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Abstract
In this article, we propose a two-step method to identify and estimate endogenous and exogenous social interactions in the Manski (Citation1993) and Brock and Durlauf's (Citation2001a,b) discrete choice model with unobserved group variables. Taking advantage of social groups with large group sizes, we first estimate a probit model with group fixed-effects, and then use the instrumental variables method to estimate endogenous and exogenous group effects via the group fixed-effect estimates. Our method is computationally simple. The method is applicable not only to the case of single equilibrium but also the multiple equilibria case without the need to specify an (arbitrary) equilibrium selection mechanism. The article provides a Monte Carlo study on the finite sample performance of such estimators.
ACKNOWLEDGMENT
We are grateful for discussion with Bruce A. Weinberg. We thank three referees, who have provided helpful suggestions. We take responsibility for all errors. The second author is grateful to the NSF for financial support for his research under the grant number SES-0519204.
Notes
1Brock and Durlauf (2007) analyze how identification is affected by the presence of UG effects. The UG variable might be correlated with various observables in the model. In that situation, parameters would not be identifiable. Brock and Durlauf discuss additional restrictions for identification or partial identification. In our model, we assume the UG variable is independent with other explanatory variables in the model so that the model can be identified with valid instrumental variables (IVs) implied by the structure of the model for the EG variable. With distributional assumptions, the ML method can be applied.
2Su and Judd (Citation2008) have recently presented a constrained optimization approach to estimate structural models with the constraint that the endogenous variable is consistent with an equilibrium for the model parameters. The approach avoids repetitive solutions of the model and can be faster than the nested fixed-point approach.
3Brock and Durlauf (2001a,b) show multiple solutions of the EG variable may exist when the coefficient of the EG variable exceeds a threshold value which depends on the distribution of the MBD model. Their analysis is based on the MBD model without a UG variable. When allowing a UG variable in the model, the UG variable may influence the magnitude of the EG variable in a way similar to observed group variables. However, it does not change the threshold value of the coefficient of the EG variable.
4In a game theoretical model on social interactions, where the observed choices of the peers instead of the expected choices appear directly as an explanatory variable, there always exist multiple equilibria. Krauth (Citation2006) and Soetevent and Kooreman (Citation2007) assume a random selection rule.
5Using this strategy, Shang (Citation2008) studies how a woman's welfare participation is affected by the participation of other women in the Public Use Microdata Area (PUMA) where she lives. The PUMA is a geographic area on Census samples with a minimum census population of 100,000. The data used in the study is a 5% sample of 1990 Census data, which is an application of large social groups we discuss here.
6However, it is implicit that decision makers would have a rule to select an equilibrium for their actual decisions.
7If group members use different equilibrium selection rules, the expectations would depend on individuals and become heterogeneous. This would contradict the homogenous specification of the endogenous group variable.
8Groups considered here are predetermined, for example, the geographical areas as in many studies on neighborhood effects and schools or classes as in the studies on peer effects. A subset of group members are randomly selected into the sample.
9Estimation without the assumption of a parametric distribution of the individual disturbance term is discussed in footnote 11. For a parametric distributional specification, a logistic distribution which gives a logit model is a close substitute for the normal distribution assumed here. As the logit and probit models are known to give similar results, we pick the probit model without any particular reason. However, for more complicated models with dynamics or in a multivariate setting, probit models may allow more flexible covariance structures.
10Shang (2008) is an example. See footnote 5 for details.
11It is possible to estimate the model semiparametrically under some circumstances. Consider Eq. (1)′. Because group sizes are large, each group can be regarded as cross-section data. Without loss of generality, consider the estimation of Eq. (1)′ for group 1, i.e., . The distribution of ϵ's is assumed to be independent of x's, continuous, and strictly increasing. According to the identification via the single index formulation in Ichimura (Citation1993), the δ can be identified after a proper normalization. The α1, which represents a location, is not identifiable in a distributional free choice model. Let u
1i
= −(α1 + ϵ1i
) and F
u
be its distribution. Because E(y
1i
|x
1i
), as a nonparametric regression function, is identifiable and δ is identified, the distribution of F
u
will be identifiable on the support of xδ via E(y
1i
|x) = F
u
(xδ). Under the assumption that the support of xδ is unbounded, the distribution F
u
is identifiable everywhere. With F
u
being identified,
, where
can be identified for the gth group. This is so because
. The identification of all α
g
's up to a location is sufficient for the second step estimation of the social interaction effects of interest.
