# Identification and Inference for Synthetic Controls with Confounding111We
thank Phillip Heiler, David Hirshberg, Jann Spiess, Jesse Shapiro, Stefan
Wager, Kaspar Wüthrich, Yinchu Zhu for helpful comments.

Guido W. Imbens222Graduate School of Business and Department of Economics,
Stanford University. Email<EMAIL_ADDRESS>Davide Viviano333Department of
Economics, Harvard University. Email<EMAIL_ADDRESS>
###### Abstract

This paper studies inference on treatment effects in panel data settings with
unobserved confounding. We model outcome variables through a factor model with
random factors and loadings. Such factors and loadings may act as unobserved
confounders: when the treatment is implemented depends on time-varying
factors, and who receives the treatment depends on unit-level confounders. We
study the identification of treatment effects and illustrate the presence of a
trade-off between time and unit-level confounding. We provide asymptotic
results for inference for several Synthetic Control estimators and show that
different sources of randomness should be considered for inference, depending
on the nature of confounding. We conclude with a comparison of Synthetic
Control estimators with alternatives for factor models.

Keywords: Synthetic Control, Panel Data, Difference-in-Differences, Causal
Inference.
JEL Codes: C33

##  1 Introduction

This paper studies identification and inference in panel or longitudinal data
settings where a (possibly small) number of units is exposed to treatment from
one period $T_{0}$ onwards. We will refer broadly to these scenarios as
Synthetic Control settings. The existing literature has proposed a variety of
methods for estimating counterfactual outcomes: controlling for the lagged
outcomes (known as horizontal regression), controlling for the control units
outcomes (vertical/Synthetic Control regression, Abadie and Gardeazabal, 2003;
Abadie et al., 2010), or using information both across time and units as for
the Synthetic DiD (Arkhangelsky et al., 2021) and matrix completion methods
and factor models (Athey et al., 2021; Gobillon and Magnac, 2016; Xu, 2017).
Most of this literature has focused on settings where the treatment assignment
process is not random, and unobservable characteristics of treated and control
units can be matched almost surely.444See for example, the discussion in
Ferman (2021).  This approach allows the researcher to interpret the task of
counterfactual estimation as a prediction problem, while inference must only
consider variation in idiosyncratic shocks. This paper studies the properties
of synthetic control-type estimators in the presence of unobservable (random)
confounders. These confounders cannot be matched exactly. We derive
identification conditions that formalize (i) which regression strategy is best
suited in the presence of different confounding mechanisms and (ii) which
sources of randomness should be considered for inference.

We study settings where potential outcomes follow an interactive factor model,
with possibly high dimensional factors and loadings and additive (nonrandom)
treatment effects. Unlike previous literature on synthetic control methods,
here, both the factors and loadings are random variables and can act as
confounders. The loadings act as unobserved confounders for whom receives the
treatment, and the factors act as confounders for when the treatment is
implemented. For instance, consider the problem of studying the effect of
state-level regulation (Abadie et al., 2010). When the regulation is
implemented may depend on aggregate factors of the economy, and which state
implements the regulation may depend on state-specific (unobserved)
characteristics. Because of confounding, conditional on the assignment
mechanism, the distribution of the factors and loadings can be arbitrary.

We provide identification restrictions corresponding to the vertical,
horizontal, and Synthetic DiD regressions within this confounding model.
Vertical regression (i.e., controlling for a weighted combination of control
units’ outcomes) allows for arbitrary time-level confounders, and it imposes
restrictions on unit confounders. It assumes that conditional on who receives
the treatment, unit-level confounders over the treated and control units match
in expectations after appropriately reweighting (for weights summing to one).
The leading case is the following: we can match all unit confounders’
conditional expectations that are endogenous. The remaining loadings are
exogenous with respect to the assignment mechanism. We refer to the latter
identification restriction as “no high dimensional unit confounders” since
these scenarios are attained only when a few loadings are endogenous. Vice-
versa, horizontal regression (i.e., controlling for a weighted combination of
lagged outcomes) allows for arbitrary endogenous unit-level confounders and
assumes “no high dimensional time confounders”, inverting the role of the
factors and loadings. This finding illustrates the trade-off between the
confounding restriction and the regression strategy.

We then turn to identification strategies robust to either high dimensional
units or time confounders. We show that the Synthetic Difference-in-
Differences in Arkhangelsky et al. (2021) is robust to either specification:
its bias depends on the product of two differences; the difference of the time
confounders’ weighted expectations between treatment and control periods, and
the difference of the unit level confounders. Therefore, its bias is zero if
there are either low dimensional unit confounders or time confounders. This
characterization of the bias formalizes double-robustness in Synthetic Control
settings with confounding, and, while building on the intuition in
Arkhangelsky et al. (2021), is novel to the literature.

Taking these results as stock, we draw their implications for inference.
Inference with time confounders for Synthetic Controls must consider the
randomness generated across units by the (exogenous) high dimensional loadings
but can condition on time variation induced by the (endogenous) factors. The
reason is that the estimator is unbiased only unconditional on the loadings.
For the horizontal regression, instead, we should account for the randomness
generated by the factors but not the loadings. Finally, inference with
Synthetic DiD must account for randomness over both unit and time dimensions
because robust to confounding occurring either across units or time,
motivating the construction of standard errors that capture both sources of
randomness.

We derive asymptotic properties of the estimators and standard errors
corresponding to confounding over time, units, or either of the two. We assume
that the post-treatment period and the number of treated units grow with the
sample but are small relative to the number of control periods and units. (We
return to the case of one or few treated units in Appendix A.) We characterize
the convergence rates of each estimator as a function of the treated units
$N_{1}$ and treatment periods $T_{1}$. The convergence rate of Synthetic
Control is between $1/\sqrt{N_{1}}$, and $1/\sqrt{N_{1}T_{1}}$, with
$1/\sqrt{N_{1}}$ if the factors are arbitrarily endogenous, and
$1/\sqrt{N_{1}T_{1}}$ if the factors are exogenous. Surprisingly, the rate of
convergence of Synthetic DiD can be faster than the one of Synthetic Control
with confounding. The key insight is that the Synthetic control’s convergence
rate is of order $1/\sqrt{N_{1}T_{1}}$ only if both unit and time confounders
concentrate around zero. However, for Synthetic DiD, a convergence rate of
order $1/\sqrt{N_{1}T_{1}}$ only requires that the time confounders before and
after the treatment concentrate around the same expectation after reweighting.

We conclude with a discussion on Synthetic Control and factor models. Under
distributional restrictions on the factors and loadings, for a Synthetic
Control regression, convergence rates depend on the rank of a matrix of (low)
dimensional unit confounders, as opposed to the rank of the (possibly high
dimensional) matrix of factors and loadings. However, sufficiently fast
convergence rates of the estimated weights also require some restrictions on
the distribution of factors before the treatment time occurs since, otherwise
we would not be able to estimate consistently such weights.

Our paper relates to an extensive literature on panel data models and
Synthetic Controls. We build on the literature on synthetic controls (Abadie
and Gardeazabal, 2003; Abadie et al., 2010; Doudchenko and Imbens, 2016;
Abadie and Vives-i Bastida, 2022), matrix completion methods and synthetic
difference-in-differences (Abadie et al., 2010; Athey et al., 2021;
Arkhangelsky et al., 2021; Liu et al., 2022; Arkhangelsky and Hirshberg,
2023). We contribute to this literature by studying confounding through the
unit and time-level confounders. Different from existing analysis with factor
models (e.g. Athey et al., 2021; Arkhangelsky et al., 2021), here, both
factors and loadings are random instead of deterministic and low rank,
motivating different identification properties and inference. Ben-Michael et
al. (2021) provide useful finite sample upper bound with fixed loadings and
factors using a balancing procedure. Here, we clarify conditions for different
estimation strategies and variance estimators in the presence of random
confounders.

We relate to studies on confounding with panel data, including Ferman and
Pinto (2016), Hahn and Shi (2017), Kellogg et al. (2021), and Agarwal and
Syrgkanis (2022) for recent works on identification with dynamic treatments.
Shi et al. (2021), Imbens et al. (2021) study instead proximal methods with
panel data. Here, we allow confounding over either or both time and units,
whereas these references implicitly condition on either (or both) the factors
and loadings. Our focus on inference and its connection to confounding is a
further distinction from these references. Shen et al. (2022) show that
horizontal and vertical regression provide the same point estimates in the
absence of an intercept and constraint on the weights. Here, we show that
these estimators would be biased without an intercept and weights summing to
one. Also, the authors consider random assignments while here we motivate the
choice of regression strategies and confidence intervals based on the
confounding mechanisms.

We complement the literature on inference with synthetic control and two-way
fixed effects, including Bottmer et al. (2021),Chernozhukov et al. (2019),
Cattaneo et al. (2021), Chernozhukov et al. (2021), Viviano and Bradic (2023),
Imai and Kim (2021). Different from the references above, here we explicitly
model the confounding mechanism in the construction of confidence intervals.
Shaikh and Toulis (2021) allow for random treatment timing but does not
consider unobserved confounders. Finally, we more broadly relate to a larger
strand of literature on factor models Bai (2009), Moon and Weidner (2017), Bai
and Ng (2019) among others, which, however, does not directly tackle the
problem of confounding for inference on causal effects. We defer an extensive
discussion of theoretical comparisons of synthetic control weights and
estimators for factor models to Section 5.3.

##  2 Setup

We consider a panel with units and periods

$i\in\\{1,\cdots,N_{0},\cdots,N_{0}+N_{1}\\},\quad
t\in\\{1,\cdots,T_{0},\cdots,T_{0}+T_{1}\\},$

respectively. Define $W_{i,t}\in\\{0,1\\}$ the treatment assignment for unit
$i$ at time $t$, with

$W_{i,t}=1\\{i\geq N_{0}\\}1\\{t\geq T_{0}\\}.$

We let both $T_{0}$ and $N_{0}$ be random variables. Denote
$\Big{(}Y_{i,t}(0),Y_{i,t}(1)\Big{)}$ the potential outcomes under treatment
and control of unit $i$ at time $t$. Throughout our discussion, we assume
constant and homogeneous treatment effects of the form

$Y_{i,t}(1)=\tau+Y_{i,t}(0),$ (1)

where $\tau$ denotes the average treatment effect. We return to heterogeneous
treatment effects in Remark 2. Researchers observe a matrix of potential
outcomes under control depicted below, and potential outcomes
$\mathbf{Y}_{i,t}(1)$ only for units $i\geq N_{0},t\geq T_{0}$.

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Our goal is to estimate the average treatment effect $\tau$, which entails
estimating the counterfactual outcomes. The literature has proposed three
approaches for this problem: horizontal regression, vertical regression (i.e.,
Synthetic Control methods, Abadie et al. (2010)), and matrix completion
methods (Athey et al., 2021). The horizontal regression uses past outcomes to
predict future outcomes of treated units. The vertical regression uses the
control units’ outcomes to predict the treated units’ outcomes in periods
$t>T_{0}$. The matrix completion method leverages the assumption that
$\mathbf{Y}(0)$ can be approximated by a low-rank factor model (Bai, 2003). In
this paper, we study how different sources of confounding justify different
regression strategies and standard errors.

###  2.1 Data generating process and confounding

We consider the following factor model.

######  Assumption 1 (High-dimensional factor model).

Assume that for all $(i,t)$,

$\displaystyle Y_{i,t}(0)$
$\displaystyle=\Big{(}\lambda_{t}+\tilde{\Lambda}_{t}\Big{)}^{\top}\Big{(}\gamma_{i}+\tilde{\Gamma}_{i}\Big{)}+\iota_{0,i}+\iota_{1,t}+\varepsilon_{i,t},$
$\displaystyle||\mathbb{E}[\tilde{\Gamma}_{i}]||_{\infty}$
$\displaystyle=||\mathbb{E}[\tilde{\Lambda}_{t}]||_{\infty}=\mathbb{E}[\varepsilon_{i,t}]=0,$

for random variables
$\tilde{\Gamma}_{i},\tilde{\Lambda}_{t}\in\mathbb{R}^{r},\varepsilon_{i,t}\in\mathbb{R}$,
and constants
$\lambda_{t},\gamma_{i}\in\mathbb{R}^{r},\iota_{0,i},\iota_{0,t}\in\mathbb{R}$,
and $r$ possibly unknown to researchers. Let
$\varepsilon_{i,t}|N_{0},T_{0}\sim_{i.i.d.}\mathcal{P}$ with
$\mathbb{E}[\varepsilon_{i,t}^{3}]<\infty,\mathbb{E}[\varepsilon_{i,t}^{2}]=\sigma_{\varepsilon}^{2}>0$.

Assumption 1 postulates that outcomes follow a factor model. The factor model
depends on factors $\tilde{\Lambda}_{t},\lambda_{t}$, and loadings
$\tilde{\Gamma}_{i},\gamma_{i}$. Here, $\varepsilon_{i,t}$ is an exogenous
shock. The main difference between the model in Assumption 1 with methods
discussed in previous literature (e.g. Athey et al., 2021; Abadie et al.,
2010; Arkhangelsky et al., 2021; Shen et al., 2022) is that that we treat the
factors and loadings as random variables, possibly acting as confounders.

Confounding over time and across units follows two independent processes.

