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Control VAR: a counterfactual based approach to inference in macroeconomics
\maketitle
\begin{abstract}
This paper addresses the challenges of giving a causal interpretation
to vector autoregressions (VARs). I show that under independence assumptions
VARs can identify average treatment effects, average causal responses,
or a mix of the two, depending on the distribution of the policy.
But what about situations in which the economist cannot rely on independence
assumptions? I propose an alternative method, defined as control-VAR,
which uses control variables to estimate causal effects. Control-VAR
can estimate average treatment effects on the treated for dummy policies
or average causal responses over time for continuous policies.
The advantages of control-based approaches are demonstrated by examining
the impact of natural disasters on the US economy, using Germany as
a control. Contrary to previous literature, the results indicate that
natural disasters have a negative economic impact without any cyclical
positive effect. These findings suggest that control-VARs provide
a viable alternative to strict independence assumptions, offering
more credible causal estimates and significant implications for policy
design in response to natural disasters.\\
\indent{\textbf{JEL Classification:}} C32, E6, Q54.\\
\noindent{\textbf{Keywords:}} Vector autoregressive models, cointegration, identification, disaster shocks.
\end{abstract}
\section{Introduction}
What would have happened to inflation had central banks around the
world not introduced unconventional monetary policy on the offset
of the Great Financial Crisis? Is monetary policy effective in controlling
inflation? What is the impact of monetary policy on the economy?
While some of these questions may appear to be too tough to tackle,
economists have them frequently in mind, and find themselves at loss
as to what methods to use to answer them. Long run, short run, sign
restrictions in VARs, LPs, or linear regressions are just some of
the hundreds tools specifically developed to address these issues.
Yet, economists find themselves in front of the difficult reality
of being unable to provide a proper causal answer to their questions.
A lot of such issues would naturally involve the formulation of a
counterfactual. When we think about the effectiveness of unconventional
monetary policy, we have in mind a counterfactual world in which the
central banks could have preferred inaction over unconventional monetary
policy in the face of a looming disinflationary pressure. When we
think about the effects of natural disasters, we have in mind a counterfactual world in which no disaster happened. Yet, we rarely
make the case that we are truly identifying the effect of unconventional
monetary policy over inaction; or the effect of the disaster.
For economists, talking about causality in macroeconomics has for a long time been
a forbidden topic.
This is an issue that exists almost exclusively in social sciences.
The lack of the possibilities of running experiments leaves economists
with the goal of making informed guesses. \citet{Nakamura2018a}
make the comparison of economists and weather forecasters, painting
a gloomy picture of economic science. They, however, express
optimism towards the direction social sciences have been aiming for.
This could be because, recently, much of the literature has focused
on the development of tools that would allow economists to make truly
causal claims that involve the formulation of a counterfactual in
models akin to the one of \citet{Rubin1980} for tools such as VARs
or Local Projections, with a particularly relevant progress coming
from the works of \citet{bojinovshephard2019}, \citet{RambachanSheppard2021}, and \citet{dube2023local}.
Another stream of literature has instead taken the road of developing
methods for the definition of clear counterfactuals by making use
of definitions of optimal policies (\citet{mckay2023can}). Both
these literatures have, at heart, the goal of clearly defining the
effects of macroeconomic policies as the difference between two alternative
assignments: one in which a causing variable is zero, one in which
it is one.
This paper has the goal of providing economists with additional tools to make truly causal claims. Identification in macroeconomics
frequently uses Impulse Response Functions (IRF) to analyze short
and long-run effects of changes in a policy variable on an outcome
variable. This is a particularly needed feature when it comes to data
with sufficiently large time components, since having an idea of possibly
long-term effects of a change in a policy can result in valuable information.
Suppose the economist is trying to identify the effect of natural disasters on the US economy. One commonly utilised approach would be the one proposed by \citet{Ludvigson2021}. They estimate a VAR that includes the cost of
natural disasters (CD), Industrial Production (IP), and Macroeconomic Uncertainty
(MU). They argue that a recursive identification scheme could be appropriate to identify the impact of natural disasters on the US economy.
From the point of view of the causal literature, this example could be the perfect one to test the theoretical framework of \citet{RambachanSheppard2021}. They show that if the natural disasters are (1) independent on the past of other policies, (2) the future of other policies, (3) other contemporaneous policies; (4) and are independent to the potential outcomes of industrial production and macroeconomic uncertainty;\footnote{Essentially, natural disasters do not happen in systematic coincidence with economic fluctuations.} VARs are causal. They could potentially identify the Average Treatment Effect (ATE): the average effect of a natural disaster hitting the US economy.
This paper therefore tries to understand what type of causal effects can be identified under the model of \citet{RambachanSheppard2021} under that specific example.
Several important issues arise. First, the majority of natural disasters in the United States happened
in August and September, which could violate the assumption of independence (4) if there are hidden vulnerabilities or strengths specific to that period of time of the US economy.\footnote{Everything that is not captured by seasonality could potentially introduce a selection bias because it is unobserved.} Second, because Industrial Production and Macroeconomic Uncertainty are non-stationary, a filter is required. This creates an issue because filters will contaminate the informativeness of the estimator, an issue also recognised by \citet{RambachanSheppard2021}, which call those effects F-ATE. Third, the distribution of the treatment variable - the cost of natural disaster - unevenly weights some disasters over others (see \citet{chavleishvili2025natural}). This last feature has been found to be particularly problematic by the broader causal literature. I show that, in this case, the effects
will be a mix of an ATE and an ACRT, similarly to \citet{callawaysantannabacon2021}.
This is particularly relevant since non-negative policies are often
the object of causal inference and the complexity of the interpretation
of a mix of two causal effects can be troublesome.
Perhaps the simplest case of a VAR that could claim causality ends up having features such as seasonality, filters, or discontinuities in the distribution which heavily impact its validity and interpretation. Essantially, independence assumptions are stronger than many of the identification schemes currently utlised by the literature, even if the link from estimator to estimand may not be as clear as the one they describe. \footnote{For instance, economists frequently use instruments - not recursive identification - to identify the effect of monetary policy because even if there is not a widely understood concept of SVAR-IV capturing a LATE, they know independence would not hold.}
As an alternative to such methods, I present a novel approach to estimating
causal effects that uses some control series that share a trend with
the outcomes to eliminate the long-run component from the inference
problem through cointegration, a system that will take the name of
CVAR (Control VAR). Such an idea can be implemented in two different
ways. The simplest one is to estimate a VAR on the difference of the
treated and control variable. The second approach is to simply estimate
a Vector Error Correction Model (VECM) that includes the control variables. Such estimation methods
will therefore be named \emph{simple difference }and \emph{VECM} respectively.
These approaches have several advantages compared to the wide range of independence assumptions previously described. First, they do not make
use of any independence or exogeneity assumption. Second, they do
not require any form of filtration\footnote{This advantage is particularly relevant if one considers that most
macroeconomic time series are non-stationary (see \citet{NelsonPlosser1982}).}. Third, they only require the user to discuss whether the assumptions
are satisfied, and have a known estimation that is readily available
in any statistical toolkit.
In the simple difference approach, the assumptions needed are fairly intuitive. Suppose we use German macroeconomic uncertainty as a counterfactual for the United States. Because the two series are cointegrated, they share common long-run trends and thus provide valuable low-frequency information to the researcher. The approach requires two conditions: first, that in the absence of a natural disaster, U.S. uncertainty would have followed the same low-frequency path as observed in Germany; and second, that in normal times, the two trends move together without diverging structurally. These assumptions are demanding, especially the first. For example, while Hurricane Katrina had devastating effects on U.S. uncertainty, Germany was not completely insulated from spillover effects on its own level of uncertainty.
For this reason, the paper takes a second approach to cases in which the simple difference assumptions may not be satisfied.
The VECM approach uses lags in place of the original variables. Essentially, say now that the researcher runs a VECM with uncertainty in the United States and in Germany. In the equation for macroeconomic uncertainty in the US at time $t$, german data is indexed at time $t-1$. In this case, the assumption required is that the common trend is sufficient to eliminate long run structural variations in the United States.
Under these assumptions, if the policy variable is binary, the CVAR identifies an Average Treatment effect on the Treated (ATT). If the policy is normally distributed it is possible to identify an ACR.
The IRFs estimated by CVAR may appear to be substantially different
from the IRFs estimated without controls. In the case of no controls,
the IRF may present long-term effects. In the case of the CVAR, the
IRF may converge to zero within short periods because the long run
component is eliminated by the control series (\citet{Johannsen1995}). This is exactly what this paper finds in its empirical application.
I show that the original system of \citet{Ludvigson2021} estimates the effect of a shock in disasters
to be negative on IP for two months and indefinitely positive after
that. The natural disaster shock impact is positive on MU for three
months and negative for the remaining two years.\footnote{This is an extension of the result of the original model which plots
logistic transformations of 6 horizon shocks.}
As an alternative, I estimate a CVAR through a VECM using
the same variables and adding German Industrial Production and Macroeconomic
Uncertainty as a control. I use German data because out of the countries
for which a MU series is available (see \citet{Meinen2017}), Germany
is the one that provides a higher rank statistic. To identify an average treatment effect on the treated, I shift from a continuous and
non-negative variable (cost of natural disasters) to a dummy equal
to one when the cost of such disasters is above the 95-\emph{th }quantile.
Therefore, the estimated effect is an ATT representing the difference
between large and non-existent or minor disasters. \footnote{Appendix \ref{subsec:Alternative-with-many-control} presents similar
results obtained using all the available countries from the \citet{Meinen2017}
dataset.}
The resulting Impulse Response Functions present a much shorter horizon
of relevance when compared to the results of the original paper. This
suggests that the control based approach may solve the classic issue
of possibly long-term effects of temporary shocks when the outcome
variables are non-stationary (\citet{StockWatson1990}).
