Lecture 11 Roy model, MTE, PRTE

Transcription

1 Lecture 11 Roy model, MTE, PRTE Economics 2123 George Washington University Instructor: Prof. Ben Williams

2 Roy Model Motivation The standard textbook example of simultaneity is a supply and demand system Y S = α S + β S P + ε S Y D = α D + β D P + ε D In equilibrium, Y S = Y D, and the equilibrium price is P = α D α S β S β D + ε D ε S β S β D The Roy model is an alternative example of simultaneity namely, self-selection that arises in micro models.

3 Roy Model Two sector economy (e.g., think college graduates and high school graduates) Wage differential between sectors in general does not give the (counterfactual) wage effect from switching sectors. Sector choice causes wage gains (or losses) but wage differentials also induces sector choice.

4 Roy Model Suppose Y d = µ d + U d for d = 0, 1 and that D = 1(Y 1 Y 0 c) Then E(Y 0 D = 1) E(Y 0 D = 0) (selection bias) and and E(Y 1 Y 0 D = 1) E(Y 1 Y 0 ) (sorting gains)

5 Roy Model Sector 1 observed earnings: E(Y 1 D = 1) = µ 1 + E(U 1 U 1 U 0 > (µ 1 µ 0 c)) not equal to µ 1 because U 1 and V = U 1 U 0 are correlated Suppose (U 1, U 0 ) are jointly normal with mean 0, covariance denoted by σ 10 and variances σ 2 d = Var(U d). Due to normality, U 1 = ρσ V V + ε where ρ correlation between U 1 and V σ V variance of V ε 1 and V are independent

6 Roy Model As a result, observed sector 1 earnings ( ) µ 1 + Var(U 1) Cov(U 1, U 0 ) λ µ 1 µ 0 c Var(U1 U 0 ) Var(U1 U 0 ) λ( ) is the inverse mills ratio λ(z) = E(Z Z > z) = φ(z)/(1 Φ(z)) for Z N(0, 1)

7 Roy Model some empirical implications: positive selection in both sectors if the two distributions are uncorrelated in general, cannot have negative selection in both sectors cost increase E(Y 1 D = 1) c = Var(U 1) Cov(U 1, U 0 ) λ (x ) Var(U 1 U 0 ) where λ > 0

8 Roy Model more results: Also, Var(Y 1 D = 1)/ c has the opposite sign. negative selection in sector 0 requires Var(U 0 ) < Cov(U 0, U 1 ) < Var(U 1 ) selection reduces inequality if the distributions are not normal these things don t have to be true

9 Roy Model Identification: If U 1, U 0 are jointly normal then the unknown parameters are µ 1, µ 0, Var(U 1 ), Var(U 0 ), Cov(U 1, U 0 ). Three different observation schemes: Observe (Y i, D i ) for iid sample i = 1,..., n Observe Y i for iid sample i = 1,..., n Observe (Ỹi, D i ) for iid sample i = 1,..., n where Ỹi is observed to be missing if D i = 0.

10 Roy Model Identification: If U 1, U 0 are jointly normal then all parameters are identified in first sampling scheme. Partial identification in the other two sampling schemes. Suppose there are regressors, µ d = β d X for d = 0, 1. Then E(Y 1 D = 1, X = x) = β 1 x + E(U 1 U 1 U 0 > z, X = x) E(Y 0 D = 0, X = x) = β 0 x + E(U 0 U 1 U 0 < z, X = x) Pr(D = 1 X = x) = Pr(U 1 U 0 > z X = x) where z = (β 1 β 0 ) x c

11 Roy Model Now suppose U 1, U 0 are jointly normal, conditional on X Then U 1 = σ2 1 σ 10 σ V (U 1 U 0 ) + ε where ε U 1 U 0 X Therefore, E(Y 1 D = 1, X = x) = β 1 x + σ2 1 σ 10 σ V λ( z) where z = (β 1 x β 0 x c)/σ V and the conditional probability of being in sector 1 is Pr(D = 1 X = x) = Φ ( z) (propensity score)

12 Roy Model Based on these two conditional moments, β 1 is identified as well as some combinations of other parameters. The direction of selection is identified β 1,k β 0,k is identified up to scale

