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Theory for identification and Inference with Synthetic Controls: A Proximal Causal Inference Framework

Synthetic control methods are commonly used to estimate the treatment effect on a single treated unit in panel data settings. A synthetic control (SC) is a weighted average of control units built to match the treated unit's pre-treatment outcome trajectory, with weights typically estimated by regressing pre-treatment outcomes of the treated unit to those of the control units. However, it has been established that such regression estimators can fail to be consistent. In this paper, we introduce a proximal causal inference framework to formalize identification and inference for both the SC weights and the treatment effect on the treated. We show that control units previously perceived as unusable can be repurposed to consistently estimate the SC weights. We also propose to view the difference in the post-treatment outcomes between the treated unit and the SC as a time series, which opens the door to a rich literature on time-series analysis for treatment effect estimation. We further extend the traditional linear model to accommodate general nonlinear models allowing for binary and count outcomes which are understudied in the SC literature. We illustrate our proposed methods with simulation studies and an application to evaluation of the 1990 German Reunification.


Introduction
Synthetic control (SC) methods are commonly used to estimate the impact of an intervention when one has observed time series data on a single treated unit and multiple untreated units in both pre-and post-treatment periods (Abadie & Gardeazabal 2003, Abadie et al. 2010).
The idea is to match the treated unit to a weighted average of control units, referred to as "synthetic control", such that the synthetic control's post-treatment outcome predicts the treated unit's unobserved potential outcome under no treatment. In practice, this is often operationalized by regressing (summaries of) the pre-treatment outcome and covariates of the treated unit on those of the control units using ordinary or weighted least squares (OLS/WLS) under the constraints that the coefficients are non-negative and sum to one, and taking the estimated regression coefficients as synthetic control weights. The treatment effect on the treated unit is then estimated as the difference of post-treatment outcomes between the treated unit and its synthetic control. The synthetic control method has become increasingly popular in recent years with a surge of new methods including methods for multiple treated units (Abadie & L'Hour 2021, Ben-Michael et al. 2021a, matrix completion and matrix estimation (Athey et al. 2021, Bai & Ng 2021, Amjad et al. 2018), penalization (Doudchenko & Imbens 2016, Ben-Michael et al. 2021b, and statistical inference (Li 2020, Chernozhukov et al. 2021, Cattaneo et al. 2021).
The original synthetic control method of Abadie et al. (2010) assumes a perfect match of the synthetic control to the treated unit in the pre-treatment period, i.e., there exists a set of weights such that the weighted average of the pre-treatment outcomes and covariates among control units is exactly equal to that of the treated unit. However, as acknowledged by Abadie et al. (2010), it is rarely the case that this assumption holds exactly in the observed data. Instead of assuming perfect match of the observed data, an alternative framework widely studied in the synthetic control literature is to assume perfect match of the underlying unobserved factor loadings (Ferman & Pinto 2019, Amjad et al. 2018, Powell 2018, Ferman 2021. In this setting, Ferman & Pinto (2019) showed that the OLS estimates of the synthetic control weights can be inconsistent. Intuitively, the pre-treatment outcome trajectories of control units are proxies of time-varying latent factors but are measured with error.
When such proxies of the latent factors are used as independent variables in OLS/WLS, estimated regression coefficients will generally be inconsistent due to the well-known errorsin-variables regression problem (Carroll et al. 2006). To address this problem, Ferman & Pinto (2019) proposed to use lagged control unit outcomes as instrumental variables to construct unbiased estimating equations. Amjad et al. (2018) proposed to de-noise the data via singular value thresholding. However, both rely on the assumption that idiosyncratic error terms are independent across units and time.
In this paper, we take a completely different view: in practice, not all control units contribute to construction of a synthetic control, and data on control units not used to form the synthetic control are typically discarded. The outcomes of such control units previously perceived as unusable can in fact be repurposed to construct identifying moment equations of synthetic control weights under certain conditions. More generally, we argue that outcomes of control units are essentially proxies of the latent factors. Rather than directly regressing the outcome on all such proxies, which as noted above fails to be consistent, one can split the set of proxies into two, thus leveraging the first set of proxies as instrumental variables for a synthetic control defined in terms of the second set of proxies. Besides outcomes of control units, measured covariates contemporaneous with outcomes of the treated and control units can also serve as candidate proxies. As we show in this paper, leveraging such proxies of latent factors can yield identification of synthetic control weights under certain conditions, which is motivated by recent work on proximal causal inference (Miao, Shi & Tchetgen Tchetgen 2018. Our method produces consistent estimators of synthetic control weights allowing for serial correlation of error terms, as well as both stationary and non-stationary latent factors (Hsiao et al. 2012, Ferman & Pinto 2019. We also establish nonparametric identification which relaxes the linear factor model assumption widely assumed in the literature. Moreover, we propose to view the difference in the post-treatment outcomes between the treated unit and the synthetic control as a time series where the treatment effect on the treated unit is captured by a deterministic time trend. This observation opens the door to a rich and extensive literature on time series analysis for estimation of the treatment effect.
Our paper is organized as follows. In Section 2 we introduce our proposed method in a simple setting of linear interactive fixed effects model, which is the most widely used conceptual framework for synthetic control methods. We discuss the potential lack of consistency of classical synthetic control methods in Section 2.1. We then propose a proximal causal inference method for consistent estimation and formal statistical inference in Sections 2.2-2.3. Specifically, we first focus on estimation of the synthetic control weights in Section 2.2.
Then we propose time series analysis of the time-varying treatment effect on the treated unit, as well as joint estimation and inference for both synthetic control weights and a constant average treatment effect on the treated unit in Section 2.3. We extend our results to allow for covariate adjustment in Section 2.4. We then show that the fixed effects model and the corresponding identification condition are a special case of a more general identification condition in Section 3. In fact, we establish nonparametric identification of the treatment effect on the treated which accommodates nonlinear models and different types of outcomes such as binary or count outcomes. We illustrate our proposed methods with simulation studies in Section 4 and an application to evaluation of the 1990 German reunification in Section 5.
We close with a brief discussion in Section 6.

