Identification and Estimation of Semi-Parametric Link Formation Models with Externalities
May 23rd, 2025
Context
Summary
Novel semi-parametric framework
Identifying link formation with externalities in socio-economic networks
Recursive estimation integrating kernel density and method of moments elements.
Motivation
Making friends is easy!
Motivation
Making friends is easy!
Motivation
Making friends was easy!
Motivation
Making friends was easy!
Evidence
- Clustering: 0.46
- Random: 0.18
Evidence
- Prison friendships (MacRae 1960): 0.31 vs. 0.0134
- Co-authorships
- Mathematics 0.15 vs. 0.00002
- Biology (Newman 2001) 0.09 vs. 0.00001
- Economics (Goyal, Van Der Leij, and Moraga-González 2006) 0.19 vs. 0.00002
- Web (Adamic 1999) 0.11 vs. 0.0002
Previous Work
Link Formation
Model
\(G_{ij}\)\(\; = {\Large 𝟙}\bigg\{\) \(\mathrm{H}_{i}\)\(\;+\;\)\(\mathrm{H}_{j}\) \(\ge\;\)\(U_{ij}\)\(\;\bigg\}\)
- \(G_{ij}\): binary link formation variable.
- \(\mathrm{H}= {(\mathrm{H}_{i})}_{i\in[n]}\): i.i.d. unobservable heterogeneity
- \(\mathrm{supp}\left(\mathrm{H}_{i}\right)=\mathcal{H}=\mathbb{R}\).
- \(U = {(U_{ij})}_{i, j\in[n]} \overset{i.i.d.}{\sim} F_{U}\): unobservable shocks
Model
\(G_{ij} = {\Large 𝟙}\bigg\{\) \(h\)\((\)\(X_{i}\)\(,\,\)\(X_{j}\)\()\;+\;\) \(\mathrm{H}_{i} + \mathrm{H}_{j}\) \(\ge U_{ij}\bigg\}\)
- \(X = {(X_{i})}_{i\in[n]} \overset{i.i.d.}{\sim} F_{X}\) observable characteristics
- \(\mathrm{supp}\left(X_{i}\right) \subseteq\mathbb{R}^{k}\).
- \(h\colon {\mathcal{X}}^{2} \to \mathbb{R}\) unknown function
- symmetric
- \(h(X_{i}, X_{j}) = h_{d}\) for all \(X_{i}=X_{j}\).
Model
\(G_{ij} = {\Large 𝟙}\bigg\{\) \(h\)\((X_{i}, X_{j}) +\;\) \(\mathrm{H}_{i} + \mathrm{H}_{j}\) \(\ge U_{ij}\bigg\}\)
\(h\colon {\mathcal{X}}^{2} \to \mathbb{R}\) examples:
- \(h(X_{i}, X_{j}) = \alpha\, \lVert X_{i} - X_{j} \rVert_{2}\)
- \(h(X_{i}, X_{j}) = \alpha\, \mathrm{e}^{-{\lVert X_{i} - X_{j} \rVert}^{2}_{2}}\)
Model
\(G_{ij} = {\Large 𝟙}\bigg\{\) \(h(X_{i}, X_{j}) +\;\) \(\mathrm{H}_{i} + \mathrm{H}_{j}\) \(+ \beta \displaystyle\sum_{k\in\gamma_{n}(i, j)} \mathrm{H}_{k}\) \(\ge U_{ij}\bigg\}\)
Model
\(G_{ij} = {\Large 𝟙}\bigg\{\) \(h(X_{i}, X_{j}) +\;\) \(\mathrm{H}_{i} + \mathrm{H}_{j}\) \(+ \beta \displaystyle\sum_{k\in{\color{red}\gamma_{\color{red}n}}(i, j)} \mathrm{H}_{k}\) \(\ge U_{ij}\bigg\}\)
- \(\gamma_{n}\colon D_{n} \to \mathcal{P}\left([n]\right)\) known functional form
- \(D_{n}=\left\{(i, j)\in{[n]}^{2}\colon i\neq j\right\}\)
- symmetric
- \(i, j \notin \gamma_{n}(i, j)\) for all \((i, j)\in D_{n}\)
Externalities
- Common Friends
- Cliques
Identification
Assumptions
- \(F_{U}\) is continuous and strictly increasing
- \((X, \mathrm{H}) \perp U\)
- \(\mathrm{supp}\left(\mathrm{H}\mid X, \gamma\right) = \mathbb{R}\)
- \(F_{U} = F_{U\mid \gamma}\) and \(F_{X} = F_{X\mid \gamma}\)
- For all \(i,j\in\mathbb{N}\), the sequence \((\gamma_{n}(i,j))_{n\ge i,j}\) is finally constant.