The discussion will be organized as follows.
Section \ref{sec:Definition-of-causal} introduces the notation and
definitions required to discuss the nature of the causal effects in
the context of VAR. Section \ref{sec:Identification-using-VAR} focuses on the first contribution
of the paper, the causal interpretation of a VAR. Subsection \ref{subsec:Dummy-variable}
discuss the case of a dummy policy variable, a result drawn from \citet{RambachanSheppard2021}
but framed under my alternative causal notation. Subsection \ref{subsec:Normally-distributed-continuous}
discusses the case of a normally distributed policy variable. Subsection
\ref{subsec:Non-negative-continuous-policy} discusses the case of
a non-negative continuous policy variable. Section \ref{subsec:The-CVAR-approach} focuses on the second contribution
of the paper. It introduces the CVAR, and is divided between subsection
\ref{subsec:Estimation-through-simple}, which introduces the simple
difference approach, and subsection \ref{subsec:Estimation-through-VECM},
which discusses the VECM approach. Section \ref{sec:The-effects-of-of natural-disasters-on-Industrial-Production-and-Macroeconomic-Uncertainty}
discusses the empirical applications and gives a context for the results. Section \ref{sec:Conclusions-3} concludes.
\section{Definition of causal effects\label{sec:Definition-of-causal}}
\subsection{Notations and definitions\label{subsec:Notations-and-definitions}}
The economist is generally interested in estimating a dynamic causal
effect of assignments on outcomes in observational time series settings.
Every time period, a unit receives a vector of assignments, and their
outcomes can be observed. There exists a causal relation between the
assignments and the outcomes through a \emph{potential outcome process},
a stochastic process that generally defines the counterfactual assignment
paths. A \emph{dynamic causal effect} is typically described as the
difference of potential outcome processes along distinct assignment
paths at a specific moment in time.
The problem for the economist is related to the inherent impossibility
of observing both the potential outcomes under treatment and the potential
outcomes in the absence of a treatment at a given time. This problem
is usually dealt with by inferring a \emph{counterfactual }which represents
what would have been the value of the outcome variable, if the treatment
variable was zero when the treatment was one. Many ways of inferring
such values have been introduced across different scientific fields
and types of available data and experiments.
Time series methods have traditionally relied on the computation of
a counterfactual using the forecast of the outcome variable. This
is the case, among others, of intervention analysis (\citet{BoxTiao1975})
and Structural VARs (\citet{Sims1980}).
I will consider the following model of the outcome variable for the
identification of causal effects:
\[
Y_{j,t}(w_{j,k})=Y_{j,t}(W_{j,1:t-1},W_{j,1:k-1,t},w_{j,k},W_{j,k+1:K,t},W_{j,t+1:t+h}),
\]
is the potential outcome process of the outcome variable $Y_{t,j}$ under the assignment $w_{j,k}$, where $k=1,..,K$ is an index that indicates
multiple possible treatments and $j=1,..,J$ indicates the generic outcome variables. For example,
if the researcher is interested on the causal effect of hurricanes
and heat waves on GDP, inflation, and unemployment, $K=2$ and $J=3$. Therefore $W_{j,k,t}$ describes the amount of treatment
$k \in \{ 1,...,K \}$ received by series $j\in \{ 1,...,J \}$ at time $t \in \{ 1,...,T \}$, with $w_{j,k,t}$ denoting
one realization from the random variable.
Moreover, this definition highlights that the potential outcome depends
on the past of the policies, their future, and the contemporaneous
policies that are not the target of analysis.
With such notation, a dynamic causal effect can be defined as $Y_{j,t}(w_{j,k,1:t})-Y_{j,t}(w_{j,k,1:t}^{\prime})$.
Often Impulse Response Functions that derive from a VAR are interpreted
as $Y_{j,t}(w_{j,k,t})-Y_{j,t}(w_{j,k,t}^{\prime})$, which means that the
interest is not on the effect of receiving a subsequent set of treatments
$w_{j,k,1:t}$ versus no treatments $w_{j,k,1:t}^{\prime}$, but an artificial
type of counterfactual that captures the effect of a non time dependent
policy variable. For the sake of simplicity, $w_{j,k}$ will be considered a 1\% shock and $w^{\prime}_{j,k}$ will be considered a 0\% shock. This substantially simplifies the notation.
Only recently \citet{RambachanSheppard2021} highlighted some restrictive
conditions under which the impulse responses may be given a causal
interpretation. They discuss two cases: (i) $Y_{j,t}$ is stationary,
(ii) $Y_{j,t}$ is not stationary. According to their interpretation,
in the first case the economist may identify an ATE, defined as $\mathbb{E}[Y_{j,t}(1)-Y_{j,t}(0)]$,
if the sources of selection bias are independent on the treatment. This means that the treatments
are required to be independent of their history and their future,
on other contemporaneous treatments, and on the potential outcome
path of the outcome variable. In the case of non-stationary variables,
this needs to hold conditionally on the filtration, so that the causal
effect is defined as $\mathbb{E}[Y_{j,t}(1)-Y_{j,t}(0)|\mathcal{F}_{t-1}]$.
However, more causal effects could be imputed to Vector Autoregressions.
For example, a VAR identified using external instruments can be interpreted
as a LATE (\citet{StockWatson2018,Olea2021,Mertens2014}). \citet{pala2024pvarcontrol}
analyzes causality in panel vector autoregressions and shows that
their interpretation depends on the distribution of the policy variable
and the assumptions the economist may put on the data and the associated
narrative. In that case, however, the system assumes the existence
of a large unit component, which may not be available in the case
of macroeconomic data.
Hence, this paper takes two aims:
\begin{enumerate}
\item clarify the causal interpretation of VAR with macroeconomic data,
and highlight the different causal effects that can be identified
using VAR with different policy distributions under the same type
of independence assumptions as \citet{RambachanSheppard2021},
\item propose using controls to eliminate the long run component of time
series data through cointegration, a method named CVAR.
\end{enumerate}
\subsection{Causal effects\label{subsec:Causal-effects}}
The potential outcome for series $j$ at time $t$ can be written as
\[
Y_{j,t}\big(W_{k,1:J,1:T}\big),
\]
indicating a dependence on the treatments received by all series at all times, both past and future.
We restrict this dependence under the following assumptions:
\begin{assumption}[SUTVA]\label{ass:sutva}
The potential outcomes for series $j$ depend only on its own treatment path. That is,
\[
Y_{j,t}\big(W_{k,1:J,1:T}\big) = Y_{j,t}\big(W_{k,j,1:T}\big).
\]
\end{assumption}
See also \citet{bojinov2021panel}.
\begin{assumption}[Homogeneity]\label{ass:homogeneity}
The treatment paths are identical across units, so that
\[
Y_{j,t}\big(W_{k,j,1:T}\big) = Y_{j,t}\big(W_{k,1:T}\big).
\]
\end{assumption}
\begin{assumption}[Absence of Anticipation]\label{ass:noanticipation}
Future treatments do not influence present outcomes. Formally, the potential outcome at time $t$ depends only on the treatment path up to $t$:
\[
Y_{j,t}\big(W_{k,1:T}\big) = Y_{j,t}\big(W_{k,1:t}\big).
\]
\end{assumption}
Under the Rubin's causal framework, I define the following assumption, which binds the observed and potential outcomes.
\begin{assumption}
\label{assu:(realised-outcomes)}(Realized outcomes): The realized
outcomes are linked to the potential outcomes through the following
assignment process:
\[
Y_{j,t}^{obs}=\begin{cases}
Y_{j,t}(1) & \text{if }W_{j,k,t}=1\\
Y_{j,t}(0) & \text{if }W_{j,k,t}=0
\end{cases}
\]
\end{assumption}
This definition binds the observed outcome at time $t$ to depend
on the value of the policy variable. It means that if the policy variable
corresponds to a treatment status $1$, then we observe the potential
outcome of the variable $Y_{t}$ under the treatment status. If the
policy variable corresponds to a non-treatment, then the observed
outcome variable can be only observed under a non-treatment. A counterfactual
is therefore defined as what we do not observe, that is:
\begin{assumption}
\label{assu:(non-realised-outcomes):}(Non realized outcomes): The
non realized outcomes are linked to the potential outcomes through
the following assignment process
\[
Y_{j,t}^{nobs}=\begin{cases}
Y_{j,t}(0) & \text{if }W_{j,k,t}=1\\
Y_{j,t}(1) & \text{if }W_{j,k,t}=0
\end{cases}
\]
\end{assumption}
\begin{rem}
\label{rem:Counterfactuals-can-only}Counterfactuals can only be generated
my making assumptions about the non-observed treated unit. For example,
the DiD literature regularly assumes that the potential outcome of
the treated units under no treatment $\mathbb{E}[Y_{t}(0)|W_{j,k,t}=1]$
can be credibly formulated by using information from the control units.
\end{rem}
According to assumptions \ref{assu:(realised-outcomes)} and \ref{assu:(non-realised-outcomes):}
it is possible to define the causal effects. The definitions that
follow indicate the outcome variable in the first part of the parentheses,
the potential outcome in the second one, and the conditioning argument
in the third one.
An ATE represents the difference of the outcome variable under a treatment
$1$ and the absence of a treatment $0$.
\[
\text{ATE}_{j,k}(Y_{j}|1,0)=\mathbb{E}[Y_{j}(1)-Y_{j}(0)].
\]
An ATE therefore represents a measure that is close to the ideal of
a dynamic causal effect, since it captures the potential outcome value
under the treatment $1$ versus the treatment $0$.
