Identification and Estimation of Semi-Parametric Link Formation Models with Externalities

December 11th, 2024

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!

Image: Generated by Microsoft Designer

Motivation

Making friends was easy!

Image: Generated by Microsoft Designer

Motivation

Making friends was easy!

Image: Generated by Microsoft Designer

Evidence

Florentine marriages and business dealings
  • Clustering: 0.46
  • Random: 0.18
Data: (Padgett and Ansell 1993), Image: Matthew Jackson

Evidence

Credit: Matthew Jackson

Previous Work

  • Mele (2017); Miyauchi (2016); Sheng (2020); Ridder and Sheng (2015); Ridder and Sheng (2020); Menzel (2015)
  • Jackson and Rogers (2007); Goldsmith-Pinkham and Imbens (2013);

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:

  • \(h(X_{i}, X_{j}) = \lVert X_{i} - X_{j} \rVert_{2}\)
  • \(h(X_{i}, X_{j}) = \mathrm{e}^{-{\lVert X_{i} - X_{j} \rVert}^{2}_{2}}\)

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

Show me your Friends…

Show me your Friends…

Show me your Friends…

Show me your Friends…

Show me your Friends…

Show me your Friends…

Show me your Friends…

Show me your Friends…

Show me your Friends…

Show me your Friends…

Show me your Friends…

Show me your Friends…

Show me your Friends…

Show me your Friends…

Show me your Friends…

Identification

Point Identification Impossibility

  • For any model \(M = \left(F_{X}, h, \mathrm{H}, F_{U}, \beta, \gamma\right)\)
    • \(h' = a h + b\)
    • \(\mathrm{H}' = a \mathrm{H}\)
    • \(U' = a U + b\)

\[\begin{align*} G_{ij}' &= {\Large 𝟙}\{h'(X_{i}, X_{j}) + \mathrm{H}_{i}' + \mathrm{H}_{j}' + \beta \sum_{k\in\gamma(i, j)} \mathrm{H}_{k}' \ge U_{ij}'\} \\ &= {\Large 𝟙}\{ah(X_{i}, X_{j}) + b + a\mathrm{H}_{i} + a\mathrm{H}_{j} + \beta \sum_{k\in\gamma(i, j)} a \mathrm{H}_{k} \ge aU_{ij} + b\} \\ &= G_{ij} \end{align*}\]

Assumptions

  1. \(F_{U}\) is continuous and strictly increasing
  2. \((X, \mathrm{H}) \perp U\)
  3. \(\mathrm{supp}\left(\mathrm{H}\mid X, \gamma\right) = \mathbb{R}\)
  4. \(F_{U} = F_{U\mid \gamma}\) and \(F_{X} = F_{X\mid \gamma}\)
  5. For all \(i,j\in\mathbb{N}\), the sequence \((\gamma_{n}(i,j))_{n\ge i,j}\) is finally constant.
  6. \(\lim\inf_{n \to \infty} \{i\in[n]\colon \forall j \neq i\; \gamma(i,j) = \emptyset\}\) is countably infinite

Observable Congruence and Copies

Observable Congruence and Copies

  • Nodes \(0\) and \(2\) have same externalities with all other nodes.

Observable Congruence and Copies

Congruent Nodes:

  • Nodes \(0\) and \(2\) have same externalities with all other nodes.
  • Nodes \(0\) and \(2\) have the same observable characteristics.

Observable Congruence and Copies

Copies:

  • Nodes \(0\) and \(2\) have same externalities with all other nodes.
  • Nodes \(0\) and \(2\) have the same observable characteristics.
  • Nodes \(0\) and \(2\) have the same unobservable characteristics.

Order Identification of Congruent Nodes

Lemma 1 Under assumptions (A1)-(A6), the order of the fixed effects \({\{\mathrm{H}_{i}\}}_{i\in\mathcal{J}(x)}\) is asymptotically identifiable for any \(\mathcal{J}(x)\) collection of congruent nodes with observable characteristics \(x \in \mathcal{X}\).