######  Assumption 2 (Independent processes).

$\Big{[}\Big{(}\tilde{\Gamma}_{i}\Big{)}_{i\geq
1},N_{0}\Big{]}\perp\Big{[}\Big{(}\tilde{\Lambda}_{t}\Big{)}_{t\geq
1},T_{0}\Big{]}$, and $\Big{[}(\tilde{\Gamma}_{i})_{i\geq
1},(\tilde{\Lambda}_{t})_{t\geq 1}\Big{]}\perp(\varepsilon_{i,t})_{i\geq
1,t\geq 1}$.

Assumption 2 states that who receives the treatment depends on individual
confounders and is independent of when the treatment is implemented.
Assumption 2 simplifies our theoretical analysis and allows us to study
independently two sources of confounding over time and across units.

######  Example 2.1 (Effects of minimum wage on wages).

For state-level regulations, such as studying the effect of minimum wage,
Assumption 2 states that which states may implement the policy depends on
state-level characteristics, whereas when the policy is implemented depends on
aggregate (time-varying) factors of the economy. ∎

######  Assumption 3.

For all $i,t$,
$||\tilde{\Gamma}_{i}||_{2},||\gamma_{i}||_{2},||\tilde{\Lambda}_{t}||_{2},||\lambda_{t}||_{2}$
are uniformly bounded almost surely. Let
$\tilde{\Gamma}_{i}\sim_{i.i.d.}\mathcal{P}_{\Gamma},\tilde{\Lambda}_{t}\sim_{i.i.d}\mathcal{P}_{\Lambda}$,
with
$\mathrm{Var}(\tilde{\Lambda}_{t})=\Sigma_{\tilde{\Lambda}},\mathrm{Var}(\tilde{\Gamma}_{i})=\Sigma_{\tilde{\Gamma}}$
being positive definite.

Assumption 3 states that (i) factors and loadings have bounded $l_{2}$-norm;
(ii) factors and loadings are $i.i.d.$ unconditional on the treatment
assignment mechanism. Assumption 3 does not impose restrictions on how factors
and loadings relate to the assignment mechanism. Therefore, it does not rule
out dependence between factors, loadings, and the assignment mechanism. The
bounded $l_{2}$-norm restriction allows for a growing number of possibly weak
factors and allows us to control the variance of potential outcomes.555The
assumption that factors are uniformly bounded can be relaxed by subgaussianity
at the expense of additional notation. Finally, note that time trends or
seasonality components can be directly incorporated in a time fixed effect,
assuming the separability of non-stationary components.

###  2.2 Research questions

Given Equation (1), it follows that

$\displaystyle(i)$
$\displaystyle\quad\quad\mathbb{E}\Big{[}Y_{N,T}\Big{|}N_{0}=N,T_{0}=T,\tilde{\Gamma}_{N},\tilde{\Lambda}_{T}\Big{]}-\mathbb{E}\Big{[}Y_{N,T}(0)\Big{|}N_{0}=N,T_{0}=T,\tilde{\Gamma}_{N},\tilde{\Lambda}_{T}\Big{]}$
$\displaystyle=\tau.$ (2) $\displaystyle(ii)$
$\displaystyle\quad\quad\mathbb{E}\Big{[}Y_{N,T}\Big{|}N_{0}=N,T_{0}=T,\tilde{\Gamma}_{N}\Big{]}-\mathbb{E}\Big{[}Y_{N,T}(0)|N_{0}=N,T_{0}=T,\tilde{\Gamma}_{N}\Big{]}$
$\displaystyle=\tau$ $\displaystyle(iii)$
$\displaystyle\quad\quad\mathbb{E}\Big{[}Y_{N,T}\Big{|}N_{0}=N,T_{0}=T,\tilde{\Lambda}_{T}\Big{]}-\mathbb{E}\Big{[}Y_{N,T}(0)|N_{0}=N,T_{0}=T,\tilde{\Lambda}_{T}\Big{]}$
$\displaystyle=\tau.$

Namely, researchers may estimate counterfactual outcomes conditional on
different information sets to recover the same estimand $\tau$. This raises
the following questions:

  * Q1

Which conditions on factors and loadings guarantee the identification of
$\tau$, and how do these conditions relate to existing regression strategies?
Since the factors and loadings are random, identification requires conditions
on how they relate to treatment assignments. In Section 3, we argue that
different regression strategies are motivated by different identification
strategies.

  * Q2

Which source of randomness should we consider for inference? Equation (2)
shows that we can take differences of expectations conditional on different
information sets and recover the same estimand $\tau$. This creates confusion
for inference on $\tau$: estimators of different conditional expectations will
present different variances, not only because such estimators are different
but also because we can condition on different information sets. In Section 4,
we relate confidence intervals (and sources of randomness) to the underlying
identification assumptions.

  * Q3

Synthetic Control or factor models? Equation (2) $(i)$ shows that it suffices
to estimate the factors and loadings to recover $\tau$. However, standard
estimators for factor models require a low-rank representation. In Section 5,
we provide a discussion and comparison with Synthetic Controls in settings
with high dimensional (weak) factors.

##  3 Identification

This section studies identification conditional on $(N_{0},T_{0})=(N,T)$ for
given $(N,T)$.

###  3.1 Vertical and horizontal identification

We first discuss identification for the horizontal regression.

######  Assumption 4 (Limited confoundedness over time).

Suppose that

  * (A)

For all $i,t$, for fixed $\gamma_{i},\lambda_{t}$,

$Y_{i,t}(0)\perp\Big{[}N_{0},T_{0}\Big{]}\Big{|}\tilde{\Gamma}_{i}.$

  * (B)

There exist weights $w_{h,1},\cdots,w_{h,T},v_{h,T+1},\cdots,v_{h,T+T_{1}}$
such that for $T_{0}=T$

$\Big{|}\Big{|}\sum_{s<T}w_{h,s}\lambda_{s}-\sum_{t\geq
T}v_{h,t}\lambda_{t}\Big{|}\Big{|}_{2}=0,\quad\sum_{s<T}w_{h,s}=1,\quad\sum_{t\geq
T}v_{h,t}=1.$

Assumption 4 states that we can match the endogenous factors ($\lambda_{t}$)
exactly for some pre and post-treatment weights $w_{h},v_{h}$. The remaining
(random) factors $\tilde{\Lambda}_{t}$ are exogenous. Because we require exact
matching for each entry of $\lambda_{t}$, we interpret the second restriction
as imposing a sparsity restriction on $\lambda_{t}$.666We can relax the second
condition in Assumption 4 up to a small error of order
$o(N_{1}^{-1/2}T_{1}^{-1/2})$. As noted in Table 3 and discussed more
extensively in Section 5.3, Assumption 4 differs from usual low-rank
restrictions in factor models, that instead typically assume that we can
consistently estimate all interactive fixed effects. Instead, Assumption 4
states that we can find a set of weights such that we can match the (low-
dimensional) endogenous factors $\lambda_{t}$, whereas the remaining ones are
exogenous.

######  Proposition 3.1 (Horizontal identification).

Let Assumptions 1, 2, 4 hold. Then for all
$i\in\\{1,\cdots,N,\cdots,N+N_{1}\\}$

$\displaystyle\sum_{t\geq
T}v_{h,t}\mathbb{E}\Big{[}Y_{i,t}(0)|N_{0}=N,T_{0}=T,\tilde{\Gamma}_{i}\Big{]}=\sum_{s<T}w_{h,s}\mathbb{E}\Big{[}Y_{i,s}\Big{|}N_{0}=N,T_{0}=T,\tilde{\Gamma}_{i}\Big{]}+\beta_{0,h}(w_{h},v_{h})$
(3)

for some constant $\beta_{0,h}(w_{h},v_{h})$, which depends on $T$,
$w_{h},v_{h}$ and any weights $w_{h},v_{h}$ satisfying Assumption 4.

###### Proof of Proposition 3.1.

See Appendix C.1 ∎

For known weights $w_{h},v_{h}$, Proposition 3.1 suggests regressing the
outcomes of the control units onto a weighted average of the outcomes in the
previous period as discussed in Appendix 5. A similar result holds also for a
Synthetic Control (vertical) regression, after inverting the role of the
factor and loadings.

######  Assumption 5 (Limited confoundedness over units).

Suppose that

  * (A)

For all $i,t$, for fixed $\gamma_{i},\lambda_{t}$,

$Y_{i,t}(0)\perp\Big{[}N_{0},T_{0}\Big{]}\Big{|}\tilde{\Lambda}_{t}.$

  * (B)

There exist weights $w_{v,1},\cdots,w_{v,N-1},v_{v,N},\cdots,v_{v,N+1}$ such
that for $N_{0}=N$,

$\Big{|}\Big{|}\sum_{j<N}w_{v,j}\gamma_{j}-\sum_{n\geq
N}v_{v,n}\gamma_{n}\Big{|}\Big{|}_{2}=0,\quad\sum_{j<N}w_{v,j}=1,\quad\sum_{n\geq
N}v_{v,n}=1.$

######  Proposition 3.2 (Vertical identification).

Let Assumptions 1, 2, 5 hold. Then for all
$t\in\\{1,\cdots,T,\cdots,T+T_{1}\\}$,

$\displaystyle\sum_{n\geq
N}v_{v,n}\mathbb{E}\Big{[}Y_{n,t}(0)|N_{0}=N,T_{0}=T,\tilde{\Lambda}_{t}\Big{]}=\sum_{j<N}w_{v,j}\mathbb{E}\Big{[}Y_{j,t}|N_{0}=N,T_{0}=T,\tilde{\Lambda}_{t}\Big{]}+\beta_{0,v}(w_{v},v_{v}).$

for some constant $\beta_{0,v}(w_{v},v_{v})$ which only depends on $N$,
$w_{v},v_{v}$, and any weights $w_{v},v_{v}$ satisfying Assumption 5.

The proof mimics the one of Proposition 3.1 once we invert the loadings and
factors.

Proposition 3.1 and 3.2 provide identification restrictions for horizontal and
vertical regression. The former assumes no high-dimensional time confounders,
and the latter assumes no high-dimensional unit confounders.777Note that
Proposition 3.1, 3.2 hold under (weaker) moment restrictions which only
require mean exogeneity of $\tilde{\Lambda}_{t},\tilde{\Gamma}_{i}$ instead of
complete exogeneity of such random variables. In that case we can interpret,
$\lambda_{t}=\mathbb{E}[\tilde{\Lambda}_{t}|T_{0}=T],\gamma_{i}=\mathbb{E}[\tilde{\Gamma}_{i}|N_{0}=N]$,
as the conditional expectations of the factors and loadings, conditional on
the treatment assignment.

In the context of Example 2.1, if we interpret $\tilde{\Gamma}_{i}$ as sectors
in the economy, Assumption 5 states that the decision to introduce the minimum
wage may only depend on a few (aggregate) sectors but not on each of the
sectors separately.

######  Remark 1 (Intercept and weights summing to one).

The presence of an intercept in both vertical and horizontal identification
strategies and the weights summing to one guarantee unbiasedness in the
presence of time and unit fixed effects. Without such conditions, we would be
unable to guarantee unbiasedness in the presence of fixed effects. This
differs from Shen et al. (2022), who show that vertical and horizontal
regressions lead to the same point estimates assuming no intercepts and
unconstrained weights. ∎

###  3.2 Double-robust identification

Next, we study settings where either unconfoundedness over time or units may
occur. Consider the population equivalent of the Synthetic Differences-in-
Difference (sDiD) in Arkhangelsky et al. (2021), here augmented with post-
treatment weights $v_{v},v_{h}$:

$\displaystyle\bar{\tau}^{dr}(w_{h},w_{v},v_{h},v_{v})$
$\displaystyle=\sum_{j\geq N,t\geq
T}v_{v,j}v_{h,t}\mathbb{E}\Big{[}\bar{\tau}_{j,t}^{dr}(w_{h},w_{v})\Big{|}N_{0}=N,T_{0}=T\Big{]}$

where

$\displaystyle\bar{\tau}_{N,T}^{dr}(w_{h},w_{v},v_{h},v_{v})$
$\displaystyle=Y_{N,T}-\Big{\\{}\sum_{s<T}w_{h,s}\mathbb{E}\Big{[}Y_{N,s}\Big{|}N_{0}=N,T_{0}=T\Big{]}+\sum_{j<N}w_{v,j}\mathbb{E}\Big{[}Y_{j,T}\Big{|}N_{0}=N,T_{0}=T\Big{]}$
$\displaystyle\quad\quad\quad\quad\quad\quad-\sum_{s<T}\sum_{j<N}w_{h,s}w_{v,j}\mathbb{E}\Big{[}Y_{j,s}\Big{|}N_{0}=N,T_{0}=T\Big{]}\Big{\\}}.$

In the following proposition, we formalize double-robustness of
$\bar{\tau}^{dr}(\cdot)$.

