Nguyen Ngoc Anh Nguyen Ha Trang Applied Econometrics Instrumental Variable Approach DEPOCEN

Topics That Will Be Covered in this Workshop Why use IV? – Discussion of endogeneity bias – Statistical motivation for IV What is an IV? – Identification issues – Statistical properties of IV estimators How is an IV model estimated? – Software and data examples – Diagnostics: IV relevance, IV exogeneity, Hausman

Review of the Linear Model (in metrix algebra) Population model: Y = α + β X + ε – Assume that the true slope is positive, so β > 0 Sample model: Y = a + bX + e – Least squares (LS) estimator of β: b LS = (X ′ X) –1 X ′ Y = Cov(X,Y) / Var(X) Under what conditions can we speak of b LS as a causal estimate of the effect of X on Y?

Review of the Linear Model Key assumption of the linear model: – E(  |x) = E(  ) = 0  Cov(x,  ) = E(x  ) = 0 – E(X ′ e) = Cov(X,e) = E(e | X) = 0 – Exogeneity assumption = X is uncorrelated with the unobserved determinants of Y Important statistical property of the LS estimator under exogeneity: E(b LS ) = β + Cov(X,e) / Var(X) plim(b LS ) = β + Cov(X,e) / Var(X) Second terms 0, so b LS unbiased and consistent

Review of the Linear Model When you regress Y on X, Y = β 0 + β 1 X + ε and the OLS estimate of β 1 can be described as But since X and ε are correlated, b OLS does not estimate β 1 but some other quantity that depends on the correlation of X and ε

Endogeneity and the Evaluation Problem When is the exogeneity assumption violated? – Measurement error → Attenuation bias – Instantaneous causation → Simultaneity bias – Omitted variables → Selection bias Selection bias is the problem in observational research that undermines causal inference – Measurement error and instantaneous causation can be posed as problems of omitted variables Potential outcome approach!!!!

When Is the Exogeneity Assumption Violated? Omitted variable (W) that is correlated with both X and Y – Classic problem of omitted variables bias Coefficient on X will absorb the indirect path through W, whose sign depends on Cov(X,W) and Cov(W,Y) XY W Things more complicated in applied settings because there are bound to be many W’s, not to mention that the “smearing” problem applies in this context also

Example #1: Police Hiring Measurement error – Mobilization of sworn officers (M.E. in X) as well as differential victim reporting or crime recording (M.E. in Y) may be correlated with police size Instantaneous causation – More police might be hired during a crime wave Omitted variables – Large departments may differ in fundamental ways difficult to measure (e.g., urban, heterogeneous)

Example #2: Delinquent Peers Measurement error – Highly delinquent youth probably overestimate the delinquency of their peers (M.E. in X), and likely underestimate their own delinquency (M.E. in Y) Instantaneous causation – If there is influence/imitation, then it is bidirectional Omitted variables – High-risk youth probably select themselves into delinquent peer groups (“birds of a feather”)

Regression Estimation Ignoring Omitted Variables Suppose we estimate treatment effect model: Y = α + β X + ε – Let’s assume without loss of generality that X is a binary “treatment” (= 1 if treated; = 0 if untreated) Least squares estimator: b LS = Cov(X,Y) / Var(X) = E(Y | X = 1) – E(Y | X = 0) – Simply the difference in means between “treated” units (X = 1) and “untreated” units (X = 0)

Estimating Treatment Effects Consider treatment assignment (dummy variable) X and outcome Y Regress Y on X Y i = β 0 + β 1 X i + ε i The estimate of β 1 is just the difference between the mean Y for X = 1 (the treatment group) and the mean Y for X = 0 (the control group) Thus the OLS estimate is = β 1 +

Estimating Treatment Effects (With Random Assignment) If the treatment is randomly assigned, then X is uncorrelated with ε (X is exogenous) If X is uncorrelated with ε if and only if But if, then the mean difference is = β 1 + = β 1 This implies that standard methods (OLS) give an unbiased estimate of β 1, which is the average treatment effect That is, the treatment-control mean difference is an unbiased estimate of β 1,

What goes wrong without randomization? If we do not have randomization, there is no guarantee that X is uncorrelated with ε (X may be endogenous) Thus the OLS estimate is still = β 1 + If X is correlated with ε, then Hence does not estimate β 1, but some other quantity that depends on the correlation of X and ε If X is correlated with ε, then standard methods give a biased estimate of β 1

Omitted Variables in applied research What variables of interest to us are surely endogenous? – Micro = Employment, education, marriage, military service, fertility, conviction, family structure,.... – Macro = Poverty, unemployment rate, collective efficacy, immigrant concentration,.... Basically, EVERYTHING! – (I’m sorry But it suck)

Potential outcome framework

Traditional Strategies to Deal with Omitted Variables Randomization (physical control) Covariate adjustment (statistical control) – Control for potential W’s in a regression model – But...we have no idea how many W’s there are, so model misspecification is still a real problem here

