1 / 39 Applied Microeconometrics Chapter 5 Instrumental Variables with Heterogeneous Causal Effect Romuald Méango & Michele Battisti LMU, SoSe 2016
2 / 39 Instrumental Variables with Heterogeneous Causal Effect Chapter Overview: Heterogeneity in Treatment Effect (Deaton, 2010) LATE - Local Average Treatment Effect (MHE Chap 4.4) How Useful is LATE? (MHE Chap 4.4) Reading for the Tutorial: Angrist (ECA, 1998)
3 / 39 Running Example Angrist (1990): Causal relationship of interest: The effects of the Vietnam-era military service on the earnings of the veterans. Selection problem: Veterans are not a random sample of the population. Identification: Draft-eligibility lottery as instrument. In the 1960s and 1970s, young men were at risk of being drafter for military service. Concerns about the fairness of US conscription policy led to the institution of a draft lottery [over birthdays] that was used to determine priority for conscription.
4 / 39 Running Example Angrist (1990): Random sequence numbers were randomly assigned to each birth date. Ceiling below which men were eligible for the draft. Men with large numbers could not be drafted. Many draft-eligible were still exempted from service for health and other reasons. Many men who were draft-exempt volunteered. Concern: decision to comply to draft-eligibility is correlated with the treatment effect intensity.
5 / 39 Heterogeneity in Treatment Effect Let Y i be the earnings of i, D i : veteran status, and Z i : the randomized draft-eligibility. where α = E(Y 0i D = 1), Y i = Y 0i + (Y 1i Y 0i )D i = α + ρd i + η i. ρ = E(Y 1i Y 0i D = 1), and η i = Y 0i E(Y 0i D i = 1) + [(Y 1i Y 0i ) E(Y 1i Y 0i D i = 1)] D i := v i + (τ τ)d i
6 / 39 Heterogeneity in Treatment Effect Condition for consistency: E(Z i η i ) = 0 E(Z i η i ) = E(Z i (v i + (τ τ)d i )) = E(Z i (τ τ)d i ), by random assignment = E(τ τ Z i = 1, D i = 1) P(Z i = 1, D i = 1) The exogeneity condition holds if: 1. The average effect of military service among the veterans induced into military service by the draft-eligibility is the same as the average effect among those who would have participated anyway. 2. or no one who is not drafted volunteers.
7 / 39 Heterogeneity in Treatment Effect In the example of Vietnam veterans, the instrument (the draft lottery number) fails to be exogenous because the error term in the earnings equation depends on each individual s rate of return to schooling, and whether or not potential draftee accepted their assignment [...] depends on the rate of return. Deaton, 2010.
8 / 39 Heterogeneity in Treatment Effect An alternative way to see it with the Wald Estimator: E (Y i Z i = 1) = E (Y 0i + (Y 1i Y 0i )D 1i Z i = 1) E (Y i Z i = 0) = E (Y 0i + (Y 1i Y 0i )D 0i Z i = 0) Take the difference: E (Y i Z i = 1) E (Y i Z i = 0) = E ((Y 1i Y 0i )(D 1i D 0i )) = E (Y 1i Y 0i D 1i > D 0i ) P (D 1i > D 0i ) E (Y 1i Y 0i D 1i < D 0i ) P (D 1i < D 0i )
9 / 39 Heterogeneity in Treatment Effect If the treatment effect is constant: E (Y 1i Y 0i D 1i > D 0i ) = E (Y 1i Y 0i D 1i > D 0i ) = ρ So that: E (Y i Z i = 1) E (Y i Z i = 0) = E ((Y 1i Y 0i )(D 1i D 0i )) = ρ (P (D 1i > D 0i ) P (D 1i < D 0i )) = ρe(d 1i D 0i ) = ρ (E (D i Z i = 1) E (D i Z i = 0))
10 / 39 Heterogeneity in Treatment Effect If the treatment effect is not constant: E (Y 1i Y 0i D 1i > D 0i ) > 0 E (Y 1i Y 0i D 1i < D 0i ) > 0 does not guarantee that the Wald estimator is positive. Treatment effect can be positive for everyone, yet, the reduced form can be zero or even negative. To solve the problem, the LATE will assume either P (D 1i < D 0i ) = 0 or P (D 1i > D 0i ) = 0
11 / 39 LATE Notations: Y i (d, z) potential outcome of i with D i = d and Z i = z. Example: Effect of veteran status on income: - Causal effect of veteran status given Z i : Y i (1, Z i ) Y i (0, Z i ) - Causal effect of draft-eligibility given D i : Y i (D i, 1) Y i (D i, 0) D 1i, i s treatment status when Z i = 1. D 0i, i s treatment status when Z i = 0.
12 / 39 LATE Observed treatment status: D i = D 0i + (D 1i D 0i )Z i = π 0 + π 1i Z i + ξ Example: Effect of veteran status on income: - D 0i whether i would serve in the military if he is draft-ineligible. - D 1i whether i would serve in the military if he is draft-eligible. π 1i heterogeneous causal effect of Z i on D i. E[π 1i ] Average Causal Effect of Z i on D i.