Identification of Copies

  • By Lemma 1, \(\{\mathrm{H}_{i}\ge \mathrm{H}_{j}\}\) for any \(i, j\in\mathcal{J}(x)\) are asymptotically identifiable.
  • Thus, \(\{\mathrm{H}_{i} = \mathrm{H}_{j}\} = \{\mathrm{H}_{i}\ge \mathrm{H}_{j}\} \cap \{\mathrm{H}_{j}\ge \mathrm{H}_{i}\}\) is also asymptotically identifiable.

Identification at a Know Error Distribution Point

  • Suppose we know \(F_{U}(u_{0}) = f_{0}\)
  • Externality-hermit nodes with observables \(x\) and uobservables \(\eta_{0} = (u_{0} - h_{d}) /2\) are asymptotically identifiable.

\[\begin{align*} \mathbb{E}\left[G_{ij}\mid \gamma(i, j)=\emptyset, \mathrm{H}_{i} = \mathrm{H}_{j} = \eta_{0} \right] &= \mathbb{P}\left(U_{ij} \le h(x, x) + u_{0} - h_{}\right) \\ &= \mathbb{P}\left(U_{ij} \le u_{0}\right) \\ &= f_{0} \end{align*}\]

Bisection (Infill)

  • \(h(X_{i}, X_{j}) = h_{d}\)
  • \(F_{U}(u_0) = f_{0}\)
  • \(F_{U}(u_1) = f_{1}\)
  • \(\eta_{0} = (u_{0} - h_{d}) / 2\), \(\eta_{1} = (u_{1} - h_{d}) / 2\)
  • \(u_{2} = (u_{0} + u_{1}) / 2 = h_{d} + (\eta_{0} + \eta_{1})\)

Bisection (Infill)

  • \(\eta_{0} = (u_{0} - h_{d}) / 2\), \(\eta_{1} = (u_{1} - h_{d}) / 2\)
  • \(u_{2} = (u_{0} + u_{1}) / 2 = h_{d} + (\eta_{0} + \eta_{1})\)

\[\begin{align*} f_{2} & = F_{U}\left(u_{2}\right) \\ &= \mathbb{P}\left(U_{ij} \le u_{2}\right) \\ &= \mathbb{P}\left(U_{ij} \le h(x, x) + \eta_{0} + \eta_{1} \right) \\ &= \mathbb{E}\left[G_{ij} \mid X_{i} = X_{j} = \bar x, \mathrm{H}_{i} = \eta_{0}, \mathrm{H}_{j} = \eta_{1}, \gamma(i, j) = \emptyset\right]. \end{align*}\]

Bisection (Infill)

  • Repeat for \(u_{0}\) and \(u_{2}\)
    • Identify on \(u_{3}\).
  • Repeat for \(u_{2}\) and \(u_{1}\)
    • Identify on \(u_{4}\).

Bisection (Infill)

  • Inductivelly

Bisection (Infill)

  • Inductivelly

Bisection (Infill)

  • Inductivelly
  • Identify \(F_{U}\) on a dense subset of \([u_{0}, u_{1}]\).

Bisection (Infill)

  • Inductivelly
  • Identify \(F_{U}\) on a dense subset of \([u_{0}, u_{1}]\).
  • By continuity, identify \(F_{U}\) on \([u_{0}, u_{1}]\).

Extrapolation

Extrapolation

Extrapolation

Extrapolation

  • \(\eta\)\(\;\in\;[\)\(\eta_{m} - 2\delta\)\(,\,\)\(\eta_{m} + 2\delta\)\(]\)
  • \(r', r'' \in [-\delta, \delta]\)
  • \(\eta\)\(\;= \eta_{m} + r' + r''\)

Extrapolation

  • \(\eta\)\(\;\in\;[\)\(\eta_{m} - 2\delta\)\(,\,\)\(\eta_{m} + 2\delta\)\(]\)
  • \(r', r'' \in [-\delta, \delta]\)
  • \(\eta\)\(\;= \eta_{m} + r' + r''\)
  • \(\eta'\)\(\;= \eta_{m} + r'\)
  • \(\eta''\)\(\;= \eta_{m} + r''\)

