Introduction to Instrumental Variables Methods

Introduction to Instrumental Variables Methods Christian Hansen Booth School of Business University of Chicago

Introduction • Many studies in social sciences interested in inferring structural/causal/treatment effects • Price elasticity of demand • Effect of smoking on birthweight • Effect of 401(k) participation on saving • Effect of job training on wages/employment • Effect of schooling on wages • … • Only have observational data Conventional statistical methods may not recover desired effect

Example 1: Supply and Demand Supply Curve Equilibrium Demand Curve

Example 1: Supply and Demand Suppose demand and supply fluctuate from day to day. We observe the market for several days and want to infer either the slope of the supply or demand curve (from which economic quantities such as demand elasticity are derived).

Example 1: Supply and Demand Suppose demand and supply fluctuate from day to day. We observe the market for several days and want to infer either the slope of the supply or demand curve (from which economic quantities such as demand elasticity are derived). Observed relationship between price and quantity reveals neither supply nor demand! (Simultaneity)

Example 2: Job Training • Observe data on earnings for people who have and have not completed job training. • Want to infer the causal effect of job training on earnings • What if people who are more “motivated” are more likely to get training and on average earn more than less “motivated”? • Difference between average earnings across the trained and untrained confounds the effects of motivation and training • Omitted variables bias: Would like to control for unobserved (and unobservable?) motivation

Example 3: Classical Measurement Error • Model: . Want to know the effect of x on y (β) • Only observe noisy signal for x: • What do you get from regressing on ? • The last term generally does not have 0 expectation or converge to 0 even in large samples • Under “classical measurement error” () the second term converges to (attenuation bias)

Common Structure: • “Structural” Model: • y – outcome of interest • x – observed “treatment” variable • – treatment/structural/causal effect (NOT regression coefficient) • (Endogeneity) Instrumental variables (IV) offers one approach to estimating (when instruments are available…)

What is an instrument? • Instrumental variable (denoted ) shifts but is unrelated to structural error (). • (relevance) • (exclusion) • Intuition: Movements in are unrelated to movements in but are related to movements in Movements in “induced” by are uncontaminated. Can be used to estimate treatment effect. Key statistical conditions

How do instruments help? • Consider supply/demand example: • is a variable that affects supply (but not demand (say the price of a factor of production). . • Valid instrument for demand.

How do instruments help? • Heuristic: When changes: • supply changes • demand remains the same (on average) Movements in the supply curve induced by changing z trace out the demand curve

How do instruments help? • Quasi-mathematically. IV model • Recall OLS estimator: • Neither unbiased nor consistent since

How do instruments help? • IV estimator: • Uses only uncontaminated variation (covariation between instrument and X and instrument and Y) • Under conditions above (plus regularity), generally consistent, asymptotically normal with estimable covariance matrix. • Software: (Also really easy to code yourself) • Stata: ivregress (and variants) • SAS: procsyslin (and others) • R: sem package • SPSS: Analyze -> Regression -> Two-stage least squares • …

Questions? • Where do instruments come from? • Intuition, subject matter knowledge, randomization • Can I assess the validity of the underlying assumptions? • Sort of, is fundamentally untestable (though aspects can be tested if extra instruments available) • is testable • What if I have extra instruments (so Z’X is not invertible)? • Two-stage least squares (2SLS). Same intuition. Default implementation in all stats packages. Other options • Are there things I should look out for? • Weak instruments. Many instruments. Reasons to doubt the exclusion restriction. What exactly is estimated when treatment effects are heterogeneous.

