A Causal Inference Framework for Data Rich Environments

A Causal Inference Framework for Data Rich Environments

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\@startsection{section}{1}{0mm}{-\baselineskip}{0.25\baselineskip}{\center\normalfont\bf}{Introduction} One of the central goals of empirical economic research is to ascertain the effects of treatments (policies, treatments) on the outcomes of interest. A fundamental challenge for the estimation of treatment effects is the pervasive presence of unobserved confounders. For example, in a study of the effects of health insurance on healthcare utilization, unobserved or latent health determinants may differ between insured and uninsured individuals, biasing treatment effect estimates. Several approaches have been put forward to estimate treatment effects in the presence of confounders, including explicit randomization of the treatment, controlling for measured confounders, and instrumental variable methods. Traditionally, these methods are not designed to operate in data-rich environments where the curse of dimensionality creates challenges for estimation and inference, and do not take advantage of the information contained in high-dimensional data to identify treatment effects.

In recent times, the availability of high-dimensional data on economic behavior has become commonplace. Modern data harvesting technologies, based on digitization and pervasive sensors, enable the collection of detailed high-frequency attribute and outcome information on individuals (or other observational units; e.g., geo-locations) concurrently undergoing different treatments. For example, electronic health records contain rich information about patients' medical history over time. Similarly, internet retailers and marketing firms use scanner data to collect high-dimensional information on customers' purchases. The goal of this article is to provide a framework for causal inference that takes advantage of modern data-rich environments to counter the effect of unobserved or latent confounders.

Given this goal, we consider a setting where we have access to data for $N$ units (e.g., individuals, sub-populations, firms, geographic locations) and $T$ measurements of outcomes per unit. Different measurements may represent the same outcome metric at different time periods, different outcome metrics (e.g., customers' expenditures in different product categories) for the same time period, or a combination of both. We argue that (high-dimensional) data-rich environments---i.e., large $N$ and large $T$---make it possible to estimate treatment effects in the presence of unobserved or latent confounding, without needing to make parametric assumptions in the manner in which unobserved confounders affect selection for treatment and the outcome metrics.

\@startsection{subsection}{2}{0mm}{-\baselineskip}{0.05\baselineskip}{\normalfont\bf}{Contributions and Related Work} {\bf Contributions.} We propose a formal model for counterfactual estimation with unobserved confounding in “data-rich” settings, i.e., where there are a large number of units and a large number of measurements per unit. We posit a general data-generating process (DGP) for how potential outcomes are defined and how treatments are assigned, allowing for unobserved confounding. We provide a structural causal model view of the conditional independence conditions required in our DGP that imply that the treatment assignments are exogenous of the potential outcomes conditional on these unobserved confounders. We establish that if the unobserved confounders are low-dimensional, relative to the number of units and measurements, and the potential outcomes are a smooth non-linear function of them, it {\em implies} that an approximate linear latent factor model of appropriate dimension holds, where the approximation error decays as the number of units and measurements increase. In doing so, we believe this model provides a formal bridge between the structural causal model view of causal inference common in the graphical models literature with that of the latent factor model view common in the potential outcomes literature.

We formalize how classic models for potential outcomes and treatment assignments fit within our framework. For the potential outcomes, we show how two-way fixed effects, interactive fixed effects, binary choice, and dictionary basis expansions fit within our framework. For treatment assignments, we show how randomized control trials (RCTs), selection on (un)observables, regression discontinuity, random utility models, and staggered adoption settings fit within our framework. Theoretically, we provide an identification argument for the average treatment effect (ATE), the average treatment effect on the treated (ATT), and the average treatment effect on the untreated (ATU). For any estimator that has a fast enough estimation error rate for a certain nuisance parameter, we establish it is consistent for these various causal parameters. We then show principal component regression (PCR) is one such estimator that leads to consistent estimation, and we analyze the minimal smoothness required of the potential outcomes function for consistency.

