On the sparsity of synthetic control method

ABSTRACT

Synthetic Control Method (SCM) is a popular approach for causal inference in panel data, where the optimal weights for control units are often sparse. But the sparsity of SCM has received little attention in the literature except Abadie (2021), which explores the sparsity from the perspective of predictor space. In this paper, we make three contributions. First, we show that if there is a unique solution, then the number of positive weights is upper-bounded by the number of covariates. Second, we offer a simple alternative explanation about the sparsity of SCM from the perspective of parameter space. Third, we conduct a meta-analysis of empirical studies using SCM in the literature, which shows that the sparsity of SCM decreases with the relative number of covariates. A practical implication is that if the number of positive weights exceeds the number of covariates, there are multiple solutions and possibly unstable weights.

KEYWORDS:

• Synthetic control method
• sparsity
• meta-analysis

1. Introduction

Synthetic control method (SCM) is a popular approach for causal inference in panel data with a single treated unit (Abadie & Gardeazabal, 2003; Abadie et al., 2010), which has found many applications in fields such as economics, politics and health. For a treated unit, SCM constructs its counterfactual outcomes via a convex combination of control units, i.e., a linear combination with nonnegative weights summed to one. In practice, the optimal weights are usually sparse such that the share of zero weights are high. Obviously, sparsity of weights makes the interpretation of SCM easy. As pointed out by Abadie (2021, p. 407), “sparsity plays an important role for the interpretation and evaluation of the estimated counterfactual”.

However, there has been little formal discussion about the sparsity of SCM until Abadie (2021), which explains the origin of sparsity from the perspective of predictor space. In this paper, we make three contributions in understanding the sparsity of SCM. First, we show that if there is a unique solution, then the number of positive weights cannot exceed the number of covariates, which yields a lower bound for the sparsity defined as the share of zero weights. Second, we offer a simple alternative explanation about the sparsity. Viewed from the perspective of parameter space, the sparsity of SCM arises simply because the feasible parameter space is a probability simplex with sharp vertices, similar to the sparsity of Lasso in origin. Third, we conduct a meta-analysis of empirical studies using SCM in the literature, and show that consistent with our theoretical finding, the sparsity of SCM decreases with the relative number of covariates as a fraction of the number of control units.

The rest of the paper is organized as follows. Section 2 reviews the methodology of SCM. Section 3 explores the sparsity of SCM from the perspective of predictor space, and derives a lower bound for sparsity when there is a unique solution. Section 4 proposes an alternative explanation of sparsity from the perspective of parameter space. Section 5 conducts a meta-analysis of empirical studies using SCM in the literature, which yields results consistent with our theoretical finding. Section 6 provides a discussion and guidance to empirical practice, and section 7 concludes.

2. Synthetic control method

Suppose there are $\mathit{N}+1$ cross-sectional units indexed by $\mathit{i}=1,\cdots ,\mathit{N}+1$ and observed over periods $\mathit{t}=1,\cdots ,T_0$ (pre-intervention) and$\mathit{t}=T_0+1,\cdots ,\mathit{T}$ (post-intervention). The first unit $(\mathit{i}=1)$ is the treated unit, while all other units $(\mathit{i}=2,\cdots ,\mathit{N}+1)$ are control units, which forms the donor pool. Let $y_{it}^1$ and $y_{it}^0$ be the outcome of unit $\mathit{i}$ in period $\mathit{t}$ with and without intervention respectively, and${\mathrm{\Delta }}_{\mathit{it}}=y_{it}^1-y_{it}^0$ is the corresponding treatment effect. SCM seeks to approximate the unknown $y_{1t}^0(\mathit{t}=T_0+1,\cdots ,\mathit{T})$ by a weighted average of control units, and the treatment effects are estimated by

(1) ${\hat{\mathrm{\Delta }}}_{1\mathit{t}}=y_{1\mathit{t}}-{\hat{y}}_{1\mathit{t}}^0=y_{1\mathit{t}}-\sum _{\mathit{i}=2}^{\mathit{N}+1}{w}_i{y}_{\mathit{it}}(\mathit{t}=T_0+1,\cdots ,\mathit{T}),$(1)

where $\mathbf{w}=(w_2\cdots w_{\mathit{N}+1}){ }^{\mathrm{\prime }}$ is a $\mathit{N}\times 1$ vector of weights (a potential synthetic control) such that $0\le w_i\le 1$ for $\mathit{i}=2,\cdots ,\mathit{N}+1$ and ${\sum }_{i=2}^{N+1}{w}_i=1$.

