Leung (2020) Treatment and spillover effects under network interference, REStat.

Model

Undirected adjacency matrix: $A$ , binary treatment: $D_i$ , unobserved heterogeneity: $\epsilon_i$ , and treatment response $Y_i$ .
仮定： $\{D_i\}_{i=1}^N$ は i.i.d.， $\{\epsilon_i, \epsilon_j, A_{ij}\}_{i \neq j}$ は identically distributed.
Nonparametric model:

$Y_i = r(D_i, T_i, \gamma_i, \epsilon_i)$ , where $T_i = \sum_{i = 1}^N A_{ij} D_j$ , and $\gamma_i = \sum_{j = 1}^N A_{ij}$ *1.

Sampling Frame

各 $i$ について $W_i = (Y_i, D_i, T_i, \gamma_i)$ が観測される．データは $\{W_i\}_{i = 1}^n$ の triangular array, where $n \le N$ is the number of "focal units"．

$\ldots$ the researcher first draws a random sample of $n \le N$ focal units and collects their treatment responses, treatment assignments, and identities of network neighbors. Then the researcher collects the treatment assignments of each neighbor.

ネットワーク $A$ 全体が観測できるわけではない．観測できる sub-network を $\tilde A$ と書く．
以下を定義する：

$\mu_h(d, t, \gamma) = \mathbb{E}\Bigl[ h(\underbrace{r(d, t, \gamma, \epsilon_1)}_{= \: \text{potential outcome}})\Bigr]$

$h(\cdot)$ が恒等関数の場合は $\mu(d,t,\gamma)$ と書く＝average structural function (ASF).
他の $h(\cdot)$ の例としては $h(x) = \mathbf{1}\{x \le q\}$ ＝quantile structural function (QSF).

Identification

Assumption (Exogeneity)

(a) $(D_1, \ldots, D_n) \perp (\tilde A, \epsilon_1, \ldots, \epsilon_n)$

(b) For all $i$ , $\epsilon_i \perp \tilde A \mid \gamma_i$

Proposition 1: Exogeneity assumption + regularity conditions

$\Longrightarrow$ $\mu_h(d, t, \gamma) = \mathbb{E}\left[ h(Y_1) \mid D_1 = d, T_1= t, \gamma_1 = \gamma \right]$

右辺は観測データのみで構成される＝identifiable

Estimation and Inference

$\mu_h(d, t, \gamma)$ の推定量として以下を考える：

$\displaystyle \hat \mu_h(d,t,\gamma) = \frac{\sum_{i=1}^n h(Y_i) \mathbf{1}\{D_i = d, T_i = t, \gamma_i = \gamma\}}{\sum_{i=1}^n \mathbf{1}\{D_i = d, T_i = t, \gamma_i = \gamma\}}$

Assumption (Correlated Effects)

(a) $A_{ij} = 0$ かつ $\max_k A_{ik}A_{jk} = 0$ ならば $\epsilon_i \perp \epsilon_j$

(b) $(\epsilon_i, \epsilon_j) \perp \tilde A \mid A_{ij}, \gamma_i, \gamma_j, \sum_k A_{ik}A_{jk}$

以下を定義する：

$G_{ij} = \mathbf{1}\{(A_{ij} =1) \lor (\max_k A_{ik}A_{jk} \ge 1) \lor (i = j) \}$

上の仮定と合わせて， $G = (G_{ij})$ が適切な dependency graph になっていることが分かる．
あとはChin (2019) 同様，Ross (2011) に基づいて Stein’s method を適用するための条件を整える↓

Assumption (Degree Distribution)

$n^{-1}\sum_{i=1}^n|\mathbf{N}_i|^3 = O_P(1)$ & $n^{-1}\sum_{i=1}^n\sum_{j \neq i}(G^3)_{ij} = O_P(1)$ , where $\mathbf{N}_i = \{j \mid G_{ij}=1\}$ .

Main Results

Assumpions (Exogeneity) + (Correlated Effects) + (Degree Distribution) + regularity conditions

$\Longrightarrow$

Theorem 1: $\hat \mu_h(d,t,\gamma) \overset{p}{\to} \lim_{N \to \infty} \mu_h(d,t,\gamma)$

Theorem 2: $\sqrt{n}\left( \hat \tau(d,t,d',t',\gamma) - \tau(d,t,d',t',\gamma) \right) \overset{d}{\to} N(0, \sigma^2)$ , where $\hat \tau(d,t,d',t',\gamma) = \hat \mu(d,t,\gamma) - \hat \mu(d',t',\gamma)$ ， $\tau(d,t,d',t',\gamma) = \mu(d,t,\gamma) - \mu(d',t',\gamma)$ .

*1:Exposure mapping を $e_i = e(T_i, \gamma_i)$ と特定化するのと同じ．最近の流れからするとやや制約的か．