Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Quantile Treatment Effect (Cont’d)
Duration outcome and dynamic treatment
Pauline Givord
INSEE-DMS
2014/2015
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Quantile treatment effect - introduction
I
Previous lecture : quick introduction to quantile regression
I
Estimation relies on the check function : ρτ
X
βτ = arg minβ
ρτ (Yi − Xi β)
I
Definition of the quantile treatment effect
δτ = Qτ (Y1 ) − Qτ (Y0 )
Horizontal “distance” between distributions of potential
outcomes FY0 and FY1
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Observed and Counterfactual Distributions of Potential
Outcomes (Treatment Group)
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Quantile Treatment Effect for the Treated
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Interpretation
I
without further assumption on the joint distribution of
potential outcomes, we estimate the difference of the quantiles
and not the quantile of the difference (i.e. the treatment
effect) Y1 − Y0
→ no individual interpretation : a finding of a treatment effect
of δ at the τ th quantile says nothing about the treatment
effect for the person at the τ th quantile of the untreated
outcome distribution
I
In many (or most) applications, this is sufficient to answer
economically meaningful questions
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Identification
I
Extension of the classical identification methods to estimate
counterfactual distributions.
I
Last lecture we have seen extension on the case of conditional
independance assumption : Firpo (2007), “Efficient
Semiparametric Estimation of Quantile Treatment Effects,”
Econometrica, vol. 75(1).
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Quantile Treatment Effect under CIA
I
(two stages) semi-parametric direct estimation of
unconditional quantile treatment effect
I
under Conditional independence assumption (CIA) :
Y0 , Y1 ⊥ T |X
with X observable characteristics and commun support
assumption we have
T
τ =E
1(Y ≤ QY1 (τ ))
p(X )
and similarly for other counterfactual distribution
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Quantile Treatment Effect under CIA
I
meaning that the τ th quantile of the distribution of potential
outcome Y1 is an implicit function of the observed (Y , T , X )
I
Can be extimated using quantile regression on the sample of
1
treated (T = 1) using weights corresponding to p(X
)
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Estimation
Two-stage procedure :
1. (non parametric) estimation of p(X )
2. Estimates of QY1 and QY0 are obtained by a reweighted
version of the standard quantile regression procedure:
ˆ
consistent
Pestimators of QYt (τ ) (t = 0, 1) are given by:
argminb ω
ˆ t,i ρτ (Yi − b)
i
with ω
ˆ 1,i = N pˆT(X
) (i.e. sample analogue of T /p(X )) and
ω
ˆ 0,i =
1−Ti
N(1−ˆ
p (Xi )) .
ˆ Y (τ ) − Q
ˆ Y (τ )
QTE = Q
1
0
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Instrumental Variable Estimate of the Quantile Treatment
Effect
Abadie, Angrist et Imbens (2002), “Instrumental Variables
Estimates of the Effect of Subsidized training on the quantiles of
Trainee Earnings”, Econometrica
I
extension of AIR framework to quantile regression
I
estimation of the QTE with an instrument
I
as Firpo, AAI show that it can be obtained as a weighted
version of the standard quantile regression
I
application to a randomized experiment evaluation
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Notation
I
Random affectation to treatment : Z = 0, 1
I
treatment T = 0, 1, depends on instrument (denoted by T0
and T1 )
I
outcome Y depends on treatment Yt (ie Y0 and Y1 )
I
observable characteristics X
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Assumptions
1. independence: (Y1 , Y0 , T1 , T0 ) indep of Z cond. X
2. non trivial assignment : 0 < P(Z = 1|X ) < 1
3. first stage E [T1 |X ] 6= E [T0 |X ]
4. monotonicity : P(T1 ≥ T0 |X ) = 1 (no defiers)
compliers still defined as individuals who change their treatment
status with instrument : T1 > T0
independence of the potential outcome with treatment for
compliers:
(Y1 , Y0 ) ⊥ T |X , T1 > T0
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Estimation of QTEc
Identifiable parameter:
Qτ (Y |X , T , T1 > T0 ) = δτ T + X βτ
i.e. estimation of the QTE for compliers
estimation of
(ˆ
ατ , δˆτ ) = argminE (ρτ (Y − δT − X β)|T1 > T0 )
Pb: population of compliers is not identified
Pauline Givord
Evaluation of Public Policies
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
I
AAD show that for all function h(Y , T , X ):
E [h(Y , T , X )|T1 > T0 ] =
I
1
E [κh(Y , T , X )]
P(T1 > T0 )
with the weight function:
κ(T , Z , X ) = 1 −
T (1 − Z )
(1 − T )Z
−
(1 − π0 (X ))
π0 (X )
with π0 (X ) = P(Z = 1|X ).
I
note that κ equals 1 if T = Z (Compliers).
