ECON Introductory Econometrics. Lecture 16: Instrumental variables

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1 ECON Introductory Econometrics Lecture 16: Instrumental variables Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 12

2 Lecture outline 2 OLS assumptions and when they are violated Instrumental variable approach 1 endogenous regressor & 1 instrument IV assumptions: instrument relevance instrument exogeneity 1 endogenous regressor, 1 instrument & control variables 1 endogenous regressor & multiple instruments multiple endogenous regressors & multiple instruments

3 Introduction 3 Y i = β 0 + β 1 X i + u i The 3 assumptions of an OLS regression model: 1 E(u i X i ) = 0 2 (X i, Y i ), i = 1,...N are independently and identically distributed 3 Big outliers are unlikely. Threats to internal validity (violation of 1st OLS assumption): Omitted variables Functional form misspecification Measurement error Sample selection Simultaneous causality

4 Introduction 4 Y i = β 0 + β 1 X i + u i We can use OLS to obtain consistent estimate of the causal effect if X Y u We can t use OLS to obtain consistent estimate of the causal effect if X Y X Y u and/or u

5 5 Instrumental variables: 1 endogenous regressor & 1 instrument Y i = β 0 + β 1 X i + u i Potential solution if E[u i X i ] 0 : use an instrumental variable (Z i ) We want to split X i into two parts: 1 part that is correlated with the error term (causing E[u i x i ] 0) 2 part that is uncorrelated with the error term If we can isolate the variation in X i that is uncorrelated with u i......we can use this to obtain a consistent estimate of the causal effect of X i on Y i

6 6 Instrumental variables: 1 endogenous regressor & 1 instrument In order to isolate the variation in X i that is uncorrelated with u i we can use an instrumental variable Z i with the following properties: 1 Instrument relevance: Z i is correlated with the endogenous regressor Cov(Z i, X i ) 0 2 Instrument exogeneity: Z i is uncorrelated with the error term Cov(Z i, u i ) = 0 and has no direct effect on Y i Z X Y u

7 7 Instrumental variables: 1 endogenous regressor & 1 instrument We can extend the linear regression model Y i = β 0 + β 1 X i + u i X i = π 0 + π 1 Z i + v i We can estimate the causal effect of X i on Y i in two steps: First stage: Regress X i on Z i & obtain predicted values X i = π 0 + π 1 Z i If Cov(Z i, u i ) = 0, X i contains variation in X i that is uncorrelated with u i Second stage: Regress Y i on X i to obtain the Two Stage Least Squares estimator ˆβ 2SLS : ˆβ 2SLS = n i=1 ( Yi Ȳ ) ( Xi X) n i=1 ( Xi X) 2

8 Application: estimating the returns to education 8 Data from the NLS Young Men Cohort collected in 1976 on (among others) wages and years of education for 3010 men. Wednesday March 5 14:31: Page 1 Data are provided by Professor David Card, he used the data in his article "Using Geographic Variation in College Proximity to Estimate (R) the Return to Schooling" 1. regress ln_wage education, robust / / / / / / / / / / / / Statistics/Data Analysis Linear regression Number of obs = 3010 F( 1, 3008) = Prob > F = R-squared = Root MSE = Robust ln_wage Coef. Std. Err. t P> t [95% Conf. Interval] education _cons OLS estimate of the returns to education likely inconsistent due to omitted variables and measurement error.

9 Application: estimating the returns to education 9 We want to isolate variation in years of education that is uncorrelated with the error term Card (1995) uses variation in college proximity as instrumental variable We have the following instrumental variable Wednesday March 5 14:46:54 1 if individual 2014 Page grew 1 up in area with a 4-year college near_college= 0 if individual grew up in area without a 4-year college (R) / / / / / Step 1: First stage regression / / / / / / / Statistics/Data Analysis 1. regress education near_college, robust Linear regression Number of obs = 3010 F( 1, 3008) = Prob > F = R-squared = Root MSE = Robust education Coef. Std. Err. t P> t [95% Conf. Interval] near_college _cons

10 Application: estimating the returns to education 10 Wednesday March 5 15:08: Page 1 Step 2: Obtain the predicted values and perform the second stage regression 1. predict pr_education, xb 2. regress ln_wage pr_education, robust (R) / / / / / / / / / / / / Statistics/Data Analysis Linear regression Number of obs = 3010 F( 1, 3008) = Prob > F = R-squared = Root MSE = Robust ln_wage Coef. Std. Err. t P> t [95% Conf. Interval] pr_education _cons