######  Proposition 3.3.

Suppose that Assumptions 1, 2 hold. Then

$\displaystyle\tau-$
$\displaystyle\bar{\tau}^{dr}(w_{h},w_{v},v_{h},v_{v})=\Big{(}\bar{\Gamma}_{pre}(w_{v})-\bar{\Gamma}_{post}(v_{v})\Big{)}^{\top}\Big{(}\bar{\Lambda}_{pre}(w_{h})-\bar{\Lambda}_{post}(v_{h})\Big{)},$
(4)

where

$\displaystyle\bar{\Gamma}_{pre}(w_{v})=\sum_{i<N}w_{v,i}\Big{(}\mathbb{E}[\tilde{\Gamma}_{i}|N_{0}=N]+\gamma_{i}\Big{)},\quad\bar{\Gamma}_{post}(v_{v})=\sum_{i\geq
N}v_{v,i}\Big{(}\mathbb{E}[\tilde{\Gamma}_{i}|N_{0}=N]+\gamma_{i}\Big{)}$
$\displaystyle\bar{\Lambda}_{pre}(w_{h})=\sum_{t<T}w_{h,t}\Big{(}\mathbb{E}[\tilde{\Lambda}_{t}|T_{0}=T]+\lambda_{t}\Big{)},\quad\bar{\Lambda}_{post}(v_{h})=\sum_{t\geq
T}v_{h,t}\Big{(}\mathbb{E}[\tilde{\Lambda}_{t}|T_{0}=T]+\lambda_{t}\Big{)}.$

###### Proof.

See Appendix C.2. ∎

Proposition 3.3 shows the robustness properties of the Synthetic DiD method:
the method is robust to imperfect match over time and units. Here,
$\bar{\Gamma}_{pre}(w_{v}),\bar{\Gamma}_{post}(v_{v})$ denote the pre and post
treatment expectation of the loadings, and $\bar{\Lambda}_{pre}(w_{h})$,
$\bar{\Lambda}_{post}(v_{h})$ of the factors (conditional on the event
$N_{0}=N,T_{0}=T$). Under Assumption 4,
$\Big{(}\bar{\Lambda}_{pre}(w_{h})-\bar{\Lambda}_{post}(v_{h})\Big{)}=0$, and
under Assumption 5,
$\Big{(}\bar{\Gamma}_{pre}(w_{v})-\bar{\Gamma}_{post}(v_{v})\Big{)}=0$.

Equation (4) connects to previous results in the double robust literature
(e.g. Farrell, 2015). However, here we intend double-robustness at the
population level for $\bar{\tau}^{dr}$, as a function of the time and unit
level mismatch, instead of a function of the estimators’ convergence rates. It
also differs from the analysis in Arkhangelsky et al. (2021) because we
characterize double robustness as a function of the distributional properties
of the factors and loadings.

###  3.3 Implications for estimators

We conclude this discussion with a short summary of our findings in this
section. Assuming that we know the weights and intercepts
$w_{h},v_{h},\beta_{h},w_{v},v_{v},\beta_{v}$ for the moment, we can construct
three estimators

$\displaystyle\hat{\tau}_{n}^{h}(w_{h},v_{h},\beta_{h})$
$\displaystyle=\sum_{t\geq
T}v_{h,t}Y_{n,t}-\sum_{s<T}w_{n,s}Y_{n,s}+\beta_{h}$ (5)
$\displaystyle\hat{\tau}_{t}^{v}(w_{v},v_{v},\beta_{v})$
$\displaystyle=\sum_{n\geq
N}v_{v,n}Y_{t,n}-\sum_{i<N}w_{v,i}Y_{t,i}+\beta_{v}$
$\displaystyle\hat{\tau}^{dr}(w_{h},w_{v}.v_{h},v_{v})$
$\displaystyle=\sum_{t\geq T,n\geq
N}v_{h,t}v_{v,n}\Big{\\{}Y_{t,n}-\sum_{s<T}w_{h,s}Y_{n,s}-\sum_{i<N}w_{v,i}Y_{i,t}+\sum_{i<N}\sum_{s<T}w_{v,i}w_{h,s}Y_{i,s}\Big{\\}}$

corresponding to the horizontal regression for unit $n$, vertical for unit
$t$, and double robust estimator. For the horizontal and vertical regression,
we take

$\displaystyle\hat{\tau}^{h}(w_{h},v_{h},\beta_{h},q_{h})$
$\displaystyle=\sum_{n\geq
N}q_{h,n}\hat{\tau}_{n}^{h}(w_{h},v_{h},\beta_{h}),\quad$
$\displaystyle\sum_{n\geq N}q_{h,n}=1$ (6)
$\displaystyle\hat{\tau}^{v}(w_{v},v_{v},\beta_{v},q_{v})$
$\displaystyle=\sum_{t\geq
T}q_{v,t}\hat{\tau}_{t}^{v}(w_{v},v_{v},\beta_{v}),\quad$
$\displaystyle\sum_{t\geq T}q_{v,t}=1,$

where $q_{h},q_{v}$ are arbitrary weights that sum to one. A simple example is
$q_{h,n}=1/N_{1},q_{v,t}=1/T_{1}$. The additional weights $q_{h},q_{v}$ and
the post-treatment weights $v_{h},v_{v}$ are motivated treatment effect
homogeneity, see Remark 2. In the following proposition, we illustrate when
each of these estimators is unbiased.

######  Proposition 3.4 (Conditioning sets).

Under Assumptions 1, 2, for $n\geq N,t\geq T$,

$\displaystyle(A)$
$\displaystyle\tau-\mathbb{E}\left[\hat{\tau}_{n}^{h}(w_{h},v_{h},\beta_{h})\Big{|}N_{0}=N,T_{0}=T,\Big{(}\tilde{\Gamma}_{i}\Big{)}_{i\geq
1}\right]=0,\text{ if Assumption }\ref{ass:unc1a}\text{ holds}.$
$\displaystyle(B)$
$\displaystyle\tau-\mathbb{E}\left[\hat{\tau}_{t}^{v}(w_{v},v_{v},\beta_{v})\Big{|}N_{0}=N,T_{0}=T,\Big{(}\tilde{\Lambda}_{t}\Big{)}_{t\geq
1}\right]=0,\text{ if Assumption }\ref{ass:unc2a}\text{ holds}.$
$\displaystyle(C)$
$\displaystyle\tau-\mathbb{E}\left[\hat{\tau}^{dr}(w_{h},w_{v},v_{h},v_{v})\Big{|}N_{0}=N,T_{0}=T\right]=0,\text{
if either Assumption }\ref{ass:unc1a}\text{ or }\ref{ass:unc2a}\text{ holds}.$

###### Proof of Proposition 3.4.

The proof follows directly from Propositions 3.1, 3.2 and 3.3. ∎

Proposition 3.4 has important implications for inference. It shows that we can
condition on the loadings for a horizontal regression and factors for a
vertical regression, but not vice-versa. Inference with the Synthetic DiD
should take into account the randomness generated by both the factors and
loadings. We formalize these intuitions in the following section.

######  Remark 2 (Heterogeneous treatment effects).

Our analysis assumes that treatment effects are constant, i.e.,
$Y_{i,t}(1)=Y_{i,t}(0)+\tau$. Consider instead settings with heterogeneous
(additive) treatment effects $\tau_{i,t}=Y_{i,t}(1)-Y_{i,t}(0)$. Effects
heterogeneity may, for example, be relevant when treatment intensity varies by
state due to different regulatory systems. Estimated treatment effects denote
weighted averages, which depend on such heterogeneity. In the context of
Synthetic Controls (vertical regression), researchers may be able to identify
a weighted combination of treatment effects $\sum_{t\geq T}q_{v,t}\sum_{n\geq
N}v_{v,n}\tau_{n,t}$ where the weights $v_{v,n}$ must satisfy the balancing
restriction in Assumption 5. Intuitively, if we can guarantee that Assumption
5 holds for $v_{v,n}=1/N_{1}$, we can recover the average effect on the
treated unit when choosing $q_{v,t}=1/T$. In contrast, if Assumption 5 holds
for some other weighting mechanism, we can only recover a weighted combination
of treatment effects. This intuition formalizes the trade-offs of matching
restrictions on the weights $(v_{h},v_{v})$ and the identified class of
estimands. On the other hand, because $q_{h},q_{v}$ can be chosen arbitrarily
by the researcher, we recommend as choices for $q_{h},q_{v}$,
$q_{h}=1/N_{1}\mathbf{1},q_{v}=1/T_{1}\mathbf{1}$, i.e., imposing equal
weights on the estimators.888This choice is recommended under the failure of
homogeneous treatment effects when the goal is to study the average effect on
the treated unit. A second possible choice is to minimize the variance of the
estimator, which, however, would require estimating the factors and loadings
(and would affect the estimand of interest in the absence of homogeneous
treatment effects). ∎

Table 1: Summary of the identification strategies of different regressions.
The “Source of randomness” denotes the residual source of variation after
conditioning on the endogenous variables (and that should be considered for
confidence intervals presented in Section 4). “Confounding effects” denote the
components whose expectation is not zero, given the assignment mechanism.

Regression | Assumption | Source of randomness | Confounding Effects
---|---|---|---
Horizontal | Restricted time confounding ($\tilde{\Lambda}_{t}\perp T_{0}$) | $(\tilde{\Lambda}_{t},\varepsilon_{i,t})$ | $(\lambda_{t},\gamma_{i}+\tilde{\Gamma}_{i})$
| \+ Exact matching of factors’ conditional expectations |  |
|  $(\sum_{s\leq T}w_{h,s}\lambda_{s}=\sum_{t\geq T}v_{h,t}\lambda_{t}$ for some $w_{h},v_{h})$ |  |
Vertical | Restricted unit confounding ($\tilde{\Gamma}_{i}\perp N_{0}$) | $(\tilde{\Gamma}_{i},\varepsilon_{i,t})$ | $(\gamma_{i},\lambda_{t}+\tilde{\Lambda}_{t})$
| \+ Exact matching of loadings’ conditional expectations |  |
|  $(\sum_{i\leq N}w_{v,i}\gamma_{i}=\sum_{j\geq N}v_{v,j}\gamma_{j}$ for some $w_{v},v_{v}$) |  |
Synthetic DiD |  Either no high dimensional unit | $(\tilde{\Gamma}_{i},\tilde{\Lambda}_{t},\varepsilon_{i,t})$ | Either $(\lambda_{t}+\tilde{\Lambda}_{t},\gamma_{i})$
| or time confounders (and matching over time or units) |  | or $(\lambda_{t},\tilde{\Gamma}_{i}+\gamma_{i})$
PCA/Least Squares | No high rank confounders | $\varepsilon_{i,t}$ | $(\tilde{\Gamma}_{i}+\gamma_{i},\tilde{\Lambda}_{t}+\lambda_{t})$

##  4 Inference over the post-treatment period

This section studies inference for vertical, horizontal regression and
Synthetic DiD. We consider the estimators in Equation (6) for the horizontal
and vertical regression and $\hat{\tau}^{dr}(\cdot)$ in Equation (5) for the
Synthetic DiD. We will condition on the event $N_{0}=N,T_{0}=T$. We study
asymptotics as $N,T,N_{1},T_{1}\rightarrow\infty,$ with $N_{1},T_{1}$ growing
at an appropriate slower rate than $N,T$. We return to settings with a finite
number of treated units in Appendix A.

We study inference given some known (not data-dependent) weights
$(w_{h}^{\star},v_{h}^{\star},\beta_{h}^{\star}),(w_{v}^{\star},v_{v}^{\star},\beta_{v}^{\star})$.
We will assume that these weights satisfy the restrictions in Assumptions 4
and 5, respectively, and regularity conditions presented below. Corollary 2
shows that estimation in the absence of estimation error is asymptotically
equivalent to inference with estimated weights – assuming that the post-
treatment period is sufficiently shorter than the pre-treatment period. We
study the estimation of the weights in Section 5.

As shown in Proposition 3.4, inference must account for different sources of
randomness depending on the nature of confounding: estimators are unbiased
only unconditional (but not necessarily conditional) on either time or unit-
level confounders, depending on the confounding assumption. We formalize this
intuition below.

###  4.1 Inference with horizontal and vertical regression

We present assumptions and results for the horizontal regression first.

######  Assumption 6 (Horizontal regression conditions).

Assume the following:

  * (A)

$T_{1}N_{1}/T_{0}^{1/3}=o(1)$;

  * (B)

$||w_{h}^{\star}||_{\infty}=\mathcal{O}(T_{0}^{-2/3}),||v_{h}^{\star}||_{\infty}=\mathcal{O}(T_{1}^{-2/3})$,
and $w^{\star},v^{\star}$ satisfy condition (B) in Assumption 4.

Condition (A) states that the post-treatment period is much shorter than the
pre-treatment period. Condition (B) assumes that the oracle weights guarantee
pre and post treatment balance in the factors. Condition (B) assumes that no
single (or few) units receive all the weights similar to Arkhangelsky and
Imbens (2018), ruling out sparse settings.

######  Theorem 4.1.

Suppose that Assumptions 1, 2, 3, 4, 6 hold. Then as $T_{1}\rightarrow\infty$,
for any $q_{h}:\sum_{n\geq N}q_{h,n}=1$,

$\displaystyle\frac{\Big{(}\hat{\tau}^{h}(w_{h}^{\star},v_{h}^{\star},\beta_{h}^{\star},q_{h})-\tau\Big{)}}{\bar{\mathbb{V}}_{h}^{1/2}(N,T,q_{h})}\rightarrow_{d}\mathcal{N}(0,1)$

where $\bar{\mathbb{V}}_{h}(N,T,q_{h})=\mathbb{V}\left(\sum_{n\geq
N}q_{h,n}\sum_{t\geq
T}v_{h,t}^{\star}Y_{n,t}\Big{|}N_{0}=N,T_{0}=T,\Big{(}\tilde{\Gamma}_{i}\Big{)}_{i\geq
1}\right)$.

