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A Relaxation Approach to Synthetic Control
A fundamental challenge in policy evaluation is that the potential outcomes in the absence of treatment are unobservable in the treated group. When resources are available to conduct randomized controlled trials, carefully designed experiments can identify the average treatment effect (ATE) by comparing outcomes between the treatment group and the control group. However, when only observational data are available, one prominent solution is the synthetic control method (SCM) abadie2003economic,abadieSyntheticControlMethods2010. A typical application involves a panel data with a single treated unit of interest and a set of control units, where they are connected by sharing latent common factors. SCM has been successfully applied to many economic studies, for example, kleven2013taxation, bohn2014did, acemoglu2016value, and cunningham2018decriminalizing. athey2017state hailed SCM as “arguably the most important innovation in the policy evaluation literature in the last 15 years.”
A pivotal step in SCM is assigning weights to the units in the donor pool to construct a “synthetic” version of the treated unit. These weights are determined by minimizing the discrepancy between the treated unit and the synthetic control in the pre-treatment period. The resulting synthetic unit is then used to extrapolate outcomes in the post-treatment period, with the estimated policy effect given by the difference between the observed outcome of the treated unit and the counterfactual prediction. In a standard SCM implementation, the weights are constrained into the simplex---that is, they must be non-negative and sum to one. This constraint often leads to sparse solutions, where only a few weights are non-zero. While such sparsity enhances interpretability, it may not always be desirable. In many practical economic settings, there is no compelling reason to assume that only a small subset of variables are relevant to the target variable giannone2021economic. When researchers invest substantial effort in collecting data from a large pool of potential controls, they may prefer dense weighting schemes that make a full use of the available information. A dense solution can also help diversify prediction risk, potentially improving the robustness of the counterfactual estimates liao2024does.
To encourage dense solutions, in this paper we propose a relaxation approach within the synthetic control framework, termed SCM-relaxation, for estimating the weights and predicting the counterfactual outcomes. Our approach is built on the $L_{2}$-relaxation method introduced by shi$ell_2$relaxationApplicationsForecast2022. It formulates the weight estimation as a constrained optimization problem, where the objective function is an information-theoretic divergence measure of the weight vector. Staying true to the spirit of SCM, we retain the simplex constraint. In addition, to address challenges arising from many potential control units, we introduce a relaxed form of the first-order condition (FOC) derived from the original SCM formulation.
In high-dimensional settings, developing asymptotic theory requires structural assumptions that enable effective dimension reduction. The search for a convex combination in SCM parallels the classical econometric problem of forecast combination, making it natural to leverage the aggregated information from multiple control units. Recent advancements in the literature on latent groups in panel data---bonhomme2015grouped, su2016identifying, and vogt2017classification---provide a useful framework to bridge the two extremes of full homogeneity (identical units) and full heterogeneity (distinct units). Under the latent group structures in the loadings of a factor model, this paper's (ref) shows that our estimated weight vector converges sufficiently fast to a desirable target, even when the number of control units ($J$) is much larger than the number of pre-treatment time periods ($T_{0}$). Once the weights are estimated, the ATE can be readily obtained through the predicted counterfactual. (ref) establishes that the prediction risk of the counterfactuals is asymptotically equal to that under the oracle weights.
Our use of divergence measures is motivated by the literature on generalized empirical likelihood (GEL) kitamuraEmpiricalLikelihoodMethods2007,neweyHigherOrderProperties2004. We develop a unified asymptotic framework that accommodates any convex divergence function with non-negative support and restricted strong convexity. This generalization, shown in (ref), allows our method to encompass several popular divergence-based estimators, including empirical likelihood owen1988empirical and entropy balancing hainmuellerEntropyBalancingCausal2012.
Our approach, as a convex optimization programming, is easy to implement numerically. We conduct Monte Carlo simulations to demonstrate the finite sample performance of our proposal in comparison with other off-the-shelf machine learning methods. Finally, as an empirical example we evaluate the impact of Brexit on the real GDP growth of the United Kingdom. We find substantial economic loss after Brexit, and the estimated weights of the control units from our method are more interpretable than those from SCM.
In the development of the asymptotics, we make three original theoretical contributions. Firstly, as mentioned above, we set up a general framework that accommodates the information-theoretic approach in formulating SCM by maintaining the simplex constraint. Secondly, we overcome a key technical challenge in handling many non-negativity constraints rising from the simplex constraint. The same issue appeared in risk management for large portfolios jagannathan2003risk, but to the best of our knowledge we are unaware of a theoretical investigation of the consequence. We provide a rigorous proof by taking the non-negativity constraints into consideration. In particular, (ref) explains our innovative idea in pushing the estimated vector to its desirable target by establishing the compatibility condition buhlmann2011statistics that connects the $L_{2}$-norm of the high-dimensional weight vector with its low-dimensional counterpart. Last but not least, while the group structures and the low-dimensional factor model are dimension-reduction assumptions, we allow both the number of groups ($K$) and the number of factors ($r$) to diverge to infinity, regardless of their relative order. It offers a flexible accommodation for approximate group features bonhomme2022discretizing and factor propagation feng2020taming, and it breaks the severe restriction $K\leq r$ in shi$ell_2$relaxationApplicationsForecast2022.
This paper stands on several strands of vast literature. We follow abadieUsingSyntheticControls2021's proposal by maintaining the simplex constraint, while in the meantime take advantage of the latent group structures. doudchenkoBalancingRegressionDifferenceDifferences2016 develop a constrained regression framework for balancing, regression, difference-in-difference, and SCM. Ours serves as an complementary method for estimating individual weights in their procedure when the donor pool exhibits latent groups. When the pre-treatment outcomes of the treated unit lie in the convex hull of the pre-treatment outcomes of the control units, the estimated weights induced by SCM may not be unique. This occurs particularly when $J$ is larger than $T_{0}$. abadiePenalizedSyntheticControl2021 utilize a penalization method to guarantee a unique solution, where the penalty is imposed on the pairwise distance of covariates between units. arkhangelsky2021synthetic and chernozhukov2021exact propose different versions of penalized SCM, including $L_{1}$- (Lasso), $L_{2}$- (Ridge), and the elastic net. Departing from the conventional penalty-based regularization, this paper takes the relaxation approach to guarantee the uniqueness of the solution.
Our method is closely related to forecast combination batesCombinationForecasts1969. Many generalizations have been proposed to improve the robustness of the optimization-based combination methods, in particular machine learning techniques such as regularization dieboldMachineLearningRegularized2019. Another popular method for constructing counterfactuals is the panel data approach (PDA) proposed by hsiao2012panel. PDA also uses linear combination of outcomes from control units to predict the counterfactual. It does not impose the simplex constraint as in SCM. The focal issue of PDA concerns the selection of control units. li2017estimation propose a Lasso method, while shi2023forward suggest forward selection.