12Effective algorithms for handling a possibly large number of incidental parameters in a nonlinear fixed effect model are described in Hall (Citation1978) and Greene (Citation2004). According to Greene (Citation2004), it is feasible to compute the full parameter vector of a nonlinear fixed effects model with a very large number of groups. Even when the number of groups is as large as tens of thousands, computation of a fixed effect probit or logit model can be obtained with modest desktop computers.
13The above derivation by the Taylor expansion is based on a specified distribution Φ. The idea might still be useful without any distributional assumption, but will involve approximations (or nonparametric estimates) of the unknown distribution and its density. An unknown distribution can in principle be approximated by a sequence of specific functions, and so is the corresponding density by the corresponding sequence of derivative functions by the theory of a Hilbert space. A possible sequence might be the Hermite polynomials where the leading term is the normal distribution. The approximation of an unknown distribution by the Hermite polynomials has been considered in Lee (Citation1982), Newey et al. (Citation1990), and Gallant and Nychka (Citation1987) for related estimation problems.
14Amemiya (1985) provides some guidelines for the best IV. For the estimation of a nonlinear simultaneous equation y
i
= f(Y
i
, z
i
, θ) + u
i
, where z
i
is a vector of exogenous variables which is independent of u
i
, Y
i
is a vector of endogenous explanatory variables, and θ is a vector of unknown parameters. Amemiya (Citation1985) shows that the best IV is the conditional expectation of the derivative of f with respect to θ, i.e., (p. 248). This conditional expectation is unknown. Amemiya suggests that a practical procedure is to predict the conditional expectation by regression with a set of independent variables. However, the optimum set of independent variables is left unknown in a nonlinear system. Newey (Citation1990) has suggested the construction of the best IV by a nonparametric kernel regression function. In our case, Eq. (Equation4) is linear in parameters. The best IV would be the conditional expectation
because
is the vector of all exogenous variables in the model. The difficulties of our best IV are due to the unobserved nature of E
g
(y), even though it can be approximated by
. In practice, it is possible that the model shall include many exogenous variables, which characterize the group. Many exogenous variables create a curse of dimensionality problem for a nonparametric regression. Our suggestions are to approximate this best IV with the empirical functions of exogenous variables in a tractable fashion, as we have recommended here.
15Because of finite population for each group, the distribution is assumed to be a counting measure.
Note:
1) Model 1: α g = β1 + E g (y)β3 + u g .
2) Approximation of .
3) IV: , and w
1, gm
(IV1.1).
4) R 2 is for the regression: Y a on 1 and IVs.
5) G = number of groups: m = group size.
16The results are based on m = 200 and G = 200. The effects of the group size m and the number of groups G on R 2 are small. For comparison, the mean (resp., SD) of the R 2 at m = 25 and G = 50 is 0.766 (0.058), and the mean (resp., SD) at m = 200 and G = 50 is 0.824 (0.045). The R 2's reported in other tables are also based on m = 200 and G = 200.
17In 5.0% of the replications, the overidentification test (equivalently, the J test) is rejected at the level of significance α = 0.05. The test results are based on m = 200 and G = 200, which is also the case for other overidentification tests reported in the article. Group size and number of the groups have little effect on the test.
Note:
1) IV1.2: ; IV1.3: w
1,gm
.
2) Approximation of E
g
(y) used in Panel 1:
3) R 2 is for the regression: Y a on 1 and IVs.
4)
5) IV used in Panel 2: (IV1.1).
18As in Panel 3 of Table , the Y a is used as an approximation of E g (y). Using the alternative approximation Y e or the true EG variable Y t , the results are very similar and are not reported.
19The vector of IVs used in the estimation is IV1.1. The results are very similar when other IVs are used instead, and are not reported in the article.
Note:
1) Model 2: α g = β1 + E g (x 1)β21 + E g (y)β3 + u g .
2) Approximation of E
g
(y) Y
a
=
3) IV: (IV2.1).
4) G = number of groups; m = group size.
Note:
1) IV2.1: ; IV2.2:
IV2.3:
; IV2.4:
.
2) R
2-1 is for the regression: Y
a
on 1, , and IVs; R
2-2 is for the regression:
.
20In 4.6% of the replications using all IVs as in IV2.1, overidentification tests are rejected at the 5% level of significance.
Note:
1) Model 3: α g = β + E g (x 1)β21 + E g (x 2)β22 + E g (y)β3 + u g .
2) Approximation of E
g
(y): .
3) IV3.1: w 1,gm , w 2, gm , and w 3, gm .
4) R
2-1 is for the regression: Y
a
on 1, , and IV3.1; R
2-2 is for the regression:
on 1 and
.
5) G = number of groups; m = group size.