###### Proof.

See Appendix C.3. ∎

Theorem 4.1 shows that the horizontal estimator, for any weighted combination
of treated units converges in distribution to a Gaussian. The convergence
rates also depend on unit-level confounders. The variance can be estimated
directly using the empirical moments of the post-treatment outcomes.

######  Corollary 1 (Rate of convergence).

Let the conditions in Theorem 4.1 hold. Then

$\displaystyle\frac{\bar{\mathbb{V}}_{h}(N,T,q_{h})}{||v_{h}^{\star}||_{2}^{2}}=\Big{(}\sum_{n\geq
N}(\tilde{\Gamma}_{n}+\gamma_{n})q_{h,n}\Big{)}^{\top}\Sigma_{\tilde{\Lambda}}\Big{(}\sum_{n\geq
N}(\tilde{\Gamma}_{n}+\gamma_{n})q_{h,n}\Big{)}+\sigma_{\varepsilon}^{2}||q_{h}||_{2}^{2},$

where
$\Sigma_{\tilde{\Lambda}}=\mathbb{V}(\tilde{\Lambda}_{t}),\sigma_{\varepsilon}^{2}=\mathbb{V}(\varepsilon_{i,t})$.

Corollary 1 shows that the variance (and rate of convergence) depends on the
norm of the weights and on the loadings. The rate of convergence is weakly
slower than $\frac{1}{\sqrt{T_{1}N_{1}}}$, and equal to
$\frac{1}{\sqrt{T_{1}N_{1}}}$ if the loadings are exogenous. Specifically, we
can write

$\bar{\mathbb{V}}_{h}(N,T,q_{H})=\mathcal{O}\Big{(}||q_{h}||^{2}||v_{h}^{\star}||_{2}^{2}\sigma_{\varepsilon}^{2}+\rho_{N_{1}}(q_{h})||v_{h}^{\star}||_{2}^{2}\Big{)},\quad\rho_{N_{1}}(q_{h})=\Big{|}\Big{|}\sum_{n\geq
N}q_{h,n}(\tilde{\Gamma}_{n}+\gamma_{n})\Big{|}\Big{|}_{2}^{2}.$

Here, $||v_{h}^{\star}||_{2}^{1}=1/T$, and $\rho_{N_{1}}$ characterizes the
strength of the unit-level confounding, with $\rho_{N_{1}}$ either bounded
away from zero, or $\rho_{N_{1}}\rightarrow 0$. For $\rho_{N_{1}}\rightarrow
0$ to converge to zero, we would need no unit-level confounding, that is
$\frac{1}{N}\sum_{n\geq N}\tilde{\Gamma}_{n}$ concentrates around its
unconditional expectation (unconditional on $N_{0}$) and $\gamma_{n}$ is local
to zero. This result illustrates properties of convergence rates that depend
on the strength of confounding.

The Synthetic Control case follows similarly to the horizontal regression
case, where we invert the role of the loadings and factors, with

$\displaystyle\frac{\Big{(}\hat{\tau}^{v}(w_{v}^{\star},v_{v}^{\star},\beta_{v}^{\star},q_{v})-\tau\Big{)}}{\bar{\mathbb{V}}_{v}^{1/2}(N,T,q_{v})}\rightarrow_{d}\mathcal{N}(0,1)$

where $\bar{\mathbb{V}}_{v}(N,T,q_{v})=\mathbb{V}\left(\sum_{t\geq
T}q_{v,t}\sum_{n\geq
N}v_{v,n}^{\star}Y_{n,t}\Big{|}N_{0}=N,T_{0}=T,\Big{(}\tilde{\Lambda}_{t}\Big{)}_{t\geq
1}\right)$. Here,

$\displaystyle\frac{\bar{\mathbb{V}}_{v}(N,T,q_{v})}{||v_{v}^{\star}||_{2}^{2}}=\Big{(}\sum_{t\geq
T}(\tilde{\Lambda}_{t}+\lambda_{t})q_{v,t}\Big{)}^{\top}\Sigma_{\Gamma}\Big{(}\sum_{t\geq
T}(\tilde{\Lambda}_{t}+\lambda_{t})q_{v,t}\Big{)}+\sigma_{\varepsilon}^{2}||q_{v}||_{2}^{2},$

where $\Sigma_{\tilde{\Gamma}}=\mathbb{V}(\tilde{\Gamma}_{i})$. With diagonal
$\Sigma_{\tilde{\Gamma}}$, the rate of convergence is of order
$||v_{v}^{\star}||_{2}^{2}||q_{v}||_{2}^{2}\sigma_{\varepsilon}^{2}+||v_{v}^{\star}||_{2}^{2}\Big{|}\Big{|}\sum_{t\geq
T}q_{v,t}(\tilde{\Lambda}_{t}+\lambda_{t})\Big{|}\Big{|}_{2}^{2}$, which
depends on time-level confounders. Therefore, whereas each regression provides
consistent estimates as $N,T\rightarrow\infty$ under their corresponding
unconfoundedness restriction (either no high dimensional time or unit level
confounders), a faster rate of convergence can be obtained if both no high
dimensional time and unit level confounders hold.

We conclude this discussion by showing how our results in Theorem 4.1 apply in
the presence of estimated weights.

######  Corollary 2.

Suppose that the conditions in Theorem 4.1 hold, and consider any data
dependent weights $(\hat{w}_{h},\hat{v}_{h},\hat{\beta}_{h})$ such that

$\hat{\tau}_{n,t}^{h}(\hat{w}_{h},\hat{v}_{h},\hat{\beta}_{h})-\hat{\tau}_{n,t}^{h}(w_{h}^{\star},v_{h}^{\star},\beta_{h}^{\star})=o_{p}\left(N_{1}^{-1/2}T_{1}^{-1/2}\right).$
(7)

Then Theorem 4.1 holds with
$\hat{\tau}^{h}(\hat{w}_{h},\hat{v}_{h},\hat{\beta}_{h})$ in lieu of
$\hat{\tau}^{h}(w_{h}^{\star},v_{h}^{\star},\beta_{h}^{\star})$.

###### Proof of Corollary 2.

See Appendix C.6. ∎

An important insight of Corollary 2 is that even if the variance is
conditional on the loadings for the horizontal regression, the estimated
weights only need to converge to their population counterpart unconditionally
on the factors and loadings. The reason is that the variance
$\bar{\mathbb{V}}_{h}$ converges to zero at a rate at most
$1/\sqrt{N_{1}T_{1}}$, while the estimated weights can converge at a faster
rate when the pre-treatment period and number of control units is larger than
the post-treatment period and number of treated units. Appendix C.6 presents
the details. Conditions on the convergence rates of the weights are discussed
in Section 5, where we review different weights estimators.

######  Remark 3 (Inference with a single treated unit).

In Appendix A, we study inference in the presence of a single treated unit. We
show how placebo tests in Abadie et al. (2010) can be used in our setting
under additional unconfoundedness restrictions on the endogenous loadings,
assuming that we can find a donor pool of “placebo treated units” such that we
can exactly match their loadings with the loadings of the remaining control
units. ∎

###  4.2 Robust inference

In the following lines, we study inference that is robust to confoundedness of
either the loadings or factors. We impose the following conditions.

######  Assumption 7 (Robust regression conditions).

Assume the following:

  * (A)

$T_{1}N_{1}/T_{0}^{1/3}=o(1),N_{1}T_{1}/N_{0}^{1/3}=o(1)$;

  * (B)

$||w_{h}^{\star}||_{\infty}=\mathcal{O}(T_{0}^{-2/3}),||w_{v}^{\star}||_{\infty}=\mathcal{O}(N_{0}^{-2/3})$,
$||v_{h}^{\star}||_{\infty}=\mathcal{O}(T_{1}^{-2/3}),||v_{v}^{\star}||_{\infty}=\mathcal{O}(N_{1}^{-2/3})$,
and either $(w_{h}^{\star},v_{h}^{\star})$ satisfy (B) in Assumption 4 or
$(w_{v}^{\star},v_{v}^{\star})$ satisfy (B) in Assumption 5.

Assumption 7 formalizes double robustness properties, for which either no high
dimensional unit confounders or no high dimensional time confounders exist.

######  Theorem 4.2.

Let Assumptions 1, 2, 3, 7 hold, and either Assumption 4 or Assumption 5 (or
both) hold. Then as $N_{1},T_{1}\rightarrow\infty$,

$\displaystyle
P\left(\frac{\Big{(}\hat{\tau}^{dr}(w_{h}^{\star},w_{v}^{\star},v_{h}^{\star},v_{v}^{\star})-\tau\Big{)}}{\max\\{\mathbb{V}_{h}(v_{h}^{\star},v_{v}^{\star}),\mathbb{V}_{v}(v_{v}^{\star},v_{h}^{\star})\\}^{1/2}}\leq
z_{1-\alpha}\Big{|}N_{0}=N,T_{0}=T\right)\geq 1-\alpha,$

where $\Phi(z_{1-\alpha})=1-\alpha$, $\Phi$ is the Gaussian CDF, and

$\displaystyle\mathbb{V}_{h}(v_{h}^{\star},v_{v}^{\star})$
$\displaystyle=\mathbb{V}\left(\sum_{t\geq T}\sum_{j\geq
N}v_{h,j}^{\star}v_{v,t}^{\star}\Big{\\{}Y_{j,t}-\sum_{i<N}w_{v,i}^{\star}Y_{i,t}\Big{\\}}\Big{|}N_{0}=N,T_{0}=T,(\tilde{\Gamma}_{i})_{i\geq
1}\right),$ $\displaystyle\mathbb{V}_{v}(v_{v}^{\star},v_{h}^{\star})$
$\displaystyle=\mathbb{V}\left(\sum_{j\geq N}\sum_{t\geq
T}v_{v,t}^{\star}v_{h,t}^{\star}\Big{\\{}Y_{j,t}-\sum_{s<T}w_{v,s}^{\star}Y_{j,s}\Big{\\}}\Big{|}N_{0}=N,T_{0}=T,(\tilde{\Lambda}_{t})_{t\geq
1}\right).$

###### Proof of Theorem 4.2.

See Appendix C.4. ∎

Theorem 4.2 shows that confidence intervals for Synthetic DiD depend on the
worst-case variance conditional on either the factors or loadings. The
variance calculation differs from settings with non-random factors and
loadings (e.g. Arkhangelsky et al., 2021), and can be computed using the
empirical moments of weighted combinations of the outcomes.999Because the
$\max\\{\\}$ operator is a continuous function, we can use the estimated
variances $\mathbb{V}_{h},\mathbb{V}_{u}$ and invoke Slutsky theorem to
provide asymptotically valid confidence intervals.  This difference is because
of the population double-robustness property we derived.

A direct corollary of Theorem 4.2 is that the estimator
$\hat{\tau}^{dr}(\cdot)$’s convergence rate depends on both $N_{1},T_{1}$, and
on the unconfoundedness restriction. The rate of converge is
$\max\\{\mathbb{V}_{h}(v_{h}^{\star},v_{v}^{\star}),\mathbb{V}_{v}(v_{v}^{\star},v_{h}^{\star})\\}$,
namely (let $\wedge$ denote the maximum operator,
$\Gamma_{i}=\gamma_{i}+\tilde{\Gamma}_{i},\Lambda_{t}=\lambda_{t}+\tilde{\Lambda}_{t}$)

$\displaystyle||v_{v}^{\star}||_{2}^{2}||v_{h}^{\star}||_{2}^{2}\sigma_{\varepsilon}^{2}+\Big{\\{}$
$\displaystyle||v_{h}^{\star}||_{2}^{2}\Big{(}\sum_{n\geq
N}\Gamma_{n}v_{v,n}^{\star}-\sum_{j<N}\Gamma_{j}w_{v,j}^{\star}\Big{)}^{\top}\Sigma_{\tilde{\Lambda}}\Big{(}\sum_{n\geq
N}\Gamma_{n}v_{v,n}^{\star}-\sum_{j<N}\Gamma_{j}w_{v,j}^{\star}\Big{)},$ (8)
$\displaystyle\quad\wedge||v_{v}^{\star}||_{2}^{2}\Big{(}\sum_{t\geq
T}v_{h,t}^{\star}\Lambda_{t}-\sum_{s<T}\Lambda_{s}w_{h,s}^{\star}\Big{)}^{\top}\Sigma_{\tilde{\Gamma}}\Big{(}\sum_{t\geq
T}v_{h,t}^{\star}\Lambda_{t}-\sum_{s<T}\Lambda_{s}w_{h,s}^{\star}\Big{)}\Big{\\}}.$

If both $\Lambda_{t}\perp T_{0},\Gamma_{i}\perp T_{0}$, the right-hand side of
Equation (8) converges almost surely to zero, with rate $1/\sqrt{N_{1}T_{1}}$.
On the other hand, under lack of either unconfoundedness restriction (over
time or units), the right-hand-side expression in Equation (8) does not
converge to zero and the rate of convergence is
$\max\\{T_{1}^{-1/2},N_{1}^{-1/2}\\}$. Convergence rates faster than
$1/\sqrt{T_{1}}$ or $1/\sqrt{N_{1}}$ do not require that factors or loadings
are exogenous. Convergence rates of order $1/\sqrt{N_{1}T_{1}}$ allow the
conditional expectations of the loadings and factors to be different from zero
but require that the expectations match before and after the treatment after
reweighting.

