The exact contents of citations.db main_text.text for this paper — one flattened LaTeX string, title through conclusion, appendix excluded, unmodified except for removing email addresses. This is what our citation measures are computed over.
43,689 characters
Identification, estimation and inference in Panel Vector Autoregressions using external instruments
\maketitle
\begin{abstract}
This paper proposes an identification inspired from the SVAR-IV literature that uses external instruments to identify PVARs, and discusses associated issues of identification, estimation, and inference.
I introduce a form of local average treatment effect - the $\mu$-LATE - which arises when a continuous instrument targets a binary treatment. Under standard assumptions of independence, exclusion, and monotonicity, I show that externally instrumented PVARs estimate the $\mu$-LATE. Monte Carlo simulations illustrate that confidence sets based on the Anderson-Rubin statistics deliver reliable convergence for impulse responses.
As an application, I instrument state-level military spending with the state's share of national spending to estimate the dynamic fiscal multiplier. I find multipliers above unity, with effects concentrated in the contemporaneous year and persisting into the following year.
\indent {\bf{JEL Classification}}: C32, C33. \\
\noindent {\bf{Keywords}}: Vector autoregressive models, panel data, identification, instruments.
\end{abstract}
\clearpage
Panel vector autoregressions (PVAR) are one of the standard tools
for the estimation of dynamic causal effects in macroeconomics. However,
their causal interpretation is frequently limited. Sometimes they
are described as identifying a temporally related causality, ``Granger
causality''; sometimes, as discussed in \citet{pala2024pvarcontrol},
they may posses a contemporaneous causal interpretation under either
exogeneity assumptions, or a suitable control type of assumption.
However, Granger causality is not contemporaneous causality\footnote{In the words of \citet{granger2014forecasting}: ``A better term
might be temporally related, but since it is such a simple term
we shall continue to use it .}; exogeneity is hard to satisfy due to the rich endogeneity among
macroeconomic variables (\citet{Nakamura2018a}); and the control
type of approach proposed by \citet{pala2024pvarcontrol} may not
be satisfied if some units cannot act as controls for some others.
However, when all the previous cases
fail, instruments could be utilised to carry causal claims.
This paper shows that it is possible to identify causal
effects by the means of instrumental variables in the case of panel
vector autoregressions. By implementing the same approach as \citet{gertler2015monetary,Mertens2013,Mertens2014,StockWatson2018,Olea2021}
- defined as Proxy-SVAR or SVAR-IV- in PVARs it is possible to retrieve a causal
estimand that I define as $\mu-\text{LATE}$. Normally, the causal literature has a clear target in mind in the case of a dummy policy and a dummy instrument (\citet{angrist1996identification}) - the LATE. LATEs can be viewed as a special case of principal stratification for which there are 4 categories of compliance status. In the case of continuous instruments the interpretation becomes troublesome because there are infinitely many possible strata (\citet{antonelli2023principal}). Contrarily, $\mu$-LATEs seemlessy move from continuous variables to a binary interpretation akin to a LATE. In particular, they compare the value of the outcome and policy under a 1\% and 0\% instrument assignment.
In this special context, the defiers are (under positive
monotonicity) units that do not observe an increase (decrease) in
their policy variable residuals when the instrument increases (decreases).
Moreover, because usually the residuals of a PVAR are assumed to be
continuous and normally distributed, the $\mu$- LATE strongly depends on the instrument and policy continuity. This latter feature is rarely discussed
by the causal literature but imposes a strong linearity assumption.\footnote{In fact, most of the IV literature has focused on either binary (\citet{angrist1996identification})
or multi-valued instruments (\citet{heckman2001policy,vytlacil2002independence}),
but rarely discussed how a LATE could emerge from the comparison of
of two predicted values of the outcome variables under a continuous
instrument case. In general, semi parametric or non parametric estimators
tend to be preferred when the data allows it. } This is, however, necessary. Because macroeconomic data is frequently
relatively small in the unit component, making use of a linearity
assumption to obtain meaningful estimates is often the only possibility.
