Discussion of “Who wins, who loses? Tools for distributional policy evaluation” James Heckman

Discussion of
“Who wins, who loses? Tools for distributional
policy evaluation”
James Heckman
September 26, 2014
Heckman
Kasy Discussion
Kasy Paper: Many Moving Parts
Claims
(1) Point or set identification of “expected” individual welfare
gains conditional on choice variables
(2) Distribution of welfare effects
(3) No restriction on dimension of the heterogeneity vis-á-vis
dimension of endogenous variables
(4) Aggregation over the population using a SWF
(5) GE effects
(6) Study of the EITC impact on welfare
Some Key Assumptions
(1) One dimensional parameterization of policies
(2) Functions differentiable in policies
(3) Strong support conditions
Heckman
Kasy Discussion
Table 1 : Comparison of Steady States Under Alternative Tax Regimes
Source: Heckman, Lochner, and Taber (1998).
Heckman
Kasy Discussion
Placing Paper in Literature on Distributional Treatment
Effects
• Two outcome model: (Y0 , Y1 )
• Observe only one coordinate and that subject to selection bias
• D = 1 if person gets treatment; D = 0 otherwise
• Y = DY1 + (1 − D)Y0
Heckman
Kasy Discussion
Two Problems
I. From data on outcomes F1 (y1 | D = 1, X ), F0 (y0 | D = 0, X ),
under what conditions can one recover F1 (y1 | X ) and
F0 (y0 | X ), respectively?
II. Construct the joint distribution F (y0 , y1 | X ) from the marginal
distributions.
Heckman
Kasy Discussion
Why Bother Identifying Joint Distributions?
Heckman
Kasy Discussion
Depends on the criterion
Pr (Y1 ≥ Y0 ) :
Pr (Y1 > Y0 |Y0 < y0 ) :
Heckman
Percentage of people voting
Gains to poor people in base state
Kasy Discussion
Solutions
• Two basic approaches in the literature to solving the problem
of identifying F (y0 , y1 | X )
(A) Bounds
(B) Solutions that postulate assumptions about dependence
between Y0 and Y1 .
(C) Solutions based on information from agent participation rules
and choice data.
Heckman
Kasy Discussion
Bounds
(1) Frechet Bounds
(2) Makarov Inequality Bounds (For Y1 − Y0 )
Heckman
Kasy Discussion
Solutions Based on Conditional Independence or Matching
• Q: Conditioning Variable
• F0 (y0 | D = 0, X , Q) = F0 (y0 | X , Q)
• F1 (y1 | D = 1, X , Q) = F1 (y1 | X , Q).
• All of the dependence between (Y0 , Y1 ) given X comes through
Q
• F (y1 , y0 | X , Q) = F1 (y1 | X , Q) F0 (y0 | X , Q).
Heckman
Kasy Discussion
Common Coefficient:
Y1 − Y0 = ∆
(1)
∆ is a constant given X .
Heckman
Kasy Discussion
Quantile Treatment Effects
• Y1 = F1−1 (F0 (Y0 )).
• This is the tight upper bound of the Fréchet bounds.
• Alternative assumption: Y1 = F1−1 (1 − F0 (Y0 )).
• Tight Fréchet lower bound.
Heckman
Kasy Discussion
Constructing Distributions from Assuming Independence of
the Gain from the Base
C-1
Y1 = Y0 + ∆
Y0 ⊥⊥ ∆ | X .
•
M-1
(Y0 , Y1 ) ⊥
⊥ D | X,
• Identify F (y0 , y1 | X ) from the cross section outcome
distributions of participants and non-participants and estimate
the joint distribution by using deconvolution.
Heckman
Kasy Discussion
Information From Revealed Preference
• e.g. Roy Model, Generalized Roy Model
Heckman
Kasy Discussion
Additional Information
• Dependence through factor structure or other assumptions on
copulas.
Heckman
Kasy Discussion
Kasy: Yd on a continuum
d = φ(α)
Same unobservables across all d
Heckman
Kasy Discussion
1
Identification (Non-parametric)
1
2
3
2
Aggregation
social welfare & distributional decompositions
1
2
3
Main challenge: γ(w , l) = E [ẇ · l|w · l, α]
Causal effect of policy
Conditional on endogenous outcomes,
welfare weights ≈ derivative of influence function
welfare impact = impact on income - behavioral correction
Inference
1
2
3
local linear quantile regressions
combined with control functions
suitable weighted averages
Heckman
Kasy Discussion
Setup
u(c, l) :
• u differentiable, increasing in c, decreasing in l, quasiconcave,
does not depend on α
• γ(α) continuous in α.