Extrapolation

  • \(\eta\)\(\;\in\;[\)\(\eta_{m} - 2\delta\)\(,\,\)\(\eta_{m} + 2\delta\)\(]\)
  • \(r', r'' \in [-\delta, \delta]\)
  • \(\eta\)\(\;= \eta_{m} + r' + r''\)
  • \(\eta'\)\(\;= \eta_{m} + r'\)
  • \(\eta''\)\(\;= \eta_{m} + r''\)
  • \(u =\;\)\(\eta'\)\(\;+\;\)\(\eta''\)\(\;+\; h_{d}\)

Extrapolation

  • \(\eta\)\(\;\in\;[\)\(\eta_{m} - 2\delta\)\(,\,\)\(\eta_{m} + 2\delta\)\(]\)
  • \(r', r'' \in [-\delta, \delta]\)
  • \(\eta\)\(\;= \eta_{m} + r' + r''\)
  • \(\eta'\)\(\;= \eta_{m} + r'\)
  • \(\eta''\)\(\;= \eta_{m} + r''\)
  • \(u =\;\)\(\eta'\)\(\;+\;\)\(\eta''\)\(\;+\; h_{d}\)

\(f\)\(\;= F_{U}(u)\)

\(\quad = \mathbb{E}\big[G_{ij} \mid X_{i} = X_{j} = x, \mathrm{H}_{i} = \;\)\(\eta'\)\(\;, \mathrm{H}_{j} = \;\)\(\eta''\)\(\;, \gamma(i, j) = \emptyset\big]\)

Extrapolation

  • \(\eta\)\(\;\in\;[\)\(\eta_{m} - 2\delta\)\(,\,\)\(\eta_{m} + 2\delta\)\(]\)
  • \(r', r'' \in [-\delta, \delta]\)
  • \(\eta\)\(\;= \eta_{m} + r' + r''\)
  • \(\eta'\)\(\;= \eta_{m} + r'\)
  • \(\eta''\)\(\;= \eta_{m} + r''\)
  • \(u =\;\)\(\eta'\)\(\;+\;\)\(\eta''\)\(\;+\; h_{d}\) \(\;=\;\)\(\eta\)\(\;+\; \eta_{m} + h_{d}\)

\(f\)\(\;= F_{U}(u)\)

\(\quad = \mathbb{E}\big[G_{ij} \mid X_{i} = X_{j} = x, \mathrm{H}_{i} = \;\)\(\eta'\)\(\;, \mathrm{H}_{j} = \;\)\(\eta''\)\(\;, \gamma(i, j) = \emptyset\big]\)

Extrapolation

  • \(\eta\)\(\;\in\;[\)\(\eta_{m} - 2\delta\)\(,\,\)\(\eta_{m} + 2\delta\)\(]\)
  • \(r', r'' \in [-\delta, \delta]\)
  • \(\eta\)\(\;= \eta_{m} + r' + r''\)
  • \(\eta'\)\(\;= \eta_{m} + r'\)
  • \(\eta''\)\(\;= \eta_{m} + r''\)
  • \(u =\;\)\(\eta'\)\(\;+\;\)\(\eta''\)\(\;+\; h_{d}\) \(\;=\;\)\(\eta\)\(\;+\; \eta_{m} + h_{d}\)

\(f\)\(\;= F_{U}(u)\)

\(\quad = \mathbb{E}\big[G_{ij} \mid X_{i} = X_{j} = x, \mathrm{H}_{i} = \;\)\(\eta\)\(\;, \mathrm{H}_{j} = \eta_{m}, \gamma(i, j) = \emptyset\big]\)

Extrapolation

Extrapolation

Extrapolation

Interquantile normalization

  • Pick \(\alpha \in (0, 2^{-1})\)

Interquantile normalization

  • Pick \(\alpha \in (0, 2^{-1})\)
  • Let \(a = (F_{U}^{-1}(1 - \alpha) - F_{U}^{-1}(\alpha))^{-1}\)
  • Let \(b = -a F_{U}^{-1}(\alpha)\)

