Learning Treatment Effects in Panels with General Intervention Patterns

∑ ij TijZij/ ∑ ij Zij . The synthetic control paradigm provides an approach to estimating τ∗ when Z places support on a single row . This paper extends that framework to allow rate-optimal recovery of τ∗ for general Z , thus broadly expanding its applicability . Our guarantees are the first of their type in this general setting . Computational experiments on synthetic and real-world data show a substantial advantage over competing estimators . 1 Introduction . Consider that we are given a sequence of T observations on each of n distinct ‘ units ’ . In an econometric context , a unit might correspond to a geographic region with the associated sequence of observations corresponding to some economic time series of interest . In e-commerce , a unit may correspond to a customer , with the associated sequence of observations corresponding to site-visits for that customer over time . For each unit , some subset of its observations are potentially impacted by an intervention . This may correspond to a new economic policy in the econometric example , or the application of a new type of promotion in the e-commerce context . The question at hand is to estimate the ‘ average treatment effect ’ of this intervention . This is a core question in econometrics . The problem above can be formalized as follows : Let M∗ ∈ Rn×T be a fixed , unknown matrix and E be a zero-mean random matrix ; we refer to M∗ + E as the ‘ counterfactual ’ matrix with each row corresponding to a distinct ‘ unit ’ . A known ‘ treatment ’ matrix Z with entries in { 0 , 1 } encodes observations impacted by an intervention . Specifically , we observe a matrix O with entries Oij = M ∗ ij + Eij + TijZij where the Tij are unknown , heterogenous treatment effects.1 Our goal is to recover the average treatment effect τ∗ = ∑ ij TijZij/ ∑ ij Zij . 1Observable covariates on each unit may also be available ; we suppress this aspect here for clarity . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . The ‘ synthetic control ’ paradigm [ 1 , 2 ] has been a transformative approach to addressing this problem . That setup addresses the special case of this problem where a single unit ( say , the first row ) is treated ; i.e . Z has support on a portion of the first row . The approach is conceptually simple : find a linear combination of the untreated units ( i.e . all rows other than the first ) that approximate the first row on its untreated entries . One then uses the same linear combination to impute counterfactual values i.e . the entires of M∗ in the support of Z . Under the assumption that M∗ is low rank ( and other assumptions ) , one can show that this approach recovers τ∗ . Applications like the e-commerce setup alluded to earlier require we allow for more general intervention patterns , i.e . allow for general Z as opposed to the single row ( or block ) support required in synthetic control . A tantalizing possibility , first raised by [ 3 ] , is treating entries of M∗ in the support of Z as missing and applying matrix completion techniques to impute these counterfactual values . This viewpoint has the benefit of making no assumptions on the treatment effects and yields algorithms that do not require any special structure for Z . However , it is unclear that counterfactual recovery – which is equivalent to matrix recovery with general missingness patterns – is actually possible via such a method , so that recovery guarantees are typically unavailable . 1.1 This Paper . Succinctly , we develop an estimator that recovers the average treatment effect under provably minimal assumptions on Z . This is made possible by a new de-biasing identity and an extension of the entrywise uncertainty quantification analysis of [ 4 , 5 ] to general non-random missingness patterns that is of independent interest . In providing context for our contributions , it is worth asking what one can hope for in this problem . In addition to requiring that M∗ have low rank ( say r ) , it is clear that we can not in general expect to recover τ∗ absent assumptions on the treatment matrix Z . For instance , we must rule out the existence of a rank r matrix M ′ , distinct fromM∗ for whichM ′ = M∗+γZ for some γ 6= 0 , or else identifying τ∗ is impossible even if E is identically zero ( we could rule this out if Z were not in the tangent space of M∗ ) . Separately , unlike matrix completion , we actually observe M∗ij + Tij +Eij on treated entires . If the heterogeneity in treatment effects is too large , however , it is unclear that these observations are of much value . Thus , in addition to benign assumptions on M∗ and E , we make a set of assumptions on ( A ) the projection of Z onto the tangent space of M∗ and ( B ) the heterogeneity in treatment