A variety of asymptotic frameworks have appeared to justify SCM. abadieSyntheticControlMethods2010 derive bounds on the SCM estimator when perfect pre-treatment fit is met under fixed $J$ and diverging $T_{0}$. ferman2021synthetic and fermanPropertiesSyntheticControl2021 analyze the asymptotic property of estimated weights under the imperfect pre-treatment fit assumption when $T_{0}$ goes to infinity while $J$ is held fixed and diverges, respectively. A key message is that when $J$ is fixed, the SCM estimator will be biased even $T_{0}$ goes to infinity. If $J$ diverges, the estimator will be unbiased as weights spread out across the units. In a panel of two-way fixed effects, arkhangelsky2023large show consistency and asymptotic normality of SCM where the treatment is selected on permanent and time-varying components. Our theory adds to this literature by demonstrating the consistency of SCM-relaxation when $J$ diverges into the high-dimensional regime.
Finally, wang2020minimal, hainmuellerEntropyBalancingCausal2012 and zheng2024dynamic estimate the weights by minimizing an information-theoretic divergence measure. While the first paper is for general causal inference, the latter two focus on SCM. hainmuellerEntropyBalancingCausal2012 works with the entropy, and zheng2024dynamic use the empirical likelihood (EL). Our estimator is different from theirs in two aspects. First, we do not attempt to exactly match the synthetic control's outcomes with those of the treated unit but instead take the route of relaxation. Second, our estimator could be adapted to a class of general convex functions including both entropy and EL. We provide in-depth theoretical analysis about these specific choices.
Organization. The rest of the paper is organized as the following. Section (ref) sets up the data generating process (DGP) and motivates the SCM-relaxation method. Section (ref) then establishes the asymptotic theory of the SCM-relaxation estimators under the group structures of factor loadings. We conduct Monte Carlo experiments in Section (ref), and in Section (ref) we apply our method to evaluate the Brexit shock. Proofs of all theoretical results are relegated to the online appendix.
Notations. For a positive integer $N$, we use $[N]$ as shorthand for $\{1,2,\dots,N\}$. Let $a\wedge b:=\min(a,b)$. Absolute constants are positive finite numbers independent of sample size. Vectors and matrices are in bold case. For a generic $n\times1$ vector $\bm{a}$, let $\left\Vert \bm{a}\right\Vert _{p}=(\sum_{i\in[n]}|a_{i}|^{p})^{1/p}$ and $\left\Vert \bm{a}\right\Vert _{\infty}=\max_{i\in[n]}|a_{i}|$ be its $L_{p}$- and $L_{\infty}$-norm, respectively. We denote $\bm{1}_{n}$ and $\bm{0}_{n}$ as the $n\times1$ vector of ones and zeros, respectively. The $n$-dimensional simplex is $\Delta_{n}=\{\bm{a}\in\mathbb{R}^{n}\colon\min_{i\in[n]}a_{i}\geq0,\bm{a}'\bm{1}_{n}=1\}$. For an $m\times n$ matrix $\bm{A}$, let $\varsigma_{\max}(\bm{A})$ be its largest singular value and $\varsigma_{\min}(\bm{A})$ its minimum nonzero singular value. Let $\|\bm{A}\|_{2}=\varsigma_{\max}(\bm{A})$ be the spectral norm. For a symmetric matrix $\bm{B}$, let $\phi_{\min}(\bm{B})$ and $\phi_{\max}(\bm{B})$ be its minimum and maximum eigenvalues, respectively. The symbol “$\to_{p}$” signifies convergence in probability.
A researcher possesses data of $J+1$ cross-sectional units and $T_{0}+T_{1}$ time periods, where $T_{0}$ is the number of pre-treatment periods and $T_{1}$ the number of post-treatment periods. Let $\mathcal{T}_{0}:=[T_{0}]$ denote the index set of the pre-treatment periods, $\mathcal{T}_{1}:=\{T_{0}+1,\dots,T_{0}+T_{1}\}$ denote that of the post-treatment periods, and $\mathcal{T}:=\mathcal{T}_{0}\cup\mathcal{T}_{1}$. Unit $j=0$ is the sole individual that received a treatment during $t\in\mathcal{T}_{1}$. The remaining $J$ units, indexed by $j\in[J]$, make the donor pool of potential controls. In the potential outcome framework, let $y_{jt}^{I}$ be the outcome of unit $j$ being treated at time $t$, and $y_{jt}^{I}$ if untreated.\footnote{Following the literature, the superscripts “$N$” and “$I$” stand for “non-intervened” (interchangeable for “untreated”) and “intervened” (interchangeable for “treated”), respectively.} In reality, we observe $y_{jt}=d_{jt}y_{jt}^{I}+(1-d_{jt})y_{jt}^{N}$, where $d_{jt}=1$ for treatment and $d_{jt}=0$ otherwise. We consider the simple case that only one individual is treated after an event, and therefore the treatment indicator is $d_{jt}=1\left\{ j=0,t\in\mathcal{T}_{1}\right\} $, where $1\left\{ \cdot\right\} $ is the indicator function. The treatment effect at time $t\in\mathcal{T}_{1}$ is \[ \tau_{0t}=y_{0t}^{I}-y_{0t}^{N}. \] Since $y_{0t}^{I}$ is observable while $y_{0t}^{N}$ is unobserved in the post-treatment periods, the counterfactual $y_{0t}^{N}$ must be estimated from the data.