Heckman
Kasy Discussion
Objects of Interest: W : conditioning variables not affected by α
Sets of winners and losers:
W := {(y , W ) : γ(y , W ) ≥ 0}
L := {(y , W ) : γ(y , W ) ≤ 0}
Heckman
Kasy Discussion
Question:
Treatment of heterogeneity among winners and losers?
Want to compute
Pr (γ(y , W ) ≥ 0)
Need more than E (γ(y , W ))
• Given y , W what is the distribution of γ?
Heckman
Kasy Discussion
Identification of disaggregated welfare effects
• Goal: identify γ(y, W) = E[ė|y, W, α]
• Simplified case:
no change in prices, or unearned income
no covariates, just tax change
• Then
γ(y ) = E [l · (1 − ∂z t) · ẇ |l · w , α]
z = lw
• Denote x = (l, w ).
Seek to identify
g (x, α) = E [ẋ|x, α]
(2)
from
f (x|α).
• Made necessary by combination of
1 utility-based social welfare
2 heterogeneous wage response.
Heckman
Kasy Discussion
Question: Do we really know f (x|α)? (Random assignment with a
continuum of treatments)
If so, can do table look up for policy effects.
How dense is the set of policies?
Heckman
Kasy Discussion
Assumptions:
1
2
3
x = x(α, ), x ∈ Rk
α ⊥ (really “⊥⊥”)
x(., ) differentiable ( explicitly introduced for the first time)
Heckman
Kasy Discussion
Analogy from fluid dynamics:
• x(α, ): position of particle at time α
• f (x|α): density of gas / fluid at time α, position x
• f˙ change of density
• h(x, α) = E [ẋ|x, α] · f (x|α): “flow density”
Heckman
Kasy Discussion
• Knowledge of f (x|α)
• identifies ∇ · h =
Pk
j=1 ∂x j h
j
(k = number of endogenous variables)
• where h = E [ẋ|x, α] · f (x|α)
• identifies nothing else.
• Add to h
• h̃ such that ∇ · h̃ ≡ 0
• cannot identify true h(= h0 ) from h̃ perturbations
Heckman
Kasy Discussion
Density and flow
ḟ = −∇ · h
Heckman
(3)
Kasy Discussion
• This relationship is a property of differentiability of functions.
Heckman
Kasy Discussion
Question: Additional restrictions from properties of expenditure
function? Properties of demand functions?
Heckman
Kasy Discussion
Theorem
The identified set for h is given by
h0 + H
(4)
where
H = {h̃ : ∇ · h̃ ≡ 0}
0j
h (x, α) = f (x|α) · ∂α Q(v j |v 1 , . . . , v j−1 , α)
v j = F (x j |x 1 , . . . , x j−1 , α) Heckman
Kasy Discussion
Theorem
1
2
Suppose k = 1. Then
H = {h̃ ≡ 0}.
(5)
H = {h̃ : h̃ = A · ∇H for some H : X → R}.
(6)
Suppose k = 2. Then
where
A=
3
0 1
−1 0
.
Suppose k = 3. Then
H = {h̃ : h̃ = ∇ × G }.
where G : X → R3 . Heckman
Kasy Discussion
(7)
Question: Initial Conditions? Boundary values? How
identified?
Heckman
Kasy Discussion
Point identification
Theorem
Assume
∂
E [ẋ i |x, α] = 0 for j > i.
∂x j
Then h is point identified, and equal to h0 as defined before.
In particular
g j (x, α) = E [ẋ j |x, α]
= ∂α Q(v j |v 1 , . . . , v j−1 , α). Triangularity (e.g. as in Blundell and Matzkin, 2014).
Heckman
Kasy Discussion
(8)
Multiple component policies?
(a) Does result generalize to PDEs?
(b) Convert to ODE? Method of characteristic curves?
(c) What restrictions give linear and hyperbolic PDEs?
Heckman
Kasy Discussion
Aggregation
• Relationship
social welfare ⇔ distributional decompositions?
• public finance welfare weights
≈ derivative of dist decomp influence functions
˙
• Alternative representations of SWF
˙ :
⇒ alternative ways to estimate SWF
1
2
3
weighted average of individual welfare effects ė, γ
distributional decomposition for counterfactual income ỹ
(holding labor supply constant)
distributional decomposition of realized income
minus behavioral correction
Heckman
Kasy Discussion
Estimation
1
First estimate the disaggregated welfare impact
γ(y , W ) = E [ė|y , W , α]
= E [l · ẇ · (1 − ∂z t) − ṫ + y˙0 − c · ṗ|y , W , α]
2
(9)
Then estimate other objects by plugging in γ
b:
c = {(y , W ) : γ
W
b(y , W ) ≥ 0}
c= {(y , W ) : γ
L
b(y , W ) ≤ 0}
[
˙ = EN [ωi · γ
SWF
b(yi , Wi )].