- \(\lim\inf_{n \to \infty} \{i\in[n]\colon \forall j \neq i\; \gamma(i,j) = \emptyset\}\) is countably infinite
Identification up to Normalization
Theorem 1 Under assumptions (A1)-(A6) and a know hyper-diagonal value \(h_{d}\) the fixed effects, the error distribution, the externality parameter, and the homophily function are asymptotically uniquely identifiable up to an interquantile normalization.
Estimation
Initialization
- Notation: \(v_{ij}=\eta_i+\eta_j+\beta \gamma_{ij}\)
- Issues:
- We need \(F_u\) to estimate \(v_{ij}\) and \(v_{ij}\) to estimate \(F_u\)
- The functional form of \(v_{ij}\) is different for each pair (due to \(\gamma_{ij}\))
Initialization
- Select pairs without externalities \(\mathcal{M} \subseteq \mathcal{L}\)
- Pairs in \(\mathcal{M}\) satisfy \(v_{ij}=\eta_i+\eta_j\)
- And the sufficency condition \[\mathbb{P}\left(g_{ij}=1 \mid \boldsymbol{\eta}\right) =F_u(v_{ij})\]
- \(F_u\) and \(\boldsymbol{\eta}\) can be estimated simultaneously (KS or DBMM estimators)
Initialization
- \(F_u\) and \(\boldsymbol{\eta}\) can be estimated simultaneously (KS or DBMM estimators)
\[\begin{align} F_u(v) &= \mathbb{P}\left(G_{ij}=1|\boldsymbol{\eta},\beta\right) \\ &= \frac{\mathbb{P}\left(G_{ij}=1 \right) f_{v|G_{ij=1}}(v)}{\mathbb{P}\left(G_{ij}=1 \right) f_{v|G_{ij=1}}(v) + \mathbb{P}\left(G_{ij}=0 \right) f_{v|G_{ij=0}}(v)} \\ &\overset{def}{=} \frac{p_{1}(v)}{p_{1}(v) + p_{0}(v)} \end{align}\]
Initialization
\(F_u\) and \(\boldsymbol{\eta}\) can be estimated simultaneously (KS or DBMM estimators)
Pick normalized candidate \(\boldsymbol{\eta}\), and bandwidth \(b_{0}\)
\(p_{1} (v_{ij};\, \boldsymbol{\eta})= \frac{1}{b^0(|\mathcal{M}|-1)} \displaystyle{\sum_{km \in\{\mathcal{M} - \{ij\}\}}} {\Large 𝟙}_{\{g_{km}=1\}} K \left(\frac{v_{ij}-v_{km}}{b^0} \right)\)
\(p_{0} (v_{ij};\, \boldsymbol{\eta})= \frac{1}{b^0(|\mathcal{M}|-1)} \displaystyle{\sum_{km \in\{\mathcal{M} - \{ij\}\}}} {\Large 𝟙}_{\{g_{km}=0\}} K \left(\frac{v_{ij}-v_{km}}{b^0} \right)\)
Initialization
- \(F_u\) and \(\boldsymbol{\eta}\) can be estimated simultaneously (KS or DBMM estimators)
- \(\displaystyle\max_{\boldsymbol{\eta}} \mathbb{P}\left(G \mid \boldsymbol{\eta}, p_{1}(\cdot;\,\boldsymbol{\eta}), p_{0}(\cdot;\,\boldsymbol{\eta})\right)\)
- Obtain \(\boldsymbol{\hat\eta^{0}}\) and \(\hat F_u^{0}\)