For example, it could represent the effect of a natural disaster versus
no natural disaster.
An ATT is defined as the same difference, with the conditioning argument
that the times observed are only ones in which the treated unit is
subject to the treatment, i.e.
\[
\text{ATT}_{j,k}(Y_{j}|1,0|1,1)=\mathbb{E}[Y_{j}(1)|W_{j,k,t}=1]-\mathbb{E}[Y_{j}(0)|W_{j,k,t}=1].
\]
An ATT is generally the result of inference conducted using a direct
counterfactual, so that the treated units are compared to control
units. It may represent the effect of a natural disaster versus no
natural disaster, under a conditioning argument that depends on the
time extracted to be one in which there was a natural disaster.
I will also consider the ACR, Average Causal Response, which represent
the effect of marginal changes in the treatments on the outcome variable.
The concept of ACR was first discussed by \citet{AngristImbensACR1995},
which make the case for the advantages of a more comprehensive measure
of causal effects that may not be dummies. Hence, it considers the
effect of variations in the value of $W_{j,k,t}$ on the outcome variable
$Y_{j,t}$,
\[
\text{ACR}_{j,k}(Y_{j}|w_{j,k})=\mathbb{E}\left[\frac{\delta Y_{j}(w_{j,k})}{\delta w_{j,k}}\right],
\]
where $w_{j,k}$ is a value of a continuous $W_{j,k,t}$. Ideally, the researcher would be able to identify the impact of moving along different values of the treatment variable $W_{j,k,t}$ without assuming linearity. In practice, however, linearity is frequently a useful assumption in cases in which the data is not sufficiently dense around some target values. If linearity holds, the ACR is the most informative quantity the economist could capture.
One way to think about the ACR is akin to the most traditional definition of the linear regression
coefficient under the assumptions of exogeneity, and its derivative
form makes it possible to generate credible counterfactual comparisons.
For example, through the ACR, the economist could answer questions
regarding the difference between the effect of a natural disaster
that causes damages for 1000 billion dollars versus 100 billion dollars.
This is possible through the comparison of the predicted values of
the outcome variable under a plug-in quantity of the policy variable.
They also have a conditional counterpart, defined as ACRT.
\[
\text{ACRT}_{j,k}(Y_{j}|w_{j,k}|w_{j,k})=\mathbb{E}\left[\frac{\delta Y_{j}(w_{j,k})}{\delta w_{j,k}}|W_{j,k,t}=w_{j,k}\right].
\]
In this case, such quantity includes an additional conditioning argument
that is related to the time being extracted being equal to $w_{j,k}$. In this case, the ACRT implements a selection bias that depends on the treatment time. For example, if natural disasters are more frequent in some periods of the year, the ACRT indicates that the esitimates of the impact of natural disasters on the economy will be biased by the fact that natural disasters are concentrated in specific seasons.
There are conditions under which it is possible to move from the ACRT
to the ACR, and therefore pin the selection bias to zero. The most
common one is the independence of the treatment assignments to the
potential outcomes.
Finally, all such causal effects may have conditional counterparts
which may be useful in cases in which the outcome variable is non
stationary. Hence, the filtered causal effects are as follows.
\begin{enumerate}
\item $\text{F-ATE}_{j,k}(Y_{j}|1,0)=\mathbb{E}[Y_{j}(1)-Y_{j}(0)|\mathcal{F}_{t-1}]$,
\item $\text{F-ATT}_{j,k}(Y_{j}|1,0|1,1)=\mathbb{E}[Y_{j}(1)|W_{j,k,t}=1,\mathcal{F}_{t-1}]-\mathbb{E}[Y_{j}(0)|W_{j,k,t}=1,\mathcal{F}_{t-1}]$,
\item $\text{F-ACR}_{j,k}(Y_{j}|w_{j,k})=\mathbb{E}\left[\frac{\delta Y_{j}(w_{j,k})}{\delta w_{j,k}}|\mathcal{F}_{t-1}\right]$,
\item $\text{F-ACRT}_{j,k}(Y_{j}|w_{j,k}|w_{j,k})=\mathbb{E}\left[\frac{\delta Y_{j}(w_{j,k})}{\delta w_{j,k}}|W_{j,k,t}=w_{j,k},\mathcal{F}_{t-1}\right].$
\end{enumerate}
\begin{rem}
The literature is ambiguous about the interpretability of any filtered
treatment effect. In particular, the macroeconometrics traditional
approach has been to first difference or filter the series before
estimating a VAR to avoid violations of the Wold theorem. The appeal
to such procedures is evident. Making any type of causal claim while
using a non-stationary variable would be impossible due to the non-finiteness
of their moments and filtering and first differencing avoids the burden
of stationarity. However, it is important to recognize the limitations
of any kind of causal estimation procedure that results from a process
of filtration, since it adds an additional form of uncertainty.\footnote{The first uncertainty comes from the filtration procedure. The second
from the AR estimation. The third from the way the variance-covariance
matrix of the residuals is utilized to carry inference. Often standard
confidence intervals only embed the second type of uncertainty.}
\end{rem}
\section{Identification using VAR\label{sec:Identification-using-VAR}}
A VAR can be identified in several ways. In the case of non-stationarity,
the most commonly utilized procedure is including all the variables
in a first differenced vector of the type that follows.\footnote{The results of this section hold even in the case of a VECM estimated
in the original variables.}
\begin{equation}
\Delta X_{t}=(\Delta W_{1,t}^{\prime},..,\Delta W_{j,k,t}^{\prime},\Delta Y_{1,t}^{\prime},..,\Delta Y_{J,t}^{\prime})^{\prime}.\label{eq:DeltaXis:}
\end{equation}
The goal of the economist is to estimate a causal effect for each
combination of the policies with every outcome variable. However,
the general idea of VAR models is that there are autoregressive components
which may need to be modelled. Hence, the VAR equation is as follows.
\begin{equation}
\Delta X_{t}=\sum_{l=1}^{p}A_{l}\Delta X_{t-l}+\epsilon_{t}\label{eq:VAR}
\end{equation}
Here, $\epsilon_{t}=(\widetilde{W}_{1}^{\prime},..,\widetilde{W}_{k}^{\prime},\widetilde{Y}_{1}^{\prime},..,\widetilde{Y}_{J}^{\prime})^{\prime}$ is the vector of errors.
Normally the causal effects are represented as the contemporaneous
effect of each variable within $X_{t}$. This is represented through
the matrix $A_{0}$.
\begin{equation}
A_{0}\Delta X_{t}=\sum_{l=1}^{p}A_{l}^{\ast}\Delta X_{t-l}+\epsilon_{t}^{\ast}, \quad \epsilon_{t}\sim\mathcal{D}_{p}(0,\Omega). \label{eq:SVAR}
\end{equation}
Here $A_{l}^{\ast}=A_{0}A_{l}$ for all $l$, $\epsilon_{t}^{\ast}=A_{0}\epsilon_{t}$
and $\epsilon_{t}^{\ast}\sim\mathcal{D}_{p}^{\ast}(0,\Omega^{\ast}),\Omega^{\ast}=A_{0}^{*}\Omega A_{0}^{\ast\prime}$,
where $\mathcal{D}$ is a generic distribution (not necessarily normal).
\begin{example}
Consider the case of one outcome variable and one policy variable,
so that $j=1$ and $k=1$. In that case,
\[
\Omega=\left[\begin{array}{cc}
var(\widetilde{W}_{t}) & cov(\widetilde{W}_{t},\widetilde{Y}_{t})\\
cov(\widetilde{W}_{t},\widetilde{Y}_{t}) & var(\widetilde{Y}_{t})
\end{array}\right]
\]
is the variance-covariance matrix used for the causal estimation.
The associated Choleksy decomposition is
\[
chol(\Omega)=\left[\begin{array}{cc}
\sqrt{var(\widetilde{W}_{t})} & 0\\
\frac{cov(\widetilde{W}_{t},\widetilde{Y}_{t})}{\sqrt{var(\widetilde{W}_{t})}} & \sqrt{var(\widetilde{W}_{t})-(\frac{cov(\widetilde{W}_{t},\widetilde{Y}_{t})}{\sqrt{var(\widetilde{W}_{t})}})^{2}}
\end{array}\right]=\left[\begin{array}{cc}
o_{11} & 0\\
o_{12} & o_{22}
\end{array}\right].
\]
Inference is usually carried out by normalizing $chol(\Sigma_{E})$
through an array $(1/\sqrt{var(\widetilde{W}_{t})},0)^{\prime}$, which
makes the response to a unitary shock in $\widetilde{W}_{t}$ become
$(1,\gamma)^{\prime}$, where $\gamma=\frac{cov(\widetilde{W}_{t},\widetilde{Y}_{t})}{var(\widetilde{W}_{t})}$.
\end{example}
One important message can be drawn from the example. Inference is,
in general, conducted on $\widetilde{Y}_{t}$, and not directly on
$Y_{t}$. This is particularly relevant because an argument for making
use of identification through VAR is that, if the disturbances are
symmetrically distributed, the estimation error of the autoregressive
coefficients goes to zero. Such argument has been alternatively put
forth by \citet{bruggemann2016inference} in Theorem 2.1 or
\citet{LewisTimeVarying2021} in footnote 15. It has also created
some debates about whether GLS may be a better alternative than the
traditional 2 step procedures for the estimation of VAR since there
is no guarantee that the errors are going to be symmetric (see \citet{Kilian2017}).
\begin{rem}
An alternative to the Cholesky decomposition could be put forth in
the case of a not autocorrelated policy variable. In this case, an
alternative system of the type
\[
\Delta Y_{t}=\sum_{l=1}^{p}A_{l}\Delta Y_{t-l}+\epsilon_{t},
\]
where $\Delta Y_{t}=(\Delta Y_{1,t}^{\prime},..,\Delta Y_{J,t}^{\prime})$. Then, inference can be carried out by simply running
a set of linear regressions of the disturbances on the policy variables
$W_{j,k,t}$\footnote{Basically this system would consist on (1) estimating the VAR, (2)
running a set of $K\times J$ regressions, (3) plugging the results
in the covariance matrix and using them to generate the Impulse Response
Functions.}. While the following discussion will focus on the first case, this
second type of system could provide some advantages as it may be more
parsimonious in the number of estimated coefficients. Moreover, while
all the results that follow will be discussed in terms of the distribution
of the innovation of the policy variable, they can be easily extended
to be representative of the effects estimated using $W_{j,k,t}$, the
distribution of the original policy variable.
