07350015%2E2013%2E792262
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Charles F. Manski
To cite this article: Charles F. Manski (2013) Comment, Journal of Business & Economic
Statistics, 31:3, 273-275, DOI: 10.1080/07350015.2013.792262
To link to this article: http://dx.doi.org/10.1080/07350015.2013.792262
Published online: 22 Jul 2013.
Submit your article to this journal
Article views: 295
View related articles
Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=ubes20
Download by: [Universitas Maritim Raja Ali Haji]
Date: 11 January 2016, At: 22:11
Manski: Comment
273
strategic network formation are still in their infancy (Christakis
et al. 2010; Mele 2011; Chandrasekhar and Jackson 2012; Leung
2013). The very important lesson that we should take from the
analysis of Goldsmith-Pinkham and Imbens is that accounting
for the endogeneity of relationships in analyses of peer effects is
feasible, and can provide substantial new insights, for example,
into unobserved characteristics that might correlate both with
behavior and friendship formation. The specifications that are
needed to properly model both network formation and peer effects require careful additional analysis in context, and provide
us with a rich agenda going forward.
Downloaded by [Universitas Maritim Raja Ali Haji] at 22:11 11 January 2016
REFERENCES
Aral, S., Muchnik, L., and Sundararajan, A. (2009), “Distinguishing Influence Based Contagions from Homophily Driven Diffusion in Dynamic Networks,” Proc. Natl. Acad. Sci., 106, 21544–21549. [270]
Bala, V., and Goyal, S. (2000), “A Noncooperative Model of Network Formation,” Econometrica, 68, 1181–1229. [272]
Banerjee, A., Chandrasekhar, A., Duflo, E., and Jackson, M. (2012), “Diffusion of Microfinance,” NBER Working Paper 17743. Available at
http://www.stanford.edu/jacksonm/diffusionofmf.pdf. [272]
Blume, L., Brock, W., Durlauf, S., and Ioannides, Y. (2011), “Identification of
Social Interactions,” in The Handbook of Social Economics, ed. J. Benhabib,
A. Bisin, and M. Jackson, San Diego: North Holland. [270]
Bramoull´e, Y., Djebbari, H., and Fortin, B. (2009), “Identification of Peer Effects
Through Social Networks,” Journal of Econometrics, 150, 41–55. [270]
Centola, D. (2010), “The Spread of Behavior in an Online Social Network
Experiment,” Science, 32, 1194–1197, doi: 10.1126/science.1185231. [270]
Chandrasekhar, A., and Jackson, M. (2012), “Tractable and Consistent Random Graph Models,” SSRN Working Paper. Available at
http://ssrn.com/abstract=2150428. [270,273]
Christakis, N., Fowler, J., Imbens, G., and Kalyanaraman, K. (2010), “An Empirical Model for Strategic Network Formation,” NBER Working Paper.
[273]
Currarini, S., Jackson, M., and Pin, P. (2009), “An Economic Model of Friendship: Homophily, Minorities, and Segregation,” Econometrica, 77, 1003–
1045. [271]
Currarini, S., Jackson, M., and Pin, P. (2010), “Identifying the Roles of RaceBased Choice and Chance in High School Friendship Network Formation,”
Proceedings of the National Academy of Sciences, 107, 4857–4861. [271]
Duflo, E., and Saez, E. (2003), “The Role of Information and Social Interactions
in Retirement Plan Decisions: Evidence From a Randomized Experiment,”
Quarterly Journal of Economics, 118, 815–842. [270]
Goldsmith-Pinkham, P., and Imbens, G. (2013), “Social Networks and the Identification of Peer Effects,” Journal of Business and Economic Statistics, 31,
253–264. [271]
Jackson, M. (2008), Social and Economic Networks, Princeton, NJ: Princeton
University Press. [270,272]
Jackson, M., Barraquer, T., and Tan, X. (2012), “Social Capital and Social
Quilts: Network Patterns of Favor Exchange,” American Economic Review,
102, 1857–1897. [272]
Jackson, M., and Wolinsky, A. (1996), “A Strategic Model of Social and Economic Networks,” Journal of Economic Theory, 71, 44–74. [272]
Leung, M. (2013), “Two-Step Estimation of Network-Formation Models With
Incomplete Information,” Working Paper, Stanford University. [273]
Manski, C. (1993), “Identification of Endogenous Social Effects: The Reflection
Problem,” The Review of Economic Studies, 531–542. [270]
Mele, A. (2011). “A Structural Model of Segregation in Social Networks,”
Working Paper, Johns Hopkins University. [273]
Comment
Charles F. MANSKI
Department of Economics and Institute for Policy Research, Northwestern University, Evanston, IL 60201
(cfmanski@northwestern.edu)
To begin, I will express mixed feelings about the decision
by Goldsmith-Pinkham and Imbens to attach my name to the
linear-in-means models that are the concern of their article. I
will not object strenuously to their invention of the acronym
MLIM model, if only because I am aware of the adage that
“any publicity is good publicity, as long as they spell your name
right.” However, I will note that Manski (1993) did not originate linear-in-means models, whose use in empirical research
goes back at least to the 1960s and perhaps earlier. Nor did my
article advocate empirical application of these models to study
social interactions—it also studied nonparametric models and
parametric nonlinear ones, taking no position on the realism
of any of them. My objective in writing the article was to analyze identification of models with endogenous social effects,
recognizing that the outcomes in such models solve equilibrium
conditions. For this purpose, linear-in-means models provided
an analytically tractable case that illustrates well some basic
issues.
