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Supplement to ``Uncertainty Quantification in Synthetic Controls with Staggered Treatment Adoption''
\title{Supplement to ``Uncertainty Quantification in Synthetic Controls with Staggered Treatment Adoption''\medskip }
\author{Matias D. Cattaneo\thanks{Department of Operations Research and Financial Engineering, Princeton University.} \and
Yingjie Feng\thanks{School of Economics and Management, Tsinghua University.} \and
Filippo Palomba\thanks{Department of Economics, Princeton University.} \and
Roc{\'i}o Titiunik\thanks{Department of Politics, Princeton University.}}
\maketitle
\begin{abstract}
\noindent This supplement contains all proofs, additional results, other technical details about data preparation and estimation, and the code used to reproduce the results.
\end{abstract}
\vspace{-.1in}
\clearpage
\tableofcontents
\clearpage
\section{Notation}
We summarize the notation used throughout the paper in the following table.
\begin{table}[!ht]
\centering
\caption{Summary of Notation}
\label{tab: notation}
\doublespacing
\resizebox{.95\textwidth}{!}{\begin{tabular}{p{0.35\linewidth} p{0.65\linewidth}}
\toprule\toprule
\multicolumn{1}{l}{\textbf{Quantity}} & \multicolumn{1}{l}{\textbf{Description}} \\
\midrule
$\tau_{ik}$ & time-specific unit-specific predictand\\
$\tau_{i\cdot}$ & time-averaged unit-specific predictand\\
$\tau_{\mathcal{Q}k}$ & time-specific unit-averaged predictand\\
$\tau_{\cdot\cdot}$& time-averaged unit-averaged predictand\\
$\mathcal{N}$ & set of never-treated units\\
$\mathcal{E}$ & set of ever-treated units\\
$\mathcal{W}^{[i]}$ or $\mathcal{W}$ &
constraint set imposed in SC construction \\
$J_0$ & number of never-treated units\\
$J_1$ & number of ever-treated units\\
$Q$ & number of units in $\mathcal{Q}$\\
$Y_{it}$ & real GDP per capita of country $i$ in year $t$\\
$\mathbf{Y}_{\mathcal{N}t}$& a vector of outcomes of never-treated (donor) units\\
$u_{it}$ & out-of-sample error in decomposition of $\hat{Y}_{it}$\\
$\varrho_\ell^{[i]}$ & tuning parameter used to check if the $\ell$-th constraint is binding\\
$\mathcal{M}_{\widehat{\boldsymbol{\gamma}}}^{[i]}$ or
$\mathcal{M}_{\widehat{\boldsymbol{\gamma}}}$ & feasible set for SC weights deduced from optimization\\
$\mathcal{M}_{\mathbf{G}}^{[i]\star}$ or $\mathcal{M}_{\mathbf{G}}^{\star}$&
feasible set used for in-sample uncertainty quantification\\
$\Delta^{[i]}$ or $\Delta$ & centered constraint set\\
$\mathrm{dist}(\mathbf{a}, \mathcal{A})$ &
distance between a point $\mathbf{a}$ and a set $\mathcal{A}$, i.e., $\inf_{\bm{\lambda}\in\mathcal{A}}\|\mathbf{a}-\bm{\lambda}\|$\\
$\mathcal{A}_\varepsilon$ & $\varepsilon$-enlargement of $\mathcal{A}$, i.e., $\{\mathbf{a}:\mathrm{dist}(\mathbf{a}, \mathcal{A})\leq\varepsilon\}$\\
$|\mathcal{A}|$ & number of elements in a set $\mathcal{A}$\\
$\times_{i\in\mathcal{S}}\mathcal{A}_i$ & Cartesian product of sets $\mathcal{A}_i$'s for $i\in\mathcal{S}$\\
$\|\cdot\|_1$, $\|\cdot\|_2$ & L1 and L2 vector norms\\
$s_{\min}(\mathbf{A})$, $s_{\max}(\mathbf{A})$ & minimum and maximum singular values of a matrix $\mathbf{A}$\\
$(v_j: j\in\mathcal{A})$ & a vector consisting of all $v_j$'s with $j\in\mathcal{A}$\\
$\otimes$ & Kronecker product \\
subscripts $\mathtt{in}$, $\mathtt{out}$ & ``in-sample'', ``out-of-sample''\\
superscript $[i]$ & indicate a quantity is related to a treated unit $i$\\
lower bar, upper bar & ``lower bound'', ``upper bound''\\
$\star$ & indicate a quantity is related to simulation
\\\bottomrule\bottomrule
\end{tabular}
}
\end{table}
\section{Other Strategies for Uncertainty Quantification}
\subsection{Out-of-Sample Error}
\label{sec: other out-of-sample UQ approaches}
In Section 4.2 we discuss the approach for quantifying the out-of-sample uncertainty based on the non-asymptotic bounds. We briefly describe two strategies below.
\begin{itemize}[leftmargin=*]
\item \textit{Location-scale model}. Suppose that $u_{it}=\mathbb{E}[u_{it}|\mathscr{H}]+(\mathbb{V}[u_{it}|\mathscr{H}])^{1/2}\nu_{it}$ with $\nu_{it}$ statistically independent of $\mathscr{H}$. The bounds on $u_{it}$ can now be set to
$\underline{M\mkern-4mu}\mkern4mu _{\mathtt{out}}=\mathbb{E}[u_{it}|\mathscr{H}]+(\mathbb{V}[u_{it}|\mathscr{H}])^{1/2}\mathfrak{c}_\nu(\alpha_{\mathtt{out}}/2)$ and $\overline{\mkern-3muM\mkern-1.5mu}_{\mathtt{out}}=\mathbb{E}[u_{it}|\mathscr{H}]+(\mathbb{V}[u_{it}|\mathscr{H}])^{1/2}\mathfrak{c}_\nu(1-\alpha_{\mathtt{out}}/2)$ where
$\mathfrak{c}_\nu(\alpha_{\mathtt{out}}/2)$ and $\mathfrak{c}_\nu(1-\alpha_{\mathtt{out}}/2)$ are $\alpha_{\mathtt{out}}/2$ and $(1-\alpha_{\mathtt{out}}/2)$ quantiles of $\nu_{it}$, respectively, and $\alpha_{\mathtt{out}}$ is the desired pre-specified level.
\item \textit{Quantile regression}. We can determine the $\alpha_{\mathtt{out}}/2$ and $(1-\alpha_{\mathtt{out}}/2)$ conditional quantiles of $u_{it}|\mathscr{H}$. Consequently, another possibility is to employ quantile regression methods to estimate those quantities using pre-treatment data.
\end{itemize}
\subsection{Simultaneous Prediction Intervals}
\label{sec: other simultaneous PI}
Section 4.4 constructs prediction intervals with simultaneous coverage. We briefly describe two other common approaches below.
\begin{itemize}[leftmargin=*]
\item \textit{Bonferroni-type correction}. There is a large literature on Bonferroni corrections
that can be used to construct multiple prediction intervals with simultaneous coverage. For example, consider a simple correction strategy: for each $k=0, \ldots, L$, use any strategy described in Section 4.2 or Section \ref{sec: other out-of-sample UQ approaches} to construct a prediction interval for $u_{i(T_i+k)}$ that has a coverage probability at least $1-(\alpha_{\mathtt{out}}/(L+1))$. Then, the simultaneous coverage probability of the $L+1$ prediction intervals
$\{\tilde{\mathcal{I}}_{k}: 0\leq k\leq L\}$ is at least $1-\alpha_{\mathtt{out}}$. Some other more sophisticated corrections are also available in the literature (see, e.g., \citealp{Ravishanker-Hochberg-Melnick_1987_TCMA}). For instance, the second-order Bonferroni-type bound implies that
\[
\begin{split}
&\mathbb{P}\big[u_{i(T_i+k)}\in\tilde{\mathcal{I}}_{k} \text{ for all } 0\leq k\leq L\,\big|\, \mathscr{H}\big]\geq
1-\sum_{k=0}^L p_k+\sum_{k=0}^{L-1} p_{k,k+1},\quad \text{where}\\
&p_k=\mathbb{P}(u_{i(T_i+k)}\in\tilde{\mathcal{I}}_{k}|\mathscr{H}), \quad
p_{k,k+1}=\mathbb{P}(u_{i(T_i+k)}\in\tilde{\mathcal{I}}_{k},\;
u_{i(T_i+k+1)}\in\tilde{\mathcal{I}}_{k+1}|\mathscr{H}).
\end{split}
\]
Then, one can construct the prediction intervals $\tilde{\mathcal{I}}_{k}$'s with corresponding coverage probabilities $p_k$ and $p_{k,k+1}$ such that
$1-\sum_{k=0}^L p_k+\sum_{k=0}^{L-1} p_{k,k+1}\geq 1-\alpha_{\mathtt{out}}$. Such bounds are usually sharper, but their implementation requires the modeling of the dependence of $(u_{it}, u_{i(t+1)})$ conditional on $\mathscr{H}$ and is computationally more burdensome.
\item \textit{Scheff\'{e}-type intervals}. An alternative approach is to construct Scheff\'{e}-type simultaneous prediction intervals, though stronger distributional assumptions need to be made. For instance, assume that $(u_{iT_i}, \cdots, u_{i(T_i+L)})'$ jointly follows a conditional Gaussian distribution with mean zero and variance $\Sigma_{\mathscr{H}}$. Then,
$$(u_{iT_i}, \cdots, u_{i(T_i+L)})\Sigma_{\mathscr{H}}^{-1}(u_{iT_i}, \cdots, u_{i(T_i+L)})'\sim\chi^2_{L+1},$$
where $\chi^2_{L+1}$ is $\chi^2$ distribution with $L+1$ degrees of freedom.
The sequence of prediction intervals $\tilde{\mathcal{I}}_k=\Big[-\sigma_{\mathscr{H},kk}\sqrt{\chi^2_{L+1}(1-\alpha_{\mathtt{out}})},\;
\sigma_{\mathscr{H},kk}\sqrt{\chi^2_{L+1}(1-\alpha_{\mathtt{out}})}\,\Big]$ have the simultaneous coverage probability at least $1-\alpha_{\mathtt{out}}$, where $\sigma^2_{\mathscr{H},kk}$ is the $k$-th diagonal element of $\Sigma_{\mathscr{H}}$ and $\chi^2_{L+1}(1-\alpha_{\mathtt{out}})$ is the $(1-\alpha_{\mathtt{out}})$-quantile of $\chi^2$ distribution with $L+1$ degrees of freedom.
\end{itemize}
\subsection{Alternative Bounds for In-sample Error}\label{sec: alternative bound in a special case}
When the causal predictand of interest depends on treatment effects on \textit{multiple} treated units, such as the time-specific unit-averaged predictand $\tau_{\mathcal{Q}k}$, there is an alternative method for in-sample uncertainty quantification that may yield tighter prediction intervals.
It relies on the fact that if the SC weights $\widehat{\boldsymbol{\beta}}^{[i]}$ for each ever-treated unit $i$ are obtained through a \textit{separate} optimization process using data $\mathbf{A}^{[i]}$, $\mathbf{B}^{[i]}$ and $\mathbf{C}^{[i]}$, then, as outlined in Remark 1, a sequence of restrictions must be obeyed:
for each $i\in\mathcal{E}$,
\begin{equation}\label{eq: basic inequality for each subvector}
(\widehat{\boldsymbol{\beta}}^{[i]}-\boldsymbol{\beta}_0^{[i]})'\widehat{\mathbf{Q}}^{[i]}(\widehat{\boldsymbol{\beta}}^{[i]}-\boldsymbol{\beta}_0^{[i]})-
2(\widehat{\boldsymbol{\gamma}}^{[i]}-\boldsymbol{\gamma}^{[i]})'(\widehat{\boldsymbol{\beta}}^{[i]}-\boldsymbol{\beta}_0^{[i]})\leq 0,\;
\widehat{\boldsymbol{\beta}}^{[i]}-\boldsymbol{\beta}_0^{[i]}\in\Delta^{[i]}.
\end{equation}
Then, analogously to (6.5) in the main text, for any predictand $\tau$ defined before, we can set
\begin{equation}\label{eq: bound on in-sample error, special case}
\begin{aligned}
\underline{M\mkern-4mu}\mkern4mu _{\mathtt{in}}(\tau)&=(\alpha_{\mathtt{in}}/2)\text{-quantile of } \inf_{\boldsymbol{\delta}\in\tilde{\mathcal{M}}_{\mathbf{G}}^\star}
-\mathbf{p}_{\tau}'\boldsymbol{\delta}
\quad\text{and}\\
\overline{\mkern-3muM\mkern-1.5mu}_{\mathtt{in}}(\tau)&=(1-\alpha_{\mathtt{in}}/2)\text{-quantile of } \sup_{\boldsymbol{\delta}\in\tilde{\mathcal{M}}_{\mathbf{G}}^\star}
-\mathbf{p}_{\tau}'\boldsymbol{\delta}
\end{aligned}
\end{equation}
conditional on the data, with
$\tilde{\mathcal{M}}_{\mathbf{G}}^\star=\{\boldsymbol{\delta}=(\boldsymbol{\delta}^{[1]\prime},\cdots, \boldsymbol{\delta}^{[J_1]\prime})': \; \boldsymbol{\delta}^{[i]}\in\Delta^{[i]^\star},\;
\boldsymbol{\delta}^{[i]\prime}\widehat\mathbf{Q}^{[i]}\boldsymbol{\delta}^{[i]}-
2(\mathbf{G}^{[i]\star})'\boldsymbol{\delta}^{[i]}\leq 0,\; i\in\mathcal{E}\}$
and $\mathbf{G}^\star=(\mathbf{G}^{[1]\prime},\cdots, \mathbf{G}^{[J_1]\prime})'|\mathsf{Data}\sim
\mathsf{N}(\mathbf{0},\widehat{\boldsymbol{\Sigma}})$.
Again, $\Delta^{[i]}$ is replaced with its feasible (enlarged) version $\Delta^{[i]\star}$, and $\widehat{\boldsymbol{\gamma}}^{[i]}-\boldsymbol{\gamma}^{[i]}$ is replaced with the approximating Gaussian vector $\mathbf{G}^{[i]\star}$.
As a reminder, this strategy is not equivalent to doing in-sample uncertainty quantification for each time-specific unit-specific prediction $\widehat{\tau}_{ik}$ and then combining their bounds together,
since the whole vector $\mathbf{G}^\star$ has the same covariance as $\widehat{\boldsymbol{\gamma}}=(\widehat{\boldsymbol{\gamma}}^{[1]\prime}, \cdots, \widehat{\boldsymbol{\gamma}}^{[J_1]\prime})'$,
thus maintaining the correlation structure among different treated units.
Technically, the above optimization procedure for constructing SC weights is equivalent to a special case of (6.1), where the weighting matrix $\mathbf{V}$ is block diagonal, taking the form $\mathbf{V}=\mathsf{diag}(\mathbf{V}^{[1]}, \cdots, \mathbf{V}^{[J_1]})$, and the feasible set $\Delta=\mathcal{W}\times \mathcal{R}$ is the Cartesian product of $\Delta^{[i]}=\mathcal{W}^{[i]}\times \mathcal{R}^{[i]}$ for each subvector $\boldsymbol{\beta}^{[i]}$ (hence, there is no cross-treated-unit constraint).
