Lecture 8. Roy Model, IV with essential heterogeneity, MTE

Transcription

1 Lecture 8. Roy Model, IV with essential heterogeneity, MTE Economics 2123 George Washington University Instructor: Prof. Ben Williams

2 Heterogeneity When we talk about heterogeneity, usually we mean heterogeneity in causal effects. The individual causal effect differs across individuals. James Heckman, among many others, has argued over the past years that this type of heterogeneity is prevalent.

3 Heterogeneity Recall the discussion from the first lecture: Y 1i Y 0i represents the individual treatment effect δ x = E(Y 1i Y 0i X i = x) is the average treatment effect conditional on x observable heterogeneity is when δ x varies with x

4 Heterogeneity Recall the discussion from the first lecture: Y 1i Y 0i represents the individual treatment effect δ x = E(Y 1i Y 0i X i = x) is the average treatment effect conditional on x observable heterogeneity is when δ x varies with x under the conditional independence assumption, OLS estimates a weighted average, x w xδ x.

5 Heterogeneity Recall the discussion from the first lecture: Y 1i Y 0i represents the individual treatment effect δ x = E(Y 1i Y 0i X i = x) is the average treatment effect conditional on x observable heterogeneity is when δ x varies with x under the conditional independence assumption, OLS estimates a weighted average, x w xδ x. note that this result allows for unobserved heterogeneity too because we do not assume that Y 1i Y 0i = δ Xi.

6 Heterogeneity What if the conditional independence assumption fails? we may use an instrumental variable strategy if there is also heterogeneity, what does IV estimate?

7 Heterogeneity What if there is unobserved heterogeneity? i.e., if Y 1i Y 0i δ Xi this could be ok a textbook example: suppose Y i = α + β i D i + u i where β i = β + η i

8 Heterogeneity What if there is unobserved heterogeneity? i.e., if Y 1i Y 0i δ Xi this could be ok a textbook example: suppose Y i = α + β i D i + u i where β i = β + η i then Y i = α + βd i + ε i where ε i = u i + η i D i if E(u i D i ) = 0 and E(η i D i ) = 0 then OLS estimates β = E(β i )

9 Heterogeneity The Roy model will be used to demonstrate a link between unobserved heterogeneity and endogeneity. The textbook example above is misleading because often η i will be correlated with D i. Moreover, even if Z i is uncorrelated with u i it will often not be uncorrelated with η i D i.

10 Roy model The Roy model is a model of comparative advantage: Potential earnings in sectors 0 and 1: Y 0, Y 1 Individuals choose sector 1 if and only if Y 1 Y 0 c where c is a nonrandom cost. Heckman and Honore (1990) studied the empirical implications and identification of this model.

11 Roy model The extended and generalized Roy model: the extended model allows for an observable cost component, D = 1(Y 1 Y 0 c(z )) where Z is a vector of covariates and c is a possibly unknown function. the generalized model allows for an unobservable cost component, D = 1(Y 1 Y 0 c(z, V )) where V is unobservable

12 Roy model In the Roy model, Y d = µ d + U d where E(U d ) = 0 for d = 0, 1. If we observe a vector of covariates X, µ d = µ d (X). Often µ d (X) = β d X. Now Y i = Y 0i + (Y 1i Y 0i )D i = µ 0 + (Y 1i Y 0i )D i + U 0i ( = α + β i D i + u i )

13 Roy model In the Roy model, Y d = µ d + U d where E(U d ) = 0 for d = 0, 1. If we observe a vector of covariates X, µ d = µ d (X). Often µ d (X) = β d X. Now Y i = Y 0i + (Y 1i Y 0i )D i = µ 0 + (Y 1i Y 0i )D i + U 0i ( = α + β i D i + u i ) What the Roy model gives us is that it adds a model for D to the potential outcomes framework and demonstrates the important link between the model for D and the model for the potential outcomes.

14 Roy model Problem #1 if µ d = µ d (X) then OLS does not identify ATE = E(Y 1i Y 0i ) generally because of nonlinearity (µ d (X) β d X) and observed heterogeneity (µ 1 (x) µ 0 (x) varies with x)

15 Roy model Problem #1 if µ d = µ d (X) then OLS does not identify ATE = E(Y 1i Y 0i ) generally because of nonlinearity (µ d (X) β d X) and observed heterogeneity (µ 1 (x) µ 0 (x) varies with x) Of course, if µ d = β d X then we solve this problem by regressing Y i on D i, X i and D i X i. Alternatively, we do matching to overcome these two problems. Or, we simply do OLS (without the interaction) which identifies a weighted average of conditional treatment effects.