11 11 Instrumental variables: 1 endogenous regressor & 1 instrument Regression Y i on X i gives the 2SLS estimator ( Yi Ȳ ) ( Xi X) ˆβ 2SLS = n i=1 n i=1 ( Xi X) 2 If we substitute X i X ( ) ( ) = ( π 0 + π 1 Z i ) π 0 + π 1 Z = π 1 Z i Z we get n ( i=1 Yi ˆβ Ȳ ) ( ) π 1 Z i Z 2SLS = ( ) 2 = 1 π n i=1 π2 1 Z i Z 1 Since π 1 is the first stage OLS estimator: ( ) 2 n i=1 Z i Z ˆβ 2SLS = ( Xi X ) ( ) Z i Z n i=1 Which gives the instrumental variable estimator ( Yi Ȳ ) ( ) Z i Z ˆβ IV = n i=1 n i=1 n ( i=1 Yi Ȳ ) ( ) Z i Z ( ) 2 n i=1 Z i Z n ( i=1 Yi Ȳ ) ( ) Z i Z ( Xi X ) ( Z i Z ) ( ) 2 n i=1 Z i Z

12 Application: estimating the returns to education 12 We can obtain the 2SLS estimator in two steps as we have seen However the standard errors reported in the second stage regression are incorrect Stata does not recognize that it is a second stage of a two stage Wednesday process, March it fails 5 15:35:04 to take2014 into account Page 1 the uncertainty in the first stage estimation. Instead obtain the 2SLS-estimator in 1 step: 1. ivregress 2sls ln_wage (education=near_college), robust (R) / / / / / / / / / / / / Statistics/Data Analysis Instrumental variables (2SLS) regression Number of obs = 3010 Wald chi2( 1) = Prob > chi2 = R-squared =. Root MSE = Robust ln_wage Coef. Std. Err. z P> z [95% Conf. Interval] education _cons Instrumented: education Instruments: near_college

13 13 Instrumental variables: 1 endogenous regressor & 1 instrument n i=1 ( Yi Ȳ ) ( Z i Z ) ˆβ IV = ( Xi X ) ( ) Z i Z n i=1 In large samples the IV-estimator converges to plim( ˆβ IV ) = Cov(Y i, Z i ) Cov(X i, Z i ) = Cov(β 0 + β 1 X i + u i, Z i ) = β 1 + Cov(u i, Z i ) Cov(X i, Z i ) Cov(X i, Z i ) If the two IV-assumptions hold 1 Instrument relevance: Cov(Z i, X i ) 0 2 Instrument exogeneity: Cov(Z i, u i ) = 0 The IV-estimator is consistent plim( ˆβ IV ) = β 1, and is normally distributed in large samples ( ) ˆβ IV N β 1, 1 Var [(Z i µ Z ) u i ] n [Cov (Z i, X i )] 2

14 14 Instrumental variables: 1 endogenous regressor & 1 instrument The Instrumental Variables estimator is not unbiased [ ] [ ni=1 ] (Y E ˆβIV = E i Ȳ)(Z i Z) ni=1 (X i X)(Z i Z) = E [ ni=1 ] ((β 0 +β 1 X i +u i ) (β 0 +β 1 X+u))(Z i Z) ni=1 (X i X)(Z i Z) [ β ni=1 = E 1 (X i X)(Z i Z)+ ] n i=1 (u i ū)(z i Z) ni=1 (X i X)(Z i Z) [ ni=1 ] [ ni=1 ] (u = β 1 + E i ū)(z i Z) u ni=1 (X i X)(Z = β i Z) 1 + E i(z i Z) ni=1 (X i X)(Z i Z) [ ni=1 ] E[u = β 1 + E i Z i,x i ](Z i Z) X,Z ni=1 (X i X)(Z i Z) β 1 Instrument exogeneity implies E[u i Z i ] = 0 but not E[u i Z i, X i ] = 0 (this would mean that E[u i X i ] = 0 and we would not need an instrument!)

15 15 Instrumental variables: 1 endogenous regressor & 1 instrument How can we know whether the IV assumptions hold? 1 Instrument relevance: Cov(Z i, X i ) 0 We can check whether instrument relevance holds. Note that π 1 = Cov(Z i,x i ) Var(Z i ) We can therefore test H 0 : π 1 = 0 against H 1 : π Instrument exogeneity: Cov(Z i, u i ) = 0 We can t check whether this assumption holds. We need to use economic theory, expert knowledge and intuition.