Methods under the interactive fixed effects model
In this section, we introduce the proximal causal inference framework under a linear interactive fixed effects model. For ease of exposition, we first introduce our framework ignoring measured covariates in Sections 2.1-2.3, then incorporate covariates in Section 2.4.

Review of classical synthetic control methods
Suppose there are N + 1 units indexed by i = 0, . . . , N , which are observed over T time periods indexed by t = 1, . . . , T . Only one unit is treated. Without loss of generality, let the treated unit be indexed by i = 0. Let T 0 (1 < T 0 < T ) be the number of pre-treatment periods for the treated unit, then T − T 0 is the number of post-treatment periods. We focus on the scenario where N is fixed and T 0 and T are large. Let X t denote the binary treatment indicator for the treated unit at time t, then X t = 0 if t ≤ T 0 and X t = 1 if t > T 0 . To distinguish the outcome trajectories of the single treated unit and multiple control units, let Y t and W it denote the observed outcome of the treated unit and the i-th control unit, respectively, at time t. Likewise, let (Y t (1), Y t (0)) and (W it (1), W it (0)) denote the pair of potential outcomes for the treated unit and the i-th control unit, respectively, one would have observed had possibly contrary to fact, treatment or control been assigned at time t.
We are interested in the treatment effect on the treated unit in the post-treatment period, We assume that the observed outcome is a realization of the potential outcome corresponding to the assigned treatment value: , for any unit i and time t.
A common synthetic control model in the literature assumes that the outcomes of the treated and control units follow the data generating mechanism (Bai 2009) below: Assumption 2 (Interactive fixed effects model). For any unit i at time t, where β t is the time-varying coefficient of treatment (assumed to be fixed) at each t; λ t ∈ R r is an r × 1 vector of unmeasured common factors (assumed to be random, can be either stationary or non-stationary); µ i ∈ R r , i = 0, . . . , N is an r × 1 vector of unit-specific factor loadings (assumed to be fixed); and ε it , i = 0, . . . , N is the error term with It is important to note that the treatment assignment is not exogenous but rather depends on µ i λ t , i = 0, . . . , N . The interactive effect µ i λ t can be viewed as the main source of unmeasured confounding because it is associated with the treatment assignment and is also predictive of the outcome. Therefore, we also refer to λ t as the unmeasured confounder. Figure 1 presents a graphical illustration of our model assumptions.
Remark 1. Our proposed method can also accommodate additive fixed effects (a special case . . , N , including unknown common factors δ t with constant factor loadings across units as considered in Abadie et al. (2010) and unknown unit-specific effect ζ i . Figure 1: A graphical illustration of the relationships among the treatment X t , the observed outcomes of the treated unit Y t and the control units W it , i = 0, and the unobserved common factor λ t at each time point t. Measured covariates are suppressed for simplicity.
Under Assumptions 1 and 2, we have (4) Eq. (5) implies no interference, i.e., the treatment assigned to the treated unit does not impact the outcomes of the control units. Eqs. (3) and (4) imply that Y t (x t ) = x t β t +µ T 0 λ t +ε 0t and thus the causal effect on the treated unit at time t is β t = Y t (1) − Y t (0). In the posttreatment period, Y t (1) = Y t is always observed while Y t (0) is unobserved. Our goal is to find a "synthetic control" such that the post-treatment outcome of the synthetic control can predict the unobserved Y t (0) of the treated unit.
A key assumption of the synthetic control methods is that the unmeasured confounding effect on the treated unit can be matched by a weighted average of the unmeasured confounding effect on a set of the control units, referred to as the "donor pool". Let D index the control units selected into the "donor pool", and let |D| denote the number of control units in the donor pool. For a sequence of unit-specific values a i ∈ R, i = 1, . . . , N , let a i,i∈D denote the |D| × 1 vector of a i 's for i ∈ D, i.e., units in the donor pool; if a i ∈ R d is a d-dimensional vector, then a i,i∈D is a d × |D| matrix. Similarly, let W it,i∈D denote the The assumption below is key to identification of a synthetic control, an extension of which to nonlinear setting is provided in Section 3.
Assumption 3 (Existence of synthetic control). There exist a set of weights α i,i∈D with Remark 2. A necessary condition for existence of a solution to the system of linear equations (6) is that the r × |D| matrix µ i,i∈D has full row rank with r ≤ |D|, which entails that the number of donors at least as large as the number of latent factors.
Remark 3. Abadie et al. (2010) additionally assume that the weights are non-negative and sum to one, such that the synthetic control lies in the convex hull of the donors' outcomes.
The normalization condition of the weights is implied by Assumption 3 when there exist unknown common factors δ t with constant factor loadings as discussed in Remark 1. The positive weights assumption is required mainly for interpretation. Because α i,i∈D is estimated by regression in Abadie et al. (2010), when the number of parameters, |D|, is larger than the sample size, the convex hull assumption can bypass the need for regularization. Relaxations of the convex hull restriction are discussed in Doudchenko & Imbens (2016). It is important to note that while α i,i∈D satisfying Assumption 3 may not be unique, i.e., there may not be a unique synthetic control, all synthetic controls satisfying Assumption 3 lead to the same E[Y t (0)] for any t as detailed in Theorem 1. As such, the treatment effect on the treated unit is identified as long as one can identify at least one set of synthetic control weights α i,i∈D satisfying Assumption 3. Furthermore, our proposed method does not preclude the possibility to further require synthetic controls to fall within the convex hull of donor outcomes, which as argued by Abadie et al. (2010) can provide an appealing interpretation.
Under Assumptions 2 and 3 we have that in the pre-treatment period, Thus a natural approach to estimate α i,i∈D is regression using pre-treatment data. Particularly, the observed outcomes of the control units W it = µ T i λ t + ε it can be viewed as proxies of the latent factors λ t up to a scale µ i with measurement error ε it , hence regression adjustment for W it allows one to adjust for mismeasured versions of the unmeasured confounder. Abadie et al. (2010) proposed the following constrained OLS α i,i∈D = arg min where A = {α i,i∈D : α i ≥ 0, i∈D α i = 1} restricts the weights to be non-negative and sum to one. Then the treatment effect in the post-treatment period is estimated with However, the (constrained) OLS weights are generally inconsistent. This is because in equation (7), the residual ε 0t − i∈D α i ε it is correlated with the regressor W it,i∈D , therefore the coefficients α i,i∈D obtained from (constrained) OLS is inconsistent unless ε it is zero, i.e., a noiseless setting. The inconsistency of α i,i∈D has been pointed out by Ferman & Pinto (2019) and Powell (2018). They showed that α i,i∈D converges to a set of weights that not only attempt to match the latent factors as in Assumption 3, but also strain to minimize the noise in the synthetic outcome.
It is important to note that this result does not necessarily conflict with that of Abadie et al. (2010), as they do not make Assumption 3. Specifically, rather than assuming perfect match of µ i , Abadie et al. (2010) assumed perfect match of the observed outcomes, that is, there exists α i,i∈D that solves (8) with the optimum equal to exactly zero, i.e., Under certain regularity conditions they established that the bias of the estimated treatment effect is bounded by a function that goes to zero as T 0 goes to infinity, or as the maximal variance of ε it goes to zero. As exact matches can rarely be enforced in practice, it is unclear whether the theoretic error bounds provide a reliable guaranty for finite horizon performance of synthetic controls.