Example: Job Training • Goal: Assess impact of job training on subsequent labor market outcomes (e.g. employment, wages) • Problem: Training receipt is not randomly assigned. May be endogenous. • E.g. maybe more motivated people more likely to receive job training and more likely to find subsequent employment/have better job performance • Instrument: Under Job Training Partnership Act (JTPA), randomized trial conducted in which individuals offered JTPA services • Use JTPA offer of services as instrument for receiving training

Example: Job Training • Plausibility of instrument: • Offer of services randomly assigned => Independent of structural error by construction. (No evidence aboutthis in data) • ≈ 60% of those offered training accepted offer => Offer strongly related to receipt. (Can look at correlation between instrument and endogenous variable to assess this) • Aside: One could simply regress outcome on offer of treatment to estimate intention-to-treat (ITT). Our goal is to estimate effect of treatment, not offer.

Example: Job Training • Structural Equation: • First-Stage Equation: • Regression of treatment on instrument and controls • Note: E[zixi] ≠ 0 => π1,z ≠ 0 • Reduced Form Equation: • Regression of outcome on instrument and controls • Note: If can’t rule out π2,z = 0, can’t rule out α = 0 • First-stage and reduced form are predictive representations, not structural . Good practice to report results from all three.

Example: Job Training • Variables: • Earn – total earnings over 30 month period following assignment of offer • Train – dummy for receipt of job training services • Offer – dummy for offer to receive training services • x– 13 additional control variables • dummies for black and Hispanic persons, a dummy indicating high-school graduates and GED holders, five age-group dummies, a marital status dummy, a dummy indicating whether the applicant worked 12 or more weeks in the 12 months prior to the assignment, a dummy signifying that earnings data are from a second follow-up survey, and dummies for the recommended service strategy

Example: Job Training • OLS Results (from Stata): regress earnings train x1-x13 , robust Linear regression Number of obs = 5102 F( 14, 5087) = 38.35 Prob > F = 0.0000 R-squared = 0.0909 Root MSE = 18659 ------------------------------------------------------------------------------ | Robust earnings | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- train | 3753.362 536.3832 7.00 0.000 2701.82 4804.904 . . . If intuition about source of endogeneity is correct, this should be an over-estimate of the effect of training.

Example: Job Training • First-Stage Results (from Stata): regress train offer x1-x13 , robust Linear regression Number of obs = 5102 F( 14, 5087) = 390.75 Prob > F = 0.0000 R-squared = 0.3570 Root MSE = .39619 ------------------------------------------------------------------------------ | Robust train | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- offer | .6088885 .0087478 69.60 0.000 .591739 .6260379 . . . Strong evidence that E[zixi] ≠ 0

Example: Job Training • Reduced-Form Results (from Stata): regress earnings offer x1-x13 , robust Linear regression Number of obs = 5102 F( 14, 5087) = 34.19 Prob > F = 0.0000 R-squared = 0.0826 Root MSE = 18744 ------------------------------------------------------------------------------ | Robust earnings | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- offer | 970.043 545.6179 1.78 0.075 -99.60296 2039.689. . . . Moderate evidence of a non-zero treatment effect (maintaining exclusion restriction)

Example: Job Training Note: Some software reports R2 after IV regression. This object is NOT meaningful and should not be used. • IV Results (from Stata): ivreg earnings (train = offer) x1-x13 , robust Instrumental variables (2SLS) regression Number of obs = 5102 F( 14, 5087) = 34.38 Prob > F = 0.0000 R-squared = 0.0879 Root MSE = 18689 ------------------------------------------------------------------------------ | Robust earnings | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- train | 1593.137 894.7528 1.78 0.075 -160.9632 3347.238 . . . Moderate evidence of a positive treatment effect (maintaining exclusion restriction). Substantially attenuated relative to OLS, consistent with intuition.