Heckman
(10)
Kasy Discussion
Question:
Can SWF depend on x?
Heckman
Kasy Discussion
Consumer Demand with Multidimensional
Nonseparable Unobserved Heterogeneity
Blundell, Kristensen, and Matzkin 2014
• Use restrictions from economic theory in the estimation of
nonparametric models of consumer behaviour
• Develop nonparametric methods that can be applied to general
systems of demands
• Can be used to construct γ.
Heckman
Kasy Discussion
• Particular attention is given to discrete prices, multiple goods
•
•
•
•
and nonseparable unobserved heterogeneity.
Demand systems are the reduced form of models of
simultaneous equations.
Develop methods for simultaneous equations.
Use them to identify the effect on any particular individuals
of a change in his/her budget set
Identifying individuals across different Engel curves allows to
impose Revealed Preference inequalities on the demand of
any particular individual
Heckman
Kasy Discussion
• Z : set of conditioning variables
• Identification when:
1 A unimodal restriction with respect to Z on the conditional
density of the vector of unobserved heterogeneity.
Or
2 Rank condition on the conditional density of the unobserved
heterogeneity given Z .
• Methods to estimate the value of the vector of unobserved
tastes of each consumer and the demand function of each
consumer.
• Methods to estimate the effect of finite and infinitesimal
changes in prices and income on the demand of each individual
consumer.
• Estimators are consistent and asymptotically normal.
Heckman
Kasy Discussion
• System of demand functions
Y1 = d 1 (p, I , ε1 , ..., εG )
Y2 = d 2 (p, I , ε1 , ..., εG )
···
YG = d G (p, I , ε1 , ..., εG )
(ε1 , ..., εG ) is independent of (p, I ) conditional on Z
• (ε1 , ..., εG ) vector of unobserved heterogeneity (tastes)
Heckman
Kasy Discussion
They show
• When Z is discrete the rank condition can still be used for
point identification results.
• When Z is discrete the unimodal condition can be used for
partial (set) identification of unobserved tastes ε.
• Methods can be used for continuously distributed and for
discrete prices.
• When prices are discrete, partial identification results for the
demand of any particular budget that had not been observed,
‘predicted demands’, can be obtained by extending the results
in Blundell, Kristensen, and Matzkin (2011).
Heckman
Kasy Discussion
• Identify the effect of a discrete change in (p, I ) when ε stays
fixed
d (p 0 , I 0 , ε) − d (p, I , ε)
• Identify the effect of an infinitesimal change in (p, I ) when ε
stays fixed
∂d (p, I , ε)
∂d (p, I , ε)
and
∂p
∂I
Heckman
Kasy Discussion
• System of demand functions
Y1 = d 1 (p, I , ε1 , ..., εG )
Y2 = d 2 (p, I , ε1 , ..., εG )
···
YG = d G (p, I , ε1 , ..., εG )
where (ε1 , ..., εG ) is independent of (p, I ) conditional on Z
Heckman
Kasy Discussion
• Key assumption (from Matzkin, 2008, and other co-authored
papers)
• Invertibility
ε1 = r 1 (Y1 , ..., YG , p, I )
ε2 = r 2 (Y1 , ..., YG , p, I )
···
εG = r G (Y1 , ..., YG , p, I )
Heckman
Kasy Discussion
EITC: Questions
• Many reforms not captured by scalar α
• Multiple margins and policy change over time not uniform
• EITC affects skill accumulation and wages at micro level
• Model of skill accumulation estimated affects estimates
• Are estimated wage effects GE effects or the effects on skill?
Heckman
Kasy Discussion
Credit
Figure 1
Earned Income Tax Credit
sb
0
a
b
c
Earnings
Does scalar α describe this policy?