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
I
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
the QTET can be estimated using
(ˆ
ατ , δˆτ ) = argminE [κρτ (Y − δT − X β)]
I
in practice, pb as κ can be negative, so they use instead the
nonnegative weight κν = E [κ|Y , T , X ] = P(T1 > T0 |Y , T , X )
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Application : JTPA
I
experimental data
I
Job Training Partnership Act (JTPA) : offer services for
individuals facing “barriers to employment”
I
random assignment (20 000 individuals), but only about 60%
of those offered training actually received JTPA
I
note that very few individuals in the control group received
JTPA services: less than 2%
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Application
I
outcome : 30-months earnings
I
random affectation to treatment gives a valid instrument
also control for individual characteristics : race, education,
age, recent job, marital status...
I
I
I
not theoretically required as the treatment was randomized,
but can improve precision
the drawback is that we will obtained conditional quantile
estimator : see Froelich et Melly (2008) for unconditional
quantile estimator in this context
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Results
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Extension of classical evaluation methods to quantile treatment
effect estimation :
I
Lamarche (2007) : fixed effects to deal with endogeneity bias
in a evaluation of the vouchers experiment (Milwaukee).
I
Froelich and Melly (2008) extension to discontinuity regression
design
I
application to duration models : Koenker and Bilias (2002),
evaluation of the “Bonus Experiment”
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Extension of classical evaluation methods to quantile treatment
effect estimation :
I
Lamarche (2007) : fixed effects to deal with endogeneity bias
in a evaluation of the vouchers experiment (Milwaukee).
I
Froelich and Melly (2008) extension to discontinuity regression
design
I
application to duration models : Koenker and Bilias (2002),
evaluation of the “Bonus Experiment”
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
definition
QTE under CIA
QTE estimation with IV
Summary on QTE
Quantile Treatment Effect - Summary
I
We define new parameters, the Quantile Treatment Effect: and
the Quantile Treatment Effect on the treated that compare
the quantiles of the distributions of potential outcomes
I
They compare distributions (no individual interpretation
without further assumption); no reason to be similar to the
quantiles of the distribution of Y1 − Y0 (the impact of the
treatment): Qτ (Y1 ) − Qτ (Y0 ) 6= Qτ (Y1 − Y0 )
I
We have to deal as usual with selection effects
I
We have seen two common identification strategies (under CIA
or when IV is available) : simple estimation procedure by using
the common quantile regression methods with an appropriate
weighting scheme
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Motivation
I
In many context, time is a dimension that we want to take
into account in the evaluation
I
The outcome of interest may be a duration : spell of
unemployment or employment, price-adjustment, “survival” of
a firm,...
Example: evaluation of the causal impact of a policy that
occurred at a specific date Illustration
I
Other (related) case: the treatment or policy has a specific
dynamic that we want to analyze
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Illustration
Policy was implemented a date τ ∗
τ : calendar date, t elapsed duration (for instance in
unemployment)
back
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Motivation (2)
Example: evaluation of a training program for unemployed
individuals, that may begin at any time tp after the beginning of
the unemployment spell
I We may want to address several different questions:
1. What is the impact of being treated on the duration of
unemployment?
2. Is the timing of the treatment important?
Is it the same to be trained at the beginning of the spell of
unemployment than later?
3. What is the impact of the duration of the treatment?
Short-term training vs long term training
4. How does the impact of the treatment evolve over the time?
I
that requires to use specific methods
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Basics - definitions
State dependance, selectivity
Identification
Estimation
Summary
Very brief description of duration model analysis
I
The variable of interest is T , the duration before an event
occurred (for instance, getting a job for an unemployed person)
I
Issue: Given that this event has not yet happened, what are
the chances it will happen subsequently?
I
Definition survival function: The probability that “up until
now” the event has not yet occurred.
S(t) = P(T > t) = 1 − F (T ), with F cdf of the duration T
I
Definition hazard function : probability that the event
occurred at t, conditional on survival until this date
)
f (t)
Hazard function : h(t) = limdt→0 P(t<T ≤t+dt|t≤T
= S(t)
dt
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Basics - definitions
State dependance, selectivity
Identification
Estimation
Summary
Observed and unobserved heterogeneity
I
As usual, we are interested in the causal impact of a policy,
program... D on duration T once controlled unobserved and
observed heterogeneity (observables covariables X and
unobservable ones v )
I
We model the hazard function and survival conditional to
,X ,v )
these covariates: h(t, X , v ) = limdt→0 P(t<T ≤t+dt|t≤T
dt
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Basics - definitions
State dependance, selectivity
Identification
Estimation
Summary
Why not performing OLS?
I
We can think of using classic linear regression
I
Assume data obtained from an “ideal” randomized experiment
(full compliance, no contamination effect...) : a training
program has been provided to some (randomly chosen)
unemployment individuals
I
We observe a duration in unemployment of a cohort of
unemployed individuals, at a date t
I
We can think of using :
Ti = X 0 β + D∆ + u
with D the fact of being treated, X observable covariates.
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Basics - definitions
State dependance, selectivity
Identification
Estimation
Summary
Why not performing OLS?