\end{rem}
\begin{rem}
While other procedures for the identification of causal effects, such
as short run, long run, and sign restrictions, have been proposed
in the macroeconometrics literature, I do not discuss the interpretation
of such methods since they often result in a system of partially identified
equations. This complicates the interpretation of the estimands and
analyzing the resulting set of impulse responses would require an
ad-hoc type of causal system that admits uncertainty over the assignment
process.
\end{rem}
The discussion that follows will focus on three different distributions
of the policy variable, and highlight the assumptions required to
conduct inference. Those cases have been selected because they may
represent likely scenarios in which to carry causal inference. Nevertheless,
from a purely mathematical perspective, any kind of feasible combination
of policy distribution/assumption can result in a different causal
estimand. For example, a policy that is distributed with a known distribution
that is not normal can still result in a causal estimand that includes
a weight times a derivative.
\subsection{Dummy variable \label{subsec:Dummy-variable}}
Consider the case of a simple linear
regression in which the independent (policy) variable is a dummy that
indicates the treatment status of the dependent (outcome) variable.
This is the case discussed by \citet{bojinovshephard2019,RambachanSheppard2021}.
It can easily be shown that $\gamma_{jk}$ estimates
effects that are equivalent to the difference in mean estimator between
$\widetilde{Y}_{j,t}(\widetilde{W}_{j,k,t}=1)$ and $\widetilde{Y}_{j,t}(\widetilde{W}_{j,k,t}=0)$, if the following assumption is satisfied.
\begin{assumption}
\label{assu:(Policy-is-a-dummy)}(Policy is a dummy variable) At each
time $t\geq 1$, either $\widetilde{W}_{j,k,t}=1$ or $\widetilde{W}_{j,k,t}=0$.
\end{assumption}
\begin{assumption}
\label{assu:(Exogeneity-of-treatments)}(Independence of treatments)
For each $k$ and $j$, the reduced form shock $\widetilde{W}_{j,k,t}$ is independent from
\end{assumption}
\begin{enumerate}
\item the other policies, i.e. $\widetilde{W}_{j,k,t}\perp\widetilde{W}_{j,1:k-1,t}$ and $\widetilde{W}_{j,k,t}\perp\widetilde{W}_{j,k+1:K,t}$
\item its past, $\widetilde{W}_{j,k,t}\perp\widetilde{W}_{j,k,1},..,\widetilde{W}_{j,k,t-1}$
\item its future, $\widetilde{W}_{j,k,t}\perp\widetilde{W}_{j,k,t+1},..,\widetilde{W}_{j,k,t}$
\item the potential outcome process, $\widetilde{W}_{j,k,t}\perp\widetilde{Y}_{j,t}(\widetilde{W}_{j,k,1:t})$
\end{enumerate}
Assumption \ref{assu:(Exogeneity-of-treatments)} states
that the assignment to the treatment must be independent of other
policies ($\widetilde{W}_{j,-k,t}$), the past of all the policies, the
future of all the other policies, and whether it happened because
it relates to a particular potential outcome of the outcome variable.
This last assumption is the reason why such an approach is almost
never used in macro economics. For example, monetary policy is especially
targeted to have an effect on the state of the economy by means
of the Taylor rule (\citet{Nakamura2018a}). The resulting ATE theorem
is:
\begin{thm}
\label{thm:(Identification-of-ATE)} (Identification of ATE) Under
assumptions \ref{assu:(Policy-is-a-dummy)} and \ref{assu:(Exogeneity-of-treatments)},
the Cholesky decomposition of a VAR identifies
\[
\gamma_{j,k}=\mathbb{E}[\widetilde{Y}_{j}(1)-\widetilde{Y}_{j}(0)]=\text{ATE}_{j,k}(\widetilde{Y}_{j}|1,0)
\]
\end{thm}
It should be noted that the ATE in this theorem is the target of a lot of the research conducted in economics. In particular, the claim in the original work of \citet{RambachanSheppard2021} is that the ATE is always the target of Structural Vector Autoregression. However, most macroeconomic shocks, such as fiscal or monetary policy, are not measured by dummy variables. For example, the first is measured in terms of government spending or taxes, and the second in terms of changes in interest rates. Neither is measured in terms of a dummy variable that credibly represents a 1\% shock versus a 0\% shock. Accordingly, subsection \ref{subsec:Normally-distributed-continuous} will discuss a more credible scenario in which the policy variable is continuously distributed.
In the case of non-stationary filtered outcome variable the estimated
effect becomes an $\text{F-ATE}_{j,k}(\widetilde{Y}_{j}|1,0)$.
\subsection{Normally distributed continuous variable\label{subsec:Normally-distributed-continuous}}
Often the researcher is interested in the effects of a policy that
is not a dummy variable that indicates treatment periods, and the
most common alternative is the one of the normality of the errors
distribution. I will show that the resulting estimand, in the case
of the same assumption as \ref{assu:(Exogeneity-of-treatments)},
is an ACR.
First, consider the following assumptions, which bind the distributions
of the policy innovations to be normal and the outcomes innovations
to be continuously differentiable.
\begin{assumption}
\label{assu:The-estimated-errors-are-normal} (Normality) The reduced form disturbances
are normally distributed around zero, so that, for every $t\geq 1$
and $k\geq 1$, $\widetilde{W}_{j,k,t}\sim\mathcal{N}(0,\sigma_{1}^{2})$.
\end{assumption}
\begin{assumption}
\label{assu:(Continuous-differentiability)-T}(Continuous differentiability)
The reduced form innovations of the outcome variable $Y_{j,t}$, for every $j\geq 1$
and every $t\geq 1$, are continuously differentiable with respect
to $W_{j,k,t}$ for any $k\geq 1$.
\end{assumption}
\begin{thm}
\label{thm:(Identification-of-ACRT)}(Identification of ACRT) Under
assumptions \ref{assu:The-estimated-errors-are-normal} and \ref{assu:(Continuous-differentiability)-T},
the Choleksy decomposition of a VAR identifies
\[
\gamma_{j,k}=\int q(w_{j,k})g_{j}^{\prime}(w_{j,k})dw_{j,k}=\frac{\delta\mathbb{E}[\widetilde{Y}_{j}(w_{j,k})|\widetilde{W}_{j,k,t}=w_{j,k}]}{\delta w_{j,k}}=\text{ACRT}_{j,k}(\widetilde{Y}_{j}|w_{j,k}|w_{j,k})
\]
where $q(w_{j,k})>0$ and $\int q(w_{j,k})dw_{j,k}=1$.
The weights are:
\[
\begin{aligned}q(w_{j,k})=\frac{1}{\sigma_{\widetilde{W}_{k}}^{2}}\int_{-\infty}^{\infty}(\mathbb{E}[\widetilde{W}_{k}]F_{\widetilde{W}_{k}}(w_{j,k})-\theta_{\widetilde{W}_{k}}(w_{j,k}))\end{aligned}
\]
and $\Theta_{\widetilde{W}_{k}}(w_{j,k})=\int_{-\infty}^{w_{j,k}}mf_{\widetilde{W}_{k}}(m)dm=F_{\widetilde{W}_{k}}(w_{j,k})\mathbb{E}[\widetilde{Y}_{j}(w_{j,k})|\widetilde{W}_{k}=w_{j,k}]$.
\end{thm}
Therefore, according to theorem \ref{thm:(Identification-of-ACRT)},
the ACRT gives information about the entire distribution of causal
effects of the policy variable on the outcome variable. However, it
still includes a conditioning component which could contaminate inference.
Under the same type of independence assumptions as \citet{RambachanSheppard2021},
it can be shown that it becomes the ACR.
\begin{thm}
\label{thm:(Identification-of-ACR)}(Identification of ACR) Under
assumptions \ref{assu:(Exogeneity-of-treatments)}, \ref{assu:The-estimated-errors-are-normal},
\ref{assu:(Continuous-differentiability)-T}, the Cholesky decomposition
of a VAR identifies
\[
\frac{\delta\mathbb{E}[\widetilde{Y}_{j}(w_{j,k})]}{\delta w_{j,k}}=\text{ACR}_{j,k}(\widetilde{Y}_{j}|w_{j,k}).
\]
\end{thm}
A few considerations are in order. First, it is possible to eliminate
the selection bias by using the same set of independence assumptions
that are used by \citet{RambachanSheppard2021}. Second, this is
not the only case in which it is possible to move from the ACRT to
the ACR. Later in the paper (in section \ref{subsec:Continuous-policy-simplediff})
I will show that it is possible to obtain a similar type of quantity
under strict conditions on counterfactual units.
The following paragraph will show that deviations from the normality
assumption, in particular in the case of a non-negative distribution,
can be harmful.
