Single proxy synthetic control

2.1 Setup

Let us consider a setting where N + 1 units are observed over T time periods. Units and time periods are indexed by i ∈ { 0 , 1 , … , N } and t ∈ { 1 , … , T } , respectively. Following the standard synthetic control setting, we suppose that only the first unit with index i = 0 is treated, whereas the latter N units with index i ∈ { 1 , … , N } are untreated control units; these untreated control units are also referred to as donors. Consider a binary treatment indicator A t which encodes whether time t is in the pretreatment period, in which case A t = 0 for t ∈ { 1 , … , T 0 } , or the posttreatment period, in which case A t = 1 for t ∈ { T 0 + 1 , … , T } , respectively. Thus, T 0 is the number of pretreatment periods and T 1 = T − T 0 is the number of posttreatment periods. Unless otherwise stated, we assume that N is fixed and T 0 and T 1 are large with similar order of magnitude. Let Y t and W i t denote observed outcomes of the treated unit and the i th control unit, respectively, for i ∈ { 1 , … , N } . We define W t = ( W 1 t , … , W N t ) ⊺ ∈ R N as the N -dimensional vector of the untreated units’ outcome at time t . We define O t = ( Y t , W t ⊺ , A t ) as the observed data at time t . Let Y t ( a ) and W i t ( a ) denote the potential outcomes of the treated and i th control units, respectively, which one would have observed had, possibly contrary to fact, the treatment been set to A t = a at time t .

For illustrative purposes, we will consider the following two examples throughout:

Throughout, let 1 ( ℰ ) denote the indicator function of an event ℰ , i.e., 1 ( ℰ ) = 1 if ℰ is satisfied and 1 ( ℰ ) = 0 otherwise. Let R be the set of real numbers. Let V 1 ⊥ ⊥ V 2 ∣ V 3 denote that V 1 and V 2 are conditionally independent given V 3 . Conversely, we use V 1 ⊥̸ ⊥ V 2 ∣ V 3 to denote that V 1 and V 2 are conditionally dependent given V 3 . Let 0 p × d , 1 p × d , and I p × p denote the ( p × d ) -dimensional zero matrix, ( p × d ) -dimensional matrix with ones, and ( p × p ) -dimensional identity matrix, respectively.

2.2 Review of existing synthetic control framework

A common target estimand in the synthetic control setting is the average treatment effect on the treated unit (ATT) at time t in the posttreatment periods, i.e.,

Note that, by definition, Y t ( 1 ) − Y t ( 0 ) = τ t * + ν t for t ∈ { T 0 + 1 , … , T } , where ν t is a mean-zero idiosyncratic residual error and, therefore, τ t * may be viewed as a deterministic function of time capturing the expected effect of the treatment experienced by the treated unit if one were to average over the residual ν t . In Section 3.5, we describe an approach for constructing prediction intervals for Y t ( 1 ) − Y t ( 0 ) by appropriately accounting for the idiosyncratic error term ν t . To proceed, we make the consistency assumption:

In addition, we assume no interference, i.e., the treatment has no causal effect on control units.

In the context of Example 1, Assumption 2.2 means that Proposition 99 does not have a causal effect on other states’ cigarette sales; a similar interpretation applies to Example 2.

Under Assumptions 2.1 and 2.2, we have the following result almost surely for t ∈ { 1 , … , T } :

Therefore, for the posttreatment period, Y t ( 1 ) matches the observed outcome Y t while Y t ( 0 ) is unobserved, implying that an additional assumption is required to establish identification of the ATT.