I similarly interpret Goldsmith-Pinkham and Imbens not to
be advocating linear-in-means models but rather to be focusing on these models to illustrate some basic issues. Their article
extends the literature in several directions. One that I find particularly interesting is their development of some relatively simple
economic models of network formation and integration of these
models with study of interactions within the formed networks.
Another is their use of Bayesian inferential methods to circumvent conceptually and technically difficult issues that arise in
performing frequentist inference in settings where a population
does not partition into a large set of separate networks. When
making these contributions, the authors maintain parametric
modeling assumptions that may be too restrictive to provide a
realistic basis for credible empirical analysis. Yet, as with the
linear-in-means model itself, I can appreciate the illustrative
value of focusing attention on tractable special cases.
SPECIFYING THE OBJECT OF INTEREST
I will focus my comment on one passage in the article by
Goldsmith-Pinkham and Imbens, where they consider the object
of interest for empirical analysis of social interactions. They
write
© 2013 American Statistical Association
Journal of Business & Economic Statistics
July 2013, Vol. 31, No. 3
DOI: 10.1080/07350015.2013.792262
274
Journal of Business & Economic Statistics, July 2013
Downloaded by [Universitas Maritim Raja Ali Haji] at 22:11 11 January 2016
The main object of interest is the effect of peers’ outcomes on own
outcomes. . . . Also of interest is the exogenous peer effect. . . .
Here we interpret the endogenous effect as the average change we
would see in an individual’s outcome if we changed their peer’s
outcomes directly. . . . . In some cases this may be difficult to
envision. In our example, we will think of this as something along
the lines of the direct, causal, effect of providing special tutoring
to one’s peers on one’s own outcome. . . . . The exogenous effect
is interpreted as the causal effect of changing the peer’s covariate
values. For some covariates, this thought experiment may be
difficult, but for others, especially lagged values of choices, it
may be feasible to consider interventions that would change those
values for the peers.
It is accurate that the extant literature has taken the main
object of interest to be the endogenous effect, with secondary
interest in the exogenous effect. However, my own thinking on
this matter has evolved over the past 20 years. Whereas Manski
(1993) focused primarily on the endogenous effect, I have long
been concerned that this rarely is the object of interest from
the perspective of policy formation. The reason is that a policy
maker can rarely manipulate peer outcomes directly. Moreover,
if a policy maker somehow is able to manipulate peer outcomes,
then doing so breaks the equilibrium conditions of endogenous
effects models.
Motivated by policy concerns, Manski (2013) studied settings
with endogenous effects as problems of analysis of treatment
response with social interactions. The treatments are variables
that a policy maker can manipulate, which may jointly affect the
outcomes of all persons in a network. An endogenous effects
model specifies a mechanism through which treatment response
may propagate through the network.
The analysis in the published version of Manski (2013) is
entirely nonparametric, but earlier working paper versions used
a linear-in-means model to illustrate abstract ideas. Writing a
comment on the article by Goldsmith-Pinkham and Imbens provides me with an opportunity to place this illustration in print.