The general in-sample uncertainty quantification strategy presented earlier in (6.5) of the main text relies on the fact that
$(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0)'\widehat{\mathbf{Q}}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0)-
2(\widehat{\boldsymbol{\gamma}}-\boldsymbol{\gamma})'(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0)=
\sum_{i\in\mathcal{E}}[(\widehat{\boldsymbol{\beta}}^{[i]}-\boldsymbol{\beta}_0^{[i]})'\widehat{\mathbf{Q}}^{[i]}(\widehat{\boldsymbol{\beta}}^{[i]}-\boldsymbol{\beta}_0^{[i]})-
2(\widehat{\boldsymbol{\gamma}}^{[i]}-\boldsymbol{\gamma}^{[i]})'(\widehat{\boldsymbol{\beta}}^{[i]}-\boldsymbol{\beta}_0^{[i]})]\leq 0$
and $\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0\in\Delta=\times_{i\in\mathcal{E}}\Delta^{[i]}$, which are immediately implied by \eqref{eq: basic inequality for each subvector}. In this sense, the restriction \eqref{eq: basic inequality for each subvector} is stricter, making the alternative bounds in \eqref{eq: bound on in-sample error, special case} tighter (or at least no looser) than those in (6.5) in the main text. Table \ref{tab: prediction intervals length reduction} quantifies the gains from this restriction in terms of the length of the prediction intervals in our leading empirical application.
\begin{table}[!ht]
\centering
\caption{Achieved reduction in the prediction intervals length with separate optimization.}
\resizebox{.95\textwidth}{!}{\begin{tabular}{ccccccccc}
\toprule
\toprule
$\mathcal{W}$ & \multicolumn{2}{c}{\textit{Ridge}} & & \multicolumn{2}{c}{\textit{Simplex}} & & \multicolumn{2}{c}{\textit{L1-L2}} \\
\cmidrule{2-3}\cmidrule{5-6}\cmidrule{8-9}\multicolumn{1}{l}{Predictand} & $M=1$ & $M=2$ & & $M=1$ & $M=2$ & & $M=1$ & $M=2$ \\
\midrule
\multicolumn{1}{l}{\textit{All treated units }} & & & & & & & & \\
\multirow{2}[0]{*}{$\tau_{ik}$, Figure 3(a)} & 41.98 & 55.05 & & 59.68 & 60.63 & & 60.08 & 60.38 \\
& $[16.80;92.28]$ & $[33.15;80.68]$ & & $[17.70;93.56]$ & $[36.61;92.64]$ & & $[11.66;92.92]$ & $[37.44;92.97]$ \\
& & & & & & & & \\
\multirow{3}[0]{*}{$\tau_{i\cdot}$, Figure 4(a)} & 42.78 & 56.54 & & 61.32 & 61.13 & & 60.41 & 60.17 \\
& & & & & & & & \\
& $[21.08;87.87]$ & $[35.36;73.31]$ & & $[22.20;93.66]$ & $[40.56;81.27]$ & & $[15.28;91.55]$ & $[42.20;81.02]$ \\
& & & & & & & & \\
\multirow{2}[0]{*}{$\tau_{\cdot k}$, Figure 6(a)} & 94.21 & 94.09 & & 94.49 & 94.12 & & 95.03 & 94.33 \\
& $[94.06;94.24]$ & $[94.04;94.16]$ & & $[94.40;95]$ & $[94.07;94.25]$ & & $[94.77;95.11]$ & $[94.3;94.41]$ \\
& & & & & & & & \\
\multicolumn{1}{l}{\textit{Countries Liberalized Before 1987}} & & & & & & & & \\
\multirow{2}[0]{*}{$\tau_{\mathcal{Q}_1 k}$, Figure 5(a)} & 76.65 & 78.91 & & 78.04 & 78.04 & & 79.92 & 76.64 \\
& $[76.55;76.92]$ & $[77.61;79.84]$ & & $[77.64;78.92]$ & $[77.73;78.72]$ & & $[79.78;80.13]$ & $[76.45;77.03]$ \\
& & & & & & & & \\
\multicolumn{1}{l}{\textit{Countries Liberalized in 1987-1991}} & & & & & & & & \\
\multirow{2}[0]{*}{$\tau_{\mathcal{Q}_2 k}$, Figure 5(c)} & 84.86 & 84.76 & & 85.74 & 86.28 & & 85.51 & 85.12 \\
& $[84.47;85.30]$ & $[84.41;84.94]$ & & $[85.45;86.44]$ & $[86.08;86.54]$ & & $[84.86;85.92]$ & $[84.98;85.23]$ \\
& & & & & & & & \\
\multicolumn{1}{l}{\textit{Countries Liberalized After 1991}} & & & & & & & & \\
\multirow{2}[1]{*}{$\tau_{\mathcal{Q}_3 k}$, Figure 5(e)} & 76.65 & 78.91 & & 78.04 & 78.04 & & 79.92 & 76.64 \\
& $[76.55;76.92]$ & $[77.61;79.84]$ & & $[77.64;78.92]$ & $[77.73;78.72]$ & & $[79.78;80.13]$ & $[76.45;77.03]$ \\
\bottomrule
\bottomrule
\end{tabular}
}
\label{tab: prediction intervals length reduction}
\begin{center}
\parbox[1]{\textwidth}{\footnotesize \textit{Notes:} For each target predictand we report the median percentage change in the length of the prediction intervals, whereas in brackets we report the minimum and maximum change, respectively. The reported statistics are computed across horizons and/or treated units. We computed the percentage change of the prediction intervals without out-of-sample uncertainty.}
\end{center}
\end{table}
Due to the complexity of the feasibility set $\tilde{\mathcal{M}}_{\mathbf{G}}^\star$,
the alternative bounds \eqref{eq: bound on in-sample error, special case} may not be theoretically justified using the same argument for (6.5) in the main text. However, other techniques, such as strong approximations, can be employed to establish the validity of \eqref{eq: bound on in-sample error, special case}, albeit at the expense of additional technical complexity. We do not pursue further justification of \eqref{eq: bound on in-sample error, special case} and leave it for future research.
\section{Linear Factor Model Justification}\label{sec: alternative justification}
There are different ways to justify the synthetic control method, including the cointegrated system motivated by our empirical application. In this section, we briefly discuss an alternative justification that assumes the data are generated through a linear factor model:
\[
Y_{it}(\infty)=\lambda_t\mu_i + \nu_{it}, \qquad 1\leq i\leq N,\; 1\leq t\leq T.
\]
For simplicity, assume that only the first unit is treated, the common factor $\lambda_t$ is i.i.d. over $1\leq t\leq T$, $\nu_{it}$ is i.i.d. across $1\leq i\leq N$ and over $1\leq t\leq T$, and $\{\lambda_t:1\leq t\leq T\}$ and $\{\nu_{it}:1\leq i\leq N, 1\leq t\leq T\}$ are independent of each other.
In addition, we assume that $\mathbb{E}[\nu_{it}]=0$, the factor loadings $\{\mu_i:1\leq i\leq N\}$ are fixed, and only the pre-intervention outcomes are used in the SC construction.
Then, $\mathscr{H}=\{(Y_{2t}(\infty), \cdots, Y_{Nt}(\infty)): 1\leq t\leq T\}$. Accordingly, $\mathbf{w}_0$ is given by the following expression:
\[
\mathbf{w}_0=\underset{\mathbf{w}\in\mathcal{W}}{\operatorname*{arg\,min}}\; \frac{1}{T_0}\sum_{t=1}^{T_0}\mathbb{E}\Big[\Big((\mu_1-\boldsymbol{\mu}_c'\mathbf{w})\lambda_t+(\nu_{1t}-\bm{\nu}_{t,c}'\mathbf{w})\Big)^2\Big|\mathscr{H}\Big],
\]
where $\boldsymbol{\mu}_c=(\mu_2, \cdots, \mu_N)'$ and $\bm{\nu}_{t,c}=(\nu_{2t}, \cdots, \nu_{Nt})'$, assuming the expectation (exists and) is finite.
Define $\mathscr{H}_t=\{Y_{2t}(\infty), \cdots, Y_{Nt}(\infty)\}$.
Then, $\mathbf{w}_0$ can be further written as
\begin{align*}
\mathbf{w}_0=\underset{\mathbf{w}\in\mathcal{W}}{\operatorname*{arg\,min}}\; \Big\{
&\frac{1}{T_0}\sum_{t=1}^{T_0}\mathbb{E}[\lambda_t^2|\mathscr{H}_t](\mu_1-\boldsymbol{\mu}_c'\mathbf{w})^2 + \mathbb{E}[\nu_{11}^2]+\mathbf{w}'\Big(\frac{1}{T_0}\sum_{t=1}^{T_0}\mathbb{E}[\bm{\nu}_{t,c}\bm{\nu}_{t,c}'|\mathscr{H}_t]\Big)\mathbf{w}\\
&\qquad - 2(\mu_1-\boldsymbol{\mu}_c'\mathbf{w})\Big(\frac{1}{T_0}\sum_{t=1}^{T_0}\mathbb{E}[
\lambda_t\bm{\nu}_{t,c}'|\mathscr{H}_t]\Big)\mathbf{w}
\Big\},
\end{align*}
assuming these expectations are finite. Given our assumptions on $\lambda_t$ and $\nu_{it}$, we expect that
$$\frac{1}{T_0}\sum_{t=1}^{T_0}\mathbb{E}[\lambda_t^2|\mathscr{H}_t] \approx \mathbb{E}[\lambda_t^2], \quad
\frac{1}{T_0}\sum_{t=1}^{T_0}\mathbb{E}[\bm{\nu}_{t,c}\bm{\nu}_{t,c}'|\mathscr{H}_t] \approx \mathbb{E}[\nu_{1t}^2]\mathbf{I}_{N-1}, \quad\text{and}\quad
\frac{1}{T_0}\sum_{t=1}^{T_0}\mathbb{E}[\lambda_t\bm{\nu}_{t,c}'|\mathscr{H}_t]\approx \bm{0}
$$
with high probability when $T_0$ is large, which can be shown under additional moment conditions.
As a consequence, our expression for $\mathbf{w}_0$ is similar to that in \cite{Ferman-Pinto_2021_wp}.
Then, the in-sample error for the prediction of the TSUS effect $\tau_{1k}$ is given by
\[
\mathsf{InErr}(\tau_{1k})=-\mathbf{Y}_{\mathcal{N}t}'(\widehat{\mathbf{w}}-\mathbf{w}_0),
\]
which represents the error in estimating the weights. The out-of-sample error is given by
\[
\mathsf{OutErr}(\tau_{1k})=Y_{1(T_1+k)}-\mathbf{Y}_{\mathcal{N}(T_1+k)}'\mathbf{w}_0=
\nu_{1(T_1+k)}-\bm{\nu}_{T_1+k,c}'\mathbf{w}_0+
(\mu_1-\boldsymbol{\mu}_c'\mathbf{w}_0)\lambda_{T_1+k}.
\]
The first term $\nu_{1(T_1+k)}$ represents the treated unit's innovation in the post-treatment period $T_1+k$;
the second term $\bm{\nu}_{T_1+k,c}'\mathbf{w}_0$ represents the weighted average of the innovations for the donor units in the post-treatment period $T_1+k$;
and the third term $(\mu_1-\boldsymbol{\mu}_c'\mathbf{w}_0)\lambda_{T_1+k}$ can be thought of as the impact of the ``bias'' of the SC weights in $T_1+k$, if one thinks the target weight in this context is the one recovering the true factor loading $\mu_1$ of the treated unit. Given the conditions imposed, our method is still applicable to this scenario, and importantly, the ``bias'' due to $(\mu_1-\boldsymbol{\mu}_c'\mathbf{w}_0)\lambda_{T_1+k}$ is taken into account.
\section{Proofs}
\subsection{Proof of Corollary 1}
\begin{proof}
We only need to verify the conditions in Theorem 1. The proof is divided into several steps.
\textbf{Step 0:} We first give several useful facts. The SC construction considered in this corollary is a special case of that considered in Theorem 1, where the weighting matrix is an identity matrix, and there is only one feature to be matched (GDP per capita) and no additional covariates.
As shown in the main paper, for any $\widehat\tau\in\{\widehat\tau_{ik}, \widehat\tau_{i\cdot}, \widehat\tau_{\mathcal{Q}k}, \widehat\tau_{\cdot\cdot}\}$, the in-sample error can always be expressed as $\mathbf{p}_\tau'(\widehat{\mathbf{w}}-\mathbf{w}_0)$.
Note that $\Delta^{[i]}$ and $\Delta^{[i]\star}$
are convex in this case, and thus $\times_{i\in\mathcal{S}}\Delta^{[i]}$ and $\times_{i\in\mathcal{S}}\Delta^{[i]\star}$ (for some set of treated units $\mathcal{S}$) are all convex as well.
For any $\kappa$, define $\mathcal{A}_\kappa=
\{\boldsymbol{\xi}: \sup_{\boldsymbol{\delta}\in\mathcal{M}_{\boldsymbol{\xi}}}\mathbf{p}_\tau'\boldsymbol{\delta}\leq\kappa\}$, which is convex by Lemma 2 of \cite{Cattaneo-Feng-Titiunik_2021_JASA}.
Let $d_\beta=J_0J_1$, and $\mathfrak{C}_1$, $\mathfrak{C}_2$, $\cdots$ denote some constant independent of $T_0$. Throughout this proof, $\|\cdot\|$ denotes the L2 norm for vectors and the operator norm for matrices.
\medskip
\textbf{Step 1:}
We want to verify condition (i) in Theorem 1.
Without loss of generality, we only consider the upper bound in this step, and the lower bound follows similarly. So our goal is to bound
$|\mathbb{P}(\widehat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}\in\mathcal{A}_\kappa|\mathscr{H})-\mathbb{P}(\mathbf{G}\in\mathcal{A}_\kappa|\mathscr{H})|$.
Note that by Assumption 1, $u_{it}$ is independent conditional on $\mathscr{H}$.
To simplify expressions, assume in this proof that $\boldsymbol{\gamma}=0$, and define $\tilde{u}_{it}=u_{it}$ if $t\leq T_i-1$ and $\tilde{u}_{it}=0$ if $t>T_i-1$. So we can write
$\widehat{\boldsymbol{\gamma}}=(\sum_{t=1}^{T_1-1}\mathbf{Y}_{\mathcal{N}t}'u_{1t}, \cdots, \sum_{t=1}^{T_{J_1}-1}\mathbf{Y}_{\mathcal{N}t}'u_{J_1t})'=
\sum_{t=1}^{T_{J_1}}(\mathbf{I}_{J_1}\otimes\mathbf{Y}_{\mathcal{N}t})\tilde{u}_t$ where
$\tilde{u}_t=(u_{1t}, \cdots, u_{J_1t})'$.
Applying the Berry-Esseen theorem for convex sets \citep{Raivc_2019_Bernoulli},
\[
|\mathbb{P}(\widehat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}\in\mathcal{A}_\kappa|\mathscr{H})-
\mathbb{P}(\mathbf{G}\in\mathcal{A}_\kappa|\mathscr{H})|
\leq 42(d_\beta^{1/4}+16)\sqrt{J_1}\|\boldsymbol{\Sigma}^{-\frac{1}{2}}\|^3\sum_{t=1}^{T_{J_1}} \mathbb{E}\Big[\sum_{i=1}^{J_1}\Big\|\mathbf{Y}_{\mathcal{N}t}u_{it}\Big\|^3\Big|\mathscr{H}\Big].
\]
Given condition (i) in the corollary, we only need to bound
$\sum_{t=1}^{T_{J_1}}
\|\mathbf{Y}_{\mathcal{N}t}\|^3$ and $\|\boldsymbol{\Sigma}^{-1}\|$.
First note that
\[
\frac{1}{T_0^3}\sum_{t=1}^{T_{J_1}}\|\mathbf{Y}_{\mathcal{N}t}\|^3
\leq\frac{\sqrt{J_0}}{T_0^{3/2}}\sum_{t=1}^{T_{J_1}}\sum_{j\in\mathcal{N}}|Y_{jt}/\sqrt{T_0}|^3,
\]
By Assumption 1, each $\mathbf{Y}_{\mathcal{N}t}$ can be understood as a multivariate partial sum process indexed by $t$.
By strong approximation of partial sum processes (e.g., \citealp[Lemma 2.2 of][]{Chang-Park-Song_2006_JoE}),
\[
\mathbb{P}\Big(\max_{1\leq t\leq T_0}
\|T_0^{-1/2}\mathbf{Y}_{\mathcal{N}t}-\tilde\mathbf{G}(t/T_0)\|
\geq T_0^{-0.1} \Big)\leq \mathfrak{C}_1T_0^{-1},
\]
where $\tilde\mathbf{G}(\cdot)$ is a $J_0$-dimensional Brownian motion on $[0,1]$ with the variance $\mathbb{E}[\mathbf{v}_t\mathbf{v}_t']$. On the other hand, it is well known that for each $1\leq j\leq J_0$, for any $m>0$,
\[
\mathbb{P}(\max_{0\leq r\leq 1}|\tilde{G}_{j}(r)|>m)\leq 2\mathbb{P}(\max_{0\leq r\leq 1}\tilde{G}_{j}(r)>m)=2\mathbb{P}(|G_{j}(1)|>m),
\]
where $\tilde{G}_{j}(\cdot)$ is the $j$-th element of $\mathbf{G}(\cdot)$.
Using the tail bound for Gaussian distributions, we can set $m=\sqrt{2\log (2J_0T_0)\sigma_{\max}^2}$ where $\sigma_{\max}^2$ is the largest variance of $\{v_{jt}:1\leq j\leq J_0\}$, which leads to
$\max_{0\leq r\leq 1}|G_{j}(r)|\leq m$
with probability over $\mathscr{H}$ at least $1-(J_0T_0)^{-1}$.
Therefore,
\[
\frac{1}{T_0^3}\sum_{t=1}^{T_0}\|\mathbf{Y}_{\mathcal{N}t}\|^3
\leq J_0^{3/2}T_0^{-1/2}\Big(\sqrt{2\log (2J_0T_0)\sigma^2_{\max}}+
T_0^{-0.1}\Big)^3
\]
with probability over $\mathscr{H}$ at least $1-T_0^{-1}-\mathfrak{C}_1T_0^{-1}$.
In addition, note that
$\frac{1}{T_0^3}\sum_{t=T_0+1}^{T_{J_1}}\|\mathbf{Y}_{\mathcal{N}t}\|^3\leq
\frac{4}{T_0^3}\sum_{t=T_0+1}^{T_{J_1}}(\|\mathbf{Y}_{\mathcal{N}T_0}\|^3+\|\sum_{s=T_0+1}^{t}\mathbf{v}_s\|^3)$. Then, by the previous result and sub-Gaussianity of $\mathbf{v}_t$, we conclude that with probability at least $1-\mathfrak{C}_\pi T_0^{-1}$ for some constant $\mathfrak{C}_\pi>0$,
\[
\frac{1}{T_0^3}\sum_{t=1}^{T_{J_1}}\|\mathbf{Y}_{\mathcal{N}t}\|^3
\leq J_0^{3/2}(T_0^{-1/2}+\mathfrak{C}_2T_0^{-3/2}+\mathfrak{C}_3)\Big(\sqrt{2\log (2J_0T_0)\sigma^2_{\max}}+
T_0^{-0.1}\Big)^3.
\]
Finally, we consider $\boldsymbol{\Sigma}$. By assumption in the corollary, $s_{\min}(\boldsymbol{\Sigma})\geq\eta\widehat{\mathbf{Q}}$. Recall that
$\widehat{\mathbf{Q}}=\operatorname*{diag}(\widehat{\mathbf{Q}}^{[1]}, \cdots, \widehat{\mathbf{Q}}^{[J_1]})$.