This paper presents the simple forms of SCM with the outcome variable only, as in ferman2021synthetic, fermanPropertiesSyntheticControl2021, and chenSyntheticControlOnline2023. SCM takes a linear combination $\sum_{j\in[J]}w_{j}y_{jt}$ of the control units' outcomes, where the weights $w_{j}$ fall into the $J$-dimensional simplex $\Delta_{J}$---they are non-negative and sum to one. The synthetic control estimator for $\bm{w}=(w_{1},\dots,w_{J})'$ is obtained by minimizing the $L_{2}$-distance between the treated unit and synthetic control in the pre-treatment periods, i.e.,
where $\bm{y}_{j}=(y_{j1},\dots,y_{jT_{0}})'$, $j\in\{0\}\cup[J]$ and $\bm{Y}=(\bm{y}_{1},\dots,\bm{y}_{J})$.\footnote{When additional covariates are available, the synthetic control can incorporate them into the quadratic form. abadie2003economic minimizes $(\bm{x}_{0}-\bm{X}\bm{w})'\bm{V}(\bm{x}_{0}-\bm{X}\bm{w})$, where $\bm{X}=(\bm{x}_{1},\cdots,\bm{x}_{J})$ consists of individual characteristics $\bm{x}_{j}$ which can include $\boldsymbol{y}_{j}$, and $\bm{V}$ is a $T_{0}\times T_{0}$ symmetric and positive semi-definite matrix.} abadieSyntheticControlMethods2010 motivate ((ref)) by a factor model. Suppose that the untreated potential outcomes $y_{jt}^{N}$ of each individual are generated from the linear factor model
where $u_{jt}$ is the idiosyncratic error, $\bm{f}_{t}$ is an $r\times1$ vector of latent common factors at time $t$, and it is multiplied by an $r\times1$ vector $\bm{\lambda}_{j}$ of factor loadings for individual $j$.\footnote{The factor model ((ref)) can incorporate individual- and time-specific fixed effects by setting $\bm{\lambda}_{j}=(1,\alpha_{j},\tilde{\bm{\lambda}}_{j}')'$ and $\bm{f}_{t}=(\gamma_{t},1,\tilde{\bm{f}}_{t}')'$.} Throughout this paper, we view the factors $f_{t}$ as random whereas factor loadings $\bm{\lambda}_{j}$ as deterministic. The common factors shared by the treated unit and the untreated ones generate correlations to be utilized for counterfactual prediction. abadieUsingSyntheticControls2021 stresses the importance of the simplex constraint. It ensures that all members in the donor pool make non-negative contributions as a weighted average, and in practice the solution $\hat{\bm{w}}^{\mathrm{SC}}$ is often sparse with many zeros.
In practical observational studies with panel data, researchers may have many potential control units in the donor pool, while the time dimension, mostly collected in low frequency, is not too long. In abadieSyntheticControlMethods2010's example of California tobacco control program $(J,T_{0})=(16,30)$, and abadie2015comparative's example of German reunification has $(J,T_{0})=(38,19)$. The number of control units $J$ is often comparable to, or even exceeds, the number of pre-treatment periods $T_{0}$. To prevent in-sample over-fitting that may occur when $J>T_{0}$, we propose the SCM-relaxation scheme.
To motivate SCM-relaxation, we consider the Lagrangian of ((ref)). When no confusion arises, we suppress the superscript $N$ for the pre-treatment potential outcome $y_{jt}^{N}$ for $t\in\mathcal{T}_{0}$ as they are observable. We temporarily ignore the non-negativity constraint and write the Lagrangian as \[ \frac{1}{2}\left\Vert \bm{y}_{0}-\bm{Y}\bm{w}\right\Vert _{2}^{2}+\gamma(\bm{w}'\bm{1}_{J}-1), \] where $\gamma$ is the Lagrangian multiplier associated with the equality constraint $\bm{w}'\bm{1}_{J}=1$. This is doudchenkoBalancingRegressionDifferenceDifferences2016 and li2020statistical's modified SCM. The corresponding Karush--Kuhn--Tucker (KKT) conditions are
where $\hat{\bm{\Sigma}}:=T_{0}^{-1}\bm{Y}'\bm{Y}$ is an $J\times J$ cross moment matrix of the control units, and $\hat{\bm{\Upsilon}}:=T_{0}^{-1}\bm{Y}'\bm{y}_{0}$. A unique closed-form solution to ((ref)) is \[ \hat{\bm{\Sigma}}^{-1}\biggl(\hat{\bm{\Upsilon}}-\frac{\bm{1}_{J}'\hat{\bm{\Sigma}}^{-1}\hat{\bm{\Upsilon}}-1}{\bm{1}_{J}'\hat{\bm{\Sigma}}^{-1}\bm{1}_{J}}\bm{1}_{J}\biggr) \] when $\hat{\bm{\Sigma}}$ is invertible, but if $J$ is close to $T_{0}$, some eigenvalues of $\hat{\bm{\Sigma}}$ are close to zero, leading to numerical instability. Moreover, the invertibility of $\hat{\bm{\Sigma}}$ must be violated if $J>T_{0}$.
To stabilize the solution, we estimate the weights by solving the following convex optimization problem
where $\eta=\eta_{T_{0},J}$ is a user-specified tuning parameter that depends on $T_{0}$ and $J$ but we suppress the subscript for conciseness. Here we follow abadieUsingSyntheticControls2021 to keep the simplex constraint $\bm{w}\in\Delta_{J}$. We call ((ref)) the $L_{2}$-SCM-relaxation problem, for $\Vert\hat{\bm{\Sigma}}\bm{w}-\hat{\bm{\Upsilon}}+\gamma\bm{1}_{J}\Vert_{\infty}\leq\eta$ is a “relaxation” of the FOC ((ref)).
The problem ((ref)) is strictly convex. In operational research, the set
is called the feasible set. If $S_{\eta}\neq\emptyset$, then the solution to ((ref)) is unique. The $L_{2}$-norm encourages diversified weights across similar control units to lower the prediction variance. It is inspired by the Dantzig selector candes2007dantzig which minimizes the $L_{1}$-norm of parameters for sparsity, and $L_{2}$-relaxation shi$ell_2$relaxationApplicationsForecast2022 in the forecast combination problem.
Let $(\hat{\bm{w}},\hat{\gamma})$ be the the solution to ((ref)) if $S_{\eta}\neq\emptyset$, which holds if $\eta$ is sufficiently large. On the one hand, if $\eta\geq$$\Vert\hat{\bm{\Sigma}}\boldsymbol{1}_{J}/J-\hat{\bm{\Upsilon}}\Vert_{\infty}$, then $\left(\hat{\bm{w}}=J^{-1}\boldsymbol{1}_{J},\widehat{\gamma}=0\right)$ is feasible and the objective function reaches the lowest bound.\footnote{The seemingly naive equal weight solution is a surprisingly robust estimator in empirical applications in economics and finance, leading to the “forecast combination puzzle” Stock2004combination and equal-weight portfolio demiguel2009optimal.} On the other hand, if $S_{\eta=0}$ is nonempty, then ((ref)) entails exact satisfaction of the FOC. With a proper choice of $\eta\in[0,\Vert\hat{\bm{\Sigma}}\boldsymbol{1}_{J}/J-\hat{\bm{\Upsilon}}\Vert_{\infty}]$, the relaxation scheme balances the bias and variance and mitigates the sensitivity of SCM weights to the sampling noise in $(\hat{\bm{\Sigma}},\hat{\bm{\Upsilon}})$, effectively preventing in-sample over-fitting.
Next, we explore the asymptotic properties of the relaxation scheme.