effects . Against this backdrop , we make the following contributions : Rate Optimal Estimator : We construct an estimator that achieves rate optimal guarantees for the recovery of the average treatment effect τ∗ ( Theorem 1 ) with general treatment patterns . We show under additional assumptions that our estimator is asymptotically normal ( Theorem 2 ) . Minimality of Assumptions : Should the conditions we place on the projection of Z onto the tangent space of M∗ be violated by an amount that grows small with problem size , we show no algorithm can recover τ∗ even with homogeneous treatments ( Proposition 2 ) . Our assumptions on heterogeneity are also shown to be minimal , and satisfied by extant models in the synthetic control literature . General Treatment Patterns Work : We show that our assumptions on Z are satisfied by general treatment patterns , e.g. , requiring that at least a constant fraction of entries are not treated or matrices Z with rank polynomially larger than the rank of M∗ . Empirical Performance : We show both for synthetic and real data that our estimator provides a material improvement in empirical performance relative to available alternatives , including matrix completion based estimators and , where applicable , state-of-the-art synthetic control estimators . 1.2 Related Literature . The synthetic control literature pioneered by [ 1 , 2 ] has grown to encompass sophisticated learning and inferential methods ; see [ 6 ] for a review . [ 7 , 8 , 9 ] consider a variety of regularized regression techniques to learn the linear combination of untreated units that yields a synthetic control . [ 10 , 11 , 12 ] consider instead the use of principal component regression techniques . [ 13 ] propose alternative approaches to imputing counterfactuals by averaging across both untreated units ( rows ) and time ( untreated columns ) . [ 14 , 15 ] address inferential questions that arise in synthetic control with the latter providing a permutation test that is generally applicable . Matrix completion methods present a means to allow for inference with general treatment patterns . [ 3 ] are among the first to study this , but provide no guarantees on recovering the average treatment effect . Alternatively methods that do provide guarantees on the recovery of treatment effects via matrix completion tend to make strong assumptions : [ 16 ] effectively assume that Z has support on a block ( so that traditional synthetic control techniques could also apply ) , [ 17 ] make stationarity assumptions on M∗ and require it to be zero-mean and [ 18 ] assume that Z has i.i.d . entries ( wherein a trivial estimator of the average treatment effect is also applicable ) . A distinct ( and common ) setting for treatment effect estimation presents us with a single observation per unit and a rich set of observed covariates on these units . These observed covariates roughly ensure exchangeability of units across interventions within covariate strata . ML has been broadly used in this setting ; eg . [ 19 , 20 , 21 ] . In contrast for our problem , one could view these covariates as latent . Our estimation procedure begins with first computing a ‘ rough ’ estimate of the treatment effect via a natural convex estimator ; [ 22 , 23 , 24 ] are empirical studies that have uses this estimator . Crucially , we provide a new de-biasing technique that allows for recovery guarantees and exhibits a significant performance improvement relative to this rough estimate . It is also worth noting that this convex estimator also finds application in the related problem of panel data regression ; see [ 25 , 26 , 27 ] . State of the art methods there effectively require that Z is dense . Whereas work on matrix completion with non-standard observation patterns [ 28 , 29 , 30 , 31 ] exists this is by and large not obviously useful or applicable to our problem . Instead , we build on a recent program to bridge convex and non-convex formulations for matrix completion [ 4 ] and RobustPCA [ 5 ] . That work has provided a pioneering analysis of entry-wise guarantees and uncertainty quantification for convex matrix completion estimators wherein entries remain missing at random . Our work here may be viewed as extending that program to a broad class of non-random missingness patterns , a contribution of important independent interest . 