\footnote{This point, and its many drawbacks, are developed in \citet{kolesar2024dynamic}.}
On the other hand, inference can be carried by simply translating
the conclusions of \citet{Olea2021} regarding VAR-IVs to their panel counterpart. This
implies that the optimal approach to inference is to make use of the
Anderson Rubin test statistic (henceforth defined as AR - see \citet{Fieller1944,Anderson1949}
for its definition) in the presence of one instrument, and, at the
current state of the literature, the conditional likelihood ratio
test (henceforth defined as CLR - see \citet{andrews2006optimal}
for its definition) in the presence of multiple instruments.
I give a context to the proposed causal framework through an original
application in which I compute the dynamic fiscal military multiplier in the United States. Using local level data can be advantageous for several reasons. First, \citet{Nakamura2014}
argue that in a monetary union, the central bank (the Federal Reserve) cannot
raise interest rates in some states relative to others, and federal tax policy is
common across states in the union. This means that the open economy multiplier identifies the measure of fiscal policy that does not
depend on monetary movements.
Second, the advantage of using military spending is that the DD-350 military procurement forms
made available from the US Department of Defense provide a clean, direct, and localised measure of
government spending. Unfortunately, data of such quality, that starts from such an early age - 1966 -
is not available for any other form of spending.
For this reason, I consider a PVAR that includes military state-level
spending as instrumented by the states fraction of total national
military spending, with GDP growth as the outcome variable. Such IV
approach is generally defined as Bartik instrument and is becoming
increasingly popular in social sciences, where the use of local-level
data is intensifying (see \citet{bartik1991benefits,goldsmith2020bartik}).
My findings indicate that the fiscal multiplier may be about $\sim1.7$
on impact, a value not dissimilar from the ones previously observed
by the state-multiplier literature (\citet{chodorow2019geographic}).
The advantage of using PVARs over simple panel linear regressions becomes clear when considering the usefulness of Impulse Response Functions.
In this sense, my findings suggest a multiplier of about $1.5$ one
year ahead and negligible after, suggesting the existence of large cumulative multipliers.
This paper contributes to several streams of the literature. One natural
contribution is at the intersection of the causal inference and the
time series fields. In this sense, it complements \citet{pala2024pvarcontrol}
in the research agenda oriented to give a causal interpretation to
panel vector autoregressions and, more broadly, the literature that
uses the Rubin causal model to motivate the causal interpretation
of time series models (\citet{menchettibojinov2022,menchetticipollinimealli2022,RambachanSheppard2021,bojinovshephard2019}).
Second, it contributes to the econometric literature by extending
the identification of structural VARs through an external instrument
to panel VARs. Third, it indicates a potentially interesting recipe
for the interpretation of any estimator in the case of continuous
instruments that is different from the current literature. Fourth,
it extends the common knowledge of the good coverage properties of
the Anderson-Rubin statistics for the impulse response functions generated
by a PVAR-IV, indicating a venue for the construction of reliable
confidence sets. Fifth, it contributes to the applied literature for
the estimation of the dynamic fiscal multipliers.
The discussion is organized as follows. Section \ref{sec:Identification}
introduces the potential outcome framework and the $\mu-\text{LATE}$.
Section \ref{sec:Assumptions} introduces the assumptions required
for the identification of the causal effects. Section \ref{sec:Estimation}
introduces the estimation procedure of the PSVAR-IV. Section \ref{sec:Inference}
discusses the Anderson Rubin statistics\footnote{In appendix \ref{sec:Appendix-B:-Monte} I provide the main theorems
and simulations that showcase the good coverage properties of the
statistic.}. Section \ref{sec:Fiscal-multiplier} discusses the estimation of
the dynamic fiscal multiplier using the PSVAR-IV.
Finally, allow me to introduce some useful notation. A set of observations
of variable $s$ for state $i$ at time $t$, $x_{s,it}$, is part
of a larger matrix of variables $\boldsymbol{x}_{s}$, and $\boldsymbol{x}_{2:S}$
indicates the partition from the second to the $S$ variable of such
matrix. $\mathbb{E}(X)$ will indicate the expected value of the variable
$X$, $var(X)$ will indicate its variance, $cov(X,Y)$ the covariance
between variables $X$ and $Y$, and $\boldsymbol{cov}(X,Y)$ their
variance-covariance matrix, which contains their variance on the diagonal
and their covariance in the off diagonal. For $R$ being a square
matrix, $R_{a,b}$ indicates the entry of $R$ in the row $a$ and
column $b$. $X\perp Y$ will indicate independence between $X$ and
$Y$.
