Rambachan and Roth (2020) Design-based uncertainty for quasi-experiments, arXiv.

Section 2: A finite population model for quasi-experiments

サイズ $N$ の有限母集団を考える*1．
Binary treatment: $D_i$ , potential outcomes: $Y_i(1), Y_i(0)$ . Potential outcome は固定値と仮定する．
Observed outcome: $Y_i = D_i Y_i(1) + (1 - D_i)Y_i(0)$ .
各 $i$ が独立にトリートメントを受ける確率を $p_i$ とする． $p_i$ は未知で，potential outcome と相関しても良い: $p_i = g(Y_i(1), Y_i(0), W_i)$ , where $W_i$ は pre-treatment covariates.
Treatment group と control group のサイズをそれぞれ $N_1$ と $N_0$ とする．
$\mathbf{D} = (D_1, \ldots, D_N)$ とすれば，

$\Pr\left( \mathbf{D} = \mathbf{d} \mid \sum_{i = 1}^N D_i = N_1 \right) \propto \prod_{i = 1}^N p_i^{d_i} (1 - p_i)^{1 - d_i}$

s.t. $\sum_{i = 1}^N d_i = N_1$ (zero otherwise)

また， $\pi_i = \Pr(D_i = 1 \mid \sum_{i = 1}^N D_i = N_1)$ と定義する．

Section 3: Simple difference-in-means

Simple difference-in-means (SDIM) estimator:

$\displaystyle \hat t \equiv \frac{1}{N_1} \sum_{i = 1}^N D_i Y_i - \frac{1}{N_0} \sum_{i = 1}^N (1 - D_i) Y_i$

このとき， $t_i = Y_i(1) - Y_i(0)$ と定義すれば，

\begin{align} \mathbb{E} \left[ \hat t \mid \sum_{i = 1}^N D_i = N_1 \right] & = \frac{1}{N_1} \sum_{i = 1}^N \pi_i (Y_i(0) + t_i) - \frac{1}{N_0} \sum_{i = 1}^N (1 - \pi_i) Y_i(0) \\& = \frac{1}{N_1} \sum_{i = 1}^N \pi_i t_i + \frac{N}{N_0 N_1} \sum_{i = 1}^N \pi_i Y_i(0) - \frac{1}{N_0} \sum_{i = 1}^N Y_i(0) \end{align}

ここで，

$\displaystyle \frac{1}{N_1} \sum_{i = 1}^N \pi_i t_i = \mathbb{E} \left[ \frac{1}{N_1} \sum_{i = 1}^N D_i t_i \mid \sum_{i = 1}^N D_i = N_1 \right] \equiv t_{ATT}$

= expected SATT (sample average treatment effect on the treated)

\begin{align} \frac{N}{N_0 N_1} \sum_{i = 1}^N \pi_i Y_i(0) - \frac{1}{N_0} \sum_{i = 1}^N Y_i(0) & = \frac{N}{N_0 N_1} \sum_{i = 1}^N \pi_i Y_i(0) - \frac{N}{N_0 N_1} \sum_{i = 1}^N \frac{N_1}{N} Y_i(0) \\ & = \frac{N N}{N_0 N_1} \left( \frac{1}{N}\sum_{i = 1}^N \left[ \pi_i Y_i(0) - \frac{N_1}{N} Y_i(0) \right] \right) \\ & = \frac{N N}{N_0 N_1} [\text{$\pi_i$ と $Y_i(0)$ の共分散}] \end{align}

したがって， $\mathbb{E} [ \hat t \mid \sum_{i = 1}^N D_i = N_1 ] = t_{ATT} + Bias$ で， $Bias$ は全ての個人が等しい確率でトリートメントを受けるときには（ $\pi_i = N_1/N$ for all $i$ ）ゼロになる．
以降は variance bound や $\hat t- \mathbb{E} [ \hat t \mid \sum_{i = 1}^N D_i = N_1 ]$ の漸近正規性について．

