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The inclusive Synthetic Control Method
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Keywords: Causal inference, iSCM, Multiple treated, Spillover effects, Synthetic Control Method.\\ JEL classification: C21, C23, C31, C33.
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The synthetic control method (SCM), introduced by Aba2003 and further developed in Aba2010 and Aba2015, estimates the causal effect of a policy intervention in settings with few treated and control units observed over a long time period. SCM constructs a linear combination of control units to mimic what would have happened to the treated units had the intervention not occurred. The weights for each control unit are chosen to minimize differences in pre-intervention characteristics between the treated units and the synthetic control. The causal effect is then estimated as the difference between the observed outcome of the treated units and that of the synthetic control in the post-intervention period.
A key assumption of SCM is that only units not affected by the intervention are included in the control group, often called the donor pool. This can be problematic in scenarios where (i) some treated units should ideally be included in the donor pool to improve pre-intervention fit, or (ii) some control units are indirectly affected by the intervention, but excluding them from the donor pool substantially deteriorates the pre-intervention fit.
As a motivating example, consider the German reunification study by Aba2015 (see also Aba2020). The authors suggest that reunification may have had spillover effects on neighboring countries like Austria. Since Austria receives a substantial weight (42%) in constructing "synthetic West Germany," any significant spillover effect could introduce considerable bias. Excluding Austria from the donor pool substantially reduces the quality of the match between the treated and synthetic units. This is also the case when using the penalized SCM (see, Section (ref)). Moreover, excluding West Germany from Austria's donor pool when estimating the spillover effect yields implausible positive spillover effects on Austria's GDP.
Our main contribution is the introduction of the inclusive synthetic control method (iSCM), a novel procedure that allows us to eliminate post-intervention effects from control units and safely include them in the donor pool. We exploit the fact that synthetic control weights are estimated using only pre-intervention data, during which no unit is treated. By adding "potentially affected" units to the donor pool, the synthetic control becomes a weighted average of all donor units' outcomes, including any intervention effects on these units, weighted by their synthetic weights. Comparing the "main treated" unit and its synthetic control provides an estimate of the true effect minus the effects on other units, each weighted by their synthetic weights.
By creating synthetic control versions of all "potentially affected" units (including the "main treated" unit in their respective donor pools), we obtain estimates of the true effects on those units plus the effects on other affected units, weighted by their synthetic weights. This process yields a system of equations with "m" unknowns—the treatment effect on the "main treated" unit and the "m-1" effects on the "potentially affected" units—which can be solved using Cramer's rule. Importantly, our procedure does not require any modification of the synthetic control estimator. Any SCM-type method that creates a synthetic control as a weighted average of donor units' outcomes can be used (e.g., Aba2019, Amj2018, Ben2020, Ben2019, Dou2016, Fer2019, Kel2020, Xu2017). In addition to the assumptions of the chosen SCM estimator, our iSCM requires prior knowledge of which units may be "potentially affected" by the treatment.
While iSCM requires at least one "pure control" unit receiving non-zero weight, the method becomes less reliable as the number of "potentially affected" units increases. Therefore, it is advisable to impose assumptions that limit this number, aligning with the spillover effects literature, which often assumes interactions only within certain groups (Cer2017, For2018, Hub2019, Vaz2017). Existing extensions of SCM that account for spillover effects often reduce the donor pool to unaffected units. For instance, grossi2024direct estimate effects for the treated unit using SCM with a restricted donor pool and estimate spillover effects by comparing spillover-affected units with the restricted donor pool. Their method works well when the restricted donor pool generates a good synthetic control. However, when including spillover-affected units in the donor pool improves pre-intervention fit, as in the German reunification example, their method may be less reliable.
Cao2019 provide a different identification and estimation strategy focusing more on the spillover effects structure. In contrast, our approach can also be used in applications where one wants to include treated units in the donor pool. Another difference between the two approaches is that we allow for using any SC-type estimator as soon as the final estimator is based on a weighted average of the outcome of the units in the donor pool.
Although our paper is related to the literature on spillover effects, our results also apply when there are multiple treated units and no spillover effects. Moreover, our identification relies on being able to observe the potential outcome under non-treatment of every unit in the pre-intervention period and it differs from the identification results in the spillover/peer effect literature.