In the classical synthetic control setting, a further assumption relates the observed outcomes of the untreated units with the treatment-free potential outcome of the treated unit. Specifically, following Abadie et al. [2] and Ferman and Pinto [6], suppose that units’ outcomes are generated from the following IFEM [9] for t ∈ { 1 , … , T } :

Here, τ t * is the fixed, nonrandom treatment effect at time t , λ t ∈ R r is a random r -dimensional vector of latent factors that are known a priori to causally impact the treated and donor units, despite being unobserved, and can potentially be nonstationary over time, μ i ∈ R r is a time-fixed r -dimensional vector of unit-specific factor loadings, and e i t is a random error. For identification, it is typically assumed that the number of latent factors r is no larger than the number of donor units N and the pretreatment period length T 0 . Combined with Assumptions 2.1 and 2.2, IFEM (1) implies Y t ( 0 ) = μ 0 ⊺ λ t + e 0 t and Y t ( 1 ) = τ t * A t + Y t ( 0 ) , where the ATT is represented as τ t * = Y t ( 1 ) − Y t ( 0 ) for t ∈ { T 0 + 1 , … , T } ; note that Y t ( 1 ) − Y t ( 0 ) is nonrandom under model (1). In addition, if there were a donor whose factor loading matched that of the treated unit, i.e., μ i = μ 0 for some i ∈ { 1 , … , N } , then W i t would be unbiased for Y t ( 0 ) and, therefore, Y t ( 1 ) − W i t would be unbiased for the ATT. This suggests that confounding bias of the treatment effect on the treated unit’s outcome reflects the extent to which donors’ factor loadings differ from the treated unit’s.

Next, following Ferman and Pinto [6] and Shi et al. [8], suppose that a set of weights γ † = ( γ 1 † , … , γ N † ) ⊺ satisfies

Equations (1) and (2) imply that there exists a synthetic control W t ⊺ γ † = ∑ i = 1 N γ i † W i t satisfying

In the context of Example 1, equation (3) means that:

A similar interpretation holds for Example 2. Therefore, τ t * = E { Y t ( 1 ) − W t ⊺ γ † } for t ∈ { T 0 + 1 , … , T } , i.e., Y t − W t ⊺ γ † is unbiased for the ATT. Unfortunately, it is impossible to obtain γ † from equation (2) because the factor loadings μ i are unknown. Importantly, the synthetic control weights satisfying (2) naturally accommodate an imperfect pretreatment fit as shown in (3), i.e., the synthetic control can significantly deviate from the observed pretreatment fit; however, the corresponding error is mean zero.

Based on (3), one may consider estimating γ † via penalized least squares minimization, say:

where ℛ ( γ ) is a penalty that constraints γ . For instance, Abadie et al. [2] restricts the weight to lie within a simplex, meaning that they are nonnegative and sum to one, Doudchenko and Imbens [18] uses elastic-net penalization, and Robbins et al. [19] uses entropy penalization. In words, γ ^ PLS is obtained by fitting a possibly constrained ordinary least squares (OLS) regression of Y t on W i t . Importantly, without penalization, the moment restriction solving (5) reduces to E { Ψ OLS ( O t ; γ ) } = 0 for t ∈ { 1 , … , T 0 } , where Ψ OLS ( O t ; γ ) = W t ( Y t − W t ⊺ γ ) are standard least squares normal equations.

However, as discussed by Ferman and Pinto [6] and Shi et al. [8], the OLS weights obtained from (5) are generally inconsistent under (2) as T 0 tends to infinity, which can result in biased estimation of the treatment effect unless e i t is exactly zero for all i and t ; see Supplementary Material S1.1 for details. We remark that this result does not conflict with Abadie et al. [2] because their synthetic control weights are assumed to satisfy a perfect pretreatment fit; specifically, there exist values γ # = ( γ 1 # , … , γ N # ) ⊺ satisfying

In the context of Example 1, equation (6) means that:

Example 2 follows a similar interpretation. Note that (6) is distinct from condition (2) of Ferman and Pinto [6] and Shi et al. [8], as reflected in their interpretations (4) and (7). Moreover, as discussed in Ferman and Pinto [6], (6) can be expected to hold approximately under (2) when the variance of the error e i t in (1) becomes negligible as T 0 becomes large. Specifically, in a noiseless setting where e i t = 0 almost surely for all i ∈ { 0 , 1 , … , N } , (1) and (2) imply (6) because (3) becomes equivalent to (6); see Abadie et al. [2] for related results, and Sections 1 and 3.1 of Ferman and Pinto [6], and Section 2 Shi et al. [8] for detailed discussions.

