In denisagniel/synthate: Synthetic Causal Inference

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Here, we document all candidate estimators we will use in building synthetic causal estimators. For the most part, we write the estimators in the form \begin{align} \sumin \omega_i y_i, \end{align} where $\omega_i$ is a weight function that can in general depend on all the data. Let $d_i$ be the treatment indicator, and let $\bx_i$ denote covariates. We will denote the propensity score as $e_i$ and the prognostic score as $g_i$.

IPW: $$\omega_i = \frac{d_i}{e_i \sumin \frac{d_i}{e_i}} - \frac{1-d_i}{(1-e_i)\sumin \frac{1-d_i}{1-e_i}}$$
Regression $$ \omega_i = \left{(X'X)^{-1}X'\right}_{ji} $$ where this refers to the appropriate entry of the hat matrix and $X$ is the design matrix including covariates and treatment.
Regression within $K$ strata $$ \omega_i = \sum_{k=1}^KK^{-1}\frac{n_k}{n}I{e_i \in E_k}\left{(X(k)'X(k))^{-1}X(k)'\right}{ji} $$ where ${E_k}{k = 1, ..., K}$ is a partition of (0,1) and $X(k)$ is the design matrix including only those individuals for whom $e_i \in E_k$.
Doubly robust $$ \omega_i = n^{-1}\left{ \frac{d_i}{e_i} - \frac{1-d_i}{1-e_i} - \sum_{j=1}^n\left( \frac{d_j - e_j}{e_j}h_{j1i} + \frac{d_j - e_j}{1-e_j}h_{j0i}\right) \right} $$ where $h_{jdi}$ corresponds to the $i$th entry in the matrix $X_{jd}(X'X)^{-1}X'$ and $X_{jd}$ corresponds to the $j$th row of $X$ with treatment set to value $d$.
Stratified $$ \omega_i = K^{-1}\sum_{k=1}^K\frac{n_k}{n}I{e_i \in E_k}\left{\frac{d_i}{\sumin d_iI{e_i \in E_k}} - \frac{1-d_i}{\sumin (1-d_i)I{e_i \in E_k}}\right} $$ where $n_k = \sumin I{e_i \in E_k}$.
Matching ($M = 5$) with replacement on propensity $$ \omega_i = n^{-1}(2d_i - 1)\left(1 + M^{-1}\sum_{j: d_j \neq d_i}I\left{\sum_{k=1}^n I\left{|e_i - e_j| > |e_k - e_j|\right}\right) < M\right} $$
Matching ($M = 5$) with replacement on prognostic $$ \omega_i = n^{-1}(2d_i - 1)\left(1 + M^{-1}\sum_{j: d_j \neq d_i} I\left{\sum_{k=1}^n I\left{|g_i - g_j| > |g_k - g_j|\right} < M\right}\right) $$
Matching ($M = 5$) with replacement on both scores $$ \omega_i = n^{-1}(2d_i - 1)\left(1 + M^{-1}\sum_{j: d_j \neq d_i} I\left{\sum_{k=1}^n I\left{\|z_i - z_j\| > \|z_k - z_j\|\right} < M\right}\right) $$ where $z_i$ is the vector of both scores.
Caliper ($\delta = 0.05$) matching ($M = 5$) with replacement on propensity score $$ \omega_i = n^{-1}(2d_i - 1)I{|e_i - e_j| < \delta \quad \exists j}\left(1 + M^{-1}\sum_{j: d_j \neq d_i} I{|e_i - e_j| < \delta}I\left{\sum_{k=1}^n I\left{|e_i - e_j| > |e_k - e_j|\right} < M\right}\right) $$
Caliper ($\delta = 0.05$) matching ($M = 5$) with replacement on prognostic score $$ \omega_i = n^{-1}(2d_i - 1)I{|g_i - g_j| < \delta \quad \exists j}\left(1 + M^{-1}\sum_{j: d_j \neq d_i} I{|g_i - g_j| < \delta}I\left{\sum_{k=1}^n I\left{|g_i - g_j| > |g_k - g_j|\right} < M\right}\right) $$
Caliper ($\delta = 0.2$) matching ($M = 5$) with replacement on both scores $$ \omega_i = n^{-1}(2d_i - 1)I{\|z_i - z_j\| < \delta \quad \exists j}\left(1 + M^{-1}\sum_{j: d_j \neq d_i} I{\|z_i - z_j\| < \delta}I\left{\sum_{k=1}^n I\left{\|z_i - z_j\| > \|z_k - z_j\|\right} < M\right}\right) $$
Balancing $$ \omega_i = d_ip_K(\bx_i) - (1-d_i)q_K(x_i), $$ where $p_K(\bx_i)$ and $q_K(\bx_i)$ are weights that ensure that $\sumin d_ip_K(\bx_i)u_K(\bx_i) = \sumin (1-d_i)p_K(\bx_i)u_K(\bx_i) = n\inv\sumin u_K(\bx_i)$ for some contrasts encoded in $u_K(\bx_i)$.
Approximate residual balancing $$ \bxbar\trans(\bbetahat_t - \bbetahat_c) + \sumin d_i\gamma_{t,i}(y_i - \bx_i\trans\bbetahat_t) - \sumin (1-d_i)\gamma_{c,i}(y_i - \bx_i\trans\bbetahat_c),$$ where $$\bgamma_t = \argmin{\bgamma} \left{(1-\zeta)\|\bgamma\|2^2 + \zeta\|\bxbar - \bx_t\trans\bgamma\|\infty^2\right}$$ subject to $\sum_{i : d_i = 1} \gamma_i = 1$ and $0 \leq \gamma_i \leq n_t^{-2/3}$ and $\bgamma_c$ is defined similarly.