Let ${\mathbf{x}}_1$ be the $\mathit{K}\times 1$ vector containing the pretreatment covariates (also known as predictors) of the treated unit,¹ and ${\mathbf{X}}_0$ be the $\mathit{K}\times \mathit{N}$ matrix containing the pretreatment covariates of the $\mathit{N}$ control units. SCM selects the optimal ${\mathbf{w}}^{\ast }$ such that the pretreatment characteristics of the synthetic control are most similar to that of the treated unit. Specifically, the optimal synthetic control ${\mathbf{w}}^{\ast }$ is obtained by solving the following minimization problem:

(2) ${\mathbf{w}}^{\ast }=\underset{\mathbf{w}}{argmin}{\left({\mathbf{x}}_1-{\mathbf{X}}_0\mathbf{w}\right)}^{\mathrm{\prime }}\mathbf{V}\left({\mathbf{x}}_1-{\mathbf{X}}_0\mathbf{w}\right),$(2)

where $\mathbf{V}$ is a $\mathit{K}\times \mathit{K}$ diagonal matrix with nonnegative elements on its diagonal, which measures the relative importance of each covariate in predicting the outcome. The optimal $\mathbf{V}$ can be determined by minimizing the mean squared prediction error (MSPE) of the outcome variable during the pretreatment period. See Abadie et al. (2010) for details of implementation.

3. The sparsity of SCM viewed from the predictor space

Consider the minimization problem in Equation (2)(2) ${\mathbf{w}}^{\ast }=\underset{\mathbf{w}}{argmin}{\left({\mathbf{x}}_1-{\mathbf{X}}_0\mathbf{w}\right)}^{\mathrm{\prime }}\mathbf{V}\left({\mathbf{x}}_1-{\mathbf{X}}_0\mathbf{w}\right),$(2) along with constraints on the weights:

(3) $\begin{array}{rl}& \underset{\mathbf{w}}{min}{\left({\mathbf{x}}_1-{\mathbf{X}}_0\mathbf{w}\right)}^{\mathrm{\prime }}\mathbf{V}\left({\mathbf{x}}_1-{\mathbf{X}}_0\mathbf{w}\right),\\ & \mathit{\hspace{0.17em}}\mathit{s}.\mathit{t}.\mathbf{w}\ge 0,\mathit{\iota }{ }^{\mathrm{\prime }}\mathbf{w}=1\end{array}$(3)

where $\mathit{\iota }=(1\cdots 1){ }^{\mathrm{\prime }}$ is a $\mathit{N}$ dimensional vector consisting of all ones. Abadie (2021) proposes an explanation about the sparsity of SCM from the perspective of predictor space. Define $\mathit{H}({\mathbf{X}}_0)$ as the convex hull of the columns of ${\mathbf{X}}_0$, i.e., the smallest convex set containing the columns of ${\mathbf{X}}_0$; or equivalently, the intersection of all convex sets containing the columns of ${\mathbf{X}}_0$. Specifically, $\mathit{H}({\mathbf{X}}_0)$ can be defined as

(4) $\mathit{H}({\mathbf{X}}_0)\equiv \left\{{\mathbf{X}}_0\mathbf{w}:\mathit{\hspace{0.17em}}\mathbf{w}\ge 0,\mathit{\iota }{ }^{\mathrm{\prime }}\mathbf{w}=1\right\}.$(4)

If ${\mathbf{x}}_1$ is in the convex hull, i.e., ${\mathbf{x}}_1\in \mathit{H}({\mathbf{X}}_0)$, then the objective function achieves a minimum of zero. In this case, we can find optimal weights ${\mathbf{w}}^{\ast }$ that perfectly reproduces the pretreatment characteristics of the treat unit, i.e., ${\mathbf{x}}_1-{\mathbf{X}}_0\mathbf{w}=\mathbf{0}$. We call the optimal weights ${\mathbf{w}}^{\ast }$ an “exact solution”.

For illustration, consider a simple case with $\mathit{K}=2$ (two covariates) and $\mathit{N}=5$ (five control units); see Figure 1. In Figure 1, ${\mathbf{x}}_1\in int\left(\mathit{H}({\mathbf{X}}_0)\right)$ (located in the interior of the convex hull) represents the covariates of the treated unit, while ${\mathbf{x}}_2,\cdots ,{\mathbf{x}}_6$ represent the covariates for units 2 through 6, and ${\mathbf{X}}_0=({\mathbf{x}}_2\cdots {\mathbf{x}}_6)$. In this case, it is easy to see that exact solutions ${\mathbf{w}}^{\ast }$ are not unique, since there are different convex combinations of ${\mathbf{x}}_2,\cdots ,{\mathbf{x}}_6$ that can perfectly reproduce ${\mathbf{x}}_1$. Therefore, the number of positive weights is only limited by the number of control units in general when there are exact solutions and ${\mathbf{x}}_1\in int\left(\mathit{H}({\mathbf{X}}_0)\right)$.

Figure 1. Multiple exact solutions.