ANALYSIS OF TREATMENT RESPONSE
IN A LINEAR-IN-MEANS MODEL
Here is the basic setup considered in Manski (2013). Let
J be a population and (J, , P) be a probability space. Let T be
a set of feasible treatments and let T J ≡ × j ∈ J T be the space of
treatment vectors potentially assigned to the entire population.
For each j ∈ J, let response function yj (·): T J → Y map treatment
vectors into potential outcomes. Thus, yj (tJ) is the outcome for
j under a specified treatment vector tJ ≡ (tk , k ∈ J). Person j
has realized treatment zj and outcome yj ≡ yj (zJ). Observation
of [(yj , zj ), j ∈ J] reveals P(y, z), hence P[y(zJ)]. The objective is
to learn the distribution of treatment response under tJ, that is,
P[y(tJ)].
A simple linear-in-means model emerges if the population
partitions into symmetric reference groups characterized by values for an observed covariate x. Each group contains a continuum of persons. The linear-in-means model assumes that for
each person j
yj (t J ) = α + β1 tj + β2 E(t|xj ) + γ E[y(t J )|xj ] + uj .
(1)
Here, parameter β 2 measures the exogenous effect and γ the
endogenous effect. Taking expectations conditional on xj yields
E[y(t J )|xj ] = α + (β1 + β2 )E(t|xj ) + γ E[y(t J )|xj ]
+ E(u|xj ).
(2)
Unless γ = 1, the unique equilibrium value of E[y(tJ)|xj ] is
E[y(t J )|xJ ] =
E(u|xj )
α
β1 + β2
+
E(t|xj ) +
1−γ
1−γ
1 − γ.
(3)
Insertion of the right-hand side of Equation (3) into the structural function (1) yields the response function
yj (t j ) =
γβ1 + β2
α
+ β1 tj +
E(t|xj )
1−γ
1−γ
γ
+
E(u|xj ) + uj .
1−γ
(4)
Thus, the response function is the reduced form of the structural function.
The model thus far does not pin down the structural or response functions. The reason is that it does not restrict the unobserved covariates (uj , j ∈ J). Assume that E(u|z, x) = 0. This
implies a linear mean regression relating realized treatments and
outcomes:
E(y|z, x) =
α
γβ1 + β2
+ β1 z +
E(z|x)
1−γ
1−γ
≡ ϕ0 + ϕ1 z + ϕ2 E(z|x)
(5)
Observation of realized treatments and outcomes reveals
E(y|z, x) on the support of (z, x). Hence, ϕ is point-identified
if the support of [1, z, E(z|x)] is not contained in a linear subspace of R3. Knowledge of ϕ and the empirical evidence imply
knowledge of (uj , j ∈ J). Finally, knowledge of ϕ and (uj , j ∈ J)
implies knowledge of the response functions [yj (·), j ∈ J].
Note that point-identification of the response-function parameters ϕ does not imply point-identification of the structural
parameters (α, β 1 , β 2 , γ ). β 1 is point-identified but (α, β 2 , γ )
are not. Thus, one cannot distinguish exogenous from endogenous effects under the maintained assumptions. Nevertheless,
the assumptions fully reveal the population vector of response
functions.
WHEN IS IT IMPORTANT TO IDENTIFY
THE STRUCTURAL PARAMETERS?
The above shows that point-identification of the structural
parameters is not a prerequisite for point-identification of treatment response. Yet inference on structural parameters has been
the dominant theme of modern econometric analysis of social
interactions. It also has been the dominant theme of the classical literature on identification of linear simultaneous equations.
A rare exception was voiced by Arthur Goldberger in his ET
Interview. Responding to a question from Nick Kiefer, Goldberger said (Kiefer and Goldberger 1989, p. 150): “Well, that’s
one position, that the entire content in a structural model is
simply in the restrictions, if any, that it implies on the reduced
form—that’s true. That gives priority to the reduced form.”
Sacerdote: Comment
275
One reason for inference on structural parameters may be
“science.” Researchers may want to characterize reality, as an
end in itself. A different reason is to predict treatment response
when a regime change (an uber treatment) alters part of a structural model in a known way, leaving other parts invariant. Then
response functions change in a way that can be predicted with
knowledge of the structural model but not otherwise.