{\bf Related work.} This model builds upon the latent factor model literature studied in the growing literature on causal panel data models (chamberlain_factor, SC, bai2009interactive, athey2021matrix, bai2021matrix, SDID, CMC, SI, dwivedi2022doubly, synth_combo). The model we propose can be viewed as a generalization of the exact linear factor model studied in these works to a non-linear factor model. Our model allows for both panel data and cross-sectional data, and combinations thereof. Importantly, we argue that beginning from a general structural causal model, if the outcomes are a smooth function of the unobserved confounders, then it {\em implies} that a factor model of appropriate dimension holds if there are large number of units and measurements, thereby hopefully providing a bridge between these two frameworks. In terms of the estimator we propose, to the best of our knowledge, this is also the first theoretical analysis of consistency for the ATE with a (smooth) non-linear factor model. It is also the first analysis of PCR (PCR1, PCR2, CICD, PCR3) for such target causal estimands with unobserved confounding. This requires dealing with the novel technical challenge of error-in-variables, only an approximate low-rank model holding on the noiseless covariates, and linear misspecification error.

\@startsection{subsection}{2}{0mm}{-\baselineskip}{0.05\baselineskip}{\normalfont\bf}{Notation} For a matrix $\bA \in \Rb^{a \times b}$, we denote its transpose as $\bA^T \in \Rb^{b \times a}$. We denote the operator (spectral) and Frobenius norms of $\bA$ as $\|\bA\|_{\text{op}}$ and $\|\bA\|_F$, respectively. The columnspace (or range) of $\bA$ is the span of its columns, which we denote as $\Rc(\bA) = \{v \in \Rb^a: v = \bA x, x \in \Rb^b\}$. The rowspace of $\bA$, given by $\Rc(\bA^T)$, is the span of its rows. Recall that the nullspace of $\bA$ is the set of vectors that are mapped to zero under $\bA$. For any vector $v \in \Rb^a$, let $\|v\|_p$ denote its $\ell_p$-norm, and let $\|v\|_\infty$ denote its max-norm. The inner product between vectors $v, x \in \Rb^a$ is $\langle v, x \rangle = \sum_{\ell=1}^a v_\ell x_\ell$. If $v$ is a random variable, we denote its sub-Gaussian (Orlicz) norm as $\|v\|_{\psi_2}$. For any positive integer $a$, we use the notation $[a] = \{1, \dots, a\}$.

Let $f$ and $g$ be two real-valued functions defined on $\mathcal X$, an unbounded subset of $[0,\infty)$. We say that $f(x)$ = $O(g(x))$ if and only if there exists a positive real number $M$ and $x_0\in \mathcal X$ such that, for all $x \ge x_0$, we have $|f (x)| \le M|g(x)|$. Analogously, we say that $f (x) = \Theta(g(x))$ if and only if there exist positive real numbers $m, M$ and $x_0\in \mathcal X$ such that for all $x \ge x_0$, we have $m|g(x)| \le |f(x)| \le M|g(x)|$; $f (x) = o(g(x))$ if for any $m > 0$, there exists $x_0\in \mathcal X$ such that for all $x \ge x_0$, we have $|f(x)| \le m|g(x)|$.