Again, by strong approximation used previously, with probability over $\mathscr{H}$ at least $1-\mathfrak{C}_\pi T_0^{-1}$,
$$
\bigg\|\frac{1}{T_0^2}\sum_{t=1}^{T_0}\mathbf{Y}_{\mathcal{N}t}\mathbf{Y}_{\mathcal{N}t}'-
\frac{1}{T_0}\sum_{t=1}^{T_0}\tilde\mathbf{G}\Big(\frac{t}{T_0}\Big)\tilde\mathbf{G}\Big(\frac{t}{T_0}\Big)'\bigg\|\leq
2\sqrt{J_0}T_0^{-0.1}(m+T_0^{-0.1}).
$$
Then, by the condition in the corollary, with probability over $\mathscr{H}$ at least $1-\pi_0-\mathfrak{C}_\pi T_0^{-1}$,
$s_{\min}(\widehat{\mathbf{Q}}^{[1]})\geq \frac{2}{3}(\log T_0)^{-1/5}T_0^2$ for $T_0$ large enough.
For other blocks $\widehat\mathbf{Q}^{[j]}$ with $j\neq 1$, using the previous results about the bound on $\mathbf{Y}_{\mathcal{N}t}$, we have with probability $1-\mathfrak{C}_\pi T_0^{-1}$,
$\|\widehat{\mathbf{Q}}^{[j]}-\widehat{\mathbf{Q}}^{[1]}\|\leq \mathfrak{C}_4(m+T_0^{-0.1})^2T_0$. Therefore, we can conclude that with probability over $\mathscr{H}$ at least $1-\pi_0-\mathfrak{C}_\pi T_0^{-1}$,
$s_{\min}(\widehat{\mathbf{Q}})\geq (\log T_0)^{-1/5}T_0^2/2$ for $T_0$ large enough.
Therefore, we can take $\pi_\gamma=\pi_0+\mathfrak{C}_\pi T_0^{-1}$ and $\epsilon_{\gamma}=\mathfrak{C}_\epsilon (\log T_0)^2T_0^{-1/2}$ for some non-negative finite constant $\mathfrak{C}_\epsilon$ implied by the previous calculations.
\medskip
\textbf{Step 2}: Consider condition (ii) in Theorem 1. We have the following basic inequality hold:
\[
\lambda_{\min}(\mathfrak{D}_T^{-1}\widehat{\mathbf{Q}}\mathfrak{D}_T^{-1})\|\mathfrak{D}_T\boldsymbol{\delta}\|^2\leq
\boldsymbol{\delta}\widehat{\mathbf{Q}}\boldsymbol{\delta}\leq
2\mathbf{G}'\boldsymbol{\delta}\leq
2\|\mathfrak{D}_T^{-1}\mathbf{G}\|\|\mathfrak{D}_T\boldsymbol{\delta}\|.
\]
where $\mathfrak{D}_T=\operatorname*{diag}((T_1-1)\mathbf{I}_{T_1-1}, \cdots, (T_{J_1}-1)\mathbf{I}_{T_{J_1}-1})$.
By condition (ii) imposed in the corollary and the argument given in Step 1,
$s_{\min}(\mathfrak{D}_T^{-1}\widehat{\mathbf{Q}}\mathfrak{D}_T^{-1})\geq(\log T_0)^{-1/5}/2$ and
$\lambda_{\max}(\mathfrak{D}_T^{-1}\widehat{\mathbf{Q}}\mathfrak{D}_T^{-1})\leq 2(\log T_0)^{1/5}$, with probability over $\mathscr{H}$ at least $1-\pi_\gamma$.
Then, by the Gaussian tail bound, we can take
$\varpi_{\delta}^{\star}=\mathfrak{C}_5(\log T_0)^{0.9}/T_0$,
$\pi_\delta^{\star}=\pi_\gamma$, and
$\epsilon_{\delta}^{\star}=T_0^{-1}$.
\medskip
\textbf{Step 3:} Consider condition (iii) in Theorem 1. Given the specific choices of $\varrho_\ell^{[i]}$'s and the argument in Step 2, the conditions in Lemma 1 are satisfied for $T_0$ large enough, and we can differentiate the binding and nonbinding constraints with high probability. For each treated unit $i$, an L1-L2 constraint is imposed on the corresponding SC weights. The set $\Delta^{[i]}$ then can be written as $\Delta^{[i]}=\Delta_{\tt l}^{[i]}\cap \Delta_{\tt nl}^{[i]}$, where $\Delta_{\tt l}^{[i]}$ and $\Delta_{\tt nl}^{[i]}$ denote the feasibility sets defined by the L1 (simplex) and L2 (ridge) constraints respectively. Similarly, $\widehat{\Delta}$ in (6.7) for this special case can be written as
$\widehat{\Delta}^{[i]}=\widehat{\Delta}_{\tt l}^{[i]}\cap \widehat{\Delta}_{\tt nl}^{[i]}$. The L1 constraint is linear, and thus by Lemma 1, $\Delta^{[i]}_{\tt l}\cap\mathcal{B}(\bm{0},\varpi_\delta^\star)\subseteq\widehat{\Delta}_{\tt l}^{[i]}$. For the L2 constraint, Lemma 1 implies that $\Delta^{[i]}_{\tt nl}\cap\mathcal{B}(\bm{0},\varpi_\delta^\star)\subseteq\{\boldsymbol{\delta}:\mathrm{dist}(\boldsymbol{\delta}, \widehat{\Delta}_{\tt nl}^{[i]}\leq \mathfrak{C}(\varpi_\delta^\star)^2)\}$ for some constant $\mathfrak{C}>0$.
Then, by the specified adjustment for the L2 constraint, condition (iii) holds for every $\Delta^{[i]\star}$ as well as $\times_{i\in\mathcal{S}}\Delta^{[i]\star}$, for $\mathcal{S}=\mathcal{Q}$ or $\mathcal{E}$. In this case, we can set
$\pi_\Delta^\star=\pi_\gamma$ and $\epsilon_\Delta^\star=T_0^{-1}+\epsilon_\gamma$.
\medskip
\textbf{Step 4:}
Finally, consider condition (iv) in Theorem 1. Note that
\[
\operatorname*{tr}\Big[(\boldsymbol{\Sigma}^{-1/2}\widehat{\boldsymbol{\Sigma}}\boldsymbol{\Sigma}^{-1/2}-\mathbf{I}_{d_\beta})^2\Big]
\leq d_\beta s_{\min}(\boldsymbol{\Sigma})^{-2}\|\widehat{\boldsymbol{\Sigma}}-\boldsymbol{\Sigma}\|^2.
\]
Since $\lambda_{\min}(\boldsymbol{\Sigma})\geq \eta\lambda_{\min}(\widehat{\mathbf{Q}})\geq\eta(\log T_0)^{-1/5}T_0^2/2$, with probability over $\mathscr{H}$ at least $1-\pi_{\delta}^{\star}$,
it follows that $\pi_\gamma^\star=\pi_\gamma$, $\epsilon_{\gamma,2}^\star=\epsilon_{\Sigma,2}^\star$ and $\epsilon_{\gamma,1}^\star=2\sqrt{d_\beta}\epsilon_{\Sigma,1}^\star(\log T_0)^{1/5}/(T_0^2\eta)$.
\medskip
Given all results above, we finally set $\pi=\pi_\gamma$ and $\epsilon=2\mathfrak{C}_\epsilon (\log T_0)^2T_0^{-1/2}
+4\sqrt{d_\beta}\epsilon_{\Sigma,1}^\star(\log T_0)^{1/5}/(T_0^2\eta)
+\epsilon_{\Sigma,2}^\star+3T_0^{-1}$. Then, the proof is complete.
\end{proof}
\subsection{Proof of Theorem 1}
\begin{proof}
Let
\begin{align*}
&\ell(\boldsymbol{\delta})=\boldsymbol{\delta}'\widehat{\mathbf{Q}}\boldsymbol{\delta}-2\mathbf{G}'\boldsymbol{\delta}\quad \text{with }\mathbf{G}|\mathscr{H}\sim\mathsf{N}(\bm{0}, \boldsymbol{\Sigma}),\\
&\ell^\star(\boldsymbol{\delta})=\boldsymbol{\delta}'\widehat{\mathbf{Q}}\boldsymbol{\delta}-2(\mathbf{G}^\star)'\boldsymbol{\delta}\quad \text{with }\mathbf{G}^\star|\mathsf{Data}\sim\mathsf{N}(\bm{0}, \widehat\boldsymbol{\Sigma}).
\end{align*}
Accordingly, define
\[
\begin{split}
&\bar{\varsigma}^{\star}=\sup\Big\{-\mathbf{p}_{\tau}'\boldsymbol{\delta}: \boldsymbol{\delta}\in\Delta^{\star},\,\ell^{\star}(\boldsymbol{\delta})\leq 0\Big\}\\
&\bar{\varsigma}_\mathtt{r}^{\mathsf{int}}=\sup\Big\{-\mathbf{p}_{\tau}'\boldsymbol{\delta}: \boldsymbol{\delta}\in\Delta,\,\|\boldsymbol{\delta}\|\leq \varpi_{\delta}^{\star},\,
\ell^{\star}(\boldsymbol{\delta})\leq 0 \Big\},\\
&\bar{\varsigma}_\mathtt{r}=\sup\Big\{-\mathbf{p}_{\tau}'\boldsymbol{\delta}: \boldsymbol{\delta}\in\Delta,\,\|\boldsymbol{\delta}\|\leq \varpi_{\delta}^{\star},\,
\ell(\boldsymbol{\delta})\leq 0 \Big\}.
\end{split}
\]
The subscript $\mathtt{r}$ indicates the quantity is a supremum over a further restricted region for $\boldsymbol{\delta}$ (due to the constraint $\|\boldsymbol{\delta}\|\leq \varpi_\delta^\star$).
For any $\alpha_0\in[0,1]$, let
$\bar{\mathfrak{c}}_{\mathtt{r}}(\alpha_0)$ be the $\alpha_0$-quantile of $\bar{\varsigma}_{\mathtt{r}}$ conditional on $\mathscr{H}$.
Similarly, define
\[
\begin{split}
&\underline{\varsigma}^{\star}:=\inf\Big\{-\mathbf{p}_{\tau}'\boldsymbol{\delta}: \boldsymbol{\delta}\in\Delta^{\star},\,\ell^{\star}(\boldsymbol{\delta})\leq 0\Big\}\\
&\underline{\varsigma}_\mathtt{r}^{\mathsf{int}}=\inf\Big\{-\mathbf{p}_{\tau}'\boldsymbol{\delta}: \boldsymbol{\delta}\in\Delta,\,\|\boldsymbol{\delta}\|\leq \varpi_{\delta}^{\star},\,
\ell^{\star}(\boldsymbol{\delta})\leq 0 \Big\},\\
&\underline{\varsigma}_\mathtt{r}=\inf\Big\{-\mathbf{p}_{\tau}'\boldsymbol{\delta}: \boldsymbol{\delta}\in\Delta,\,\|\boldsymbol{\delta}\|\leq \varpi_{\delta}^{\star},\,
\ell(\boldsymbol{\delta})\leq 0 \Big\}.
\end{split}
\]
Let
$\underline{\mathfrak{c}}_{\mathtt{r}}(\alpha_0)$ be the $\alpha_0$-quantile of $\underline{\varsigma}_{\mathtt{r}}$ conditional on $\mathscr{H}$.
Let $\mathbb{P}_1=\mathsf{N}(\bm{0},\boldsymbol{\Sigma})$ and $\mathbb{P}_2=\mathsf{N}(\bm{0},\widehat{\boldsymbol{\Sigma}})$. By condition (iv), on an event with $\mathbb{P}$-probability at least $1-\pi_\gamma^\star$,
with $\mathbb{P}(\cdot|\mathscr{H})$-probability at least $1-\epsilon_{\gamma,2}^\star$, the Kullback-Leibler divergence
$\mathbb{KL}(\mathbb{P}_1, \mathbb{P}_2)\leq 2(\epsilon_{\gamma,1}^\star)^2$,
and by Pinsker’s inequality, this implies that for any
$\kappa'\leq \kappa$,
\[
|\mathbb{P}^{\star}(\bar{\varsigma}_{\mathtt{r}}^{\mathsf{int}}\leq \kappa)-\mathbb{P}^{\star}(\bar{\varsigma}_{\mathtt{r}}
\leq\kappa)|
\leq \epsilon_{\gamma,1}^{\star}\quad\text{and}\quad
|\mathbb{P}^{\star}(\underline{\varsigma}_{\mathtt{r}}^{\mathsf{int}}\geq \kappa')-\mathbb{P}^{\star}(\underline{\varsigma}_{\mathtt{r}}
\geq\kappa')|
\leq \epsilon_{\gamma,1}^{\star}.
\]
On the other hand, note that by condition (iii), on an event with $\mathbb{P}$-probability over $\mathscr{H}$ at least $1-\pi_\Delta^\star$, with $\mathbb{P}(\cdot|\mathscr{H})$-probability at least $1-\epsilon_\Delta^\star$, the event
$\{\bar{\varsigma}^{\star}\leq\kappa\}$
implies that
\[
\sup\Big\{-\mathbf{p}_{\tau}'\boldsymbol{\delta}: \boldsymbol{\delta}\in\Delta,\,\|\boldsymbol{\delta}\|\leq \varpi_{\delta}^{\star},\,
\ell^{\star}(\boldsymbol{\delta})\leq 0 \Big\}\leq
\sup\Big\{-\mathbf{p}_{\tau}'\boldsymbol{\delta}: \boldsymbol{\delta}\in\Delta^{\star},\,
\ell^{\star}(\boldsymbol{\delta})\leq 0 \Big\}\leq\kappa
\]
and
$\{\underline{\varsigma}^{\star}\geq \kappa'\}$
implies that
\[
\inf\Big\{-\mathbf{p}_{\tau}'\boldsymbol{\delta}: \boldsymbol{\delta}\in\Delta,\,\|\boldsymbol{\delta}\|\leq \varpi_{\delta}^{\star},\,
\ell^{\star}(\boldsymbol{\delta})\leq 0 \Big\}\geq
\inf\Big\{-\mathbf{p}_{\tau}'\boldsymbol{\delta}: \boldsymbol{\delta}\in\Delta^\star,\,
\ell^{\star}(\boldsymbol{\delta})\leq 0 \Big\}
\geq \kappa'.
\]
Therefore,
\[
\mathbb{P}^{\star}(\bar{\varsigma}^{\star}\leq \kappa)\leq
\mathbb{P}^{\star}(\bar{\varsigma}_{\mathtt{r}}^{\mathsf{int}}\leq \kappa)\quad \text{and}\quad
\mathbb{P}^{\star}(\underline{\varsigma}^{\star}\geq \kappa')\leq
\mathbb{P}^{\star}(\underline{\varsigma}_{\mathtt{r}}^{\mathsf{int}}\geq \kappa').
\]
Then, by definitions of $\bar{\varsigma}^{\star}$ and $\underline{\varsigma}^{\star}$, on an event with $\mathbb{P}$-probability over $\mathscr{H}$ at least $1-\pi_\gamma^\star-\pi_\Delta^\star$, with $\mathbb{P}(\cdot|\mathscr{H})$-probability at least $1-\epsilon_{\gamma,2}^\star-\epsilon_\Delta^\star$, we have
\begin{align*}
&1-\alpha_{\mathtt{in}}/2\leq \mathbb{P}^{\star}\Big(
\bar{\varsigma}^{\star}\leq
\overline{\mkern-3muM\mkern-1.5mu}_{\mathtt{in}}(\tau)\Big)
\leq
\mathbb{P}^\star\Big(
\bar{\varsigma}_{\mathtt{r}}\leq
\overline{\mkern-3muM\mkern-1.5mu}_{\mathtt{in}}(\tau)\Big)+
\epsilon_{\gamma,1}^\star\quad \text{and}\\
&1-\alpha_{\mathtt{in}}/2\leq \mathbb{P}^{\star}\Big(
\underline{\varsigma}^{\star}\geq \underline{M\mkern-4mu}\mkern4mu _{\mathtt{in}}(\tau)\Big)
\leq
\mathbb{P}^\star\Big(
\underline{\varsigma}_{\mathtt{r}}\geq \underline{M\mkern-4mu}\mkern4mu _{\mathtt{in}}(\tau)\Big)+
\epsilon_{\gamma,1}^\star.
\end{align*}
Also, by condition (ii), we have with $\mathbb{P}$-probability over $\mathscr{H}$ at least $1-\pi_\delta^\star$,
$$
\bar{\mathfrak{c}}_{\mathtt{r}}(1-\alpha_{\mathtt{in}}/2-\epsilon_{\gamma,1}^\star)\geq
\bar{\mathfrak{c}}(1-\alpha_{\mathtt{in}}/2-\epsilon_{\gamma,1}^\star-\epsilon_\delta^\star)\quad\text{and}\quad
\underline{\mathfrak{c}}_{\mathtt{r}}(\alpha_{\mathtt{in}}/2+\epsilon_{\gamma,1}^\star)\leq
\underline{\mathfrak{c}}(\alpha_{\mathtt{in}}/2+\epsilon_{\gamma,1}^\star+\epsilon_\delta^\star).
$$
Using condition (i) and all results above, we conclude that with $\mathbb{P}$-probability over $\mathscr{H}$ at least $1-\pi_\gamma-\pi_\gamma^\star-\pi_\Delta^\star-\pi_\delta^\star$,
\[
\begin{split}
&\mathbb{P}\Big(
\underline{M\mkern-4mu}\mkern4mu _{\mathtt{in}}(\tau)
\leq
-\mathbf{p}_{\tau}'(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0)
\leq \overline{\mkern-3muM\mkern-1.5mu}_{\mathtt{in}}(\tau)\Big|\mathscr{H}\Big)\\
\geq&\, \mathbb{P}\Big(
\underline{\mathfrak{c}}_{\mathtt{r}}(\alpha_{\mathtt{in}}/2+\epsilon_{\gamma,1}^{\star})
\leq-\mathbf{p}_\tau'(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0)\leq \bar{\mathfrak{c}}_{\mathtt{r}}(1-\alpha_{\mathtt{in}}/2-\epsilon_{\gamma,1}^{\star})\Big|\mathscr{H}\Big)-\epsilon_{\gamma,2}^{\star}-\epsilon_{\Delta}^{\star}\\
\geq&\, \mathbb{P}\Big(
\underline{\mathfrak{c}}(\alpha_{\mathtt{in}}/2+\epsilon_{\gamma,1}^{\star}+
\epsilon_{\delta}^{\star})\leq
-\mathbf{p}_\tau'(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0)\leq
\bar{\mathfrak{c}}(1-\alpha_{\mathtt{in}}/2-\epsilon_{\gamma,1}^{\star}-\epsilon_{\delta}^{\star})\Big|\mathscr{H}\Big)-\epsilon_{\gamma,2}^{\star}-\epsilon_{\Delta}^{\star}\\
\geq&\, 1-\alpha_{\mathtt{in}}-2\epsilon_{\gamma,1}^{\star}-2\epsilon_{\delta}^{\star}-\epsilon_{\gamma}-\epsilon_{\gamma,2}^{\star}-\epsilon_{\Delta}^{\star}.