Table 2: Convergence rates under the model in Assumption 1 with independent
factors and loadings. The convergence rate for Horizontal and Vertical
regression depends on the post-treatment weights $v^{\star}$ and on the
concentration of the loadings or factors. The convergence rate of Synthetic
DiD depends on the post-treatment weights and the mismatch between the post-
treatment and pre-treatment factors and loadings.

Regression | Bound on Standard Error
---|---
Horizontal |  $||v_{h}^{\star}||_{2}\Big{|}\Big{|}\sum_{n\geq N}q_{h,n}(\tilde{\Gamma}_{n}+\gamma_{n})\Big{|}\Big{|}_{2}$ for arbitrary $q_{h}:1^{\top}q_{h}=1$
Vertical |  $||v_{v}^{\star}||_{2}\Big{|}\Big{|}\sum_{t\geq T}q_{v,n}(\tilde{\Lambda}_{t}+\lambda_{t})\Big{|}\Big{|}_{2}$ for arbitrary $q_{v}:1^{\top}q_{v}=1$
Synthetic DiD | $||v_{h}^{\star}||_{2}\Big{|}\Big{|}\sum_{n\geq N}(\tilde{\Gamma}_{n}+\gamma_{n})v_{v,n}^{\star}-\sum_{j<N}(\tilde{\Gamma}_{j}+\gamma_{j})w_{v,j}^{\star}\Big{|}\Big{|}_{2}\wedge||v_{v}^{\star}||_{2}\Big{|}\Big{|}\sum_{t\geq T}v_{h,t}^{\star}(\tilde{\Lambda}_{t}+\lambda_{t})-\sum_{s<T}(\tilde{\Lambda}_{s}+\lambda_{s})w_{h,s}^{\star}\Big{|}\Big{|}_{2}$

##  5 Estimation of the weights and factors: a discussion

In this section, we review existing methods for estimating the weights and
discuss properties and assumptions that these methods require in the context
of our model of confounding.

###  5.1 Weights estimation with penalized $l_{2}$-norm minimization

First, we review estimation of the weights as in Arkhangelsky et al. (2021),
which are similar to those in Abadie et al. (2010) with the additional norm
constraint. For the horizontal regression, we can construct weights’
estimators as follows:

$\displaystyle(\hat{\beta}_{h},\hat{w}_{h},\hat{v}_{h})$
$\displaystyle=\mathrm{arg}\min_{w,v,\beta,1^{\top}w=1,w\geq
0,||w||_{\infty}\leq T_{0}^{-2/3},1^{\top}v=1,v\geq 0,||v||_{\infty}\leq
T_{1}^{-2/3}}\mathcal{L}_{h,N,T}(w,v,\beta),$ (9)
$\displaystyle\mathcal{L}_{h,N,T}(w,v,\beta)$
$\displaystyle=\sum_{i<N}\Big{(}\sum_{t\geq
T}v_{h,t}Y_{i,t}-\sum_{s<T}w_{s}Y_{i,s}-\beta\Big{)}^{2}+p_{h}N(||w||^{2}+||v||^{2})$

where we minimize the $l_{2}$-distance between the post-treatment average
control outcomes and a weighted combination of the pre-treatment outcomes.
Similarly, for the vertical regression

$\displaystyle(\hat{\beta}_{v},\hat{w}_{v},\hat{v}_{v})$
$\displaystyle=\mathrm{arg}\min_{w,v,\beta,1^{\top}w=1,w\geq
0,||w||_{\infty}\leq N_{0}^{-2/3},1^{\top}v=1,v\geq 0,||v||_{\infty}\leq
N_{1}^{-2/3}}\mathcal{L}_{v,N,T}(w^{\prime},v^{\prime},\beta^{\prime})$ (10)
$\displaystyle\mathcal{L}_{v,N,T}(w^{\prime},v^{\prime},\beta^{\prime})$
$\displaystyle=\sum_{s<T}\Big{(}\sum_{i\geq
N}v_{v,i}^{\prime}Y_{i,s}-\sum_{i<N}w_{i}^{\prime}Y_{i,s}-\beta^{\prime}\Big{)}^{2}+p_{v}T(||w^{\prime}||^{2}+||v^{\prime}||^{2}).$

Here, $p_{v},p_{h}$ denote choosen penalty parameters discussed in
Arkhangelsky et al. (2021).

The constraints are as discussed in Sections 3, 4. Algorithm 1 presents a
summary.

Algorithm 1 Vertical and Horizontal Regressions

1:$Y_{i,t},i<N_{0},t\in\\{1,\cdots,T\\}$

2:Minimize
$\mathcal{L}_{h,N_{0},T_{t}}(w,v,\beta),\mathcal{L}_{v,N_{0},T_{0}}(w^{\prime},v^{\prime},\beta^{\prime})$
under the constraints  subject to $\displaystyle
1^{\top}w=1,||w||_{\infty}\leq KN_{0}^{-2/3},w\geq
0,1^{\top}v=1,||v||_{\infty}\leq KN_{1}^{-2/3},v\geq 0$ $\displaystyle
1^{\top}w^{\prime}=1,||w^{\prime}||_{\infty}\leq KT_{0}^{-2/3},w^{\prime}\geq
0,1^{\top}v=1,||v^{\prime}||_{\infty}\leq KT_{1}^{-2/3},v^{\prime}\geq 0$
return $(\hat{w}_{h},\hat{w}_{v},\hat{v}_{h},\hat{v}_{v})$.

The minimization problem in Section 5.1 corresponds to a error-in-variables
optimizaton problem. By letting $\Gamma_{i}=\gamma_{i}+\tilde{\Gamma}_{i}$

$\displaystyle\sum_{i<N_{0}}\left(\sum_{t\geq
T_{0}}v_{h,t}Y_{i,t}-\sum_{s<T}w_{h,t}Y_{i,s}\right)^{2}+(\sigma^{2}_{h}-1)\zeta_{h}N_{0}(||w||^{2}+||v||^{2})=$
(11) $\displaystyle\sum_{i<N_{0}}\left(\sum_{t\geq
T_{0}}v_{h,t}\Big{(}\lambda_{t}^{\top}\Gamma_{i}+\eta_{i,t}+\iota_{1,t}\Big{)}-\sum_{s<T_{0}}w_{h,s}\Big{(}\lambda_{s}^{\top}\Gamma_{i}+\eta_{i,s}+\iota_{1,s}+\beta\Big{)}\right)^{2}$
$\displaystyle\quad\quad\quad\quad+(\sigma^{2}_{h}-1)\zeta_{h}N_{0}(||w||^{2}+||v||^{2}),$

where

$\eta_{i,t}=\tilde{\Lambda}_{t}^{\top}\Gamma_{i}+\varepsilon_{i,t},\quad\Gamma_{i}=\gamma_{i}+\tilde{\Gamma}_{i}$
(12)

Following Arkhangelsky et al. (2021), we define “oracle” counterpart of such a
problem for the horizontal regression solves the following optimization
problem.

$\displaystyle(w_{h}^{\star},v_{h}^{\star},\beta_{h}^{\star})=\mathrm{arg}\min_{w,v,\beta,1^{\top}w=1,w\geq
0,||w||_{\infty}\leq T_{0}^{-2/3},1^{\top}v=1,v\geq 0,||v||_{\infty}\leq
T_{1}^{-2/3}}\mathcal{L}_{h}^{\star}(w,v,\beta,\delta_{h}),$ (13)
$\displaystyle\mathcal{L}_{h}^{\star}(\cdot)=\sum_{i<N_{0}}\delta_{h,i}\left(\Gamma_{i}^{\top}\Big{(}\sum_{t\geq
T_{0}}v_{t}\lambda_{t}-\sum_{s<T_{0}}w_{s}\lambda_{s}\Big{)}+\sum_{t\geq
T_{0}}v_{t}\iota_{1,t}-\sum_{s<T_{0}}w_{s}\iota_{1,s}-\beta\right)^{2}+N_{0}p_{h}^{\star}(||w||^{2}+||v||^{2}).$

where $p_{h}^{\star}$ is a penalty that depends on the residuals’ variance as
in Arkhangelsky et al. (2021). The penalization of the oracle solution is
different from the penalization used for estimation as motivated in Hirshberg
(2021). We can define oracle weights for Synthetic Control
$(w_{v}^{\star},v_{v}^{\star},\beta_{v}^{\star})$ in a similar manner.

Consider estimating the horizontal regression weights, and let
$||\cdot||_{op}$ be the operator norm for a matrix,101010For a matrix $A$ the
operator norm is defined as $\sup_{u,||u||=1}||Au||$. and

$\displaystyle\Sigma_{\eta}$
$\displaystyle=\frac{1}{T_{0}+T_{1}}\mathbb{E}\Big{[}\tilde{\eta}\tilde{\eta}^{\top}\Big{|}(\tilde{\Gamma}_{i})_{i\geq
1},N_{0}=N,T_{0}=T\Big{]},\quad\mu^{2}=||\Sigma_{\eta}||_{op}.$

Under Assumptions 1, 2, 3, 4

$\displaystyle\Sigma_{\eta}^{(i,j)}=\begin{cases}\Big{(}\tilde{\Gamma}_{i}+\gamma_{i}\Big{)}^{\top}\mathrm{Var}(\tilde{\Lambda}_{t})\Big{(}\tilde{\Gamma}_{i}+\gamma_{i}\Big{)}+\mathrm{Var}(\varepsilon_{i,t}),\quad&i=j\\\
\Big{(}\tilde{\Gamma}_{i}+\gamma_{i}\Big{)}^{\top}\mathrm{Var}(\tilde{\Lambda}_{t})\Big{(}\tilde{\Gamma}_{j}+\gamma_{j}\Big{)}\quad&\text{otherwise}.\end{cases}$
(14)

It follows that conditional the treated units $N_{0}=N$, $\mu^{2}$ measures
the amount of endogeneity of $\tilde{\Gamma}_{i}$. Specifically, let

$\displaystyle A$
$\displaystyle\in\mathbb{R}^{(N_{0}-1)\times(T_{0}+T_{1})},\quad
A_{i,t}=\begin{cases}\lambda_{t}^{\top}\Gamma_{i}+\iota_{0,t}&\text{ if
}t<T_{0}\\\ -\lambda_{t}^{\top}\Gamma_{i}-\iota_{0,t}&\text{
otherwise}.\end{cases},$ $\displaystyle\tilde{\eta}$
$\displaystyle\in\mathbb{R}^{(N_{0}-1)\times(T_{0}+T_{1})},\quad\tilde{\eta}_{i,t}=\begin{cases}\eta_{i,t},&\text{
if }t<T_{0}\\\ -\eta_{i,t},&\text{ otherwise}.\end{cases}$

Here the matrix $A$ depends on the endogenous factors; $\tilde{\eta}$ is the
noise matrix for the control units. Define

$\theta_{h}=(w_{h},v_{h})\text{ and
}\theta_{h}^{\star}=(w_{h}^{\star},v_{h}^{\star}).$

In Appendix B, using rate-properties of error in variable models in Hirshberg
(2021) applied to our model of confounding, we show that the convergence rate
depends on three main components:

$\frac{\mathrm{rank}(A)\mu^{2}||\theta^{\star}||_{2}^{2}}{N_{0}},\quad\mu^{2}||\theta^{\star}||_{2}\log(T_{0}+T_{1})N_{0}^{-1/2},\quad\frac{\mu^{2}\log(T_{0}+T_{1})}{N_{0}}.$

Importantly, here, the rank of the matrix $A$ only depends on the non-zero
number of endogenous factors $\lambda_{t}$, i.e., $||\lambda_{t}||_{0}$. In
the presence of few endogenous factors, the rank of $A$ is small, even in
settings where $\tilde{\Lambda}_{t}$ is high-dimensional. The component
$\mu^{2}$ depends on the degree of endogeneity of the loadings (unit-level
confounders). The rate of convergence of $\mu^{2}$ is of order $\sqrt{N_{0}}$
from standard properties of matrix concentration inequalities (Van Handel,
2017) for exogenous $\tilde{\Gamma}_{i}$ and slower otherwise. The component
$||\theta_{h}^{\star}||$ instead converges to zero as
$T_{1}\rightarrow\infty$. The error does not necessarily converge to zero for
arbitrarly endogenous loadings $\tilde{\Gamma}_{i}$. It converges to zero
under restrictions on $\mu^{2}$ (e.g., only few loadings $\tilde{\Gamma}_{i}$
are endogenous and the remaining ones are exogenous). The same results holds
for vertical weights once we exchange the role of the factors and loadings.

This result illustrates the benefits of the Synthetic control methods in the
presence of high-rank factor models.