We adopt standard notation and definitions for stochastic convergence. We employ $\xrightarrow{d}$ and $\xrightarrow{p}$ to indicate convergence in distribution and probability, respectively. For any sequence of random vectors, $X_n$, and any sequence of positive real numbers, $a_n$, we say $X_n = O_p(a_n)$ if for every $\varepsilon>0$, there exists constants $C_\varepsilon$ and $n_\varepsilon$ such that $\mathbb{P}( \| X_n \|_2 > C_\varepsilon a_n) < \varepsilon$ for every $n \ge n_\varepsilon$; equivalently, we say $(1/a_n) X_n$ is uniformly tight or bounded in probability. $X_n = o_p(a_n)$ means $X_n/a_n\xrightarrow{p} 0$. We say a sequence of events $\Ec_n$, indexed by $n$, holds “with probability approaching one” (w.p.a.1) if $\mathbb{P}(\Ec_n) \rightarrow 1$ as $n \rightarrow \infty$, i.e., for any $\varepsilon > 0$, there exists a $n_\varepsilon$ such that for all $n > n_\varepsilon$, $\mathbb{P}(\Ec_n) > 1 - \varepsilon$. More generally, a multi-indexed sequence of events $\Ec_{n_1,\dots, n_d}$, with indices $n_1,\dots, n_d$ with $d \geq 1$, is said to hold w.p.a.1 if $\mathbb{P}(\Ec_{n_1,\dots, n_d}) \rightarrow 1$ as $\min\{n_1,\dots, n_d\} \rightarrow \infty$. We also use $\Nc(\mu, \sigma^2)$ to denote a normal or Gaussian distribution with mean $\mu$ and variance $\sigma^2$---we call it {\em standard} normal or Gaussian if $\mu = 0$ and $\sigma^2 = 1$. We use $C$ to denote a positive constant, with a value that can change across instances.

\@startsection{section}{1}{0mm}{-\baselineskip}{0.25\baselineskip}{\center\normalfont\bf}{Model} We are interested in evaluating the effect of treatments on outcomes of interest. Specifically, we observe $T$ outcomes or measurements for $N$ units. For each measurement $t \in [T]$ and unit $n \in [N]$, we observe $Y_{n, t} \in \Rb$ under treatment $A_{n, t} \in \Ac$, where $|\Ac| = A$.

Let $\bA = [A_{n, t}]_{n \in [N], t \in [T]} \in \Ac^{N \times T}$ and $\boldsymbol{Y} = [Y_{n, t}]_{n \in [N], t \in [T]} \in \Rb^{N \times T}$ collect the matrix of treatment assignments and outcomes, respectively. We now define how the treatment assignments and outcomes are generated. We define the random variables $\bU \in \mathcal{U}, \bE \in \mathcal{E}$ and functions $h: \mathcal{U} \to \Ac^{N \times T}$, $f: \Ac^{N \times T} \times \mathcal{U} \times \mathcal{E} \to \Rb^{N \times T}$ such that

We allow the functions $h, f$ and the variables $\bU, \bE$ to be unobserved; that is, we only observe the treatment assignments $\bA$ and the outcomes $\boldsymbol{Y}$. $\bU$ contains all potential confounders that can affect both the treatment assignment $\bA$ and the outcomes $\boldsymbol{Y}$. $\bE$ is the random variation in $\boldsymbol{Y}$ not explained by $\bU$. The question we study in this paper is:

\@startsection{subsection}{2}{0mm}{-\baselineskip}{0.05\baselineskip}{\normalfont\bf}{Data Generating Process} Towards answering the question above, we assume the following data-generating process (DGP). As stated earlier, we hope this DGP serves a bridge between the SCM and latent factor model view of causal inference.

\@startsection{section}{1}{0mm}{-\baselineskip}{0.25\baselineskip}{\center\normalfont\bf}{Outcome and Treatment Assignment Functions Within our Framework} Thus far, the setup has been quite general. To make progress, we impose relatively generic smoothness conditions on $f_{t, a}$, and argue this encompasses familiar models for potential outcomes considered in the econometric literature. Further, we show how various models for the treatment assignment functions studied in the literature can be encompassed within our framework.

\@startsection{subsection}{2}{0mm}{-\baselineskip}{0.05\baselineskip}{\normalfont\bf}{Outcome Functions}

We first formally define what we mean by smoothness.

In essence, Definition (ref) requires that the $(S-1)$-th derivatives of $g$ are Lipchitz continuous. For example, it is easy to verify that an analytic function with compact domain is H\"older continuous for all $S \in \Nb$.