Our setting can also be viewed as a multiple treated units scenario, inviting a comparison to methods developed for such contexts (Aba2019, Ben2019, Kel2020). These methods either modify traditional SCM by introducing a penalty term to reduce large discrepancies between the treated unit's pre-treatment characteristics and those of its synthetic control, or they combine SCM with matching estimators to balance interpolation and extrapolation biases. While these approaches are useful in high-dimensional or disaggregated data settings, they are not designed to handle scenarios where the ideal donor pool includes units potentially affected by the intervention, either directly or through spillover effects.
In contrast, our iSCM approach allows for the direct inclusion of these affected units in the donor pool without modifying the underlying SCM estimator (which could be any of the methods mentioned above). This is particularly beneficial in multiple treated units settings, where excluding treated or spillover-affected units could weaken the quality of the synthetic match, as demonstrated in our application.
In conclusion, our iSCM allows for the inclusion of units "potentially affected" directly or indirectly by the intervention in the donor pool. Given its ease of implementation, iSCM can always be used, even as a robustness check. Additionally, we provide a data-driven procedure to determine whether iSCM is preferable to a restricted SCM that excludes these "potentially affected" units from the donor pool.
The rest of the paper is organized as follows: Sections (ref) describe the iSCM; Section (ref) discuss the special case where there is only one "potentially affected" unit; Section (ref) compare the estimation errors of SCM and iSCM; in Section (ref) we compare iSCM and a "restricted" SCM; Section (ref) suggests possible inference procedures; Section (ref) shows the results of the empirical application; and Section (ref) concludes the paper.
Assume we observe $j=1,...,J$ units for $t=1,...,T$ periods and that an intervention occurs at time $T_0$. For each unit $j$ at time $t$ we observe the outcome of interest $Y_{jt}$ and a set of $k$ predictors of the outcome $X_{1j},..., X_{kj}$, which often includes pre-intervention values of $Y_{jt}$. We refer to the treated unit (unit $1$) as the "main treated". We also assume that the $J-1$ units in the donor pool, include $m< J-2$ units (units $2$ to $m$) that are directly or indirectly affected by the intervention (“potentially affected” hereafter), i.e., they are either other treated units that we would like to include in the donor pool or control units that might be affected by spillover effects from the main treated. We refer to units $m+1$ through $J$, as "pure control" units and assume that they are not affected by the intervention at all. We define the potential outcome $Y_{1t}^I$ as the outcome that the "main treated" unit would obtain under the intervention at time $t$. We define the potential outcome of "potentially affected" units $j$ at time $t$ as $Y_{jt}^S, j=2,\ldots,m$, which represents either the outcome of a unit "potentially affected" by spillover effects or simply the potential outcome under treatment if unit $j$ is a different treated unit. Finally, we define $Y_{jt}^N, j=1,\ldots,J$ as the potential outcome in the absence of the intervention. We assume there are no anticipation effects and that the standard Stable Unit Treatment Value Assumption (SUTVA) holds, except for the potential presence of spillover effects on units, who are chosen a priori. This assumption can be formalized as
Assumption 1:
Assumption 1 may be restrictive in scenarios where it is unclear which units might be "potentially affected" by spillover effects. However, it is standard in contexts where the "potentially affected" units are other treated units.
We are interested in the causal effect of the intervention on the "main treated" at time $t>T_0$, denoted by $\theta_{1t}$, and the causal effects on the other "potentially affected" units denoted by $\gamma_{jt}, j=2,\ldots, m, t>T_0$, defined as
and
To identify these causal effects, we need to recover $Y_{1t}^N$ and $Y_{jt}^N$ for $j=2, \dots, m$ in the post-intervention period.\\ Assume we use a SC-type estimator $\widehat{Y}_{1t}^N= \sum_{j=2}^J \widehat{w}_{j}Y_{jt}$, which estimates $Y_{1t}^N$ as a weighted average of the post-intervention outcomes of the units in the donor pool, e.g., the original SCM as described in Aba2010. Let the $\left((J -1)\times 1\right)$ vector of weights $\widehat{W}=(\widehat{w}_{2}, \ldots , \widehat{w}_{J})'$ be the estimated weights of the chosen SC-type estimator that includes also the "potentially affected" units in the donor pool, under Assumption 1, $\forall \ t>T_0,$ we have
Therefore, under Assumption 1, we can decompose the estimation error\footnote{We will refer to the estimation error as bias hereafter. } of $\widehat{Y}_{1t}^N$, in two parts:
Similarly, consider a generic "potentially affected" unit $i$, $i \in M\equiv [2, \ldots, m]$. Let $\widehat{L}^{i}$ the vector of weights estimated to create $\widehat{Y}_{it}^N= \sum_{j\neq i} \widehat{l}_{j}^{i}Y_{jt} $ a SC-type estimator of $Y_{it}^N$ that includes the "main treated" (unit 1), the other $m-1$ "potentially affected" units, as well as the $J-m$ "pure" control units in the donor pool. Assumption 1 implies that, $\forall \ i=2,\ldots,m, \ t>T_0$, we have
Therefore under Assumption 1 we can decompose the estimation error of $\widehat{Y}_{it}^N, i=2,\ldots,m$, in two parts:
Looking at the bias decomposition in ((ref)) and ((ref)) it is clear that we have two distinct sources of bias. The first, $B^j_{sc}, j=1,\ldots,m$, relates to how well the chosen SC-type estimators approximate the post-intervention counterfactual. This is the usual source of bias we encounter when using any SC-type estimator. The second, $B^j_{te}, j=1,\ldots,m$, is induced by the fact that we are using units that are "potentially affected" by the intervention in the donor pool. This type of bias solely depends on whether or not a given ("potentially affected") unit receives weight and its respective treatment effect but not on how "good" the chosen estimator is in recovering $Y_{jt}^N, j=1,\ldots,m$.
