In this section, I will briefly restate the semiparametric model of social interactions, identi- fication results and estimation procedure.
2.2.1 The Semiparametric Social Interaction Model
This subsection reviews the semiparametric social interaction models proposed in Chapter 1. Assume that the outcome variable of individual i in group g, Yg,i (g = 1, ..., G, i = 1, ..., ng),
is determined according to the following semiparametric model of social interactions:
Yg,i = β0Yg,−i+ h0(Xg,i, Wg,i, Wg,−i, Υg) + Ug,i; (g = 1, ..., G; i = 1, ..., ng), (2.2.1)
where Yg,iis the outcome variable of interest for individual i in group g. Yg,−i = ng1−1
P
j∈Ig,j6=i Yj denotes the leave-i-out average outcome within the gth group. Xg,i denotes the d dimen-
sional (dim(Xg,i) = d) individual-specific characteristics that only affect the outcome Yg,i
through individual level, which means the leaving-i-out group average Xg,−i does not af-
fect the outcome directly. Wg,i denotes the dW dimensional (dim(Wg,i) = dW) individual
characteristics which also induce the contextual effect. That is, the leaving-i-out group av- erage for Wg,i, Wg,−i, is allowed to affect the outcome Yg,i. Υg denotes the dΥ dimensional
(dim(Υg) = dΥ) observed group characteristics. The last term in model (2.2.1), Ug,i, is
the disturbance term that is unobserved to econometricians. This paper will work with the exogeneity condition, which requires Ug,i to be independent of all the controls.
Let ϑ0 = (β0, h0(·))0 be the true parameter vector. The parameter of interest β0 is the
1-dimensional parametric part which captures the endogenous social interaction effect. To have a stable equilibrium social interaction model, it is reasonable to require that |β0| < 1.
The unknown function h0(·) is the nonparametric part of the model which captures the
individual and contextual effects. This nonparametric function is also a nuisance parameter in this paper.
The goal of this paper is to obtain a estimator for β0, in the presence of the possibly highly
complex nuisance function h0(·) and also the endogenous effect of group social interactions.
To handle severe nonlinearity functions, higher-order interactions and more covariates in the nonparametric functions h0(·), Machine Learning methods are applied to concentrated out
h0(·), such as LASSO, Random Forest, and Neural Nets. And then the parameter of inter-
est β0 can be estimated using the MM/GMM approach with properly choose instrumental
variables for the endogenous social interaction effect.
2.2.2 Identification and Moment Conditions
Here I restate the identification results for the semiparametric social interaction model (2.2.1).
Let Xg,i = (Xg,i, Wg,i, Wg,−i, Υg, ng) denote the collection of control variables at both the
individual and group levels. The identification strategy is to partial out the nonparametric nuisance function h0(·) by subtracting the conditional expectations on both side of model
(2.2.1) in the first step,
simultaneous equations on the residulized model to identify β0. Throughout this paper, I
consider to use Xg,−i as the IV for the endogenous variable Yg,−i. Then, the parameter of
interest, β0, is identified by,
EXg,−i Yg,i − E[Yg,i|Xg,i] = β0EXg,−i Yg,−i− E[Yg,−i|Xg,i] . (2.2.3)
For notation simplicity, in the subsequent sections, I just use the notation Xg,i instead of
Xg,i. The results for the semiparametric estimation for β0 will not be affected.
Let µ0(Xg,i) , E[Yg,i|Xg,i]; ν0(Xg,i) , E[Yg,−i|Xg,i]; φ0(Xg,i) , E[Xg,−i|Xg,i] denote
the conditional expectations of Yg,i, Yg,−i, Xg,−i on Xg,i, respectively. The identification
condition for β0 can be restated using the following moment condition:
E h
Xg,−i
Yg,i− µ0(Xg,i) − β0 Yg,−i− ν0(Xg,i)
i
= 0, (2.2.4)
The parameter of interest, β0, then can be estimated using (2.2.4) by plugging in the first
step nonparametric estimators of µ0 and ν0. However, the moment function of (2.2.4),
m(Zg,i; β, µ, ν) = Xg,−i
Yg,i− µ(Xg,i) − βXg,−i Yg,−i− ν(Xg,i)
, (2.2.5)
is not orthogonal to the first step nonparametric parameters, which might lead to severe bias for the semiparametric estimation of β0 in the second step (Newey, 1994; Chernozhukov
et al., 2018b).
The robust strategy for estimating β0 is to use orthogonal moment condition instead.
Following the strategy in Newey (1994); Chernozhukov et al. (2018a), the orthogonal moment condition can be constructed by adding an adjustment term,
EhXg,−i− φ0(Xg,i)
(Yg,i − µ0(Xg,i)) − β0 Yg,−i− ν0(Xg,i)
i
= 0. (2.2.6)
orthogonal moment function in (2.2.6),
ψ(Zg,i, β, µ, ν, φ) =
Xg,−i− φ(Xg,i)
Yg,i− µ(Xg,i) − β Yg,−i− ν(Xg,i)
. (2.2.7)
It can be verified that the moment function ψ(·) is not only locally robust but also doubly robust to the first step nonparametric estimators.
In the following discussion, I will call the estimator for β0 based on moment function
(2.2.5) as the plug-in (PI) estimator, and the estimator based on the orthogonal moment function (2.2.7) as the debiasing (DB) estimator. For a detailed discussion about the identifi- cation results and moment conditions of the semiparametric social interaction model, please refer to Section 1.3 and 1.4 in Chapter 1.
2.2.3 Estimation Procedure
The goal of this paper is to obtain the estimator for β0, in the presence of the possibly
highly complex nuisance function h0(.) and also the endogenous effect of group social inter-
actions. Based on the (orthogonal) moment conditions, I propose a semiparametric two-step estimation procedure for the endogenous social interaction effect β0 for model (2.2.1).
The first step regression require the estimation of conditional expectations,
µ0(Xg,i) = E[Yg,i|Xg,i], ν0(Xg,i) = E[Yg,−i|Xg,i] and φ0(Xg,i) = E[Xg,−i|Xg,i],
which are all nonparametric functions. This paper considers to apply the Machine Learning methods, such as LASSO, Random Forest and Neural Nets, in the first step estimation. The Machine Learning methods are widely used for estimating the conditional expectations and are able to handle highly complex function forms. Section 2.3 will give a brief review of the widely used Machine Learning methods for the estimation of conditional expectations and also discuss the application of these methods for our social interaction model.
moment condition (2.2.4) or orthogonal moment condition (2.2.6). Due to the regulariza- tion bias of the first step Machine Learning methods, the plug-in estimator for β0 based
on moment condition (2.2.4) could be severely biased and cannot obtain the root-n consis- tency. Following Chernozhukov et al. (2018a), this paper make use the idea of orthogonal moment condition (2.2.6) to remove the regularization bias. Section 2.4 will discuss the semiparametric MM/GMM estimation for β0 in detail.