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: The paper considers the problem of average treatment effect estimation in panel data. The authors consider a setup where the observed outcomes form a matrix with a special structure: it is the sum of a counterfactual matrix and a treatment matrix, where the counterfactual matrix is assumed to be a “noisy observation” of a low rank matrix. The paper proposes a debiased estimator, shows that it is rate optimal, derives its asymptotic normality and demonstrate its performance through numerical experiments.