13 / 39 LATE s Assumptions Assumption (Independence) ({Y i (d, z); d, z}, D 1i, D 0i ) = Z i The instrument is as good as randomly assigned. Implications: E (Y i Z i = 1) E (Y i Z i = 0) = E (Y i (D 1i, 1) Z i = 1) E (Y i (D 0i, 0) Z i = 0) = E (Y 1i ) E (Y 0i ) E (D i Z i = 1) E (D i Z i = 0) = E (D 1i ) E (D 0i )
14 / 39 LATE s Assumptions Assumption (Exclusion restriction) Y i (d, 1) = Y i (d, 0) := Y di Potential outcomes are only a function of d, not of z. The instrument operates through a single known channel. Y i = Y i (0, Z i ) + (Y i (1, Z i ) Y i (0, Z i )) D i (1) = Y 0i + (Y 1i Y 0i ) D i (2) = α 0 + ρ i D i + η i (3) The traditional error-term notation E(Z i η i ) does not clearly distinguish between independence and exclusion restrictions.
15 / 39 LATE s Assumptions Example: Effect of veteran status on income Educational draft deferments would have led men with low lottery numbers to stay in college longer than they would have otherwise desired. If so, draft lottery numbers are correlated with earnings for at least two reasons: an increased likelihood of military service and an increased likelihood of college attendance. The fact that the lottery number is randomly assigned (and therefore satisfies the independence assumption) does not make this possibility less likely.
16 / 39 LATE s Assumptions Assumption (Monotonicity / Uniformity) Either D 1i D 0i or D 1i D 0i for all i. Equivalently, there exists a function µ(z i ) and random variable V such that: D i = I(µ(Z i ) V i ) Example: Effect of veteran status on income (D 1i D 0i ) no one who was actually kept out of the military by being draft-eligible.
17 / 39 LATE Theorem Theorem (LATE) (A1, Independence) ({Y i (d, z); d, z}, D 1i, D 0i ) Z i ; (A2, Exclusion) Y i (d, 1) = Y i (d, 0) := Y di, for d = 0, 1; (A3, First-stage) E(D 1i D 0i ) 0 (A4, Monotonicity) D 1i D 0i 0, i, or vice-versa; Then: = E (Y i Z i = 1) E (Y i Z i = 0) E (D i Z i = 1) E (D i Z i = 0) = E (Y 1i Y 0i D 1i > D 0i ) (4) = E(ρ i π 1i > 0)
18 / 39 LATE Theorem Proof: Note that: E (Y i Z i = 1) = E (Y 0i + (Y 1i Y 0i )D 1i Z i = 1) E (Y i Z i = 0) = E (Y 0i + (Y 1i Y 0i )D 0i Z i = 0) So that the difference is: E ((Y 1i Y 0i )(D 1i D 0i )) = E ((Y 1i Y 0i ) D 1i > D 0i ) P (D 1i > D 0i ) A similar argument shows that: E (D i Z i = 1) E (D i Z i = 0) = P (D 1i > D 0i )
19 / 39 LATE Theorem Interpretation:...an instrument which is as good as randomly assigned, affects the outcome through a single known channel, has a first-stage, and affects the causal channel of interest only in one direction, can be used to estimate the average causal effect on the affected group. [IV ] estimate[s] the effect of military service on men who served because they were draft-eligible, but would not otherwise have served.
20 / 39 IV and Causality: Wald Estimator Angrist (1990):
21 / 39 LATE: Why Monotonicity? Without monotonicity: E ((Y 1i Y 0i )(D 1i D 0i )) = E (Y 1i Y 0i D 1i > D 0i ) P (D 1i > D 0i ) E (Y 1i Y 0i D 1i < D 0i ) P (D 1i < D 0i ) Treatment effect can be positive for everyone, yet, the reduced form is zero or even negative. With heterogeneity, the LATE can fail to identify the sign of the effect.
22 / 39 How useful is the LATE? No theorem answers this question, but it s always worth discussing. 1. Whose effect is identified? 2. Special cases where LATE = ATT or LATE = ATNT. 3. Counting and characterizing the compliers.
23 / 39 LATE: effect identified Definition 1. Compliers. D 1i = 1 and D 0i = 0. 2. Defiers. D 1i = 0 and D 0i = 1. 3. Always-Takers. D 1i = 1 and D 0i = 1. 4. Never-Takers. D 1i = 0 and D 0i = 0. LATE is the effect of treatment on the population of compliers. LATE rules out defiers.