In the case of non-stationarity, the estimated effects on the filtered
outcome become respectively an $\text{F-ACRT}_{j,k}(\widetilde{Y}|w_{j,k}|w_{j,k})$
and an $\text{F-ACR}_{j,k}(\widetilde{Y}|w_{j,k})$ depending on
whether the economist relies on an independence type of assumption.
\begin{rem}
In the case of a non-normal but symmetric distribution, it is possible
to recover the ACRT and ACR by making use of carefully selected weights.
This is partially discussed in \citet{Yitazaki1996} and belongs
to a stream of the biometrics and microeconometrics literature that
dates back to \citet{horowitz1998censoring,manski2002inference}.
\end{rem}
\subsection{Non-negative continuous policy variable\label{subsec:Non-negative-continuous-policy}}
Often the researcher is interested in the effects of a non-negative
continuous policy. This is, for example, the effect of natural disasters
measured by their damage on the economy.
A recently developed stream of research has evolved around isolating
the kind of effects that such policies may identify. This case is
frequently defined as ``continuous treatment''. In a TWFE with a
continuous non-negative policy the effects identified are a weighted
sum of an ATE and an ACRT (see \citet{callawaysantannabacon2021}).
While it is possible to generate scenarios and restrictions to bind
such causal effects to a more interpretable quantity, it is particularly
hard to do so in macro economics due to the lack of credible restrictions.
For example, the micro econometrics literature makes the argument
that, by using controls, such effects could be disentangled, especially
using non-parametric types of estimators. Since the focus of the attention
of this paper is on VARs, this result will be considered especially
negative because it implicitly shows that to obtain sensible causal
estimates the researcher would need to make use of non-parametric
estimators, which is a type of estimator that is rarely utilized in
the context of Vector Autoregressions.
\begin{assumption}
\label{assu:(Non-negative-policy)-For}(Non-negative policy) For each
$t$, the policy of interest $k$ is such that $W_{j,k,t}\geq0$.
\end{assumption}
\begin{thm}
\label{thm:(Identification-of-ATE+ACR)}(Identification of ATE+ACR)
Under assumptions \ref{assu:(Exogeneity-of-treatments)}, \ref{assu:(Continuous-differentiability)-T},
\ref{assu:(Non-negative-policy)-For}, the Cholesky decomposition
of a VAR identifies
\[
\gamma_{j,k}=\int_{d_{L}}^{d_{U}}q_{1}(w_{j,k})(\text{ACRT}_{j,k}(\widetilde{Y}_{j}|w_{j,k})+\frac{\delta\text{ATT}(\widetilde{Y}_{j}|w_{j,k},w_{j,k}|a,0)}{\delta a}|_{a=w_{j,k}})+q_{0}\frac{\text{ATE}_{j,k}(\widetilde{Y}_{j}|d_{L},0)}{d_{L}}
\]
where
\[
\begin{aligned}q_{1}(w_{j,k}):=\frac{\mathbb{E}[\widetilde{W}_{k}|\widetilde{W}_{k}\geq w_{j,k}]-\mathbb{E}[\widetilde{W}_{k}])\mathbb{P}(\widetilde{W}_{k}\geq w_{j,k})}{var(\widetilde{W}_{k})} & \text{ and } & q_{0}:=\frac{(\mathbb{E}[\widetilde{W}_{k}|\widetilde{W}_{k}>0]-\mathbb{E}[\widetilde{W}_{k}])\mathbb{P}(\widetilde{W}_{k}>0)d_{L}}{var(\widetilde{W}_{k})}\end{aligned}
.
\]
\end{thm}
The conclusion that can be drawn from Theorem \ref{thm:(Identification-of-ATE+ACR)}
is that the identified causal effects are a mix of different distributions
with weights that depend on the density of the policy variable, and
specifically the frequency of it being above zero and/or above $w_{j,k}$.
The general conclusion is that if the weights are non-negatively distributed,
the estimated effects will hardly be interpretable.
In the case of non-stationarity of the outcome variable, the estimated
effects become
\[
\int_{d_{L}}^{d_{U}}q_{1}(w_{j,k})(\text{F-ACRT}_{j,k}(\widetilde{Y}|w_{j,k})+\frac{\delta F-\text{ATT}(\widetilde{Y}_{j}|w_{j,k},w_{j,k}|a,0)}{\delta a}|_{a=w_{j,k}})+q_{0}\frac{\text{F-ATE}_{j,k}(\widetilde{Y}|d_{L},0)}{d_{L}}.
\]
\section{The CVAR approach \label{subsec:The-CVAR-approach}}
The results of the previous section may appear to be troublesome.
In particular, I have highlighted some relevant problems in the causal
macroeconomics literature:
\begin{enumerate}
\item To obtain quantities like the ATE and the ACR, the required assumptions
on VARs are unrealistic, since we need to advocate the independence
of the treatment on their past, future, other policies, and the potential
outcomes.
\item Even in the most advocated case of normally distributed residuals
and independence assumptions, the causal effects could be difficult
to interpret if normality is violated.
\end{enumerate}
This section will consider a different type of approach to causal
inference that is inspired from the DiD literature. The idea is to
use one or many control variables to provide an alternative to causal
inference procedures that suffer from the first two of the problems
above, a setting that I will define CVAR (Control VAR).
I will discuss two ways by which controls can be implemented in structural
vector autoregressions. The first method will essentially discuss
the case of a VAR estimated on the difference between the treatment
and control outcome variable, that will therefore be named \emph{simple
difference}. Its interpretation is closely connected to the one of
DiD, but its estimation presents some disadvantages:
\begin{enumerate}
\item It is vulnerable to the high probability of spillover effects when
dealing with macroeconomic variables.
\item It does not allow for the estimation of a rescaling coefficient that
relates the control and treated variable, which means it assumes a
1:1 relationship between the trends of the treated and controls.
\item It can only be estimated with one control variable.\footnote{This problem could be solved by using a weighted average of the control
variables, with a procedure similar to synthetic controls (\citet{Abadie2010,Abadie2015}).
This type of approach, while possibly convenient, could require a
strict set of assumptions on the estimation of weights such as the
presence of a common data generating process in the absence of the
intervention in a factor model type of system.}
\end{enumerate}
In the second part, I will propose a method/assumption scheme that
may appear to be more credible in macroeconomic setting, which will
make use of standard VECM estimation procedures to estimate the CVAR.
This alternative approach will therefore be defined as \emph{VECM}.
\subsection{Estimation through simple difference\label{subsec:Estimation-through-simple}}
Define $X_{t}=(W_{1,t}^{\prime},..,W_{j,k,t},(Y_{1,t}^{1}-Y_{1,t}^{0})^{\prime},..,(Y_{J,t}^{1}-Y_{J,t}^{0})^{\prime})^{\prime}$,
where the superscript is $1$ for treated units and $0$ for the non
treated ones. Under the following alternative system, estimate the VAR:
\[
X_{t}=\sum_{l=1}^{p}X_{t-l}+\epsilon_{t},
\]
and run the Cholesky decomposition on $\epsilon_{t}$. First, I will
describe the assumption required for the identification procedure
in the case of a dummy policy in section \ref{subsec:Dummy-policy-simplediff}.
Then, I will move to the case of a continuous policy in section \ref{subsec:Continuous-policy-simplediff}.
\subsubsection{Dummy policy in the simple difference case \label{subsec:Dummy-policy-simplediff}}
I will use the following assumptions:
\begin{assumption}
\label{assu:The-potential-outcome}The observed outcome of the treated
and untreated units is
\[
Y_{t}^{1,obs}=\begin{cases}
Y_{j,t}(1) & \text{if }\widetilde{W}_{j,k,t}=1\\
Y_{j,t}(0) & \text{if }\widetilde{W}_{j,k,t}=0
\end{cases}\qquad Y_{t}^{0,obs}=\begin{cases}
Y_{j,t}(0) & \text{if }\widetilde{W}_{j,k,t}=1\\
Y_{j,t}(0) & \text{if }\widetilde{W}_{j,k,t}=0
\end{cases}
\]
and the non observed outcome of the treated units is
\[
\begin{aligned}Y_{t}^{1,nobs}=\begin{cases}
Y_{j,t}(0) & \text{if }\widetilde{W}_{j,k,t}=1\\
Y_{j,t}(1) & \text{if }\widetilde{W}_{j,k,t}=0
\end{cases}\end{aligned}
\]
\end{assumption}
\begin{assumption}
\label{assu:(Difference-stationarity)}(Difference stationarity):
For each $j\geq 1$, $k\geq 1$, $t\geq 1$, $Y_{j,t}^{1}-Y_{j,t}^{0}$
is stationary.
\end{assumption}
\begin{assumption}
\label{assu:(Deviations-from-the-controls-in-treated-times)}(Deviations
from the controls in treated times are causal). For each $j\geq 1$,
$k\geq 1$,$t\geq 1$,
\[
\mathbb{E}[Y_{j,t}^{1}(0)-Y_{j,t}^{0}(0)|\widetilde{W}_{j,k,t}=1]=0.
\]
\end{assumption}
\begin{assumption}
\label{assu:(Deviations-from-controls-in-non-treated-times)}(Deviations
from controls in non-treated times are zero): For each $j\geq 1$,
$k\geq 1$,$t\geq 1$,
\[
\mathbb{E}[Y_{j,t}^{1}(0)-Y_{j,t}^{0}(0)|\widetilde{W}_{j,k,t}=0]=0.