Statistical analysis of high-dimensional problems typically postulates certain structures on the DGP for dimension reduction. For example, variable selection methods such as Lasso tibshirani1996regression and SCAD fan2001variable are suitable for linear regressions with sparse coefficients, meaning most of the parameters are either exactly zero or approximately zero. Similarly, in large covariance matrix estimation, various structures have been proposed: bickel2008covariance impose sparse covariance entries, and tong2025cluster assume a block correlation matrix.
The group pattern in panel data has been widely adopted as a bridge between homogeneous coefficients and fully heterogeneous coefficients bonhomme2015grouped,su2016identifying,vogt2017classification. Were the factors $\boldsymbol{f}_{t}$ observable, the factor loadings play the role as the slope coefficients. Based on this resemblance, we assume latent groups across the factor loadings of the control units.\footnote{In analysis of large factor models, he2024penalized also borrows this setting for dimension reduction. } The donor pool is partitioned into $K$ disjoint groups $\mathcal{G}_{1},\dots,\mathcal{G}_{K}$, where $\mathcal{G}_{k}\cap\mathcal{G}_{\ell}=\emptyset$ for any $k\neq\ell$ and $\cup_{k=1}^{K}\mathcal{G}_{k}=[J]$. The factor loadings are homogeneous within each group, that is, for each $k\in[K]$, for any $i,j\in\mathcal{G}_{k}$, $\bm{\lambda}_{i}=\bm{\lambda}_{j}=\bm{\lambda}_{\mathcal{G}_{k}}$. To encode the group identities, we introduce a $J\times K$ membership matrix $\bm{Z}$, where $\bm{Z}_{jk}=1\{j\in\mathcal{G}_{k}\}$. Then the group structure can be written as
where $\bm{\Lambda}=(\bm{\lambda}_{1},\dots,\bm{\lambda}_{J})'$ is the matrix of factor loadings and $\bm{\Lambda}^{\mathrm{co}}=(\bm{\lambda}_{1}^{\mathrm{co}},\dots,\bm{\lambda}_{K}^{\mathrm{co}})^{\prime}$ collects all group-specific loadings; here the superscript “co” stands for “core.” Let $J_{k}:=|\mathcal{G}_{k}|$ be the number of controls in the $k$-th group; obviously $J=\sum_{k\in[K]}J_{k}$.
While the literature mostly assumes finite groups and finite factors, here we allow $K$ and $r$ to be either fixed or diverging, which accommodates “many groups” and “many factors.” In asymptotic statements, we will explicitly send $T_{0}\to\infty$, whereas $K$, $r$, $J$, and $T_{1}$ are viewed as deterministic functions of $T_{0}$ that go to infinity. We impose the following regularity conditions.
Due to their multiplicative form, the factors and loadings cannot be uniquely identified. (ref) requires the existence of desirable factors and the loadings. Part (ref) is standard in factor models bai2002determining,bai2003inferential. The invertibility condition and the boundedness of eigenvalues ensure that the factors are non-degenerate. We require $\hat{\bm{\Omega}}_{\bm{F}}$ to be invertible itself, not just have an invertible limit, because the expression of oracle weight (introduced later) implicitly relies on a nonsingular $\hat{\bm{\Omega}}_{\bm{F}}$, though in an asymptotic sense, an invertible limit would suffice. Part (ref) requires bounded loadings $\boldsymbol{\lambda}_{0}$ for the treated unit. Regarding the control units, a sufficient condition for a bounded $\|\bm{\lambda}_{k}^{\mathrm{co}}\|_{2}$ is assuming the maximum loading of order $O(1/\sqrt{r})$, as what liDeterminingNumberFactors2017 do, to accommodate a diverging number of factors. The rank condition implies that $\bm{\Lambda}^{\mathrm{co}}$ has $K\wedge r$ nonzero singular values, and thus $\sum_{k\in[K\wedge r]}\varsigma_{k}^{2}(\bm{\Lambda}^{\mathrm{co}})=\mathrm{trace}(\bm{\Lambda}^{\mathrm{co}\prime}\bm{\Lambda}^{\mathrm{co}})=\sum_{k\in[K]}\|\bm{\lambda}_{k}^{\mathrm{co}}\|_{2}^{2}=O(K)$. The assumption implies all singular values are of the same order. Intuitively, these conditions make sure that the core loadings are immune from collinearity and no single group makes dominant contribution to the variances of the outcome variables.
Under the latent group structure ((ref)), the sample covariance matrix for $\bm{Y}$ can be decomposed as $\hat{\bm{\Sigma}}=\hat{\bm{\Sigma}}^{*}+\hat{\bm{\Sigma}}^{\mathrm{e}}$, where \[ \hat{\bm{\Sigma}}^{*}:=\frac{1}{T_{0}}\bm{Z}\bm{\Lambda}^{\mathrm{co}}\bm{F}'\bm{F}\bm{\Lambda}^{\mathrm{co}}{}'\bm{Z}', \] and thus $\hat{\bm{\Sigma}}^{\mathrm{e}}:=(\bm{Z}\bm{\Lambda}^{\mathrm{co}}\bm{F}'\bm{U}+\bm{U}'\bm{F}\bm{\Lambda}^{\mathrm{co}}{}'\bm{Z}'+\bm{U}^{\prime}\bm{U})/T_{0}$, with $\bm{U}=(\bm{u}_{1},\dots,\bm{u}_{J})$ and $\bm{u}_{j}=(u_{j1},\dots,u_{jT_{0}})'$ for $j\in[J]$. Here the superscript “$*$” denotes the leading component---the sample covariance of the infeasible signal $\boldsymbol{\Lambda}\bm{f}_{t}$, whereas “e” stands for the remaining idiosyncratic error. Similarly, the covariance between $\bm{Y}$ and $\bm{y}_{0}$ can be written as $\hat{\bm{\Upsilon}}=\hat{\bm{\Upsilon}}^{*}+\hat{\bm{\Upsilon}}^{\mathrm{e}}$, where \[ \hat{\bm{\Upsilon}}^{*}:=\frac{1}{T_{0}}\bm{Z}\bm{\Lambda}^{\mathrm{co}}\bm{F}'\bm{F}\bm{\lambda}_{0}, \] and $\hat{\bm{\Upsilon}}^{\mathrm{e}}:=(\bm{Z}\bm{\Lambda}^{\mathrm{co}}\bm{F}'\bm{u}_{0}+\bm{U}^{\prime}\bm{F}\bm{\lambda}_{0}+\bm{U}^{\prime}\bm{u}_{0})/T_{0}$ with $\bm{u}_{0}=(u_{01},\dots,u_{0T_{0}})^{\prime}$. Next, we specify an asymptotic target for the $L_{2}$-SCM-relaxation estimator $\hat{\bm{w}}$. Consider the oracle problem of $L_{2}$-SCM-relaxation with infeasible data $y_{jt}^{*}\coloneqq\bm{\lambda}_{j}'\bm{f}_{t}$ that is free of the idiosyncratic errors:
Denote the solution to the above problem as $\bm{w}^{*}$.