We will now assume that we have chosen a SC-type estimator to recover $Y_{1t}^N$ for which the first source of bias is negligible, formally
We state Assumption 2 in terms of estimated weights without any loss of generality and to simplify the discussion. Alternatively we can assume that the set of weights $\widehat{W}$ converges to a set of weights $W^*$ such that $\sum_{j=2}^J w^*_{j}{Y_{jt}^N}= Y_{1t}^N+ o_p(1)$. When using the original SC-estimator, Assumption 2 amounts to assuming an (approximately) perfect fit.\footnote{See the discussion in Aba2020, Fer2019, and Pow2022 about the lack of a perfect fit.} This assumption can be relaxed by using an SC-type estimator that is robust to violation of the perfect fit assumption. We state assumption 2 in very general terms as our results apply to any SC-type estimator that can be written as a weighted average of the control units post-intervention outcomes. For example, for the original SCM one can use the results in Zha2022, and state Assumption 2 accordingly. Depending on the chosen estimator, one can state Assumption 2 more precisely. A non-exhaustive list of possible estimators includes the one proposed in: Aba2019, Amj2018, Ben2020, Ben2019, Dou2016, Fer2019, Kel2020, Xu2017.
\paragraph{Lemma 1:} Under Assumption 1 and 2
\paragraph{Proof of Lemma 1:} Under Assumption 1, we have
Thus Assumption 2 immediately implies $$ \widehat{\theta}_{1t} ={Y}_{1t}^I- \widehat{Y}_{1t}^N= \theta_{1t} - \sum_{j=2}^m \widehat{w}_{j}{\gamma_{jt}}+o_p(1). $$ $\square$
\paragraph{Remark 1:} It is important to notice that for each unit $j=2,\ldots,m$, if either $\gamma_{jt}$ or $\widehat{w}_{j}$ is zero, that unit does not induce any extra bias in $\widehat{\theta}_{1t}$. This implies that units that receive a low estimated weight need to have an extremely large effect to induce a non-negligible bias in $\widehat{\theta}_{1t}$. For this reason, units that receive a low weight can be relatively safely treated as "pure controls" when estimating $\theta_{1t}$ in empirical applications. When using SC-type estimators that allow for negative weights one needs to check the magnitude of the weights given to "potentially affected" units.
\paragraph{Remark 2:} For estimators that use a bias correction, one needs to subtract the estimated bias from the outcome of all units before running our iSCM (see equation (16) in Aba2020).