Now consider the situation in Figure 2, where ${\mathbf{x}}_1$ is located on the boundary of the convex hull, i.e., ${\mathbf{x}}_1\in \mathrm{\partial }\left(\mathit{H}({\mathbf{X}}_0)\right)$. Clearly, there is a unique exact solution with positive weights for ${\mathbf{x}}_5$ and ${\mathbf{x}}_6$ respectively. When there is a unique exact solution and $\mathit{K}=2$ (as presented in Figure 2), there can be at most 2 positive weights, since two vertices in a two-dimensional space form an edge. Similarly, when there is a unique exact solution and $\mathit{K}=3$, there can be at most 3 positive weights, since three vertices in a three-dimensional space form a face. By the same reasoning, when there is a unique exact solution, the number of positive weights cannot exceed the number of covariates. We state this formally in Proposition 1 below with a proof.

Figure 2. A unique exact solution.

However, in empirical practice, ${\mathbf{x}}_1$ usually falls outside of the convex hull, i.e., ${\mathbf{x}}_1\notin \mathit{H}({\mathbf{X}}_0)$, which results in an “approximate solution”. In this case, the optimal weights ${\mathbf{w}}^{\ast }$ provides a prediction ${\mathbf{X}}_0{\mathbf{w}}^{\ast }$, which is a linear projection of ${\mathbf{x}}_1$ to $\mathit{H}({\mathbf{X}}_0)$. Thus, the sparsity of SCM arises because projection to a convex hull typically do not rely on all vertices, where each vertex represents a control unit, i.e., a column in ${\mathbf{X}}_0$; see Figure A1 in Abadie (2021), which is reproduced in Appendix 1 for convenience. Those control units with vertices making no contribution to the linear projection thus receive zero weights.

Consider again the case with $\mathit{K}=2$ (two covariates) and $\mathit{N}=5$ (five control units), but now there is a unique approximate solution on an edge (see Figure 3). In Figure 3, ${\mathbf{x}}_1$ falls outside the convex hull of ${\mathbf{X}}_0$. The linear projection to $\mathit{H}({\mathbf{X}}_0)$ results in two positive weights for ${\mathbf{x}}_2$ and ${\mathbf{x}}_6$ respectively, whereas ${\mathbf{x}}_3,{\mathbf{x}}_4,{\mathbf{x}}_5$ all receive zero weights.

Figure 3. A unique approximate solution on an edge.

In Figure 3, the number of positive weights and the number of covariates happen to be equal. Of course, the number of positive weights could be smaller than the number of covariates. For example, if the projection of ${\mathbf{x}}_1$ to $\mathit{H}({\mathbf{X}}_0)$ lands on a vertex, instead of any line segment connecting two vertices, then the number of positive weights drops to just one; see Figure 4.

Figure 4. A unique approximate solution on a vertex.

A further question is whether the number of positive weights can surpass the number of covariates. Geometrically, this appears to be impossible from Figure 3. In this case, the projection of ${\mathbf{x}}_1$ to $\mathit{H}({\mathbf{X}}_0)$ lies on the line segment connecting ${\mathbf{x}}_2$ and ${\mathbf{x}}_6$. Note that in any plane, a straight line passing through two different points cuts the plane into two half-planes. In our case, the straight line passing through ${\mathbf{x}}_2$ and ${\mathbf{x}}_6$ cuts the plane into two half-planes, such that the other vertices (i.e., ${\mathbf{x}}_3,{\mathbf{x}}_4,{\mathbf{x}}_5$) are in the open half-plane further away from ${\mathbf{x}}_1$ by definition (thus receiving zero weights), since the unique approximate solution lies on the line segment connecting ${\mathbf{x}}_2$ and ${\mathbf{x}}_6$ (thus receiving positive weights). Therefore, the number of positive weights cannot be more than 2 in this case.

The same reasoning applies to higher dimensions in general. For any fixed $\mathit{K}$, consider the $\mathit{K}$dimensional predictor space. When there is a unique approximate solution with $\mathit{K}$ positive weights, then the hyperplane passing through the corresponding $\mathit{K}$ vertices cuts the predictor space into two half-spaces, such that the other vertices are in the open half-space further away from ${\mathbf{x}}_1$ by definition (thus receiving zero weights), since the unique approximate solution lies on the face connecting the $\mathit{K}$ vertices with positive weights. Therefore, the number of positive weights cannot be larger than $\mathit{K}$ in this case. We state this result formally below, and give a rigorous proof by contradiction.

Proposition 1.

Consider a synthetic control problem with $\mathit{K}$ covariates and $\mathit{N}$ control units, and $\mathit{N}>\mathit{K}$. If there is a unique solution (either exact or approximate), then the number of positive weights are less than or equal to $\mathit{K}$.

Proof.