Recently, Shi et al. [8] introduced a proximal causal inference framework for synthetic controls. Specifically, they assume that they have also observed proxy variables Z t = ( Z 1 t , … , Z M t ) ⊺ a priori known to satisfy the following condition in the pretreatment period:

A reasonable candidate for Z t maybe the outcome of units excluded from the donor pool; see Shi et al. [8] for alternative choices of proxies. Then, under Assumptions 2.1 and 2.2, the IFEM (1), condition (2), and the existence of proxies satisfying (8), the synthetic control weights γ † in (2) satisfy E ( Y t − W t ⊺ γ † ∣ Z t ) = 0 and E { Ψ PSC ( O t ; γ † ) } = 0 for t ∈ { 1 , … , T 0 } where Ψ PSC ( O t ; γ ) = g ( Z t ) ( Y t − W t ⊺ γ ) ; here, g is a user-specified function of Z t with dim ( g ) ≥ d . Based on this second result, one can estimate the synthetic control weights as the solution to the generalized method of moments (GMM) [20], i.e., γ ^ PSC = ( γ ^ PSC , 1 , … , γ ^ PSC , N ) ⊺ is the minimizer of T 0 − 1 ∑ t = 1 T 0 { Ψ PSC ( O t ; γ ) } ⊺ Ω ^ { Ψ PSC ( O t ; γ ) } where Ω ^ is a user-specified symmetric and positive-definite weight matrix. Importantly, in contrast to the OLS-based estimator γ ^ PLS in (5), the proximal estimator γ ^ PSC is consistent for γ † . Under certain regularity conditions, Shi et al. [8] established that the resulting GMM estimator of the ATT is consistent and asymptotically normal. For instance, in the special case of constant ATT, i.e., τ t * = τ * for all t ∈ { T 0 + 1 , … , T } , the estimator T 1 − 1 ∑ t = T 0 + 1 T ( Y t − W t ⊺ γ ^ PSC ) is consistent for τ * ; see Section 3.2 of Shi et al. [8] for details.

3.1 Assumptions

In this section, we provide a novel synthetic control approach which obviates the need for an IFEM, and, in fact, does not necessarily postulate the existence of a latent factor λ t . At its core, the approach views the outcomes of the untreated units W i t as proxies for the treatment-free potential outcome of the treated unit Y t ( 0 ) , which is formally stated as follows:

Assumption 3.1 encodes that a function of the untreated units’ outcomes W t is associated with and, therefore, predictive of Y t ( 0 ) at time t ∈ { 1 , … , T 0 } . In terms of Example 1, Assumption 3.1 means that there exists a function of 38 states’ cigarette sales that is associated with cigarette sales in counterfactual California where Proposition 99 was not implemented; a similar interpretation also applies to Example 2. Note that Assumption 3.1 allows for the existence of irrelevant donors among the donor pool, i.e., some untreated units can be independent of Y t ( 0 ) as long as the remaining untreated units are associated with the latter. Additionally, we make the following assumption for h * :

Assumption 3.2 is the key identification assumption of the SPSC framework. It posits the existence of a synthetic control h * ( W t ) that is conditionally unbiased for Y t ( 0 ) . In words, there exists a function of donors h * , possibly nonlinear, whose conditional expectation given Y t ( 0 ) recovers Y t ( 0 ) ; the function h * is a kind of bridge functions [12,13], and we aptly refer to h * as a synthetic control bridge function in this paper. The synthetic control bridge function h * is a solution to the Fredholm integral equation of the first kind (9), and sufficient conditions for the existence of a solution are well-studied in previous related works developed under i.i.d. settings such as Miao et al. [12] and Cui et al. [21]; see Supplementary Material S2.2 for details. Importantly, Assumption 3.2 may still hold in non-i.i.d. settings, such as when ( Y t ( 0 ) , W t ) is nonstationary; see Supplementary Material S1.5 for further details.