\end{split}
\]
Finally, by condition (v), we immediately have
$\mathbb{P}(\underline{M\mkern-4mu}\mkern4mu _{\mathtt{out}}(\tau)\leq \mathsf{OutErr}(\tau)\leq \overline{\mkern-3muM\mkern-1.5mu}_{\mathtt{out}}(\tau))\geq 1-\alpha_{\mathtt{out}}$.
Then the proof is complete.
\end{proof}
\subsection{Verification of Condition (i) in Theorem 1}
\label{sec: verify condition i}
As explained in the main paper, by convexity of the constraint set $\mathcal{W}\times\mathcal{R}$ and the optimality of $\widehat{\boldsymbol{\beta}}$,
$$
\inf_{\boldsymbol{\delta}\in\mathcal{M}_{\widehat\boldsymbol{\gamma}-\boldsymbol{\gamma}}}-\mathbf{p}_\tau'\boldsymbol{\delta}
\leq -\mathbf{p}_\tau'(\widehat\boldsymbol{\beta}-\boldsymbol{\beta}_0)\leq
\sup_{\boldsymbol{\delta}\in\mathcal{M}_{\widehat\boldsymbol{\gamma}-\boldsymbol{\gamma}}}-\mathbf{p}_\tau'\boldsymbol{\delta},
$$
where $\mathcal{M}_{\widehat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}}=
\{\boldsymbol{\delta}\in\Delta:\boldsymbol{\delta}'\widehat{\mathbf{Q}}\boldsymbol{\delta}-
2(\widehat{\boldsymbol{\gamma}}-\boldsymbol{\gamma})'\boldsymbol{\delta}\}$. Thus,
condition (i) in Theorem 1 indeed requires that
$\widehat\boldsymbol{\gamma}-\boldsymbol{\gamma}$ can be approximated by a Gaussian vector $\mathbf{G}$.
Corollary 1 in the paper provided a verification of this condition in the special case of cointegrated data. In this section, we provide a more general way to verify condition (i)
by imposing a conditional independence assumption on the pseudo-true residuals. The extension that allows for weakly dependent errors can be established using the idea of Theorem A in \cite{Cattaneo-Feng-Titiunik_2021_JASA}. For simplicity, we assume that only $T_0$ pre-treatment periods are used to obtain the weights. Also, we write $\mathbf{U}^{[i]}=(u_{it,1}, \cdots, u_{it,M})'$, which is the vector of pseudo-true residuals corresponding to the treated unit $i$.
\begin{lem}\label{lem: gaussian approximation}
Assume $\mathcal{W}$ and $\mathcal{R}$ are convex, $\widehat{\boldsymbol{\beta}}$ in Equation (6.1) and
$\boldsymbol{\beta}_0$ in Equation (6.3) exist, and $\mathscr{H}=\sigma(\mathbf{B}, \mathbf{C}, \mathbf{p}_\tau)$. In addition,
for some finite nonnegative constants, the following conditions hold:
\begin{enumerate}[label=(\roman*)]
\item $\mathbf{u}_t=(u_{1t,1}, \cdots, u_{1t,M}, \cdots, u_{J_1t,1}, \cdots, u_{J_1t,M})$ is independent over $t$ conditional on $\mathscr{H}$;
\item $\mathbb{P}(\sum_{t=1}^{T_0}\mathbb{E}[\|\sum_{j=1}^{J_1}\sum_{l=1}^M\tilde{\mathbf{z}}_{t,l}^{[j]}
(u_{jt,l}-\mathbb{E}[u_{jt,l}|\mathscr{H}])\|_2^3|\mathscr{H}]\geq \epsilon_\gamma(84(d_\beta^{1/4}+16))^{-1})\geq 1-\pi_\gamma$ where
$\tilde{\mathbf{z}}_{t,l}^{[j]}$ is the $((j-1)T_0M+(l-1)T_0+t)$-th column of $\boldsymbol{\Sigma}^{-1/2}\mathbf{Z}'$.
\end{enumerate}
Then, with $\mathbb{P}$-probability over $\mathscr{H}$ at least $1-\pi_\gamma$,
\[
\mathbb{P}\Big(\underline{\mathfrak{c}}(\alpha_0)\leq
-\mathbf{p}_\tau'(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0)\leq \bar{\mathfrak{c}}(1-\alpha_0)|\mathscr{H}\Big)
\geq 1-2\alpha_0-\epsilon_\gamma.
\]
\end{lem}
\begin{proof}
Define $\mathcal{M}_{\boldsymbol{\xi}}=\{\boldsymbol{\delta}\in\Delta:\boldsymbol{\delta}'\widehat{\mathbf{Q}}\boldsymbol{\delta}-2\boldsymbol{\xi}'\boldsymbol{\delta}\}$.
Fix $\widehat{\mathbf{Q}}$ and $\mathbf{p}_\tau$.
By Lemma 2 of \cite{Cattaneo-Feng-Titiunik_2021_JASA}, for any $\kappa$,
$\mathcal{A}_\kappa:=\{\boldsymbol{\xi}\in\mathbb{R}^{d_\beta}:\sup_{\boldsymbol{\delta}\in\mathcal{M}_\xi}-\mathbf{p}_\tau'\boldsymbol{\delta}
\leq \kappa\}$ and
$\mathcal{A}_\kappa'=\{\boldsymbol{\xi}\in\mathbb{R}^{d_\beta}:\inf_{\boldsymbol{\delta}\in\mathcal{M}_\xi}-\mathbf{p}_\tau'\boldsymbol{\delta}
\geq \kappa\}$ are convex.
By Berry-Esseen Theorem for convex sets \cite{Raivc_2019_Bernoulli},
\[
|\mathbb{P}(\widehat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}\in\mathcal{A}_\kappa|\mathscr{H})-
\mathbb{P}(\mathbf{G}\in\mathcal{A}_\kappa|\mathscr{H})|
\leq 42(d_\beta^{1/4}+16)\sum_{t=1}^{T_0}\mathbb{E}\Big[\Big\|\sum_{j=1}^{J_1}\sum_{l=1}^M
\tilde{\mathbf{z}}_{t,l}^{[j]}\tilde{u}_{jt,l}\Big\|^3|\mathscr{H}\Big],
\]
where $\tilde{u}_{jt,l}=u_{jt,l}-\mathbb{E}[u_{jt,l}|\mathscr{H}]$.
By condition (ii), with $\mathbb{P}$-probability over $\mathscr{H}$ at least $1-\pi_\gamma$,
\[
|\mathbb{P}(\widehat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}\in\mathcal{A}_\kappa|\mathscr{H})-
\mathbb{P}(\mathbf{G}\in\mathcal{A}_\kappa|\mathscr{H})|\leq \epsilon_\gamma/2.
\]
Then, for any $\kappa$, with $\mathbb{P}$-probability over $\mathscr{H}$ at least $1-\pi_\gamma$,
\[
\mathbb{P}(-\mathbf{p}_\tau'(\widehat\boldsymbol{\beta}-\boldsymbol{\beta}_0)\leq \kappa|\mathscr{H})\geq
\mathbb{P}(\widehat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}\in\mathcal{A}_\kappa|\mathscr{H})
\geq\mathbb{P}(\mathbf{G}\in\mathcal{A}_\kappa|\mathscr{H})-\epsilon_\gamma/2.
\]
Similarly, we can show for any $\kappa$,
\[
\mathbb{P}(-\mathbf{p}_\tau'(\widehat\boldsymbol{\beta}-\boldsymbol{\beta}_0)\geq \kappa|\mathscr{H})\geq
\mathbb{P}(\widehat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}\in\mathcal{A}_\kappa'|\mathscr{H})
\geq\mathbb{P}(\mathbf{G}\in\mathcal{A}_\kappa'|\mathscr{H})-\epsilon_\gamma/2.
\]
Therefore, with $\mathbb{P}$-probability over $\mathscr{H}$ at least $1-\pi_\gamma$,
\[
\mathbb{P}\Big(\underline{\mathfrak{c}}(\alpha_0)\leq
-\mathbf{p}_\tau'(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0)\leq \bar{\mathfrak{c}}(1-\alpha_0)\Big|\mathscr{H}\Big)
\geq 1-2\alpha_0-\epsilon_\gamma.
\]
Then the proof is complete.
\end{proof}
\subsection{Proof of Lemma 1}
\begin{proof}
In this proof, the constant $\mathfrak{C}>0$ is a generic constant that is independent of $T_0$ and may be different in different uses.
Note that $\bm{m}_{=}(\boldsymbol{\beta}_{0})=\bm{0}$ and $\bm{m}_{\leq}(\boldsymbol{\beta}_{0})\leq \bm{0}$.
For
$\mathfrak{c}:=\max_{1\leq\ell\leq d_{\leq}}\sup_{\boldsymbol{\beta}\in\mathcal{B}(\boldsymbol{\beta}_0,\varpi_\delta^\star)}\|\frac{\partial}{\partial\boldsymbol{\beta}}m_{\leq,\ell}(\boldsymbol{\beta})\|$,
we have $\max_{1\leq\ell\leq d_{\leq}}
|m_{\leq,\ell}(\widehat\boldsymbol{\beta})-m_{\leq,\ell}(\boldsymbol{\beta}_0)|\leq \mathfrak{c}\varpi_{\delta}^\star$ with $\mathbb{P}(\cdot|\mathscr{H})$-probability at least $1-\epsilon_{\Delta}^\star$, on an event with $\mathbb{P}$-probability over $\mathscr{H}$ at least $1-\pi_\Delta^\star$.
Note that if the $\ell$-th inequality constraint is binding, i.e., $m_{\leq,\ell}(\boldsymbol{\beta}_0)=0$, then
$m_{\leq,\ell}(\widehat{\boldsymbol{\beta}})=\frac{\partial}{\partial\boldsymbol{\beta}'}m_{\leq,\ell}(\tilde\boldsymbol{\beta})(\widehat\boldsymbol{\beta}-\boldsymbol{\beta}_0)$ for some $\tilde{\boldsymbol{\beta}}$ between $\boldsymbol{\beta}_0$ and $\widehat\boldsymbol{\beta}$.
By the condition imposed on the tuning parameters $\varrho_\ell$'s, on an event with $\mathbb{P}$-probability over $\mathscr{H}$ at least $1-\pi_\Delta^\star$, with $\mathbb{P}(\cdot|\mathscr{H})$-probability at least $1-\epsilon_\Delta^\star$, $\mathcal{A}$ coincides with the set of indices for the binding inequality constraints.
Without loss of generality, we assume $\bm{m}_{\leq}(\boldsymbol{\beta}_0)=\bm{0}$ hereafter. Otherwise, the non-binding constraints can be dropped, and the proof can proceed the same way as described below.
Define
$\Gamma_{=}(\boldsymbol{\beta})=\frac{\partial}{\partial \boldsymbol{\beta}'}\bm{m}_{=}(\boldsymbol{\beta})$ and $\Gamma_{\leq}(\boldsymbol{\beta})=\frac{\partial}{\partial \boldsymbol{\beta}'}\bm{m}_{\leq}(\boldsymbol{\beta})$. Let
\[
\Gamma(\boldsymbol{\beta})=\Big(\Gamma_{=}'(\boldsymbol{\beta}), \Gamma_{\leq}'(\boldsymbol{\beta}),\Gamma_{c}'(\boldsymbol{\beta}_0)\Big)',\quad
\Gamma^0=\Gamma(\boldsymbol{\beta}_0),\quad \Gamma^\star=\Gamma(\widehat\boldsymbol{\beta}),
\]
where $\Gamma_c(\boldsymbol{\beta}_0)$ is chosen such that $\Gamma(\boldsymbol{\beta}_0)$ is non-degenerate.
By conditions (i) and (ii) imposed in the lemma, $\|\Gamma^0-\Gamma^\star\|\leq \mathfrak{C}\|\widehat\boldsymbol{\beta}-\boldsymbol{\beta}_0\|$ with $\mathbb{P}(\cdot|\mathscr{H})$-probability at least $1-\epsilon_\Delta^\star$, on an event with $\mathbb{P}$-probability at least $1-\pi_\Delta^\star$.
Let
$$\bm{m}^+(\cdot)=\Big(
\bm{m}_{=}'(\cdot),\;\bm{m}_{\leq}'(\cdot), \;(\cdot-\boldsymbol{\beta}_0)'\times\Gamma_c(\boldsymbol{\beta}_0)'\Big)'.
$$
Then, $\bm{m}^+(\boldsymbol{\beta}_0)=\bm{0}$.
For each $\boldsymbol{\beta}$ in the neighborhood around $\boldsymbol{\beta}_0$ such that $\boldsymbol{\beta}-\boldsymbol{\beta}_0\in\Delta\cap\mathcal{B}(0,\varpi_\delta^\star)$, define
\[
\boldsymbol{\lambda}^0=(\Gamma^{ 0})^{-1}\Big(\bm{m}^{+}(\boldsymbol{\beta})-\bm{m}^+(\boldsymbol{\beta}_0)\Big).
\]
Thus, $\Gamma^0_{=}\boldsymbol{\lambda}^0=\bm{0}$, $\Gamma^0_{\leq}\boldsymbol{\lambda}^0\leq \bm{0}$.
Note that by Taylor's expansion,
\[
\|\boldsymbol{\lambda}^0-(\Gamma^0)^{-1}\Gamma^0(\boldsymbol{\beta}-\boldsymbol{\beta}_0)\|\leq \mathfrak{C}\|\boldsymbol{\beta}-\boldsymbol{\beta}_0\|^2,
\]
implying that $\|\boldsymbol{\lambda}^0-(\boldsymbol{\beta}-\boldsymbol{\beta}_0)\|
\leq \mathfrak{C}(\varpi_\delta^\star)^2
$ with $\mathbb{P}(\cdot|\mathscr{H})$-probability
at least $1-\epsilon_\Delta^\star$, on an event with $\mathbb{P}$-probability at least $1-\pi_\Delta^\star$.
Next, define $\tilde{\bm{m}}(\cdot)=\bm{m}^+(\widehat\boldsymbol{\beta}+\cdot)-\bm{m}^+(\widehat\boldsymbol{\beta})$ and $\tilde{\boldsymbol{\beta}}=\boldsymbol{\phi}^\star+\widehat\boldsymbol{\beta}$ for $\boldsymbol{\phi}^\star$ defined below:
\[
\boldsymbol{\phi}^\star:=\tilde{\bm{m}}^{-1}\Big(\Gamma^\star(\boldsymbol{\lambda}^0-(\Gamma^\star)^{-1}
(\Gamma^\star-\Gamma^0)\boldsymbol{\lambda}^0)\Big).
\]
By Taylor's expansion,
\[
\begin{split}
\boldsymbol{\phi}^\star
&=\tilde{\bm{m}}^{-1}(\bm{0})+
\Big[\frac{\partial}{\partial \boldsymbol{\phi}'}\tilde{\bm{m}}(\bm{0})\Big]^{-1}
\Gamma^\star\Big(\boldsymbol{\lambda}^0-(\Gamma^\star)^{-1}
(\Gamma^\star-\Gamma^0)\boldsymbol{\lambda}^0\Big)+\mathfrak{Re}\\
&=\boldsymbol{\lambda}^0-(\Gamma^\star)^{-1}(\Gamma^\star-\Gamma^0)\boldsymbol{\lambda}^0+\mathfrak{Re},
\end{split}
\]
where $\|(\Gamma^\star)^{-1}(\Gamma^\star-\Gamma^0)\boldsymbol{\lambda}^0+\mathfrak{Re}\|\leq \mathfrak{C}\|\boldsymbol{\lambda}^0\|^2$ with $\mathbb{P}(\cdot|\mathscr{H})$-probability
at least $1-\epsilon_\Delta^\star$, on an event with $\mathbb{P}$-probability over $\mathscr{H}$ at least $1-\pi_\Delta^\star$.
That is, we actually find $\tilde{\boldsymbol{\beta}}$ such that $\|(\tilde{\boldsymbol{\beta}}-\widehat\boldsymbol{\beta})-\boldsymbol{\lambda}^0\|\leq \mathfrak{C}(\varpi_\delta^\star)^2$.
Note that
$$
\bm{m}^+(\widehat\boldsymbol{\beta}+\boldsymbol{\phi}^\star)-\bm{m}^+(\widehat\boldsymbol{\beta})=
\tilde{\bm{m}}(\boldsymbol{\phi}^\star)=
\tilde{\bm{m}}\Big(\tilde{\bm{m}}^{-1}\Big(\Gamma^\star(\boldsymbol{\lambda}^0-(\Gamma^\star)^{-1}
(\Gamma^\star-\Gamma^0)\boldsymbol{\lambda}^0)\Big)\Big).$$
Thus, $\bm{m}_{=}(\tilde{\boldsymbol{\beta}})=
\bm{0}$ and $\bm{m}_{\leq}(\tilde{\boldsymbol{\beta}})\leq\bm{m}_{\leq}(\widehat{\boldsymbol{\beta}})$, i.e., $\boldsymbol{\phi}^\star\in\widehat{\Delta}$.