Consider the linear-in-means response function
yj (t J ) =
α
γβ1 + β2
+ β1 tj +
E(t|x) + uj
1−γ
1−γ
Downloaded by [Universitas Maritim Raja Ali Haji] at 22:11 11 January 2016
+ ϕ0 + ϕ1 tj + ϕ2 E(t|xj ) + uj .
(6)
A regime change might alter some of the structural parameters
(α, β 1 , β 2 , γ ) in a known way while leaving others unchanged.
Then knowledge of (α, β 1 , β 2 , γ ) enables prediction of treatment
response but knowledge of (ϕ 0 , ϕ 1 , ϕ 2 ) does not.
REFERENCES
Kiefer, N., and Goldberger, A. (1989), “The ET Interview: Arthur S.
Goldberger,” Econometric Theory, 5, 133–160. [274]
Manski, C. (1993), “Identification of Endogenous Social Effects: The Reflection
Problem,” Review of Economic Studies, 60, 531–542. [273,274]
—— (2013), “Identification of Treatment Response with Social Interactions,”
The Econometric Journal, 16, S1–S23. [274]
Comment
Bruce SACERDOTE
Department of Economics, Dartmouth College, Hanover, NH 03755 and NBER (Bruce.Sacerdote@dartmouth.edu)
This is an innovative article and a nice addition to the literature on the estimation of endogenous and exogenous peer
effects. There are two main contributions. First, the authors
suggest that we can incorporate the endogeneity of peer choice
in a parsimonious way. The authors introduce an unobserved
individual specific parameter ξ i . For any two individuals i and
j, the distance between ξ i and ξ j affects the probability that i
and j form a link. Then this ξ i is introduced directly and linearly
(in Equation (6.1)) as a determinant of i’s outcome Yi . This is a
very clever approach and has the potential to greatly reduce the
complexity of an otherwise intractable problem.
The second advance of the article is to show that all the
model’s parameters (including the ξ i ’s) can in principle be estimated in a Bayesian framework using Monte Carlo methods.
This answers the obvious question of how we might estimate
the individual specific unobserved regressor.
The authors proceed to estimate their model using Ad Health
data and calculating exogenous and endogenous peer effects on
own Grade Point Average. The estimates seem quite plausible.
For example, an individual’s own past grades predict current
grades with a coefficient of 0.73. Peers’ past grades predict
own current grades with a coefficient of 0.11. Such estimates
are in the same ballpark as existing articles that have random
assignment to classrooms.
Interestingly the introduction of endogenous network formation (through the vector of etas) does not have a meaningful
impact on the estimated peer effects. Compare, for example,
the results in Tables 5 and 6, where the former table assumes
that peer choice is exogenous. My own experimentation with
the model found much the same result. Using data from a military academy with randomly assigned squadrons, I found that
accounting for an individual specific ξ i (which affects friend
choice) affects the outcome, but does not affect the estimated
peer effects. (My coauthors Scott Carrell and James West who
have access to the data were kind enough to run the code for
me.)
Besides being a fan of the article, I have two general comments. First, not all readers will accept the simple parameterization of friendship choice as having solved the peer selection
problem. I suspect that the authors’ formulation works particularly well in their example and in my example because in both
cases we have strong controls for own ability and peer (background) ability. Staiger and Kane have convinced me that in test
score value added models, having prior test scores does a great
deal to compensate for the selection of students into classrooms
and schools.
Second, economics researchers have become progressively
less interested in the linear-in-means model. Models beyond the
linear-in-means models allow the possibility for Pareto improving reallocations of students, such as tracking students (grouping
them into classrooms) by ability. Hoxby and Weingarth (2005)
and my own work with Imberman and Kugler finds that nonlinear models fit the data much better. I suspect that with a
minimum of tinkering the authors’ model could be extended
to a more flexible (nonlinear) formulation. Part of the beauty
of the Markov chain Monte Carlo method being used is that a
wide variety of models can be estimated even in cases where we
cannot conduct maximum likelihood estimation.
Overall I found this to be a thoughtful article and a worthwhile
contribution.
REFERENCE
Hoxby, C. M., and Weingarth, G. (2005), “Taking Race Out of the Equation:
School Reassignment and the Structure of Peer Effects,” Working Paper,
Harvard University. [275]
© 2013 American Statistical Association
Journal of Business & Economic Statistics
July 2013, Vol. 31, No. 3
DOI: 10.1080/07350015.2013.792263
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Charles F. Manski
To cite this article: Charles F. Manski (2013) Comment, Journal of Business & Economic
Statistics, 31:3, 273-275, DOI: 10.1080/07350015.2013.792262
To link to this article: http://dx.doi.org/10.1080/07350015.2013.792262
Published online: 22 Jul 2013.