Identification and estimation of synthetic control weights
In this paper, we propose a novel method to consistently estimate synthetic control weights.
Our proposal is motivated by the key observation that, in the presence of latent variables λ t , direct regression adjustment of W it,i∈D , which are valid proxies of latent variables, will generally fail to be consistent due to measurement errors ε it . However, as we establish below, consistent estimation is possible if one has access to supplemental proxies which, although not used directly to form a synthetic control, may be used to identify synthetic control weights. Specifically, suppose that additional proxies of λ t are available, which are a priori known to be associated with the treated and donor units in the pre-treatment period only through λ t . Formally, We argue that such proxy variables are often available in synthetic control settings; in fact, as mentioned in Section 1, a convenient choice of proxy variables is the outcome of Such data previously perceived as unusable can in fact be repurposed towards identification and thus continue to play an important role as valid proxy variables. Another candidate set of proxy variables is the set of measured covariates of units in the donor pool if they are contemporaneous with {Y t , W it,i∈D }, and hence do not have a causal impact on the outcomes but are associated with the outcomes to the extent of an association with λ t and X t .
We now introduce our proposed identification and estimation strategy leveraging proxy variable(s) Z t . Under Assumptions 2-3 we have, in the pre-treatment period, By Assumption 4 we have Marginalizing over λ t on each side of (10) with respect to Eq. (11) indicates that α i,i∈D can in fact be identified upon re-framing the approach in terms of proxies. Such re-framing formalizes a connection between synthetic controls and recently ing assumption under certain conditions. The synthetic control setting can be viewed as a special case of the proximal causal inference framework with W it and Z t serving as proxies of the latent factors λ t .
In order to ensure unique identification of the synthetic control weights as a solution of Eq. (11) in the pre-treatment period, we introduce the following completeness condition.