This shows that $\mathrm{dist}(\Delta\cap\mathcal{B}(\bm{0}, \varpi_\delta^\star),\widehat{\Delta})\leq \varpi_\Delta^\star$ with $\mathbb{P}(\cdot|\mathscr{H})$-probability
at least $1-\epsilon_\Delta^\star$, on an event with $\mathbb{P}$-probability over $\mathscr{H}$ at least $1-\pi_\Delta^\star$. Then, the desired conclusion holds by definition of the $\varepsilon$-enlargement $\widehat{\Delta}_\varepsilon$. Note that when constraints are linear, $\Gamma^0=\Gamma^\star$ and the second-order derivative of $\bm{m}^+(\cdot)$ is exactly zero. So the above calculation implies that $\boldsymbol{\beta}-\boldsymbol{\beta}_0=\boldsymbol{\lambda}_0=\boldsymbol{\phi}^\star$.
In the above, we make use of the fact that $\Gamma^\star$ is non-degenerate, i.e., its smallest eigenvalue is bounded away from zero. Note that by assumptions on the constraints, it is feasible to construct $\Gamma_c$ such that $\Gamma^0$ is non-degenerate (with high probability over $\mathscr{H}$). Then, by Weyl's inequality,
\[
s_{\min}(\Gamma^\star(\Gamma^\star)')\geq
s_{\min}(\Gamma^0(\Gamma^0)')-\mathfrak{C}\|\widehat\boldsymbol{\beta}-\boldsymbol{\beta}_0\|,
\]
implying $s_{\min}(\Gamma^\star)\geq s_{\min}(\Gamma^0)-\mathfrak{C}\|\widehat\boldsymbol{\beta}-\boldsymbol{\beta}_0\|>0$ with $\mathbb{P}(\cdot|\mathscr{H})$-probability
at least $1-\epsilon_\Delta^\star$, on an event with $\mathbb{P}$-probability at least $1-\pi_\Delta^\star$, where $s_{\min}(\cdot)$ here denotes the smallest eigenvalue of the symmetric matrix inside. Then, the proof is complete.
\end{proof}
\section{Discussion on Tuning Parameters}
\label{sec: SA tuning parameter}
\subsection{Determining Binding Constraints}
Our proposed strategy in Section 6.2 for constructing the constraint set in the simulation relies on the tuning parameters $\varrho_\ell$'s, which are used to differentiate the binding constraints from the non-binding ones. The recommended choice described in (6.8) and (6.9) can be rationalized as follows.
For ease of notation, suppose that there is only one treated unit, and thus we can omit the superscript $[i]$ for quantities like $\boldsymbol{\beta}_0^{[i]}$ or $\varrho^{[i]}$ with no confusion. Assume the $\ell$-th constraint $m_{\leq, \ell}(\boldsymbol{\beta})\leq 0$ is binding ($m_{\leq, \ell}(\boldsymbol{\beta}_0)=0$), and $m_{\leq, \ell}(\cdot)$ is sufficiently smooth. By the mean value theorem, $|m_{\leq, \ell}(\widehat{\boldsymbol{\beta}})|=|\frac{\partial}{\partial\boldsymbol{\beta}'}m_{\leq, \ell}(\check\boldsymbol{\beta})(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0)|\leq \|\frac{\partial}{\partial\boldsymbol{\beta}}m_{\leq, \ell}(\check\boldsymbol{\beta})\|_2\|\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0\|_2$, where $\check{\boldsymbol{\beta}}$ is some point between $\boldsymbol{\beta}_0$ and $\widehat{\boldsymbol{\beta}}$ . The parameters $\varrho_\ell^{[i]}$ are used to characterize this upper bound. For $\frac{\partial}{\partial\boldsymbol{\beta}}m_{\leq, \ell}(\check\boldsymbol{\beta})$, a natural ``estimator'' would be $\frac{\partial}{\partial\boldsymbol{\beta}}m_{\leq, \ell}(\widehat{\boldsymbol{\beta}})$. The difference between the two is bounded by $c_1\|\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0\|_2$ for some constant $c_1>0$ (with high probability). Then, it remains to characterize $\|\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0\|_2$.
Note that the basic inequality below always holds (deterministically) by optimality of $\widehat{\boldsymbol{\beta}}$:
\begin{equation}\label{eq: SA basic inequality}
\|\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0\|_2\leq
\frac{2\|\widehat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}\|_2/\sqrt{T_0}}
{s_{\min}(\widehat{\mathbf{Q}})/T_0}\times T_0^{-1/2}=:\mathcal{C}T_0^{-1/2}.
\end{equation}
We first discuss the denominator. Recall that $\widehat{\mathbf{Q}}=\mathbf{Z}'\mathbf{V}\mathbf{Z}$. Since an (unrestricted) intercept is often included in the SC construction, we assume the predictor variables included in $\mathbf{Z}$ are de-meaned (and the constant term is excluded from $\mathbf{Z}$). If $\mathbf{V}$ is an identity matrix and the predictor variables are (approximately) orthogonal to each other, then $s_{\min}(\widehat{\mathbf{Q}}/T_0)$ is the minimum diagonal entry of $\widehat{\mathbf{Q}}/T_0$, i.e., $\min_{1\leq j\leq J_0}\widehat{\sigma}_{b_j}^2$, which motivates the choice of the denominator in (6.9). On the other hand, recall that the numerator $\widehat{\boldsymbol{\gamma}}=\mathbf{Z}'\mathbf{u}$. As discussed in the paper, we can (conditionally) approximate $\widehat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}$ by the Gaussian vector $\mathbf{G}$. Then, by Gaussian tail bound, it can be shown that, for any large $\lambda>0$,
$T_0^{-1/2}\|\widehat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}\|_\infty\leq (\sqrt{2\log d_\beta}+\lambda)\times\sqrt{\max_{1\leq j\leq J_0}
\mathbb{V}[\widehat{\gamma}_j|\mathscr{H}]/T_0}$ with high probability. When $\mathbf{u}$ is independent of $\mathscr{H}$ and its components are independent over $t$, $\max_{1\leq j\leq J_0} \mathbb{V}[\widehat{\gamma}_j|\mathscr{H}]/T_0=\sigma_u^2\times \max_{1\leq j\leq J_0}\widehat{\sigma}_{b_j}^2$. Therefore, we can set
\[
\mathcal{C}=\mathcal{C}_1:= \frac{2\sqrt{d_\beta}(\sqrt{2\log d_\beta}+\lambda)\max_{1\leq j \leq J_0}\widehat{\sigma}_{b_j}\widehat{\sigma}_{u}}{\min_{1\leq j \leq J_0}\widehat{\sigma}^2_{b_j}}
\]
for any large $\lambda$. In most applications, however, the SC weights are sparse due to the simplex- or lasso-type constraints. It is known from the sparse linear regression literature that the bound $\mathcal{C}_1$ can be further improved by replacing the factor $\sqrt{d_\beta}$ by $c_3\sqrt{\|\boldsymbol{\beta}_0\|_0}$, where $\|\cdot\|_0$ denotes the number of nonzeros in a vector and $c_3>0$ is some absolute constant (see, e.g., \citealp[][Theorem 7.13]{Wainwright_2019_Book}). Assuming $\sqrt{\log T_0}\geq 4\sqrt{2}c_3$, we take $\lambda=\sqrt{\log d_\beta\log T_0}/(4c_3)$ and use $\|\widehat{\boldsymbol{\beta}}\|_0$ as a proxy for $\|\boldsymbol{\beta}_0\|_0$, which yields our recommended choice of $\mathcal{C}$ given in (6.9):
\[
\mathcal{C}=\mathcal{C}_2:= \frac{\sqrt{\|\widehat{\boldsymbol{\beta}}\|_0\log d_\beta\log T_0}\max_{1\leq j \leq J_0}\widehat{\sigma}_{b_j}\widehat{\sigma}_{u}}{\min_{1\leq j \leq J_0}\widehat{\sigma}^2_{b_j}}.
\]
Furthermore, when $\widehat{\sigma}_{b_j}^2$ is roughly the same across $j$, then $\mathcal{C}_2$ can be further simplified to
\[
\mathcal{C}=\mathcal{C}_3 = \frac{\sqrt{\|\widehat{\boldsymbol{\beta}}\|_0\log d_\beta\log T_0}\;\widehat{\sigma}_{u}}{\min_{1\leq j \leq J_0}\widehat{\sigma}_{b_j}}.
\]
While these choices are only rules of thumb justified under specific assumptions on the data generating process, they at least have the correct order of magnitude and are valid at least when $T_0$ is large. For instance, for the cointegrated data considered in our basic setup, it can be shown that, for sufficiently small $c_4>0$, $\widehat{\sigma}^2_{b_j}\geq c_4T_0$ with high probability, and $\widehat{\sigma}_u^2$ is bounded with high probability. Therefore, the suggested $\varrho$ based on $\mathcal{C}_2$ has the order of $T_0^{-1}$, which is the well-known rate of convergence for cointegrated regression.
When SC weights are obtained by matching on both stationary and non-stationary features, the contribution of the stationary components to the concentration of $\widehat{\boldsymbol{\beta}}$ is negligible. In other words, the order of magnitude for the bound on $\|\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}\|_2$ is primarily determined by the non-stationary components. To see this, assume that we match on two features, a non-stationary $\mathbf{B}_{1}=(\mathbf{b}_{1,1}, \cdots, \mathbf{b}_{T_0,1})'\in\mathbb{R}^{T_0\times J_0}$ and a stationary $\mathbf{B}_{2}=(\mathbf{b}_{1,2}, \cdots, \mathbf{b}_{T_0,2})'\in\mathbb{R}^{T_0\times J_0}$. Consider the basic inequality \eqref{eq: SA basic inequality} again. $\widehat{\mathbf{Q}}/T_0$ can be written as $\frac{1}{T_0}\sum_{t=1}^{T_0}\mathbf{b}_{t,1}\mathbf{b}_{t,1}'+\frac{1}{T_0}\sum_{t=1}^{T_0}\mathbf{b}_{t,2}\mathbf{b}_{t,2}'=:\mathrm{I}+\mathrm{II}$. As explained previously, with high probability, $s_{\min}(\mathrm{I}/T_0)$ is bounded from below by $T_0$, up to a sufficiently small constant, whereas $\mathrm{II}/T_0$ is much smaller---under mild conditions, it is bounded with high probability. For the numerator $(\widehat{\boldsymbol{\gamma}}-\boldsymbol{\gamma})/T_0$, we can similarly decompose it into a non-stationary part and a stationary part. By the argument described before, it follows that the stationary part is much smaller than the non-stationary part, leading to a deviation bound for $\widehat{\boldsymbol{\beta}}$ on the order of $T_0^{-1}$. Therefore, we recommend applying the above formulas using the non-stationary features only in practice.
Finally, as a reminder, other choices of $\varrho$ can be proposed based on a slightly different basic inequality: $\|\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0\|_2\leq
2\|\widehat{\boldsymbol{\gamma}}\|_2/s_{\min}(\widehat{\mathbf{Q}})$. It also holds deterministically by optimality of $\widehat{\boldsymbol{\beta}}$. The denominator is the same as before, but the numerator $\widehat{\boldsymbol{\gamma}}$ is not demeaned.
When the model is correctly specified ($\boldsymbol{\gamma}=\bm{0}$), the numerator may be small; however, when the model is misspecified, $\widehat{\boldsymbol{\gamma}}$ is not necessarily small, and it may be characterized by its sample analogue. Thus, we can set
\[
\varrho=\frac{2\sqrt{d_\beta}\max_{1\leq j\leq J_o}\widehat{\sigma}^2_{b_ju}}{\min_{1\leq j\leq J_0}\widehat{\sigma}^2_{b_j}}
\]
where $\widehat{\sigma}^2_{b_ju}$ is the estimated covariance between the pseudo-true residual $\mathbf{u}$ and the $j$-th column of $\mathbf{B}$ (the features of the $j$-th control unit). However, compared to the alternatives described in the paper, this bound is generally loose and thus is not recommended.
\subsection{Adjustment for Nonlinear Constraints}
As described in the main paper, we have an additional adjustment to nonlinear constraints in the simulation (the second term on the right-hand side of (6.10)). In this section we briefly discuss the necessity of this adjustment and the justification of the proposed strategy.
\subsubsection{Necessity of Adjustment}
The constraints allowed in this paper is more general than those considered in \cite{Cattaneo-Feng-Titiunik_2021_JASA}. Specifically, condition (T2.iii) in \cite{Cattaneo-Feng-Titiunik_2021_JASA} requires the constraint set used in the simulation for in-sample uncertainty quantification be locally \textit{equal to} the original constraint set in the synthetic control problem. In the notation of this paper, that condition means that, with some high probability, $\Delta^{\star}\cap\mathcal{B}(\bm{0},\varpi_\delta^\star)=\Delta\cap\mathcal{B}(\bm{0},\varpi_\delta^\star)$ for some (small) $\varpi_\delta^\star$-neighborhood of zero. This is generally true when constraints are formed by linear functions. To see this, suppose that there are two donor units, an L1 constraint $\mathcal{W}=\{(w_1, w_2): |w_1|+|w_2|\leq 1\}$ is imposed, and the pseudo-true value is $\mathbf{w}_0=(1,0)$. The constraint set $\Delta^\star$ used in the simulation relies on the estimated weights $\widehat{\mathbf{w}}$, which are generally close (but not exactly equal) to $\mathbf{w}_0$ with high probability.
For instance, let $\widehat{\mathbf{w}}=(0.99, 0.01)$, and applying the suggested strategy described in Section 6.2, one correctly finds that two linear constraints among those derived from decomposing the original L1 constraint, $w_1+w_2\leq 1$ and $w_1-w_2\leq 1$, are binding. Thus, one defines $\Delta^\star=\{(\delta_1, \delta_2): (\delta_1+0.99)+(\delta_2+0.01)\leq 0.99+0.01, (\delta_1+0.99)-(\delta_2+0.01)\leq 0.99-0.01\}$, which is exactly the same as $\Delta=\mathcal{W}-\mathbf{w}_0$ locally around $(0,0)$.
However, such an exact equality is generally not possible when constraints are formed by nonlinear functions. Still consider the same example, but an L2 constraint $\mathcal{W}=\{(w_1, w_2): w_1^2+w_2^2\leq 1\}$ is imposed instead. Applying the same strategy, one correctly finds that the constraint should be binding and thus lets $\Delta^\star=\{(\delta_1,\delta_2):(\delta_1+0.99)^2+(\delta_2+0.01)^2\leq 0.99^2+0.01^2\}$, whereas the (centered) original constraint set is $\Delta=\{(\delta_1, \delta_2): (\delta_1+1)^2+\delta_2^2=1\}$. In other words, $\Delta$ and $\Delta^\star$ are two ``circles'' passing through $(0,0)$. Although they are locally close near $(0,0)$, their boundaries have different tangents at that point, and hence the two sets are not locally identical, no matter how small the neighborhood $\mathcal{B}(\bm{0},\varpi^\star)$ is.
In this sense, the results in \cite{Cattaneo-Feng-Titiunik_2021_JASA} are more applicable to synthetic controls with linear constraints (such as simplex and Lasso). By contrast, the local approximate equality in condition (iii) of Theorem 1 is much weaker and makes the results in this paper applicable to more general cases with possibly nonlinear constraints. Given the fact described above, when there are (binding) nonlinear constraints, we propose to enlarge the feasibility set defined by nonlinear constraints to satisfy the sufficient condition (iii) of Theorem 1.
\subsubsection{Enlarging Nonlinear Constraint Sets}
In this section we briefly discuss the justification for the adjustment in (6.10). Our strategy is motivated by the proof of Lemma 1, which characterizes the distance between $\Delta\cap\mathcal{B}(\bm{0},\varpi_\delta^\star)$ and $\widehat{\Delta}$. For simplicity, assume that there is only one treated unit (so the superscript $[i]$ can be omitted with no confusion).
Suppose that we only have binding inequality constraints $\bm{m}(\boldsymbol{\beta})\leq \bm{0}$ (so $\bm{m}(\boldsymbol{\beta}_0)=\bm{0}$), where $\bm{m}(\cdot)$ is a $d_\leq$-vector of sufficiently smooth functions. For any feasible $\boldsymbol{\beta}\in\mathbb{R}^{d_\beta}$ close to $\boldsymbol{\beta}_0$ such that $\boldsymbol{\beta}-\boldsymbol{\beta}_0\in\Delta\cap\mathcal{B}(\bm{0}, \varpi_\delta^\star)$,
\[
\bm{m}(\boldsymbol{\beta})=\bm{m}(\boldsymbol{\beta})-\bm{m}(\boldsymbol{\beta}_0)=\frac{\partial}{\partial\boldsymbol{\beta}'}\bm{m}(\boldsymbol{\beta}_0)(\boldsymbol{\beta}-\boldsymbol{\beta}_0)+\frac{1}{2}(\boldsymbol{\beta}-\boldsymbol{\beta}_0)'\Big[\frac{\partial^2}{\partial\boldsymbol{\beta}\partial\boldsymbol{\beta}'}\bm{m}(\check{\boldsymbol{\beta}})\Big](\boldsymbol{\beta}-\boldsymbol{\beta}_0)
\]
for some point $\check{\boldsymbol{\beta}}$ between $\boldsymbol{\beta}$ and $\boldsymbol{\beta}_0$, where $\partial^2\bm{m}(\check{\boldsymbol{\beta}})/\partial\boldsymbol{\beta}\partial\boldsymbol{\beta}'$ is a $d_\beta\times d_\beta\times d_\leq$ array, with each sheet the second-order derivative matrix for one constraint function. Thus, $(\boldsymbol{\beta}-\boldsymbol{\beta}_0)'[\partial^2\bm{m}(\check{\boldsymbol{\beta}})/\partial\boldsymbol{\beta}\partial\boldsymbol{\beta}'](\boldsymbol{\beta}-\boldsymbol{\beta}_0)$ is a $d_\leq$-vector, with each element corresponding to $(\boldsymbol{\beta}-\boldsymbol{\beta}_0)'[\partial^2m_\ell(\check{\boldsymbol{\beta}})/\partial\boldsymbol{\beta}\partial\boldsymbol{\beta}'](\boldsymbol{\beta}-\boldsymbol{\beta}_0)$ for $1\leq \ell\leq d_\leq$.