In particular, if h * has a linear form, say h * ( W t ) = W t ⊺ γ * for some γ * ∈ R N , the assumption implies the following linear model with an error e ¯ t :

Regression model (10) essentially implies that Y t ( 0 ) falls in the linear span of E { W t ∣ Y t ( 0 ) } , up to a mean zero residual. Thus, Assumption 3.2 may be interpreted as follows for Example 1:

Assumption 3.2 plays an analogous role as condition (2) in Ferman and Pinto [6] and Shi et al. [8] and condition (6) in Abadie et al. [2] in that it establishes a relationship between Y t ( 0 ) and W t ; however, Assumption 3.2 is fundamentally different from these assumptions. In particular, condition (2) implies that the counterfactual outcome Y t ( 0 ) is equal to the synthetic control W t ⊺ γ * plus an error; in contrast, Assumption 3.2 with a linear h * implies that the synthetic control W t ⊺ γ * is equal to the couterfactual outcome Y t ( 0 ) plus a residual error. This distinction highlights that Assumption 3.2 and condition (2) can be viewed as reversed assumptions: they differ in which variable is treated as an error-prone version of the other. Finally, condition (6) is a special case of the former two cases where the residual error is assumed to be exactly zero, i.e., noiseless setting. Consequently, in the pretreatment periods, Assumption 3.2 is strictly weaker than condition (6) because e ¯ t is not necessarily zero.

Unlike condition (2), Assumption 3.2 obviates the need for latent factors, their corresponding factor loadings, IFEM (1), or any related latent factor models. Instead, Assumption 3.2 simply states that it is possible to construct a function of the control units’ outcomes h * ( W t ) , which is conditionally unbiased for the treatment-free potential outcome of the treated units Y t ( 0 ) , without requiring assumptions about how these outcomes are generated. From this viewpoint, h * in Assumption 3.2 serves as a bridge function relating W t and Y t ( 0 ) in that h * ( W t ) is an error-prone version of Y t ( 0 ) . This perspective can be illustrated in Example 1: cigarette sales in counterfactual California, had Proposition 99 not been implemented, are viewed as a variable a priori determined by an unknown mechanism, while cigarette sales in the other 38 states are seen as error-prone transformations of this counterfactual outcome. Then, Assumption 3.2 implies that cigarette sales in counterfactual California can be recovered by aggregating these latter variables up to a mean-zero error.

Moreover, this perspective aligns with existing statistical literature. In particular, model (10) is reminiscent of a nonclassical measurement model [22,23]. From a regression model perspective, the donors’ outcomes W t and the treated unit’s treatment-free potential outcome Y t ( 0 ) in model (10) can be viewed as dependent and independent variables, respectively. This may appear somewhat unconventional at first glance, as some previous synthetic control methods treat Y t ( 0 ) and W t as dependent and independent variables, respectively, in estimation of synthetic control weights. To be more precise, they use equation (5) to estimate the synthetic control weights by regressing Y t ( 0 ) on W t using standard ordinary (or weighted) least squares. However, as model (10) suggests, our framework is different from previous works in synthetic control and better aligned with regression calibration techniques in measurement error literature [22] in that we view the problem as the reverse regression model of W t on Y t ( 0 ) . From this perspective, synthetic control weights γ * are sought to make the weighted response W t ⊺ γ * as close as possible to the regressor Y t ( 0 ) .

To summarize, the SPSC framework differs from existing synthetic control frameworks in its identifying assumptions and interpretation of the synthetic control. Specifically, in the SPSC framework, the synthetic control is viewed as an error-prone outcome measurement (see (10)), eliminating the need for a generative model for Y t ( 0 ) . In contrast, existing approaches interpret the synthetic control as either the projection of the outcome onto the donor’s outcome space (see (3) and (5)) or the outcome itself (see (6)). Despite these differences, both frameworks share key similarities. In both frameworks, synthetic controls are constructed by weighting donor units to optimally match the treated unit during the pretreatment period, though the matching criteria differ, as previously noted. Furthermore, synthetic controls in both approaches serve as unbiased forecasts of the mean treatment-free potential outcome, E { Y t ( 0 ) } , enabling treatment effect estimation by comparing observed outcomes Y t to the synthetic controls over the posttreatment period. In addition, like other synthetic control methods, the SPSC framework accommodates time-varying confounders, distinguishing it from difference-in-differences approaches. Most importantly, the SPSC framework is compatible with the IFEM, as shown in the next section. Thus, while the interpretation of the synthetic control differs, most features of existing synthetic control approaches carry over to the SPSC framework.