21In spite of the difficulties in the IV estimations shown here, controlling for the UG effect is necessary in terms of reducing the bias in the estimators. Without controlling for the UG effect, the OLS estimators are severely biased. For example, the means (resp. SDs) of the OLS estimators at m = 200 and G = 200 are: 6.430 (0.279) for β3, −1.157 (0.109) for β21, −1.368 (0.124) for β22, and -3.170 (0.154) for β1. When the generated unobserved group variable u
g
is regressed on a constant, ,
and Y
a
, the means of the estimated coefficients are: 5.457 for Y
a
, −1.536 for
, −1.730 for
, and, −2.921 for the constant term. Therefore, the biases of the OLS estimates due to the lack of controlling for the UG variable are huge.
22In 8% of the replications at m = 200 and G = 200, the overidentification test is rejected at the 5% level of significance.
Note:
1) IV3.2: w 1,gm.
23Brock and Durlauf (2001a,b) show in a EG effect model, under certain conditions, with the parameter β3 of the EG variable larger than 1, there may be three equilibria in the model. In their model, the two choices are represented by y
gi
= −1 or y
gi
= 1, and the individual error term follows a logistic distribution. It corresponds to β3 > 2.5 when the error term is assumed to follow the standard normal distribution with two choices represented by y
gi
= 0 or y
gi
= 1. Let H(p) = Φ(α + β3
p), where p is the expected average choice of the group. When , H(p) is a contraction mapping, and, hence, a unique fixed point p* exists such that
. This can be shown by noting that by the mean value theorem
. For the probit model,
, where
because the mode of the standard normal density is 0. Therefore,
if
. We note that
.
24We have experimented with models in which β3 takes values larger than 2.5 but smaller than 3.5. For those models, either there are no multiple equilibria or only a few groups have multiple equilibria.
25This is based on 100,000 simulated groups with m = 200 per group.
26For comparison, in Models 1–3, the distribution of E g (y) is bell shaped around 0.5. It appears more like a normal distribution when fewer XG variables are included in the model.
27Each sample is mixed with groups with multiple equilibria (17% on average) and groups with a single equilibrium (83% on average). For samples under the low equilibrium criterion, the median of E g (y) is 0.059 and the 90th percentile is 0.962. For samples under the high equilibrium criterion, the median is 0.154 and the 90th percentile is 0.965. For samples under the mixed equilibrium criterion, the median is 0.066 and the 90th percentile is 0.965.
28There are similar situations in a few replications for Models 1–3, but they are ignorable. The maximum number of groups dropped there is 1.
Note:
1) Latent model: .
2) Low: low level stable equilibrium is selected; High: high level stable equilibrium is selected; Mixed: low or high level stable equilibrium is randomly selected with equal probability.
3) Approximation of .
4) IV4.1: and w
1,gm
; IV4.2:
; and IV4.3: w
1,gm
.
5) R 2 is for the regression: Y a on 1 and IV's.
6) G = number of groups; m = group size 200.
29Krauth (2006) and Soetevent and Kooreman (Citation2007) have estimated their game theoretical social interaction models using an arbitrary random selection rule. A MC study in Krauth (Citation2006) investigates the possible consequences when the assumed selection rule differs from the one which generates the data. The results in Krauth (Citation2006) show the inconsistency of the estimates, but the biases seem small or moderate in his model.
30Using IV4.1 at m = 200 and G = 200, the percentage of replications in which the overidentification test is rejected at the 5% level of significance is 10.2.
Note:
1) , and Y
t
= E
g
(y).
2) IV: and w
1, gm
(IV4.1).
3) G = number of groups; m = group size 200.
31IV4.1 is used as instrument.
Note:
1) Model 2: α g = β1 + E g (x 1)β21 + E g (y)β3 + u g .
2) Approximation of E
g
(y): Y
a
= .
3) IV: (IV2.1).
4) G = number of groups; m = group size.
Note:
1) Latent model: .
2) Low: low level stable equilibrium is selected; High: high level stable equilibrium is selected; Mixed: low or high level stable equilibrium is randomly selected with equal probability.
3) Approximation of E
g
(y): Y
a
= .
4) IV: (IV 4.1).
5) G = number of groups; m = group size 200.
32We have also studied Model 1 under random sampling. The conclusions of the results are very similar to what we have described here of Model 2 and, therefore, not reported.
33Hahn and Newey (2004) have focused on nonlinear panel models with n cross-sectional units and T time periods observations for each unit. The model considered in this article is not a panel data model, but it structurally fits well into a ‘panel’ setting by regarding each group as a cross-sectional unit and observations of members within a group as repeated time observations (e.g., Borjas and Sueyoshi Citation1994).