December 11th, 2024
Novel semi-parametric framework
Identifying link formation with externalities in socio-economic networks
Recursive estimation integrating kernel density and method of moments elements.
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\(G_{ij}\)\(\; = {\Large 𝟙}\bigg\{\) \(\mathrm{H}_{i}\)\(\;+\;\)\(\mathrm{H}_{j}\) \(\ge\;\)\(U_{ij}\)\(\;\bigg\}\)
\(G_{ij} = {\Large 𝟙}\bigg\{\) \(h\)\((\)\(X_{i}\)\(,\,\)\(X_{j}\)\()\;+\;\) \(\mathrm{H}_{i} + \mathrm{H}_{j}\) \(\ge U_{ij}\bigg\}\)
\(G_{ij} = {\Large 𝟙}\bigg\{\) \(h\)\((X_{i}, X_{j}) +\;\) \(\mathrm{H}_{i} + \mathrm{H}_{j}\) \(\ge U_{ij}\bigg\}\)
\(h\colon {\mathcal{X}}^{2} \to \mathbb{R}\) examples:
\(G_{ij} = {\Large 𝟙}\bigg\{\) \(h(X_{i}, X_{j}) +\;\) \(\mathrm{H}_{i} + \mathrm{H}_{j}\) \(+ \beta \displaystyle\sum_{k\in\gamma_{n}(i, j)} \mathrm{H}_{k}\) \(\ge U_{ij}\bigg\}\)
\(G_{ij} = {\Large 𝟙}\bigg\{\) \(h(X_{i}, X_{j}) +\;\) \(\mathrm{H}_{i} + \mathrm{H}_{j}\) \(+ \beta \displaystyle\sum_{k\in{\color{red}\gamma_{\color{red}n}}(i, j)} \mathrm{H}_{k}\) \(\ge U_{ij}\bigg\}\)
\[\begin{align*} G_{ij}' &= {\Large 𝟙}\{h'(X_{i}, X_{j}) + \mathrm{H}_{i}' + \mathrm{H}_{j}' + \beta \sum_{k\in\gamma(i, j)} \mathrm{H}_{k}' \ge U_{ij}'\} \\ &= {\Large 𝟙}\{ah(X_{i}, X_{j}) + b + a\mathrm{H}_{i} + a\mathrm{H}_{j} + \beta \sum_{k\in\gamma(i, j)} a \mathrm{H}_{k} \ge aU_{ij} + b\} \\ &= G_{ij} \end{align*}\]
Congruent Nodes:
Copies:
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 \(x \in \mathcal{X}\).
\[\begin{align*} \mathbb{E}\left[G_{ij}\mid \gamma(i, j)=\emptyset, \mathrm{H}_{i} = \mathrm{H}_{j} = \eta_{0} \right] &= \mathbb{P}\left(U_{ij} \le h(x, x) + u_{0} - h_{}\right) \\ &= \mathbb{P}\left(U_{ij} \le u_{0}\right) \\ &= f_{0} \end{align*}\]
\[\begin{align*} f_{2} & = F_{U}\left(u_{2}\right) \\ &= \mathbb{P}\left(U_{ij} \le u_{2}\right) \\ &= \mathbb{P}\left(U_{ij} \le h(x, x) + \eta_{0} + \eta_{1} \right) \\ &= \mathbb{E}\left[G_{ij} \mid X_{i} = X_{j} = \bar x, \mathrm{H}_{i} = \eta_{0}, \mathrm{H}_{j} = \eta_{1}, \gamma(i, j) = \emptyset\right]. \end{align*}\]
\(f\)\(\;= F_{U}(u)\)
\(\quad = \mathbb{E}\big[G_{ij} \mid X_{i} = X_{j} = x, \mathrm{H}_{i} = \;\)\(\eta'\)\(\;, \mathrm{H}_{j} = \;\)\(\eta''\)\(\;, \gamma(i, j) = \emptyset\big]\)
\(f\)\(\;= F_{U}(u)\)
\(\quad = \mathbb{E}\big[G_{ij} \mid X_{i} = X_{j} = x, \mathrm{H}_{i} = \;\)\(\eta'\)\(\;, \mathrm{H}_{j} = \;\)\(\eta''\)\(\;, \gamma(i, j) = \emptyset\big]\)
\(f\)\(\;= F_{U}(u)\)
\(\quad = \mathbb{E}\big[G_{ij} \mid X_{i} = X_{j} = x, \mathrm{H}_{i} = \;\)\(\eta\)\(\;, \mathrm{H}_{j} = \eta_{m}, \gamma(i, j) = \emptyset\big]\)
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.