\section{Identification \label{sec:Identification}}
Let us say the researcher is interested in the estimation of a dynamic
causal effect, represented as the impact of a change in a \emph{policy
}variable $W_{i,t}$ on one or many \emph{outcome }variables $Y_{j,i,t}$.
As such, the potential outcomes of the outcome variables can be represented
as follows:
\[
Y_{j,i,t}(w,z)=Y_{j,i,t}((W_{i,1:t-1},w,W_{i,t+1:T})(Z_{i,1:t-1},z,Z_{i,t+1:T})).
\]
Such definition is similar to the one introduced by \citet{RambachanSheppard2021}
and indicates that the potential outcome of the different outcome
variables $j$ depend on the policy assignment $w$ and on the instrumental
variable $z$\footnote{Such form was first introduced in LATEs by \citet{angrist1996identification}.}.
In the leading example of this paper, there will only be GDP growth
as outcome variable ($j=1$), while $W_{i,t}$ will indicate a fiscal
policy variable. Moreover, $Z_{it}$ will be the instrumental variable.
Finally, the index $i,t$ will refer to region $i$ at time $t$.\footnote{In this paper, I will solely focus on a clean case of one instrument, one policy. This choice stems from the higher likelihood of researchers to come up with identification strategies that involve a clear one insturment, one instrumented, many outcomes scenarios. It can be extended to multiple instruments and multiple instrumented variables. The analytical extensions are provided in \citet{Olea2021}.} All variables are assumed to be continuous.
I define a $\mu-\text{LATE}_{j}$ as the following
causal effect.
\begin{defn}
A $\mu\text{-LATE}_{j}$ is the LATE of receiving assignment $w$
versus assignment $w'$ for those units that comply with the assignment
\[
\mu\text{-LATE}_{j}
= \mathbb{E}\bigl[ Y_{j}(w) - Y_{j}(w') \,\big|\, W(z)=w \bigr].
\]
\end{defn}
\section{Assumptions \label{sec:Assumptions}}
The assumptions required to give a causal interpretation to a LATE
estimator are independence, exclusion and monotonicity. Here the assumptions
are put on the residuals of the PVAR. In particular, define $\widetilde{W}_{it}=\mathbb{E}[W_{it}|\omega_{it}]$ and $\widetilde{Y}_{j,it}=\mathbb{E}[Y_{j,it}|\omega_{it}]$, where $\omega_{it}=(W_{i,t-1}.^{\prime}.. W_{i,t-p}^{\prime},Y_{1,i,t-1}^{\prime},..,Y_{1,i,t-p}^{\prime},Y_{J,i,t-1}^{\prime},..,Y_{J,i,t-p}$ are the regressors. The PVAR estimates the impact of variations in $\widetilde{W}_{it}$ on $\widetilde{Y}_{j,it}$.
The treatments are assumed
to be continuous, so that $\widetilde{w}^{\circ}$ is any value extracted
from the distribution of $\widetilde{W}$ and $z^{\circ}$ is any
value extracted from the distribution of $Z$.
\begin{assumption}
\label{assu:(Independence)}\textbf{(Independence)} For all $\widetilde{w}^{\circ}\in\widetilde{W}$,
all $\widetilde{z}^{\circ}\in Z$, all $t\geq1$, all $i\geq1$, and
all $j\geq1$, it holds that
\begin{equation}
\{\widetilde{Y}_{j,i,t}(\widetilde{w}^{\circ},z^{\circ}),\widetilde{W}_{i,t}(z^{\circ})\}\perp Z_{i,t}\label{eq:Indep}
\end{equation}
\end{assumption}
Assumption \ref{assu:(Independence)} establishes that the instruments
are as good as randomly assigned with respect to any of the outcome
variable's residuals or any of the policy variable's residuals. The
researcher will normally assume that the potential outcomes are not
affected by $Z_{it}$ if not through $\widetilde{W}_{it}$, i.e.
\begin{assumption}
\label{assu:(Exclusion)}\textbf{(Exclusion)} For all $\widetilde{w},\widetilde{w}^{\prime}\in\widetilde{W}_{i,t}$,
$t\geq1$, and $i\geq1$ it holds that
\[
\{\widetilde{Y}_{j,it}(\widetilde{w},z)=Y_{j,it}(\widetilde{w},z^{\prime})\}
\]
where $z,z^{\prime}\in Z_{i,t}$ are any possible combination of values
of $Z_{i,t}$.