Submit your article to this journal
Article views: 295
View related articles
Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=ubes20
Download by: [Universitas Maritim Raja Ali Haji]
Date: 11 January 2016, At: 22:11
Manski: Comment
273
strategic network formation are still in their infancy (Christakis
et al. 2010; Mele 2011; Chandrasekhar and Jackson 2012; Leung
2013). The very important lesson that we should take from the
analysis of Goldsmith-Pinkham and Imbens is that accounting
for the endogeneity of relationships in analyses of peer effects is
feasible, and can provide substantial new insights, for example,
into unobserved characteristics that might correlate both with
behavior and friendship formation. The specifications that are
needed to properly model both network formation and peer effects require careful additional analysis in context, and provide
us with a rich agenda going forward.
Downloaded by [Universitas Maritim Raja Ali Haji] at 22:11 11 January 2016
REFERENCES
Aral, S., Muchnik, L., and Sundararajan, A. (2009), “Distinguishing Influence Based Contagions from Homophily Driven Diffusion in Dynamic Networks,” Proc. Natl. Acad. Sci., 106, 21544–21549. [270]
Bala, V., and Goyal, S. (2000), “A Noncooperative Model of Network Formation,” Econometrica, 68, 1181–1229. [272]
Banerjee, A., Chandrasekhar, A., Duflo, E., and Jackson, M. (2012), “Diffusion of Microfinance,” NBER Working Paper 17743. Available at
http://www.stanford.edu/jacksonm/diffusionofmf.pdf. [272]
Blume, L., Brock, W., Durlauf, S., and Ioannides, Y. (2011), “Identification of
Social Interactions,” in The Handbook of Social Economics, ed. J. Benhabib,
A. Bisin, and M. Jackson, San Diego: North Holland. [270]
Bramoull´e, Y., Djebbari, H., and Fortin, B. (2009), “Identification of Peer Effects
Through Social Networks,” Journal of Econometrics, 150, 41–55. [270]
Centola, D. (2010), “The Spread of Behavior in an Online Social Network
Experiment,” Science, 32, 1194–1197, doi: 10.1126/science.1185231. [270]
Chandrasekhar, A., and Jackson, M. (2012), “Tractable and Consistent Random Graph Models,” SSRN Working Paper. Available at
http://ssrn.com/abstract=2150428. [270,273]
Christakis, N., Fowler, J., Imbens, G., and Kalyanaraman, K. (2010), “An Empirical Model for Strategic Network Formation,” NBER Working Paper.
[273]
Currarini, S., Jackson, M., and Pin, P. (2009), “An Economic Model of Friendship: Homophily, Minorities, and Segregation,” Econometrica, 77, 1003–
1045. [271]
Currarini, S., Jackson, M., and Pin, P. (2010), “Identifying the Roles of RaceBased Choice and Chance in High School Friendship Network Formation,”
Proceedings of the National Academy of Sciences, 107, 4857–4861. [271]
Duflo, E., and Saez, E. (2003), “The Role of Information and Social Interactions
in Retirement Plan Decisions: Evidence From a Randomized Experiment,”
Quarterly Journal of Economics, 118, 815–842. [270]
Goldsmith-Pinkham, P., and Imbens, G. (2013), “Social Networks and the Identification of Peer Effects,” Journal of Business and Economic Statistics, 31,
253–264. [271]
Jackson, M. (2008), Social and Economic Networks, Princeton, NJ: Princeton
University Press. [270,272]
Jackson, M., Barraquer, T., and Tan, X. (2012), “Social Capital and Social
Quilts: Network Patterns of Favor Exchange,” American Economic Review,
102, 1857–1897. [272]
Jackson, M., and Wolinsky, A. (1996), “A Strategic Model of Social and Economic Networks,” Journal of Economic Theory, 71, 44–74. [272]
Leung, M. (2013), “Two-Step Estimation of Network-Formation Models With
Incomplete Information,” Working Paper, Stanford University. [273]
Manski, C. (1993), “Identification of Endogenous Social Effects: The Reflection
Problem,” The Review of Economic Studies, 531–542. [270]
Mele, A. (2011). “A Structural Model of Segregation in Social Networks,”
Working Paper, Johns Hopkins University. [273]
Comment
Charles F. MANSKI
Department of Economics and Institute for Policy Research, Northwestern University, Evanston, IL 60201
(cfmanski@northwestern.edu)
To begin, I will express mixed feelings about the decision
by Goldsmith-Pinkham and Imbens to attach my name to the
linear-in-means models that are the concern of their article. I
will not object strenuously to their invention of the acronym
MLIM model, if only because I am aware of the adage that
“any publicity is good publicity, as long as they spell your name
right.” However, I will note that Manski (1993) did not originate linear-in-means models, whose use in empirical research
goes back at least to the 1960s and perhaps earlier. Nor did my
article advocate empirical application of these models to study
social interactions—it also studied nonparametric models and
parametric nonlinear ones, taking no position on the realism
of any of them. My objective in writing the article was to analyze identification of models with endogenous social effects,
recognizing that the outcomes in such models solve equilibrium
conditions. For this purpose, linear-in-means models provided
an analytically tractable case that illustrates well some basic
issues.