Assumption 5 (Completeness). For any square integrable function
Assumption 5 is formally known as completeness condition which characterizes the informativeness of Z t about W it,i∈D , in the sense that any infinitesimal variation in W it,i∈D is captured by variation in Z t such that no information has been lost through projection of W it,i∈D on Z t in the pre-treatment period. Completeness is a fundamental statistical concept in minimum variance unbiased estimation and hypothesis testing (Lehmann & Scheffé 2012, Basu 2011) and has recently been used to establish identification condition in a variety of problems such as nonparametric instrumental variable methods (Newey & Powell 2003, Ai & Chen 2003, Chernozhukov & Hansen 2005, Hall & Horowitz 2005, Darolles et al. 2011, D'Haultfoeuille 2011, Chen et al. 2014, measurement error (Hu & Schennach 2008, Carroll et al. 2010, An & Hu 2012, and missing data (Miao & Tchetgen Tchetgen 2016). It holds in a large class of commonly-used models such as exponential families (Newey & Powell 2003) and location-scale families (Mattner 1992, Hu & Shiu 2018, and nonparametric additive models (Darolles et al. 2011, D'Haultfoeuille 2011. In practice, we recommend to measure a rich set of proxies to make the completeness assumption plausible. We refer the readers to Miao et al. (2020) for a thorough review and discussion on the completeness condition.
We now present the following result for identification of the synthetic control weights which is proved in Section A of the Supplementary Material.
Based on the above identification results, we propose to estimate α i,i∈D using pre-treatment data with the following estimating function where g(·) is a d-dimensional (d ≥ |D|) vector of user-specified functions. We show that Remark 4. Although Assumption 3 allows |D| ≥ r, Assumption 5 implicitly requires that |D|, r, and the dimension of Z t , denoted dim(Z t ), are all equal, which is a strong assumption and can in fact be relaxed. To relax Assumption 5, instead of assuming that Z t is complete with respect to W it,i∈D , one can assume that Z t is complete with respect to λ t . Under this weaker assumption, Z t is an instrumental variable for λ t with dim(Z t ) ≥ r and |D| ≥ r.
control weights that satisfy Assumption 3 and thus identify the mean potential outcome as and W it,i∈D is rank deficient and there are infinite solutions to Eq. (12). In this case, we refer the reader to Shi, Miao, Nelson & Tchetgen Tchetgen (2020) for possible strategies to estimate α i,i∈D . Under the linear interactive fixed effects model, one can further relax arguably not necessarily an easily interpretable condition.
Example 1. As discussed above, when available, the outcome of supplemental control unit(s) not in the donor pool, W jt,j∈[N ]\D may be used as proxy variables, assuming independent errors \D for all t and that Assumption 5 holds for Z t = W jt,j∈[N ]\D . Then our proposed method entails estimating α i,i∈D based on the following estimating function given λ t in the pre-treatment period, hence one can show that U PI,t (α i,i∈D ) is an unbiased estimating function with mean zero. In contrast, the moment function used in the classical OLS-based synthetic control method is the standard normal equation which does not have mean zero and therefore fails to be unbiased because W it,i∈D is correlated with the residual.
The above moment equation motivates use of the generalized method of moments (GMM) method with U t (α i,i∈D ) providing identifying moment restrictions (Hansen 1982). The GMM solves α i,i∈D = arg min is the sample moments evaluated at an arbitrary value α i,i∈D and Ω is a d × d user-specified symmetric and positive-definite weight matrix.
Previous literature such as Abadie et al. (2010) and the lagged-outcome method of Ferman & Pinto (2019) assumed that the error terms ε it are independent both across units and over time to ensure identification and consistent estimation. We do not necessarily require independence either across units or over time. Instead, we allow ε it to be serially correlated over time, e.g. MA(1) and AR(1) time series: Assumption 6 (Stationary error). For each unit i, ε it is a stationary (i.e., any sequence is identically distributed) and weakly dependent process (i.e., asymptotically uncorrelated Our weakly dependent assumption replaces the assumption of independent and identically distributed errors while ensuring that ergodic laws of large numbers and central limit theorems apply. Consistency and asymptotic normality of GMM estimators have been well established (Hansen 1982, Newey & West 1986, Hall 2005) under such conditions. The theorem below follows Theorem 3.2 of Hall (2005)

Estimation and inference of the causal effect on the treated
Recall that by Eq. (4), the treatment effect on the treated unit is β t = Y t (1) − Y t (0), for any t. Under Assumptions 2-3, we have that, given the true α i,i∈D , β t is identified by in the post-treatment period for any t = T 0 + 1, . . . , T . However, at each time point t, only a single realization of (Y t , W it,i∈D ) is available for inference, which is clearly insufficient to consistently estimate β t without pooling information across time via some form of smoothing, e.g., a parametric model encoding smooth dependence on t. For example, when appropriate, one may assume that β t = β(t/T ; γ) is a function of time indexed by a finite-dimensional parameter γ ∈ R k , such as β t = γ 0 + γ 1 t/T .
More generally, β t can be estimated quite flexibly, using standard time series analysis. This is because may be viewed as a standard time series where β t captures deterministic trends over time, including potential secular and seasonal patterns, and r t = ε 0t − i∈D α i ε it is a mean zero weakly dependent stationary process by Assumption 6. Thus, given a consistent estimator α i,i∈D based on pre-treatment data using the methods described in Section 2.2, one can estimate e t with e t = Y t − i∈D α i W it in the post-treatment period, which can in turn be analyzed as time series data. For example, one may first inspect the estimated time series e t to determine an appropriate functional form for β(t/T ; γ), which may then be used to estimate γ by standard regression techniques. Alternatively, one may estimate β t by standard time series smoothing techniques such as moving averages. When both secular and seasonal pattern exist, one may estimate both components, say using an autoregressive-moving-average (ARMA) model or possible generalizations of the latter with the goal of producing stationary residuals, a process sometimes referred to as pre-whitening a time series. As there exist a rich and extensive literature on such standard time series modeling techniques, we do not further discuss them here, and instead refer the readers to the monograph Hamilton (1994) for more detailed exposition of such techniques.
In this section, it is worth focusing on the important commonly adopted special case of β t = β, i.e., the causal effect is constant over time which we refer to as constant average treatment effect on the treated (CATT). In this case, Eq. (13) suggests using for user-specified weight v t as an estimator of β based on the post-treatment data. The most commonly used weight is equal weight, i.e., v t = 1, in which case We note that the above two-stage estimation of α i,i∈D followed by β can be achieved by stacking Eqs. (12) and (14) into a combined estimating function for simultaneous estimation of α i,i∈D and β. We illustrate in the scenario when v t = 1.
We show that E[ U t (θ)] = 0 in Section B of the Supplementary Material. Then we have that the two-stage estimation of α i,i∈D followed by β based on Eqs. (12) and (14) can be equivalently achieved as , which satisfies the following theorem proved in Section B of the Supplementary Material.
is the variance-covariance matrix of the limiting distribution of √ T m(θ).
To make inference, a consistent estimator of Σ, denoted as Σ can be computed with Σ 0 S Σ T 0 , where S is a heteroskedasticity consistent (HC) estimator of S when U t is serially uncorrelated (White 1980), and the heteroskedasticity and autocorrelation consistent (HAC) estimator of S when U t is serially correlated (Newey & West 1986), and Σ 0 is the empirical version of Σ 0 . Both estimation and inference can be implemented with off-the-shelf R package such as gmm. For instance, one can conveniently call the gmm function of the gmm package as follows gmm::gmm(g=Y∼X+W,x=∼X+Z), where Z denotes the data object for (1−X t ) * Z t , W for W it,i∈D , X for X t , and Y for Y t , and one can further specify whether U t is serially independent or correlated. When U t is serially independent and homoskedastic, as discussed in Tchetgen Tchetgen et al. (2020), our proposed estimator is equivalent to a simple two stage least squares estimator.
Remark 5. When Z t satisfies Z t ⊥ ⊥ (Y t , W it,i∈D ) | λ t , X t , a stronger assumption than As-sumption 4, we have that for any x t and any t Therefore, instead of Eq. (15) which estimates α i,i∈D using pre-treatment data and β using post-treatment data, one may jointly estimate θ using vector of userspecified functions. One could enrichg(·) byg(t, Z t , X t ) to allow for flexible parameter estimation using a subset of data within a specified time period. With appropriate choice ofg, this framework unifies both two-stage estimation approach (g = {(1 − X t )g(Z t ) T , X T t } T ) and joint estimation approach (g = {g(Z t ) T , X T t } T ). The advantage of joint estimation may be particularly notable in case of a short pre-treatment period relative to post-treatment period, as it could lead to efficiency improvements because both pre-and post-treatment data contribute to estimation of α i,i∈D . Joint estimation can also be achieved in R using gmm::gmm(g=Y∼X+W,x=∼X+Z), where Z denotes the data object for Z t .