16 Roy model Problem #2 The above solutions only work under the conditional independence assumption, (Y 0i, Y 1i ) D i X i. In the generalized Roy model, this is only satisfied if (U 0i, U 1i ) (U 1i U 0i, V i ) X i, Z i no unobserved heterogeneity and non-random or independent costs

17 Roy model Problem #2 A more general model is D = 1(E(Y 1 Y 0 c(z, V ) I) 0) where E( I) represents the expected value from the decision-maker s perspective. In this case, conditional independence can be stated in terms of the information available to the econometrician relative to what s available to the decision-maker. What if I consists of X and Z but not U 1i, U 0i or V i?

18 Roy model Problem #2 Note also that Y i = µ 0 + (µ 1 µ 0 )D i + U 0i + (U 1i U 0i )D i There is a selection on unobservables problem (D i is correlated with U 0i ) and an unobserved heterogeneity problem (U 1i U 0i 0). An exercise for you: What happens if U 1i = i + U 0i where i is independent of U 0i? What if U 1i = U 0i but U 0i is not independent of D i (perhaps because U 0i is correlated with V i )?

19 Roy model Problem #3 In the simple Roy model, no instruments are available. In the extended and generalized models, Z i is potentially a valid instrument because it is relevant but excluded from the outcome equations. However, when is E(U 0i + (U 1i U 0i )D i Z i ) = 0? even if E(U 0i Z i ) = 0, it is unlikely that 0 = E(U 1i U 0i )D i Z i ) = E ((U 1i U 0i )1(µ 1 µ 0 + U 1 U 0 c(z, V )) Z i ) unless U 1 = U 0.

20 Roy model The rest of this lecture 1. What can be identified in the various versions of the Roy model if we assume normal errors? 2. What does IV estimate when there is essential heterogeneity? 3. How can we estimate the ATE (or other similar parameters) when we have an instrument Z i such that (X i, Z i ) (U 0i, U 1i, V i )? 4. Can we estimate policy counterfactuals with such a Z i?

21 Roy model In the Roy model, Y d = µ d + U d for d = 0, 1. Suppose we observe a vector of covariates X so that µ d = β d X. Then E(Y D = 1, X = x) = β 1 x + E(U 1 U 1 U 0 z, X = x) E(Y D = 0, X = x) = β 0 x + E(U 0 U 1 U 0 < z, X = x) where z = (β 1 β 0 ) x c.

22 Roy model Assumption: (U 1, U 0 ) X = x N(0, Σ) where ( ) σ 2 Σ = 1 σ 10 σ 10 σ0 2 Let V = U 1 U 0 and σ 2 V = Var(V )

23 Roy model Assumption: (U 1, U 0 ) X = x N(0, Σ) Let z = z /σ V. Then under this assumption, E(Y D = 1, X = x) = β 1 x + σ2 1 σ 10 λ( z) σ V E(Y D = 0, X = x) = β 0 x + σ2 0 σ 10 λ( z) σ V and Pr(D = 1 X = x) = Φ( z)

24 Roy model Estimation with only sector one observed. If we only observe Y for those with D = 1 (for example, a wage-lfp model) then we can (1) estimate the probit: Pr(D = 1 X = x) = Φ(γ 0 + γ 1 x) = Φ( z) (2) compute the predicted values from (1), ˆ Z = ˆγ0 + ˆγ 1 X and plug into λ to get λ( ˆ Z ) (3) estimate a regression of Y on X, λ( ˆ Z ) for those with D = 1 This enables us to estimate β 1 but not β 0, σ 1, σ 10, σ 0.

25 Roy model Estimation with both sectors observed. If we only observe Y, D, X for everyone. (1) estimate the probit: Pr(D = 1 X = x) = Φ(γ 0 + γ 1 x) = Φ( z) (2) compute the predicted values from (1), ˆ Z = ˆγ0 + ˆγ 1 X and plug into λ to get λ( ˆ Z ) (3) estimate a regression of Y on X, λ( ˆ Z ) for those with D = 1 (4) estimate a regression of Y on X, λ(ˆ Z ) for those with D = 0 This enables us to estimate β 1, β 0, β 1 β 0 and σ V, σ2 0 σ 10 σ V σ1 2 σ 10 σ V. Thus we get σ V too From the variance of the residuals from the two regressions we can also identify σ1 2, σ2 0 and σ 10.