The following lemma characterizes the oracle solution $\bm{w}^{*}$.
(ref) demonstrates that for $j\in\mathcal{G}_{k}$, the oracle weights $w_{j}^{*}=J_{k}^{-1}w_{\mathcal{G}_{k}}^{*}$ are evenly distributed within group $\mathcal{G}_{k}$. We allow the group weight $w_{\mathcal{G}_{k}}^{*}=0$ for some $k$. Such group does not contribute to the prediction of $\bm{y}_{0}$.
We will show that $\hat{\bm{w}}$ and $\bm{w}^{*}$ are sufficiently close in both $L_{1}$ and $L_{2}$ distance. Define $\phi_{J,T_{0}}:=\sqrt{(\log J)/T_{0}}$, which will serve as an upper bound of sampling errors and average cross-sectional dependence in the following assumption. Note that $\phi_{J,T_{0}}$ can shrink to 0 even if $J$ is much larger than $T_{0}$; for example, if $J=T_{0}^{2}$, then $\phi_{J,T_{0}}=\sqrt{2(\log T_{0})/T_{0}}\to0$ as $T_{0}\to\infty$.
(ref) imposes high-level conditions on idiosyncratic errors. Specifically, Condition (ref) means that $\mathbb{E}(\bm{f}_{t}u_{jt})=\bm{0}_{r\times1}$ and Condition (ref) ensures that the corresponding sampling errors is controlled by $\phi_{J,T_{0}}$ uniformly for all $j$ and $t$. Condition (ref) allows for mild cross-sectional dependence in idiosyncratic errors of the control units; note that this is a weak assumption since it only requires that on average $\mathbb{E}(T_{0}^{-1}\bm{u}_{i}^{\prime}\bm{u}_{j})$ should be controlled by $\phi_{J,T_{0}}$. On the other hand, the weak correlation of the idiosyncratic errors between the treated unit and the controls is for simplicity, so that the predictability is dominantly due to the factors. Sampling errors are restricted in Condition (ref). The order $\phi_{J,T_{0}}=\sqrt{(\log J)/T_{0}}$ for the sup-norm of the sampling errors is commonplace in high-dimensional statistics, and can be established from low-level assumptions; see, for instance, fan2013large and wainwright2019high.
Part (ref) of (ref) ensures that none of the groups is negligible in terms of its size; otherwise a tiny group contributes little in prediction but its associate weight can be disproportionately large. Part (ref) requires that the tuning parameter $\eta$ should shrink to 0 in a rate faster than $1/(K\wedge r)^{2}$ to guarantee the convergence of $\hat{\bm{w}}$, but slower than $K\sqrt{r}\phi_{J,T_{0}}$ so that the oracle weight $\bm{w}^{*}$ satisfies the relaxation $\Vert\hat{\bm{\Sigma}}\bm{w}^{*}-\hat{\bm{\Upsilon}}+\gamma^{*}\bm{1}_{J}\Vert_{\infty}\leq\eta$ with high probability.
The tuning parameter $\eta$ plays an important role in the finite sample behavior of the estimator. In asymptotic analysis we specify the rate for a proper $\eta$ as the sample size increases. In practice, the tuning parameter is chosen by cross validation; this is what we do in the numerical work throughout this paper.
The assumptions in the previous section are prepared for the consistency of $L_{2}$-SCM-relaxation.
(ref) shows that the estimator $\hat{\bm{w}}$ converges to $\bm{w}^{*}$ under both $L_{1}$- and $L_{2}$-norm with desirable rates of convergence. Its proof deviates substantially from that in shi$ell_2$relaxationApplicationsForecast2022, which resorts to the dual problem (unconstrained optimization) of their primal $L_{2}$-relaxation (constrained optimization). Had we mimic their proof, as many as $J$ non-negativity constraints from the simplex would make the dual formulation intractable. Therefore, here we must come up with a new strategy that directly works with the primal problem.
An unexpected windfall of this strategy of proof is that the asymptotic analysis can be carried over into general convex objective functions, to be elaborated in Section (ref). Moreover, it overcomes a longstanding technical challenge of establishing asymptotic results with a high-dimensional simplex constraint, which was also encountered as the no-short-sell constraint in large portfolios jagannathan2003risk.
(ref) has established that SCM-relaxation diversifies prediction risk over many control units. Next, we continue with the empirical risk. For a generic weight $\bm{w}\in\Delta_{J}$, define the in-sample empirical risk as $R_{\mathcal{T}_{0}}(\bm{w}):=T_{0}^{-1}\sum_{t\in\mathcal{T}_{0}}(\bm{w}'\bm{y}_{t}-y_{0t})^{2}$ and similarly the out-of-sample empirical risk as $R_{\mathcal{T}_{1}}(\bm{w})=T_{1}^{-1}\sum_{t\in\mathcal{T}_{1}}(\bm{w}'\bm{y}_{t}-y_{0t})^{2}$.
(ref) (ref) is an in-sample oracle equality, and (ref) is an out-of-sample oracle equality, where the extra condition $r\log(J)/T_{1}=O(1)$ is mild in that it allows $T_{1}$ to diverge at a much slower speed than $T_{0}$. This theorem shows that empirical risk under the weights estimated by $L_{2}$-SCM-relaxation from the training data is asymptotically as low as the oracle $\bm{w}^{*}$, up to an asymptotically negligible $o_{p}(1)$ gap. It highlights the effectiveness of the relaxation scheme: even though the relaxation method does not seek to identify the group membership, the risk of our estimator would be as good as if we were informed of the infeasible oracle group identities of the control units.
The $L_{2}$-norm objective in ((ref)) is one of the information-theoretic divergence measures. Shall we prefer the $L_{2}$-norm, or it is equivalent if another member of the family is summoned? We will provide an in-depth analysis in this section. Denote a generic divergence measure as $g(\cdot)$, which is strictly convex and sufficiently smooth, and then the associated $g$-SCM-relaxation problem solves
where $S_{\eta}$ is the feasible set defined by ((ref)), and the $\boldsymbol{w}$-part of the solution is denoted as $\hat{\bm{w}}_{(g)}$. Conformably, the oracle weights, denoted $\bm{w}_{(g)}^{*}$, are the solution to
Similar to (ref), it is easy to show that $\bm{w}_{(g)}^{*}$ is within-group equal, though a closed-form expression is unavailable for a generic $g$ function.