Two-Stage Least Squares • May have more instruments than endogenous variables • In principle, many IV estimators can be constructed • 2SLS is the minimum variance (under homoskedasticity) linear combination of the potential IV estimators (otherwise may use GMM) • 2SLS is the GMM estimator using the full set of orthogonality conditions implied by • 2SLS and IV are numerically equivalent when # of endogenous variables = # of instruments • Aside: Some jargon • r = # of instruments, k = # of endogenous variables • r = k, “just-identified” • r > k, “over-identified”

2SLS • 2SLS estimator: • PZ = Z(Z’Z)-1Z’, projection matrix onto Z • Uses only uncontaminated variation (covariation between instrument and X and instrument and Y) • Under conditions above (plus regularity), generally consistent, asymptotically normal with estimable covariance matrix. • Can be viewed as two-step procedure where endogenous variables and outcomes are first projected onto Z and then projections are used in OLS • Software: (Also really easy to code yourself) • Stata: ivregress (and variants) • SAS: procsyslin (and others) • R: sem package • SPSS: Analyze -> Regression -> Two-stage least squares • …

Testing Overidentifying Restrictions • Have more instruments than we need to estimate the treatment effect • If all instruments satisfy exclusion restriction, all subsets should (asymptotically) return the same estimate of the treatment effect • Idea: Obtain multiple estimates of the treatment effect and test that they are the same. • Rejection implies some subset of exclusion restrictions may be invalid

Hansen’s Over-Identification Test • Also called the “J-Test”, Sargan test, “S-Test” • Based on GMM criterion function • For IID, homoskedastic data: (More generally, J is GMM objective function evaluated at GMM point-estimate)

Overidentification Tests: • Can never tell you that the exclusion restriction ( is satisfied • Failure to reject does not imply true • Even if it did, only learn that probability limits of various IV estimators are the same. Maybe all the same and wrong. • Heterogeneous treatment effects? • Rejection indicates that some subset of instruments may be invalid • Does not indicate which subset • Does not mean all exclusion restrictions are invalid • Heterogeneous treatment effects?

Example: Returns to Schooling • Goal: Estimate the value added of additional years of schooling in terms of wages • Problem: Years of completed schooling is not randomly assigned. May be endogenous. • E.g. maybe academic ability is related to qualities that relate to job performance/salary (motivation, intelligence, task orientation, etc.) • Instrument: Quarter of birth (Angrist and Krueger, 1991)

Example: Returns to Schooling • Plausibility of instrument: • Compulsory schooling laws in the U.S. are typically based on age, not number of years of school. People born at different times of the year can drop out after receiving different amounts of school. (Can look at correlation between instrument and endogenous variable to assess this) • When a person is born is unrelated to inherent traits (e.g. motivation, intelligence, …) and so should not have a direct effect on wages but only affect wages through the relationship to completed schooling induced by compulsory education laws. • Untestable, but we do have overidentifying restrictions coming from different birth quarters. • Validity has been questioned. E.g. winter birth may be correlated to increased exposure to early health problems; more conscientious parents may respond by timing birth; …

Example: Returns to Schooling • Structural Equation: • First-Stage Equation: • Note: E[zixi] ≠ 0 => π1,1 ≠ 0 or π1,2≠ 0 or π1,3≠ 0 • Reduced Form Equation:

Example: Returns to Schooling • Data from 1980 Census for men aged 40-49 in 1980 • Variables: • Wage – hourly wage • School – reported years of completed schooling • Q1-Q3 – dummies for quarter of birth • x– 59 control variables. Dummies for state of birth and year of birth

Example: Returns to Schooling • OLS Results (from Stata): xi: reglwageeduci.yobi.sob , robust i.yob _Iyob_30-39 (naturally coded; _Iyob_30 omitted) i.sob _Isob_1-56 (naturally coded; _Isob_1 omitted) Linear regression Number of obs = 329509 F( 60,329448) = 649.29 Prob > F = 0.0000 R-squared = 0.1288 Root MSE = .63366 ------------------------------------------------------------------------------ | Robust lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- educ | .067339 .0003883 173.40 0.000 .0665778 .0681001 . . . If intuition about source of endogeneity is correct, this should be an over-estimate of the effect of schooling.