In addition to Assumption 2, we assume that, $\forall \ i=2,\ldots,m$, we have chosen a SC-type estimator to recover $Y_{it}^N$ for which the bias $B_{sc}^i$ is negligible, formally
\paragraph{Lemma 2:} Under Assumption 1 and 3
\paragraph{Proof of Lemma 2:} Under Assumption 1 we have
with $M= \{2, ... , m\}$. Under assumption 3 it follows that $$ \widehat{\gamma}_{it}=Y_{it}^I-\widehat{Y}_{it}^N= \gamma_{it} - \sum_{j \in M \backslash \{{i} \}} \widehat{l}_{j}^{i}{\gamma_{jt}}- \widehat{l}_{1}^{i}{\theta_{1t}}+ o_p(1), \ \forall \ i=2,\ldots,m. $$ $\square$
Combining the results of Lemma 1 and Lemma 2, and ignoring the estimation biases $B^j_{sc}, j=1,\ldots,m$, without loss of generality, the following system of equations holds
After some simple manipulations, we obtain\footnote{One can add the biases $B^j_{sc}, j=1,\ldots,m$, on the left-hand side of each equation, however, they are negligible under assumptions 2 and 3. We show how these biases impact our iSCM estimator in the special case with $m=1$ in Section (ref) below. }:
This is a system of $m$ equations with $m$ unknowns, i.e., the treatment effect on the "main treated" and the $m-1$ effects on the "potentially affected" units.
We can write this system in matrix form, denoted by $\vartheta_t$ the ($m \times 1$) vector of unknown quantities (our effects of interest), by $\widehat{\Omega}$ the ($m \times m$) matrix of known quantities (our estimated weights) that has 1 on the main diagonal and by $\beta_t$ the ($m \times 1$) vector of known quantities (biased estimated effects), as
We now assume that $\widehat{\Omega}$ is invertible, namely
It is easy to show that $\widehat{\Omega}$ is always invertible, if $m< J-2$, except for some extreme cases. For example, $\widehat{\Omega}$ is not invertible if two units give weight 1 to each other and/or every single weight associated with the "pure control" units is zero (see the appendix (ref) for more details). Note that, one can easily check the invertibility of $\widehat{\Omega}$ in the data.
We now state our main result in the following theorem.
\paragraph{Theorem 1:} Under Assumptions 1, 2, 3, and 4, we have $$\widehat{\vartheta}^{iSCM}_t=\widehat{\Omega}^{-1}\widehat{\beta}_t= \vartheta_t+o_p(1) .$$ \paragraph{Proof of Theorem 1:} The result immediately follows from lemma 1 and 2 and using the fact that under Assumption 4 $\widehat{\Omega}$ is invertible. $\square$
The result in Theorem 1 can be readily used to identify our effects of interest by simply applying Cramer's rule:
where $\widehat{\Omega}_{j,t}$ is the matrix obtained by replacing the $j$-th column of $\widehat{\Omega}$ by the vector $\widehat{\beta}_t$.
The expression above makes it very easy to construct estimators for our causal effects of interest that only require very basic linear algebra operations together with the preferred SC-type estimator for the weight matrix $\widehat{\Omega}$ and the vector $\widehat{\beta}_t$.
To further illustrate our results, it is useful to consider the special case in which, together with the "main treated" unit, only one additional unit is "potentially affected" by the intervention ($m=1$). First, we show the standard case and then the simplified case in which including the "main treated" in the donor pool of the "potentially affected" unit is not necessary.
With only one "potentially affected" unit, the system of equation defined in Section (ref), again ignoring the estimation errors, simplifies to
where ($\widehat{\theta}_{t}$) is the estimated effect for the "main treated"; ($\widehat{\gamma}_{t}$) is the estimated effect for the "potentially affected" unit; ($\widehat{w}_{2}$) and ($\widehat{l}_{1}$) are the estimated weights; ($\theta_{t}$) and ($\gamma_{t}$) are the unknown effects.
Therefore, we have
To derive expressions for our estimators, we need to find $\det(\widehat{\Omega})$, $\det(\widehat{\Omega}_{1,t})$, and $\det(\widehat{\Omega}_{2,t})$, which are given by
\\
Following Cramer's rule, we obtain
In this case, it is easy to see that $\det(\widehat{\Omega})$ is always different from zero, except when $\widehat{w}_{2}$ = $\widehat{l}_{1}= 1$. Thus, our effects of interest are always identified unless the "main treated" gives weight 1 to the other "potentially affected" unit, which in turn gives weight 1 to the "main treated". This would be the case, for example, if there are no "pure control" units.
An interesting special case is when we do not need to include the "main treated" in the donor pool of the "potentially affected" unit. In this case, the system of equations further simplifies to
Thus, estimating $\theta_t$ and $\gamma_t$ becomes substantially easier.