I similarly interpret Goldsmith-Pinkham and Imbens not to
be advocating linear-in-means models but rather to be focusing on these models to illustrate some basic issues. Their article
extends the literature in several directions. One that I find particularly interesting is their development of some relatively simple
economic models of network formation and integration of these
models with study of interactions within the formed networks.
Another is their use of Bayesian inferential methods to circumvent conceptually and technically difficult issues that arise in
performing frequentist inference in settings where a population
does not partition into a large set of separate networks. When
making these contributions, the authors maintain parametric
modeling assumptions that may be too restrictive to provide a
realistic basis for credible empirical analysis. Yet, as with the
linear-in-means model itself, I can appreciate the illustrative
value of focusing attention on tractable special cases.
SPECIFYING THE OBJECT OF INTEREST
I will focus my comment on one passage in the article by
Goldsmith-Pinkham and Imbens, where they consider the object
of interest for empirical analysis of social interactions. They
write
© 2013 American Statistical Association
Journal of Business & Economic Statistics
July 2013, Vol. 31, No. 3
DOI: 10.1080/07350015.2013.792262
274
Journal of Business & Economic Statistics, July 2013
Downloaded by [Universitas Maritim Raja Ali Haji] at 22:11 11 January 2016
The main object of interest is the effect of peers’ outcomes on own
outcomes. . . . Also of interest is the exogenous peer effect. . . .
Here we interpret the endogenous effect as the average change we
would see in an individual’s outcome if we changed their peer’s
outcomes directly. . . . . In some cases this may be difficult to
envision. In our example, we will think of this as something along
the lines of the direct, causal, effect of providing special tutoring
to one’s peers on one’s own outcome. . . . . The exogenous effect
is interpreted as the causal effect of changing the peer’s covariate
values. For some covariates, this thought experiment may be
difficult, but for others, especially lagged values of choices, it
may be feasible to consider interventions that would change those
values for the peers.
It is accurate that the extant literature has taken the main
object of interest to be the endogenous effect, with secondary
interest in the exogenous effect. However, my own thinking on
this matter has evolved over the past 20 years. Whereas Manski
(1993) focused primarily on the endogenous effect, I have long
been concerned that this rarely is the object of interest from
the perspective of policy formation. The reason is that a policy
maker can rarely manipulate peer outcomes directly. Moreover,
if a policy maker somehow is able to manipulate peer outcomes,
then doing so breaks the equilibrium conditions of endogenous
effects models.
Motivated by policy concerns, Manski (2013) studied settings
with endogenous effects as problems of analysis of treatment
response with social interactions. The treatments are variables
that a policy maker can manipulate, which may jointly affect the
outcomes of all persons in a network. An endogenous effects
model specifies a mechanism through which treatment response
may propagate through the network.
The analysis in the published version of Manski (2013) is
entirely nonparametric, but earlier working paper versions used
a linear-in-means model to illustrate abstract ideas. Writing a
comment on the article by Goldsmith-Pinkham and Imbens provides me with an opportunity to place this illustration in print.