We study the asymptotic properties of $g$-SCM-relaxation, represented by two popular choices of $g(\cdot)$. The function $g(x)=-\log x$ implicitly keeps weights positive. This choice is closely related to the empirical likelihood owen1988empirical. We call the associate problem EL-SCM-relaxation, and the solution is denoted as $\hat{\bm{w}}^{\mathrm{EL}}$. Alternatively, the entropy function $g(x)=x\log x$ also ensures $x>0$, and it leads to entropy-SCM-relaxation with the solution $\hat{\bm{w}}^{\mathrm{entr}}$. In SCM with fixed $J$, zheng2024dynamic use EL and hainmuellerEntropyBalancingCausal2012 employs entropy in association with moment equalities. Their theory cannot be generalized to our high-dimensional contexts, where the moment conditions must be relaxed by inequalities.
To carry out analysis in a unified framework, we introduce a few notions that characterize the shape of function $g$. A differentiable function $g\colon\mathcal{D}\to\mathbb{R}$ is called $\alpha_{g}$-strongly convex on $\mathcal{D}\subseteq\mathbb{R}$ if for any $x,y\in\mathcal{D}$, we have $g(x)-g(y)\geq\frac{\mathrm{d}g(y)}{\mathrm{d}y}(x-y)+\frac{\alpha_{g}}{2}(x-y)^{2}$. Since a function $f$ is convex if and only if $f(x)-f(y)\geq\frac{\mathrm{d}f(y)}{\mathrm{d}y}(x-y)$ for any $x$ and $y$, an equivalent condition for $\alpha_{g}$-strong convexity is that $f(x)=g(x)-\frac{\alpha_{g}}{2}x^{2}$ is convex. Clearly, an $\alpha_{g}$-strongly convex function has its Bregman divergence bounded from below by $\frac{\alpha_{g}}{2}(x-y)^{2}$. A function $g\colon\mathcal{D}\to\mathbb{R}$ is called $\beta_{g}$-Lipschitz on $\mathcal{D}\subseteq\mathbb{R}$ if for any $x,y\in\mathcal{D}$, we have $|g(x)-g(y)|\leq\beta_{g}|x-y|$. A differentiable $\beta_{g}$-Lipschitz function $g$ must posses bounded derivative $|\mathrm{d}g(x)/\mathrm{d}x|\leq\beta_{g}$ for all $x\in\mathcal{D}$.
Typically, the weights $\hat{w}_{(g),j}$ and $w_{(g),k}^{*}$ lie in some interval $[\underline{c}/J,\bar{c}K/J]$. (ref) basically requires the ratio $\beta_{g}/\alpha_{g}$ to be small enough in that interval. We discuss this for $g(x)=-\log x$ and $g(x)=x\log x$.
Given a proper choice of $\eta$, the consistency of $\hat{\bm{w}}_{(g)}$ follows.
(ref) guarantees that $\hat{\bm{w}}_{(g)}$ behaves as well as the oracle weight $\bm{w}^{*}$ in large sample, as in the following corollary.
Although (ref) shows SCM-relaxation with the EL or entropy objective function can also achieve the out-of-sample oracle performance parallel to the $L_{2}$-norm objective as in (ref), we recommend $L_{2}$-SCM-relaxation for practical use. The EL and entropy cousins have the following drawbacks. First, to achieve the same convergence rate, both EL and entropy rely on extra technical conditions to ensure $\left\Vert \hat{\bm{w}}_{(g)}\right\Vert _{\infty}\leq\bar{c}K/J$.\footnote{This upper bound, determined by $K$ and $J$, is proven to hold for the $L_{2}$-norm objective.} Second, EL and entropy implicitly require that the weights be strictly positive, which incurs bias when some true group weights $\boldsymbol{w}^{*}$ are zero. This is particularly undesirable, as the true weight can lie on the boundary of the simplex, which is stressed by abadieUsingSyntheticControls2021.
Before we conclude this section, we would like to mention that feature engineering matters for all machine learning methods, including ours. If the scales of the empirical data is heterogeneous across the individuals, then a scale-standardization step can be desirable prior to feeding the data into the the SCM-relaxation algorithm. All theoretical results discussed above can be extended to the standardized version in a straightforward fashion; see Appendix (ref) for details.
In this section, we conduct Monte Carlo simulations to study the finite sample performance. We consider two sets of simulations: (i) the factor loadings follow an exact group structure as ((ref)) specifies; and (ii) the factor loadings fluctuate around the group means so that ((ref)) is violated. We estimate the weights by the relaxation methods proposed in this paper, including $L_{2}$-, EL-, and entropy-SCM-relaxation, and compare them with SCM. We also compute the weights by the off-the-shelf Lasso and Ridge methods:
where $\lambda^{\mathrm{Lasso}}$ and $\lambda^{\mathrm{Ridge}}$ are tuning parameters, and the weights are shrunken toward the simple average for a fair comparison. We further report the performance of shi2023forward's forward-selected PDA (fsPDA). For every method, the tuning parameters are selected via a two-fold cross-validation (CV) if $T_{0}<50$, and four-fold CV otherwise. For relaxation methods, the grid for $\eta$ spans $[0,\bar{\eta}]$, where the upper bound $\bar{\eta}=\min_{\gamma\in\mathbb{R}}\Vert\hat{\bm{\Sigma}}\bm{1}_{J}/J-\hat{\bm{\Upsilon}}+\gamma\bm{1}_{J}\Vert_{\infty}$ so that $J^{-1}\boldsymbol{1}_{J}$ is a feasible solution.
In the two sets of experiments, $T_{1}$ is fixed at $50$, $J\in\left\{ 50,100,200\right\} $ and $T_{0}\in\{J/2,J,2J\}$. To account for diverging number of factors and groups, we set $r=\lfloor\log T_{0}\rfloor$ and $K\in\{\lfloor0.8r\rfloor,r,\lfloor1.2r\rfloor+1\}$ which corresponds to $K<r$, $K=r$, and $K>r$, respectively. The experiment is repeated 1000 times in each DGP. We calculate the oracle weights $\bm{w}^{*}$ following ((ref)) (or ((ref)) for a generic objective function). Define the oracle synthetic control as $y_{0t}^{N,*}:=\sum_{j=1}^{J}w_{j}^{*}y_{jt}^{N}$. For a synthetic control estimator $\hat{\bm{w}}$ that yields the outcome estimate $\widehat{y}_{0t}^{N}$, the post-treatment prediction error is $\sum_{t\in\mathcal{T}_{1}}(\widehat{y}_{0t}^{N}-y_{0t}^{N,*})^{2}$. For ease of comparison, we treat SCM's $\hat{\bm{w}}^{\mathrm{SC}}$ as the benchmark; with the corresponding estimated counterfactual $\widehat{y}_{0t}^{N,\mathrm{SC}}$, we assess the out-of-sample performance of each estimator by the ratio \[ \sum_{t\in\mathcal{T}_{1}}\left(\widehat{y}_{0t}^{N}-y_{0t}^{N,*}\right){}^{2}\bigg/\sum_{t\in\mathcal{T}_{1}}\left(\widehat{y}_{0t}^{N,\mathrm{SC}}-y_{0t}^{N,*}\right){}^{2}. \] Moreover, we present the the $L_{1}$- and $L_{2}$-distance ratios, $\left\Vert \hat{\bm{w}}-\bm{w}^{*}\right\Vert _{1}/\Vert\hat{\bm{w}}^{\mathrm{SC}}-\bm{w}^{*}\Vert_{1}$ and $\left\Vert \hat{\bm{w}}-\bm{w}^{*}\right\Vert _{2}/\Vert\hat{\bm{w}}^{\mathrm{SC}}-\bm{w}^{*}\Vert_{2}$, to check the quality of the weight estimation.