\]
\end{assumption}
Assumption \ref{assu:The-potential-outcome} binds the control unit
to always be not treated, and allows the potential outcome of the
treated unit to depend on their treatment status. Moreover, it assumes
that the potential outcomes are common to the units, so that the potential
outcomes of the non-treated units under a not treatment scenario,
and the one of the treated units under a non-treatment scenario, are
the same. Assumption \ref{assu:(Difference-stationarity)} prevents
violations of the Wold theorem. Assumption \ref{assu:(Deviations-from-the-controls-in-treated-times)}
is used similarly to a parallel trend condition. It means that the
unobserved outcome of the treated units in treated time can be reasonably
compared to the realized one of the control units. Assumption \ref{assu:(Deviations-from-controls-in-non-treated-times)}
is similar to a no anticipation condition, in that the units are also
assumed to have similar behavior in the untreated times after conditioning
on their past difference.
\begin{thm}
\label{thm:(Identification-of-ATT-direct-control)-1}(Identification
of ATT in the direct control case). For each $j\geq 1$, $k\geq 1$,$t\geq 1$,
if the policy variable is a dummy and assumptions \ref{assu:The-potential-outcome},
\ref{assu:(Difference-stationarity)}, \ref{assu:(Deviations-from-the-controls-in-treated-times)},
and \ref{assu:(Deviations-from-controls-in-non-treated-times)}, are
satisfied, the Cholesky decomposition of the direct control approach
of the CVAR identifies
\[
\gamma_{j,k}=\mathbb{E}[Y_{j,t}(1)-Y_{j,t}(0)|\widetilde{W}_{j,k,t}=1]+\Delta_{AR}=\text{ATT}_{j,k}(Y_{j}|1,0|1)+\Delta_{AR},
\]
where
\[
\Delta_{AR}=\mathbb{E}[Y_{j,t}^{1}-Y_{j,t}^{0}|\widetilde{W}_{j,k,t}=0,Y_{j,t-1,j}^{1}-Y_{j,t-1}^{0},...]-\mathbb{E}[Y_{j,t}^{1}-Y_{j,t}^{0}|\widetilde{W}_{j,k,t}=1,,Y_{j,t-1,j}^{1}-Y_{j,t-1}^{0},...]
\]
is a residual component that depends on the prediction of the VAR
autoregressive components.
\end{thm}
The presence of a residual component in the interpretation of the
causal estimand should not necessarily be an element of concern. It
indicates the presence of a selection bias that depends on the prediction
of the autoregressive component of the VAR model that becomes zero
in the case of common predictions of the treated and control units.
One way to interpret this additional component is to think about it
as the idea that treated and control variables may have common autoregressive
components in equilibrium, and deviations from such long term equilibrium
come from the residual impact of interventions, which negatively weights
the causal estimand. Symmetrically, it is negatively impacted by potential
spillover effects.
Most importantly, the theorem implies that using a control variable
can result in a causal estimand without making use of any exogeneity
assumption.
\subsubsection{Continuous policy in the simple difference case \label{subsec:Continuous-policy-simplediff}}
In the case of a continuous policy, the CVAR estimated through the
direct control approach will require assumption \ref{assu:(Difference-stationarity)}
and the following assumptions to hold:
\begin{assumption}
\label{assu:The-potential-outcome-1}The observed outcome of the treated
and untreated units
\[
Y_{t}^{1,obs}=\begin{cases}
Y_{j,t}(w_{j,k}) & \text{if }\widetilde{W}_{j,k,t}=w_{j,k}\\
Y_{j,t}(0) & \text{if }\widetilde{W}_{j,k,t}=0
\end{cases}\qquad Y_{t}^{0,obs}=\begin{cases}
Y_{j,t}(0) & \text{if }\widetilde{W}_{j,k,t}=w_{j,k}\\
Y_{j,t}(0) & \text{if }\widetilde{W}_{j,k,t}=0
\end{cases}
\]
and the non observed outcome of the treated units under no treatment
are
\[
\begin{aligned}Y_{t}^{1,nobs}=\begin{cases}
Y_{j,t}(0) & \text{if }\widetilde{W}_{j,k,t}=w_{j,k}\\
Y_{j,t}(w_{j,k}) & \text{if }\widetilde{W}_{j,k,t}=0
\end{cases}\end{aligned}
\]
\end{assumption}
\begin{assumption}
\label{assu:(Strong-Parallel-Trends)}(Strong Parallel Trends). For
each $j\geq 1$, $k\geq 1$,$t\geq 1$,
\[
\mathbb{E}[Y_{j,t}^{1}(w_{j,k})-Y_{j,t}^{0}(0)|\widetilde{W}_{j,k,t}=w_{j,k}]=\mathbb{E}[Y_{j,t}^{1}(w_{j,k})-Y_{j,t}^{0}(0)].
\]
\end{assumption}
Assumption \ref{assu:The-potential-outcome-1} in this case indicates
that we observe the treatment assignment of treated units at each
time, but we are incapable of observing them under a no treatment.
The underlying assumption is that the control units have identical
potential outcomes to what would be the one of the treated under no
treatment. Assumption \ref{assu:(Strong-Parallel-Trends)} is the
same as the one of strong parallel trends by \citet{callawaysantannabacon2021}
but must be read differently. In this case, it indicates that the
control unit eliminates the time selection bias. This lead to the
following theorem:
\begin{thm}
\label{thm:(Identification-of-ACR-direct-control)}(Identification
of ACR in the direct control case). For each $j\geq 1$, $k\geq 1$,$t\geq 1$,
if the policy variable is normally distributed and assumptions \ref{assu:The-potential-outcome-1},
\ref{assu:(Difference-stationarity)}, and \ref{assu:(Strong-Parallel-Trends)},
are satisfied, the Cholesky decomposition of the direct control approach
to the CVAR identifies
\[
\gamma_{j,k}=\frac{\delta\mathbb{E}[Y_{j}(w_{j,k})-Y_{j}(0)]}{\delta w_{j,k}}+\Delta_{AR}=\text{ACR}_{j,k}(Y_{j}|w_{j,k},0)+\Delta_{AR},
\]
where $\Delta_{AR}$ is akin to Theorem \ref{thm:(Identification-of-ATT-direct-control)-1}.
\end{thm}
This result should be read exactly like the previous one for the dummy
policy variable case. Even in the case of a continuous policy variable,
it is possible to obtain the ACR without relying on independence assumptions.
Section \ref{subsec:Estimation-through-VECM} will present an alternative
to this set of assumptions with similar benefits to the dummy case.
\subsection{Estimation through VECM\label{subsec:Estimation-through-VECM}}
The C-VAR I propose is the following:
\[
X_{t}=(W_{1,t}^{\prime},..,W_{j,k,t}^{\prime},Y_{1,t}^{1\prime},..,Y_{J,t}^{1\prime},Y_{1,t}^{0\prime},..,Y_{J,t}^{0\prime})^{\prime}.
\]
The VECM estimates the long run relationship between variables through
$\Pi=\alpha\beta^{\prime}$, so that $X_{t}\sim\mathcal{I}(1)$ but
$X_{t}-\beta X_{t}\sim\mathcal{I}(0)$. Traditionally, the model is
written as follows
\begin{equation}
\Delta X_{t}=\Pi X_{t-1}+\sum_{l=1}^{p}A_{l}\Delta X_{t-l}+\epsilon_{t},\label{eq:VECM}
\end{equation}
where $\Delta X_{t}$ represents the first difference of the non-stationary
variables. For such estimation to be credible, the following assumption
needs to hold.
\begin{assumption}
\label{assu:(The-roots-of-1}(The roots of the characteristic polynomial
lie outside the unit circle). \label{assu:(The-roots-of}The roots
of the characteristic polynomial
\[
\Pi(z)=(1-z)I_{p}-\alpha\beta'z-\sum_{l=1}^{p-1}A_{l}(1-z)z^{l},
\]
are outside the complex unit circle or at 1. Moreover, the matrices
$\alpha$ and $\beta$ have full column rank $r$. Further, define
$\Psi=I_{k+(j\times2)}-\sum_{l=1}^{p-1}A_{l}$ and assume full rank
of the matrix
\[
\alpha'_{\bot}\Psi\beta_{\bot}.
\]
\end{assumption}
Hence, Theorem 4.2 of \citet{Johannsen1995} holds as follows:
\begin{thm}
(Granger's Respresentation Theorem). \label{thm:(Granger's-Respresentation-Theor}Suppose
assumption \ref{assu:(The-roots-of} is satisfied. Then, there exists
a matrix $\Pi=\alpha\beta^{\prime}$. Moreover, a necessary and sufficient
condition that $\Delta X_{t}-\mathbb{E}[\Delta X_{t}]$ and $\beta^{\prime}X_{t}-\mathbb{E}[\beta^{\prime}X_{t}]$
can be given initial distributions such that they become $I(0)$ is
that
\begin{equation}
|-\alpha_{\bot}^{\prime}A(1)\beta_{\bot}|=|\alpha_{\bot}^{\prime}\Psi\beta_{\bot}|\neq0,\label{eq:first_part_granger}
\end{equation}
In this case the solution of the VECM in Equation \ref{eq:VECM} can
be written with the following representation
\begin{equation}
X_{t}=C\sum_{l=1}^{t}\epsilon_{l}+C_{1}(L)\epsilon_{t}+A,\label{eq:second_part_granger}
\end{equation}
where $A$ depends on the initial values, such that $\beta'A=0$,
and where $C=\beta_{\bot}(\alpha_{\bot}^{\prime}\Gamma\beta_{\bot})^{-1}\alpha_{\bot}^{\prime}$.
It follows that $X_{t}$ is a cointegrated $I(1)$ process with cointegrating
vectors $\beta$. The function $C_{1}(z)$ satisfies
\begin{equation}
A^{-1}(z)=C\frac{1}{1-z}+C_{1}(z),\quad z\neq1,\label{eq:last_part_granger}
\end{equation}
where the power series for $C(z)$ is convergent for $|z|<1+\delta$
for some $\delta>0$.