2 Model and Algorithm . We begin by formally defining our problem ; we in fact present a generalization to the problem described in the previous section , allowing for multiple treatments . Let M∗ ∈ Rn×n be a fixed rank-r matrix with singular value decomposition ( SVD ) denoted by M∗ = U∗Σ∗V ∗ > , where U∗ , V ∗ ∈ Rn×r have orthonormal columns , and Σ∗ ∈ Rr×r is diagonal with diagonal entries σ1 ≥ · · · ≥ σr > 0.2 Let σmax : = σ1 , σmin : = σr and κ : = σmax/σmin be the condition number of M∗ . There are k treatments that can be applied to each entry , and for each treatment m ∈ { 1 , . . . , k } , we are given a treatment matrix Zm ∈ { 0 , 1 } n×n which encodes the entries which have received the mth treatment ( 0 meaning no treatment , and 1 meaning being treated ) .3 We then observe a single matrix of outcomes : O : = M∗ + E + k∑ m=1 Tm ◦ Zm ( ◦ is the Hadamard or ‘ entrywise ’ product ) , where each Tm ∈ Rn×n is an unknown matrix of treatment effects , and E ∈ Rn×n is a ( possibly heterogeneous ) random noise matrix . Finally , let τ∗ ∈ Rk be the vector of average treatment effects , whosemth value is defined as τ∗m : = 〈Tm , Zm〉/‖Zm‖1 and let δm = Tm ◦ Zm − τ∗mZm be associated ‘ residual ’ matrices . Our problem is to estimate τ∗ after having observed O and Z1 , . . . , Zk . It is worth noting that the representation above is powerful : for instance , it subsumes the setting where the intervention on any entry is associated with a { 0 , 1 } k-valued covariate vector , and the treatment effect observed on that entry is some linear function of this covariate vector plus idiosyncratic noise . Recovery of τ∗ is then equivalent to recovering covariate dependent heterogeneous treatment effects . Our problem also subsumes the synthetic control setting where k = 1 and Z1 must place support on a single row ; the focus of our later analysis will be the case where Z1 is allowed to be general . 2Note that in contrast to the previous section , we are now assuming square matrices ( i.e . n = T in the notation of the previous section ) . This is purely to simplify the notation – for a rectangular n-by-T matrix , all of our theoretical guarantees hold if one swaps n with min { n , T } . 3We allow for multiple treatments to be applied to an entry . The assumptions that we will need to impose in order to state meaningful results can be divided into two groups . The first are assumptions on M∗ and E that are , by this point , canonical in the matrix completion literature : Assumption 1 ( Random Noise ) . The entries of E are independent , mean-zero , sub-Gaussian random variables with sub-Gaussian norm bounded by σ : that is , ‖Eij‖ψ2 ≤ σ for every i , j ∈ [ n ] . Assumption 2 ( Incoherence ) . M∗ is µ-incoherent : ‖U∗‖2 , ∞ ≤ √ µr/n and ‖V ∗‖2 , ∞ ≤ √ µr/n , where ‖ · ‖2 , ∞ denotes the maximum ` 2-norm of the rows of a matrix . In addition to these standard conditions onM∗ andE which we will assume throughout this paper , we will also need to impose conditions on the relationship between M∗ and the Zm ’ s . Loosely speaking , these conditions preclude treatment matrices Zm that can be ‘ disguised ’ within M∗ , in the sense that their projections onto the tangent spaces of M∗ are large . Specifically , the formal statements relate to a particular decomposition of the linear space of n× n matrices , Rn×n = T ∗ ⊕ T ∗⊥ , where T ∗ is the tangent space of M∗ in the manifold consisting of matrices with rank no larger than rank ( M∗ ) : T ∗ = { U∗A > +BV ∗ > | A , B ∈ Rn×rank ( M ∗ ) } . Equivalently , the orthogonal space of T ∗ , denoted T ∗⊥ , is the subspace of Rn×n whose columns and rows are orthogonal , respectively , to the spaces U∗ and V ∗ . Let PT∗⊥ ( · ) denote the projection operator onto T ∗⊥ : PT∗⊥ ( A ) = ( I − U∗U∗ > ) A ( I − V ∗V ∗ > ) . We will defer the formal statements of the additional conditions to the next section , but suffice to say for now that they assume PT∗⊥ ( Zm ) to be sufficiently large ( we will show that lower bounds on the size of PT∗⊥ ( Zm ) , in the the precise form of our own conditions , are nearly necessary ) . In the remainder of this section , we will outline the core contribution of this paper , which is an estimator for τ∗ with a provably rate-optimal guarantee .

Summary: This paper considers a general problem of applying multiple treatments (heterogenous) to multiple units across multiple time-steps, where the outcome is of this treatment has a linear relationship. Given the treatment decision matrix and the outcome matrix, the goal of the algorithm is to obtain the treatment effect (or more specifically a sufficient statistic that can be used to derive the treatment effect given the model). In particular, the outcome matrix O for n units across T steps is determined by a linear system M* + E + sum_{k} treatment_effect_matrix_k * decision_matrix_k. The algorithm sees the outcome matrix O and knows the decision_matrix_k for each of the k treatments and it needs to recover the treatment_effect_matrix_k for each of the k treatments. The paper identifies assumptions under which such an estimator is possible, shows that when violating one of the assumptions no algorithm can obtain such an estimator and that they have good empirical performance on semi-synthetic datasets.