Comments on “Partial
Identification by Extending
Subdistributions” by Alexander
Torgovitsky
Frank A. Wolak
Department of Economics
Director, Program on Energy and Sustainable Development
Stanford University
Stanford, CA 94305-6072
wolak@zia.stanford.edu
http://www.stanford.edu/~wolak
Motivation for Research
• Obtaining point identification of economic magnitude of
interest often requires difficult-to-defend distributional
assumptions or functional form assumptions on
econometric model
• Partial identification modeling framework provides
alternative approach to estimating economic magnitude of
interest without imposing these assumptions
– Advantage—Researcher only imposes assumptions on
distribution of unobservables and functional forms of econometric
model that he/she finds credible
– Disadvantages--Researcher can typically only estimate identified
set that contains true economic magnitude of interest
– Extremely challenging numerical problem to compute estimate of
identified set
– Computationally intensive procedures for testing hypotheses
about characteristics of identified set or points in identified set
Summary of Results
• Main Result---Partial Identification by extending subdistributions (PIES)
• General econometric model
– Y = h(X,U)
– Y = vector of outcome variables
– X = vector of explanatory variables
– U = L-dimensional vector of latent variables with conditional distribution
U|X = x given by F(u|x)
– h(u,x) = structural function
• Researcher makes assumptions about F(u|x) and
h(x,u) that identifies the set that contains these
magnitudes from conditional distribution of Y given X
– h and F that satisfy these assumptions are called admissible values
• By definition, h is in the identified set if and only if there exists an
admissible F that generates the observed distribution of Y given
X
Summary of Results
• This requirement constrains the behavior F(u|x) on subset
Ux(h) of 𝑅𝐿
– Author calls restriction of F(u|x) to Ux(h) a subdistribution
– Has properties of distribution function on Ux(h)
• Main Theoretical Result --Fix admissible h, if there exists a
subdistribution function, 𝐹 𝑢 𝑥 , on Ux(h) for each x that satisfies
observational equivalence condition, then subdistribution can be
extended to a distribution function F(u|x) on 𝑅 𝐿 that yields observed
distribution of Y|X=x
• Importance of result is that restrictions that determine whether
a function is a subdistribution are linear constraints on the
values of the function 𝐹 𝑢 𝑥
Summary of Results
• Paper applies result to ordered discrete response model
• 𝑌=
𝐽
𝑗=1 𝑦𝑗
1[𝑔𝑗−1 𝑋 < 𝑈 ≤ 𝑔𝑗 𝑋 ]
•
•
•
•
•
{g0,g1, …, gJ} = a vector of functions X
U = scalar latent variable
{y1, …, yJ} = discrete support of Y
{x1, …, xK} = discrete support of X
Computes values of identified set for binary response
model with g1(X) = β0 + β1X1 + β2X2
• Considers case that X1 exogenous and X1 is endogenous
– X1 exogenous cases considered—(1) X and U are independent,
(2) median of U given X is zero, (3) U given X is symmetric around
zero
– X1 endogenous, same cases considered as well additional cases
for latent variable in second equation of model determining value
of indicator Y2 (endogenous X1) that depends on instrument X3
Summary of Results
• Extends PIES framework to compute identified set for
average structural function (ASF) for binary response
model and average treatment effect (ATE)
– E(Y1 |X2,Y2 = 1) and E(Y1 |X2,Y2 = 0)
– Average Treatment Effect is difference of ASFs
– For some assumptions on binary choice model with endogenous
right-hand side variable identified set for ATE is not connected
• Results in Table 2
• PIES framework extended to derive subdistribution
extension lemma for general modeling framework
– PIES applied to two-sector Roy model in abstract form but no
identified sets where computed
– Applying procedure to compute identified sets for more general
models likely to be challenging
Comments on Paper
• General comment on partial identification
literature
– Theory-based empirical researchers are very
sympathetic to this approach, but it is hard to convince
other empirical researchers of its merits given the lack
of examples demonstrating empirical content
• Are there simple examples to illustrate how to
use estimation and inference procedures on an
important empirical question?
– Example that demonstrates that assumptions needed
for point identification can yield estimates that are
outside identified set for more credible assumptions
• Can computer software or detailed instructions on
how to implement procedures be provided for a
class of empirical problems?
Comments on Paper
• Can realistic Monte Carlo studies be run
illustrating
– Biases in common parametric approaches that are not
present in partial identification approaches
– Credible identifying assumptions that can still yield
informative answers about economic magnitudes
interest from identified set
• Identified set of demand price elasticity
• Identified set of compensating variation associated with price
change
• Partial identification approach offers way for
economic theory to be used to measure
magnitudes of economic interest without
“incredible assumptions” needed for point
identification
Comments on Paper
• Specific comments/questions about paper
• More details on procedure used to solve for
identified sets would be very informative
• More discussion of cause of results in Table 2
– Disconnected identified sets
• More discussion of specific classes of models
PIES approach could be applied to would be
useful
• More discussion of ways to relax linear functional
form assumption on g(X) function
– Particularly for binary response models, linear index seems more
restrictive than distributional assumption on latent variable
• Amemiya (1981) derives approximate relationships between probability limit of
slope coefficients in linear index model for probit, logit and linear probability
models
Concluding Comments
• Partial identification methods have potential to “put the
economics back into econometrics”
• To do so researchers must
• Show practical usefulness of partial identification
methods to empirical researchers
• Illustrate relationship between assumptions
researcher is willing to make and form of identified
set for some commonly employed model
• Provide software and more details on how to
implement methods
• Simple to implement rules-of-thumb may be
preferable to rigorous, but difficult to implement
approaches
Thank You