3.2 A generative model

While, in principle, Assumptions 3.1 and 3.2 do not require a generative model, it is instructive to consider a model compatible with these assumptions. In this vein, suppose that Y t ( 0 ) and W i t are generated from the following nonparametric structural equation model [24] for t ∈ { 1 , … , T } :

Here, f 0 and f i are structural equations for Y t ( 0 ) and W i t , respectively, λ t = ( λ 1 t , … , λ r t ) ⊺ is an r -dimensional latent factor, and the errors satisfy e i t ⊥ ⊥ λ t for i ∈ { 0 , 1 , … , N } and e 0 t ⊥̸ ⊥ e i t , where the latter condition further strengthens Assumption 3.1 in the sense that W i t is relevant for Y t ( 0 ) even beyond λ t . Figure 1 provides graphical representations compatible with Assumption 3.1 and model (12).

Figure 1

Graphical illustrations for (a) Assumption 3.1, (b) model (12) with correlated errors, and (c) model (12) with independent errors. The dashed bow arcs depict the association between two variables. For illustration, we consider N = 1 .

Under model (12), Y t ( 0 ) is determined by λ t and e 0 t . Given this relationship, it is natural to consider a sufficient condition of Assumption 3.2 characterized in terms of λ t and e 0 t , say:

Condition 3.1 is a sufficient condition for Assumption 3.2 because, under Condition 3.1, we obtain E { h * ( W t ) ∣ Y t ( 0 ) } = E [ E { h * ( W t ) ∣ λ t , e 0 t } ∣ Y t ( 0 ) ] = Y t ( 0 ) .

Under model (12) and Condition 3.1, consider the special case where f i is the IFEM (1):

Here, ω i is a regression coefficient obtained from regressing the i th donor’s error e i t on the treated unit’s error e i t . We remark that ω 0 = 1 and ω i ≠ 0 for some i ∈ { 1 , … , N } , encoding e 0 t ⊥̸ ⊥ e i t . Under the IFEM, Condition 3.1 holds with h * ( W t ) = W t ⊺ γ * if γ * = ( γ 1 * , … , γ N * ) ⊺ solves the following linear system:

A sufficient condition for the existence of the weight γ * is that the matrix A is of full row rank, which is satisfied under the following sufficient (but not necessary) conditions: (i) r < N , i.e., the number of donors N is greater than the number of latent factors, and (ii) the factor loadings μ i are linearly independent. If the matrix A is square and invertible, γ * is uniquely determined. This observation informs that a linear synthetic control satisfying Condition 3.1, and thus Assumption 3.2, is likely to exist when the errors are correlated, and there are sufficient number of donors, regardless of the distribution of the latent factors and errors.

Since equation (14) is based on the IFEM, it has interesting connections with previous works that also rely on this model. In order to elucidate these connections, we consider the following alternative representation of equation (14):