Initialization
- Obtain \(\boldsymbol{\hat\eta^{0}}\) and \(\hat F_u^{0}\)
\(\hat p_{1} (v;\, \boldsymbol{\hat\eta^{0}})= \frac{1}{b^0|\mathcal{M}|} \displaystyle{\sum_{km \in \mathcal{M}}} {\Large 𝟙}_{\{g_{km}=1\}} K \left(\frac{v-\hat\eta_{k}^{0} -\hat\eta_{m}^{0}}{b^0} \right)\)
\(\hat p_{0} (v;\, \boldsymbol{\hat\eta^{0}})= \frac{1}{b^0|\mathcal{M}|} \displaystyle{\sum_{km \in\mathcal{M}}} {\Large 𝟙}_{\{g_{km}=0\}} K \left(\frac{v-\hat\eta_{k}^{0} -\hat\eta_{m}^{0}}{b^0} \right)\)
Step
- The first step estimates were consistent but inefficient
- With an estimate of \(F_u\), we can use all links in \(\mathcal{L}\)
- \(\displaystyle\max_{\boldsymbol{\eta},\beta} \mathbb{P}\left(G \mid \boldsymbol{\eta},\hat{F}^{0}_u(\boldsymbol{\eta},\beta)\right)\)
- Obtain \(\boldsymbol{\hat\eta}^1,\hat\beta^1\) and \(\hat{F}_u^{1}\)
Step
- Obtain \(\boldsymbol{\hat\eta}^1,\hat\beta^1\), and \(\hat{F}_u^{1}\)
\(\hat p_{1} (v;\, \boldsymbol{\hat\eta^{1}}, \hat\beta^1)= \frac{1}{b^1|\mathcal{L}|} \displaystyle{\sum_{km \in \mathcal{L}}} {\Large 𝟙}_{\{g_{km}=1\}} K \left(\frac{v-\hat\eta_{k}^{1} - \hat\eta_{m}^{1} - \hat\beta^1\sum_{j\in\gamma(k,m)}\hat\eta_{j}^{1} }{b^1} \right)\)
\(\hat p_{0} (v;\, \boldsymbol{\hat\eta^{1}}, \hat\beta^1)= \frac{1}{b^1|\mathcal{L}|} \displaystyle{\sum_{km \in\mathcal{L}}} {\Large 𝟙}_{\{g_{km}=0\}} K \left(\frac{v-\hat\eta_{k}^{1} - \hat\eta_{m}^{1} - \hat\beta^1\sum_{j\in\gamma(k,m)}\hat\eta_{j}^{1} }{b^1} \right)\)
Recursion
- Iteration between estimation of \(F_u\) and estimation of \(\beta, \boldsymbol{\eta}\)
- Each step: efficiency gain
- Iteration until convergence
Simulation
- \(n=100\) draws
- \(\eta \overset{i.i.d.}{\sim} \mathcal{U}(-2,2)\)
- \(\mathrm{e}^u \overset{i.i.d.}{\sim} \mathcal{N}(0,1)\)
- \(\beta = 0.25\)
- \(h\equiv 0\)
- Erdos-Renyi past network with \(p=0.3\)
- Common friends selection
Simulation
Identification and Estimation of Semi-Parametric Link Formation Models with Externalities
- Novel externality formulation accomodating many commonly occuring patterns in socio-economic networks
- Semi-parametric approach relying on spartan a set of assumptions
- Identification up to normalization of link formation models with externalities
- Recursive estimation combining kernel density and method of moments components