07350015%2E2013%2E792260
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Rejoinder
Paul Goldsmith-Pinkham & Guido Imbens
To cite this article: Paul Goldsmith-Pinkham & Guido Imbens (2013) Rejoinder, Journal of
Business & Economic Statistics, 31:3, 279-281, DOI: 10.1080/07350015.2013.792260
To link to this article: http://dx.doi.org/10.1080/07350015.2013.792260
Published online: 22 Jul 2013.
Submit your article to this journal
Article views: 273
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
Goldsmith-Pinkham and Imbens: Rejoinder
Goldberger, A. (1991), A Course in Econometrics, Cambridge, MA: Harvard
University Press. [277]
Goldsmith-Pinkham, P., and Imbens, G. W. (2013), “Social Networks and the
Identification of Peer Effects,” Journal of Business and Economic Statistics,
31, 253–264. [276,278]
Graham, B. (2008), “Identifying Social Interactions Through Conditional Variance Restrictions,” Econometrica, 76, 643–660. [276]
Graham, B., and Hahn, J. (2005), “Identification and Estimation of the
Linear-in-Means Model of Social Interactions,” Economics Letters, 88,
1–6. [276]
Kline, B. (2012), “Identification of Complete Information Games,” Working
Paper. [277]
279
Kline, B., and Tamer, E. (2012), “Bounds for Best Response Functions in Binary
Games,” Journal of Econometrics, 166, 92–105. [277]
Lee, L. (2007), “Identification and Estimation of Econometric Models With
Group Interactions, Contextual Factors and Fixed Effects,” Journal of
Econometrics, 140, 333–374. [276]
Manski, C. (1993), “Identification of Endogenous Social Effects: The Reflection
Problem,” Review of Economic Studies, 60, 531–542. [276]
Tamer, E. (2003), “Incomplete Simultaneous Discrete Response Model
With Multiple Equilibria,” Review of Economic Studies, 70, 147–
165. [277]
Wooldridge, J. (2001), Econometric Analysis of Cross Section and Panel Data,
Cambridge, MA: The MIT Press. [277]
Downloaded by [Universitas Maritim Raja Ali Haji] at 22:11 11 January 2016
Rejoinder
Paul GOLDSMITH-PINKHAM
Department of Economics, Harvard University, Cambridge, MA 02138 ([email protected])
Guido IMBENS
Graduate School of Business, Stanford University, Stanford, CA 94305, and NBER ([email protected])
First of all, we thank the Coeditors again for inviting us
to present the article and for organizing such a distinguished
group of researchers to comment on our work. We would also
like to thank these individuals for their thoughtful discussion.
These comments contain a great number of interesting questions
and suggestions for future research, more than we can hope to
address in our response. This partly reflects the fertility of this
area of study, and we hope and suspect that the comments will
stimulate further research.
A general issue raised in three of the comments concerns the
focus on model parameters versus specific policy interventions.
Jackson stresses the importance of being very explicit about the
specific effects of interest, while Manski takes issue with our
claim that “the main object of interest is the effect of peers’
outcomes on own outcomes” and points out that identification
of the structural parameters is not the same as identification of
the treatment responses. Relatedly, Kline and Tamer raise the
issue of the interpretation of the parameters and their link to
policy interventions.
Here, we were less careful than we should have been. Ultimately, we agree wholeheartedly with these comments and
withdraw the claim to which Manski objects. Manski’s linear-inmeans example illustrates nicely the pitfalls of focusing solely
on identifying parameters rather than policies, and we believe
this is an important issue. Too often econometricians focus
solely on the particular parameters in their models without relating them to interpretable and feasible interventions. With
this objective in mind, Kline and Tamer focus on the effect of a
change in either one’s own or others’ covariate values on the outcome of interest. They demonstrate how these measured effects
are related in potentially complicated ways to the parameters of
the model.
In the area of industrial organization, it is often the norm
to report the effects of specific counterfactuals or policy interventions, while in other areas of empirical economic research,
it is much less common. In peer effects models, there are di-
rect policy interventions to consider, but there are also other
policies whose effects are complicated functions of the model
parameters. For example, in the context of a network formation
model, Christakis et al. (2010) considered the effect of changing features of the assignment to classes on the formation of
friendships. Given our setup, one could envision a policy affecting friendship formation that was enacted with students’
grades in mind. In sum, we agree with the recommendation
that researchers should routinely assess the effects of specific
and relevant interventions rather than simply report parameter
estimates.