So far we have ignored the estimation biases $B_{sc}^i$. When $m=1$, $$Bias\left(\widehat{\theta}_{t}\right)=-B_{sc}^1-\widehat{w}_{2}\gamma_t.$$
We can derive the bias of $\widehat{\theta}^{iSCM}_{t}$ by adding those biases into equation ((ref))
The bias of $\widehat{\theta}_{t}^{iSCM}$ approaches zero as the estimation biases $B_{sc}^1$ and $B_{sc}^2$ approach zero. In contrast, if we include "potentially affected" units, $Bias\left(\widehat{\theta}_{t}\right) =- \widehat{w}_{2}\gamma_t$, even when $B_{sc}^1$ approaches zero.
Note that $Bias\left(\widehat{\theta}^{iSCM}_{t}\right)$ is proportional to $-B_{sc}^1 - \widehat{w}_{2}B_{sc}^2$ by a factor of $a = \frac{1}{1 - \widehat{w}_{2}\widehat{l}_{1}} > 1$. To compare the two biases, we can express $Bias\left(\widehat{\theta}^{iSCM}_{t}\right)$ as $-aB_{sc}^1 - a\widehat{w}_{2}B_{sc}^2$ and compare each term with the one of the bias of $Bias\left(\widehat{\theta}_{t}\right)$ which is $-B_{sc}^1 - \widehat{w}_{2}\gamma_t$. Clearly, the first term of $Bias\left(\widehat{\theta}^{iSCM}_{t}\right)$, $-aB_{sc}^1$, is larger in magnitude than the first term of $Bias\left(\widehat{\theta}_{t}\right)$, $-B_{sc}^1$. We can easily determine the exact difference by calculating $a$. For example, in our application where $\widehat{w}_{2} = 0.42$ and $\widehat{l}_{1} = 0.33$, $a \approx 1.16$. Thus, $-aB_{sc}^1$ is about 16% larger in magnitude than $-B_{sc}^1$. However, the second terms of the biases of $Bias\left(\widehat{\theta}^{iSCM}_{t}\right)$ and $Bias\left(\widehat{\theta}_{t}\right)$ are driven by $B_{sc}^2$ and $\gamma_t$, respectively. Note that, $B_{sc}^2$ represents the estimation error in estimating $\gamma_t$. Therefore, unless the chosen SC-type estimator of $\gamma_t$ performs poorly, $-a\widehat{w}_{2}B_{sc}^2$ will be significantly smaller in magnitude than $-\widehat{w}_{2}\gamma_t$, implying that $Bias\left(\widehat{\theta}^{iSCM}_{t}\right)$ is generally much smaller than $Bias\left(\widehat{\theta}_{t}\right)$.
In case it is not necessary to include the "main treated" in the donor pool of the "potentially affected" unit, we have $$ Bias\left(\widehat{\theta}^{iSCM}_{t}\right)=-B_{sc}^1 - \widehat{w}_{2}B_{sc}^2. $$ Thus, as soon as we have chosen a SC-type estimator such that its bias ($B_{sc}^2$) is smaller in magnitude than its target parameter ($\gamma_t$), $\widehat{\theta}^{iSCM}_{t}$ will always be smaller than $\widehat{\theta}_{t}$.
When using SC-type estimators, it is advisable to include in the donor pool units with similar characteristics and those possibly affected by similar shocks as the treated unit. Often, these units are either directly or indirectly affected by the intervention. For instance, it is likely that other treated units or units "potentially affected" by spillover effects are the closest (geographically and/or economically) to the "main treated" unit. For example, Aba2020 proposes including units "potentially affected" by spillover in the donor pool, even if they induce a bias, which our iSCM eliminates. \footnote{In the appendix (Section (ref)), we show that iSCM may still be superior to "restricted" SCM even in scenarios where both "main treated" and "potentially affected" units are within the "pure control" units' convex hull. In this case, a "restricted" SCM will potentially work, however, if the “potentially affected” units are the closest to the “main treated”, excluding them might increase interpolation bias. Additionally, we explore cases more common in empirical settings, where either or both the "main treated" and "potentially affected" units fall outside the convex hull.} For example, if $m=1$, as we have shown in Section (ref), iSCM bias is $\frac {-B_{sc}^1-\widehat{w}_{2}B_{sc}^2}{1-\widehat{w}_{2}\widehat{l}_{1}}$. Let the approximation error of $Y_{1t}^N$ obtained with a the "restricted" SCM be $B_{rSC}$. Then the "restricted" SCM bias will be $-B_{rSC}$. It is hard a priori to judge which method has a larger bias as this depends on the sign of $-B_{rSC}-\frac {-B_{sc}^1-\widehat{w}_{2}B_{sc}^2}{1-\widehat{w}_{2}\widehat{l}_{1}}$ which is equal to the sign of $(B_{sc}^1+\widehat{w}_{2}B_{sc}^2)-(1-\widehat{w}_{2}\widehat{l}_{1})B_{rSC}$. Thus the difference in the biases of iSCM and "restricted" SCM is a function of three unknown approximation biases.