We prove by contradiction. Denote $N_p$ as the number of positive weights. Suppose there is a unique solution (either exact or approximate), but $N_p>\mathit{K}$. Without loss of generality, assume that the first $N_p$ control units $\left\{{\mathbf{x}}_2,\cdots ,{\mathbf{x}}_{N_p+1}\right\}$ receive positive weights, whereas all the rest control units $\left\{{\mathbf{x}}_{N_p+2},\cdots ,{\mathbf{x}}_N\right\}$ get zero weights.

Denote ${\mathbf{X}}_0^{(1)}\equiv ({\mathbf{x}}_2\cdots {\mathbf{x}}_{N_p+1})$, and note that ${\mathbf{X}}_0^{(1)}$ is a $\mathit{K}\times N_p$ matrix with row rank less than or equal to $\mathit{K}$. Thus the column rank of ${\mathbf{X}}_0^{(1)}$ is also less than or equal to $\mathit{K}$, which in turn is strictly less than $N_p$ by assumption:

$\mathit{colrank}\left({\mathbf{X}}_0^{(1)}\right)=\mathit{rowrank}\left({\mathbf{X}}_0^{(1)}\right)\le \mathit{K}<N_p.$

Therefore, the $\mathit{K}\times N_p$ matrix ${\mathbf{X}}_0^{(1)}$ does not have a full column rank, so its columns $\left\{{\mathbf{x}}_2,\cdots ,{\mathbf{x}}_{N_p+1}\right\}$ are strictly multicollinear. Hence, the solution to synthetic control problem is not unique, which leads to a contradiction.

An immediate implication of Proposition 1 is that if the number of positive weights exceeds the number of covariates, then the solutions to the synthetic control problem under consideration are not unique. Therefore, applied researchers should be concerned if the number of positive weights is larger than the number of covariates, which is symptomatic of the existence of multiple solutions. As a way out, one may change the model specification by adding more covariates or dropping multicollinear control units (more on this below), until the number of positive weights are less than or equal to the number of covariates.

Proposition 1 gives an upper bound on the number of positive weights, which naturally yields a lower bound on the number of zero weights. Since this paper is primarily concerned with the sparsity of SCM, denote $N_z$ as the number of zero weights, such that $\mathit{N}=N_p+N_z$. Moreover, define the sparsity of SCM as the share of zero weights, i.e., $N_z/\mathit{N}$. From Proposition 1, we immediately have the following proposition.

Proposition 2.

Consider a synthetic control problem with $\mathit{K}$ covariates and $\mathit{N}$ control units, and $\mathit{N}>\mathit{K}$. If there is a unique solution, then the sparsity of SCM satisfies the following inequality:

(5) $1-\frac{K}{N}\le \mathit{sparsity}\le 1-\frac{1}{N}.$(5)

Proof.

First, according to Proposition 1, $N_p\le \mathit{K}$, which implies $N_z\ge \mathit{N}-\mathit{K}$. Divide both sides of this inequality by $\mathit{N}$, and rearranging, we have $\mathit{sparsity}=\frac{N_z}{\mathit{N}}\ge 1-\frac{K}{N}$.

Second, since there needs to be at least one positive weight, the largest possible value of $N_z$ is $(\mathit{N}-1)$, thus $\mathit{sparsity}=\frac{N_z}{\mathit{N}}\le 1-\frac{1}{N}$

From the above discussion, it is clear that when there is a unique solution, the lower bound for sparsity given by $\left(1-\frac{K}{N}\right)$ could be tight in some cases, as long as the optimal solution resides on a regular face of the convex hull (which is usually a polyhedron or polytope), instead of on a lower-dimensional edge or vertex.

Also note that an important condition for Proposition 1 is the uniqueness of solution. On the contrary, if the solutions are not unique, then the number of positive weights can be larger than the number of covariates, and it is only limited by the number of control units; see Figure 5. In Figure 5, ${\mathbf{x}}_2,{\mathbf{x}}_3,{\mathbf{x}}_5,{\mathbf{x}}_6$ line up in a straight line. Clearly, multiple solutions are now available. In particular, it is possible to assign positive weights to control units ${\mathbf{x}}_2,{\mathbf{x}}_3,{\mathbf{x}}_5,{\mathbf{x}}_6$ as an optimal solution. In practice, this is the optimal solution that is often found by quadratic programming algorithms in used (say, the algorithm used by the Stata command synth), which results in many small positive weights that are far from being sparse. In fact, even if ${\mathbf{x}}_2,{\mathbf{x}}_3,{\mathbf{x}}_5,{\mathbf{x}}_6$ does not perfectly line up in a straight line, as long as the iterative algorithm thinks they are, the same result would follow. In the presence of (near) multicollinearity, a simple fix is to drop control units that are most responsible for (near) multicollinearity.² For example, dropping ${\mathbf{x}}_2$ and ${\mathbf{x}}_6$ in Figure 5 would solve the problem, and obtain a unique solution.