26 Roy model Concerns: If λ( z) is approximately linear then we will have a serious collinearity problem. If U 1, U 0 is not normal then the model is misspecified and identification is not transparent. If there are variable costs the model is misspecified.

27 Roy model In the extended Roy model, Y d = µ d + U d for d = 0, 1 but D = 1(Y 1 Y 0 γ 1 X + γ 2 Z ) Cost of participation varies with X and also with other variables Z. Then E(Y D = 1, X = x, Z = z) = β 1 x E(Y D = 0, X = x, Z = z) = β 0 x where z = (β 1 β 0 ) x γ 1 x γ 2 z. + E(U 1 U 1 U 0 z, X = x) + E(U 0 U 1 U 0 < z, X = x)

28 Roy model Assumption: (U 1, U 0 ) X = x, Z = z N(0, Σ) Under this assumption, if z = z /σ V, E(Y D = 1, X = x) = β 1 x + σ2 1 σ 10 λ( z) σ V E(Y D = 0, X = x) = β 0 x + σ2 0 σ 10 λ( z) σ V and Pr(D = 1 X = x, Z = z) = Φ( z) β 1 and β 0 are still identified Σ only identified if there is an exclusion: a component of X that does not affect costs

29 Roy model What do we need/want to identify? The ATE is E(Y 1 Y 0 ) = (β 1 β 0 )E(X)

30 Roy model What do we need/want to identify? The ATE is E(Y 1 Y 0 ) = (β 1 β 0 )E(X) The distribution of gains: Y 1 Y 0 X = x N((β 1 β 0 ) x, σv 2 ) various other counterfactuals need Σ to go beyond mean treatment effects

31 Roy model In the generalized Roy model, Y d = µ d + U d for d = 0, 1 but D = 1(γ 1 X + γ 2 Z V ) V includes an unobservable component of cost Then E(Y D = 1, X = x, Z = z) = β 1 x E(Y D = 0, X = x, Z = z) = β 0 x where z = γ 1 x + γ 2 z. + E(U 1 V z, X = x) + E(U 0 V > z, X = x)

32 Roy model Assumption: (U 1, U 0, V ) X = x, Z = z N(0, Σ) Under this assumption, β 1 and β 0 are identified σ V is identified under the exclusion restriction but Var(U 1 U 0 ) σv 2 (key ingredient needed for distribution of Y 1 Y 0 ) is not identified

33 Generalized roy model without normality Assumption: (U 1, U 0, V ) X, Z and lim z Pr(D = 1 X = x, Z = z) = 1 The first assumption is essentially the same one used by Imbens and Angrist (1994) The second assumption is called identification at infinity

34 Roy model without normality Under these assumptions Let P(x, z) = Pr(D = 1 X = x, Z = z) E(Y D = 1, X = x, Z = z) = β 1 x + K 1(P(x, z)) and lim z E(Y D = 1, X = x, Z = z) = β 1 x selection on unobservables goes away in the limit Using the same argument for D = 0, we can identify ATE(x) = (β 1 β 0 ) x We ve traded normality for identification at infinity.

35 Some preliminaries Let D = 1(γ 1 X + γ 2Z V ) and assume that (U 1, U 0, V ) (X, Z ) Let U D = F V (V ) where V has distribution function F V ( ). Then D = 1(P(X, Z ) U D ) where P(X, Z ) = F V (γ 1 X + γ 2Z ) is the propensity score and U D Uniform(0, 1).

36 Definition of the MTE Then the marginal treatment effect (MTE) is defined as MTE(x, u) = E(Y 1 Y 0 X = x, U D = u) This demonstrates (observable and unobservable) heterogeneity in Y 1 Y 0.

37 Definition of the MTE MTE(x, u) can be interpreted as the effect of participation for those individuals who would be indifferent if we assigned them a new value of P = P(x, z) equal to u. Someone with a large value of u (close to 1) will participate only if P is quite large; this person will be indifferent if P is equal to u. These are the high unobservable cost individuals. Someone with a small value of u (close to 0) will participate even if P is quite small; this person will be indifferent if P is equal to u. These are the low unobservable cost individuals.