######  Remark 4 (Balancing).

We can gain further intuition if we interpret the weights’ estimator in
Section 5.1 as the dual of a balancing problem. For given constraint
$\nu=o(1)$ (as a function of $N,T$) we can formulate the dual of Equation (9)
as

$\displaystyle\min_{w_{h},v_{h},\beta}||w_{h}||^{2}+||v_{h}||_{2}^{2},\quad\text{
s.t. }$ $\displaystyle\underbrace{\frac{1}{N}\sum_{i<N}\Big{(}\sum_{t\geq
T_{0}}v_{h,t}Y_{i,t}-\sum_{s>T}w_{h,s}Y_{i,s}+\beta\Big{)}^{2}}_{:=\frac{1}{N}||(A+\tilde{\eta})\theta_{h}+\beta||_{2}^{2}}\leq\nu,\quad(w_{h},v_{h})\geq
0,$ $\displaystyle||w_{h}||_{\infty}\leq T_{0}^{-2/3},||v_{h}||_{\infty}\leq
T_{1}^{-2/3},1^{\top}w_{h}=1^{\top}v_{h}=1.$

The dual formulation clarifies the role of the constraint on the weights
$w_{h}$: The constraint on the weights norm guarantees the variance of the
error component $\tilde{\eta}\theta_{h}$ converges to zero (its variance is of
order $||\theta_{h}||^{2}$). Intuitively, the Synthetic Control method
averages over the component $\tilde{\eta}$ to obtain consistent weight’s
estimators without estimating the factors directly.111111It is possible to
consider alternative balancing estimators. For example, we might balance the
means of treated and control units by imposing
$\Big{|}\frac{1}{N}\sum_{i<N}\sum_{t\geq
T}v_{h,t}Y_{i,t}-\frac{1}{N}\sum_{i<N}\sum_{s>T}w_{h,s}Y_{i,s}\Big{|}\leq\nu^{\prime},$
for some constraint $\nu^{\prime}=o(1)$. Imposing such balancing restriction
assumes that we can find a set of weights $v_{h}^{\star},v_{h}^{\star}$ such
that $\sum_{t<T}v_{h,t}^{\star}\lambda_{t}=\sum_{t\geq
T}w_{h,t}^{\star}\lambda_{t}$ and
$\sum_{t<T}v_{h,t}^{\star}\iota_{1,t}=\sum_{t\geq
T}w_{h,t}^{\star}\iota_{1,t}$, i.e., we can also match fixed effects before
and after the treatment. ∎

###  5.2 Proximal methods for weights estimation

It is possible to use proximal methods to estimate the weights in the spirit
of Shi et al. (2021), Imbens et al. (2021). Consider estimating synthetic
control weights first. Suppose we can find variables $Z_{t}$ independent of
$\varepsilon_{i,t}$ such that for all $w_{v},v_{v}$

$\mathbb{E}\Big{[}Z_{t}\Big{(}\sum_{j<N}w_{v,j}\tilde{\Gamma}_{j}^{\top}-\sum_{i\geq
N}v_{v,i}\tilde{\Gamma}_{i}\Big{)}^{\top}\tilde{\Lambda}_{t}\Big{|}N_{0}=N,T_{0}=T,\tilde{\Lambda}_{t}\Big{]}=0,\quad\forall
t\leq T_{0}.$ (15)

Assuming that the number of such variables $Z_{t}$ is sufficiently larger than
the number of parameters (weights) to be estimated, we can use such moment
restrictions to estimate the weights. The main advantage of the proximal
method is that it does not impose restrictions on the distribution of the
endogenous factors $\tilde{\Lambda}_{t}$. However, it requires finding a set
of proximal variables that satisfy Equation (15). A similar approach follows
for horizontal weights, where we should find a (recenter) instrument $X_{i}$
such that

$\mathbb{E}\Big{[}X_{i}\Big{(}\sum_{t<T}w_{h,t}\tilde{\Lambda}_{t}^{\top}-\sum_{s\geq
T}v_{h,s}\tilde{\Lambda}_{s}\Big{)}^{\top}\tilde{\Gamma}_{i}\Big{|}N_{0}=N,T_{0}=T,\tilde{\Gamma}_{i}\Big{]}=0,\quad\forall
i\leq N_{0}.$

###  5.3 Factor models estimators

We conclude this section with a discussion on factor models. We write our
model as (for $\Gamma_{i}=\tilde{\Gamma}_{i}+\gamma_{i}$)

$Y_{i,t}(0)=\lambda_{t}^{\top}\Gamma_{i}+\underbrace{\tilde{\Lambda}_{t}^{\top}\Gamma_{i}+\varepsilon_{i,t}}_{=\eta_{i,t}},$
(16)

where $\lambda_{t}^{\top}\Gamma_{i}$ is low rank when $||\lambda_{t}||_{0}$ is
finite, and $\tilde{\Lambda}_{t}^{\top}\Gamma_{i}$ can be an high rank
component.

Our discussion above suggests that horizontal regression (and similarly
vertical regression after exchanging the factors with the loadings) leverage
the assumption that endogenous factors $\lambda_{t}$ are low dimensional
(e.g., $||\lambda_{t}||_{0}$ is uniformly bounded).

It is interesting to contrast the convergence rate of the weights of Synthetic
controls (described in Section 5.1 and formally presented in Proposition B.1)
with those that we would obtain using standard methods for estimating factor
models. Two main differences from properties of least squares estimators as in
Bai (2009) is that Synthetic Control does not require a (i) low-rank
assumption on the factors, but only a low-rank assumption on the endogenous
factors; (ii) it does not require to specify or estimate the number of
(endogenous) factors. For example, if we consider a factor model where the
factors are aggregate sectoral shocks and the loadings are sectoral weights, a
sparsity assumption on loadings implies that the policy implementation may
only depend on a subset of sectoral weights.

More closely related to the results in Proposition B.1, Moon and Weidner
(2017) show that conditions for least squares factor model can be expressed as
a function of the operator norm ($||\eta||_{op}$). The main distinction from
Proposition B.1, however, is that the number of factors (the degree of
sparsity of $\lambda_{t}$ in our framework) must be known for Moon and Weidner
(2017)’s results to hold. With an unknown number of factors (as in the case of
Proposition B.1), stronger conditions on the error term $\eta$, such as rank
restrictions, are imposed (Moon and Weidner, 2015).

The model in Equation (16) also connects to the approximate factor model
discussed in Chamberlain and Rothschild (1982). The main distinction, however,
is that low dimensional factor structure $\lambda_{t}^{\top}\gamma_{i}$ cannot
be necessarily separated by the (exogenous and high dimensional) structure
$\tilde{\Gamma}_{i}^{\top}\tilde{\Lambda}_{t}$, different from Chamberlain and
Rothschild (1982). This is an important distinction also from work in the
literature of causal inference with panel data as in Athey et al. (2021) (see,
e.g., Section 8.2).

Interestingly, Equation (16) may justify alternative estimators if additional
conditions are imposed on the loadings and factors. For example, under low
rank $\lambda_{t}^{\top}\Gamma_{i}$ and sparse
$\tilde{\Lambda}_{t}^{\top}\Gamma_{i}$ we could use estimators in Candès et
al. (2011), and for bounded eigenvalues of
$\tilde{\Lambda}_{t}^{\top}\Gamma_{i}$ as in Bai and Ng (2019).

Finally, Table 3 presents numerical studies that support the benefit of
Synthetic control methods over least square regression.

Table 3: Illustrative example where we report the mean-squared error using data from CPS on state earnings (Arkhangelsky et al., 2021), after removing the treated units. Here, $T_{0}=30,T_{1}=10,N=42$. We simulate an environment where either two or three units (originally under control in the original study) are exposed to treatment, with treatment effects equal to zero. We compute the root-mean-squared error of the estimated average treatment effect for each estimator, after averaging the mean-squared error averaged over all pairs or triads of placebo treated units in the sample. Here, vertical denotes a vertical regression and similarly horizontal for the horizontal regression. PCA denotes a regression where we first estimate the principal components via PCA using all control units. We then run a regression for each placebo-treated unit on the principal components and compute the counterfactuals. The weights for Synthetic DiD, vertical and horizontal regression are all computed as in Section 5, with a small penalty $\lambda=0.01$ to guarantee stability, a constraint that sum to one, and constraints on the $l_{\infty}$-norm as in Section 5. For PCA, the number of factors is computed using the BIC.  Root-mean-squared-error | $N_{1}=3$ | $N_{1}=2$
---|---|---
Synthetic DiD | $0.020$ | $0.025$
Vertical | $0.020$ | $0.025$
Horizontal | $0.026$ | $0.031$
PCA | $0.674$ | $0.674$

##  6 Conclusion

This paper studies inference on treatment effects in panel data settings in
the presence of confounding. We model confounding through unobserved factors –
which might affect when the treatment occurs – and unobserved loadings – which
might affect which units receive the treatment. We illustrate the existence of
a trade-off between assuming no (high dimensional) confounding across units or
time and introducing notions of double robustness in this setting. We relate
notions of confounding to the source of randomness for confidence intervals.

This paper opens new questions on trade-offs between the choice of the
estimator and robustness to confounding. Different sources of confounding
justify different estimators, including Synthetic DiD, Synthetic Control,
factor models, or proxy variable methods. A comprehensive comparison of such
estimators remains an open question. Future research should also study trade-
offs between weak factors and the choice of the estimator. Synthetic control
methods implicitly leverage the low-rank representation of the confounders,
whereas the least-squares method estimates all such confounders. Finally, a
further avenue for future avenue of research is to study augmented inverse
probability weight estimators in the spirit of Robins et al. (1994) in
contexts with synthetic controls and high dimensional factor models.

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##  Appendix A Randomization inference with a single treated unit

In this section, we revisit placebo tests in Abadie et al. (2010) and
illustrate how such tests can be used for inference. Consider conducting
inference under the null hypothesis

$H_{0}:\tau=0,$

which corresponds to a sharp null hypothesis by the assumption of additive and
homogeneous treatment effects. We construct a set of “placebo” treated units,
where we pool with the treated unit few control units “as if” they were
treated.121212The larger the set of placebo-treated units, the better the
asymptotic approximation through randomization inference at the expense of
reducing the size of the control pool. We study inference with few treated
units for Synthetic Controls. Algorithm 2 describes the procedure.

Algorithm 2 Randomization inference for Synthetic DiD with a single treated
unit

1:$N_{1}$ (number of placebo treated units), $N$ (number of control units
after excluding the placebo treated units)

2:Estimate a placebo treatment effect estimator for each unit $j\geq N$ as in
Equation (17), after excluding the other units in the same set $j\geq N$ when
estimating the placebo treatment effect.

3:Construct confidence intervals by permuting which unit receives the
treatment in the set $j\geq N$ and using the empirical distribution obtained
from such permutations (after excluding the treated unit).

4:return P-value for testing the null hypothesis of no treatment effect
$\tau=0$.

######  Assumption 8 (Identification condition for Placebo tests).

Suppose that

  * (A)

For all $i,t$,

$Y_{i,t}(0)\perp\Big{[}N_{0},T_{0}\Big{]}\Big{|}\tilde{\Lambda}_{t}$

  * (B)

There exist weights $w_{v}^{\star}(i)$, for units
$i\in\\{N,\cdots,N+N_{1}\\}$, for $N,N_{1}$ as in Algorithm 2, satisfying
$||w_{v}^{\star}(i)||_{\infty}=\mathcal{O}(N^{-2/3})$ and

$\Big{|}\Big{|}\gamma_{i}-\sum_{j\neq
i}w_{v,j}^{\star}(i)\gamma_{j}\Big{|}\Big{|}_{2}=0,\quad\sum_{j<N}w_{v,j}^{\star}(i)=1.$

Assumption 8 requires that the high dimensional loadings $\tilde{\Gamma}_{i}$
are exogenous and that we can match the endogenous loadings $\gamma_{i}$ of
each unit $i$ in the treated pool (where the treatment pool can be arbitrary).
For given unit $j$, we construct a “placebo” Synthetic Difference-in-
Differences by taking

$\displaystyle\hat{\tau}^{pl}(j;w_{v}^{\star},w_{h}^{\star},v_{h}^{\star})=\sum_{t\geq
T}v_{h,t}^{\star}\Big{\\{}Y_{j,t}-\sum_{s<T}w_{h,s}^{\star}Y_{j,s}-\sum_{i\neq
j}w_{v,i}^{\star}(j)Y_{i,t}+\sum_{i\neq
j,s<T}w_{v,i}^{\star}(j)w_{h,s}^{\star}Y_{i,s}\Big{\\}}.$ (17)

The estimator $\hat{\tau}^{pl}(j;w_{v}^{\star},w_{h}^{\star},v_{h}^{\star})$
uses the same horizontal weights for each placebo units, and different
vertical weights for each unit $j$.

######  Theorem A.1.

Suppose that Assumptions 1, 2, 3, 8 hold. Let $\delta_{h}=\mathbf{1}$. Then
under the null hypothesis $H_{0}:\tau=0$, for $N_{1}$ fixed, $T_{1}\geq
1,x\in\mathbb{R}$, as $T,N\rightarrow\infty$, with $T_{1}/N^{1/3}=o(1)$

$\displaystyle\Big{|}P\Big{(}\frac{\hat{\tau}^{pl}(n;w_{v}^{\star},w_{h}^{\star},v_{h}^{\star})}{||v_{h}^{\star}||_{2}}\leq
x\Big{|}(\tilde{\Lambda}_{t})_{t\geq 1},T_{0}=T,N_{0}=N\Big{)}$
$\displaystyle\quad\quad\quad-P\Big{(}\frac{\hat{\tau}^{pl}(N;w_{v}^{\star},w_{h}^{\star},v_{h}^{\star})}{||v_{h}^{\star}||_{2}}\leq
x\Big{|}(\tilde{\Lambda}_{t})_{t\geq
1},T_{0}=T,N_{0}=N\Big{)}\Big{|}\rightarrow 0,$

with
$\\{\hat{\tau}^{pl}(n;w_{v}^{\star},w_{h}^{\star},v_{h}^{\star})/||v_{h}^{\star}||_{2}\\}_{n\geq
N}$ asymptotically independent across $n$, conditional on
$(\tilde{\Lambda}_{t})_{t\geq 1},T_{0}=T,N_{0}=N$.