Kline and Tamer also study the interpretation of the model
parameters themselves and show how they can be interpreted
as “best responses” in a game. This greatly improves the interpretability of these parameters, although ultimately we would
still stress the importance of focusing on the effect of interventions rather than the parameters themselves. In some cases, it
seems likely that the peer effects model is attempting to capture
the effects of a “social multiplier” through a specification that
reflects the researcher’s ignorance about the particular channels by which this interaction occurs. However, we agree with
both Jackson and Sacerdote’s view that a researcher should be
modeling the specific type of social interaction she hopes to
find. Interestingly, as Manski points out in his comment and
in Manski (2013), even researchers fortunate enough to have
randomized interventions will have to think hard about how to
model the social interaction.
Bramoullé raises important issues regarding the type of endogeneity we allow for in our model, discusses some identification questions, and offers some suggestions on estimation and
on modeling heterogeneity in peer effects. It is interesting as a
© 2013 American Statistical Association
Journal of Business & Economic Statistics
July 2013, Vol. 31, No. 3
DOI: 10.1080/07350015.2013.792260
Downloaded by [Universitas Maritim Raja Ali Haji] at 22:11 11 January 2016
280
general matter that the econometrics literature lumps together
various sources of endogeneity without a well-established terminology to distinguish between them. In our model, we introduce
into the network literature a form of “omitted-variable endogeneity,” as opposed to “simultaneous equations endogeneity.”
If we were to observe the ξi that are unobserved in our model,
the endogeneity we are concerned with would disappear.
A very different, and arguably more challenging, form of
endogeneity would arise if decisions concerning the formation
of links and outcomes were taken simultaneously. The clearest analog in the established econometrics literature would be
a nontriangular system of equations. What type of plausible information and restrictions would allow researchers to establish
the presence of that type of simultaneity is a very interesting
and, at first sight, very difficult question that should be pursued
in future research.
Bramoullé also offers some comments on our Bayesian approach. He suggests that advances in computational graph theory
may facilitate the calculation of maximum likelihood estimators.
This is an interesting suggestion, and there may well be computational advantages to such an approach. However, our Bayesian
approach was not solely motivated by computational reasons.
It was also motivated by the lack of large sample results for
maximum likelihood estimators in the current setting, whose
difficulties are mentioned by Manski (which we will discuss
shortly). If the suggestions regarding the computation of maximum likelihood estimators would be effective, we would most
certainly incorporate them in our computational algorithms but
maintain the focus on posterior distributions rather than maximum likelihood estimates.
Graham presents a very interesting set of new results, focusing on a novel problem where the unobserved components are
analyzed as fixed effects rather than random effects. He studies
a setting with observations on many small networks and derives
novel identification results for such settings. In particular, he
focuses on transitivity and state dependence in the dynamic
paths of the network. Using an example with observations
on many networks with three individuals, followed for three
periods, he shows how the presence of state dependence (where
having a link between two individuals in the current period
affects the changes of the link in the next period, similar to our
D0,ij ), and transitivity (where links between two individuals
are more likely if they had friends in common in the previous
period, an analog to our F0,ij ) can be established under weak
nonparametric conditions. This identification result, and the
generalization to the case with larger networks, is a very
interesting finding and shows the scope for identification results
that can genuinely improve our understanding of what can be
learned from observations on networks.
More generally, identification is a difficult issue in network
settings. In conventional cross-section settings, identification
questions are often formulated as the ability to infer the parameters of interest from the joint distribution of some variables (Y, X). The hope is that, with sufficiently large samples,
we can approximate this joint distribution accurately and if we
can infer the parameters of interest from this distribution, we
should be able to accurately estimate them as well. However,
in the network setting, it is not clear what distribution we can
Journal of Business & Economic Statistics, July 2013
estimate in large samples. Clarification is needed in what defines a large sample and how the current sample differs from a
larger sample. In Graham’s discussion, a larger sample is taken
to mean more small networks are sampled; however, in many
cases, this asymptotic approximation does not seem appropriate. Goldsmith-Pinkham and Imbens (2013) and Boucher and
Mourifie (2012) made some progress toward furthering this research agenda, but much more is needed. We should note that
related issues come up in the context of asymptotics in discrete
choice models with large numbers of choices and large numbers
of markets (Berry, Linton, and Pakes 2004; Athey and Imbens
2007).