Figure 5. Approximate solutions that are not unique.

To illustrate the above results, we revisit the case study on the effect of California’s tobacco control program (Abadie et al., 2010). We use the Stata command synth for synthetic control estimation,³ while incrementally increasing the number of covariates from one to seven, which is the same as specified in Abadie et al. (2010). The results are reported in Table 1.

Table 1. A case study of California’s tobacco control program.

$\mathit{N}$	$\mathit{K}$	$N_p$	$N_z$	Covariates
38	1	38	0	retprice
38	2	2	36	retprice, lnincome
38	3	3	35	retprice, lnincome, age15to24
38	4	4	34	retprice, lnincome, age15to24, beer
38	5	5	33	retprice, lnincome, age15to24, beer, cigsale(1975)
38	6	4	34	retprice, lnincome, age15to24, beer, cigsale(1975), cigsale(1980)
38	7	4	34	retprice, lnincome, age15to24, beer, cigsale(1975), cigsale(1980), cigsale(1988)

For variable definitions, refer to Abadie et al. (2010).

In Table 1, when there is only one covariate ($\mathit{K}=1$), the algorithm returns all small positive weights (ranging from 0.016 to 0.11) without any zero weight. Apparently, when $\mathit{K}=1$, ${\mathbf{X}}_0$ is just a $1\times \mathit{N}$ matrix such that all columns of ${\mathbf{X}}_0$ are perfectly collinear, which line up in a straight line. Clearly, multiple solutions are available, and the algorithm just spews out a solution in a seemingly random way. On the other hand, when $2\le \mathit{K}\le 7$, the number of positive weights $N_p$ is always upper-bounded by $\mathit{K}$, where this upper bound is sometimes obtained. While this example is somewhat artificial,⁴ this is not a trivial case in practice. In fact, in the meta-analysis in Section 5, we find that among 658 empirical studies using SCM in the sample, 4.56% of them have weights that are all positive.⁵

Furthermore, since unique solution appears to be frequent in practice, while the lower bound $\left(1-\frac{K}{N}\right)$ in Proposition 2 may be tight in some cases, we propose the following hypothesis to be tested by the meta-analysis in Section 5.

Hypothesis 1.

Other things being equal, the sparsity of SCM decreases with the relative number of covariates as a fraction of the number of control units (i.e., $\mathit{K}/\mathit{N}$) in empirical studies.

4. The sparsity of SCM viewed from the parameter space

Before moving to the meta-analysis in Section 5, we propose a simple alternative explanation of the sparsity of SCM from the perspective of parameter space in this section. First, note that the feasible set of weights is actually a probability simplex defined as $\left\{\mathbf{w}:\mathbf{w}\ge 0,\mathit{\iota }{ }^{\mathrm{\prime }}\mathbf{w}=1\right\}$. Second, the objective function is a quadratic form with a nonnegative diagonal matrix as its matrix. Thus, the contour sets of the objective function are ellipses or ellipsoids in high dimensions. For example, when there are only two control units (i.e., $\mathit{N}=2$), the minimization problem can be geometrically represented in Figure 6.

Figure 6. Optimal synthetic control in the 2D parameter space.

In Figure 6, the contour sets are ellipses with axes parallel to the coordinate axes, since no cross-terms are involved in the quadratic form of the objective function. Due to the constraint$\left\{\mathbf{w}:\mathbf{w}\ge 0,\mathit{\iota }{ }^{\mathrm{\prime }}\mathbf{w}=1\right\}$, the feasible parameter space is the line segment connecting $(1,0)$ and $(0,1)$. ${\mathbf{w}}^{\ast \ast }$ is the unconstrained optimum, whereas ${\mathbf{w}}^{\ast }$ is the constrained optimum, which is a sparse solution with $w_2^{\ast }=0$ and $w_3^{\ast }=1$. Apparently, the sparsity of SCM arises simply because the feasible parameter set is a probability simplex with sharp vertices (in this case pointed ends). In a way, the origin of the sparsity of SCM is reminiscent of the sparsity of Lasso, where the feasible parameter space is of a diamond shape with sharp vertices. Of course, it is still possible for the optimal solution to lie in the middle of the line segment connecting $(1,0)$ and $(0,1)$, but the presence of sharp vertices makes it less likely. Similarly, when there are only three control units (i.e., $\mathit{N}=3$), the minimization problem can be geometrically represented in Figure 7.

Figure 7. Optimal synthetic control in the 3D parameter space.

In Figure 7, the contour sets are ellipsoids with axes parallel to the coordinate axes. The feasible parameter space is the face of a polyhedron connecting three vertices $(1,0,0)$, $(0,1,0)$, and$(0,0,1)$. The optimal synthetic control is a sparse solution with $w_2^{\ast }=w_4^{\ast }=0$ and $w_3^{\ast }=1$. Again, the sparsity of SCM arises because of the sharp vertices of the feasible parameter space. Apparently, similar situations hold in higher dimensions, although we cannot draw graphs when there are more than three control units.