Example: Returns to Schooling • First-Stage Results (from Stata): xi: regress educi.qobi.sobi.yob , robust Linear regression Number of obs = 329509 F( 62,329446) = 292.87 Prob > F = 0.0000 R-squared = 0.0572 Root MSE = 3.1863 ------------------------------------------------------------------------------ | Robust educ | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- _Iqob_2 | .0455652 .015977 2.85 0.004 .0142508 .0768797 _Iqob_3 | .1060082 .0155308 6.83 0.000 .0755683 .136448 _Iqob_4 | .1525798 .0157993 9.66 0.000 .1216137 .1835459 . . . testparm _Iqob* ( 1) _Iqob_2 = 0 ( 2) _Iqob_3 = 0 ( 3) _Iqob_4 = 0 F( 3,329446) = 36.06 Prob > F = 0.0000 First-stage F-statistic.

Example: Returns to Schooling • Reduced-Form Results (from Stata): xi: regress lwagei.qobi.sobi.yob , robust Linear regression Number of obs = 329509 F( 62,329446) = 147.83 Prob > F = 0.0000 R-squared = 0.0290 Root MSE = .66899 ------------------------------------------------------------------------------ | Robust lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- _Iqob_2 | .0028362 .0033445 0.85 0.396 -.0037188 .0093912 _Iqob_3 | .0141472 .0032519 4.35 0.000 .0077736 .0205207 _Iqob_4 | .0144615 .0033236 4.35 0.000 .0079472 .0209757 . . testparm _Iqob* ( 1) _Iqob_2 = 0 ( 2) _Iqob_3 = 0 ( 3) _Iqob_4 = 0 F( 3,329446) = 10.43 Prob > F = 0.0000

Example: Returns to Schooling • 2SLS Results (from Stata): xi: ivregress 2sls lwage (educ = i.qob) i.yobi.sob , robust Instrumental variables (2SLS) regression Number of obs = 329509 Wald chi2(60) = 9996.12 Prob > chi2 = 0.0000 R-squared = 0.0929 Root MSE = .64652 ------------------------------------------------------------------------------ | Robust lwage | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- educ | .1076937 .0195571 5.51 0.000 .0693624 .146025 . . . Bigger than OLS?

Example: Returns to Schooling • GMM Results (from Stata, efficient under heteroskedasticity): xi: ivregressgmmlwage (educ = i.qob) i.yobi.sob , robust Instrumental variables (GMM) regression Number of obs = 329509 Wald chi2(60) = 9992.90 Prob > chi2 = 0.0000 R-squared = 0.0927 GMM weight matrix: Robust Root MSE = .64658 ------------------------------------------------------------------------------ | Robust lwage | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- educ | .1077817 .0195588 5.51 0.000 .0694472 .1461163 . . . estatoverid Test of overidentifying restriction: Hansen's J chi2(2) = 3.10009 (p = 0.2122) Bigger than OLS? Fail to reject over-id test

Heterogeneous Treatment Effects: ATE • Previous discussion posits a constant treatment/structural effect • Estimated constant treatment effect = average treatment effect (ATE) sometimes • independent of and , => • where • i.e. IV model holds interpreting parameter as ATE • Note: Exclusion restriction will not hold in general if heterogeneous effect is not independent of instrument or treatment!

Heterogeneous Treatment Effects: LATE • When heterogeneous effect is not independent, can still estimate a causal effect among a subpopulation • Strip model down to binary treatment and binary instrument • Four subpopulations: • Always-takers: zi= 1, xi = 1 and zi= 0, xi = 1 • Never-takers: zi = 1, xi = 0 and zi = 0, xi = 0 • Complier: zi = 1, xi = 1 and zi = 0, xi = 0 • Defier: zi = 1, xi = 0 and zi = 0, xi = 1 • Note: Never observe individuals at both instrument states. Can’t determine observation’s subpopulation