The simulated data consist of $J+1$ units and $T_{0}+T_{1}$ time periods. The treatment to the unit $j=0$ occurs immediately after time $T_{0}$ and sustains from $T_{0}+1$ on. The potential outcome, if untreated, follows a factor model, as ((ref)):
The idiosyncratic errors $u_{jt}\stackrel{\text{i.i.d.}}{\sim}\mathcal{N}(0,1)$ and are independent of the common factors. The $r$ factors $f_{\ell t}$ ($\ell\in[r]$) are mutually independent, and each follows an AR(1) process: \[ f_{\ell t}=0.5f_{\ell t-1}+u_{\ell t}^{f},\qquad\ell\in[r],\ t\in[T_{0}+T_{1}], \] where $u_{\ell t}^{f}\stackrel{\text{i.i.d.}}{\sim}\mathcal{N}(0,1)$. Each entry in the core factor loadings $\bm{\Lambda}^{\mathrm{co}}$ is independently drawn from $\mathcal{N}(0,3/r)$. The loadings for the control units are given by $\bm{\Lambda}=\bm{Z}\bm{\Lambda}^{\mathrm{co}}$. The loading for the treated united is generated as $\bm{\lambda}_{0}=\bm{\Lambda}^{\mathrm{co}\prime}\bm{w}_{\mathcal{G}}^{*}+\bm{\varepsilon}$, where $\bm{\varepsilon}$ have entries drawn independently from $\text{Uniform}(-0.1/\sqrt{r},0.1/\sqrt{r})$ and $\bm{w}_{\mathcal{G}}^{*}=(w_{\mathcal{G}_{1}}^{*},\dots,w_{\mathcal{G}_{K}}^{*})^{\prime}$ has its first entry set to be zero and the other $K-1$ entries jointly generated from a Dirichlet distribution so that $\bm{w}_{\mathcal{G}}^{*}$ is assured to live in the simplex. The noise $\bm{\varepsilon}$ implies that $\bm{\lambda}_{0}$ may not be perfectly recovered as a convex combination of the core loadings. The loadings and weights, once drawn, are fixed over the replications.
Table (ref) reports the average (over the replications) post-treatment prediction error ratio. It reveals that all methods except fsPDA outperform the canonical SCM for all combinations of $K$ and $r$. The unfavorable performance of fsPDA is anticipated, as the oracle weights are dense and constrained on the simplex while the estimated weights of fsPDA do not lie on the simplex, and are typically sparse. Lasso and Ridge surpass SCM substantially by penalizing solutions toward the simple average. Moreover, Ridge also encourages dense solutions, which gains an edge over Lasso. Leveraging latent group structures and dense weights, the SCM-relaxation methods further improve on Ridge in all settings. Among the relaxation approaches, the $L_{2}$norm objective excels when $K\leq r$, due to its advantage of allowing for exact zero weights. Furthermore, entropy is uniformly better than EL in estimating weights close to zero, in view of the fact that $\lim_{x\to0^{+}}x\log x=0$ while $\lim_{x\to0^{+}}\log x=-\infty$. When $K>r$, the oracle weights vary with the choice of objective function. As a result, weight estimates by $L_{2}$, EL, and entropy can have different oracle targets. Overall, $L_{2}$-SCM-relaxation stands out as the best performer and is recommended in practice.
The performance of the estimated weights is displayed in (ref). Lasso and Ridge are better in recovering the oracle weights than SCM, and Ridge outperforms Lasso in terms of $L_{2}$-distance as it tends to produce weights with a smaller $L_{2}$-norm. Among SCM-relaxation methods, $L_{2}$-SCM-relaxation consistently excels in estimating the weights for $K\leq r$. Again, because of potentially different oracle targets when $K>r$, the $L_{2}$-objective is not guaranteed to yield the best performance compared with EL and entropy. For evaluation in terms of $L_{1}$-distance, the three relaxation methods perform comparably well. These findings align with our theory.
To see how well these methods recover the group structure of oracle weights, (ref) illustrates the estimated weights in a typical simulation with 3 groups, 3 factors, 200 control units and 100 pre-treatment periods. The oracle weights (the black dash line) are clustered by groups for visualization. As is clear in the figure, SCM produces sparse weights, concentrating on groups with nonzero oracle weights but deviating far from the within-group equal oracle weight. Lasso tends to collapse its weights to the simple average, the target of shrinkage. Ridge, with the denser solution, better approximates the group structure. As predicted by our theory, $L_{2}$-SCM-relaxation nearly perfectly traces the group pattern with the smallest estimation errors.
While EL and entropy objectives partially recover the group patterns, they exhibit large fluctuations in estimating weights, particularly for groups with higher oracle weights. This variability stems from the nonuniform curvature of their objective functions’ penalty terms, as measured by second derivatives wainwright2019high. For example, the second derivative of the entropy function $x\log x$ is $1/x$, which decreases from $+\infty$ to 100 as the oracle weight increases from 0 to 0.01. Consequently, the entropy function imposes disproportionately heavy penalties on groups with near-zero oracle weights compared to those with higher weights. This is why, in the last subgraph of (ref), the volatility of weights keeps growing from the first group to the third group. The same reasoning applies to the “EL” subgraph. In contrast, the constant curvature of the $L_{2}$ penalty ensures uniform penalization across all groups.
In this section, we consider an approximate group design to check the robustness of the estimation methods when the exact group structures do not hold. All the settings are the same as those in the previous subsection except that the loadings are generated as $\bm{\Lambda}=\bm{Z}\bm{\Lambda}^{\mathrm{co}}+\bm{\Xi}$ where $\bm{\Xi}$ consists of entries drawn independently from $\text{Uniform}(-0.2/\sqrt{r},0.2/\sqrt{r})$ and is independent of other random variables. (ref) presents the average post-treatment prediction error ratios. The patterns are similar to those in (ref) in exact group settings: the relaxation methods outperform substantially SCM as well as Lasso and Ridge; moreover, $L_{2}$SCM-relaxation is the best when $K\leq r$.