Section 4: Difference-in-differences

省略

Section 5: Instrumental variables

$Z_i \in \{0,1\}$ を操作変数として，potential treatment を $D_i (z) \in \{0,1\}$ と書く．観測できる outcome は $Y_i = Y_i(D_i(Z_i))$ .
$Z = 1$ を満たすグループと $Z = 0$ を満たすグループのサイズをそれぞれ $N^Z_1$ と $N^Z_0$ とする．
$\mathbf{Z} = (Z_1, \ldots, Z_N)$ とすれば，

$\Pr\left( \mathbf{Z} = \mathbf{z} \mid \sum_{i = 1}^N Z_i = N_1^Z \right) \propto \prod_{i = 1}^N p_i^{z_i} (1 - p_i)^{1 - z_i}$ s.t. $\sum_{i = 1}^N z_i = N_1^Z$

単調性を仮定： $D_i(1) \ge D_i(0)$ for all $i$ .
2SLS $\hat \beta_{2SLS} \equiv \hat t_{RF}/ \hat t_{FS}$ with

\begin{align}\hat t_{RF} & \equiv \frac{1}{N^Z_1} \sum_{i=1}^N Z_i Y_i - \frac{1}{N^Z_0} \sum_{i=1}^N (1 - Z_i) Y_i \\\hat t_{FS} & \equiv \frac{1}{N^Z_1} \sum_{i=1}^N Z_i D_i - \frac{1}{N^Z_0} \sum_{i=1}^N (1 - Z_i) D_i \\\end{align}

ここで， $\hat t_{RF}$ は $Z_i$ の SDIM なので，上と同じ計算から

\begin{align} \mathbb{E} \left[ \hat t_{RF} \mid \sum_{i = 1}^N Z_i = N_1^Z \right] & = \frac{1}{N_1^Z} \sum_{i = 1}^N \pi_i^Z (Y_i(D_i(1)) - Y_i(D_i(0))) \\ & \quad + \frac{N N}{N_0^Z N_1^Z} \underbrace{[\text{$\pi_i^Z$ と $Y_i(D_i(0))$ の共分散}]}_{= \: N^{-1} \sum_i (\pi_i^Z - N_1^Z/N)Y_i(D_i(0))} \end{align}

さらに，Complier の集合を $\mathcal{C} = \{i \mid D_i(1) \gt D_i(0) \}$ と定義すれば，単調性の仮定から，

$\displaystyle \frac{1}{N_1^Z} \sum_{i = 1}^N \pi_i^Z (Y_i(D_i(1)) - Y_i(D_i(0))) = \frac{1}{N_1^Z} \sum_{i \in \mathcal{C}} \pi_i^Z (Y_i(1) - Y_i(0))$

同様に，

\begin{align} \mathbb{E} \left[ \hat t_{FS} \mid \sum_{i = 1}^N Z_i = N_1^Z \right] & = \frac{1}{N_1^Z} \sum_{i \in \mathcal{C}}^N \pi_i^Z + \frac{N N}{N_0^Z N_1^Z} \underbrace{[\text{$\pi_i^Z$ と $D_i(0)$ の共分散}] }_{= \: N^{-1} \sum_i (\pi_i^Z - N_1^Z/N) D_i(0)}\end{align}

したがって， $\pi_i^Z = N_1^Z /N$ for all $i$ ならば以下が成り立つ：

$\displaystyle \beta_{2SLS} \equiv \frac{\mathbb{E} \left[ \hat t_{RF} \mid \sum_{i = 1}^N Z_i = N_1^Z \right] }{\mathbb{E} \left[ \hat t_{FS} \mid \sum_{i = 1}^N Z_i = N_1^Z \right]} =$ ” $\pi_i^Z$ -weighted” LATE.

以降は $\sqrt{N}(\hat \beta_{2SLS} - \beta_{2SLS})$ の漸近正規性について．

*1:州や県レベルのデータなどを想定．