Identification and Estimation of a Semiparametric Logit Model using Network Data

The way we interact and communicate with people in our social groups (e.g.; friends, family, colleagues) has a substantial impact on how we behave. We often adapt our behaviors based on social cues and feedback. Factors such as personal opinions, consumer preferences, decisions to invest, and even the inclination towards illicit economic activities are considerably influenced by social networks comprised of friends and acquaintances. For example, the choice of college major or the decision to switch major might be influenced by the attitudes and expectations of that student’s friends, neighborhood, and family (Jackson et al.,, 2017; Sacerdote,, 2001; Pu et al.,, 2021; Feld and Zölitz,, 2022). These social influences can play a significant role in an individual’s choice or decision. This means that it is important to consider social influences in economic models to accurately capture and understand the effects of various factors and interventions on outcomes and if not accounted for, social influences might distort or confound the effect of another variable being studied. For instance in the previous example, if a researcher is studying the effect of a new orientation program on students’ choice of college major without considering social influences, the results might be misleading. The true effect of the orientation program might be masked or exaggerated by the unconsidered influence of peer attitudes (AbdulRaheem et al.,, 2017; Rusli et al.,, 2021).

In practice, these social influences are not observed by the researcher, making it hard to include them as a covariate in the model. For example, in the above example, even if the researcher controls for observable individual characteristics such as gender, age, race, and parents’ education, it is likely to omit factors that influence both students’ choice of friends and their choice of college major (e.g.; family expectations, effort, motivation, psychological disorders, or unreported substance use.). These unobserved factors affect both student’s choice of friends and the choice of a specific field. In the literature, a common approach to addressing this problem is to collect network data under the assumption that the latent social influence is revealed by linking behavior in the network (Blume et al.,, 2011; Graham,, 2015; Auerbach,, 2022). That is, in practice, when a researcher observes a link between two agents, they often have similar social characteristics. Therefore, network data can potentially be used to account for unobserved social influences.

In the setting of a binary response model where a correlation arises between regressors and errors due to an omitted vector of unobserved social characteristics, this paper seeks to address two primary research questions. First, how can binary response models with endogenous networks be effectively identified? And second, how can the observed data on network links be utilized to mitigate the effects of such endogeneity? The endogeneity in this framework is due to unobservable individual characteristics that influence both link formation in the network and the binary outcome of interest. To the best of my knowledge, this is the first paper that addresses these questions specifically for binary models. Given the endogenous peer group formation, we show that we can identify the peer effects by controlling the unobserved individual heterogeneity of the network formation model. To this end, this paper specifies a joint semiparametric binary regression and nonparametric model of network formation where the unobserved social characteristics determine both the social influence in the regression and links in the network. The binary regression model is assumed to have an unknown function of the unobserved index of social characteristics as a covariate, and the probability that two agents link is an unknown function of their unobserved social characteristics.

The remainder of the paper is organized as follows. In the next section, we present the literature review. In Section 3, we formally present the model including the identification and estimation of the parameters of interest.In Section 4, we present the results of Monte Carlo simulations. The remaining work to complete this paper is presented in section LABEL:remaining.

Many decisions made by agents are influenced by the behaviors and characteristics of other agents. Valuable insights about risky behaviors or emerging technologies are often acquired from friends, as evidenced by the research by Manski, (2004), Christakis and Fowler, (2008) and Banerjee et al., (2013). There is a vast literature of analyzing the social peer effect. Betts and Morell, (1999) find that the characteristics of the high school peer group affect the average of the undergraduate grade point. Case and Katz, (1991) find large peer effects on youth criminal behavior and drug use. A rich literature on neighborhood effects including Jencks et al., (1990), Rosenbaum, (1991), and Katz et al., (2001) shows that neighborhood peers can have profound effects on both adults and children.

Despite the extensive research available on the subject, the identification of social interactions remains problematic because of two well-known issues: endogeneity, due either to peers’ self-selection or to common group effects, and reflection, a particular case of simultaneity (Manski,, 1993; Moffitt et al.,, 2001; Soetevent,, 2006). Some authors explore models with endogenous networks(Goldsmith-Pinkham and Imbens,, 2013; Hsieh and Lee,, 2016; Arduini et al.,, 2015). Yet, their models introduce certain parameter limitations on the network formation model to determine and evaluate the relevant parameters. Consequently, the reliability of their estimators is often tied to the validity of these assumptions, which might not adequately reflect the diverse connecting patterns seen in numerous real-world networks.

Auerbach, (2022) and Johnsson and Moon, (2021) provide identification conditions that do not require parametric restrictions on the network model. Auerbach, (2022) shows that one can control network endogeneity by pairwise differencing the observations of the two agents whose network formation distributions are the same, and proposes a semiparametric estimator based on matching pairs of agents with similar columns of the squared adjacency matrix. This approach has provided direct inspiration for this research.

In spite of the abundant studies on the topic, these authors have not addressed the identification and estimation of semiparametric binary models using network data. This paper aims to bridge this gap in the literature.