###### Proof of Theorem A.1.

See Appendix C.5. ∎

As a direct corollary of results in Canay et al. (2017), we can use placebo
tests as in Abadie et al. (2010) to construct confidence intervals. Here, the
validity of the placebo tests relies on Assumption 8 which, together with
exogeneity of $\tilde{\Gamma}_{i}$ imposes a symmetry restriction on the units
used for the placebo test.131313See also Hahn and Shi (2017) for a discussion
on placebo tests. Results for placebo tests (studied in the different context
of conformal inference) are also discussed in Chernozhukov et al. (2021).

##  Appendix B Convergence rates of the weights

In the following proposition, we assume that the penalties for estimating the
weights are
$p_{h}=(\sigma_{h}^{2}-1)\zeta_{h},p_{h}^{\star}=\sigma_{h}^{2}\zeta_{h}$,
where
$\sigma_{h}^{2}=\frac{1}{N-1}\sum_{i=1}^{N-1}\Big{(}(\tilde{\Gamma}_{i}+\gamma_{i})\Sigma_{\tilde{\Lambda}}(\tilde{\Gamma}_{i}+\gamma_{i})\Big{)}+\mathrm{Var}(\varepsilon_{i,t})$.
Note that $\sigma_{h}^{2}$ does not need to be known by the researcher, since
the penalty multplies by an arbitrary parameter $\zeta_{h}$. Properties of
$\zeta_{h}$ affects the guarantees in the following proposition.

######  Proposition B.1 (Weights estimation error).

Suppose that Assumptions 1, 2, 3, 4 hold. Let
$\sigma_{h}^{2}>c\times\mathrm{rank}(A)/N_{0}$ for a finite constant $c$
independent of $\theta^{\star},N_{0},T_{0},N_{1},T_{1}$. Take any $k\geq 1$.
Then conditional on $N_{0}=N,T_{0}=T$ and $(\tilde{\Gamma}_{i})_{i\geq 1}$,

$\displaystyle\zeta_{h}^{1/2}||(\hat{\theta}_{h}-\theta_{h}^{\star})||_{2}\leq
c_{0}s,\quad||\hat{\beta}_{h}-\beta_{h}^{\star}+A(\hat{\theta}_{h}-\theta_{h}^{\star})||_{2}\leq
c_{0}^{\prime}\zeta_{h}N_{0}^{1/2}s$

with probability at least $1-c^{\prime}\exp(-c^{\prime\prime}u(k,s))$, for any
$s$ satisfying

$\displaystyle s^{2}$
$\displaystyle\geq\frac{c^{\prime}}{\zeta_{h}-c\times\mathrm{rank}(A)/N_{0}}\Big{[}\frac{k^{2}\mu^{2}\log(T_{0}+T_{1})}{N_{0}}+\frac{(k^{2}\mu^{2}\mathrm{rank}(A))||\theta^{\star}||_{2}^{2}}{N_{0}}+\frac{||\theta^{\star}||_{2}(k^{2}\mu^{2}\mathrm{rank}(A))^{1/2}}{N_{0}}$
$\displaystyle\quad\quad\quad+\frac{k\mu||A\theta_{h}^{\star}+\beta_{h}^{\star}||\log(T_{0}+T_{1})+\mu^{2}k||\theta^{\star}||_{2}N_{0}^{1/2}\log(T_{0}+T_{1})}{N_{0}}\Big{]}.$

where
$u(v,s)=\min\\{v^{2}\mu^{2}\log(T_{0}+T_{1})/s^{2},k^{2}\mathrm{rank}(A),N_{0}\\}$,
and $c^{\prime},c^{\prime\prime},c_{0},c_{0}^{\prime}<\infty$ are finite
constant independent of $\theta^{\star},N_{0},T_{0},N_{1},T_{1}$.

The proof follows verbatim from Theorem 1 in Hirshberg (2021), once, in our
framework, we condition on $\Gamma_{i}=\gamma_{i}+\tilde{\Gamma}_{i}$.
Proposition B.1 is stated conditional on $N_{0}=N,T_{0}=T$ and
$\tilde{\Gamma}_{i}$. The estimation error depends on the rank of the matrix
$A$ (i.e., the number of endogenous factors), but not on the rank of
$\Gamma_{i}^{\top}\tilde{\Lambda}_{t}$.

##  Appendix C Proofs

Let $\Gamma_{i}=\tilde{\Gamma}_{i}+\gamma_{i}$,
$\Lambda_{t}=\tilde{\Lambda}_{t}+\lambda_{t}$. Recall that
$(\gamma_{i})_{i\geq 1},(\lambda_{t})_{t\geq 1}$ are deterministic.

###  C.1 Proof of Proposition 3.1

Assumption 1 guarantees that the potential outcome under control follows a
factor model. Assumption 4 guarantees that we can match $\sum_{t\geq
T}v_{h,t}\lambda_{t}=\sum_{s<T}w_{h,s}\lambda_{s}$ and that
$\mathbb{E}[\tilde{\Lambda}_{t}|T_{0}=T]=\mathbb{E}[\tilde{\Lambda}_{t}]$.
Assumption 2 guarantees that $\tilde{\Lambda}_{t}$ is independent of $N_{0}$.
The restriction that the weights sum to one in Assumption 4 guarantees that
$\sum_{t\geq T}v_{h,t}\iota_{0,i}=\sum_{s<T}w_{h,s}\iota_{0,i}$ (the weighted
combination of unit fixed effects) is the same on the right and left-hand side
of Equation (3). Proposition 3.1 holds for
$\beta_{0,h}(w_{h},v_{h})=\sum_{t\geq
T}v_{h,t}\iota_{1,t}-\sum_{s<T}w_{h,s}\iota_{1,s}$.

###  C.2 Proof of Proposition 3.3

#### Error for unit $(N,T$)

We first focus on unit $(N,T)$ whereas the reasoning applies to all other
units $i\geq N,t\geq T$. Define

$\displaystyle\bar{\Lambda}_{T}(w_{v})=\sum_{s<T}w_{v,j}\mathbb{E}\Big{[}\Lambda_{s}\Big{|}T_{0}=T,N_{0}=N,\Lambda_{T}\Big{]},\bar{\Gamma}_{N}(w_{h})=\sum_{j<N}w_{v,j}\mathbb{E}\Big{[}\Gamma_{j}\Big{|}\Gamma_{N},T_{0}=T,N_{0}=N\Big{]}.$

Because of Assumptions 1 and 2, we can write

$\displaystyle\sum_{s<T}w_{h,s}\mathbb{E}\Big{[}Y_{N,s}\Big{|}T_{0}=T,N_{0}=N,\Gamma_{N},\Lambda_{T}\Big{]}$
$\displaystyle=\Gamma_{N}\sum_{s<T}w_{h,s}\mathbb{E}\Big{[}\Lambda_{s}|\Lambda_{T},T_{0}=T,N_{0}=N\Big{]}+\sum_{s<T}w_{h,s}\iota_{1,s}+\iota_{0,N}$
$\displaystyle=\Gamma_{N}\bar{\Lambda}_{T}(w_{h})+\sum_{s<T}w_{h,s}\iota_{1,s}+\iota_{0,N}$
$\displaystyle\sum_{j<N}w_{v,j}\mathbb{E}\Big{[}Y_{j,T}\Big{|}T_{0}=T,N_{0}=N,\Gamma_{N},\Lambda_{T}\Big{]}$
$\displaystyle=\Lambda_{T}\sum_{j<N}w_{v,j}\mathbb{E}\Big{[}\Gamma_{j}\Big{|}\Gamma_{N},T_{0}=T,N_{0}=N\Big{]}+\sum_{j<N}w_{v,j}\iota_{0,j}+\iota_{1,T}$
$\displaystyle=\Lambda_{T}\bar{\Gamma}_{N}(w_{v})+\sum_{j<N}w_{v,j}\iota_{0,j}+\iota_{1,T}.$

In addition,

$\displaystyle\sum_{s<T}\sum_{j<N}w_{s}w_{v,j}\mathbb{E}\Big{[}Y_{j,s}\Big{|}T_{0}=T,N_{0}=N,\Gamma_{N},\Lambda_{T}\Big{]}$
$\displaystyle=\sum_{j<N}w_{v,j}\mathbb{E}\Big{[}\Gamma_{j}\Big{|}\Gamma_{N},T_{0}=T,N_{0}=N\Big{]}\sum_{s<T}w_{h,s}\mathbb{E}\Big{[}\Lambda_{s}\Big{|}\Lambda_{T},T_{0}=T,N_{0}=N\Big{]}\quad(\because\text{
Assumption }\ref{ass:inda})$
$\displaystyle+\sum_{s<T}w_{h,s}\iota_{1,s}+\sum_{j<N}w_{v,j}\iota_{0,j}=\bar{\Gamma}_{N}(w_{v})\bar{\Lambda}_{T}(w_{h})+\sum_{s<T}w_{h,s}\iota_{1,s}+\sum_{j<N}w_{v,j}\iota_{0,j}.$

It follows that

$\displaystyle\sum_{s<T}w_{h,s}\mathbb{E}\Big{[}Y_{N,s}\Big{|}T_{0}=T,N_{0}=N,\Gamma_{N},\Lambda_{T}\Big{]}+\sum_{j<N}w_{v,j}\mathbb{E}\Big{[}Y_{j,T}\Big{|}T_{0}=T,N_{0}=N,\Gamma_{N},\Lambda_{T}\Big{]}$
$\displaystyle-\sum_{s<T}\sum_{j<N}w_{s}w_{v,j}\mathbb{E}\Big{[}Y_{j,s}\Big{|}T_{0}=T,N_{0}=N,\Gamma_{N},\Lambda_{T}\Big{]}-\Big{(}\Lambda_{T}\Gamma_{N}+\iota_{0,N}+\iota_{1,T}\Big{)}$
$\displaystyle=\Gamma_{N}\bar{\Lambda}_{T}(w_{h})+\bar{\Gamma}_{N}(w_{v})\Lambda_{T}-\bar{\Gamma}_{N}(w_{v})\bar{\Lambda}_{T}(w_{h})-\Lambda_{T}\Gamma_{N}$
$\displaystyle=(\Gamma_{N}-\bar{\Gamma}_{N}(w_{v}))(\bar{\Lambda}_{T}(w_{h})-\Lambda_{T}).$

The equation above directly extends to any unit $(i,t),i\geq N,t\geq T$. The
final result follows by first taking expectations of
$(\Gamma_{i}-\bar{\Gamma}_{i}(w_{v}))(\bar{\Lambda}_{i}(w_{h})-\Lambda_{i}),i\geq
N,t\geq T$ over $\tilde{\Gamma}_{i},\tilde{\Lambda}_{t}$ and using Assumption
2; and then averaging over treated units and treatment periods, with
corresponding weights $v_{v},v_{h}$.

###  C.3 Proof of Theorem 4.1

We break the proof into multiple steps, where we define
$\Gamma_{i}=\tilde{\Gamma}_{i}+\gamma_{i}$,
$\sigma_{\varepsilon}^{2}=\mathbb{E}[\varepsilon_{i,t}^{2}]$. Recall that
$\mathbb{E}\Big{[}\hat{\tau}^{h}(w_{h}^{\star},v_{h}^{\star},\beta_{h}^{\star},q_{h})-\tau\Big{|}T_{0}=T,N_{0}=N,(\Gamma_{i})_{i\geq
1}\Big{]}=0$ by Proposition 3.4.