Adjustment for measured covariates
In practice, one may wish to incorporate available covariate data measured across units and over time, either to account for endogeneity or to improve efficiency. Thus we generalize Assumption 2 as follows Assumption 2 . For any unit i at time t, where C it ∈ R p is a p × 1 vector of measured covariates, ξ ∈ R p is a p × 1 vector of coefficients shared by all units, and the error terms satisfy E[ε it | X t , λ t , C it ] = E[ε it ] = 0 for all i = 0, . . . , N and t.
In the special case where β t = β, Assumption 2 is the same model considered by Xu (2017). To simplify notation, let C t = (C 0t , C it,i∈D , C jt,j∈[N ]\D ) denote all measured covariates of the treated unit and the control units in the donor pool. We modify Assumptions 4-5 to further include C t as potential common causes of treatment assignment and outcome.
Assumption 4 . We have observed Z t such that Z t ⊥ ⊥ (Y t , W it,i∈D ) | C t , λ t , X t = 0 for any t ≤ T 0 .
Assumption 5 . For any square integrable function q, if E[q(W it,i∈D , C t ) | X t = 0, Z t = z t , C t = c t ] = 0 for all z t and c t then q{W it,i∈D , C t } = 0.
To simplify notation, We have the following identification results.
Corollary 1 (Identification). Under Assumptions 1, 2 , and 3 for any x t and any t.
Under Assumptions 1, 2 , 3, 4 , and 5 , α i,i∈D and ξ are identified by solving E[ Y t − i∈D α i W it | X t = 0, Z t , C t ] = 0 for any t ≤ T 0 , and given α i,i∈D and ξ, β t is uniquely identified by Example 1 . Continuing with Example 1, where the outcomes of control unit(s) not included in the donor pool W jt,j∈[N ]\D are selected as supplemental proxy variable Z t . We have the following estimating function where g(·) is a (|D| + p)-dimensional vector of user-specified functions. One can show that Furthermore, similar to Section 2.3, one may first infer α i,i∈D and ξ using GMM, then one may implement a standard time series analysis of the corresponding estimate of e t = Y t − i∈D α i W it = β t + r t to obtain inference about the treatment effect as a function of t.
Under the CATT model, i.e., β t = β, an alternative approach is to jointly estimate θ using GMM; details are omitted but easily inferred from the exposition.

Nonparametric identification and estimation
The proximal causal inference framework is in fact far more general than the interactive fixed effects model thus far highlighted. It allows for nonparametric identification of causal effects in the sense that one can uniquely express the causal effect as a function of the observed data distribution without necessarily assuming a linear model (Miao, Shi & Tchetgen Tchetgen 2018). In this section, we establish conditions for nonparametric identification. Our framework allows for nonlinear models that can be applied to e.g., binary or count outcomes, which are rarely studied in the synthetic control literature. For notational simplicity, similar to Sections 2.2-2.3, we again suppress measured covariates, although the results below allow for adjustment of measured covariates.
We first extend previous Assumptions 2-3. In order to establish nonparametric identification we require the no interference assumption previously discussed in Eq (5), as well as the latent ignorability assumption for the potential outcomes under control Assumption 2 (Latent ignorability and no interference). For any i and t Assumption 2 implies that W it ⊥ ⊥ X t | λ t . We further extend Assumption 3 to the following assumption that the mechanism by which latent factors affect the outcome can be adequately captured by an unknown function of a sufficiently rich set of proxies W it,i∈D .
Assumption 3 . There exist a function h(W it,i∈D ) such that the outcome model for the treated unit is equivalent to a model for h(W it,i∈D ) in the pre-treatment period: Assumption 3 states that the effect of λ t on Y t can be uncovered by the effect of λ t on Example 2 (Classical setting). Assumption 3 is a generalization of Assumptions 2-3 to the nonparametric setting. Specifically, Assumptions 2-3 imply Eq. (9), i.e., which is a special case of Eq. (21) with h(W it,i∈D ) = i∈D α i W it .
Example 3 (Count outcome). With count outcomes, one often fits a multiplicative model.
In particular, if f (·) is linear in W it , then this condition involves the moment generating function and may hold for Normal and Poisson distributions.
Because Y t (1) is observed in the post-treatment period, one only needs to identify E[Y t (0) | X t = 1] to obtain the average treatment effect on the treated unit. As we prove in Section D of the Supplementary Material, the mean potential outcome under control in the post-treatment period is identified by Therefore, identification of the causal effect comes down to identification of the confounding bridge function. One cannot directly identify h(·) from Eq. (21) because λ t is unobserved.
Nevertheless, when supplemental proxy variables Z t satisfying Assumption 4 are available, h(·) is nonparametrically identified. We have the following theorem which is proved in Section D of the Supplementary Material.
Theorem 4 (Nonparametric identification). Under Assumptions 1, 2 , 3 , we have E[Y t (0) | X t = 1] = E[h(W it,i∈D ) | X t = 1] for any x t and any t. Under Assumptions 1, 2 , 3 , 4-5, h(W it,i∈D ) is identified by solving E[Y t − h(W it,i∈D ) | X t = 0, Z t ] = 0 for any t ≤ T 0 , and the causal effect on the treated unit at time t is identified by Theorem 1 (and Corollary 1) is a special case of Theorem 4 under more stringent assumptions, i.e., Assumptions 2-3. Estimation and inference can be obtained as described