38 Identification of the MTE The identifying equation: E(Y X = x, P(X, Z ) = p) p = MTE(x, p) This only works if Z is continuous. The effect for the high unobserved cost individuals is identified by the effect of a marginal increase in participation probability on Y at a high participation rate.

39 Other treatment parameters and methods ATE(x) = 1 0 MTE(x, u)du

40 Other treatment parameters and methods ATE(x) = 1 0 MTE(x, u)du TT (x) = 1 0 MTE(x, u)ω TT (x, u)du where ω TT (x, u) disproportionately weights smaller values of u OLS and IV can also be written as weighted averages of MTE(x, u).

41 Other treatment parameters and methods Consider the IV estimand IV (x) = The weight here is Cov(J(Z ),Y X=x) Cov(J(Z ),D X=x) ω IV (x, u) = E(J E(J) X = x, P u)pr(p u X = x) Cov(J, P X = x) This and many interesting implications are discussed in Heckman, Vytlacil, and Urzua (2006).

42 LATE This is a limiting version of Imbens and Angrist s LATE formulation. Ig nore X and let D z denote the value of D when Z is fixed at z.

43 LATE This is a limiting version of Imbens and Angrist s LATE formulation. Ig nore X and let D z denote the value of D when Z is fixed at z. (i.e., D z = 1(γ 2 z V )) Imbens and Angrist consider a binary Z and show that E(Y Z = 1) E(Y Z = 0) E(D Z = 1) E(D Z = 0) = E(Y 1 Y 0 D 1 > D 0 ) Thus, IV (lhs) identified the local average treatment effect (LATE, rhs), which is the average effect for those induced to participate by Z. This population is sometimes called the compliers.

44 LATE assumptions The assumptions in Theorems and in MHE are equivalent to those used by Heckman, Urzua, Vytlacil (2006). Imbens and Angrist s formulation highlights an assumption that is implicit in HUV s mode for treatment, D = 1(γ 1 X + γ 2 Z V ). monotonicity: The ceteris paribus effect of changing Z on D has the same sign for everyone, i.e., either D 1i D 0i for all i or D 1i D 0i for all i.

45 LATE assumptions The assumptions in Theorems and in MHE are equivalent to those used by Heckman, Urzua, Vytlacil (2006). Imbens and Angrist s formulation highlights an assumption that is implicit in HUV s mode for treatment, D = 1(γ 1 X + γ 2 Z V ). monotonicity: The ceteris paribus effect of changing Z on D has the same sign for everyone, i.e., either D 1i D 0i for all i or D 1i D 0i for all i. Really this is a uniformity assumption. If Z takes more than two values there is no need for monotonicity, only that D changes in the same direction for everyone as Z changes. The assumption is implied by the equation D = 1(γ 1 X + γ 2 Z V ) but it would fail if γ 2 was a random coefficient. MHE interpret the assumption as requiring no defiers.

46 LATE assumptions The independence assumption when there are covariates. Really only need (U 1, U 0, V ) Z X. Estimation of MTE is simplified under the stronger assumption that (U 1, U 0, V ) Z X. See the discussion in Carneiro, Heckman, Vytlacil (2011).

47 More on LATE When Z is continuous, we can estimate the MTE and various weighted averages of the MTE. The LATE framework is useful in understanding what we are able to learn when Z is discrete. Cases where LATE = TT or LATE = TUT Characterizing compliers. LATE with covariates

48 Special cases The TT can be written as a weighted average of LATE and the average effect for the always-takers. In some cases, D must be equal to 0 when Z = 0. The Bloom example Z is a random assignment and D a treatment and there is one-way noncompliance. One-way noncompliance means that some with Z = 1 choose D = 0 (refuse treatment) but no one with Z = 0 can have D = 1. In these cases, IV estimates TT.

49 Special cases The TUT can be written as a weighted average of LATE and the average effect for the never-takers. In some cases, D must be equal to 1 when Z = 1. Suppose D indicates having a third child (as opposed to only 2) and Z indicates whether the second birth was a multiple birth. Then if Z = 1 we must have D = 1. There are no never-takers. In these cases, IV estimates TUT.