ANALYSIS OF TREATMENT RESPONSE
IN A LINEAR-IN-MEANS MODEL
Here is the basic setup considered in Manski (2013). Let
J be a population and (J, , P) be a probability space. Let T be
a set of feasible treatments and let T J ≡ × j ∈ J T be the space of
treatment vectors potentially assigned to the entire population.
For each j ∈ J, let response function yj (·): T J → Y map treatment
vectors into potential outcomes. Thus, yj (tJ) is the outcome for
j under a specified treatment vector tJ ≡ (tk , k ∈ J). Person j
has realized treatment zj and outcome yj ≡ yj (zJ). Observation
of [(yj , zj ), j ∈ J] reveals P(y, z), hence P[y(zJ)]. The objective is
to learn the distribution of treatment response under tJ, that is,
P[y(tJ)].
A simple linear-in-means model emerges if the population
partitions into symmetric reference groups characterized by values for an observed covariate x. Each group contains a continuum of persons. The linear-in-means model assumes that for
each person j
yj (t J ) = α + β1 tj + β2 E(t|xj ) + γ E[y(t J )|xj ] + uj .
(1)
Here, parameter β 2 measures the exogenous effect and γ the
endogenous effect. Taking expectations conditional on xj yields
E[y(t J )|xj ] = α + (β1 + β2 )E(t|xj ) + γ E[y(t J )|xj ]
+ E(u|xj ).
(2)
Unless γ = 1, the unique equilibrium value of E[y(tJ)|xj ] is
E[y(t J )|xJ ] =
E(u|xj )
α
β1 + β2
+
E(t|xj ) +
1−γ
1−γ
1 − γ.
(3)
Insertion of the right-hand side of Equation (3) into the structural function (1) yields the response function
yj (t j ) =
γβ1 + β2
α
+ β1 tj +
E(t|xj )
1−γ
1−γ
γ
+
E(u|xj ) + uj .
1−γ
(4)
Thus, the response function is the reduced form of the structural function.
The model thus far does not pin down the structural or response functions. The reason is that it does not restrict the unobserved covariates (uj , j ∈ J). Assume that E(u|z, x) = 0. This
implies a linear mean regression relating realized treatments and
outcomes:
E(y|z, x) =
α
γβ1 + β2
+ β1 z +
E(z|x)
1−γ
1−γ
≡ ϕ0 + ϕ1 z + ϕ2 E(z|x)
(5)
Observation of realized treatments and outcomes reveals
E(y|z, x) on the support of (z, x). Hence, ϕ is point-identified
if the support of [1, z, E(z|x)] is not contained in a linear subspace of R3. Knowledge of ϕ and the empirical evidence imply
knowledge of (uj , j ∈ J). Finally, knowledge of ϕ and (uj , j ∈ J)
implies knowledge of the response functions [yj (·), j ∈ J].
Note that point-identification of the response-function parameters ϕ does not imply point-identification of the structural
parameters (α, β 1 , β 2 , γ ). β 1 is point-identified but (α, β 2 , γ )
are not. Thus, one cannot distinguish exogenous from endogenous effects under the maintained assumptions. Nevertheless,
the assumptions fully reveal the population vector of response
functions.
WHEN IS IT IMPORTANT TO IDENTIFY
THE STRUCTURAL PARAMETERS?
The above shows that point-identification of the structural
parameters is not a prerequisite for point-identification of treatment response. Yet inference on structural parameters has been
the dominant theme of modern econometric analysis of social
interactions. It also has been the dominant theme of the classical literature on identification of linear simultaneous equations.
A rare exception was voiced by Arthur Goldberger in his ET
Interview. Responding to a question from Nick Kiefer, Goldberger said (Kiefer and Goldberger 1989, p. 150): “Well, that’s
one position, that the entire content in a structural model is
simply in the restrictions, if any, that it implies on the reduced
form—that’s true. That gives priority to the reduced form.”
Sacerdote: Comment
275
One reason for inference on structural parameters may be
“science.” Researchers may want to characterize reality, as an
end in itself. A different reason is to predict treatment response
when a regime change (an uber treatment) alters part of a structural model in a known way, leaving other parts invariant. Then
response functions change in a way that can be predicted with
knowledge of the structural model but not otherwise.