13 Roy Model If we have complete data E(Y 0 D = 0, X = x) = β 0 x + σ2 0 σ 10 σ V λ( z) where z = (β 1 x β 0 x c)/σ V everything is identified

14 Roy Model Counterfactuals the distribution of potential wage gains Y 1 Y 0 the proportion of the population who benefits Pr(Y 1 > Y 0 ) the effect of a policy of subsidizing cost for those with Y 0 below a cutoff value, y 0

15 Generalized Roy Model Generalized Roy Model: Let Y d = µ d (X) + U d and D = 1(µ D (X, Z ) V ) where (U 1, U 0, V ) (X, Z ) special case µ D (X, Z ) = µ 1 (X) µ 0 (X) µ C (X, Z ) V = U 1 U 0 U C what is identified in this case?

16 Generalized Roy Model Generalized Roy Model: Assumptions HV1 (U 1, U 0, V ) (X, Z ) HVN1 (U 1, U 0, V ) is normally distributed Then ( E(Y 1 D = 1, X = x, Z = z) = β 1 x Cov(U 1, V ) λ µ ) D(x, z) σ V σ V

17 Generalized Roy Model Identification β 1, β 0 are identified with data on (Y, D, X, Z ) suppose µ D (x, z) = β 1 x β 0 x + γ xx + γ zz if there is a component of x with γ x set to 0 then σ V, γ x, γ z are identified regardless, the sorting gains and the selection bias can be identified

18 Generalized Roy Model Generalized Roy Model without normality Heckman and Honore (1990) study the Roy model with and without regressors under nonnormality Let s consider the generalized Roy model under nonnormality In this case, E(Y 1 D = 1, X = x, Z = z) = β 1 x + E(U 1 µ D (x, z) V )

19 Generalized Roy Model Generalized Roy Model without normality The propensity score: P(x, z) := Pr(D = 1 X = x, Z = z) = F V (µ D (x, z)) Index sufficiency: E(U 1 µ D (x, z) V ) = K 1 (P(x, z)) K 1 is called a control function

20 Generalized Roy Model Generalized Roy Model without normality β 1 is identified if lim z P(x, z) = 1 (or lim z P(x, z) = 1) because lim P(x,z) K 1 (P(x, z)) = 0 identification at infinity HV1 is needed for this argument but HVN1 is replaced by the support condition

21 MTE Marginal Treatment Effect Define U D = F V (V ) Then D = 1(P(X, Z ) U D ) what is the distribution of U D? MTE(x, u) = E(Y 1 Y 0 X = x, U D = u) (concept originates with Bjorklund and Moffitt (1987))

22 MTE Identification E(Y X = x, P(X, Z ) = p) p + E(D(Y 1 Y 0 ) X = x, P(X, Z ) = p) p = 0 + p MTE(x, u)du p 0 = MTE(x, p) = E(Y 0 X = x, P(X, Z ) = p) p

23 MTE Marginal Treatment Effect many parameters of interest can be written as 1 0 MTE(x, u)ω(x, u)du, 1 0 ω(x, u)du = 1 for example, ATE(x) := E(Y 1 Y 0 X = x), ω ATE (x, u) = 1 [0,1] TT (x) := E(Y 1 Y 0 D = 1, X = x), ω TT (x, u) Pr(P(x, Z ) > u X = x)

24 MTE weights for IV Cov(J(Z ),Y X=x) Cov(J(Z ),D X=x) where the IV estimand is IV (x) = J = J(Z ) is some function of the instruments that may depend implicitly on X as well. then IV (x) = 1 0 MTE(x, u)ω IV (x, u)du where ω IV (x, u) = E(J E(J) X = x, P u)pr(p u X = x) Cov(J, P X = x)

25 MTE weights for IV what if J = P? if Z is scalar and binary then IV (x) = p1 1 MTE(u, x) du p 0 p 1 p 0 where p s = Pr(D = 1 Z = s, X = x) for s = 0, 1 in general, weights are not always positive!!