24 / 39 LATE: effect identified LATE is not informative about effects on never-takers and always-takers because, by definition, treatment status for these two groups is unchanged by the instrument. In general: ATE LATE, ATT LATE, ATNT LATE: - ATE: whole population; - ATT : compliers (with instrument switched on) and always-takers; - ATNT : compliers (with instrument switched off) and never-takers;
25 / 39 LATE: effect identified ATT: E (Y 1i Y 0i D i = 1) = E (Y 1i Y 0i D 0i = 1, D i = 1) P(D 0i = 1 D i = 1) +E (Y 1i Y 0i D 1i > D 0i, Z i = 1) P(D 1i > D 0i, Z i = 1 D i = 1) = E (Y 1i Y 0i D 0i = 1) P(D }{{} 0i = 1 D i = 1) effect on always-takers + E (Y 1i Y 0i D 1i > D 0i ) P(D }{{} 1i > D 0i, Z i = 1 D i = 1) effect on compliers
26 / 39 LATE: effect identified ATNT: ATE: E (Y 1i Y 0i D i = 0) = E (Y 1i Y 0i D 1i = 0) P(D }{{} 0i = 0 D i = 0) effect on never-takers + E (Y 1i Y 0i D 1i > D 0i ) P(D }{{} 1i > D 0i, Z i = 0 D i = 0) effect on compliers E (Y 1i Y 0i ) = E (Y 1i Y 0i D i = 1) P(D i = 1) +E (Y 1i Y 0i D i = 0) P(D i = 0).
27 / 39 LATE: Special cases Twin instrument (Angrist and Evans, 1998): Causal relationship of interest: Effect of fertility on labor force participation. Treatment D i : Third birth. Instrument Z i : Multiple second birth. Assumptions: - Z i is randomly assigned, - Multiple births affect outcomes only by increasing fertility, - No one has a lower fertility because of a multiple birth.
28 / 39 LATE: Special cases Twin instrument (Angrist and Evans, 1998): LATE = ATNT = E (Y 1i Y 0i D i = 0) all women who have a multiple second birth end up with three children, i.e., there are no never-takers in response to the twins instrument.
29 / 39 LATE: Special cases Angrist and Evans (1998):
30 / 39 LATE: effect identified Chattopadhyay and Duflo (2004): Causal relationship of interest: Effect of women as local ruler on development outcome. Treatment D i : women as a local chief. Instrument Z i : reservation seat. Assumptions: - Z i is randomly assigned, - Reservation outcomes only through the woman position, - No village would have man local chief because the seat was reserved to a women.
31 / 39 LATE: Special Cases Chattopadhyay and Duflo (2004): LATE = ATNT ATT ATE
32 / 39 LATE: Special Cases Randomized trials with one sided non-compliance. Causal relationship of interest: Effect of treatment on outcome. Treatment D i : treatment. Instrument Z i : randomized treatment assignment. One sided non-compliance: - Randomized trial. - Voluntary participation induces imperfect compliance. - No one in the control group has access to the treatment, E (D i Z i = 0) = 0.
33 / 39 LATE: Special Cases Randomized trials with one sided non-compliance. LATE = ATT = Example: See MHE Section 4.4.3. ITT Compliance rate = E (Y i Z i = 1) E (Y i Z i = 0) E (D i Z i = 1)
34 / 39 LATE: Counting and Characterizing the Compliers How representative are compliers of the whole population? Proportion of compliers: P(D 1i > D 0i ) = E(D i Z i = 1) E(D i Z i = 0) Proportion of compliers among treated: P(D 1i > D 0i D i = 1) = P(D i = 1 D 1i > D 0i )P(D 1i > D 0i ) P(D i = 1) = P(Z i = 1)(E(D i Z i = 1) E(D i Z i = 0)) P(D i = 1)
LATE: Counting and Characterizing the Compliers 35 / 39
36 / 39 LATE: Counting and Characterizing the Compliers How representative are compliers of the whole population? Proportion of individuals with characteristics x i among compliers compared to the proportion in the population: P(x i = 1 D 1i > D 0i ) P(x i = 1) = P(D 1i > D 0i x i = 1) P(D 1i > D 0i ) = (E(D i Z i = 1, x i = 1) E(D i Z i = 0, x i = 1)) (E(D i Z i = 1) E(D i Z i = 0))
LATE: Counting and Characterizing the Compliers 37 / 39
38 / 39 Selected Bibliography I Angrist, J. D. (1990). Lifetime earnings and the vietnam era draft lottery: evidence from social security administrative records. The American Economic Review, 80(3):313 336. Angrist, J. D. (1998). Estimating the labor market impact of voluntary military service using social security data on military applicants. Econometrica, 66(2):249 288. Angrist, J. D., Imbens, G. W., and Rubin, D. B. (1996). Identification of causal effects using instrumental variables. Journal of the American statistical Association, 91(434):444 455.
39 / 39 Selected Bibliography II Deaton, A. (2010). Instruments, randomization, and learning about development. Journal of Economic Literature, 48(2):424 455. Imbens, G. W. (2010). Better late than nothing. Journal of Economic Literature, 48(2):399 423.