Therefore, we suggest implementing our iSCM and comparing it to the "restricted" SCM, i.e., excluding potentially affected units. To determine which of the two methods should be used as the main specification, we advise taking the following steps. \footnote{We recommend implementing and reporting the results of both methods to avoid potential pretest bias (see JR22).}
If ($X_1-X_0^{res}\widehat{W}^{res})\approx (X_1-X_0\widehat{W}^{unres}$) and $RMSPE^{res}\approx RMSPE^{unres}$, then the “restricted” SCM is preferable \footnote{Alternatively, one could divide the pre-intervention period into a training period and a validation period, estimate the two models during the training period, and compare their RMSPEs in the validation period. Note that the "restricted" SCM does not have by construction an RMSPE that is greater than or equal to that of the "unrestricted" one, see for example Section (ref).}. If ($X_1-X_0^{res}\widehat{W}^{res})>(X_1-X_0\widehat{W}^{unres}$) and/or $RMSPE^{res}>RMSPE^{unres}$, we advise using our iSCM as main specification. Note that, unlike a regression model, the "restricted" SCM might still achieve a lower RMSPE than the "unrestricted" one (see, for example, our empirical application), making this comparison meaningful.
Repeat these steps for each "potentially affected" unit as if it were the "main treated". In case there is no substantial gain from including the treated in the donor pool of "potentially affected" units, our iSCM becomes easier to implement, as shown in Equation (ref). Finally, it is worth comparing the results of iSCM and the "restricted" SCM with those of the "unrestricted" SCM, bearing in mind that the latter can be biased if any of the units in the donor pool receiving non-negligible weight are affected by the treatment.
Dealing with only a small number of units makes inference for synthetic control-based methods like ours complicated. We can, however, easily adapt existing methods to our setting. The most popular choice is to implement permutation tests. Aba2010 and Aba2015 propose placebo tests in time, i.e., reassigning the intervention artificially before its real implementation, and placebo tests in-space, i.e., reassigning the intervention artificially for units in the control group. Placebo tests in space measure the statistical significance of the effect through the ratio between the RMSPE in the post-treatment period and in the pre-treatment period.
To run the in-space placebo test for $\widehat\theta_t^{iSCM}$, we simply need to subtract the estimated effects from the outcomes of "potentially affected" units. Specifically, we replace $Y_{it}$ with $Y_{it} - \widehat\gamma_{it}^{iSCM}$ for all $i = 2, \ldots, m$, and $t > T_0$, as follows:
Similarly, to construct a placebo test for $\widehat\gamma_{it}^{iSCM}$ for a generic "potentially affected" unit $i$, we need to subtract the estimated effects from the outcomes of all other "potentially affected" units as well as the "main" treated unit, leaving the outcome of unit $i$ untouched. For $i=2,\ldots, m$, we have:
{
}
This idea can easily be applied to other inference procedures available in the literature (see, e.g., Cao2019,Che2019, Fir2018,Gob2016,Li2019). For example, the inference procedure of Che2019 can be easily adapted to our framework by taking into account that the post-intervention observed outcome of each "potentially affected" unit includes its treatment effect. This has to be considered both in the first step, i.e., in the construction of the data under the null hypothesis and in the second step to compute the SCM residuals.
In this section, we use iSCM to estimate the effect of German reunification on West Germany's GDP per capita. In October 1990, less than a year after the fall of the Berlin wall in November 1989, the German Democratic Republic (“East Germany”) and the Federal Republic of Germany (“West Germany”) were officially reunified. German reunification, defined as one of the most important historical milestones of European history after 1945, most likely affecting not only the German economy but also the economies of other European countries.
As discussed in Aba2015 and Aba2019, German reunification could have had negative spillover effects on Austria's economic growth because West Germany diverted demand and investment from Austria to East Germany. Austria has historically had tight links with Germany: the two countries share the same language and, to a great extent, a common history. In 1938, Austria was annexed by the Third Reich, which benefited from Austria's raw materials and labor to complete German rearmament. In 1945, Austria was separated from Germany. However, the economic cooperation between Austria and West Germany continued during the Cold War. Given these strong cultural and economic ties, it is arguably important to include Austria in the donor pool when constructing a synthetic version of West Germany. Therefore, our iSCM is very well suited for estimating the impact of the German reunification not only on West Germany but also on Austria.