Heckman
Kasy Discussion
0
1
2
3
4
5
6
period (5 year age period)
7
8
9
0
10
1
2
3
4
5
6
period (5 year age period)
7
8
9
10
Figure
1 :3:Simulated
EITC on
on Hours
HoursWorked
Worked(OJT
(OJTmodel)
model)
Figure
Simulated Effects
Eects of
of EITC
Non−White Females − Less than 10th grade
White Females − HS graduates
0.255
0.3
No EITC
EITC
0.25
No EITC
EITC
0.29
0.245
proportion of time working (h)
proportion of time working (h)
0.28
0.24
0.235
0.23
0.225
0.27
0.26
0.25
0.24
0.22
0.23
0.215
0.21
1
2
3
4
5
6
period (5 year age period)
7
8
9
10
0.22
1
2
3
4
5
6
period (5 year age period)
Source: Heckman, Lochner and Cossa (2003)
Heckman
Kasy Discussion
7
8
9
10
7.5
1
2
3
4
5
6
period (5 year age period)
7
8
9
9
10
1
2
3
4
5
6
period (5 year age period)
7
8
9
10
Figure
2 :7:Simulated
of EITC
EITC on
onHours
HoursWorked
Worked(LBD
(LBDmodel)
model)
Figure
Simulated Effects
Eects of
Non−White Females − Less than 10th grade
White Females − HS graduates
0.225
0.3
No EITC
EITC
0.28
No EITC
EITC
proportion of time working (h)
proportion of time working (h)
0.22
0.215
0.21
0.205
0.2
0.26
0.24
0.22
1
2
3
4
5
6
period (5 year age period)
7
8
9
10
0.2
1
2
3
4
5
6
period (5 year age period)
7
Source: Heckman, Lochner and Cossa (2003)
Heckman
Kasy Discussion
8
9
10
Figure
: Simulated
of EITC
EITC on
onHuman
HumanCapital
Capital(OJT
(OJTmodel)
model)
Figure3 4:
Simulated Effects
Eects of
Non−White Females − Less than 10th grade
White Females − HS graduates
4
5.5
No EITC
EITC
3.9
No EITC
EITC
3.8
5
human capital
human capital
3.7
3.6
3.5
4.5
3.4
3.3
3.2
1
2
3
4
5
6
period (5 year age period)
7
8
9
10
4
1
2
3
4
5
6
period (5 year age period)
7
8
Source: Heckman, Lochner and Cossa (2003)
Figure 5: Simulated Eects of EITC on Wage Rates (OJT model)
Non−White Females − Less than 10th grade
4
White Females − HS graduates
5.4
Heckman
Kasy Discussion
9
10
Figure 8: Simulated Eects of EITC on Human Capital (LBD model)
Figure 4 : Simulated Effects of EITC on Human Capital (LBD model)
Non−White Females − Less than 10th grade
White Females − HS graduates
3.7
5.2
3.6
5
3.5
4.8
No EITC
EITC
human capital
human capital
3.4
3.3
3.2
4.6
4.4
3.1
4.2
3
No EITC
EITC
4
2.9
2.8
1
2
3
4
5
6
period (5 year age period)
7
8
9
10
3.8
1
2
3
4
5
6
period (5 year age period)
7
8
Source: Heckman, Lochner and Cossa (2003)
9
Figure 9: Changes in Hourly Wages due to Increases in Hours Worked
(LBD model)
Non−White Females − Less than 10th grade
0.07
Heckman
White Females − HS graduates
0.1
Kasy Discussion
10
Figure
Effects ofofEITC
EITCon
onWage
WageIncome
Income(OJT
(OJT
model)
Figure5 6:: Simulated
Simulated Eects
model)
Non−White Females − Less than 10th grade
White Females − HS graduates
10
16
No EITC
EITC
15
9.5
thousand of dollars
thousand of dollars
14
9
8.5
13
12
11
8
No EITC
EITC
7.5
1
2
3
4
5
6
period (5 year age period)
7
8
9
10
10
9
1
2
3
4
5
6
period (5 year age period)
7
8
9
Source: Heckman, Lochner and Cossa (2003)
Figure 7: Simulated Eects of EITC on Hours Worked (LBD model)
Non−White Females − Less than 10th grade
0.225
White Females − HS graduates
0.3
Heckman
Kasy Discussion
No EITC
10
Figure 10: Simulated Eects of EITC on Wage Income (LBD model)
Figure 6 : Simulated Effects of EITC on Wage Income (LBD model)
Non−White Females − Less than 10th grade
White Females − HS graduates
8.5
16
15
8
14
thousand of dollars
thousand of dollars
No EITC
EITC
7.5
7
13
12
11
10
6.5
No EITC
EITC
9
6
1
2
3
4
5
6
period (5 year age period)
7
8
9
10
8
1
2
3
4
5
6
period (5 year age period)
Source: Heckman, Lochner and Cossa (2003)
Heckman
Kasy Discussion
7
8
9
10