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1 Instrumental Variables Kosuke Imai Harvard University STAT186/GOV2002 CAUSAL INFERENCE Fall 2018 Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
2 Instrumental Variables From randomized encouragement design to general instrumental variabes approach: U Y Z T Instruments in the nature natural experiments 1 random assignment of Z 2 no direct effect of Z on Y Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
3 Classical Instrumental Variables Estimator Linear model (in matrix notation): Y = Xβ + ɛ where E(ɛ) = 0 n and X is n K Endogeneity: E(ɛ i X) 0 Instruments Z is n L 1 Exogeneity: E(ɛ i Z) = 0 2 Exclusion restriction: Z i does not belong to the outcome model 3 Rank condition: Z X and Z Z have full rank Experimental setting: X i = the treatment and pre-treatment covariates Z i = the randomized encouragement and pre-treatment covariates Identification 1 K = L: just-identified 2 K < L: over-identified 3 K > L: under-identified Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
4 Geometry of Instrumental Variables Projection matrix (onto S(Z)): P Z = Z(Z Z) 1 Z Purge endogeneity: X = P Z X Since P Z = P Z and P ZP Z = P Z, we have ˆβ IV = ( X X) 1 X Y = (X Z(Z Z) 1 Z X) 1 X Z(Z Z) 1 Z Y Two stage least squares: 1 Regress X on Z and obtain the fitted values X 2 Regress Y on X We do not assume the linearity of X in Z Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
5 Asymptotic Inference Estimation error: ˆβ IV β = (X Z(Z Z) 1 Z X) 1 X Z(Z Z) 1 Z ɛ ( ) ( ) 1 ( ) 1 n = X i Z 1 n i Z i Z 1 n i Z i X i n n n ( 1 n i=1 n X i Z i i=1 ) ( 1 n i=1 n Z i Z i i=1 p Thus, ˆβ IV β Under the homoskedasticity, V(ɛ Z) = σ 2 I n : ) 1 ( 1 n i=1 ) n Z i ɛ i n( ˆβIV β) d N (0, σ 2 [E(X i Z i ){E(Z i Z i )} 1 E(Z i X i )] 1 ) i=1 1 Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
6 Residuals and Robust Standard Error ˆɛ = Y X ˆβ IV and not ˆɛ Y X ˆβ IV Under homoskedasticity: ˆσ 2 = ˆɛ 2 n K p σ 2 V( ˆβ IV ) = ˆσ 2 {X Z(Z Z) 1 Z X} 1 = ˆσ 2 ( X X) 1 Sandwich heteroskedasticity consistent estimator: bread = {X Z(Z Z) 1 Z X} 1 X Z(Z Z) 1 ( ) n meat = Z diag(ˆɛ 2 i )Z = ˆɛ 2 i Z iz i bread meat bread = ( X X) 1 X diag(ˆɛ 2 i ) X( X X) 1 Robust standard errors for clustering, auto-correlation, etc. i=1 Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
7 Multi-valued Treatment (Angirst and Imbens J. Am. Stat. Assoc) Two stage least squares regression: Y i = α + βt i + η i, T i = δ + γz i + ɛ i Binary encouragement and binary treatment, ˆβ = ĈATE (no covariate) ˆβ p CATE (with covariates) Binary encouragement multi-valued treatment Monotonicity: T i (1) T i (0) Exclusion restriction: Y i (1, t) = Y i (0, t) for each t = 0, 1,..., K Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
8 Estimator ˆβ TSLS p Cov(Y i, Z i ) Cov(T i, Z i ) = E(Y i(1) Y i (0)) E(T i (1) T i (0)) K K ( Yi (1) Y i (0) ) = w jk E T i (1) = j, T i (0) = k j k k=0 j=k+1 where w jk is the weight, which sums up to one, defined as, w jk = (j k) Pr(T i (1) = j, T i (0) = k) K k =0 K j =k +1 (j k ) Pr(T i (1) = j, T i (0) = k ). Easy interpretation under the constant additive effect assumption for every complier type Assume encouragement induces at most only one additional dose Then, w k = Pr(T i (1) = k, T i (0) = k 1) Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
9 Quarter of Birth (Angrist and Krueger Q. J. Econ.) Instrument for educational attainment to address ability bias Outcome: men s log weekly earnings in 1980 Compulsory education law in US: students must attend school until they reach age 16 Those born in the third or fourth quarter typically finish tenth grade before reaching age 16 Instrument at most decreases years of education by one year Weak instrument: first quarter vs. 2nd to 4th quarter 1920s cohorts: est. = 0.126, s.e. (HC) = 0.016, corr = s cohorts: est. = 0.109, s.e. (HC) = 0.013, corr = Wald estimates: 1920 cohorts: est. = 0.072, s.e. (HC) = cohorts: est. = 0.102, s.e. (HC) = OLS estimates: 1920 cohorts: est. = 0.080, s.e. (HC) = cohorts: est. = 0.071, s.e. (HC) = Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
10 CDFs for First and Fourth Quarter of Birth C D F I. n YEARS OF SCHOOLING Figure 2. Schooling CDF by Quarter of Birth (Men Born ; Data From the 1980 Census). Quarter of birth:, first; - -, fourth. Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