Sacerdote raises concerns with the linearity of the model. This
is likely to be a very important issue in practice, as nonlinearity
of the outcome equation in both covariates and peer effects
can generate complex responses to policy interventions. It is
a challenge, however, to introduce such nonlinear effects in
a flexible and yet parsimonious manner. For example, simply
allowing the effects of the own covariates to be nonlinear would
be straightforward, but allowing the exogenous and endogenous
peer effects to be nonlinear may lead to concerns regarding
the identification of the models. These are clearly issues that
need to be investigated further. The type of experimental data
from the Air Force Academy that Sacerdote has studied (Carell,
Sacerdote, and West 2013) may be useful to this end.
Jackson outlines the broad set of econometric issues that
applied researchers studying peer effects face, including identification, the distinction between endogeneity and homophily in
unobserved characteristics, computational challenges, measurement error in links, and misspecification, specifically that of the
relevant set of peers. He also warns against the temptation to
focus on simple models as suitable for many different settings.
We agree with these concerns and the research agenda implictly
laid out.
Another issue raised by Jackson concerns the potential inadequacy of the two-mass-point distribution of the unobserved
component of the individual characteristics. We agree that the
assumption of two mass points is limiting, although in other
areas, such as the literature on duration models, approximations
based on simple discrete distributions have been found to be
fairly accurate in simulations. More research is needed here to
assess the restrictiveness of such distributional assumptions.
We see the area of peer effects, especially in settings where
the peer groups are not obviously exogenous, as one with great
challenges, and, as a result, as a very fertile one for new research.
We recognize as well that our article raises more questions than
it answers. As a general matter, it appears clear to us that there
will not be simple solutions to each of these problems, and
there will not be a simple model that can incorporate all these
concerns.
Nevertheless, there are many areas where much progress can
be made. We see the role of econometricians in this area primarily as one of developing models that address the complications
listed by Jackson. A key concern in addressing these problems
is maintaining tractability. Jackson, here and in other work, has
stressed the severe computational difficulties facing researchers
attempting to address the econometric issues. We feel that an
important contribution of our current article is the multiple
Downloaded by [Universitas Maritim Raja Ali Haji] at 22:11 11 January 2016
Goldsmith-Pinkham and Imbens: Rejoinder
networks approach with the peer effects from two or more networks as in Equation (7.1). Such multiple network models are
natural generalizations of single network models that maintain
tractability but allow researchers to investigate a range of issues
related to endogeneity of networks, measurement error in links,
and heterogeneity in peer effects. These models also begin to
connect the peer effects models with spatial dependence models more generally, allowing more flexible forms of dependence
than the simple models with equal effects from all peers.
Additionally, we feel that econometricians must push to incorporate better datasets of networks and peer interaction. Graham’s results are an excellent example of how econometricians
can help to guide applied researchers in their collection of new
network data connected with outcomes. Larger and more robust
datasets will allow researchers to answer the research questions
regarding nonlinearities and nonparametric identification raised
by the comments.
281
ADDITIONAL REFERENCES
Athey, S., and Imbens, G. (2007), “Discrete Choice Models With Multiple
Unobserved Characteristics,” International Economic Review, 48, 1159–
1192. [280]
Berry, S., Linton, O., and Pakes, A. (2004), “Limit Theorems for Estimating
the Parameters of Differentiated Product Demand Systems,” Review of Economic Studies, 71, 613–654. [280]
Boucher, V., and Mourifie, I. (2012), “My Friend Far Far Away: Asymptotic Properties of Pairwise Stable Networks,” Working Paper. Available
at http://ssrn.com/abstract=2170803. [280]
Carrell, S. E., Sacerdote, B. I., and West, J. E. (2013), “From Natural Variation
to Optimal Policy? The Importance of Endogenous Peer Group Formation,”
Econometrica, 81, 855–882. [280]
Christakis, N. A., Imbens, G. W., Fowler, J. H., and Kalyanaraman, K. (2010),
“An Empirical Model for Strategic Network Formation,” NBER Working
Paper no. w16039, Cambridge, MA. [279]
Goldsmith-Pinkham, and Imbens. (2013), “Large-Sample Asymptotics for Network Statistics,” Working Paper, Harvard University. [280]
Manski, C. F. (2013), “Identification of Treatment Response With Social Interactions,” The Econometrics Journal, 16, S1–S23. doi: 10.1111/j.1368423X.2012.00368.x. [279]
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Rejoinder
Paul Goldsmith-Pinkham & Guido Imbens
To cite this article: Paul Goldsmith-Pinkham & Guido Imbens (2013) Rejoinder, Journal of
Business & Economic Statistics, 31:3, 279-281, DOI: 10.1080/07350015.2013.792260
To link to this article: http://dx.doi.org/10.1080/07350015.2013.792260
Published online: 22 Jul 2013.