5. A meta-analysis

In this section, we conduct a meta-analysis of empirical studies using SCM in the literature. We search JSTOR and ScienceDirect with the keyword “synthetic control”, and obtain a sample of 107 empirical papers using SCM published during 2003–2022 across disciplines such as economics, political science, health and environment.⁶ Note that some papers using SCM do not report the number of positive weights, and are thus not included in the sample. While these papers certainly do not exhaust all empirical studies using SCM to date, they are sufficient for our purpose. Among the 107 papers in the sample, 61 of them conducted multiple SCM studies, thus we end up with 658 observations. Among these 61 papers, 39 out of them conducted SCM studies for different treated units, while the rest 22 papers applied SCM to different outcome variables.⁷

As before, denote$\mathit{N}$, $\mathit{K}$, $N_p$ and $N_z$ as the number of control units, the number of covariates, the number of positive weights, and the number of zero weights respectively. The dependent variable sparsity is defined as the share of zero weights, i.e., sparsity = $N_z/\mathit{N}$. The summary statistics are reported in Table 2. As revealed by Table 2, the number of control units ($\mathit{N}$) ranges from as few as 4 to as many as 326. On the other hand, the number of covariates ($\mathit{K}$) has much less variation, ranging from 3 to 27. Our key explanatory variable $\left(\mathit{K}/\mathit{N}\right)$ has a mean of 0.522, but with a relatively large standard deviation of 0.369. The number of positive weights ranges from 1 to 54, whereas the number of zero weights ranges from as few as 0 to as many as 319.

Table 2. Summary statistics.

Variable	Observations	Mean	Std. Dev.	Min	Max
$\mathit{N}$	658	26.588	23.975	4	326
$\mathit{K}$	658	9.336	4.479	3	27
$\mathit{K}/\mathit{N}$	658	0.522	0.369	0.0215	3.25
$N_p$	658	5.784	6.491	1	54
$N_z$	658	20.792	23.959	0	319
sparsity ($N_z/\mathit{N}$)	658	0.698	0.257	0	0.991

The dependent variable sparsity ($N_z/\mathit{N}$) has a mean of 0.698, implying that the share of zero weights is 69.8% on average. However, the range of sparsity is quite wide, spanning from 0 to 0.991. Moreover, Figure 8 presents a histogram and a kernel density plot of sparsity. Clearly, most empirical studies using SCM have sparse weights. However, there are also some studies characterized by “dense” weights, i.e., there are more positive weights than zero weights. In particular, there are 30 studies (4.56% of the sample) with zero sparsity, i.e., all weights are positive and there is no zero weight.

Figure 8. The distribution of sparsity.

Next, we explore the relation between the number of positive weights ($N_p$) and the number of covariates ($\mathit{K}$). Figure 9 presents a scatterplot along with a 45 degree line. It is clear that most points are located below the 45 degree line satisfying $N_p\le \mathit{K}$, which is largely consistent with Proposition 1. In fact, among 658 SCM studies in the sample, 88.9% of them (585 studies) satisfy $N_p\le \mathit{K}$. For the rest 11.1% of them (73 studies) with $N_p>\mathit{K}$, there are multiple and potentially unstable weights according to Proposition 1.

Figure 9. The numbers of positive weights and covariates.

Before conducting regression analysis, Figure 10 provides a scatterplot and a linear fit between the dependent variable sparsity and the key explanatory variable $\left(\mathit{K}/\mathit{N}\right)$. Consistent with Hypothesis 1, they are negatively correlated. The coefficient of correlation is −0.397, which is significant at 1%.

Figure 10. The number of positive weights and the number of covariates.

To test Hypothesis 1 more rigorously, we consider the following regression specification:

(6) $\mathit{sparsit}{y}_{\mathit{ij}}={\beta }_0+{\beta }_1\left(K_{\mathit{ij}}/N_{\mathit{ij}}\right)+{\beta }_2{K}_{\mathit{ij}}+{\beta }_3{N}_{\mathit{ij}}+{\beta }_4\left(K_{\mathit{ij}}\times N_{\mathit{ij}}\right)+u_i+{\epsilon }_{\mathit{ij}},$(6)