Simulation
We investigate the finite sample performance of our proposed method under various conditions. We simulate time series data on N control units and one treated unit over T 0 = 200, 500, or 1000 time period pre-treatment and the same time length post-treatment, i.e., T = 2T 0 . We generate 10000 samples under the following data generating mechanism where X t = 1 if t > T 0 , and X t = 0 otherwise; C it i.i.d ∼ N (0, 1); ε it i.i.d ∼ N (0, 1); β = 2; and ξ = 1 or 0 corresponding to scenarios with and without measured covariates. We simulate a vector of latent factors λ t = (λ t1 , . . . , λ tr ) T , where λ tk i.i.d ∼ N (log(t), 1), k = 1, . . . , r, and r = 1, 5 or 10. That is, we generate three settings with one, five, or ten latent factors. For each setting, we assume the number of control units N = 2r, and the first half of the control units (i = 1, . . . , r) constitute the donor pool with |D| = r. We specify factor loadings µ i , i = 0, . . . , N as follows. For the treated unit, the factor loading is µ 0 = (1, . . . , 1) T . For the control units, the first half of the control units (i = 1, . . . , r) have factor loadings µ i and corresponding synthetic control weights α i,i∈D satisfying Assumption 3 as follows for r = 1, 5, 10.
The second half of the control units (i = r + 1, . . . , N ) have identical factor loadings as the first half. That is, µ r+k = µ k , for k = 1, . . . , r.

From the above model we know that
To estimate β, we implement our proposed method taking the first half of control units as donors W it,i∈D and second half of control units as supplemental proxies W jt,j∈[N ]\D . When there exists a measured covariate (i.e., ξ = 1) which is predictive of the outcome, we implement our method with and without covariate adjustment to investigate whether there is an efficiency gain from adjusting for such a predictor of the outcome. For inference, we compute an estimate of HC variance-covariance matrix as detailed in Section 2.3.
To compare with the classical synthetic control method, we implement both the constrained OLS proposed in Abadie et al. (2010) and the unconstrained OLS, taking all N control units as donors. For the unconstrained OLS method, we fit the following linear re- . . , T , where ν it is some random error. When ξ = 0, i.e., when there is no measured covariates, this is equivalent to the classical synthetic control method without the constraint that α i ≥ 0, i ∈ D and N i=1 α i = 1. When ξ = 1, i.e., with measured covariates, this model is an unconstrained version of the estimator proposed by Li (2020) for covariate adjustment in synthetic control methods. For the constrained OLS method, we use the Synth package (Abadie et al. 2011) with pre-treatment covariates and outcomes as training data, leaving all hyperparameters as the default value.
We extract the estimated synthetic control weights to predict Y t (0) in the post-treatment period and then estimate β. The performance of the constrained OLS in terms of bias is generally worse than the bias of the unconstrained OLS, therefore we only present results for the latter.

Figure 2 presents bias and 95% Monte Carlo confidence interval (CI) of the β estimates
based on the unconstrained regression method (OLS) and our proposed proximal inference method (PI). As expected from theory, the negligible bias of PI approach decreased with increasing T 0 and T . In contrast, the unconstrained OLS regression estimates were substantially biased. This is also consistent with our theoretical expectation, as pointed out in Section 2.1, the regression based estimator solves an estimating equation that is biased, and thus does not consistently estimate β. We also note that such a bias does not decrease as the number of control units increases. This is because, as discussed in Ferman (2021), OLSbased weight estimate α i,i∈D converges to weights that attempt to minimize the variance of i∈D α i ε it , which does not vanish in our setting as α i,i∈D does not decrease with increasing N . In addition, our proposed method successfully adjusts for measured covariates with similar bias but higher efficiency as indicated by narrower Monte Carlo CI compared to the PI method without covariate adjustment. Table 1 presents Monte Carlo coverage probabilities for PI and unconstrained OLS methods when ξ = 0 or 1, with covariate adjustment in the ξ = 1 scenario. The OLS method had poor coverage across all scenarios, nearly equal to zero for a large T 0 . In contrast, the PI method had coverage probabilities close to the nominal level of 95% across all settings considered. We conduct additional simulation studies under a weakly dependent residual errors setting when ε it is an AR(1) process with coefficient 0.1, i.e., ε it = 0.1ε i,t−1 + ν it with  Among the 16 control countries, five (Austria, Japan, Netherlands, Switzerland, USA) were eventually given non-zero weights in Abadie et al. (2015) to construct a synthetic West Germany, while the other 11 states were excluded with zero weights. We hence take outcome trajectories of the five countries as W it,i∈D and leverage the rest of countries as supplemental proxies, i.e., Z t = W jt,j∈[N ]\D . We further adjust for three predictors of GDP, denoted as in Assumption 2 , we estimate α i,i∈D and ξ using the pre-treatment data in 1960-1990 via GMM with a pre-specified identity weighting matrix. We then use the estimated coefficients to compute the synthetic West Germany that aims to replicate the GDP that would have evolved in the absence of the reunification.   (3) the unconstrained OLS method, and (4) the SC method in Abadie et al. (2015) which further restricts the weights to be non-negative and sum to one. In the pre-treatment period, the difference in per-capita GDP comparing West Germany and its synthetic counterpart is close to zero from all methods except for the simple average. After 1990 the trends of per-capita GDP diverged. We inferred that, in the post-reunification period, per capita GDP was reduced, with estimated causal effect -1200 USD (95%CI: -2021, -380 (HC); -2369, -32 (HAC)), suggesting a negative impact of reunification on West Germany's GDP. The corresponding estimate in Abadie et al. (2015) was somewhat larger, approximately -1600 USD, although formal statistical inference was not available.
To validate the results, we further conducted a falsification study by restricting estimation to the pre-treatment period, during which the causal effect is expected to be null. We artificially define a placebo-reunification time of 1975, which falls in the middle of the pretreatment period. Figure 3 (b) presents the per-capita GDP trajectories in West Germany and synthetic West Germany computed using the placebo period. The estimated placebo effect was 125 USD (95%CI: -289, 539 (HC); -282, 532 (HAC)) using the PI method and -243 USD using regression-based method in Abadie et al. (2015). That is, the estimated bias of the SC method was nearly double that of the PI method. These results provide some evidence of the potential validity of the PI approach and suggests robustness improvements of the PI method compared to traditional SC methods in this analysis.