Submit your article to this journal
Article views: 273
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
Goldsmith-Pinkham and Imbens: Rejoinder
Goldberger, A. (1991), A Course in Econometrics, Cambridge, MA: Harvard
University Press. [277]
Goldsmith-Pinkham, P., and Imbens, G. W. (2013), “Social Networks and the
Identification of Peer Effects,” Journal of Business and Economic Statistics,
31, 253–264. [276,278]
Graham, B. (2008), “Identifying Social Interactions Through Conditional Variance Restrictions,” Econometrica, 76, 643–660. [276]
Graham, B., and Hahn, J. (2005), “Identification and Estimation of the
Linear-in-Means Model of Social Interactions,” Economics Letters, 88,
1–6. [276]
Kline, B. (2012), “Identification of Complete Information Games,” Working
Paper. [277]
279
Kline, B., and Tamer, E. (2012), “Bounds for Best Response Functions in Binary
Games,” Journal of Econometrics, 166, 92–105. [277]
Lee, L. (2007), “Identification and Estimation of Econometric Models With
Group Interactions, Contextual Factors and Fixed Effects,” Journal of
Econometrics, 140, 333–374. [276]
Manski, C. (1993), “Identification of Endogenous Social Effects: The Reflection
Problem,” Review of Economic Studies, 60, 531–542. [276]
Tamer, E. (2003), “Incomplete Simultaneous Discrete Response Model
With Multiple Equilibria,” Review of Economic Studies, 70, 147–
165. [277]
Wooldridge, J. (2001), Econometric Analysis of Cross Section and Panel Data,
Cambridge, MA: The MIT Press. [277]
Downloaded by [Universitas Maritim Raja Ali Haji] at 22:11 11 January 2016
Rejoinder
Paul GOLDSMITH-PINKHAM
Department of Economics, Harvard University, Cambridge, MA 02138 ([email protected])
Guido IMBENS
Graduate School of Business, Stanford University, Stanford, CA 94305, and NBER ([email protected])
First of all, we thank the Coeditors again for inviting us
to present the article and for organizing such a distinguished
group of researchers to comment on our work. We would also
like to thank these individuals for their thoughtful discussion.
These comments contain a great number of interesting questions
and suggestions for future research, more than we can hope to
address in our response. This partly reflects the fertility of this
area of study, and we hope and suspect that the comments will
stimulate further research.
A general issue raised in three of the comments concerns the
focus on model parameters versus specific policy interventions.
Jackson stresses the importance of being very explicit about the
specific effects of interest, while Manski takes issue with our
claim that “the main object of interest is the effect of peers’
outcomes on own outcomes” and points out that identification
of the structural parameters is not the same as identification of
the treatment responses. Relatedly, Kline and Tamer raise the
issue of the interpretation of the parameters and their link to
policy interventions.
Here, we were less careful than we should have been. Ultimately, we agree wholeheartedly with these comments and
withdraw the claim to which Manski objects. Manski’s linear-inmeans example illustrates nicely the pitfalls of focusing solely
on identifying parameters rather than policies, and we believe
this is an important issue. Too often econometricians focus
solely on the particular parameters in their models without relating them to interpretable and feasible interventions. With
this objective in mind, Kline and Tamer focus on the effect of a
change in either one’s own or others’ covariate values on the outcome of interest. They demonstrate how these measured effects
are related in potentially complicated ways to the parameters of
the model.
In the area of industrial organization, it is often the norm
to report the effects of specific counterfactuals or policy interventions, while in other areas of empirical economic research,
it is much less common. In peer effects models, there are di-
rect policy interventions to consider, but there are also other
policies whose effects are complicated functions of the model
parameters. For example, in the context of a network formation
model, Christakis et al. (2010) considered the effect of changing features of the assignment to classes on the formation of
friendships. Given our setup, one could envision a policy affecting friendship formation that was enacted with students’
grades in mind. In sum, we agree with the recommendation
that researchers should routinely assess the effects of specific
and relevant interventions rather than simply report parameter
estimates.