26 MTE LATE Let D(z) denote the value D takes when Z takes the value z. Imbens and Angrist showed that the IV estimand in the binary case takes the form E(Y 1 Y 0 D(z) D(z ) = 1) This is called the local average treatment effect represents the average effect of treatment for individuals induced to receive treatment when Z changes from 0 to 1 MTE(P(z)) = lim z z LATE(z, z )

27 MTE Carneiro, Heckman, and Vytlacil (2011) data from NLSY Y is log age in 1991 (individuals are between 28 and 34), D represents college attendance, X contains usual controls instruments: (i) distance to college, (ii) local wage, (iii) local unemployment, (iv) average local public tuition

28 MTE VOL. 101 NO. 6 carneiro et al.: estimating marginal returns to education MTE U S Figure 4. E( Y 1 Y 0 X, U S ) with 90 Percent Confidence Interval Locally Quadratic Regression Estimates Notes: To estimate the function plotted here, we first use a partially linear regression of log wages on polynomials Economics 379 George in X, interactions Washington of polynomials University in X and P, and K(P), a locally quadratic function of P (where P is the predicted probability of attending college), with a bandwidth of 0.32; X includes experience, current average earnings in the county of residence, current average unemployment in the state of residence, AFQT, mother s education, number of

29 MTE MTE assumptions: HV1 the distribution of µ D (x, Z ) conditional on X = x is not degenerate (exclusion restriction) 0 < Pr(D = 1 X = x) < 1 for each x X is invariant to counterfactual manipulations (X 1 = X 0 ) Vytlacil shows that these are equivalent to the conditions of Imbens and Angrist (1994). uniformity/monotonicity: Pr(D(z) D(z )) is equal to 1 or 0 for each pair z, z implied by separability in HV model.

30 heterogeneity Summarizing, In models with essential heterogeneity,iv does not estimate an economically interesting parameter. Instead, IV estimates a LATE, or a weighted average of the MTE. Under the more restrictive assumptions of Imbens and Angrist (1994), this is the the treatment effect for those induced to switch by an increase in Z. More generally, the weights may be negative. Different instruments identify different parameters. The MTE itself can be identified so we can do more.

31 LIV estimator Carneiro, Heckman, and Vytlacil (2011) Recall the definition of MTE from Heckman, Uruzua, and Vytlacil (2006) S = 1(µ S (Z ) V ) = 1(P(Z ) U S ) where U S Uniform(0, 1). For each x, u, MTE(x, u) = E(Y 1 Y 0 X = x, U S = u) The MTE is identified by MTE(x, p) = E(Y X = x, P(Z ) = p) p Remember that MTE(x, u) is the average effect of treatment for those that would be indifferent if they have P(z) = u. Increasing P(z) induces precisely this group to choose S = 1.

32 LIV estimator Carneiro, Heckman, and Vytlacil (2011) Standard treatment effect parameters take the form 1 0 ω(x, u)mte(x, u)du. ATE(x) = 1 MTE(x, u)du 0 TT (x) weights proportional to Pr(P(Z ) > u X = x) TUT (x) weights proportional to Pr(P(Z ) < u X = x) Each of these requires P(Z ) to have support at 0 and/or 1 (conditional on X)!

33 LIV estimator Carneiro, Heckman, and Vytlacil (2011) Consider policies that affect P(Z ) but not Y 1, Y 0, V. Chooses S and 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 baselin ] du

34 LIV estimator Carneiro, Heckman, and Vytlacil (2011) 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 S (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 S (Z )) X)

35 LIV estimator Carneiro, Heckman, and Vytlacil (2011) Estimating the MTE - normal model Option 1. Estimate using MLE. Option 2. Two stage estimation: Probit to estimate P(Z i ) Regress Y on X i and ˆλ 1i = φ(φ 1 (ˆP(Z i ))) for S ˆP(Z i ) i = 1 Regress Y on X i and ˆλ 0i = φ(φ 1 (ˆP(Z i ))) for S 1 ˆP(Z i ) i = 0 Then MTE(x, u) = x ( ˆβ 1 ˆβ 0 ) + (ˆρ 1 ˆρ 0 )Φ 1 (u)

36 LIV estimator Carneiro, Heckman, and Vytlacil (2011) Estimating the MTE semiparametric model The outcome equation can be written 1 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