where the subscripts “$\mathit{ij}$” refer to paper $\mathit{i}$’s jth SCM study, and $u_i$ represents paper fixed effects, which can be controlled by adding dummies for all but the first papers.⁸ The key explanatory variable $\left(K_{\mathit{ij}}/N_{\mathit{ij}}\right)$ is theoretically motivated by Hypothesis 1, while the rest regressors $K_{\mathit{ij}}$, $N_{\mathit{ij}}$ and $\left(K_{\mathit{ij}}\times N_{\mathit{ij}}\right)$ are typical terms used in reduced-form regressions. We take a general-to-specific approach to model selection, also known as the “sequential t-rule”.⁹ Specifically, we first estimate Equation (6)(6) $\mathit{sparsit}{y}_{\mathit{ij}}={\beta }_0+{\beta }_1\left(K_{\mathit{ij}}/N_{\mathit{ij}}\right)+{\beta }_2{K}_{\mathit{ij}}+{\beta }_3{N}_{\mathit{ij}}+{\beta }_4\left(K_{\mathit{ij}}\times N_{\mathit{ij}}\right)+u_i+{\epsilon }_{\mathit{ij}},$(6) by OLS, then drop the most insignificant variable each time, until all regressors are significant. The results are reported in Table 3.

Table 3. Results of OLS regressions.

Dependent Variable: sparsity	(1)	(2)	(3)	(4)
$\mathit{K}/\mathit{N}$	−0.243**	−0.238***	−0.258***	−0.224***
	(0.0937)	(0.0855)	(0.0902)	(0.0705)
$\mathit{K}$	0.0228	0.0199	0.0175
	(0.0183)	(0.0137)	(0.0134)
$\mathit{N}$	0.00285	0.00199
	(0.00332)	(0.00172)
$\mathit{K}\times \mathit{N}$	−0.000126
	(0.000320)
constant	0.762***	0.783***	0.859***	1.043***
	(0.161)	(0.134)	(0.106)	(0.0529)
paper fixed effects	Yes	Yes	Yes	Yes
N	658	658	658	658
R^2	0.571	0.571	0.570	0.567
Adjusted R^2	0.485	0.486	0.486	0.483

Standard errors clustered at the paper level are in parentheses.

*p < 0.1, **p < 0.05, ***p < 0.01.

As shown in Table 3, the coefficient of $\left(\mathit{K}/\mathit{N}\right)$ is negatively significant at 5% or 1%, and the magnitudes are very stable across different specifications. This lends support to Hypothesis 1, i.e., the sparsity of SCM decreases with the relative number of covariates as a fraction of the number of control units. On the other hand, all other regressors (except the constant term) are insignificant throughout.

However, since the dependent variable sparsity is bounded between 0 and 1, this is an example of limited dependent variable. Thus, a linear regression model may be misspecified, since it places no restriction on the dependent variable. For example, the predicted outcome could take values outside the interval $[0,1]$. Therefore, as a robustness check, we next consider fractional response models such as fractional probit and fractional logit (Papke & Wooldridge, 1996, 2008; Wooldridge, 2010), which are fit by quasi maximum likelihood estimation (QMLE).

The likelihood function for fractional probit (or fractional logit) is exactly the same as that of probit (or logit), except that the dependent variable may take on values in the interval $[0,1]$ instead of just 0 and 1. Moreover, as QMLE, fractional response models are consistently estimated as long as the conditional mean is correctly specified. The results of fractional probit and fractional logit regressions are reported in Tables 4 and 5 respectively, where the results are qualitatively similar to the results from OLS regressions reported in Table 3. Overall, the meta-analysis in this section confirms the negative relation between sparsity and the relative number of covariates, which supports Propositions 1 and 2, and the derived Hypothesis 1.

Table 4. Results of fractional probit regressions.

Dependent Variable: sparsity	(1)	(2)	(3)	(4)
$\mathit{K}/\mathit{N}$	−0.804**	−0.786***	−0.867***	−0.744***
	(0.320)	(0.283)	(0.306)	(0.223)
$\mathit{K}$	0.0700	0.0618	0.0572
	(0.0574)	(0.0423)	(0.0433)
$\mathit{N}$	0.00884	0.00645
	(0.0100)	(0.00494)
$\mathit{K}\times \mathit{N}$	−0.000377
	(0.00116)
constant	0.844*	0.895**	1.115***	1.708***
	(0.441)	(0.371)	(0.331)	(0.167)
paper fixed effects	Yes	Yes	Yes	Yes
N	658	658	658	658
Pseudo R^2	0.151	0.151	0.151	0.150

Standard errors clustered at the paper level are in parentheses.

*p < 0.1, **p < 0.05, ***p < 0.01.

Table 5. Results of fractional logit regressions.

