Hong, Leung & Li (2020) Inference on finite-population treatment effects under limited overlap, Econometrics J.

Average Treatment Effect

$n$ observed units, $D_i$ : treatment assignment, $Y_i(d)$ : potential outcome under $D_i = d$ , and $W_i$ : vector of baseline covariates.
$D_i$ 以外は全て固定の個人属性とする．
観測データ: $\{Y_i, D_i, W_i\}$ , where $Y_i = Y_i(1)D_i + Y_i(0)(1-D_i)$ .
Stratum of $i$ = $X_i \equiv S(W_i) \in \mathbb{X}$ , where $\mathbb{X}$ は有限集合．
Conditionally independent randomization: 各 stratum $x$ において，確率 $p_x$ で独立にトリートメントを assign.
Stratified block randomization: 各 stratum $x$ において，全体 $n_x$ 人中ランダムに $m_x$ 人にトリートメントを assign.
Propensity score: $p_n(x) \equiv \mathbb{E}[D \mid X = x]$ .
Conditionally independent randomization の場合 $p_n(x) = p_x$ , stratified block randomization の場合 $p_n(x) = m_x / n_x$ .

定義：limited overlap $\iff$ $a_n \equiv \min_{x \in \mathbb{X}} p_n(x) (1 - p_n(x)) \to 0$ .
Finite-population conditional ATE:

\begin{align}t_n(x) & = \frac{\sum_{i = 1}^n Y_i(1)\mathbf{1}\{X_i = x\}}{\sum_{i = 1}^n \mathbf{1}\{X_i = x\}} - \frac{\sum_{i = 1}^n Y_i(0)\mathbf{1}\{X_i = x\}}{\sum_{i = 1}^n \mathbf{1}\{X_i = x\}}\end{align}

Finite-population ATE:

\begin{align}t_n & =\sum_{x \in \mathbb{X}} \hat f (x) t_n(x) \;\; \text{where} \;\; \hat f (x) = \frac{1}{n}\sum_{i = 1}^n \mathbf{1}\{X_i = x\}.\end{align}

Estimator:

\begin{align}\hat t_n(x) & = \frac{\sum_{i = 1}^n Y_i D_i \mathbf{1}\{X_i = x\}}{\sum_{i = 1}^n D_i \mathbf{1}\{X_i = x\}} - \frac{\sum_{i = 1}^n Y_i(1 - D_i)\mathbf{1}\{X_i = x\}}{\sum_{i = 1}^n (1 - D_i) \mathbf{1}\{X_i = x\}} \\ \hat t_n & = \sum_{x \in \mathbb{X}} \hat f (x) \hat t_n(x)\end{align}

Theorem 2.1: Conditionally independent or stratified block randomization + $na_n \to \infty$ + regularity conditions $\Longrightarrow$ $\sqrt{na_n}(\hat t_n - t_n)/ \sigma_n \overset{d}{\to} N(0,1)$ .
収束 rate は通常の $\sqrt{n}$ よりも遅くなり得る．
Finite-population model において， $\sigma_n$ は potential outcome に依存した推定不可能な term ( $\beta_n$ ) を含む．Conservative な信頼区間ならば作れる（Remark 2.6）.

Local Average Treatment Effect

$Z_i$ : binary instrument, $D_i(z)$ : potential treatment choice under $Z_i = z$ . $D_i(\cdot)$ は固定値， $Z_i$ のみランダム．
観測データ: $\{Y_i, D_i, Z_i, W_i\}$ , where $D_i = D_i(1)Z_i + D_i(0)(1-Z_i)$ .
Finite-population LATE:

$\displaystyle \lambda^* = \frac{\sum_{i = 1}^n(Y_i(1) - Y_i(0)) \mathbf{1}\{D_i(1) \gt D_i(0)\}}{\sum_{i = 1}^n \mathbf{1}\{D_i(1) \gt D_i(0)\}}$

Let $Y_i^*(z) = Y_i(1) D_i(z) + Y_i(0)(1 - D_i(z))$ ,

\begin{align} \mu_n^*(z, x) & = \frac{\sum_{i = 1}^n Y_i^*(z) \mathbf{1}\{X_i = x\}}{\sum_{i = 1}^n \mathbf{1}\{X_i = x\}}, \quad \mu^*_n(z) = \sum_{x \in \mathbb{X}} \hat f (x) \mu_n^*(z, x) \\ \gamma_n(z, x) & = \frac{\sum_{i = 1}^n D_i(z) \mathbf{1}\{X_i = x\}}{\sum_{i = 1}^n \mathbf{1}\{X_i = x\}}, \quad \gamma_n(z) = \sum_{x \in \mathbb{X}} \hat f (x) \gamma_n(z, x) \end{align}

and

$\displaystyle \lambda_n = \frac{\mu^*_n(1) - \mu^*_n(0)}{\gamma_n(1) - \gamma_n(0)}$ .

Proposition 3.1: (Monotonicity): $\frac{1}{n}\sum_{i = 1}^n \mathbf{1}\{D_i(0) \gt D_i(1)\} \to 0$ + (Compliers): $\lim\inf_{n \to \infty}\frac{1}{n}\sum_{i = 1}^n \mathbf{1}\{D_i(1) \gt D_i(0)\} \gt 0$ $\Longrightarrow$ $|\lambda_n^* - \lambda_n| \to 0$ .
Estimator:

\begin{align}\hat \mu_n^*(z, x) & = \frac{\sum_{i = 1}^n Y_i Z_i^z(1-Z_i)^{1-z} \mathbf{1}\{X_i = x\}}{\sum_{i = 1}^n Z_i^z(1-Z_i)^{1-z} \mathbf{1}\{X_i = x\}}, \quad \hat \mu^*_n(z) = \sum_{x \in \mathbb{X}} \hat f (x) \hat \mu_n^*(z, x) \\ \hat \gamma_n(z, x) & = \frac{\sum_{i = 1}^n D_i Z_i^z(1-Z_i)^{1-z} \mathbf{1}\{X_i = x\}}{\sum_{i = 1}^n Z_i^z(1-Z_i)^{1-z} \mathbf{1}\{X_i = x\}}, \quad \hat \gamma_n(z) = \sum_{x \in \mathbb{X}} \hat f (x) \hat \gamma_n(z, x) \end{align}

and

$\displaystyle \hat \lambda_n = \frac{\hat \mu^*_n(1) - \hat \mu^*_n(0)}{\hat \gamma_n(1) - \hat \gamma_n(0)}$ .

Theorem 3.1: $\{Z_i\}$ 's are generated according to conditionally independent or stratified block randomization + $na_n \to \infty$ + regularity conditions $\Longrightarrow$ $\sqrt{na_n}(\hat \lambda_n - \lambda^*_n)/ \sigma_{\lambda,n} \overset{d}{\to} N(0,1)$ .
ATE 同様， $\sigma_{\lambda, n}$ には potential outcome に依存した推定不可能な term ( $\beta_{\lambda, n}$ ) が含まれる．