#### Order of convergence for post-treatment period

First, we claim that

$\sum_{s<T}w_{n,s}^{\star}\Big{\\{}Y_{n,s}-\mathbb{E}\left[Y_{n,s}|N_{0}=N,T_{0}=T,(\Gamma_{i})_{i\geq
1}\right]\Big{\\}}=O_{p}(||w_{h}^{\star}||_{2})=O_{p}(T_{0}^{-1/6}).$ (18)

To show this claim we use Assumption 1 and Assumption 6 (A). First, we write

$\sum_{s<T}w_{n,s}^{\star}\Big{\\{}Y_{n,s}-\mathbb{E}\left[Y_{n,s}|N_{0}=N,T_{0}=T,(\Gamma_{i})_{i\geq
1}\right]\Big{\\}}=\Gamma_{n}^{\top}\sum_{s<T}w_{n,s}^{\star}\tilde{\Lambda}_{s}+\sum_{s<T}w_{n,s}^{\star}\varepsilon_{n,s}.$

By Assumptions 2, 3

$\displaystyle\mathbb{V}\left(\Gamma_{n}^{\top}\sum_{s<T}w_{n,s}^{\star}\tilde{\Lambda}_{s}+\sum_{s<T}w_{n,s}^{\star}\varepsilon_{n,s}\Big{|}N_{0}=N,T_{0}=T,(\Gamma_{i})_{i\geq
1}\right)$
$\displaystyle\leq||\Gamma_{n}||_{2}^{2}\mathbb{E}[||\tilde{\Lambda}_{t}||_{2}^{2}]||w_{h}^{\star}||_{2}^{2}+||w_{h}^{\star}||_{2}^{2}\sigma_{\varepsilon}^{2}.$

Therefore since $||w_{h}^{\star}||_{\infty}\leq T_{0}^{-2/3}$, and from
Assumption 3,

$\sum_{s<T}w_{n,s}^{\star}\Big{\\{}Y_{n,s}-\mathbb{E}\left[Y_{n,s}|N_{0}=N,T_{0}=T,(\Gamma_{i})_{i\geq
1}\right]\Big{\\}}=O_{p}(||w_{h}^{\star}||_{2})=O_{p}(T_{0}^{-1/6}).$

#### CLT for fixed unit $n$

We can write

$\displaystyle\sum_{t\geq T}v_{h,t}^{\star}Y_{n,t}-\sum_{t\geq
T}v_{h,t}^{\star}\mathbb{E}[Y_{n,t}|\Gamma_{n},N_{0}=N,T_{0}=T]=\Gamma_{n}^{\top}\sum_{t\geq
T}v_{h,t}^{\star}\tilde{\Lambda}_{t}+\sum_{t\geq
T}v_{h,t}^{\star}\varepsilon_{n,t}.$

We can use the Lyaponuv’s central limit theorem here. In particular, we have
under Assumption 2

$\displaystyle\mathbb{E}\Big{[}\Big{(}\Gamma_{n}^{\top}\sum_{t\geq
T}v_{h,t}^{\star}\tilde{\Lambda}_{t}+\sum_{t\geq
T}v_{h,t}^{\star}\varepsilon_{n,t}\Big{)}^{3}\Big{|}(\Gamma_{i})_{i\geq
1},N_{0}=N,T_{0}=T\Big{]}$
$\displaystyle=\mathbb{E}\Big{[}\Big{(}\Gamma_{n}^{\top}\sum_{t\geq
T}v_{h,t}^{\star}\tilde{\Lambda}_{t}\Big{)}^{3}+\Big{(}\sum_{t\geq
T}v_{h,t}^{\star}\varepsilon_{n,t}\Big{)}^{3}\Big{|}(\Gamma_{i})_{i\geq
1},N_{0}=N,T_{0}=T\Big{]}$ $\displaystyle=\sum_{t\geq
T}v_{h,t}^{\star,3}\mathbb{E}[(\Gamma_{n}^{\top}\tilde{\Lambda}_{t})^{3}|\Gamma_{n}]+\sum_{t\geq
T}v_{h,t}^{\star,3}\mathbb{E}[\varepsilon_{i,t}^{3}]\quad(\because\text{
Assumption }\ref{ass:unc1a})$
$\displaystyle=\mathcal{O}\Big{(}||v_{h}^{\star}||_{\infty}||v_{h}^{\star}||_{2}^{2}\Big{)}\quad(\because\text{
Assumption }\ref{ass:iid})$

Also, by Assumption 2

$\mathrm{Var}\left(\Gamma_{n}\sum_{t\geq
T}v_{h,t}\tilde{\Lambda}_{t}+\sum_{t\geq
T}v_{h,t}\varepsilon_{n,t}\Big{|}(\Gamma_{i})_{i\geq
1},T_{0}=T,N_{0}=N\right)\geq||v_{h}^{\star}||_{2}^{2}\sigma_{\varepsilon}^{2}.$

It follows

$\displaystyle\frac{\mathbb{E}\Big{[}\Big{(}\Gamma_{n}\sum_{t\geq
T}v_{h,t}\tilde{\Lambda}_{t}+\sum_{t\geq
T}v_{h,t}\varepsilon_{n,t}\Big{)}^{3}\Big{|}(\Gamma_{i})_{i\geq
1},N_{0}=N,T_{0}=T\Big{]}}{\mathrm{Var}\left(\Gamma_{n}\sum_{t\geq
T}v_{h,t}\tilde{\Lambda}_{t}+\sum_{t\geq
T}v_{h,t}\varepsilon_{n,t}\Big{|}(\Gamma_{i})_{i\geq
1},T_{0}=T,N_{0}=N\right)^{3/2}}=\mathcal{O}(T_{1}^{-2/3}||v_{h}^{\star}||_{2}^{-1}).$

Because $||v_{h}^{\star}||_{2}\geq 1/\sqrt{T}_{1}$ for any
$1^{\top}v_{h}^{\star}=1$, by Lyapunov’s central limit theorem for any $n$,

$\frac{1}{\sqrt{||v_{h}^{\star}||_{2}^{2}\Gamma_{n}^{\top}\Sigma_{\tilde{\Lambda}}\Gamma_{n}+\sigma_{\varepsilon}^{2}||v_{h}^{\star}||_{2}^{2}}}\Big{(}\sum_{t\geq
T}v_{h,t}^{\star}Y_{n,t}-\sum_{t\geq
T}v_{h,t}^{\star}\mathbb{E}[Y_{n,t}|\Gamma_{n},N_{0}=N,T_{0}=T]\Big{)}\rightarrow\mathcal{N}(0,1),$

as $T_{1}\rightarrow\infty$.

#### Central limit theorem after summing over $N_{1}$ treated units

Consider now summing $N_{1}$ treated units for finite $N_{1}$, taking

$\sum_{n\geq N}q_{h,n}\Big{\\{}\sum_{t\geq
T}v_{h,t}^{\star}Y_{n,t}-\sum_{t\geq
T}v_{h,t}^{\star}\mathbb{E}[Y_{n,t}|\Gamma_{n},N_{0}=N,T_{0}=T]\Big{\\}}.$

It follows that as $T_{1}\rightarrow\infty$, for given $N_{1}$, we can show
using the same technique as above that where $\Gamma_{n}$ is replaced by
$\sum_{n\geq N}q_{h,n}\Gamma_{n}$ and $\varepsilon_{n,t}$ by $\sum_{n\geq
N}q_{h,n}\varepsilon_{n,t}$,

$\displaystyle\frac{\sum_{n\geq N}q_{h,n}\Big{\\{}\sum_{t\geq
T}v_{h,t}^{\star}Y_{n,t}-\sum_{t\geq
T}v_{h,t}^{\star}\mathbb{E}[Y_{n,t}|\Gamma_{n},N_{0}=N,T_{0}=T]\Big{\\}}}{\sqrt{||v_{h}^{\star}||_{2}^{2}\Big{(}\sum_{n\geq
N}q_{h,n}\Gamma_{n}\Big{)}^{\top}\Sigma_{\tilde{\Lambda}}\Big{(}\sum_{n\geq
N}q_{h,n}\Gamma_{n}\Big{)}+||v_{h}^{\star}||_{2}^{2}||q_{h}||_{2}^{2}\sigma_{\varepsilon}^{2}}}\rightarrow_{d}\mathcal{N}\Big{(}0,1\Big{)}.$
(19)

#### Lower bound on the variance component

Finally, we show that the component in Equation (18) converges at a faster
rate than the component in Equation (19). In particular,

$\displaystyle||v_{h}^{\star}||_{2}^{2}\Big{(}\sum_{n\geq
N}q_{h,n}\Gamma_{n}\Big{)}^{\top}\Sigma_{\tilde{\Lambda}}\Big{(}\sum_{n\geq
N}q_{h,n}\Gamma_{n}\Big{)}+||v_{h}^{\star}||_{2}^{2}||q_{h}||_{2}^{2}\sigma_{\varepsilon}^{2}\geq\frac{\sigma_{\varepsilon}^{2}}{N_{1}T_{1}}$

since the weights sum to one, and therefore $||v_{h}^{\star}||_{2}^{2}\geq
1/T_{1},||q_{h}||_{2}^{2}\geq 1/N_{1}$. As a result, because
$N_{1}T_{1}/T_{0}^{1/3}=o(1)$ by Assumption 6, and $T_{0}^{-1/3}$ is the order
of convergence of the variance of the component in Equation (18), the final
result directly follows.

###  C.4 Proof of Theorem 4.2

We study asymptotic normality if either Assumption 4 or Assumption 5 hold.
Suppose Assumption 4 holds first, and therefore we condition on
$(\Gamma_{i})_{i\geq 1}$, with $\Gamma_{i}=\tilde{\Gamma}_{i}+\gamma_{i}$.

#### Preliminaries when Assumption 4 (and we can condition on $\Gamma_{i}$)

Under Assumption 4, we have

$\displaystyle\mathbb{E}\Big{[}\hat{\tau}(w_{h}^{\star},w_{v}^{\star},v_{h}^{\star},v_{v}^{\star})-\tau\Big{|}N_{0}=N,T_{0}=T,(\Gamma_{i})_{i\geq
1}\Big{]}=0.$

We now decompose
$\hat{\tau}(w_{h}^{\star},w_{v}^{\star},v_{h}^{\star},v_{v}^{\star})$ in
different components and study their convergence rates.

#### Asymptotically negligible components of
$\hat{\tau}(w_{h}^{\star},w_{v}^{\star},v_{h}^{\star},v_{v}^{\star})$

First we study two components of
$\hat{\tau}(w_{h}^{\star},w_{v}^{\star},v_{h}^{\star},v_{v}^{\star})$ and show
that their convergence rate is of order $\mathcal{O}_{p}(T_{0}^{-1/6})$. The
first component is

$\displaystyle\sum_{i<N}\sum_{s<T}w_{v,i}^{\star}w_{h,s}^{\star}\Big{\\{}Y_{i,s}-\mathbb{E}\Big{[}Y_{i,s}|N_{0}=N,T_{0}=T,(\Gamma_{i})_{i\geq
1}\Big{]}\Big{\\}}$
$\displaystyle=\sum_{i<N}w_{v,i}^{\star}\Gamma_{i}\sum_{s<T}w_{h,s}^{\star}\tilde{\Lambda}_{s}+\sum_{i<N}\sum_{s<T}w_{v,i}^{\star}w_{h,s}^{\star}\varepsilon_{i,s}.$

We have from Assumption 3

$\displaystyle\mathrm{Var}\left(\sum_{i<N}w_{v,i}^{\star}\Gamma_{i}\sum_{s<T}w_{h,s}^{\star}\tilde{\Lambda}_{s}+\sum_{i<N}\sum_{s<T}w_{v,i}^{\star}w_{h,s}^{\star}\varepsilon_{i,s}\Big{|}T_{0}=T,N_{0}=N,(\Gamma_{i})_{i\geq
1}\right)$
$\displaystyle\leq\sigma_{\varepsilon}^{2}||w_{v}^{\star}||_{2}^{2}||w_{h}^{\star}||_{2}^{2}+||w_{h}^{\star}||_{2}^{2}\mathcal{O}(1)=\mathcal{O}(||w_{h}^{\star}||_{2}^{2}).$

Because $||w_{h}^{\star}||_{2}^{2}\leq T^{-1/3}$, it follows that

$\sum_{i<N}\sum_{s<T}w_{v,i}^{\star}w_{h,s}^{\star}\Big{\\{}Y_{i,s}-\mathbb{E}\Big{[}Y_{i,s}|N_{0}=N,T_{0}=T,(\Gamma_{i})_{i\geq
1}\Big{]}\Big{\\}}=\mathcal{O}_{p}(T^{-1/6}).$

The second component is

$\sum_{s<T}w_{h,s}^{\star}\Big{\\{}Y_{n,s}-\mathbb{E}\Big{[}Y_{n,s}\Big{|}T_{0}=T,N_{0}=N,(\Gamma_{i})_{i\geq
1}\Big{]}\Big{\\}}.$

We write

$\displaystyle\mathrm{Var}\left(\sum_{s<T}w_{h,s}^{\star}Y_{n,s}\Big{|}(\Gamma_{i})_{i\geq
1},T=T_{0},N=N_{0}\right)=\mathcal{O}(||w_{h}^{\star}||_{2}^{2})=\mathcal{O}(T^{-1/3}).$

Therefore, it follows that

$\displaystyle\sum_{s<T}w_{h,s}^{\star}\Big{\\{}Y_{n,s}-\mathbb{E}\Big{[}Y_{n,s}|(\Gamma_{i})_{i\geq
1},T_{0}=T,N_{0}=N\Big{]}\Big{\\}}=\mathcal{O}_{p}(T^{-1/6}).$

#### Asymptotic normality for fixed $n$

Consider now, for given $n$,

$\displaystyle\sum_{t\geq
T}v_{v,t}^{\star}\Big{\\{}Y_{n,t}-\mathbb{E}[Y_{n,t}|(\Gamma_{i})_{i\geq
1},T_{0}=T,N_{0}=N]-\sum_{i<N}w_{v,i}^{\star}(Y_{i,t}-\mathbb{E}[Y_{i,s}|(\Gamma_{i})_{i\geq
1},N_{0}=N,T_{0}=T])\Big{\\}}$ (20)
$\displaystyle=\Big{(}\Gamma_{n}-\sum_{i<N}w_{v,i}^{\star}\Gamma_{i}\Big{)}\sum_{t\geq
T}v_{v,t}^{\star}\tilde{\Lambda}_{t}+\sum_{t\geq
T}v_{v,t}^{\star}\varepsilon_{n,t}.$

We can write