Comments on “Partial Identification by Extending

Subdistributions” by Alexander Torgovitsky

Frank A. WolakDepartment of Economics

Director, Program on Energy and Sustainable Development Stanford University

Stanford, CA 94305-6072wolak@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 – 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,, on Ux(h) for each x that satisfies observational equivalence condition, then subdistribution can be extended to a distribution function F(u|x) on 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

Summary of Results• Paper applies result to ordered discrete response model

• {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