Heterogeneous Treatment Effects: LATE • Generally, may estimate ATE among subpopulation of compliers • Termed Local Average Treatment Effect (LATE) • Conditions: • Independence – instrument independent of all unobservables affecting outcome and treatment/endogenous variable state • Exclusion – instrument only affects outcome though treatment receipt. (No direct effect of the instrument.) • First-stage – instrument predicts treatment receipt • Monotonicity – effect of instrument on probability of receiving treatment is ≥ 0 for everyone or ≤ 0 for everyone. (No defiers)

Heterogeneous Treatment Effects: LATE • LATE can be generalized (with varying degrees of difficulty) to multi-valued/continous treatments or instruments, over-identified models, models with controls • 2SLS coefficient estimates (or approximates) a weighted average of different LATEs • Each instrument/value of the instrument potentially gives a different set of compliers => a different LATE • Rejection of over-id test does not mean instruments are invalid as all could be valid but give different complier populations

Heterogeneous Treatment Effects: LATE • Can learn some things about compliers from data • A couple simple ones: • Size of complier population: • Proportion of treated who are compliers: Probability complier or always-taker Probability always-taker. No defiers

Example: JTPA • Recall treatment = dummy for receipt of job training services • Instrument = dummy for offer to receive training services • Size of complier group ≈ 61% • Proportion of treated (those who received training) who are compliers ≈ (.67/.42)*.61 ≈ .97 • IV estimate of training effect: 1593.14 (894.75) • Seems plausible that there are no defiers • Compliers = people who would not receive training if not offered but choose to have training when offered • Presumably, the remaining 39% of the population are mostly never-takers

Example: Returns to Schooling • Returns to schooling more complicated • Multi-valued treatment • Multiple instruments • IV estimand with one binary instrument = weighted average of effects of increasing schooling by one year across all schooling levels among compliers • 2SLS estimand= weighted average of individual IV estimands • Monotonicity condition => change in instrument leads to weakly increasing (decreasing) levels of schooling for everyone

Example: Returns to Schooling • Consider dummy for being born quarter j (qj) • Maintain exclusion restriction as before • First-stage: Regress schooling on qj • Can estimate fraction of compliers at each schooling level as P(si < s|qji = 0) – P(si< s|qji= 1) (Assuming monotonicity such that changing instrument from 0 to 1 increases schooling) • Can estimate weight given to each schooling value as (fraction of compliers)/first-stage

Example: Returns to Schooling Fractions of compliers for different quarter of birth instruments.

Example: Returns to Schooling Weighting functions for different quarter of birth instruments. Note this is the fraction of compliers scaled by the first-stage coefficient.

Example: Returns to Schooling • OLS Estimate: .067 (.0004) • IV Estimates: • Q2: .166 (.071) • Q3: .209 (.076) • Q4: .085 (.026) • 2SLS Estimate: .108 (.020) [Weighted average of the individual IV estimates] • Compliers: People who got more school due to being born later in the year. Margin is mostly in 10-12 years of education.

Many Instruments • IV Estimates often imprecise • Only use the variation induced by the instrument • Often plausible instruments (in the sense of satisfying the exclusion restriction) have weak predictive power for outcome • One approach to increasing precision is to use more instruments • Potentially allow extraction of more signal, adding information helps • But… • Are all the instruments really excludable? • Overfitting is bad

Many Instruments: Overfitting • Extreme case, # instruments (K) = # observations (n) • Recall where PZ = Z(Z’Z)-1Z’ • K=n => (Z’Z)-1 = Z-1(Z’)-1 => PZ = I (the identity matrix) • So • Get back original, contaminated object since perfectly fit both signal AND noise. • Overfittingfits contaminated noise as well as signal

Example: Returns to Schooling • Rather than just use 3 quarter of birth effects, could use quarter of birth interacted with year of birth x state of birth as instruments (1527 instruments) • 2SLS estimate: .0712 (.0049) • Recall OLS gives .067 (.0004) and 2SLS with 3 quarter of birth dummies gives .108 (.020) • Theory, simulation evidence, the intuition above suggests 2SLS is strongly biased towards OLS when many instruments are used

Introduction to Instrumental Variables Methods