Consider the linear-in-means response function
yj (t J ) =
α
γβ1 + β2
+ β1 tj +
E(t|x) + uj
1−γ
1−γ
Downloaded by [Universitas Maritim Raja Ali Haji] at 22:11 11 January 2016
+ ϕ0 + ϕ1 tj + ϕ2 E(t|xj ) + uj .
(6)
A regime change might alter some of the structural parameters
(α, β 1 , β 2 , γ ) in a known way while leaving others unchanged.
Then knowledge of (α, β 1 , β 2 , γ ) enables prediction of treatment
response but knowledge of (ϕ 0 , ϕ 1 , ϕ 2 ) does not.
REFERENCES
Kiefer, N., and Goldberger, A. (1989), “The ET Interview: Arthur S.
Goldberger,” Econometric Theory, 5, 133–160. [274]
Manski, C. (1993), “Identification of Endogenous Social Effects: The Reflection
Problem,” Review of Economic Studies, 60, 531–542. [273,274]
—— (2013), “Identification of Treatment Response with Social Interactions,”
The Econometric Journal, 16, S1–S23. [274]
Comment
Bruce SACERDOTE
Department of Economics, Dartmouth College, Hanover, NH 03755 and NBER (Bruce.Sacerdote@dartmouth.edu)
This is an innovative article and a nice addition to the literature on the estimation of endogenous and exogenous peer
effects. There are two main contributions. First, the authors
suggest that we can incorporate the endogeneity of peer choice
in a parsimonious way. The authors introduce an unobserved
individual specific parameter ξ i . For any two individuals i and
j, the distance between ξ i and ξ j affects the probability that i
and j form a link. Then this ξ i is introduced directly and linearly
(in Equation (6.1)) as a determinant of i’s outcome Yi . This is a
very clever approach and has the potential to greatly reduce the
complexity of an otherwise intractable problem.
The second advance of the article is to show that all the
model’s parameters (including the ξ i ’s) can in principle be estimated in a Bayesian framework using Monte Carlo methods.
This answers the obvious question of how we might estimate
the individual specific unobserved regressor.
The authors proceed to estimate their model using Ad Health
data and calculating exogenous and endogenous peer effects on
own Grade Point Average. The estimates seem quite plausible.
For example, an individual’s own past grades predict current
grades with a coefficient of 0.73. Peers’ past grades predict
own current grades with a coefficient of 0.11. Such estimates
are in the same ballpark as existing articles that have random
assignment to classrooms.
Interestingly the introduction of endogenous network formation (through the vector of etas) does not have a meaningful
impact on the estimated peer effects. Compare, for example,
the results in Tables 5 and 6, where the former table assumes
that peer choice is exogenous. My own experimentation with
the model found much the same result. Using data from a military academy with randomly assigned squadrons, I found that
accounting for an individual specific ξ i (which affects friend
choice) affects the outcome, but does not affect the estimated
peer effects. (My coauthors Scott Carrell and James West who
have access to the data were kind enough to run the code for
me.)
Besides being a fan of the article, I have two general comments. First, not all readers will accept the simple parameterization of friendship choice as having solved the peer selection
problem. I suspect that the authors’ formulation works particularly well in their example and in my example because in both
cases we have strong controls for own ability and peer (background) ability. Staiger and Kane have convinced me that in test
score value added models, having prior test scores does a great
deal to compensate for the selection of students into classrooms
and schools.
Second, economics researchers have become progressively
less interested in the linear-in-means model. Models beyond the
linear-in-means models allow the possibility for Pareto improving reallocations of students, such as tracking students (grouping
them into classrooms) by ability. Hoxby and Weingarth (2005)
and my own work with Imberman and Kugler finds that nonlinear models fit the data much better. I suspect that with a
minimum of tinkering the authors’ model could be extended
to a more flexible (nonlinear) formulation. Part of the beauty
of the Markov chain Monte Carlo method being used is that a
wide variety of models can be estimated even in cases where we
cannot conduct maximum likelihood estimation.
Overall I found this to be a thoughtful article and a worthwhile
contribution.
REFERENCE
Hoxby, C. M., and Weingarth, G. (2005), “Taking Race Out of the Equation:
School Reassignment and the Structure of Peer Effects,” Working Paper,
Harvard University. [275]
© 2013 American Statistical Association
Journal of Business & Economic Statistics
July 2013, Vol. 31, No. 3
DOI: 10.1080/07350015.2013.792263