16 Instrument relevance & weak instruments 16 Clearly, an irrelevant instrumental variable has problems, recall that ˆβ 2SLS Cov(Y i, Z i ) Cov(X i, Z i ) In case of an irrelevant (but exogenous) instrumental variable both the denominator and numerator are 0. If instrument is not irrelevant but Cov(X i, Z i ) is close to zero The sampling distribution of ˆβ 2SLS is not normal ˆβ2SLS can be severely biased, in the direction of the OLS estimator, even in relatively large samples! We should therefore always check whether an instrument is relevant enough.

17 Instrument relevance & weak instruments 17 Let F first be the F-statistic resulting from the test H 0 : π 1 = 0 against H 1 : π Staiger & Stock (Econometrica, 1997) show that in a simple model F first provides approximate estimate of finite sample bias of β 2SLS relative to β OLS Stock & Yogo (2005) argue that instruments are weak if the IV Bias is more than 10% of the OLS Bias. Rule of thumb: the F-statistic for (joint) significance of the instrument(s) in the first-stage should exceed 10.

18 Application: estimating the returns to education 18 Do the instrumental Thursday March variable 6 16:46:53 assumptions 2014 Page hold 1 for college proximity as an instrument to estimate the returns to education? 1 Instrument relevance/weak instruments (R) / / / / / / / / / / / / Statistics/Data Analysis Robust education Coef. Std. Err. t P> t [95% Conf. Interval] near_college _cons test near_college ( 1) near_college = 0 F( 1, 3008) = Prob > F = Instrument exogeneity: Is there a direct effect of living near a 4 year college on earnings? Is college proximity related to omitted variables that affect earnings? What about area characteristics, such as living in a big city instead of a small village?

19 1 endogenous regressor, 1 instrument & control variables 19 We can weaken the instrument exogeneity assumption by including area characteristics as control variables The Instrumental variables model is extended by including the control variables W 1i,..., W ri Y i = β 0 + β 1 X i + δ 1 W 1i +,..., +δ r W ri + u i X i = π 0 + π 1 Z i + γ 1 W 1i γ r W ri + v i The Instrument exogeneity condition is now conditional on the included regressors W 1i,..., W ri Cov (Z i, u i W 1i,..., W ri ) = 0 In the returns to education example we will include the following control variables: age and age squared south equals 1 if an individuals lives in the southern part of the U.S. smsa equals 1 if an individual lives in a Standard Metropolitan Statistical Area

20 Friday March 7 14:45: Page 1 Application: estimating the returns to education Control variables must also be included in the first stage regression: 1. regress education near_college age age2 south smsa, robust 20 (R) / / / / / / / / / / / / Statistics/Data Analysis Linear regression Number of obs = 3010 F( 5, 3004) = Prob > F = R-squared = Root MSE = Robust education Coef. Std. Err. t P> t [95% Conf. Interval] near_college age age south smsa _cons test near_college ( 1) near_college = 0 F( 1, 3004) = Prob > F = Don t use the overall F-statistic, this also tests whether the coefficients on the control variables equal zero!

21 Application: Friday March estimating 7 14:50: thepage returns 1 to education IV estimates with control variables (R) / / / / / / / / / / / / Statistics/Data Analysis 1. ivregress 2sls ln_wage (education=near_college) age age2 south smsa, robust Instrumental variables (2SLS) regression Number of obs = 3010 Wald chi2( 5) = Prob > chi2 = R-squared = Root MSE = Robust ln_wage Coef. Std. Err. z P> z [95% Conf. Interval] education age age south smsa _cons Instrumented: education Instruments: age age2 south smsa near_college Estimated return to an additional year of education is now 9.5% Do we believe that instrument exogeneity holds now that we have included control variables?

22 1 endogenous regressor, multiple instruments 22 Instead of 1 instrument we can also use M > 1 instruments We could calculate M different IV-estimates of β Since any linear combination of the Z mi is again a valid instrument: combine the Z mi to get a more efficient estimator of β 1 Y i = β 0 + β 1 X i + δ 1 W 1i +,..., +δ r W ri + u i X i = π 0 + π 1 Z 1i +... π M Z Mi + γ 1 W 1i γ r W ri + v i Instrumental variable assumptions: 1 Instrument relevance: at least one of the instruments Z 1i,..., Z Mi should have a nonzero coefficient in the population regression of X i on Z 1i,..., Z Mi. 2 Instrument exogeneity: Cov(Z 1i, u i ) = Cov(Z 2i, u i ) =... = Cov(Z Mi, u i ) = 0

23 Application: estimating the returns to education 23 The data set contains two potential instruments for years of education: near_2yrcollege= 1 if individual grew up in area with a 2-year college 0 if individual grew up in area without a 2-year college near_4yrcollege= 1 if individual grew up in area with a 4-year college 0 if individual grew up in area without a 4-year college To check for instrument relevance we should estimate the first stage regression, including both instruments And use an F-test to test for the joint significance of the two instruments.