Recursive estimator for \(\boldsymbol{\eta}\), \(\beta\), and \(F_{U}\)
\[\begin{align} F_u(v) &= \mathbb{P}\left(G_{ij}=1|\boldsymbol{\eta},\beta\right) \\ &= \frac{\mathbb{P}\left(G_{ij}=1 \right) f_{v|G_{ij=1}}(v)}{\mathbb{P}\left(G_{ij}=1 \right) f_{v|G_{ij=1}}(v) + \mathbb{P}\left(G_{ij}=0 \right) f_{v|G_{ij=0}}(v)} \\ &\overset{def}{=} \frac{p_{1}(v)}{p_{1}(v) + p_{0}(v)} \end{align}\]
\(F_u\) and \(\boldsymbol{\eta}\) can be estimated simultaneously (KS or DBMM estimators)
Pick normalized candidate \(\boldsymbol{\eta}\), and bandwidth \(b_{0}\)
\(p_{1} (v_{ij};\, \boldsymbol{\eta})= \frac{1}{b^0(|\mathcal{M}|-1)} \displaystyle{\sum_{km \in\{\mathcal{M} - \{ij\}\}}} {\Large 𝟙}_{\{g_{km}=1\}} K \left(\frac{v_{ij}-v_{km}}{b^0} \right)\)
\(p_{0} (v_{ij};\, \boldsymbol{\eta})= \frac{1}{b^0(|\mathcal{M}|-1)} \displaystyle{\sum_{km \in\{\mathcal{M} - \{ij\}\}}} {\Large 𝟙}_{\{g_{km}=0\}} K \left(\frac{v_{ij}-v_{km}}{b^0} \right)\)
\(\hat p_{1} (v;\, \boldsymbol{\hat\eta^{0}})= \frac{1}{b^0|\mathcal{M}|} \displaystyle{\sum_{km \in \mathcal{M}}} {\Large 𝟙}_{\{g_{km}=1\}} K \left(\frac{v-\hat\eta_{k}^{0} -\hat\eta_{m}^{0}}{b^0} \right)\)
\(\hat p_{0} (v;\, \boldsymbol{\hat\eta^{0}})= \frac{1}{b^0|\mathcal{M}|} \displaystyle{\sum_{km \in\mathcal{M}}} {\Large 𝟙}_{\{g_{km}=0\}} K \left(\frac{v-\hat\eta_{k}^{0} -\hat\eta_{m}^{0}}{b^0} \right)\)
\(\hat p_{1} (v;\, \boldsymbol{\hat\eta^{1}}, \hat\beta^1)= \frac{1}{b^1|\mathcal{L}|} \displaystyle{\sum_{km \in \mathcal{L}}} {\Large 𝟙}_{\{g_{km}=1\}} K \left(\frac{v-\hat\eta_{k}^{1} - \hat\eta_{m}^{1} - \hat\beta^1\sum_{j\in\gamma(k,m)}\hat\eta_{j}^{1} }{b^1} \right)\)
\(\hat p_{0} (v;\, \boldsymbol{\hat\eta^{1}}, \hat\beta^1)= \frac{1}{b^1|\mathcal{L}|} \displaystyle{\sum_{km \in\mathcal{L}}} {\Large 𝟙}_{\{g_{km}=0\}} K \left(\frac{v-\hat\eta_{k}^{1} - \hat\eta_{m}^{1} - \hat\beta^1\sum_{j\in\gamma(k,m)}\hat\eta_{j}^{1} }{b^1} \right)\)