Informally, Assumption (ref) is a continuity condition that posits that if latent variables $U_{n_1}$ and $U_{n_2}$ for any two units $n_1$ and $n_2$ are close ($U_{n_1}\approx U_{n_2}$), then their average potential outcomes are close as well, ($\mathbb{E}[Y^{(a)}_{n_1, t}] \approx \mathbb{E}[Y^{(a)}_{n_2, t}]$, for all $t \in [T], a \in \Ac$), where the expectation is taken with respect to $\varepsilon^{(a)}_{n_1, t}$ and $\varepsilon^{(a)}_{n_2, t}$, respectively.

A linear factor model, $f_{t, a}(U_n) = \left\langle U_n, \tilde{U}_{t, a} \right\rangle$, is a special case of Assumption (ref) and one can verify it satisfies Definition (ref) for all $S \in \mathbb{N}$ . Proposition (ref) establishes that linear factor models of sufficiently large dimension also provide a “universal” representation for smooth non-linear factor models.

Proposition (ref) establishes that if $f_{t, a}$ has a H\"older smooth latent variable representation, then it is uniformly well-approximated by a linear factor model of finite dimension, $r$. For $\delta < 1$ we have that as the latent dimension $q$ of the confounder $U_n$ increases, the bound on the rank $r$ increases, and as smoothness $S$ of $f_{t, a}$ increases, the bounds on the approximation error $\Delta_E$ decreases. If we take $\delta = (\mbox{min}\{N, T\})^{-c}$ for some constant $c$, such that $0<c<1/q$, we obtain $r \ll \mbox{min}\{N, T\}$ and $\Delta_E = o(1)$ as $N, T \to \infty$.

\@startsection{subsubsection}{3}{0mm}{-\baselineskip}{0.05\baselineskip}{\normalfont}{Classic Econometric Models that Fit our Framework}

Our framework nests some classical econometric models, which we describe below.

\@startsection{subsection}{2}{0mm}{-\baselineskip}{0.05\baselineskip}{\normalfont\bf}{Treatment Assignment Functions} Without loss of generality, we will let $\Ac = \{0, 1\}$. Below we show how various models for treatment assignment functions studied in the literature fit within our framework. In particular, our proposed DGP requires unconfoundedness conditional on $U_n$. We formally establish how and when this condition holds for commonly studied treatment assignment functions.

We denote $h_{n, t}(\bU)$ as the $(n, t)$-th output of $h(\bU)$, i.e., the treatment $A_{n, t}$.

\@startsection{section}{1}{0mm}{-\baselineskip}{0.25\baselineskip}{\center\normalfont\bf}{Identification and Estimation of Treatment Effects}

{\bf Causal parameters of interest.} We restrict our attention to the binary treatment setting where for all $t \in [T]$, we let $\Ac = \{0, 1\}$. Our analysis easily extends for any finite $\Ac$. We focus on the estimation of the average treatment effect for a given measurement $t^*$, and for a subset of units $\Mc \subset [N]$, with $|\Mc| = M$:

where expectations are taken over the distribution $\varepsilon^{(a)}_{n, t^*}$. We note our target causal parameter is defined conditional on the unobserved confounders $\bU$.

Let

That is, $\Ic^{(a)}$ is the set of units that received intervention $a$ for measurement $t^*$, and $N_a$ is the number of units in that set. For different subsets $\Mc$, $\mathsf{ATE}_{\Mc}$ nests a variety of causal parameters of interest:

For concreteness, we focus our estimation results on $\mathsf{ATT}, \mathsf{ATU}$, and $\mathsf{ATE}$. However, our results easily extend to any set of units $\Mc \subset [N]$.

\@startsection{subsection}{2}{0mm}{-\baselineskip}{0.05\baselineskip}{\normalfont\bf}{Identification} We now show how the model for treatment assignment and potential outcomes, summarized in Section (ref) leads to a novel identification argument for $\mathsf{ATE}_{\Mc}$. Motivated by Proposition (ref), we define the linear factor model approximation error to $f_{t, a}(U_n)$ as follows.