The Monte Carlo simulation evidence shows that the relaxation methods are robust against mild deviation from group structure, manifesting their practical usefulness.
In this section, we examine the impact of the 2016 Brexit referendum on the real GDP of the United Kingdom (UK). On June 23, 2016, a referendum on the UK's membership in the European Union (EU) resulted in 51.89% of voters favoring departure, triggering a withdrawal process that would conclude on January 31, 2020. The referendum's outcome was largely unexpected by markets. We designate the third quarter of 2016 as the starting point of the treatment.
We collect quarterly real GDP data for all available economies from the CEIC database, a subsidiary of Caixin providing business information. The completeness of the quarterly GDP time series varies across the countries and regions, with a fraction of them having records only in recent years. We shape the available data into a balanced panel, with a comprehensive donor pool of 57 countries from 2002Q4 to 2024Q2. To remove time trends and seasonality, we construct the year-over-year (YoY) GDP growth rate $\tilde{y}_{jt}=y_{jt}/y_{j(t-4)}-1$ for all economies. Figure (ref) plots UK's quarterly real GDP (blue line) and its YoY growth rate (red line).
We fit and predict the GDP growth rate by SCM and the three relaxation methods. Shown in Figure (ref), all methods exhibit similar patterns with pre-treatment fit. SCM has the smallest in-sample empirical risk, which is exactly the objective function it minimizes; it leads to an aggressive post-treatment extrapolation that visibly deviates from the realization. The relaxation approaches are more conservative in both the in-sample fit and the out-of-sample prediction.
Given that the counterfactuals are constructed by weighting the control units, we report in Figure (ref) the weights from SCM and those of $L_{2}$-SCM-relaxation.\footnote{The weights from the EL and entropy objectives are similar to those of $L_{2}$ and thus omitted.} The country/region names are sorted by the size of the $L_{2}$ weights, according to which we manually classified into Group 1 ($\geq0.04$, dark blue), Group 2 ($[0.01,0.04)$, light blue), Group 3 ($(0,0.01)$, red), and Group 4 (exactly zero, brown). Group 1 includes major EU countries Italy, Germany, Denmark, and France, as well as big English-speaking countries Canada and the United States. They are important economic partners of the UK. Group 2 further covers several middle-sized EU economies. Those in Group 3 and 4 are mostly geographically distant from the UK. In contrast, the sparse weights produced by SCM are less interpretable. While USA alone takes 30% of the weight, the estimate excludes Germany, France, and Italy, the top 3 EU countries by GDP, but instead places non-trivial weights on Lithuania and Luxembourg, countries that play a minor economic role.
To assess the GDP loss in levels, we use the estimated GDP growth rate counterfactuals to reconstruct the counterfactual real GDP. Let $\widehat{\tilde{y}}_{0t}^{N}=\sum_{j\in[J]}\hat{w}_{j}\tilde{y}_{jt}$ in the post-treatment period, and we compute the level GDP as \[ \hat{y}_{0t}^{N}=(1+\widehat{\tilde{y}}_{0t}^{N})z_{0(t-4)},\quad t\in\mathcal{T}_{1}, \] where the base $z_{0t}=y_{0t}$ if $t\leq T_{0}$ and $z_{0t}=\hat{y}_{0t}^{N}$ for $t>T_{0}$.
We calculate $\sum_{t\in\mathcal{T}_{1}}(y_{0t}^{I}-\hat{y}_{0t}^{N})/\sum_{t\in\mathcal{T}_{1}}y_{0t}^{I}$, the treatment effects relative to the UK's total GDP after the treatment. SCM predicts a substantial loss of $4.93\%$ of the total realized GDP. $L_{2}$-, EL- and entropy-SCM-relaxation estimate $3.85\%$, $3.64\%$ and $3.90\%$, respectively. All numbers suggest that Brexit has shrunken the UK's economy. In Figure (ref), the gap between the counterfactual and the realization is visible from 2016Q3 to 2020Q1, and the negative impact quickly worsens after 2020Q1. This underscores the delayed but substantial economic consequences of Brexit, which were not immediately evident in the aftermath of the referendum but emerged clearly following the formal exit.\footnote{The first quarter of the official departure coincided with the breakout of Covid-19, which triggered a worldwide economic recession. The pandemic was a global event that affected all countries. Given that the UK's public health system was relatively well developed compared to other countries, the GDP gap could be even larger in absence of the pandemic.} The substantial effect after official departure is huge; if we compute the corresponding ratio after 2020Q1, SCM yields a shocking loss of $7.74\%$, whereas the relaxation method outcomes range from $5.58\%$ to $5.98$%.
While the outcome of referendum was largely unexpected, the economic decoupling has been working in progress after 2016, with 2020 in anticipation. As a robustness check, we carry out re-estimation by setting the treatment starting point at 2020Q1. In Figure (ref), the in-sample empirical risks are similar to those of Figure (ref), whereas the variation of SCM weights in Figure (ref) is substantial relative to those in Figure (ref). For the loss of total GDP, SCM estimates $5.22\%$ of the UK's total GDP after 2020Q1, while the $L_{2}$, EL and entropy estimate $4.42\%$, $3.61\%$ and $4.02\%$, respectively. These enlarged effects are due to the drop of the “mild loss” period between 2016 and 2020 as shown in Figure (ref). They are smaller than the numbers for the same time period reported at the end of the last paragraph. This is due to the fact that this exercise uses the 2016--2020 data for in-sample fit, where the downward anticipation is leaked into the pre-treatment period and contaminates the weight estimation. Therefore we view that the results are cleaner with 2016Q1 as the treatment point. Overall, regardless of the choices of treatment point, the departure of the UK has inflicted a profound economic loss.
We propose a machine learning algorithm, termed SCM-relaxation, to estimate the weights for counterfactual prediction of a treated unit at the presence of many potential control units. Given a factor model with the latent group structures, $L_{2}$-SCM-relaxation produces weights that converge asymptotically to the oracle weights. The convergence rate is sufficiently fast to ensure oracle prediction accuracy as if the group membership were known. If one chooses the EL or the entropy objective function to conduct the relaxation, under extra regularization conditions they possess similar properties as the $L_{2}$ counterpart if the true weight is non-zero. Our theoretical study suggests that $L_{2}$-SCM-relaxation enjoys desirable properties under fewer conditions and is thus preferred. We apply them to evaluate the impact of Brexit on the UK's real GDP, and find that the estimated counterfactual without Brexit would be substantially higher than the realized values.
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