where μ ˜ i = ( μ 1 i , … , μ r i , ω i ) ⊺ for i ∈ { 0 , 1 , … , N } . As the expression itself indicates, condition (15) is similar to condition (2), a condition used in Ferman and Pinto [6] and Shi et al. [8], but there is a notable difference between (2) and (15) in how they handle errors e i t . Specifically, in condition (15), one can address the residual errors e i t by accommodating the regression coefficients ω i as a component of the unit-specific factor loadings μ ˜ i . In contrast, condition (2) does not account for these errors. Consequently, (15) implies (2) because μ i is a subvector of μ ˜ i , indicating that (15) is a stronger condition than (2). However, as stated in Theorem 3.1 in Section 3.3, it is crucial to note that this stronger condition is offset by not requiring an additional condition for establishing identification of the synthetic control weight γ * . In other words, condition (15) alone is sufficient for identification of γ * . On the other hand, condition (2) fails to do so, necessitating additional assumptions for identification of γ * , as exemplified by Ferman and Pinto [6] and Shi et al. [8]. Specifically, Ferman and Pinto [6] requires either (i) Var ( e i t ) = 0 for all i ∈ { 0 , 1 , … , N } , meaning a noiseless setting, or (ii) γ * is a minimizer of V ( γ ) = E { ( e 0 t − ∑ i = 1 N γ i e i t ) 2 } , the variance of a linear combination of error terms appearing in (3); see Propositions 1 and 2 of Ferman and Pinto [6] for details. Interestingly, under (i), all ω i t can be taken as zero, and (15) becomes equivalent to (2), the assumption made by Ferman and Pinto [6] and Shi et al. [8]. Finally, in the degenerate case where Y t ( 0 ) and W t share the same error, i.e., e 0 t = e 1 t = ⋯ = e N t almost surely, condition (15) implies the perfect fit condition, i.e., condition (6), in which case the unconstrained OLS weights (5) are consistent as T 0 tends to infinity.

While the IFEM with correlated errors in (13) is useful for motivating the SPSC framework, the standard IFEM typically assumes no correlation among errors, i.e., ω i = 0 for all i ∈ { 1 , … , N } in (13). When the errors are uncorrelated, the solution to equation (14) may not exist, implying that no linear SPSC bridge function satisfies Condition 3.1. This may suggest that the SPSC framework may not be compatible with a standard IFEM. However, a linear synthetic control bridge function satisfying Assumption 3.2 may still exist under the IFEM with uncorrelated errors, while Condition 3.1 is violated; this is because Condition 3.1 is not a necessary condition of Assumption 3.2. With additional assumptions regarding the latent factors and errors, it is possible to conceive of a reasonable scenario where the SPSC framework remains valid within the standard IFEM. For instance, if λ t and e t = ( e 0 t , e 1 t , … , e N t ) ⊺ follow multivariate normal distributions with homoskedastic variances, specifically λ t ∼ N r ( ν t , Σ λ ) and e t ∼ N N + 1 ( 0 ( N + 1 ) × 1 , Σ e ) , then a linear synthetic control satisfying Assumption 3.2 exists even when Σ e is a diagonal matrix; see Supplementary Material S1.5 for details. In essence, such circumstances may arise because, despite the uncorrelated errors, W t and Y t ( 0 ) remain associated through the latent factors λ t , allowing for the possibility of a linear SPSC to exist; see Figure 1 (c) for a graphical illustration. In summary, while uncorrelated errors may undermine the plausibility of the SPSC framework, it can still be valid if certain conditions on ( λ t , e t ) are met such as the normality assumption.

3.3 Identification of the synthetic control and the treatment effect

As a direct consequence of Assumptions 2.1, 2.2, 3.1, and 3.2, the synthetic control bridge function h * can be represented as a solution to the moment equation given in the following result:

The proof of the theorem, as well as all other proofs, are provided in Supplementary Material S3. Theorem 3.1 motivates our approach for estimating the synthetic control bridge function h * , as it only involves the observed data. Another consequence of Assumptions 2.1, 2.2, 3.1, and 3.2, is that, as formalized in Theorem 3.2 below, the synthetic control bridge function h * ( W t ) can be used to identify τ t * :

Theorem 3.2 provides a theoretical basis for the use of the synthetic control method to estimate the ATT. Specifically, following Abadie and Gardeazabal [1] and Shi et al. [8], we use Y t − h * ( W t ) in a standard time series regression where the ATT is identified as the deterministic component of the decomposition Y t − h * ( W t ) = τ t * + ε t , with ε t representing a mean-zero error. The following sections elaborate on this approach, first describing how the identification result leads to an estimator of the synthetic control.

