you can use the STATA command hausman IV OLS where the commands est store OLS and est store IV have followed each estimation command

31 Econ Econometric Review 31. hausman IV OLS, constant sigmamore; Note: the rank of the differenced variance matrix (1) does not equal the number of coefficients being tested (16); be sure this is what you expect, or there may be problems computing the test. Examine the output of your estimators for anything unexpected and possibly consider scaling your variables so that the coefficients are on a similar scale Coefficients ---- (b) (B) (b-b) sqrt(diag(v_b-v_b)) IV OLS Difference S.E educ exper expersq black south smsa smsa reg reg

32 Econ Econometric Review 32 reg reg reg reg reg reg _cons b = consistent under Ho and Ha; obtained from ivreg B = inconsistent under Ha, efficient under Ho; obtained from regress Test: Ho: difference in coefficients not systematic chi2(1) = (b-b) [(V_b-V_B)^(-1)](b-B) = 2.92 Prob>chi2 = (V_b-V_B is not positive definite) We conclude that educ is correlated with the error terms at the 10% level of significance.

33 Econ Econometric Review 33 Now is our instrument correlated with the error term? If there is just one instrument for our endogenous variable, we cannot test to see whether the instrument is uncorrelated with the error: the model is just identified However, if we have multiple instruments, it is possible to test the overidentifying restrictions to see if some of the instruments are correlated with the error if we are pretty sure that one instrument is excluded With ivreg2, STATA will perform the test for you

34 Econ Econometric Review 34 The idea is to regress the predicted residual û 1 on all exogenous variables, including the instrumental variables If the instruments are in fact exogenous, the coefficients on the instruments and on the included exogenous variables in a regression on û 1 should all be zero This is formally tested with either a chi-square statistics or a J-statistic to test H 0 that all IVs are uncorrelated with u 1.. ivreg2 lwage (educ=nearc2 nearc4) exper expersq black south smsa smsa66 reg661-reg668 ; Instrumental variables (2SLS) regression

35 Econ Econometric Review 35 Number of obs = 3010 F( 15, 2993) = Prob > F = Total (centered) SS = Centered R2 = Total (uncentered) SS = Uncentered R2 = Residual SS = Root MSE =.4 lwage Coef. Std. Err. z P> z [95% Conf. Interval] educ exper expersq black south smsa smsa reg reg reg reg

36 Econ Econometric Review 36 reg reg reg reg _cons Sargan statistic (overidentification test of all instruments): Chi-sq(1) P-val = Instrumented: educ Instruments: exper expersq black south smsa smsa66 reg661 reg662 reg663 reg664 reg665 reg666 reg667 reg668 nearc2 nearc4. predict res1, res; */doing it the long way/*. regress res1 exper expersq black south smsa smsa66 reg661-reg668 nearc4 nearc2 ; Source SS df MS Number of obs = F( 16, 2993) = 0.08 Model Prob > F =

37 Econ Econometric Review 37 Residual R-squared = Adj R-squared = Total Root MSE = res1 Coef. Std. Err. t P> t [95% Conf. Interval] exper expersq -3.43e black south smsa smsa reg reg reg reg reg reg reg reg

38 Econ Econometric Review 38 nearc nearc _cons gen overid2=_n*e(r2);. di overid2; gen pval=chi2tail(1,overid2);. di pval; test nearc4=nearc2=0; ( 1) nearc4 - nearc2 = 0 ( 2) nearc4 = 0 F( 2, 2993) = 0.62

39 Econ Econometric Review 39 Prob > F = gen Jstat=r(df)*r(F);. di Jstat; Therefore the variables of proximity to four and two-year college pass the overidentification test, but may not be the preferred specification since nearc2 is a weak instrument