Discussion
In this paper, we have proposed a new framework for evaluating the impact of an intervention when time series data on a single treated unit and multiple untreated units are observed, in pre-and post-treatment periods. Our proposal is motivated by recent work on proximal causal inference (Miao, Shi & Tchetgen Tchetgen 2018, which is closely related to negative control methods recently developed for analysis of observational data subject to potential unmeasured confounding bias (Lipsitch et al. 2010, Shi, Miao, Nelson & Tchetgen Tchetgen 2020.
Our proposed framework not only provides a remedy of the traditional synthetic control methods which can be inconsistent under certain conditions, but also formalizes estimation and inference of the average treatment effect on the treated unit via the generalized method of moments. In addition, our observation that the post-treatment outcome differences between the treated unit and the synthetic control is a time series with treatment effect captured by a deterministic time trend opens the door to a rich literature on time series analysis for estimation of the treatment effect. We further extend the traditional linear interactive effect model to more general cases such as nonlinear models allowing for binary and count outcomes rarely studied in the synthetic control literature. Our methods can conveniently be extended to accommodate multiple treated units (Abadie & L'Hour 2021, Ben-Michael et al. 2021a.
A simplest approach might be to do separate PI analyses for each treated unit allowing for different confounding bridge functions and possibly different donor pools. Then unit specific treatment effects could be aggregated in a meta analytic fashion either by assuming, when appropriate, a shared common effect by inverse variance weighted estimation or by allowing for heterogeneous unit-specific effects pooled via random effect approach.

Supplementary Material for "Theory for identification and Inference with Synthetic Controls: A Proximal Causal Inference Framework"
A Proof of Theorem 1 , for any x t and any t. Under Assumptions 1-3 we have for any x t and any t. By Assumption 2, E[ε it ] = 0 for any i, t, thus we have for any x t and any t.
, that is, under Assumptions 1-2, any α i,i∈D that satisfy Assumption 3 identifies E[Y t (0)] as E[ i∈D α i W it ]; , for any t. Therefore the causal effect on the treated unit at time t is Under Assumptions 2-3 we have Setting x t = 0 we have that in the pre-treatment period that is, Eq. (9) holds. Setting x t = 1 we have that in the pre-treatment period Marginalizing over f (λ t | X t = 1) we have that is, Eq. (13) holds.
We now show that Assumptions 1-4 imply E[Y t − i∈D α i W it | X t = 0, Z t ] = 0 for any t ≤ T 0 as follows.
Therefore under Assumptions 1-2 and 4, any α i,i∈D that satisfies Assumption 3, i.e., µ 0 = Lastly, we show that under Assumptions 1-5, there is a unique solution to (31) which satisfies Assumption 3 and hence identifies E[Y t (0)] as E[ i∈D α i W it ]. Suppose that both α i,i∈D and α i,i∈D satisfy Eq. (31) but α i,i∈D = α i,i∈D . Then by (31) we have that at any t, = 0 for all z t . By the completeness assumption, i.e., Assumption 5, it must be that α i − α i = 0 for all i ∈ D, which leads to a contradiction.
Therefore, there is a unique solution α i,i∈D to Eq. (31). Finally, with a uniquely identified α i,i∈D , we have that β t defined by Eq. (30), i.e., is unique for any t > T 0 .

B Proof of Theorems 2 and 3
We first show that E[U t (α i,i∈D ) | X t = 0] = 0 and E[ U t (θ)] = 0 at the true value under Assumptions 2-4. We have and Now we focus on proving Theorem 3. Similar arguments apply to Theorem 2. We follow Chapter 3 of Hall (2005) to prove the asymptotic normality of our estimator. To summarize, we have a population moment condition E[ U t (θ)] = 0 with a moment function denote the parameter space of θ, and let O ⊆ R 2+dim(Zt)+|D| denote the sample space of O t .  Assumption B.7 is insufficient to allow the application of Laws of Large Numbers and Central Limit Theorem. Therefore we impose the following assumption.
Assumption B.14 (Properties of the variance of the sample moment). Let θ * = (γ * , α i,i∈D * )  C Proof of existence of the confounding bridge function To simplify notation, for each t, let W = W it,i∈D , X = X t , Y = Y t , and λ = λ t . Then Eq.
which is a Fredholm integral equation of the first kind. Conditions for existence of a solution has been considered in Miao, Geng & Tchetgen Tchetgen (2018) and Cui et al. (2020).
Let L 2 {F (s)} denote the space of all square-integrable functions of s with respective to a cumulative distribution function F (s), which is a Hilbert space with inner product < and let (τ n , φ n , ψ n ) ∞ n=1 denote a singular value decomposition of K. We assume the following regularity conditions: Then by Picard's theorem (Kress 1989), there exist a solution to Eq. (34), which then satisfies Eq. (21).