To facilitate the exposition, hereafter in the main text, we restrict attention to inference under a linear bridge function, i.e., h * ( W t ) = W t ⊺ γ * , while allowing for the possibility for γ * not to be unique. In Supplementary Material S2, we present the more general case where h * is nonparametric.

3.4 Estimation and inference of the treatment effect under a linear bridge function

We first discuss estimation of the synthetic control weights γ * . We consider the following time-invariant estimating function for the pretreatment periods:

Here, ϕ : R → R p is a p -dimensional user-specified function of the treated unit’s outcome. Theorem 3.1 implies that the estimating function Φ pre satisfies E { Φ pre ( O t ; γ * ) } = 0 for t ∈ { 1 , … , T 0 } , indicating that the estimating function Φ pre can be used to obtain an estimator of γ * . An important remark on ϕ is that the dimension of ϕ can be smaller than the number of donors, i.e., p < N . Therefore, ϕ can be specified as a simple function, e.g., ϕ ( y ) = y .

It is instructive to note that solving the estimating equation E { Φ pre ( O t ; γ ) } = 0 has a close connection to performing an instrumental variable regression. To illustrate this, consider a simple setting where ϕ ( y ) = y and N = 1 , along with an alternative form of model (10) for t ∈ { 1 , … , T 0 } :

One might attempt to interpret the model on the right-hand side as a standard regression model, treating Y t as the response variable and W t as the explanatory variable. However, such an interpretation would not be correct, as the error term − e ¯ t is orthogonal to the response variable Y t . Instead, the right-hand side model exhibits the following properties: (i) the error term − e ¯ t is correlated with W t (as induced from the left-hand side model), making W t an endogenous explanatory variable on the right-hand side model; (ii) the error − e ¯ t is orthogonal to Y t ; and (iii) Y t is correlated with W t under Assumption 3.1. Thus, Y t can serve as an instrumental variable for W t , allowing for an instrumental variable regression estimator, where Y t and W t are used as the instrument and the endogenous explanatory variable, respectively. This estimator is given by γ ^ IV = ( T 0 − 1 ∑ t = 1 T 0 Y t W t ) − 1 ( T 0 − 1 ∑ t = 1 T 0 Y t 2 ) . Notably, γ ^ IV is consistent for γ * = { T 0 − 1 ∑ t = 1 T 0 E ( Y t W t ) } − 1 { T 0 − 1 ∑ t = 1 T 0 E ( Y t 2 ) } under some conditions, which is the solution to the estimating equation E { Φ pre ( O t ; γ ) } = 0 . The case for a general ϕ and multiple donors can be understood in a similar manner, with the main difference being the use of multiple instrumental variables ϕ ( Y t ) ∈ R p and multiple explanatory variables W t ∈ R N .

The choice of ϕ affects the efficiency of the corresponding estimator of γ * and the treatment effect parameter β * , which we later define in this section; see Section 2 of Donald et al. [25] for a similar discussion. Therefore, one could theoretically select the optimal ϕ from a set of candidates that minimizes the asymptotic variance of the estimators, thereby maximizing efficiency. For example, ϕ can be selected from basis functions such as polynomials up to the p th power, where p is determined to minimize the asymptotic variance of the estimators of ( γ * , β * ) ; other examples of basis functions include truncated polynomial bases, Fourier basis functions, splines, or wavelets such as the Haar basis; see Chen [26] the references therein for more details on how to choose the optimal ϕ over a basis function space. However, selecting the optimal ϕ can be computationally intensive, and despite this burden, it may yield only marginal gains in efficiency. From a practical standpoint, we use a simple specification for ϕ , namely, the identity function ϕ ( y ) = y , leading to p = 1 . In the simulation studies and data analysis, this simple choice of ϕ performs well and produces reasonable results compared to competing methods in settings we consider, although we cannot guarantee this to be the case in all settings one might face in practice.