Recall that if $f_{t, a}$ is H\"older continuous, then Proposition (ref) implies that both $r$ and $\Delta_E$ can be simultaneously controlled.

We define two subsets of $\Mc$: for $a \in \{0, 1\}$,

Note that $\Mc^{(a)} \subset \Ic^{(a)}$. We are now equipped to define the key assumption we require for identification of the causal parameter of interest.

This assumption implicitly adds a restriction on the treatment assignment. For example, Assumption (ref) does not allow for a treatment assignment mechanism such that the latent factors associated with the units in $\Ic^{(a)}$ and $\Mc^{(1 - a)}$ live in orthogonal spaces. Hence the assignment mechanism needs to be diverse enough, so that the latent factors associated with the units in different treatments are linearly expressible in terms of each other. We only require the weaker condition that this linear span inclusion holds for the {\em sum} of the unit latent factors associated with $\Mc^{(1 - a)}$, as opposed to it holding for {\em each} latent factor $\lambda_n$ for $n \in \Mc^{(1 - a)}$.

We note that an explicit representation of $\beta^{(a)}$ in terms of the observed data is given in (ref) below, and we establish $\widehat{\beta}^{(a)}$ as given in (ref) is a consistent estimator for it in Proposition (ref). Next, we establish how Theorem (ref) helps establish identification of our causal parameter of interest.

\@startsection{subsection}{2}{0mm}{-\baselineskip}{0.05\baselineskip}{\normalfont\bf}{Estimator} The identification result in Corollary (ref) suggests an estimator of the form

where $\widehat{\beta}^{(1)}_j$ and $\widehat{\beta}^{(0)}_j$ are estimates of $\beta^{(1)}_j$ and $\beta^{(0)}_j$, respectively. That is, $\widehat{\mathsf{ATE}}_{\Mc}$ imputes the sums of the unobserved potential outcomes with and without treatment by $\sum_{n \in \Ic^{(1)}} \widehat{\beta}^{(1)}_n Y_{n, t^*}$ and $\sum_{n \in \Ic^{(0)}} \widehat{\beta}^{(0)}_n Y_{n, t^*}$, respectively.

Below, we provide sufficient conditions on any estimator $\widehat{\beta}^{(a)}$ of $\beta^{(a)}$, which establish the finite-sample consistency of $\widehat{\mathsf{ATE}}_{\Mc}$. Hence, we denote

In Section (ref), we provide explicit conditions for consistency and normality when the estimator for $\widehat{\beta}^{(a)}$ is PCR.

\@startsection{subsection}{2}{0mm}{-\baselineskip}{0.05\baselineskip}{\normalfont\bf}{Finite-Sample Consistency} To establish consistency, we make the (mild) assumption that $\varepsilon^{(a)}_{n, t}$ has a sub-Gaussian distribution.

\@startsection{section}{1}{0mm}{-\baselineskip}{0.25\baselineskip}{\center\normalfont\bf}{Estimation results using Principal Component Regression (PCR)}

In Section (ref) below, we provide explicit bounds on $\Delta_{\beta^{(a)}}$ for the case when PCR is used to estimate the coefficients $\widehat{\beta}^{(a)}$. Theorem (ref) collects sufficient conditions for consistency of $\widehat{\mathsf{ATE}}_{\Mc}$.

\@startsection{subsection}{2}{0mm}{-\baselineskip}{0.05\baselineskip}{\normalfont\bf}{Estimating Linear Weights via PCR} We now show how to estimate $\widehat{\beta}^{(a)}$ via PCR, and subsequently control $\Delta_{\beta^{(a)}}$ for the case when there are many measurements of all units under a common set of interventions.