24 Application: estimating the returns to education / / / / / / / Statistics/Data Analysis 1. regress education near_4yrcollege near_2yrcollege age age2 south smsa, robust Linear regression Number of obs = 3010 F( 6, 3003) = Prob > F = R-squared = Root MSE = Robust education Coef. Std. Err. t P> t [95% Conf. Interval] near_4yrcollege near_2yrcollege age age south smsa _cons test near_4yrcollege=near_2yrcollege=0 ( 1) near_4yrcollege - near_2yrcollege = 0 ( 2) near_4yrcollege = 0 F( 2, 3003) = 5.09 Prob > F = The first-stage F-statistic is well below 10, which indicates that we have weak instrument problems! It is better to drop the weakest instrument, near_2yrcollege, and use only 1 instrument near_4yrcollege

25 Overidentifying restrictions test (Sargan test, J-test) 25 With more instruments than endogenous regressors we can test whether a subset of the instrument exogeneity conditions is valid. Suppose we have two instruments. Given our structural equation Y i = β 0 + β 1 X i + δ 1 W 1i +,..., +δ r W ri + u i and assuming that Cov(Z 1i, u i ) = 0 we can test whether Cov(Z 2i, u i ) = 0 (or vice versa, but not both!) Intuition is as follows: since Cov(Z 1i, u i ) = 0 : ˆβ (Z 1) 2SLS β 1 IF Cov(Z 2i, u i ) = 0 then also ˆβ (z 2) 2SLS β 1 Testing whether Cov(Z 2i, u i ) = 0 is equivalent to testing ˆβ (z 2) 2SLS = ˆβ (z 1) 2SLS

26 Overidentifying restrictions test (Sargan test, J-test) 26 We can implement the test is as follows 1 Estimate Y i = β 0 + β 1 X i + δ 1 W 1i +,..., +δ r W ri + u i by 2SLS using Z 1i and Z 2i as instruments 2 Obtain the residuals û 2SLS i = Y i β 0 + β 1 X i + δ 1 W 1i +,..., + δ r W ri Note: use the true X i and not the predicted value X i 3 Estimate the following regression û 2SLS i = η 0 + η 1 Z 1i + η 2 Z 2i + +ϕ 1 W 1i +,..., +ϕ r W ri + e i 4 And obtain the F-statistic of the test H 0 : η 1 = η 2 = 0 versus H 1 : η 1 0 and/or η Compute the J-test statistic J = mf χ 2 q where q is number of instruments minus number of endogenous regressors (in this case 1)

27 Application: Friday March estimating 7 16:11: the Page returns 1 to education 27 (R) / / / / / / / / / / / / Statistics/Data Analysis 1. ivregress 2sls ln_wage (education=near_4yrcollege near_2yrcollege) age age2 south smsa, Instrumental variables (2SLS) regression Number of obs = 3010 Wald chi2( 5) = Prob > chi2 = R-squared = Root MSE = Robust ln_wage Coef. Std. Err. z P> z [95% Conf. Interval] education age age south smsa _cons Instrumented: education Instruments: age age2 south smsa near_4yrcollege near_2yrcollege 2. predict residuals, resid

28 Application: estimating the returns to education (R) 28 / / / / / / / / / / / / Statistics/Data Analysis 1. regress residuals near_4yrcollege near_2yrcollege age age2 south smsa, robust Linear regression Number of obs = 3010 F( 6, 3003) = 0.42 Prob > F = R-squared = Root MSE = Robust residuals Coef. Std. Err. t P> t [95% Conf. Interval] near_4yrcollege near_2yrcollege age age south smsa _cons test near_4yrcollege=near_2yrcollege=0 ( 1) near_4yrcollege - near_2yrcollege = 0 ( 2) near_4yrcollege = 0 F( 2, 3003) = 1.24 Prob > F = J = mf = = < 2.71 (critical value of χ 2 1 at 10% significance level) So we do not reject the null hypothesis of instrument exogeneity.