Let $\Gamma_0=\frac{\partial}{\partial\boldsymbol{\beta}}\bm{m}(\boldsymbol{\beta}_0)$ and $\widehat{\Gamma}=\frac{\partial}{\partial\boldsymbol{\beta}}\bm{m}(\widehat{\boldsymbol{\beta}})$. Assume $\Gamma_0$ is invertible in the neighborhood of $\boldsymbol{\beta}_0$. (When $d_\leq< d_\beta$, we can complement $\Gamma_0$ with additional rows and construct an invertible one, which is formalized in the proof of Lemma 1). Then,
\[
\Gamma_0^{-1}\bm{m}(\boldsymbol{\beta})-(\boldsymbol{\beta}-\boldsymbol{\beta}_0)
=\frac{1}{2}\Gamma_0^{-1}\Big[(\boldsymbol{\beta}-\boldsymbol{\beta}_0)'\frac{\partial^2}{\partial\boldsymbol{\beta}\partial\boldsymbol{\beta}'}\bm{m}(\check{\boldsymbol{\beta}})(\boldsymbol{\beta}-\boldsymbol{\beta}_0)\Big]=:\mathscr{L}
\]
It can be shown that $\boldsymbol{\lambda}:=\Gamma_0^{-1}\bm{m}(\boldsymbol{\beta})$ is (approximately) in $\widehat{\Delta}$. To see this,
$$
\bm{m}(\widehat{\boldsymbol{\beta}}+\boldsymbol{\lambda})\approx\bm{m}(\widehat{\boldsymbol{\beta}})+\frac{\partial}{\partial\boldsymbol{\beta}'}\bm{m}(\boldsymbol{\beta}_0)\boldsymbol{\lambda}=\bm{m}(\widehat{\boldsymbol{\beta}})+(\bm{m}(\boldsymbol{\beta})-\bm{m}(\boldsymbol{\beta}_0))\leq \bm{m}(\widehat{\boldsymbol{\beta}}).
$$
This is formalized in the proof of Lemma 1. For the moment, assume $\boldsymbol{\lambda}\in\widehat{\Delta}$. Then,
\begin{align*}
\bm{m}(\boldsymbol{\beta}-\boldsymbol{\beta}_0+\widehat{\boldsymbol{\beta}})-
\bm{m}(\widehat{\boldsymbol{\beta}})&=\bm{m}(\widehat{\boldsymbol{\beta}}+\boldsymbol{\lambda}-\mathscr{L})-\bm{m}(\widehat{\boldsymbol{\beta}})\\
&\approx
\bm{m}(\widehat{\boldsymbol{\beta}}+\boldsymbol{\lambda})-\Gamma_0\mathscr{L}-\bm{m}(\widehat{\boldsymbol{\beta}})\\
&\leq -\Gamma_0\mathscr{L}\\
&\approx-\frac{1}{2}(\boldsymbol{\beta}-\boldsymbol{\beta}_0)'\frac{\partial^2}
{\partial\boldsymbol{\beta}\partial\boldsymbol{\beta}'}\bm{m}(\boldsymbol{\beta}_0)(\boldsymbol{\beta}-\boldsymbol{\beta}_0).
\end{align*}
The errors introduced by ``$\approx$'' are of smaller order under mild conditions. The above calculation implies that any $\boldsymbol{\beta}-\boldsymbol{\beta}_0\in\Delta\cap\mathcal{B}(\bm{0},\varpi_\delta^\star)$ (approximately) belongs to the following adjusted version of $\widehat{\Delta}$ in (6.7):
\[
\Big\{\boldsymbol{\delta}: \bm{m}(\widehat{\boldsymbol{\beta}}+\boldsymbol{\delta})\leq
\bm{m}(\widehat{\boldsymbol{\beta}})+\frac{1}{2}(\boldsymbol{\beta}-\boldsymbol{\beta}_0)'\frac{\partial^2}
{\partial\boldsymbol{\beta}\partial\boldsymbol{\beta}'}\bm{m}(\boldsymbol{\beta}_0)(\boldsymbol{\beta}-\boldsymbol{\beta}_0)\Big\}.
\]
Then, for each inequality constraint $m_{\leq,\ell}(\widehat{\boldsymbol{\beta}}+\boldsymbol{\delta})\leq m_{\leq, \ell}(\widehat{\boldsymbol{\beta}})$, the ``adjustment'' to the upper bound is at most
\[
\frac{1}{2}s_{\max}\Big(\frac{\partial^2}
{\partial\boldsymbol{\beta}\partial\boldsymbol{\beta}'}m_{\leq,\ell}(\boldsymbol{\beta}_0)\Big)\|\boldsymbol{\beta}-\boldsymbol{\beta}_0\|_2^2.
\]
Given the high-probability bound $\varrho$ for $\|\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0\|_2$, we propose the adjustment in (6.10).
\bigskip
\noindent\textit{\textbf{Notes:}}
Our enlargement strategy used to define $\Delta^\star$ in Section 6.2 can be conceptually understood as follows.
A constraint set $\Delta$ can be generally written as $\Delta=\cap_{j=1}^L A_j$ for a sequence of convex sets $A_j$. Each $A_j$ is defined by one inequality. Accordingly, $\widehat{\Delta}$ in (6.7) (with no enlargement) can be written as $\widehat{\Delta}=\cap_{j=1}^L\widehat{A}_j$.
Let $\widehat{A}_{j,\varepsilon_j}$ be an $\varepsilon_j$-enlargement of $\widehat{A}_j$. We can apply Lemma 1 to each pair of $A_j$ and $\widehat{A}_j$, and then $A_j\subseteq\widehat{A}_{j,\varepsilon_j}$ for an appropriate (small) $\varepsilon_j$. For any point $g\in\Delta$, $g\in A_j$ for every $j$. So $g\in\widehat{A}_{j,\varepsilon_j}$. Then, $g\in\cap_{j=1}^L \widehat{A}_{j,\varepsilon_j}$. Since it holds for any $g$, we have $\Delta\subseteq\Delta^\star:=\cap_{j=1}^L \widehat{A}_{j,\varepsilon_j}$. The sufficient condition (iii) in Theorem 1 holds.
\clearpage\section{Second Order Cone Programming} \label{sec: SA SOCP}
In this section, we first define three types of convex optimization problems that we will be relying on (for background knowledge and technical details, see \cite{Boyd_2004_BOOK}). Second, we illustrate the link between these three families of convex problems. Third, we show that the optimization problems underlying the prediction/estimation and uncertainty quantification problems for SC presented in Section 6 of the main text are quadratically constrained quadratic program (QCQP) and quadratically constrained linear problem (QCLP), respectively, and show how to represent them as second-order cone program (SOCP). Finally, we provide two examples by showing how to write the L1-L2-type and Lasso-type constraints in conic form. These approaches are implemented in our companion general-purpose software \citep*{Cattaneo-Feng-Palomba-Titiunik_2025_JSS}, where we show that they lead to remarkable speed and scalability improvements.
\subsection{Families of Convex Optimization Problems}\label{subsec: families cvx problems}
\underline{\textbf{\textit{QCQPs and QCLPs.}}}
A quadratically constrained quadratic program is an optimization problem of with the following form
\begin{align}
\min_{\mathbf{x}}\quad & \mathbf{x}' \mathbf{P}_0 \mathbf{x} + \mathbf{q}_0'\mathbf{x} + w \label{eq: qcqp formulation} \\
\text{subject to} \quad & \mathbf{x}' \mathbf{P}_j\mathbf{x} +\mathbf{q}_j'\mathbf{x} +r_j\leq 0, \quad j=1,\ldots,m, \tag{Quadratic inequality constraint}\\
& \mathbf{F}\mathbf{x} = \mathbf{g}, \tag{Linear equality constraint}
\end{align}
where $\mathbf{P}_0, \mathbf{P}_1, \ldots, \mathbf{P}_m \in \mathcal{M}_{n\times n}(\operatorname{\mathbb{R}})$, $ \mathbf{q}_0,\mathbf{q}_1,\ldots,\mathbf{q}_m\in \operatorname{\mathbb{R}}^n$, $\mathbf{x}\in \operatorname{\mathbb{R}}^n$, $\mathbf{F} \in \mathcal{M}_{m \times n}(\operatorname{\mathbb{R}})$, $\mathbf{g} \in \operatorname{\mathbb{R}}^m$, and $r_0,r_1, \ldots, r_m, w \in \operatorname{\mathbb{R}}$. If all the matrices $\mathbf{P}_0,\mathbf{P}_1,\ldots, \mathbf{P}_m$ are positive semi-definite the QCQP is convex. Moreover, if $\mathbf{P}_0=\mathbf{0}$ the QCQP becomes a QCLP. For this reason, in what follows we will restrict our attention to QCQPs as they naturally embed QCLPs. \bigskip
\underline{\textbf{\textit{SOCPs.}}} To define a SOCP, it is necessary to first give the definition of a second-order cone and then introduce the notion of associated \textit{generalized inequality}. \smallskip
\uline{Second-order cone definition.} A set $\mathcal{C}$ is called a \textit{cone} if for every $\mathbf{x} \in \mathcal{C}$ and $\alpha\geq 0$ we have $\alpha \mathbf{x}\in \mathcal{C}$. A set $\mathcal{C}$ is a \textit{convex cone} if it is convex and a cone, i.e. if $\forall\, \mathbf{x}_1,\mathbf{x}_2 \in \mathcal{C}, $and $\forall\, \alpha_1,\alpha_2 \geq 0$, we have
\[\alpha_1 \mathbf{x}_1 + \alpha \mathbf{x}_2 \in \mathcal{C}.\]
Now, consider any norm $||\cdot||$ defined on the Euclidean space $\operatorname{\mathbb{R}}^n$. The \textit{norm cone} associated with the norm $||\cdot ||$ is defined to be the set
\[\mathcal{C}=\{(\mathbf{x},t)\in \operatorname{\mathbb{R}}^{n+1}: ||\mathbf{x}|| \leq t\}\]
and it is a convex cone by the standard properties of the norms. A \textit{second-order cone} is the associated norm cone for the Euclidean norm and it is typically defined as
\begin{align*}
\mathcal{C} &=\left\{(\mathbf{x}, t) \in \operatorname{\mathbb{R}}^{n+1} :\|\mathbf{x}\|_2 \leq t\right\} =\left\{\left[\begin{array}{c}
\mathbf{x} \\
t
\end{array}\right] : \left[\begin{array}{c}
\mathbf{x} \\
t
\end{array}\right]'\left[\begin{array}{cc}
\mathbf{I} & 0 \\
0 & -1
\end{array}\right]\left[\begin{array}{c}
\mathbf{x} \\
t
\end{array}\right] \leq 0, \: t \geq 0\right\}
\end{align*}
\uline{Generalized inequality.} A cone $\mathcal{C}$ is \textit{solid} if it has non-empty interior and it is \textbf{pointed} if $\mathbf{x}\in \mathcal{C}, -\mathbf{x}\in \mathcal{C}$ implies that $\mathbf{x}=0$. We say that a cone $\mathcal{C}$ is \textit{proper} if is it convex, closed, solid, and pointed. Proper cones in the Euclidean space $\operatorname{\mathbb{R}}^n$ are useful because they induce a partial ordering that enjoys almost all the properties of the basic one in $\operatorname{\mathbb{R}}$. Therefore, given a cone $\mathcal{C}\subseteq \operatorname{\mathbb{R}}^n$, we can define the generalized inequality $\preceq_\mathcal{C}$ for any two vectors $\mathbf{x},\mathbf{y}\in \operatorname{\mathbb{R}}^n$
\[\mathbf{x}\preceq_{\mathcal{C}} \mathbf{y} \iff \mathbf{y} -\mathbf{x} \in \mathcal{C}.\]
From this definition we can see that quadratic constraints such as $||\mathbf{x}||_2 \leq h$ can be re-written as second-order cone constraints of the form $\mathbf{x}\preceq_\mathcal{C} h$ for some second-order cone $\mathcal{C}$. Note that if $\mathcal{C}=\operatorname{\mathbb{R}}_+^m$ then if $m=1$, $\preceq_\mathcal{C}$ is the standard inequality $\leq$ in $\operatorname{\mathbb{R}}$, whereas if $m>1$, $\preceq_\mathcal{C}$ is the component-wise inequality in $\operatorname{\mathbb{R}}^m$.\smallskip
\uline{Second-order cone program.} Let $\mathcal{K}$ be a cone such that $\mathcal{K} = \mathbb{R}_+^m \times \mathcal{K}_1 \times \mathcal{K}_2 \times\cdots\times \mathcal{K}_L$ where $\mathcal{K}_l := \{(k_0,\mathbf{k}_1)\in \operatorname{\mathbb{R}}\times \operatorname{\mathbb{R}}^{l}: ||\mathbf{k}_1||_2\leq k_0\}, l= 1,\ldots, L$. Let $\preceq_\mathcal{K}$ be the \textit{generalized inequality} associated with the cone $\mathcal{K}$. An optimization problem is called \textit{second-order cone program} if it has the following form
\begin{align}
\min_{\mathbf{x}} \quad & \mathbf{c}'\mathbf{x}, \label{eq: SA socp formulation} \\
\text{subject to}\quad & \mathbf{G}\mathbf{x} \preceq_\mathcal{K} \mathbf{h}, \tag{Second-order cone constraint} \\
& \mathbf{A}\mathbf{x}=\mathbf{b}. \tag{Linear equality constraint}
\end{align}
\subsection{Link Between QCQP and SOCPs}
Any QCQP can be converted to a SOCP \citep{Boyd_2004_BOOK}. In other words, we can always rewrite an optimization problem such as \eqref{eq: qcqp formulation} in the form of \eqref{eq: SA socp formulation}. First, we present the general result and then we explain all the necessary steps to reformulate QCQPs as SOCPs. Without loss of generality, assume that $w=0$ in \eqref{eq: qcqp formulation} and, to ease notation, let $m=1$ so that there is only a single quadratic inequality constraint. Moreover, given any positive semi-definite matrix $\mathbf{P}$, let $\mathbf{P}^{1/2}$ be the square root of $\mathbf{P}$, that is the unique symmetric positive semi-definite matrix $\mathbf{R}$ such that $\mathbf{R}\mathbf{R}=\mathbf{R}'\mathbf{R}=\mathbf{P}$. Then for any QCQP the following two formulations are equivalent
\begin{multicols}{2}
\begin{align*}
& \uline{\textbf{QCQP}} \\
\min_{\mathbf{x}}\quad & \mathbf{x}' \mathbf{P}_0 \mathbf{x} + \mathbf{q}_0'\mathbf{x} \\
\text{subject to} \quad & \mathbf{F}\mathbf{x} = \mathbf{g}, \\
& \mathbf{x}' \mathbf{P}_1\mathbf{x} +\mathbf{q}_1'\mathbf{x} +r_1\leq 0. \\
\end{align*}
\vfill\null
\columnbreak
\begin{align*}
& \uline{\textbf{SOCP}} \\
\min_{\mathbf{x}, v, t, s}\quad & v + \mathbf{q}_0'\mathbf{x} \\
\text{subject to} \quad &\mathbf{F}\mathbf{x} = \mathbf{g}, \\
& t+\mathbf{q}_1'\mathbf{x} +r_1\preceq_{\operatorname{\mathbb{R}}_+} 0, \\
& \mathbf{P}^{1/2}_0\mathbf{x} \preceq_{K_{1+n}} v, \\
& \mathbf{P}^{1/2}_1\mathbf{x} \preceq_{K_{1+n}} s.
\end{align*}
\end{multicols}
We can see that the logic beneath the conversion of a QCQP into a SOCP is to ``linearize'' all the non-linear terms appearing either in the objective function or in the inequality constraints. The ``linearization'' step does come at a cost, as it requires the introduction of a slack variable every time we rely on it. Indeed, above we linearized the objective function and the quadratic inequality constraint by introducing two auxiliary slack variables.
More formally, let $\mathbf{x}'\mathbf{P}\mathbf{x}$ be a symmetric positive semi-definite quadratic form and consider the constraint $\mathbf{x}'\mathbf{P}\mathbf{x}\leq y$. Then
\begin{enumerate}[label=(\roman*)]
\item Since $\mathbf{P}$ is symmetric positive semi-definite the epigraph $\mathbf{x}' \mathbf{P} \mathbf{x}\leq y$ is a convex set and $\mathbf{P}^{1/2}$ is well-defined.
\item Write the inequality constraint as a constraint involving the Euclidean norm $\|\cdot\|_2$
\[y\geq \mathbf{x}'\mathbf{P}\mathbf{x} = \mathbf{x}'\mathbf{P}^{1/2}\mathbf{P}^{1/2}\mathbf{x} = ||\mathbf{P}^{1/2}\mathbf{x}||_2^2.\]
\item Note that
\begin{align}
||\mathbf{P}^{1/2}\mathbf{x}||_2^2\leq y \iff \left\|\left[\begin{array}{c}
1-y \\ 2 \mathbf{P}^{1 / 2} \mathbf{x}
\end{array}\right]\right\|_2 \leq 1+y,
\label{eq: socp trick}
\end{align}
which can be verified by squaring the two sides of the last inequality and expand the norm.