Kline and Tamer also study the interpretation of the model
parameters themselves and show how they can be interpreted
as “best responses” in a game. This greatly improves the interpretability of these parameters, although ultimately we would
still stress the importance of focusing on the effect of interventions rather than the parameters themselves. In some cases, it
seems likely that the peer effects model is attempting to capture
the effects of a “social multiplier” through a specification that
reflects the researcher’s ignorance about the particular channels by which this interaction occurs. However, we agree with
both Jackson and Sacerdote’s view that a researcher should be
modeling the specific type of social interaction she hopes to
find. Interestingly, as Manski points out in his comment and
in Manski (2013), even researchers fortunate enough to have
randomized interventions will have to think hard about how to
model the social interaction.
Bramoullé raises important issues regarding the type of endogeneity we allow for in our model, discusses some identification questions, and offers some suggestions on estimation and
on modeling heterogeneity in peer effects. It is interesting as a
© 2013 American Statistical Association
Journal of Business & Economic Statistics
July 2013, Vol. 31, No. 3
DOI: 10.1080/07350015.2013.792260
Downloaded by [Universitas Maritim Raja Ali Haji] at 22:11 11 January 2016
280
general matter that the econometrics literature lumps together
various sources of endogeneity without a well-established terminology to distinguish between them. In our model, we introduce
into the network literature a form of “omitted-variable endogeneity,” as opposed to “simultaneous equations endogeneity.”
If we were to observe the ξi that are unobserved in our model,
the endogeneity we are concerned with would disappear.
A very different, and arguably more challenging, form of
endogeneity would arise if decisions concerning the formation
of links and outcomes were taken simultaneously. The clearest analog in the established econometrics literature would be
a nontriangular system of equations. What type of plausible information and restrictions would allow researchers to establish
the presence of that type of simultaneity is a very interesting
and, at first sight, very difficult question that should be pursued
in future research.
Bramoullé also offers some comments on our Bayesian approach. He suggests that advances in computational graph theory
may facilitate the calculation of maximum likelihood estimators.
This is an interesting suggestion, and there may well be computational advantages to such an approach. However, our Bayesian
approach was not solely motivated by computational reasons.
It was also motivated by the lack of large sample results for
maximum likelihood estimators in the current setting, whose
difficulties are mentioned by Manski (which we will discuss
shortly). If the suggestions regarding the computation of maximum likelihood estimators would be effective, we would most
certainly incorporate them in our computational algorithms but
maintain the focus on posterior distributions rather than maximum likelihood estimates.
Graham presents a very interesting set of new results, focusing on a novel problem where the unobserved components are
analyzed as fixed effects rather than random effects. He studies
a setting with observations on many small networks and derives
novel identification results for such settings. In particular, he
focuses on transitivity and state dependence in the dynamic
paths of the network. Using an example with observations
on many networks with three individuals, followed for three
periods, he shows how the presence of state dependence (where
having a link between two individuals in the current period
affects the changes of the link in the next period, similar to our
D0,ij ), and transitivity (where links between two individuals
are more likely if they had friends in common in the previous
period, an analog to our F0,ij ) can be established under weak
nonparametric conditions. This identification result, and the
generalization to the case with larger networks, is a very
interesting finding and shows the scope for identification results
that can genuinely improve our understanding of what can be
learned from observations on networks.
More generally, identification is a difficult issue in network
settings. In conventional cross-section settings, identification
questions are often formulated as the ability to infer the parameters of interest from the joint distribution of some variables (Y, X). The hope is that, with sufficiently large samples,
we can approximate this joint distribution accurately and if we
can infer the parameters of interest from this distribution, we
should be able to accurately estimate them as well. However,
in the network setting, it is not clear what distribution we can
Journal of Business & Economic Statistics, July 2013
estimate in large samples. Clarification is needed in what defines a large sample and how the current sample differs from a
larger sample. In Graham’s discussion, a larger sample is taken
to mean more small networks are sampled; however, in many
cases, this asymptotic approximation does not seem appropriate. Goldsmith-Pinkham and Imbens (2013) and Boucher and
Mourifie (2012) made some progress toward furthering this research agenda, but much more is needed. We should note that
related issues come up in the context of asymptotics in discrete
choice models with large numbers of choices and large numbers
of markets (Berry, Linton, and Pakes 2004; Athey and Imbens
2007).