{\bf Necessary notation to define PCR.} We introduce additional notation that we will use to discuss PCR estimation of $\widehat{\beta}^{(a)}$. Let $\bar{\mathcal{T}} \subset [T]$ be defined as follows:

That is, $\bar{\mathcal{T}}$ is the of measurements for which all units are seen under the same intervention. Let $a_t$ be the common treatment value for $t \in \bar{\mathcal{T}}$. For $a \in \{0, 1\}$, define

where to reduce notational burden we suppress dependence on $a$ in the notation for $\boldsymbol{Y}$, $\bZ$, and $\bX$. $\boldsymbol{Y}$ is a vector of summed outcomes of the units in $\Mc^{(1 - a)}$ for the measurements in $\bar{\mathcal{T}}$, $\bZ$ is a matrix of outcomes for the units in $\Ic^{(a)}$, and measurements in $\bar{\mathcal{T}}$, and $\bX$ is defined analogously to $\bZ$, but with respect to the expected observed outcomes. $\bX^{\text{lr}}$ is the low-rank approximation of $\bX$; note $\bX - \bX^{\text{lr}} = [\eta^{(a_t)}_{n, t}]_{t \in \bar{\mathcal{T}}, n \in \Ic^{(a)}}$.

{\bf PCR estimator for $\widehat{\beta}^{(a)}$.} Define the singular value decomposition (SVD) of $\bZ$ as

where $\hat{s}_\ell, \hat{u}_\ell, \hat{v}_\ell$ refer to the $\ell$-th singular value, left singular vector, and right singular vector, respectively. For any SVD, we order the singular values by decreasing magnitude. Given hyper-parameter $k \in [\min(\bar{T}, N_a)]$, we define $\widehat{\bX}^{\text{lr}}$ as follows:

That is, $\widehat{\bX}^{\text{lr}}$ is a low-rank approximation of $\bZ$. $\widehat{\beta}^{(a)}$ is then estimated by simply doing ordinary least squares (OLS) on $\boldsymbol{Y}$ and $\widehat{\bX}^{\text{lr}}$ as follows:

Here $\Big(\widehat{\bX}^{\text{lr}})^{+}$ denotes the Moore-Penrose pseudoinverse of $\widehat{\bX}^{\text{lr}}$ defined as

That is, PCR can be seen as doing ordinary least squares (OLS) on the best $k$-rank approximation of $\bZ$, given by $\Big(\widehat{\bX}^{\text{lr}})^{+}.$ If the ordinary least squares problem has multiple solutions, it is well-known that $\widehat{\beta}^{(a)}$ in equation (ref) is the minimum $\ell_2$-norm solution. We can then use $\widehat{\beta}^{(a)}$, to estimate $\widehat{\mathsf{ATE}}_{\Mc}$ as shown in (ref).

{\bf Interpreting PCR.} Using Assumption (ref) and Definition (ref), we have that for all $n \in [N], t \in \bar{\mathcal{T}}$,

Hence, using (ref) and the definitions of $\boldsymbol{Y}, \bZ, \bX, \bX^{\text{lr}}$, we have

By Assumption (ref), we have $\lambda_{\Mc^{(1 - a)}} = \sum_{n \in \Ic^{(a)}} \beta^{(a)}_{n} \lambda_n$. Hence, we can write

where $\phi^{\text{lr}} = \left[\sum_{n \in \Mc^{(1 - a)}} \eta^{(a_t)}_{n, t} \right]_{t \in \bar{\mathcal{T}}} \ $, $\bar{\varepsilon} = \left[\sum_{n \in \Mc^{(1 - a)}} \varepsilon^{(a_t)}_{n, t} \right]_{t \in \bar{\mathcal{T}}} \ $, $\bE^{\text{lr}} = [\eta^{(a_t)}_{n, t}]_{t \in \bar{\mathcal{T}}, n \in \Ic^{(a)}} \ $, $\bH = [\varepsilon^{(a_t)}_{n, t}]_{t \in \bar{\mathcal{T}}, n \in \Ic^{(a)}}$. Thus we have reduced our problem of estimating $\beta^{(a)}$ to that of linear regression where: (i) the covariates are noisily observed (i.e., error-in-variables regression), i.e. (ref) holds; (ii) the noiseless covariate matrix has an approximate low-rank representation, i.e. (ref) holds; (iii) an approximate linear model holds between the approximate low-rank representation of the noiseless covariates and the response variable, i.e. (ref) holds.