Dependent Variable: sparsity	(1)	(2)	(3)	(4)
$\mathit{K}/\mathit{N}$	−1.276**	−1.257***	−1.405***	−1.217***
	(0.537)	(0.469)	(0.519)	(0.373)
$\mathit{K}$	0.108	0.100	0.0937
	(0.100)	(0.0736)	(0.0758)
$\mathit{N}$	0.0144	0.0120
	(0.0192)	(0.00967)
$\mathit{K}\times \mathit{N}$	−0.000400
	(0.00227)
constant	1.449*	1.495**	1.876***	2.859***
	(0.781)	(0.674)	(0.589)	(0.280)
paper fixed effects	Yes	Yes	Yes	Yes
N	658	658	658	658
Pseudo R^2	0.151	0.151	0.150	0.150

Standard errors clustered at the paper level are in parentheses.

*p < 0.1, **p < 0.05, ***p < 0.01.

6. Discussion

Proposition 1 implies that a necessary condition for a unique solution of the SCM problem is that the number of positive weights is less than or equal to the number of covariates. For applied researchers, this provides an easy check on the possibility of multiple solutions. Specifically, if the number of positive weights exceeds the number of covariates, then one should be concerned with the existence of multiple solutions, which may result in unstable and random weights.

A possible way to resolve the issue of multiple solutions is to change the model specification by adding more covariates, since too few covariates could result in unstable weights, which are symptomatic of multiple solutions. However, too many covariates are not necessarily helpful, since irrelevant covariates with little predictive power over the outcome variable would just get zero or very small weights in the quadratic form. Apparently, getting more relevant covariates is always helpful. Therefore, not only the number of covariates matters, but also their quality.

Another way to fix the problem of multiple solutions is to drop control units that are (nearly) multicollinear. As shown in Figure 5, (nearly) multicollinear control units could also result in multiple solutions. Specifically, one may conduct regressions among $\left\{{\mathbf{x}}_2,\cdots ,{\mathbf{x}}_{\mathit{N}+1}\right\}$ and drop the variable with the highest $R^2$, and so on. Hopefully, by adding more covariates or dropping multicollinear control units, the number of positive weights no longer exceeds the number of covariates, thus relieving the concern over multiple solutions.

7. Conclusion

The optimal weights for synthetic control estimation are often sparse in practice, but little explanation has been offered in the literature except Abadie (2021). To the best of our knowledge, this is the first paper devoted entirely to understanding the sparsity of SCM. Specifically, we make three contributions. First, from the perspective of predictor space, we show that if there is a unique solution, then the number of positive weights are upper-bounded by the number of covariates. Second, we offer a simple alternative explanation about the sparsity of SCM from the perspective of parameter space. Last but not least, we conduct a meta-analysis of empirical studies using SCM in the literature. Consistently with our theoretical finding, regression analysis shows that the sparsity of SCM decreases with the relative number of covariates as a fraction of the number of control units.

An important take-away from this paper is that if the number of positive weights exceeds the number of covariates (which happened for 11.1% SCM studies in our sample), then there are multiple solutions to the synthetic control problem, which may result in unstable and random weights. In this case, applied researchers are recommended to change the model specification by adding more covariates or dropping multicollinear control units until multiple solutions are no longer a concern.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Qiang Chen

Qiang Chen is a professor at the School of Economics, Shandong University.

Wenjun Li

Wenjun Li is a PhD student at the School of Economics, Shandong University.

Notes

¹ In this paper, we use the terms “covariate” and “predictor” interchangeably.

² One may regress among $\left\{{\mathbf{x}}_2,\cdots ,{\mathbf{x}}_{\mathit{N}+1}\right\}$ and drop the variable with the highest $R^2$, and so on.

³ For illustration purpose, we do not use the options “nested allopt” for more accurate numerical computation, otherwise the algorithm would fail when there is only one covariate (K = 1).

⁴ Note that Abadie et al. (2010) never estimates the one-covariate case.

⁵ Among these 30 studies with all positive weights, the number of covariates ranges from 3 to 14 with a mean of 9, and the number of control units ranges from 4 to 54 with a mean of 17.43.

⁶ See Appendix 2 for a complete list of these papers used in the meta-analysis.

⁷ Note that all 61 papers with multiple SCM studies used the same donor pools throughout, and 49 of them used the same covariates as well, but 12 of them used different covariates due to missing values.

⁸ The paper fixed effects term $u_i$ captures unobserved heterogeneity across papers, which is analogous to the individual fixed effects in the panel data setting. Since $u_i$ may be correlated with the regressors, its omission could result in inconsistent estimation. Moreover, since some paper dummies are significant in the regressions (unreported), the paper fixed effects specification is preferred over the pooled OLS.

⁹ The opposite specific-to-general approach starts from the smallest model possible (i.e., a model with only the constant term), and proceeds by adding the most significant regressor at each step. However, the specific-to-general approach may suffer from omitted variable bias when the model is too small.

Appendix 1.

Figure A1 in Abadie (2021)

Figure A1. Projecting $X_1$ on the convex hull of $X_0$.

Figure A1 in Abadie (2021) is reproduced here for convenience.

Appendix 2.

A list of papers used in the meta-analysis

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