Consider a finite set of agents $I=\{1,2,\cdots,n\}$, where each agent is identified by an observed vector of explanatory variables $X_{i}\in\mathbb{R}^{k}(k>0)$, with a binary outcome $y_{i}\in\{0,1\}$, and an unobserved index of social characteristics $\omega_{i}\in[0,1]$. These variables are related by the following model.

where $\varepsilon_{i}$ is independent and identically distributed (i.i.d.) idiosyncratic error with a cumulative distribution function (cdf) $F$, $\beta\in\mathbb{R}^{k}$ is an unknown slope parameter. The social influence function $\lambda:[0,1]\to\mathbb{R}$ is an unknown measurable function with $\lambda(\omega_{i})$ being the realized social influence for agent $i$. The social influence term $\lambda(\omega_{i})$ is the direct effect of interacting with a particular collection of communities.

In addition to this, the researcher also observes a binary adjacency matrix $n\times n$, $D$. The element $D_{ij}=1$ if there is a direct link between agents $i$ and $j$, and $D_{ij}=0$ otherwise. By convention, any agent $i$ is not allowed to be linked to itself $D_{ii}=0$. We assume that all links are undirected, so that the adjacency matrix $D$ is symmetric, that is, $D_{ij}=D_{ji}$. The existence of a link between agents $i$ and $j$ is determined by

where $f$ is a symmetric measurable function and $\left\{\eta_{ij}\right\}^{n}_{i,j=1}$ is a symmetric matrix of unobserved scalar disturbances with independent upper diagonal entries that are mutually independent of $\{x_{i},\omega_{i},\varepsilon_{i}\}_{i=1}^{n}$.

The unobserved individual index of social characteristic $\omega_{i}$ can be interpreted as the social capital that increases the likelihood of forming a link (e.g.; socioeconomic status, social ability, family expectation, trust). $f(\omega_{i},\omega_{j})-\eta_{ij}$ is interpreted as the utility agents $i$ and $j$ receive from forming a link. This implies that $D_{ij}$ and $D_{st}$ are independent conditionally on $\omega_{i},\omega_{j},\omega_{s}$ and $\omega_{t}$ i.e. the utility agents receive from forming a link does not depend on the existence of other links in the sample.

The following examples illustrate applications of the model to the literature.

We present in this section the results of the simulation using the estimators developed in the previous section. We generate simulated data using the model described in section 3 with $\beta=1$ and $\lambda(w)=1.5w^{2}+\log(w)$. The estimators will be assessed using three different linking functions: the stochastic blockmodel $f_{1}$, the beta model $f_{2}$, and the homophily model $f_{3}$ defined below:

These will be used to define the adjacency matrix $D$ as defined by (2). We run 500 simulations of $n=$50, 100, 200, and 300 individuals. For each 500 simulations, we draw a random sample of $n$ observations $\{\xi_{i}\}_{i=1}^{n}$ from a standard univariate normal distribution, $\{\varepsilon_{i}\}_{i=1}^{n}$ from a standard logistic distribution, $\{\omega_{i}\}_{i=1}^{n}$ and the lower diagonal entries of the symmetric matrix $\{\eta_{ij}\}_{i,j=1}^{n}$ from a standard uniform distribution. We generate dependency between $x_{i}$ and $\omega_{i}$ as follows $x_{i}=\xi_{i}+\sqrt{\omega_{i}}$. We use the Epanechnikov kernel $K(x)=\frac{3}{4}(1-x^{2})\mathbbm{1}(x^{2}<1)$ and the bandwidth sequence $h=n^{-1/9}/10$.

The performance of our proposed estimators will be evaluated in comparison to the estimators of the following models:

• •

  Naive Logit: $y_{i}=\mathbbm{1}\left\{\alpha+\beta_{1}x_{i}-\varepsilon_{i}\geq 0\right\}$

• •

  Infeasible Logit: $y_{i}=\mathbbm{1}\left\{\alpha+\beta_{2}x_{i}+\theta\lambda(\omega_{i})-\varepsilon_{i}\geq 0\right\}$

• •

  Logit with controls: $y_{i}=\mathbbm{1}\left\{\alpha+\beta_{3}x_{i}+\mu\sum_{j=1}^{d}D_{ij}x_{j}+\gamma\sum_{j=1}^{d}D_{ij}y_{j}-\varepsilon_{i}\geq 0\right\}$

For each model, link function, and sample size, we report the average bias over the 500 simulations.

Table 1 shows the results of the 500 simulations using the stochastic blockmodel.

The presented results in Table 1 illustrate the average bias across 500 simulations for various models (Naive Logit, Infeasible Logit, Logit with controls) at different sample sizes within a stochastic blockmodel, beta model, and homophily model framework. Across the models, the average bias of my estimate generally diminishes with larger sample sizes, indicating that increased data volume enhances estimation accuracy. However, specific parameter biases ($\beta_{1}$ and $\beta_{3}$) showcase a nuanced pattern, fluctuating inconsistently across sample sizes within each model. Notably, certain parameters display higher biases at smaller sample sizes, suggesting potential sensitivity to data volume. Notably, the Infeasible Logit estimator performs better than the other models because it assumes that $\lambda(w_{i})$ is observed. Despite this, our estimator remains consistent regardless of the link function used. These results reinforce the assumption that the link function does not need to be known for the estimator to be effective.

This paper proposes the identification and estimation of a semiparametric binary response model in which one covariate is an unknown function of an unobserved individual characteristic that influences both link formation in the network and the binary outcome. To achieve this, the estimation is based on matching pairs of agents with similar columns of the squared adjacency matrix. The proposed estimators are consistent and asymptotically normal.

The unicity of $\lambda(\omega_{i})$ for all $i$ follows from the unicity of $\beta.$ $\blacksquare$