\end{assumption}
Notice that one of the consequences of assumption \ref{assu:(Exclusion)}
is that it implies that the potential outcomes of $\widetilde{Y}_{j,it}$
- the residual of the outcome variable - do not depend on the realized
value of the instruments, if not by the means of the policy. Such
assumption, coupled with assumption \ref{assu:(Independence)}, allows
for a seamless transition from conditional expected values and realized outcomes
to potential outcomes frameworks. In the empirical example, this would
mean that the assignment of military expenditures at the federal-level
are independent with respect to the potential outcome process of any
GDP growth innovations and with respect to any military spending innovations
at the state-level.
Finally, monotonicity as in \citet{angrist1996identification} is
required, so that
\begin{assumption}
\label{assu:(Monotonicity)-For-all}\textbf{(Monotonicity)} For all
$z,z^{\prime}\in Z_{i,t}$, and all $t\geq1$ and $i\geq1$, it holds
that either $\widetilde{W}_{i,t}(z)\geq\widetilde{W}_{i,t}(z^{\prime})$
or $\widetilde{W}_{i,t}(z)\leq\widetilde{W}_{i,t}(z^{\prime})$.
\end{assumption}
Notice that this type of monotonicity assumption needs to hold for
every couple of instrument assignments $z,z^{\prime}$. For example,
in the case of fiscal multipliers, it means that if there is an increase
in total national military spending, the residuals of state-level
military spending are either increasing or decreasing for each unit
$i$ at each time $t$. Such assumption, differently from a non parametric
case, does not allow any discountinuity of the mapping of $\widetilde{W}$
on $Z$.
\begin{rem}
\textbf{(Weakness of a fully parametric monotonicity assumption).}
Let us say that $Z$ is multivalued, such that it can only take three integer values $Z=\{z^{0},z^{1},z^{2}\}$.
A parametric estimator needs to assume $W(z^{2})\geq W(z^{1})\geq W(z^{0})$.
A non parametric estimator could potentially solve this issue by estimating
two separate quantities, one for $z^{1}$ and $z^{2}$, and assuming
that $W(z^{1})\geq W(z^{0})$, $W(z^{2})\geq W(z^{0})$, but the ordering
of $W(z^{2})\underline{?}W(z^{1})$ does not need to be assumed. This
comes at the cost of requiring the data to be dense enough around
the quantities.
\end{rem}
Finally, the instrument needs to be a predictor of the instrumented
variable, a condition frequently defined as \emph{instrument relevance
}or \emph{instrument strength.}
\begin{assumption}
\textbf{\label{assu:(Relevance).}(Relevance).} The instrument satisfies
$\mathbb{E}[\widetilde{W}_{i,t},Z_{i,t}]\neq0$.
\end{assumption}
\begin{rem}
Frequently, SVAR-IV are thought to estimate a causal effect only under
a relevance and an non correlation condition. In macro economics,
the non correlation condition is frequently stated as $\mathbb{E}[\widetilde{Y}_{j,it},\widetilde{W}_{i,t}]=0$\footnote{See, among many others, \citet{StockWatson2018,Olea2021,bruns2024testing,brignone2023robust}.}.
Yet, such imposition would only refer to a statistical relationship
among the two, and would ignore the benefits of having a potential
outcome representation\footnote{In this sense, the condition usually stated is potentially testable
(see \citet{bruns2024testing}), but does not allow to claim the identification
of a LATE.}. In practice, this means that SVAR-IV would not be able to claim
that a meaningful causal estimand has been identified unless the econometrician identifies a set of assumptions that maps the estimator to an estimand. Typically, the two conditions that are sought for are insufficient to achieve any causal identification.
\end{rem}
\section{Estimation \label{sec:Estimation}}
Consider several known outcome variables and one intervention variable
aggregated in a process of the kind
\[
x_{i,t}=(W_{i,t}^{\prime},Y_{j=1,i,t}^{\prime},Y_{j=2,i,t}^{\prime},..,Y_{j=J,i,t}^{\prime})'.