\end{thm}
This theorem implies that any deviation from the cointegrating relationship
is temporary and the processes inside the vector $X_{t}$ revert to
their common equilibrium. This implies that $\epsilon_{t}=\Delta X_{t}-\Pi X_{t-1}+\sum_{l=1}^{p}A_{l}\Delta X_{t-l}$
has a solution for each $t$.
Notice that such representation has a S-VECM equivalent representation
that also includes the causal effects of interest, which are defined
as $A_{0}$. Therefore, the structural error correction model equivalent
(S-VECM) of the VECM representation in Equation \ref{eq:VECM} would
be:
\begin{equation}
A_{0}\Delta X_{t}=\alpha^{\ast}\beta'X_{t-1}+\sum_{l=1}^{p}A_{l}^{\ast}\Delta X_{t-l}+\epsilon_{t}^{\ast},\quad \epsilon_{t}^{\ast}\sim\mathcal{D}^{\ast}_{p}(0,\Omega^{\ast}) \label{eq:SVECM}
\end{equation}
where $\alpha^{\ast}=A_{0}\alpha$, $A_{l}^{\ast}=A_{0}A_{l}$, $\mathcal{D}$ is a generic distribution (not necessarily normal),
for all $i$, $\epsilon_{t}^{\ast}=A_{0}\epsilon_{t}$ and $\Omega^{\ast}=A_{0}^{*}\Omega A_{0}^{\ast\prime}$.
According to \citet{Johannsen1995}, inference can be carried out
as usual and the rank conditions apply, as well as the formulations
in connection with the identification of $\beta$. The conclusion
of this is that the presence of non-stationary variables allows two
distinct identification problems to be formulated. First, the long-run
relations must be identified uniquely in order to estimate and interpret
them and then the short-run parameters $(\alpha^{\ast},A_{0}^{\ast},..,A_{k-1}^{\ast})$
must be identified uniquely in the usual way.
\subsubsection{Dummy policy in the VECM case \label{subsec:Dummy-policy}}
For the Choleksy decomposition to identify ATT, the trend will be
assumed to be a credible counterfactual, in that the difference between
the treated variable and the trend would be zero if the treated variable
was to not be treated.
\begin{assumption}
\label{assu:(Deviations-from-trend-in-treated-times-are-causal):}(Deviations
from trend in treated times are causal): For each $j\geq 1$, $k\geq 1$,$t\geq 1$,
\[
\mathbb{E}[\Delta Y_{j,t}^{1}(0)|\widetilde{W}_{j,k,t}=1]=\mathbb{E}[f_{K+j}(\Pi X_{t-1})+f_{K+j}(\Sigma_{l=1}^{p}A_{l}\Delta X_{t-l})|\widetilde{W}_{j,k,t}=1].
\]
\end{assumption}
\begin{assumption}
\label{assu:(Deviations-from-trend-in-non-treated-times-are-zero-1}(Deviations
from trend in non-treated times are zero): For each $j\geq 1$, $k\geq 1$,$t\geq 1$,
\[
\mathbb{E}[\Delta Y_{j,t}^{1}(0)|\widetilde{W}_{j,k,t}=0]=\mathbb{E}[f_{K+j}(\Pi X_{t-1})+f_{K+j}(\Sigma_{l=1}^{p}A_{l}\Delta X_{t-l})|\widetilde{W}_{j,k,t}=0].
\]
\end{assumption}
Here $K+j$ indicates the row specific to the outcome variable of
the treated country or unit. Notice that assumption \ref{assu:(Deviations-from-trend-in-treated-times-are-causal):}
involves an untestable hypothesis since it involves counterfactuals
assignment.\footnote{This is a property shared by many causal models. For example, Regression
Discountinuity designs and DiD designs exploit respectively units
that are right before the cutoff and other unit's realized outcomes
as counterfactuals.} However, since it involves some estimated parameters, it only requires
that a combination of $\Pi$ and $A_{l}$ that satisfies assumption
\ref{assu:(Deviations-from-trend-in-non-treated-times-are-zero-1}
exists. That is because Theorem \ref{thm:(Granger's-Respresentation-Theor},
implies that $\text{\ensuremath{\hat{\Pi}}}$ and $\hat{A}_{l}$ are
specifically chosen to minimize the difference in the assumption \ref{assu:(Deviations-from-trend-in-non-treated-times-are-zero-1}.
The asymptotics can be found in Ch. 13.4 of \citet{Johannsen1995}.
\begin{rem}
Notice that assumptions \ref{assu:(Deviations-from-trend-in-treated-times-are-causal):}
and \ref{assu:(Deviations-from-trend-in-non-treated-times-are-zero-1}
have an interpretation under the potential outcome framework in \ref{assu:The-potential-outcome}.
For example, consider the trivariate system $X_{t}=(W_{t}^{\prime},Y_{t}^{1},Y_{t}^{0})^{\prime}$
where $Y^0_t$ serves as a counterfactual trend for $Y^1_t$.
Suppose the long-run behaviour is characterized by a single cointegrating relation that involves only $Y^1_t$ and $Y^0_t$.
Using the standard factorization $\Pi = \alpha \beta'$ with
\[
\beta = (0,\,1,\,-\lambda)'
\quad \text{and} \quad
\alpha = (0,\,\alpha_2,\,0)',
\]
we obtain
\[
\Pi = \alpha \beta' =
\begin{pmatrix}
0 & 0 & 0\\[3pt]
0 & \alpha_2 & -\alpha_2 \lambda\\[3pt]
0 & \ast & \ast
\end{pmatrix}.
\]
Writing $c_1 = \alpha_2$ and $c_0 = -\alpha_2 \lambda$,
the long-run part of the $Y^1$-equation can be written
(ignoring short-run dynamics and shocks for expositional clarity) as
\begin{equation} \label{eq: CVAR - example}
\Delta Y^1_t(0) = c_1 Y^1_{t-1}(0) + c_0 Y^0_{t-1}(0).
\end{equation}
\end{rem}
From such alternative definition it is possible to see that there
is a potential outcome representation of the controls, but that is
different from the traditional causal inference for two reasons. First,
the counterfactual (right hand side of equation \ref{eq: CVAR - example}) is essentially lagged. Second, the counterfactual
is a linear transformation (through $c_{1}$ and $c_{2}$) of the realized outcomes of the control and treated
variables.
An advantage of cointegration is that it avoids the potential effect
of violations of the underlying no spillover assumptions. This is
because generating the counterfactual using lagged values of the controls
avoids the potential invalidating effect of spillovers altogether.
Under such restrictions, the following theorem holds true.
\begin{thm}
\label{thm:(Identification-of-ATT).}(Identification of ATT in the
VECM case). For each $j\geq 1$, $k\geq 1$,$t\geq 1$, if the policy
variable is a dummy and assumptions \ref{assu:(Deviations-from-trend-in-treated-times-are-causal):}
and \ref{assu:(Deviations-from-trend-in-non-treated-times-are-zero-1}
are satisfied, the Cholesky decomposition estimated with the VECM
approach to the CVAR identifies
\[
\gamma_{j,k}=\mathbb{E}[\Delta Y_{j,t}(1)-\Delta Y_{j,t}(0)|\widetilde{W}_{j,k,t}=1]=\text{ATT}_{j,k}(\Delta Y_{j}|1,0|1).
\]
\end{thm}
\subsubsection{Continuous policy in the VECM case\label{subsec: Continuous policy VECM}}
For the Choleksy decomposition to identify the ACR in the normally
distributed policy case, the trend will be assumed to be a credible
counterfactual, in that the difference between the treated variable
and the trend would be zero if the treated variable was to not be
treated by any amount $w_{j,k}$.
\begin{assumption}
\label{assu:(Continuous-deviations-from-trend-in-treated-times-are-causal)}(Continuous
deviations from trend in treated times are causal): For each $j\geq 1$,
$k\geq 1$,$t\geq 1$,
\[
\mathbb{E}[\Delta Y_{j,t}^{1}(0)|\widetilde{W}_{j,k,t}=w_{j,k}]=\mathbb{E}[f_{K+j}(\Pi X_{t-1})+f_{K+j}(\Sigma_{l=1}^{p}A_{l}\Delta X_{t-l})|\widetilde{W}_{j,k,t}=w_{j,k}].
\]
\end{assumption}
\begin{assumption}
\label{assu:(Continuous-strong-parallel}(Continuous strong parallel
trends): For each $j\geq 1$, $k\geq 1$, $t\geq 1$,
\[
\mathbb{E}[\Delta Y_{j,t}^{1}(w_{j,k})-\Delta Y_{j,t}^{1}(0)|\widetilde{W}_{j,k,t}=w_{j,k}]=\mathbb{E}[\Delta Y_{j,t}^{1}(w_{j,k})-\Delta Y_{j,t}^{1}(0)]
\]
\end{assumption}
Assumption \ref{assu:(Continuous-deviations-from-trend-in-treated-times-are-causal)}
is required to define the counterfactual generated from the control
unit to be credible at any time, a condition similar to assumption
\ref{assu:(Deviations-from-trend-in-treated-times-are-causal):}.
Assumption \ref{assu:(Continuous-strong-parallel} instead is used
to eliminate the selection bias component, a condition which credibility
depends on the endogeneity of the residuals of the policy.
This restriction is sufficient to lead to the following theorem
\begin{thm}
\label{thm:(Identification-of-ACR-VECM)}(Identification of ACR in
the VECM case). For each $j\geq 1$, $k\geq 1$,$t\geq 1$, if the policy
variable is normally distributed and assumptions \ref{assu:(Continuous-deviations-from-trend-in-treated-times-are-causal)}
and \ref{assu:(Continuous-strong-parallel} are satisfied, the Cholesky
decomposition of the VECM of the CVAR identifies
\[
\gamma_{j,k}=\frac{\delta\mathbb{E}[\Delta(Y_{j}^{1}(w_{j,k})-\Delta Y_{j}^{1}(0)]}{\delta w_{j,k}}=\text{ACR}_{j,k}(\Delta Y_{j}|w_{j,k},0).