A time-invariant specification of ϕ may sometimes lead to poorly behaved estimates of synthetic control weights, particularly in scenarios where the outcomes exhibit nonstationary behavior. To address this, the estimating function can be adapted to accommodate secular trends as follows:

Here, D t ∈ R d is a d -dimensional vector of basis functions to de-trend nonstationary behaviors of the outcomes. We assume that D t is selected such that there exists a unique vector η * satisfying E { Y t ( 0 ) } = D t ⊺ η * for t ∈ { 1 , … , T 0 } , meaning that the time trend of Y t ( 0 ) over the pretreatment period is correctly specified by the regression model spanned by D t . The selection of D t can be evaluated by examining the residuals from regressing Y t on D t over the pretreatment period. For instance, to account for a linear trend, one might select D t = ( 1 , t ⁄ T 0 ) ⊺ , where these terms account for an intercept and the drift of a nonstationary process. Alternatively, one could choose D t = ℬ d ( t ) , the d -dimensional cubic B-spline function, to capture nonlinear trends. While D t could also be specified as a dummy vector – allowing each component of η * to represent time fixed effects for each pretreatment period – this approach may result in an inconsistent ATT estimator. To ensure valid inference of the ATT while reducing the risk of misspecification, we recommend using a cubic B-spline basis of small to moderate dimension. In both our simulation studies and real-world analysis (Sections 4 and 5), we used a six-dimensional cubic B-spline basis, which demonstrated reasonable performance.

The function g : [ 0 , ∞ ) ⊗ R → R d + p can be seen as a basis function for both outcome and time period. Note that the dimension of g may be smaller than N , which may arise from simple specifications of D t and ϕ . For instance, we specify ϕ ( y ) = y and D t = ( 1 , t ⁄ T 0 ) ⊺ , the dimension of g is then equal to three, which may be substantially smaller than the number of untreated units N .

The time-varying estimating function Ψ pre satisfies E { Ψ pre ( O t ; η * , γ * ) } = 0 for t ∈ { 1 , … , T 0 } under Assumptions 2.1, 2.2, 3.1, and 3.2. This ensures that an estimator of γ * can be obtained by using the time-varying estimating function Ψ pre rather than the time-invariant estimating function Φ pre . In fact, incorporating the time-varying term can potentially enhance the finite sample performance of the proposed estimator in the presence of nonstationary behavior. For instance, under the IFEM (13), we show that incorporating time-varying components reduces the bias of the estimator of γ * when the latent factor λ t exhibits a secular trend; see Supplementary Material S1.6 for this result. Moreover, simulation studies in Section 4 suggest that including time-varying components can help reduce bias in the presence of a time trend. Therefore, in the remainder of the paper, we use the time-varying estimating function Ψ pre unless stated otherwise.

An estimator of γ * can in principle be obtained based on the empirical counterpart of the moment condition. Because the dimension of Ψ pre is allowed to be smaller than the dimension of ( η * , γ * ) , the standard GMM theory [20] does not readily apply; typically, standard GMM requires the number of moment equations to be greater or equal to the number of unknown parameters. To regularize the problem, we include a ridge penalty term for γ * . Specifically, the regularized GMM estimator γ ^ ρ with regularization penalty ρ ∈ ( 0 , ∞ ) is defined as the solution to the following minimization problem:

Here, Ψ ^ pre ( η , γ ) = T 0 − 1 ∑ t = 1 T 0 Ψ pre ( O t ; η , γ ) is the empirical mean of the estimating function over the pretreatment periods evaluated at ( η , γ ) . Also, Ω ^ pre = diag ( I d × d , Ω ^ g ) ∈ R ( 2 d + p ) × ( 2 d + p ) is a user-specified symmetric positive definite block-diagonal matrix with Ω ^ g ∈ R ( d + p ) × ( d + p ) , which can simply be set to the identity matrix. Since the first block of Ω ^ pre is the identity matrix, η ^ reduces to the OLS estimator, i.e., η ^ = ( ∑ t = 1 T 0 D t D t ⊺ ) − 1 ( ∑ t = 1 T 0 D t Y t ) .