\item More is true, as the right-most inequality in \eqref{eq: socp trick} defines the following second-order cone for given $\mathbf{P}^{1/2}$
\[\mathcal{C} = \left\{ \left(1-y, 2\mathbf{P}^{1/2}\mathbf{x}, 1+y \right): \left\|\left[\begin{array}{c}
1-y \\ 2 \mathbf{P}^{1 / 2} \mathbf{x}
\end{array}\right]\right\|_2 \leq 1+y \right\},\]
which in turn induces the generalized inequality $\mathbf{P}^{1/2}\mathbf{x} \preceq_\mathcal{C} y$.
\end{enumerate}
\subsection{Specific Synthetic Control Problems as SOCPs}
We illustrate the approach described above for the L1-L2 constraint and the Lasso constraint. To ease notation, we do not consider the regularization to the local geometry of $\Delta$. Note that simplex, ridge, or least squares are particular cases of L1-L2. Throughout this section, we assume that $T_i\equiv T_0$ for all $i\in\mathcal{E}$.
\paragraph{L1-L2-type $\mathcal{W}$.} Consider first the prediction/estimation SC optimization problem, which relies on the following program:
\begin{align}
\min_{\mathbf{w},\mathbf{r}}\quad &(\mathbf{A} - \mathbf{B}\mathbf{w} - \mathbf{C}\mathbf{r})' \mathbf{V} (\mathbf{A} - \mathbf{B}\mathbf{w} - \mathbf{C}\mathbf{r}) \label{eq: qcqp pred l1l2}\\
\text{subject to} \quad & ||\mathbf{w}^{[i]}||_1 = 1, \quad i=1,\ldots,J_1, \tag{L1 equality constraints}\\
& ||\mathbf{w}^{[i]}||_2 \leq Q^{[i]}, \quad i=1,\ldots,J_1,\tag{L2 inequality constraints}\\
& \mathbf{w} \geq \mathbf{0},\tag{non-negativity constraint}
\end{align}
where, as always, $\geq$ is understood as a component-wise inequality for vectors ($\mathbf{w}\in\operatorname{\mathbb{R}}^{J_0\cdot J_1}$). First, notice that the non-convex constraints $||\mathbf{w}^{[i]}||_1=1,i=1,\ldots,J_1$ can be replaced with the convex constraints $\mathbf{1}' \mathbf{w}^{[i]}=1,i=1,\ldots,J_1$ because of the non-negativity constraint on the elements of $\mathbf{w}$. Then, we can cast \eqref{eq: qcqp pred l1l2} as a SOCP as follows
\begin{align*}
\min_{\mathbf{w},\mathbf{r}, v, \{s_i\}_{i=1}^{J_1}}\quad & v \\
\text{subject to} \quad &\mathbf{1}' \mathbf{w}^{[i]}=1, \tag{L1 equality constraints}\\
&-\mathbf{w} \preceq_{\mathcal{C}_1} \mathbf{0}, \tag{cone in $\operatorname{\mathbb{R}}^{J_0\cdot J_1}$}\\
& \left[\begin{array}{c}
1-v \\ 2\mathbf{V}^{1/2}(\mathbf{A}-\mathbf{B}\mathbf{w}-\mathbf{C}\mathbf{r})
\end{array}\right] \preceq_{\mathcal{C}_2} 1+v,\tag{cone in $\operatorname{\mathbb{R}}^{2+\tilde{T}\cdot M}$}\\
& s_i \preceq_{\mathcal{C}_3} Q^{[i]}, \quad i=1,\ldots,J_1, \tag{$J_1$ cones in $\operatorname{\mathbb{R}}$} \\
& \left[\begin{array}{c}
1-s_i \\ 2\mathbf{w}^{[i]}
\end{array}\right] \preceq_{\mathcal{C}_4} 1+s_i,\quad i=1,\ldots,J_1,\tag{$J_1$ cones in $\operatorname{\mathbb{R}}^{2+J_0}$}
\end{align*}
where $\mathcal{K}=\mathcal{C}_1\times\mathcal{C}_2\times\mathcal{C}_3^{J_1}\times\mathcal{C}_4^{J_1}=\mathbb{R}_+^{J_0\cdot J_1} \times \mathcal{K}_{\tilde{T}\cdot M+1}\times \operatorname{\mathbb{R}}_+^{J_1} \times \mathcal{K}_{J_0 +1}^{J_1}$ is the conic constraint for this program.
For uncertainty quantification, we need to solve the optimization problem underlying (6.5) in the main text. We discuss the lower bound only. Recalling that $\boldsymbol{\beta}=(\mathbf{w}',\mathbf{r}')'$, we have
\begin{align}
\inf_{\boldsymbol{\beta}=(\mathbf{w}',\mathbf{r}')'}\quad & \mathbf{p}_\tau'(\boldsymbol{\beta} - \widehat{\boldsymbol{\beta}}) \label{eq: SA qcqp unc l1l2}\\
\text{subject to} \quad & ||\mathbf{w}^{[i]}||_1 = 1,\quad i=1,\ldots,J_1, \tag{L1 equality constraints}\\
& ||\mathbf{w}^{[i]}||_2 \leq Q^{[i]},\quad i=1,\ldots,J_1, \tag{L2 inequality constraints}\\
& \mathbf{w} \geq \mathbf{0}, \tag{non-negativity constraint}\\
& (\boldsymbol{\beta}-\widehat{\boldsymbol{\beta}})'\widehat{\mathbf{Q}}(\boldsymbol{\beta}-\widehat{\boldsymbol{\beta}})- 2(\mathbf{G}^\star)'(\boldsymbol{\beta}-\widehat{\boldsymbol{\beta}})\leq 0. \tag{constrained least squares}
\end{align}
We can cast the SC optimization problem in \eqref{eq: SA qcqp unc l1l2} in conic form as follows:
\begin{align*}
\min_{\mathbf{w},\mathbf{r}, \{s_i\}_{i=1}^{J_1}, t}\quad & \mathbf{p}_\tau'\boldsymbol{\beta} \\
\text{subject to} \quad &\mathbf{1}' \mathbf{w}^{[i]}=1,\quad i=1,\ldots,J_1, \tag{L1 equality constraints}\\
&t+\mathbf{a}' \boldsymbol{\beta} +f \preceq_{\mathcal{C}_1} 0, \tag{cone in $\operatorname{\mathbb{R}}$} \\
&-\mathbf{w} \preceq_{\mathcal{C}_2} \mathbf{0}, \tag{cone in $\operatorname{\mathbb{R}}^{J_0\cdot J_1}$}\\
& s_i \preceq_{\mathcal{C}_3} Q^{[i]}, \quad i=1,\ldots,J_1, \tag{$J_1$ cones in $\operatorname{\mathbb{R}}$} \\
& \left[\begin{array}{c}
1-s_i \\ 2\mathbf{w}^{[i]}
\end{array}\right] \preceq_{\mathcal{C}_4} 1+s_i,\quad i=1,\ldots,J_1,\tag{$J_1$ cones in $\operatorname{\mathbb{R}}^{2+J_0}$} \\
& \left[\begin{array}{c}
1-t \\ 2\mathbf{Q}^{1/2}\boldsymbol{\beta}
\end{array}\right] \preceq_{\mathcal{C}_5} 1+t,\tag{cone in $\operatorname{\mathbb{R}}^{2+(J_0+KM)\cdot J_1}$}
\end{align*}
where $\mathcal{K}=\mathcal{C}_1\times\mathcal{C}_2\times\mathcal{C}_3\times\mathcal{C}_4\times\mathcal{C}_5=\operatorname{\mathbb{R}}_+\times \operatorname{\mathbb{R}}_+^{J_0\cdot J_1}\times\operatorname{\mathbb{R}}_+^{J_1}\times \mathcal{K}_{1+J_0}^{J_1} \times \mathcal{K}_{1+(J_0+KM)\cdot J_1}$ is the conic constraint for this program, $\mathbf{a}=-2('\mathbf{Q}\widehat{\boldsymbol{\beta}} + \mathbf{G}^\star)'$, and $f=\widehat{\boldsymbol{\beta}}'\mathbf{Q}\widehat{\boldsymbol{\beta}} + 2\mathbf{G}^\star\widehat{\boldsymbol{\beta}}$.
\paragraph{Lasso-type $\mathcal{W}$.} We show how to write the QCQP as a SOCP when $\mathcal{W}$ has a lasso-type constraint. In this case, the SC weight construction (3.1) has the form:
\begin{align}
\min_{\mathbf{w},\mathbf{r}}\quad &(\mathbf{A} - \mathbf{B}\mathbf{w} - \mathbf{C}\mathbf{r})' \mathbf{V} (\mathbf{A} - \mathbf{B}\mathbf{w} - \mathbf{C}\mathbf{r}) \label{eq: qcqp pred lasso}\\
\text{subject to} \quad & ||\mathbf{w}^{[i]}||_1 \leq 1,\quad i=1,\ldots,J_1. \tag{L1 inequality constraints}
\end{align}
We can write the optimization problem in \eqref{eq: qcqp pred lasso} as a SOCP of the following form
\begin{align*}
\min_{\mathbf{w},\mathbf{r}, \{\mathbf{z}_i\}_{i=1}^{J_1},v}\quad & v \\
\text{subject to} \quad & \left[\begin{array}{c}
1-v \\ 2\mathbf{V}^{1/2}(\mathbf{A}-\mathbf{B}\mathbf{w}-\mathbf{C}\mathbf{r})
\end{array}\right] \preceq_{\mathcal{C}_1} 1+v,\tag{cone in $\operatorname{\mathbb{R}}^{2+T_0\cdot M \cdot J_1}$} \\
&\mathbf{1}' \mathbf{z}_i \preceq_{\mathcal{C}_2} 1,\quad i=1,\ldots,J_1, \tag{$J_1$ cones in $\operatorname{\mathbb{R}}$}\\
&-\mathbf{w} \preceq_{\mathcal{C}_3} \mathbf{z}, \tag{cone in $\operatorname{\mathbb{R}}^{J_0\cdot J_1}$}\\
&\quad \:\,\mathbf{w} \preceq_{\mathcal{C}_4} \mathbf{z},\tag{cone in $\operatorname{\mathbb{R}}^{J_0\cdot J_1}$}
\end{align*}
where $\mathcal{K}=\mathcal{C}_1\times\mathcal{C}_2^{J_1}\times\mathcal{C}_3\times\mathcal{C}_4=\mathcal{K}_{1+T_0\cdot M \cdot J_1}\times \operatorname{\mathbb{R}}_+^{J_1}\times \mathbb{R}_+^{J_0\cdot J_1}\times \mathbb{R}_+^{J_0\cdot J_1}$ is the conic constraint for this program and $\mathbf{z}:=(\mathbf{z}_1',\ldots,\mathbf{z}_{J_1}')'$.
For uncertainty quantification, we need to solve the optimization problem underlying (6.5) in the main text. Here we discuss the lower bound only for brevity. Recalling that $\boldsymbol{\beta}=(\mathbf{w}',\mathbf{r}')'$, we have
\begin{align}
\inf_{\boldsymbol{\beta}=(\mathbf{w}',\mathbf{r}')'}\quad & \mathbf{p}_\tau'(\boldsymbol{\beta} - \widehat{\boldsymbol{\beta}}) \label{eq: qcqp unc lasso}\\
\text{subject to} \quad & ||\mathbf{w}^{[i]}||_1 \leq 1,\quad i=1,\ldots,J_1, \tag{L1 inequality constraints}\\
& (\boldsymbol{\beta}-\widehat{\boldsymbol{\beta}})'\widehat{\mathbf{Q}}(\boldsymbol{\beta}-\widehat{\boldsymbol{\beta}})- 2(\mathbf{G}^\star)'(\boldsymbol{\beta}-\widehat{\boldsymbol{\beta}})\leq 0. \tag{constrained least squares}
\end{align}
We can cast the SC optimization problem in \eqref{eq: qcqp unc lasso} in conic form as follows:
\begin{align*}
\min_{\mathbf{w},\mathbf{r}, \{\mathbf{z}_i\}_{i=1}^{J_1}, t}\quad & \mathbf{p}_\tau'\boldsymbol{\beta} \\
\text{subject to} \quad &t+\mathbf{a}' \boldsymbol{\beta} +f \preceq_{\mathcal{C}_1} 0, \tag{cone in $\operatorname{\mathbb{R}}$} \\
&\mathbf{1}' \mathbf{z}^{[i]} \preceq_{\mathcal{C}_2} 1,\quad i=1,\ldots,J_1, \tag{$J_1$ cones in $\operatorname{\mathbb{R}}$}\\
&-\mathbf{w} \preceq_{\mathcal{C}_3} \mathbf{z}, \tag{cone in $\operatorname{\mathbb{R}}^{J_0\cdot J_1}$}\\
&\quad \:\,\mathbf{w} \preceq_{\mathcal{C}_4} \mathbf{z},\tag{cone in $\operatorname{\mathbb{R}}^{J_0\cdot J_1}$}\\
& \left[\begin{array}{c}
1-t \\ 2\mathbf{Q}^{1/2}\boldsymbol{\beta}
\end{array}\right] \preceq_{\mathcal{C}_5} 1+t,\tag{cone in $\operatorname{\mathbb{R}}^{2+(J_0+KM)\cdot J_1}$}
\end{align*}
where $\mathcal{K}=\mathcal{C}_1\times\mathcal{C}_2^{J_1}\times\mathcal{C}_3\times\mathcal{C}_4\times\mathcal{C}_5=\operatorname{\mathbb{R}}_+\times\operatorname{\mathbb{R}}_+^{J_1}\times \mathbb{R}_+^{J_0\cdot J_1}\times \mathbb{R}_+^{J_0\cdot J_1} \times \mathcal{K}_{1+(J_0+KM)\cdot J_1}$ is the conic constraint for this program, $\mathbf{a}=-2('\mathbf{Q}\widehat{\boldsymbol{\beta}} + \mathbf{G}^\star)'$, and $f=\widehat{\boldsymbol{\beta}}'\mathbf{Q}\widehat{\boldsymbol{\beta}} + 2\mathbf{G}^\star\widehat{\boldsymbol{\beta}}$.
\clearpage\section{Data Preparation and Software Implementation} \label{sec: SA data prep}
In this section, we first describe the variables in the \cite{Billmeier-Nannicini_2013_RESTAT} (BN, henceforth) dataset and then go through the details of our empirical specification.
\subsection{Data Description}
The original BN dataset contains data on some economic and political variables for 180 countries, over a period of time spanning from 1960 to 2005.\footnote{We downloaded the dataset from the Harvard Dataverse at \url{https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi:10.7910/DVN/28699}.} In detail, the variables available in the dataset are:
\begin{itemize}
\item real GDP per capita in 2002 US dollars.
\item enrollment rate in secondary schooling.
\item population growth.
\item yearly inflation rate.
\item the investment ratio (the investment of a country as a percentage of GDP).
\item an indicator that captures whether the country is a democracy (1) or not (0).
\item an indicator that captures whether the economy of the country is considered closed (0) or not (1) as developed in \cite*{Sachs-Warner_1995_BP} (henceforth, Sachs-Warner indicator). In particular, the indicator takes value 0 if any of the following conditions is verified:
\begin{enumerate}[label=\roman*)]
\item the average tariff is above 40\%;
\item non-tariff barriers are imposed on a volume of imports larger than 40\%;
\item the country has a socialist economic system;
\item the exchange rate black market premium is above 20\%;
\item state monopolies control most of the country exports.
\end{enumerate}
\end{itemize}
Despite having six candidate variables to match on, we end up matching on at most two variables because of missing data. Specifically, we always match on real GDP per-capita, whereas in the robustness exercise in Supplemental Appendix \ref{sec: SA robustness} we also match on investment-to-GDP ratio.
A second difference of our final dataset from the original one used in BN is the final pool of countries - treated and donors - on which we conduct the analysis. In particular, we adopt the following criteria to select the countries to be included in our final dataset as either donors or treated units:
\begin{enumerate}
\item We restrict the analysis to countries in Sub-Saharan Africa. We define this group by excluding from the analysis North African countries according to the United Nations (UN) classification (Algeria, Egypt, Libya, Morocco, Sudan, Tunisia, and Western Sahara) as well as any additional country that is not covered by the UN classification but is a member of the Arab League (Djibouti, Mauritania, Somalia).
\item As in \cite{Bratton1VandeWalle1997-book}, we exclude Namibia because the Sachs-Werner indicator is missing for such a country.