50 Compliers A few results: Pr(D 1 > D 0 ) = E(D Z = 1) E(D Z = 0) for any W such that (D 1, D 0 ) is independent of Z conditional on W, E(W D 1 > D 0 ) = E(κW ) E(κ) where κ = 1 D(1 Z ) (1 D)Z 1 Pr(Z = 1 W ) Pr(Z = 1 W ) and, more generally, f W D1 >D 0 (w) is equal to E(D Z = 1, W = w) E(D Z = 0, W = w) f W (w) E(D Z = 1) E(D Z = 0)

51 LATE with covariates The LATE story gets quite a bit more complicated with covariates. Let λ(x) = E(Y 1 Y 0 D 1 > D 0, X = x) denote the LATE conditional on X. We could estimate these directly using the Wald formula conditional on X.

52 LATE with covariates The LATE story gets quite a bit more complicated with covariates. Let λ(x) = E(Y 1 Y 0 D 1 > D 0, X = x) denote the LATE conditional on X. We could estimate these directly using the Wald formula conditional on X. If we do 2SLS where the first stage is fully saturated and the second stage is saturated in X we get a weighted average of the λ(x). The weights are larger for values of x such that Var(E(D X = x, Z ) X = x) is larger.

53 LATE with covariates The LATE story gets quite a bit more complicated with covariates. Let λ(x) = E(Y 1 Y 0 D 1 > D 0, X = x) denote the LATE conditional on X. We could estimate these directly using the Wald formula conditional on X. If we do 2SLS where the first stage is fully saturated and the second stage is saturated in X we get a weighted average of the λ(x). The weights are larger for values of x such that Var(E(D X = x, Z ) X = x) is larger. if Pr(Z = 1 X) is a linear function of X then 2SLS gives the minimum MSE approximation to E(Y D, X, D 1 > D 0 ). This is useful because E(Y D = 1, X, D 1 > D 0 ) E(Y D = 0, X, D 1 > D 0 ) = λ(x). Abadie (2003) proposes a way to estimate this same minimum MSE approximation when Pr(Z = 1 X) is not linear.

54 An example Carneiro, Heckman, Vytlacil (2011) use data from the NLSY Y is log wage in 1991 (ages 28-34), D represents college attendance, X a vector of controls vector Z : (i) distance to college, (ii) local wage, (iii) local unemployment, (iv) average local public tuition

55 VOL. 101 NO carneiro et al.: estimating marginal returns to education An example MTE U S

56 Estimating the MTE - normal model Option 1. Estimate using MLE. Option 2. Two stage estimation: Probit to estimate P(Z i ) (to simplify, define Z i to include X i and instrument(s)) Regress Y on X i and ˆλ 1i = φ(φ 1 (ˆP(Z i ))) for D ˆP(Z i ) i = 1 Regress Y on X i and ˆλ 0i = φ(φ 1 (ˆP(Z i ))) 1 ˆP(Z i ) Then for D i = 0 MTE(x, u) = x ( ˆβ 1 ˆβ 0 ) + (ˆρ 1 ˆρ 0 )Φ 1 (u)

57 Estimating the MTE semiparametric model The outcome equation can be written (see II.B in Carneiro et al. (2011)) as E(Y X = x, P(Z ) = p) = x δ 0 + px (δ 1 δ 0 ) + K (p) There are several ways to estimate this perhaps the simplest is a series/spline/sieve estimator. Estimate P(Z i ) (logit). Choose a set of basis functions (polynomials) and an order, K. Run the regression: Y i = X i δ 0 + ˆP(Z i )X i (δ 1 δ 0 ) + γ 1 ˆP(Zi ) γ K ˆP(Zi ) K + η i An important sacrifice here is that MTE(x, u) is only identified for u in the support of P. (Recall identification at )

58 Consider policies that affect P(Z ) but not Y 1, Y 0, V. Propensity score P under new policy. It can be shown that the effect of shifting to this new policy is given by 1 0 [ MTE(x, u) F P X=x(u) F P X=x (u) E(P X = x) E(P X = x) This will still require large support for P(Z ). define a continuum of policies consider marginal change from baseline ] du

59 MPRTE Consider increasing tuition (a component of Z ) by an amount α: tuition = tuition + α. Corresponding propensity score, P α. Define the MPRTE as 1 lim α 0 0 [ MTE(x, u) F Pα X=x(u) F P0 X=x(u) E(P α X = x) E(P 0 X = x) ] du This is also equal to lim e 0 E(Y 1 Y 0 µ D (X, Z ) V < e). And it can be written as 1 MTE(x, u)ω(x, u) where 0 ω(x, u) = f P X (u)f V X (F 1 V X (u)) E(f V X (µ D (X, Z )) X)