We use the same country-level panel data of Aba2015. The data cover the period 1960-2003, with the post-intervention period starting in 1990. In addition to Austria, the remaining "pure control" countries in the donor pool include 15 other OECD countries: Australia, Belgium, Denmark, France, Greece, Italy, Japan, the Netherlands, New Zealand, Norway, Portugal, Spain, Switzerland, the United Kingdom, and the United States. The outcome variable is real GDP at Purchasing Power Parity (PPP) per capita measured in 2002 USD.
We replicate the SCM estimate of Aba2015 and check the weight assigned to Austria.\footnote{We refer to Aba2015 for a detailed discussion of the estimation procedure.} The fact that Austria is potentially affected by spillover effects implies that including it in the donor pool might induce bias.\footnote{Given that other European countries receive very little weight, the impact of potential spillover effects on those countries would likely be negligible, as shown in Lemma 1. Results for the case in which Switzerland and the Netherlands are considered as "potentially affected" units are available from the authors upon request.} As we suggest in Section (ref), to decide which of the two methods is preferable, we need to implement both our iSCM and the "restricted" SCM, i.e., excluding Austria from the donor pool, keeping the same specification and estimation procedure used for the "unrestricted" SCM. Austria receives the highest weight (42%) thus iSCM and `unrestricted” SCM might give different results. Table (ref) suggests that the “unrestricted” synthetic version of West Germany (second column) is much closer in terms of observable characteristics to West Germany (first column) than the “restricted” version (third column). Next, we compare the pre-intervention RMSPEs of the “unrestricted” and “restricted” SCM. The RMSPE of the latter (270.74) is larger than the one of the former (119.07). Therefore, we expect iSCM to perform better than “restricted” SCM. We now repeat the same procedure to decide whether West Germany should be included in synthetic Austria's donor pool. First, we check whether West Germany receives a non-negligible weight. To construct synthetic Austria, we must use a slightly different specification than the one used for creating synthetic West Germany. In particular, we are not able to choose the weights assigned to the predictors using the sample splitting methods described in Aba2015. As described in Geh2018, in 1980, not long before the sample split cut-off, Austria provided several loans to East Germany, and in return, Austrian nationalized industries received large-scale orders. This most likely stimulated Austria’s exports and contributed to job creation in its industries. Thus, the sample split procedure might catch the effect of this economic shock. This is corroborated by the fact that using this method to choose the predictor weights leads to a poor pre-intervention fit. For this reason, we decided to follow the data driven procedure suggested by Aba2010 instead.
As shown in Table (ref), Synthetic Austria gives the highest weight to West Germany (33%). Thus, we can proceed to checking how well "restricted" (excluding West Germany) and "unrestricted" synthetic Austria matches real Austria's observable characteristics and comparing the pre-intervention RMPEs of the two specifications.
Table (ref) suggests that the “unrestricted” synthetic Austria (second column) does a better job of reproducing Austria's (first column) pre-reunification predictors than the “restricted” version (third column), except for GDP per capita. The pre-intervention RMSPE of the “restricted” SCM (181.22) is slightly lower than the one of the “unrestricted” version (194.67). However, the difference is rather small. Taking everything into account, we argue that iSCM is preferable to the “restricted” SCM.
Now we can use the "unrestricted" SCM estimates of $\widehat{\theta_{t}}$, $\widehat{\gamma_{t}}$, the weight assigned to Austria $\hat{w}_A=0.42$, and the weight assigned to West Germany $\hat{l}_{WG}=0.33$ to estimate $\theta_t$, and $\gamma_t$ as $$\widehat{\theta}_{t}^{iSCM}=\frac{\widehat{\theta}_{t}+\hat{w}_A\widehat{\gamma}_{t}}{1-\hat{w}_A\hat{l}_{WG}}=1.16\left(\widehat{\theta}_{t}+0.42\widehat{\gamma}_{t}\right), \quad \textrm{ and } \quad \widehat{\gamma}_{t}^{iSCM}=\frac{\widehat{\gamma}_{t}+\hat{l}_{WG}\widehat{\theta}_{t}}{1-\hat{w}_A\hat{l}_{WG}}=1.16\left(\widehat{\gamma}_{t}+0.33\widehat{\theta}_{t}\right).$$
We can also construct the matrix $\widehat{\Omega}$, and check whether it is non-singular as required by Assumption 4. Given the estimated weights we get
As $det(\widehat{\Omega})=0.86$, Assumption 4 holds in this application, and we can safely use our iSCM estimators.