D Proof of Theorem 4
We first show that given the bridge function, the mean potential outcome is identified as where the last line holds because W it ⊥ ⊥ X t | λ t by Assumption 2 . Taking expectation over λ t | X t = 1 on both sides of (35) we have for any t > T 0 . It is important to note that to identify the average treatment effect on the treated, one only needs to identify h( in the post-treatment period. In fact, given h(W it,i∈D ), the average treatment effect on the treated unit in the post-treatment period is identified by We now study identification of h. We first show that any h that satisfies Assumption 3 is Therefore We then show uniqueness of the solution to Eq. (38). Suppose that both h(W it,i∈D ) and h {W it,i∈D } satisfy (38) but h = h . Then by (38), we have that E[h{W it,i∈D } − h {W it,i∈D } | X t = 0, Z t = z t ] = 0 for all z t . By Assumption 5, it must be that h{W it,i∈D }−h {W it,i∈D } = 0, which leads to a contradiction. Therefore, h is a unique solution to Eq. (38).

E Additional simulation studies E.1 Weakly dependent errors
Below we present the simulation results bias, variance, and coverage probability in Figure 4 and    Table 2: Coverage probability based on the unconstrained regression method (OLS) and our proposed proximal inference method (PI), with a range of number of control units N = 2, 10, or 20 and pre-and post-treatment time period T 0 = T − T 0 = 200, 500, or 1000.

E.2 Time-varying treatment effect
We investigate the finite sample performance of our proposed method under various conditions. We simulate time series data on N control units and one treated unit over T 0 = 500, 1000, or 5000 time period pre-treatment and the same time length post-treatment, i.e., T = 2T 0 . We generate samples under the following data generating mechanism where ε it i.i.d ∼ N (0, 2) and β(t/T ; γ) = γ 0 + γ 1 t/T with γ = (1, 1) T . The rest of the data generating setting is the same as Section 4. We focus on estimation and inference of γ using the same approach as detailed in Section 4. Below we present the simulation results in terms of bias, variance, and coverage probability in Figure 5 and Table 3. For estimation of γ 0 , the relative performance comparing OLS and PI methods are similar to Section 4; For estimation of γ 1 , OLS approach still has relatively large bias compared to PI method, although coverage probabilities are closer than the results in Section 4. We also conducted a simulation study when ε it is AR (1)  With measured covariates (ξ = 1) OLS, T0 = 500 OLS, T0 = 1000 OLS, T0 = 5000 PI (w/o adj), T0 = 500 PI (w/ adj), T0 = 500 PI (w/o adj), T0 = 1000 PI (w/ adj), T0 = 1000 PI (w/o adj), T0 = 5000 PI (w/ adj), T0 = 5000 Figure 5: Bias and 95%Monte Carlo confidence interval of the γ 0 (first row) and γ 1 (second row) estimates based on the unconstrained regression method (OLS) and our proposed proximal inference method (PI), with a range of number of control units N = 2, 10, or 20 and pre-and post-treatment time period T 0 = T − T 0 = 500, 1000, or 5000.  Table 3: Coverage probability based on the unconstrained regression method (OLS) and our proposed proximal inference method (PI), with a range of number of control units N = 2, 10, or 20 and pre-and post-treatment time period T 0 = T − T 0 = 500, 1000, or 5000.

E.3 Nonlinear model
We investigate the finite sample performance of our proposed method under various conditions. We simulate time series data on N control units and one treated unit over T 0 = 500, 1000, or 5000 time period pre-treatment and the same time length post-treatment, i.e., T = 2T 0 . We generate Y t and W it i = 1, . . . , N from Poisson distribution with E[Y t | X t , λ t , C 0t ] = exp{X t β+µ T 0 λ t + C T 0t ξ}, where X t = 1 if t > T 0 , and X t = 0 otherwise; C it i.i.d ∼ Uniform(0, 1); β = −2; and ξ = 0.1 or 0 corresponding to scenarios with and without measured covariates. We set β to −2 to avoid extremely large variance of the simulated outcomes due to the mean-variance relationship of Poisson distribution. We simulate a vector of latent factors λ t = (λ t1 , . . . , λ tr ) T , where λ tk i.i.d ∼ Uniform(0, 1), k = 1, . . . , r, and r = 1, 2 or 3. That is, we generate three settings with one, two, or three latent factors. We reduce the number of latent factors compared to previous simulation studies to achieve a more stable result for illustration purpose. For each setting, we assume the number of control units N = 2r, and the first half of the control units (i = 1, . . . , r) constitute the donor pool with |D| = r. We specify factor loadings µ i , i = 0, . . . , N as follows µ 0 µ 1 · · · µ r µ r+1 · · · µ 2r         When there exists a measured covariate (i.e., ξ = 0.1) which is predictive of the outcome, we implement our method with and without covariate adjustment to investigate whether there is an efficiency gain from adjusting for such a predictor of the outcome. From Eq. (40), we derived that the confounding bridge function satisfying Assumption 3 is given by For inference, we compute an estimate of HC variance-covariance matrix as detailed in Section 2.3. Below we present the simulation results in terms of bias, variance, and coverage probability in Figure 6 and Table 4.   Table 4: Coverage probability based on the unconstrained regression method (OLS) and our proposed proximal inference method (PI), with a range of number of control units N = 2, 4, or 6 and pre-and post-treatment time period T 0 = T − T 0 = 500, 1000, or 5000.