29 Overidentifying restrictions test (Sargan test, J-test) 29 Can we conclude that the two instruments satisfy instrument exogeneity? NO! Although the J-test seems a useful test there are 3 reasons to be very careful when using this test in practice 1 When we don t reject the null hypothesis this does not mean that we can accept it! 2 The power of the J-test can be low (probability of rejecting when H o does not hold) 3 The J-test tests the joint hypothesis of instrument validity and correct functional form 1 if the test rejects, the instruments might be valid but the functional form is wrong 2 if the test rejects, the instruments might be valid but the effect of the regressor of interest is heterogeneous β 1i β 1

30 The general IV regression model 30 So far we considered the case with 1 endogenous variable, but we can extend the model to multiple endogenous variables Y i = β 0 + β 1 X 1i β K X Ki + δ 1 W 1i +,..., +δ r W ri + u i X 1i = π0 1 + π1z 1 1i πmz 1 Mi + γ1w 1 1i +,..., +γr 1 W ri + vi 1. X Ki = π0 K + π1 K Z 1i πmz K Mi + γ1 K W 1i +,..., +γr K W ri + vi K The general IV regression model has 4 types of variables 1 The dependent variable Y i 2 K (possibly) endogenous regressors X 1i,..., X Ki 3 r control variables W 1i,..., W ri (not the variables of interest) 4 M instrumental variables Z 1i,..., Z Mi

31 The general IV regression model 31 When there are multiple endogenous regressors the 2SLS algoritm is similar except that each endogenous regressor requires its own first stage. For IV regression to be possible there should be at least as many instruments as endogenous regressors The model is said to be Underidentified if M < K, we cannot estimate the model, the number of instruments is then smaller that the number of endogenous regressors Exactly identified if M = K, the number of instruments equals the number of endogenous regressors Overidentified if M > K, the number of instruments exceeds the number of endogenous regressors

32 The general IV regression model 32 Assumptions of the general IV-model 1 Instrument exogeneity: 2 Instrument relevance: Cov(Z 1i, u i ) = Cov(Z 2i, u i ) =... = Cov(Z Mi, u i ) = 0 for each endogenous regressor X 1i,..., X Ki, at least one of the instruments Z 1i,..., Z Mi should have a nonzero coefficient in the population regression of the endogenous regressor on the instruments. The predicted values and the control variables ( X 1i,..., X Ki, W 1i,..., W ri, 1) should not be perfectly multicollinear. 3 (X 1i,..., X Ki, W 1i,..., W ri, Z 1i,..., Z Mi, Y i ) should be iid draws from their joint distribution. 4 Large outliers are unlikely: the X s, W s, Z s and Y have finite fourth moments.

33 Application: estimating the returns to education 33 Summary of results using college proximity as instrument: OLS 1 IV 1 IV 2 IV s without controls with controls with controls IV results, log(earnings) as dependent variable Education 0.052*** 0.188*** 0.095** 0.093* (0.003) (0.021) (0.048) (0.048) First stage regression near 4yr college 0.829*** 0.357*** 0.357*** (0.107) (0.112) (0.112) near 2yr college (0.098) First stage F * significant at 10%, ** significant at 5%, *** significant at 1% Is college proximity a valid instrument?

34 Application: estimating the returns to education 34 Another possible instrument for education is compulsory schooling laws Between 1925 and 1970 there were quite some changes in the minimum school leaving age in the US these changes varied between states Oreopoulos (AER,2006) uses variation in minimum school leaving age as instrument for years of schooling Main assumptions Changes in minimum school leaving age uncorrelated with unobserved variables affecting education (such as ability) No direct effect of changes in minimum school leaving age on wages Minimum school leaving age has a nonzero impact of years of education

35 Estimating returns to education 35 Oreopoulos estimates the following first stage and second stage equations: Y ist = βx ist + γ s + γ t + V ist θ + W st λ + ε ist X ist = πz st + δ s + δ t + V ist ρ + W st κ + µ ist Y ist is log wage of individual i living in state s in year t at age 14 X ist is years of schooling of individual i living in state s in year t at age 14 Z st is the minimum school leaving age in state s in year t γ s and δ s are state fixed effects, γ t and δ t are year fixed effects V ist are individual characteristics and W st are state characteristics

36 Estimating returns to education 36 Results from Oreopoulos (2006) OLS First stage IV ln(earnings) Education ln(earnings) Years of education 0.078*** 0.142*** (0.0005) (0.012) Minimum school leaving age 0.110*** (0.007) First stage F-statistic: F first = t 2 = ( ) 2 = IV estimate almost twice as high as OLS estimate, not what we expect on basis of positive ability bias story Possible explanations: downward bias in OLS due to measurement error heterogeneity in the returns to education (IV estimates local average treatment effect)

ECON Introductory Econometrics. Lecture 5: OLS with One Regressor: Hypothesis Tests

ECON Introductory Econometrics. Lecture 5: OLS with One Regressor: Hypothesis Tests ECON4150 - Introductory Econometrics Lecture 5: OLS with One Regressor: Hypothesis Tests Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 5 Lecture outline 2 Testing Hypotheses about one