I
OLS raises two main issues :
1. censoring problem : at the end of the observation period, some
may have not find a job; what to do these observations?
we can think of treating censored spell as completed → will
result in biased estimates
we can think of excluding them → at the best inefficient
estimates
2. Time-varying covariates : some of the covariates change over
time (for instance, unemployment benefits, the training
program that may occur at different times for different
unemployed) : how to introduce them in the model - at the
beginning, the end, average? No natural way
I
This requires to use methods that take into account the
specific nature of duration outcomes
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Basics - definitions
State dependance, selectivity
Identification
Estimation
Summary
Remarks
Remarks:
1. framework can be extended to multiple-exit models
(multistates) : competing risks models
2. several other practical issues have to be considered, that we
will not discuss today (for details, see for instance Lancaster,
1990)
I
I
stock sampling vs inflows sampling
discrete vs continuous survival data
our emphasis : estimation of causal impact
How to deal with selection effects due to individual
heterogeneity
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Basics - definitions
State dependance, selectivity
Identification
Estimation
Summary
state dependance, selectivity
I
I
Issue of selectivity : usually re-employment hazard decreases
with unemployment duration
can be due to
1. state dependance : the unemployment reduces human capital
(“employability”)
2. dynamic sorting : those with the highest propensity to find a
job are the first to disappear
I
can we identify both separately?
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Basics - definitions
State dependance, selectivity
Identification
Estimation
Summary
Illustration
I
Assume that we observe a sample of individuals who begun at
unemployment spell at period t = 0
I
We have two types of individuals, with respectively high and
low employability
I
For the sake of illustration, assume that both types the
hazards are constant (no state dependance) : we have
hL (t) = hL ≤ hH (t) = hH ∀t
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Basics - definitions
State dependance, selectivity
Identification
Estimation
Summary
Illustration
I
We observe the marginal (aggregate) hazard that corresponds
to h(t) = pt hL + (1 − pt )hH with pt proportion of sample with
hL
I
At the first period : we observe h(t = 1) = p1 hL + (1 − p1 )hH
I
At the second period : in the remaining sample of unemployed
individuals, the proportion p2 with hL is higher than in the first
period (formally : Proof )
I
we thus observe h(t = 2) ≤ h(t = 1) meaning a spurious state
dependance
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Basics - definitions
State dependance, selectivity
Identification
Estimation
Summary
Mixed proportional hazard
I
Most popular duration model : mixed proportional hazard
(MPH) h(t, X , v ) = λ(t)φ(x)v
t duration, x observable covariates, v unobservables
I
λ(t) baseline function; φ(x) function of observables (usually
exp(X 0 β), Cox specification)
I
Assumption : normalization condition E (v ) = 1; v is assumed
independent of x
I
Very commonly used because of its simplicity, interpretability
and results on identification
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Basics - definitions
State dependance, selectivity
Identification
Estimation
Summary
Identification
I
seminal paper by Elbers and Ridders (82) : when there is some
time-varying regressors x, the MPH model is identified
(meaning there are no two combinations of heterogeneity and
time dependance function giving the same duration
distribution) under mild assumption on G (v )
I
Remark : without x, we cannot distinguish the case with no
state dependance (λ(t) = 1, any g (v )) from a case with no
unobserved heterogeneity (g (v ) = 1, any λ(t)), as illustrated
in the sample example before
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Basics - definitions
State dependance, selectivity
Identification
Estimation
Summary
Estimation of duration models
I
Estimation usually relies on likelihood maximization methods
I
Using specification for the distribution of the duration (for
instance MPH, explicit function for λ and φ), we may define
the likelihood of observing each observations (including those
censuring), that depends on X , t, unobservable v
I
That requires to specify a functional form for the baseline
hazard rate λ(t), but also on the unobservable distribution
G (v )
I
by definition v is unobservable, we “integrate out” over the
distribution G (v ): we observe
R in data the average survival
function given by S(t, x) = v h(t, x, v )dG (v )
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Basics - definitions
State dependance, selectivity
Identification
Estimation
Summary
Summary
I
duration outcomes cannot be modelized by OLS (censoring,
time-varying covariate)
I
common modelisation relies on the specification of the hazard
rates and survival function
I
identification of the time dependance is blurred by selectivity
(dynamic sorting)
I
identification results : popular mixed proportional hazard
h(t, X , v ) = λ(t)φ(x)v is identified provided that it includes
time-varying covariates
I
Estimation usually relies on parametric specifications (for the
baseline hazard and the distribution of the unobservables)
Pauline Givord
Evaluation of Public Policies
Quantile Treatment Effect
Duration outcome and dynamic treatment - introduction
Duration model
Basics - definitions
State dependance, selectivity
Identification
Estimation
Summary
Formal calculations:
p2 =
=
p1 (1 − hL )
p1 (1 − hL ) + (1 − p1 )(1 − hH )
p1
1+
< p1
(1−p1 )(hH −hL )
(1−hL )
As surviving population of type L at date t = 2 corresponds to
p1 (1 − hL ), of the whole population : p1 (1 − hL ) + (1 − p1 )(1 − hH )
back
Pauline Givord
Evaluation of Public Policies