Hence, we can interpret PCR as follows: (1) the first step of doing PCA given in (ref) creates an estimate of the approximate low-rank approximation $\bX^{\text{lr}}$; (2) the second step of doing OLS given in (ref) creates an estimate of $\beta^{(a)}$ by regressing $\boldsymbol{Y}$ on $\widehat{\bX}^{\text{lr}}$, which is motivated by (ref).

The novel technical challenge in analyzing this setting is that there are four sources of error: the noise on the covariates given by $\bH$, the low-rank approximation error given by $\bE^{\text{lr}}$, the linear model approximation error given by $\phi^{\text{lr}}$, and the error on the response given by $\bar{\varepsilon}$.

\@startsection{subsection}{2}{0mm}{-\baselineskip}{0.05\baselineskip}{\normalfont\bf}{Additional Assumptions for Estimation Results with PCR} We make the following additional assumptions to state our consistency results. Note by Assumption (ref) and Proposition (ref), for all $\delta > 0$, \[ \text{rank}(\bX^{\text{lr}}) := \bar{r} \le r \le \bar{C} \cdot \delta^{-q}, \quad \|\bX - \bX^{\text{lr}} \|_{\infty} = \Delta_E \le C_H \cdot \delta^S. \]

An interpretation of Assumption (ref) is as follows. Suppose that each entry of $\bX^{\text{lr}} \ge c > 0$, i.e. is bounded below by an absolute constant $c$. Then since $\bX^{\text{lr}} \in \Rb^{\bar{T} \times N_a}$ we have that $\sum^{\bar{r}}_{\ell = 1} s^2_\ell = \| \bX^{\text{lr}} \|^2_F \ge C \bar{T} N_a$. If all the singular values of $\bX^{\text{lr}}$ are of the same order of magnitude, i.e., $\frac{s_{\bar{r}}}{s_1} \ge C$, this immediately implies that $s_r \ge C \sqrt{\frac{ \bar{T} N_a }{\bar{r}}}$.

Note a sufficient condition for Assumption (ref) is for $a \in \{0, 1\}$

Hence, intuitively, we require that the target measurement $t^*$ for which we wish to compute $\mathsf{ATE}_{\Mc}$, the latent factors $\rho_{t^*, 0}, \rho_{t^*, 1}$, are linearly expressible in terms of the latent factors $\{\rho_{t, a_t} \}_{t \in \bar{\mathcal{T}}}$ corresponding to the measurements under which all units are under a common intervention. This is the key condition that lets us {\em generalize} from the measurements $\bar{T}$ we learn on to the measurement $t^*$ we make counterfactual predictions on.

Below, we provide exact conditions if the linear estimator is PCR, the appropriateness of which was motivated in Section (ref).

\@startsection{subsection}{2}{0mm}{-\baselineskip}{0.05\baselineskip}{\normalfont\bf}{Finite-sample Consistency using PCR}

Theorem (ref) establishes exact conditions such that PCR is a consistent estimate for $\widehat{\mathsf{ATT}}, \widehat{\mathsf{ATU}}$, and $\widehat{\mathsf{ATE}}$, which are quantified by: the smoothness of $f_{t, a}$; the dimension of $U_n$; the number of common measurements $\bar{T}$; the number of units undergoing interventions $a \in \{0, 1\}$ given by $N_0, N_1$; and the magnitude of the linear weights $\beta^{(a)}$. We recall from Section (ref) that if $f_{t, a}$ is an analytic function, then we can take $S$ to be an arbitrarily large integer, i.e., it is H\"older continuous for all $S \in \Nb$.

Below we provide a natural sufficient condition for which the disperse weights condition holds.

Proposition (ref) establishes that if for any given set of units of sufficient size, their associated latent factors space the entire space $\Rb^r$, then the disperse weights condition must hold.