\end{enumerate}
Table \ref{tab: SA final units} shows the final set of countries we select, together with their treatment date.
\begin{table}[H]
\centering
\caption{List of all countries included in our analysis}
\begin{tabular}{lcclc}
\toprule\toprule
Angola & \multicolumn{1}{l}{$\infty$} & & Lesotho & $\infty$ \\
Benin & 1990 & & Madagascar & \multicolumn{1}{r}{1996} \\
Botswana & 1979 & & Malawi & $\infty$ \\
Burkina Faso & 1998 & & Mali & \multicolumn{1}{r}{1988} \\
Burundi & 1999 & & Mauritius & \multicolumn{1}{r}{1968} \\
Cabo Verde & 1991 & & Mozambique & \multicolumn{1}{r}{1995} \\
Cameroon & 1993 & & Niger & \multicolumn{1}{r}{1994} \\
Chad & \multicolumn{1}{l}{$\infty$} & & Nigeria & $\infty$ \\
Congo & \multicolumn{1}{l}{$\infty$} & & Rwanda & $\infty$ \\
Ethiopia & 1996 & & Senegal & $\infty$ \\
Gabon & \multicolumn{1}{l}{$\infty$} & & Sierra Leone & $\infty$ \\
Gambia & 1985 & & South Africa & \multicolumn{1}{r}{1991} \\
Ghana & 1985 & & Tanzania & \multicolumn{1}{r}{1995} \\
Guinea & 1986 & & Togo & $\infty$ \\
Guinea-Bissau & 1987 & & Uganda & \multicolumn{1}{r}{1988} \\
Ivory Coast & 1994 & & Zambia & \multicolumn{1}{r}{1993} \\
Kenya & 1993 & & Zimbabwe & $\infty$ \\
\bottomrule\bottomrule
\end{tabular}
\label{tab: SA final units}
\begin{center}
\parbox[1]{\textwidth}{\small \textit{Notes}: $\infty$ denotes that a country has never experienced liberalization during the observed time span.}
\end{center}
\end{table}
\subsection{Implementation Details}
In this section, we describe all the details of the empirical application presented in the main text.
\paragraph{Constraint type.} Our preferred specification uses the L1-L2 constraint, i.e.,
\[\mathcal{W}_{\mathtt{L1-L2}} = \bigtimes_{i=1}^{J_1} \left\{\mathbf{w}^{[i]} \in \mathbb{R}_+^{J_0}:||\mathbf{w}^{[i]} ||_1 = 1, ||\mathbf{w}^{[i]} ||_2 \leq Q^{[i]}\right\},\]
whereas the results in the Supplemental Appendix Section \ref{subsec: simplex-type constraint} and \ref{subsec: ridge-type constraint} use simplex- and Ridge-type constraints of the form
\[\mathcal{W}_{\mathtt{S}} = \bigtimes_{i=1}^{J_1} \left\{\mathbf{w}^{[i]} \in \mathbb{R}_+^{J_0}:||\mathbf{w}^{[i]} ||_1 = 1\right\},\qquad \mathcal{W}_{\mathtt{R}} = \bigtimes_{i=1}^{J_1} \left\{\mathbf{w}^{[i]} \in \mathbb{R}^{J_0}: ||\mathbf{w}^{[i]} ||_2 \leq Q^{[i]}\right\}.\]
Table \ref{table: regularization parameters values} shows the effective values for $Q^{[i]}, i=1,\ldots, J_1$ that we compute in our empirical application. Further below we explain in greater detail how these regularization parameters are computed in practice.
\paragraph{Selected features.} Our main specification uses only one feature ($M=1$)--the logarithm of real GDP per-capita--, uses the identify weighting matrix, and includes a constant term, that is
\[\mathbf{B}^{[i]} = \begin{bmatrix}
\mathbf{Y}_{1} & \cdots & \mathbf{Y}_{J_0}
\end{bmatrix}, \quad \mathbf{C}^{[i]} = \mathbf{1}_{T_i}, \quad \mathcal{R}=\operatorname{\mathbb{R}}^{J_1}, \quad \mathbf{V}^{[i]} = \mathbf{I}_{T_i}, \quad i = 1,\ldots, J_1,\]
where $\mathbf{Y}_j = (Y_{j1},\ldots, Y_{jT_i})', j=1,\ldots,J_0$ is the pre-treatment log-GDP per-capita of the $j$-th donor, $\mathbf{1}_{T_i}$ is a $T_i\times 1$ vector of ones, and $\mathbf{I}_{T_i}$ is the $T_i\times T_i$ identify matrix.
In Supplemental Appendix Section \ref{subsec: multiple features}, we present results using two features ($M=2$), where we also match on the investment-to-GDP ratio and control for a feature-specific constant term and linear trend, i.e.
\[\mathbf{B}^{[i]} = \begin{bmatrix}
\mathbf{Y}_{1} & \cdots & \mathbf{Y}_{J_0} \\
\mathbf{IR}_{1} & \cdots & \mathbf{IR}_{J_0}
\end{bmatrix},
\quad \mathbf{C}^{[i]} = \begin{bmatrix}
1 & 1\\
1 & 2 \\
\vdots & \vdots \\
1 & T_i
\end{bmatrix}, \quad \mathcal{R}=\operatorname{\mathbb{R}}^{2\cdot J_1}, \quad \mathbf{V}^{[i]} = \begin{bmatrix}
\mathbf{S}^{[i]}_Y & \mathbf{0} \\
\mathbf{0} & \mathbf{S}^{[i]}_{IR}
\end{bmatrix},\quad i = 1,\ldots, J_1,\]
where $\mathbf{IR}_j = (IR_{j1},\ldots, IR_{jT_i})', j=1,\ldots,J_0$ is the pre-treatment investment-to-GDP ratio of the $j$-th donor,
$\mathbf{S}^{[i]}_Y = \mathrm{diag}(\widehat{\sigma}^{-1}_{Y,1},\ldots, \widehat{\sigma}^{-1}_{Y,T_i})$, $\mathbf{S}^{[i]}_{IR} = \mathrm{diag}(\widehat{\sigma}_{IR,1}^{-1},\ldots, \widehat{\sigma}^{-1}_{IR,T_i}),$ with
\[\widehat{\sigma}_{W,t} = \left(\frac{1}{J_0-1} \sum_{j=1}^{J_0}(W_{jt} - \overline{W}_t)^2\right)^{1/2}, \quad \overline{W}_t = \frac{1}{J_0}\sum_{i=1}^{J_0}W_{jt}, \quad t=1,\ldots, T_i, \quad W\in\{Y,IR\},\]
and $\mathrm{diag}(\mathbf{x})$ yields a square diagonal matrix with the elements of $\mathbf{x}$ on its main diagonal.
\paragraph{Tuning parameters.} Regarding the choice of $Q^{[i]}$, it is well-established that the Ridge regression problem can be equivalently expressed both as an unconstrained penalized optimization problem and as a constrained optimization problem. For simplicity, assume \( \mathbf{C} \) is not included and \( M = 1 \). The two Ridge-type optimization formulations are as follows:
\[
\widehat{\mathbf{w}}^{[i]} = \arg \min_{\mathbf{w}^{[i]} \in \mathcal{W}} (\mathbf{A}^{[i]} - \mathbf{B}^{[i]}\mathbf{w}^{[i]})' \mathbf{V}^{[i]} (\mathbf{A}^{[i]} - \mathbf{B}^{[i]}\mathbf{w}^{[i]}) + \lambda^{[i]} ||\mathbf{w}^{[i]}||_2^2,
\]
where \( \lambda^{[i]} \geq 0 \) is a regularization parameter, and
\[
\widehat{\mathbf{w}}^{[i]} = \arg \min_{\mathbf{w}^{[i]} \in \mathcal{W}, ||\mathbf{w}^{[i]}||_2^2 \leq \left(Q^{[i]}\right)^2} (\mathbf{A}^{[i]} - \mathbf{B}^{[i]}\mathbf{w}^{[i]})' \mathbf{V}^{[i]} (\mathbf{A}^{[i]} - \mathbf{B}^{[i]}\mathbf{w}^{[i]}),
\]
where \( Q^{[i]} \geq 0 \) is an explicit upper bound on the norm of \( \mathbf{w}^{[i]} \). Under the assumption of Gaussian errors, an optimal choice of the regularization parameter \( \lambda^{[i]} \) for risk minimization, as suggested by \cite{Hoerl-Kannard-Kent_1975_ridge}, is:
\[
\lambda^{[i]} = \frac{J_0 (\widehat{\sigma}^{[i]}_{\text{OLS}})^2}{\|\widehat{\mathbf{w}}^{[i]}_{\text{OLS}}\|_2^2},
\]
where \( (\widehat{\sigma}^{[i]}_{\text{OLS}})^2 \) and \( \widehat{\mathbf{w}}^{[i]}_{\text{OLS}} \) are the estimates of the residual variance and the coefficients from the ordinary least squares (OLS) regression of $\mathbf{A}^{[i]}$ onto $\mathbf{B}^{[i]}$, respectively. Given the two optimization problems above, there exists a one-to-one correspondence between \( \lambda^{[i]} \) and \( Q^{[i]} \). For example, assuming the columns of \( \mathbf{B}^{[i]} \) are orthonormal, the closed-form solution for the Ridge estimator is:
\[
\widehat{\mathbf{w}}^{[i]} = (\mathbf{I} + \lambda^{[i]} \mathbf{I})^{-1} \widehat{\mathbf{w}}^{[i]}_{\text{OLS}},
\]
and if the constraint on the \( \ell_2 \)-norm is active, we have \( Q^{[i]} = \|\widehat{\mathbf{w}}^{[i]}\|_2 = \|\widehat{\mathbf{w}}^{[i]}_{\text{OLS}}\|_2 / (1 + \lambda^{[i]}) \). When more than one feature is considered (i.e., \( M > 1 \)), we compute the constraint size \( Q^{[i]}_\ell \) for each feature \( \ell = 1, \ldots, M \), and then choose \( Q^{[i]} \) as the most restrictive constraint to promote shrinkage of \( \mathbf{w}^{[i]} \):
\[
Q^{[i]} := \min_{\ell=1, \ldots, M} Q^{[i]}_\ell.
\]
\begin{table}[!ht]
\centering
\caption{Pre-treatment length and values of the regularization parameters $Q^{[i]}, i=1,\ldots, J_1$.}
\begin{tabular}{lcccccc}
\toprule\toprule
\textit{Treated Unit} & $Q^{[i]}, M=1$ & $Q^{[i]}, M=2$& $\mathcal{T}_i$ & $|\mathcal{T}_i|$ \\
\midrule
Benin & 2.041 & 0.699 & 1963 - 1989 & 27 \\
Botswana & 2.119 & 0.771 & 1963 - 1978 & 16 \\
Cabo Verde & 0.877 & 0.5 & 1963 - 1990 & 28 \\
Cameroon & 2.336 & 0.786 & 1963 - 1992 & 30 \\
Gambia & 1.401 & 1.035 & 1963 - 1984 & 22 \\
Ghana & 0.858 & 0.972 & 1963 - 1984 & 22 \\
Guinea & 0.591 & 0.65 & 1963 - 1985 & 23 \\
Guinea-Bissau & 0.942 & 0.718 & 1963 - 1986 & 24 \\
Ivory Coast & 1.212 & 0.838 & 1963 - 1993 & 31 \\
Kenya & 0.959 & 0.569 & 1963 - 1992 & 30 \\
Mali & 1.445 & 0.65 & 1963 - 1987 & 25 \\
Mauritius & 1.705 & 1.145 & 1963 - 1967 & 5 \\
Niger & 2.252 & 1.793 & 1963 - 1993 & 31 \\
South Africa & 1.169 & 0.5 & 1963 - 1990 & 28 \\
Uganda & 1.797 & 0.5 & 1963 - 1987 & 25 \\
Zambia & 2.554 & 0.5 & 1963 - 1992 & 30 \\
\bottomrule\bottomrule
\end{tabular}
\label{tab: SA Q1Q2}
\par
\begin{center}
\parbox[1]{\textwidth}{\small \textit{Notes:} the rule of thumb to compute $Q^{[i]}$ is the same for $L1$-$L2$ and Ridge. Moreover, the values for $Q^{[i]}$ reported are the ones obtained when predicting $\tau_{it}$ and $\tau_{i\cdot}$ which are identical by construction. Finally, $Q^{[i]}$ is lower-bounded at 0.5 to avoid excessive shrinkage. More details on the rules of thumb used can be found in \cite*{Cattaneo-Feng-Palomba-Titiunik_2025_JSS}, Section 3.1.}
\end{center}
\label{table: regularization parameters values}
\end{table}
\paragraph{In-sample Uncertainty.} In order to quantify the in-sample uncertainty from estimating the SC weights, we need to construct the bounds $\underline{M\mkern-4mu}\mkern4mu _{\mathtt{in}}$ and $\overline{\mkern-3muM\mkern-1.5mu}_{\mathtt{in}}$ on $\mathbf{p}_\tau'(\widehat\boldsymbol{\beta}-\boldsymbol{\beta}_0)$. The following strategy is adopted. First, we treat the synthetic control weights as possibly misspecified, thus estimating both the first and second conditional moments of the pseudo-true residuals $\mathbf{u}$. The conditional first moment $\mathbb{E}[\mathbf{u}\,|\,\mathscr{H}]$ is estimated feature-by-feature using a linear-in-parameters regression of the residual $\widehat{\mathbf{u}} = \mathbf{A} - \mathbf{B}\widehat{\mathbf{w}} -\mathbf{C}\widehat{\mathbf{r}}$ on $\mathbf{B}$ and the first lag of $\mathbf{B}$, whereas the conditional second moment $\mathbb{V}[\mathbf{u}\,|\, \mathscr{H}]$ is estimated with an HC1-type estimator. We then draw $S=200$ i.i.d. random vectors from the Gaussian distribution $\mathsf{N}(0,\widehat{\boldsymbol{\Sigma}})$, conditional on the data, to simulate the criterion function ${\ell^\star_{(s)}(\boldsymbol{\beta}-\boldsymbol{\beta}_0):=(\boldsymbol{\beta}-\boldsymbol{\beta}_0)'\widehat{\mathbf{Q}}(\boldsymbol{\beta}-\boldsymbol{\beta}_0)-2\mathbf{G}_{(s)}'(\boldsymbol{\beta}-\boldsymbol{\beta}_0)}$, $s=1,\ldots, 200$, and solve the following optimization problems
\begin{align*}
l_{(s)}:=\inf_{\begin{array}{c}\scriptstyle \boldsymbol{\beta}-\boldsymbol{\beta}_0 \in \Delta^\star, \\[1pt]
\scriptstyle \ell_{(s)}^{\star}(\boldsymbol{\beta}-\boldsymbol{\beta}_0)\leq 0
\end{array}} \;\mathbf{p}_\tau'(\boldsymbol{\beta}-\boldsymbol{\beta}_0) \qquad\text{and}\qquad
u_{(s)}:=\sup_{\begin{array}{c}\scriptstyle \boldsymbol{\beta}-\boldsymbol{\beta}_0 \in \Delta^\star, \\[1pt]
\scriptstyle \ell_{(s)}^{\star}(\boldsymbol{\beta}-\boldsymbol{\beta}_0)\leq 0
\end{array}} \;\mathbf{p}_\tau'(\boldsymbol{\beta}-\boldsymbol{\beta}_0),
\end{align*}
where $\Delta^\star$ is constructed as explained in Section 6.1. Finally, $\underline{M\mkern-4mu}\mkern4mu _{\mathtt{in}}$ is the $(\alpha_1/2)-$quantile of $\{l_{(s)}\}_{s=1}^S$ and $\overline{\mkern-3muM\mkern-1.5mu}_{\mathtt{in}}$ is the $(1-\alpha_1/2)-$quantile of $\{u_{(s)}\}_{s=1}^S$, where $\alpha_1$ is set to 0.05.
\paragraph{Out-of-sample Uncertainty.} In order to quantify the out-of-sample uncertainty from the stochastic error in the post-treatment period, we need to construct the bounds $\underline{M\mkern-4mu}\mkern4mu _{\mathtt{out}}$ and $\overline{\mkern-3muM\mkern-1.5mu}_{\mathtt{out}}$ on the out-of-sample error $e_\tau$ (associated with the $\tau$ prediction). We employ the non-asymptotic bounds described in (6.6), assuming that $e_\tau-\mathbb{E}[e_\tau|\mathscr{H}]$ is sub-Gaussian conditional on $\mathscr{H}$.
Then, we take
\[\underline{M\mkern-4mu}\mkern4mu _{\mathtt{out}}:=\mathbb{E}[e_\tau|\mathscr{H}]-\sqrt{2\sigma_{\mathscr{H}}^2\log(2/\alpha_2)}
\qquad\text{and}\qquad
\overline{\mkern-3muM\mkern-1.5mu}_{\mathtt{out}}:=\mathbb{E}[e_\tau|\mathscr{H}]+\sqrt{2\sigma_{\mathscr{H}}^2\log(2/\alpha_2)},\]
We set $\alpha_2=0.05$, and the conditional mean $\mathbb{E}[e_\tau|\mathscr{H}]$ and the sub-Gaussian parameter $\sigma_{\mathscr{H}}$ are parametrized and estimated by a linear-in-parameters regression of the pre-treatment residuals on $\mathbf{B}$.
\bigskip
\noindent Finally, the prediction intervals for the counterfactual outcome and the treatment effect of interest are given by
\[\left[\mathbf{p}_\tau'\widehat{\boldsymbol{\beta}}-\overline{\mkern-3muM\mkern-1.5mu}_{\mathtt{in}}+\underline{M\mkern-4mu}\mkern4mu _{\mathtt{out}}; \:
\mathbf{p}_\tau'\widehat{\boldsymbol{\beta}}-\underline{M\mkern-4mu}\mkern4mu _{\mathtt{in}}+\overline{\mkern-3muM\mkern-1.5mu}_{\mathtt{out}}\right]
\quad\text{and} \quad
\left[\widehat{\tau}+\underline{M\mkern-4mu}\mkern4mu _{\mathtt{in}}-\overline{\mkern-3muM\mkern-1.5mu}_{\mathtt{out}};\:
\widehat{\tau}+\overline{\mkern-3muM\mkern-1.5mu}_{\mathtt{in}}-\underline{M\mkern-4mu}\mkern4mu _{\mathtt{out}}\right],\]
respectively.
\paragraph{Other assumptions.} Throughout all our specifications, we maintain the assumptions that (\textit{i}) there is no anticipation of the treatment and (\textit{ii}) $\mathbf{A}$ and $\mathbf{B}$ form a cointegrated system. When (\textit{i}) is relaxed to allow for anticipation results remain qualitatively the same.
\clearpage
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