Sacerdote raises concerns with the linearity of the model. This
is likely to be a very important issue in practice, as nonlinearity
of the outcome equation in both covariates and peer effects
can generate complex responses to policy interventions. It is
a challenge, however, to introduce such nonlinear effects in
a flexible and yet parsimonious manner. For example, simply
allowing the effects of the own covariates to be nonlinear would
be straightforward, but allowing the exogenous and endogenous
peer effects to be nonlinear may lead to concerns regarding
the identification of the models. These are clearly issues that
need to be investigated further. The type of experimental data
from the Air Force Academy that Sacerdote has studied (Carell,
Sacerdote, and West 2013) may be useful to this end.
Jackson outlines the broad set of econometric issues that
applied researchers studying peer effects face, including identification, the distinction between endogeneity and homophily in
unobserved characteristics, computational challenges, measurement error in links, and misspecification, specifically that of the
relevant set of peers. He also warns against the temptation to
focus on simple models as suitable for many different settings.
We agree with these concerns and the research agenda implictly
laid out.
Another issue raised by Jackson concerns the potential inadequacy of the two-mass-point distribution of the unobserved
component of the individual characteristics. We agree that the
assumption of two mass points is limiting, although in other
areas, such as the literature on duration models, approximations
based on simple discrete distributions have been found to be
fairly accurate in simulations. More research is needed here to
assess the restrictiveness of such distributional assumptions.
We see the area of peer effects, especially in settings where
the peer groups are not obviously exogenous, as one with great
challenges, and, as a result, as a very fertile one for new research.
We recognize as well that our article raises more questions than
it answers. As a general matter, it appears clear to us that there
will not be simple solutions to each of these problems, and
there will not be a simple model that can incorporate all these
concerns.
Nevertheless, there are many areas where much progress can
be made. We see the role of econometricians in this area primarily as one of developing models that address the complications
listed by Jackson. A key concern in addressing these problems
is maintaining tractability. Jackson, here and in other work, has
stressed the severe computational difficulties facing researchers
attempting to address the econometric issues. We feel that an
important contribution of our current article is the multiple
Downloaded by [Universitas Maritim Raja Ali Haji] at 22:11 11 January 2016
Goldsmith-Pinkham and Imbens: Rejoinder
networks approach with the peer effects from two or more networks as in Equation (7.1). Such multiple network models are
natural generalizations of single network models that maintain
tractability but allow researchers to investigate a range of issues
related to endogeneity of networks, measurement error in links,
and heterogeneity in peer effects. These models also begin to
connect the peer effects models with spatial dependence models more generally, allowing more flexible forms of dependence
than the simple models with equal effects from all peers.
Additionally, we feel that econometricians must push to incorporate better datasets of networks and peer interaction. Graham’s results are an excellent example of how econometricians
can help to guide applied researchers in their collection of new
network data connected with outcomes. Larger and more robust
datasets will allow researchers to answer the research questions
regarding nonlinearities and nonparametric identification raised
by the comments.
281
ADDITIONAL REFERENCES
Athey, S., and Imbens, G. (2007), “Discrete Choice Models With Multiple
Unobserved Characteristics,” International Economic Review, 48, 1159–
1192. [280]
Berry, S., Linton, O., and Pakes, A. (2004), “Limit Theorems for Estimating
the Parameters of Differentiated Product Demand Systems,” Review of Economic Studies, 71, 613–654. [280]
Boucher, V., and Mourifie, I. (2012), “My Friend Far Far Away: Asymptotic Properties of Pairwise Stable Networks,” Working Paper. Available
at http://ssrn.com/abstract=2170803. [280]
Carrell, S. E., Sacerdote, B. I., and West, J. E. (2013), “From Natural Variation
to Optimal Policy? The Importance of Endogenous Peer Group Formation,”
Econometrica, 81, 855–882. [280]
Christakis, N. A., Imbens, G. W., Fowler, J. H., and Kalyanaraman, K. (2010),
“An Empirical Model for Strategic Network Formation,” NBER Working
Paper no. w16039, Cambridge, MA. [279]
Goldsmith-Pinkham, and Imbens. (2013), “Large-Sample Asymptotics for Network Statistics,” Working Paper, Harvard University. [280]
Manski, C. F. (2013), “Identification of Treatment Response With Social Interactions,” The Econometrics Journal, 16, S1–S23. doi: 10.1111/j.1368423X.2012.00368.x. [279]