Interquantile normalization

  • Pick \(\alpha \in (0, 2^{-1})\)
  • Let \(a = (F_{U}^{-1}(1 - \alpha) - F_{U}^{-1}(\alpha))^{-1}\)
  • Let \(b = -a F_{U}^{-1}(\alpha)\)
  • Define \(U' = aU + b\)

Interquantile normalization

  • Pick \(\alpha \in (0, 2^{-1})\)
  • Let \(a = (F_{U}^{-1}(1 - \alpha) - F_{U}^{-1}(\alpha))^{-1}\)
  • Let \(b = -a F_{U}^{-1}(\alpha)\)
  • Define \(U' = aU + b\)
  • \(F_{U'}(0) = \alpha\) and \(F_{U'}(1) = 1 - \alpha\)

Identification up to Normalization

Theorem 1 Under assumptions (A1)-(A6), a know hyper-diagonal value \(h_{d}\), and an \(\alpha \in (0, 2^{-1})\), the fixed effects, the error distribution, the externality parameter, and the homophily function are asymptotically uniquely identifiable.

Estimation

Summary

Recursive estimator for \(\boldsymbol{\eta}\), \(\beta\), and \(F_{U}\)

  • Initialization:
    • Select links without externalities
    • Kernel density estimation (KDE): \(\boldsymbol{\hat\eta}^0\) and \(\hat F_u^0\)
  • Step (\(k-1 \to k\)):
    • GMM with \(\hat F_u^{k-1}\) to get \(\boldsymbol{\hat\eta}^{k}\) and \(\hat\beta^{k}\)
    • KDE using \(\boldsymbol{\hat\eta}^{k}\) and \(\hat\beta^{k}\) to get \(\hat F_u^{k}\)

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 \(\hat{F}_u^{1}\)

\(\hat p_{1} (v;\, \boldsymbol{\hat\eta^{1}}, \hat\beta^1)= \frac{1}{b^1|\mathcal{L}|} \displaystyle{\sum_{km \in \mathcal{L}}} {\Large 𝟙}_{\{g_{km}=1\}} K \left(\frac{v-\hat\eta_{k}^{1} - \hat\eta_{m}^{1} - \hat\beta^1\sum_{j\in\gamma(k,m)}\hat\eta_{j}^{1} }{b^1} \right)\)

\(\hat p_{0} (v;\, \boldsymbol{\hat\eta^{1}}, \hat\beta^1)= \frac{1}{b^1|\mathcal{L}|} \displaystyle{\sum_{km \in\mathcal{L}}} {\Large 𝟙}_{\{g_{km}=0\}} K \left(\frac{v-\hat\eta_{k}^{1} - \hat\eta_{m}^{1} - \hat\beta^1\sum_{j\in\gamma(k,m)}\hat\eta_{j}^{1} }{b^1} \right)\)

Trimming

  • Hessian has tendency to be sparse (each fixed effect coefficient appears only in some equations)
  • Probability trimming:
    • Prevent linking probabilities to be too close zero or one
    • Down-weight observation that have been subject to trimming

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

Initialization

Step 1

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
Pantelis Karapanagiotis
p.karapanagiotis@rug.nl
Sanna Stephan
l.s.stephan@rug.nl

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Goldsmith-Pinkham, Paul, and Guido W Imbens. 2013. “Social Networks and the Identification of Peer Effects.” Journal of Business & Economic Statistics 31 (3): 253–64. https://doi.org/10.1080/07350015.2013.801251.
Goyal, Sanjeev, Marco J. Van Der Leij, and José Luis Moraga-González. 2006. “Economics: An Emerging Small World.” Journal of Political Economy 114 (2): 403–12. https://doi.org/10.1086/500990.
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———. 2020. “Two-Step Estimation of a Strategic Network Formation Model with Clustering.” arXiv Preprint arXiv:2001.03838. https://doi.org/10.48550/arXiv.2001.03838.
Sheng, Shuyang. 2020. “A Structural Econometric Analysis of Network Formation Games Through Subnetworks.” Econometrica 88 (5): 1829–58. https://doi.org/10.3982/ECTA12558.