Quasi-Experimental Strategies to Deal with Omitted Variables Difference in differences (fixed-effects model) – Requires panel data Propensity score matching – Requires a lot of measured background variables Similar to covariate adjustment, but only the treated and untreated cases which are “on support” are utilized Instrumental variables estimation – Requires an exclusion restriction

Instrumental Variables Estimation Is a Viable Approach An “instrumental variable” for X is one solution to the problem of omitted variables bias Requirements for Z to be a valid instrument for X – Relevant = Correlated with X – Exogenous = Not correlated with Y but through its correlation with X Z XY W e

Important Point about Instrumental Variables Models I often hear...“A good instrument should not be correlated with the dependent variable” – WRONG!!! Z has to be correlated with Y, otherwise it is useless as an instrument – It can only be correlated with Y through X – (trong X có 2 phần, 1 phần dính với e một phần với Y, muốn tận dụng phần dính với Y) A good instrument must not be correlated with the unobserved determinants of Y

Important Point about Instrumental Variables Models Not all of the available variation in X is used – Only that portion of X which is “explained” by Z is used to explain Y XY Z X = Endogenous variable Y = Response variable Z = Instrumental variable

Important Point about Instrumental Variables Models XY Z Realistic scenario: Very little of X is explained by Z, or what is explained does not overlap much with Y XY Z Best-case scenario: A lot of X is explained by Z, and most of the overlap between X and Y is accounted for

Important Point about Instrumental Variables Models The IV estimator is BIASED – In other words, E(b IV ) ≠ β (finite-sample bias) – The appeal of IV derives from its consistency “Consistency” is a way of saying that E(b) → β as N → ∞ So…IV studies often have very large samples – But with endogeneity, E(b LS ) ≠ β and plim(b LS ) ≠ β anyway Asymptotic behavior of IV plim(b IV ) = β + Cov(Z,e) / Cov(Z,X) – If Z is truly exogenous, then Cov(Z,e) = 0

Instrumental Variables Terminology Three different models to be familiar with – First stage: X = α 0 + α 1 Z + ω – Structural model: Y = β 0 + β 1 X + ε – Reduced form: Y = δ 0 + δ 1 Z + ξ

More on the Method of Two-Stage Least Squares (2SLS) Step 1: X = a 0 + a 1 Z 1 + a 2 Z 2 +  + a k Z k + u – Obtain fitted values (X̃) from the first-stage model Step 2: Y = b 0 + b 1 X̃ + e – Substitute the fitted X̃ in place of the original X – Note: If done manually in two stages, the standard errors are based on the wrong residual e = Y – b 0 – b 1 X̃ when it should be e = Y – b 0 – b 1 X Best to just let the software do it for you

Some examples

Including Control Variables in an IV/2SLS Model Control variables (W’s) should be entered into the model at both stages – First stage: X = a 0 + a 1 Z + a 2 W + u – Second stage: Y = b 0 + b 1 X̃ + b 2 W + e Control variables are considered “instruments,” they are just not “excluded instruments” – They serve as their own instrument

Functional Form Considerations with IV/2SLS Binary endogenous regressor (X) – Consistency of second-stage estimates do not hinge on getting first-stage functional form correct Binary response variable (Y) – IV probit (or logit) is feasible but is technically unnecessary In both cases, linear model is tractable, easily interpreted, and consistent – Although variance adjustment is well advised

Technical Conditions Required for Model Identification Order condition = At least the same # of IV’s as endogenous X’s – Just-identified model: # IV’s = # X’s – Overidentified model: # IV’s > # X’s Rank condition = At least one IV must be significant in the first-stage model – Number of linearly independent columns in a matrix E(X | Z,W) cannot be perfectly correlated with E(X | W)

Instrumental Variables and Randomized Experiments Imperfect compliance in randomized trials – Some individuals assigned to treatment group will not receive T x, and some assigned to control group will receive T x Assignment error; subject refusal; investigator discretion – Some individuals who receive T x will not change their behavior, and some who do not receive T x will change their behavior A problem in randomized job training studies and other social experiments (e.g., housing vouchers)

Durbin-Wu-Hausman (DWH) Test Balances the consistency of IV against the efficiency of LS – H 0 : IV and LS both consistent, but LS is efficient – H 1 : Only IV is consistent DWH test for a single endogenous regressor: DWH = (b IV – b LS ) / √(s 2 b IV – s 2 b LS ) ~ N(0,1) – If |DWH| > 1.96, then X is endogenous and IV is the preferred estimator despite its inefficiency

Durbin-Wu-Hausman (DWH) Test A roughly equivalent procedure for DWH: 1. Estimate the first-stage model 2. Include the first-stage residual in the structural model along with the endogenous X 3. Test for significance of the coefficient on residual Note: Coefficient on endogenous X in this model is b IV (standard error is smaller, though) – First-stage residual is a “generated regressor”

Software Considerations Basic model specification in Stata ivreg y (x = z) w [weight = wtvar], options y = dependent variable x = endogenous variable z = instrumental variable w = control variable(s)