11 Analysis of Weak Instruments Recall the Wald estimator: ˆβ IV = Cov(Y i, Z i ) Cov(T i, Z i ) ˆβ IV does not exist if the instrument is irrelevant Consider the following model: Y i = α + βt i + ɛ i, T i = γ Z }{{} i + η i, where E(ɛ i Z i ) = E(η i Z i ) = 0 0 where (ɛ i, η i ) follows a bivariate normal with mean zero. Then, n i=1 ɛ iz i d n i=1 η Corr(ɛi, η i ) iz i ˆβ iv β Asymptotic analysis for weak instruments V(ɛ i ) V(η i ) + W i }{{} Cauchy Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
12 Simulated Instruments (Bound et al J. Am. Stat. Assoc) Simulation exercise: 1 Simulate Z i from Bernoulli with success probability equal to its empirical estimate 2 Compute the Wald estimate as before 1920s cohorts: Estimates: min = , 1st Qu. = 0.093, median = , 3rd Qu. = 0.260, max = Std. Errors: min = 0.057, 1st Qu. = 0.185, median = 0.393, 3rd Qu. = 1.467, max = s cohorts: Estimates: min = , 1st Qu. = 0.117, median = 0.078, 3rd Qu. = 0.284, max = Std. Errors: min = 0.064, 1st Qu. = 0.197, median = 0.421, 3rd Qu. = 1.814, max = Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
13 Randomization Inference (Imbens and Rosenbaum J. R. Stat. Soc. A.) Constant additive treatment effect model for the QoB example: Randomization test: Y i (t) = Y i (0) + β t for t = 0, 1,... 1 Null hypothesis: H 0 : β = β 0 2 Test statistic: S i = f (Y i β 0 T i, Z i ) 3 Assume Z i Bernoulli(Z n ) to obtain the reference distribution Application: S i = N i=1 Z i rank(y i β 0 T i ) 95% confidence intervals: 1920 cohorts: [0.036, 0.106], [0.028, 0.115] (Wald) 1930 cohorts: [0.049, 0.122], [0.055, 0.149] (Wald) Simulation (rejection rates of 0.05 level tests with 1000 simulations): 1920 cohorts: 0.048, (Wald) 1930 cohorts: 0.051, (Wald) Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
14 Violations of IV Assumptions 1 Violation of exclusion restriction: bias = ITT noncomplier Pr(noncomplier) Pr(complier) Weak encouragement (instruments) Direct effects of encouragement; failure of randomization, alternative causal paths 2 Violation of monotonicity: bias = {CATE + ITT defier} Pr(defier) Pr(complier) Pr(defier) Proportion of defiers Heterogeneity of causal effects Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
15 Bounding the Average Treatment Effect (Manski. (1990). Am. Econ. Rev.) Instrumental variable estimator does not point-identify the ATE Partial identification (Manski Identification Problems in the Social Sciences. Harvard UP) Consider a binary outcome with the randomized encouragement and exclusion restriction: Pr(Y i (1) = 1) = Pr(Y i (1) = 1 Z i = 1) = µ 11 π 1 + Pr(Y i (1) = 1 D i = 0, Z i = 1)(1 π 1 ) Pr(Y i (0) = 1) = µ 00 (1 π 0 ) + Pr(Y i (0) = 1 D i = 1, Z i = 0)π 0 where µ dz = Pr(Y i = 1 D i = d, Z i = z), π z = Pr(D i = 1 Z i = z) Bounds on the ATE: µ 11 π 1 µ 00 (1 π 0 ) π 0 τ µ 11 π 1 µ 00 (1 π 0 ) + 1 π 1 where the width equals 1 (π 1 π 0 ) Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
16 Sharp Bounds (Balke and Pearl J. Am. Stat. Assoc) The previous bounds are not sharp: only consider four latent types based on mapping from Z to D there are four additional mappings from D to Y a total of 16 latent types: (D i (1), D i (0), Y i (1), Y i (0)) equal to Manki s bounds under monotonicity Linear programming problem: maximize/minimize u Pr(Y i (d) = 1 U i = u) Pr(U i = u) subject to Pr(Y i = y, D i = d Z i = z) = Pr(Y i (d) = y D i = d, U i = u) Pr(D i = d Z i = z, U i = u) Pr(Z i = z) Pr(U i = u) A general strategy for a discrete potential outcome case Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
17 Revisiting the Habitual Voting Example Effect of voting in 2006 election on the turnout in the 2008 election: est = 0.128, s.e. = Potential bias of estimated CATE due to exclusion restriction: ITT noncomplier = ITT noncomplier Inference for the Average Treatment Effect exclusion restriction + monotonicity: [ 0.315, 0.602] exclusion restriction alone: [ 0.315, 0.602] Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
18 Summary Instrumental variables as a general strategy for coping with selection bias randomization of instruments monotonicity exclusion restriction Extensions to multi-valued treatment Weak instruments and randomization inference ATE vs. CATE partial identification, method of bounds Suggested readings: IMBENS AND RUBIN. Chapter 25 ANGRIST AND PISCHKE. Chapter 4 Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 18
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