\]
\end{thm}
One potential issue for macroeconomists would be the need to restrict
the control series to a conservative set. This is because adding all
the available controls to the VAR systematically increases the number
of coefficients to be estimated. Therefore, the applied researcher
should be careful about finding an appropriate set of series to be
used as counterfactuals. Many possible criteria could be used for
such minimization. In the following paragraph I will simply order
the candidate controls according to the Likelihood Ratio test of the
resulting VECM using different sets of controls. Another approach
to the problem could be using many different counterfactuals when
the time component is large enough.
\section{The effects of natural disasters on Industrial Production and Macroeconomic
Uncertainty\label{sec:The-effects-of-of natural-disasters-on-Industrial-Production-and-Macroeconomic-Uncertainty}}
\citealp{Ludvigson2021} (henceforth LMN) conducted an estimation
of an S-VAR model with 6 lags encompassing the Cost of natural Disasters
(CD), Industrial Production (IP), and Macroeconomic Uncertainty (MU)
index proposed by \citealp{Jurado2015}. I plot the original variables and the German counterpart in Figure \ref{tab:Industrial-Production,-Macroecon}.
From the figure, several relevant features emerge. The series suggest
a certain degree of contemporaneous comovements, which are especially
relevant in moments in which the variables appear to change during
the great financial crisis. Therefore, even outside the scope of using
Germany as a counterfactual for inferential purposes, adding such
data to the system may provide many modeling advantages: it may take
care of common breaks, common structural factors, and many different
elements that perturbate the two economies. In the original application,
however, the focus is only on US data. There, the causal assumption
is that the cost of natural disaster is considered exogenous. Such
an assumption allows the authors to employ a Cholesky decomposition
on a VAR composed by the CD first, followed by IP and MU. Their objective
is to simulate a shock equivalent to the magnitude of the COVID-19
pandemic and utilize the simulation to predict the evolution of IP
and MU.
\begin{figure}[H]
\centering{}\includegraphics{Descriptive_F1_DID.pdf}\caption{Industrial Production, Macroeconomic Uncertainty and Cost of Natural
Disasters in the United States and Germany.\label{tab:Industrial-Production,-Macroecon}}
\end{figure}
Although not explicitly mentioned in the paper, it is worth noting
that the selected outcome variables exhibit non-stationarity. To address
this, LMN apply the detrendization method proposed by \citealp{muller2018long}
to filter out the trend component. While such an approach is not necessarily
problematic in the view of the most classical identification in macroeconomics
literature, it presents two challenges. The first one relates to the
interpretability of the model. Since the CD variable is continuous
and non-negative, its effect will be, as discussed in Theorem \ref{thm:(Identification-of-ATE+ACR)},
a weighted average of a filtered ATE and ACRT. Therefore, the estimated
Impulse Response Function is capturing both the deviation with respect
to zero, and the deviation with respect to progressively different
values. Intuitively, the pitfall of the identification is represented
by the high discontinuity of the causing variable.\\
The second one relates to the likelihood of the exogeneity assumption
to hold. In fact the authors notice that the most relevant events
are the Heat Wave of 1980, September 11, Hurricane Katrina, Hurricane
Maria, and Hurricane Harvey. The fact that all those episodes happened
in August and September may contaminate inference due to underlying
unobservables. Many other instances could invalidate inference when
the underlying assumption is the complete randomness of the natural
disaster series, which may motivate approaches to causal inference
that do not rely on independence. \\
The three left panels of figure \ref{fig:Impulse-Response-Functions-original}
plot the original results for horizons extended to 20 periods.\\
\begin{figure}
\centering{}\includegraphics[width=8cm]{Figure_website.pdf}\caption{Impulse Response Functions of the filtered variables to a 1\% shock
in the CD. Confidence intervals are bootstrapped at the $95\%$ level
using wild bootstrap.
The left panel are obtained through a simple VAR with filtered US
data. The right panels result from a VECM estimated using also German
data, where the disasters are transformed to a dummy indicating the
$95$-th quantile of the disaster variable.\label{fig:Impulse-Response-Functions-original}}
\end{figure}
The effects of the CD on IP are negative for the first two periods
and positive in the following ones. The effects of the CD on MU are
positive for three periods and negative afterward. Those responses
may indicate the filtering procedure's ineffectiveness in changing
the impulse response's shape. \footnote{Subsection \ref{subsec:Non-filtered-series} provides some evidence
in support of the fact that the Impulse Responses are not affected
by the filtering.} \\
To solve the first issue (the interpretability of the mix of
filtered effects of a non-negative continuous variable), an alternative
to such procedure could be using a dummy variable that is equal to
1 when a particularly disruptive disaster happens and 0 otherwise,
with the goal of identifying the effect of a high cost disaster happening.
This other approach could move the interpretation of the causal effects
from the ones of a non-negative policy to the ones of a dummy policy.
\footnote{Appendix \ref{subsec:Dummy-variable-but-no-controls} provides evidence
in support of the fact that such modification may not substantially
change the IRF. In that case, I use a VAR model where the first variable
is a dummy that is equal to 1 in periods in which the CD variable
is above its $95$\emph{-th }quantile.}. Until now, VARs did not have the means to solve the second issue
(exogeneity) within a proper causal framework that declaratively assigns
a counterfactual.\\
My alternative modeling approach consists of two deviations from the
original model. The first one is using a dummy that represents when
the Cost of Natural Disasters (CD) are greater than the $95$-\emph{th}
quantile of the time series -or above a cost of 150 million- in place of the CD series itself. The
second one is using a control for the United States data. I use the
MU data from \citealp{Meinen2017}, which collects the same type of
variable as the work of \citet{Jurado2015}. Out of the countries
available in their dataset, Germany appears to be the one that provides
the best-identified system according to the battery of tests discussed
in Appendix \ref{subsec:Additional-results-for-Germany}.\footnote{I also source data on the cost of natural disasters in Germany from
the International Disasters database to control for the possible presence
of natural disasters in Germany. The results are invariant to the
addition of a dummy variable that controls for Germany disasters.}
Accordingly, I estimate a VECM for the following system:
\[
X_{t}=(CD_{US,t}^{95\prime},IP_{US,t}^{\prime},MU_{US,t}^{\prime},IP_{DEU,t}^{\prime},MU_{DEU,t}^{\prime})^{\prime},
\]
where $CD_{US,t}^{95\prime}$ represents the $95$-\emph{th }quantile
of the CD series, $IP_{US,t}^{\prime}$ represents IP in the US, $MU_{US,t}^{\prime}$
represents MU in the US, $IP_{DEU,t}^{\prime}$ represents IP in Germany,
$MU_{DEU,t}^{\prime}$ represents MU in Germany. This method relies
on assumptions \ref{assu:(Deviations-from-trend-in-treated-times-are-causal):}
and \ref{assu:(Deviations-from-trend-in-non-treated-times-are-zero-1}
instead of the independence of the policy and captures the effect
of a natural disaster of high proportion versus a null or not costly
disaster.\\
The resulting Impulse Response Functions are displayed in the right-hand
side of Figure \ref{fig:Impulse-Response-Functions-original}. Two
particular features emerge from the picture. The first one is that
the horizon of the impact of a natural disaster shock on IP is short-lived
and always negative. This result is particularly relevant for policymakers,
since it clarifies the ambiguous role that natural disasters play
in the economy. A similar feature emerges from the response of MU
to the disasters dummy, but it should be noted that uncertainty seems
to exhibit persistence but no cyclicality. The second one is that
the confidence intervals shrink to zero together with the causal responses.\\
Both the results come from a particular feature of my alternative
model. Adding another source of information (German data) results
in a preferred system that only includes one lag, which makes the
response of the outcome variables to a disaster shock shorter. Short
lived impulse responses are a property of VARs with low lags and AR(1)
coefficients far from the unit root.\\
The overall conclusion is that such an approach to modeling presents
two different advantages. The first one is related to the causal interpretability
of the model. An ATT may be more interpretable than an ACRT+ATE, and
may be preferred over an ATE because no independence assumption would
be required. The second is related to a pure macroeconomic interpretation.
A property of VECM models is that they tend to reject lags higher
than one, which may be an advantage if the objective is the interpretation
of the short-run effects of a policy.\\
In appendix \ref{sec:Additional-results-for-empirical-application}
I discuss more details about the empirical application, the lag choice,
and discuss alternative specifications with many different controls.
\section{Conclusions \label{sec:Conclusions-3}}
Traditional VAR estimators may have a broader range of causal interpretations
than the ones discussed by \citealp{RambachanSheppard2021}. I show
that VAR may identify other types of causal effects. In the case of
normal distribution of the policy innovations, the effect identified
becomes an ACR under their same independence assumptions. In the case
of a non-negative distribution of the policy the effect identified
is the sum of an ACRT and an ATE.\\
If controls are available, cointegration can be exploited, and the
resulting coefficient may still have a causal interpretation. In the
case of a dummy policy variable, they can identify an ATT. In the
case of a continuous and normal distribution, they identify an ACR.
This approach provides two substantial advantages. First, it relaxes
potentially hard to satisfy independence assumptions. Second, it may
deliver more credible impulse responses.\\
I discuss all of these advantages in the context of the work of \citealp{Ludvigson2021},
which analyzes the causal effects of natural disasters on the US economy.\clearpage{}
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