Figure (ref) shows our main results. In Panel (ref) and (ref) we report the gaps estimated by the "unrestricted" SCM, the "restricted" SCM, and our iSCM, for West Germany and Austria, respectively\footnote{Notice that the “unrestricted” SCM and the iSCM are identical in the pre-treatment period because the spillover effect due to German reunification only happens after the intervention.}. In Panel (ref) we plot the GDP per capita trajectories (1960–2003) of Real West Germany, its "unrestricted" synthetic version, its “restricted” synthetic version, and its inclusive synthetic version. Panel (ref) produces an analogous plot for Austria.
For West Germany, we can observe that both the standard and the inclusive synthetic version of West Germany in the pre-reunification period almost perfectly reproduce West Germany's per capita GDP. Meanwhile, excluding Austria substantially worsens the pre-reunification fit. This confirms the importance of including Austria in the donor pool. Aba2015 find a negative effect of the reunification on West Germany's per capita GDP, which was reduced by approximately 7.67% per year on average relative to the 1990 baseline level. Our iSCM results are not very different from those of Aba2015 and confirm their expectation about the potential direction of the bias, implying an even more negative effect of reunification. However, the difference between the trends in per capita GDP between iSCM and SCM is generally small: our iSCM estimate implies a negative effect that is up to 1.50% larger than the one estimated with a standard SCM.
If we look at the estimated spillover effects for Austria we draw similar conclusions. In the post-reunification period, iSCM estimates a decrease in per capita GDP of only up to 708 USD. However, this effect is unlikely to be statistically significant (see Figure (ref) Panel (ref) in the appendix). It is important to notice that both the "unrestricted" SCM and the "restricted" SCM estimate a positive spillover effect for Austria, up to 894 and 1350 USD, respectively. However, we know a priori that the "unrestricted" SCM is biased and its bias is $-\hat{l}_{WG}\theta_{t}$. Given we have strong evidence that $\theta_{t}$ is negative, we do expect the "unrestricted" SCM to overestimate $\gamma_{t}$. Thus, the fact that both the "unrestricted" and the "restricted" SCM estimate positive spillover effects underscores the necessity of including West Germany in the donor pool and employing our iSCM in this application.
In this section, we re-estimate the effects on West Germany and Austria using the Penalized Synthetic Control Method.
We adopt the same specification used in the original SCM analysis, selecting the penalty parameter, $\lambda$, through cross-validation. We retain the same weight matrix $V$. Although one might expect that a method improving pre-intervention fit, such as penalized SCM, would mitigate the impact of excluding Austria from the donor pool, the results suggest otherwise. Figure (ref) shows that the pre-treatment fit is substantially worse in the restricted penalized SCM version for both countries, especially for West Germany.
Notably, for Austria, the penalty parameter is nearly zero, rendering the penalized SCM nearly identical to the original SCM. Overall, iSCM and penalized iSCM yield very similar results.
We introduce iSCM, a modification of the SCM that allows the inclusion of units "potentially affected" by an intervention in the donor pool. Our method is useful in applications where it is either important to include other treated units in the donor pool or where some of the units in the donor pool are affected indirectly by the intervention (spillover effects). Our iSCM requires that we choose a priori which units are "potentially affected" by the treatment and that the assumptions of the chosen SC-type estimator are valid. In addition, our methods can only be used if at least one “pure” control unit receives non-zero weight. A major advantage of iSCM is that it can be easily implemented using the synthetic control estimator or many of the new estimation methods available in the literature. Moreover, we demonstrate that iSCM is almost certain to enhance the performance compared to the "unrestricted" SCM. Additionally, we introduce a data-driven approach to determine whether iSCM could potentially outperform the "restricted" SCM. Even in situations where excluding "potentially affected" units from the donor pool does not seem to be harmful, our iSCM can serve as a robustness check. Finally, we illustrate the use of iSCM by re-estimating the impact of German reunification on GDP per capita. Using both the original and penalized SCM, we confirm Aba2015 expectations about the direction of the spillover effect from West Germany to Austria, findings small negative spillover effects on Austria. This implies that the negative treatment effect on West Germany might be larger than previously estimated.