\]
Here $W_{i,t}$ could indicate military procurement spending in region
$i$ at time $t$, and $j=1,..,J$ could indicate output and other
outcomes of interest in region $i$ at time $t$.
PVARs are generally represented as processes that depend on their
past, a series of unit-specific characteristics, and some random disturbances,
which leads to:
\begin{equation}
\begin{aligned}x_{i,t}=(I_{m}-\Phi)\mu_{i}+\Phi x_{i,t-1}+\tilde{x}_{i,t}\qquad & i=1,..,N\quad t=1,..,T\end{aligned}
\label{eq:main_PVAR}
\end{equation}
Here $\Phi$ denotes an $m\times m$ matrix of slope coefficients\footnote{This framework could be extended to random effects instead of fixed
effects. However, such change would have no impact on the nature of
the causal effects estimated.}, $\mu_{i}$ is an $m\times1$ vector of individual-specific effects,
$\tilde{x}_{i,t}$ is an $m\times1$ vector of disturbances, and $I_{m}$
denotes the identity matrix of dimension $m\times m$. The model can
be extended to include higher lags, but to keep the notation compact
I will use a one lag representation.
The focus of the following section will be on the disturbances
\[
\tilde{x}_{i,t}=(\widetilde{W}_{i,t}^{\prime},\widetilde{Y}_{j=1,i,t}^{\prime},\widetilde{Y}_{j=2,i,t}^{\prime},..,\widetilde{Y}_{j=J,i,t}^{\prime})',
\]
where the tilde represents the specific disturbance related to the
original variable. Such disturbances are distributed according to
$\widetilde{x}_{i,t}\sim\mathcal{N}(\boldsymbol{0}_{J+1},\Sigma)$
where $\mathcal{N}$ is a normal distribution. In the rest of the
paper will assume that the policy variable goes first\footnote{This is a standard assumption in the SVAR-IV literature (see \citet{StockWatson2018})
.}, and the outcome variables follow. For example, in the case of fiscal
multipliers this would mean that $\widetilde{W}_{i,t}$ refers to
the innovations of military spending growth in each region of the
US, and $\widetilde{Y}_{1,i,t}$ refers to the innovations of GDP
growth in those same regions. Exactly like vector autoregressions,
panel vector autoregressions have a contemporaneous causal representation
that is commonly defined as panel structural vector autoregression
(PSVAR), as follows
\begin{equation}
\begin{aligned}Rx_{i,t}=R(I_{m}-\Phi)\mu_{i}+R\Phi x_{i,t-1}+R\tilde{x}_{i,t}\qquad & i=1,..,N\quad t=1,..,T\end{aligned}
\label{eq:main_PVAR-structural}
\end{equation}
Generally, the estimation of the contemporaneous causal effects is
carried on transformations of $\Sigma=R^{-1}R^{-1\prime}$, where
$R^{-1}$ is the unique lower-triangular Cholesky factor with non-negative
diagonal elements. The reduced-form innovations $\widetilde{x}_{it}$
are related to the SPVAR shocks $\eta_{t}$ by an invertible matrix
$H$:
\[
\widetilde{x}_{i,t}=H\Gamma\eta_{i,t}=R^{-1}\eta_{i,t},\qquad\eta_{i,t}\sim(0,I_{J+1}),\qquad diag(H)=1,
\]
where $R^{-1}=H\Gamma$, and $\Gamma$ is a diagonal matrix with variance
of the shocks in the diagonal entries. The structural shocks $\eta_{i,t}$
are mean zero with unit variance, serially and mutually uncorrelated.
Since the autoregressive parameters $\hat{\Phi}$ can be consistently
estimated under regularity conditions, the sample residuals $\hat{\widetilde{x}}_{i,t}$
are consistent estimates of $\widetilde{x}_{i,t}$. The empirical
SPVAR problem reduces to finding $R$ from $\hat{\Phi}$. But there
are $(J+1)^{2}$ parameters in $R$ and the sample covariance of $\hat{\widetilde{x}}_{i,t}$
only provides $(J+1)((J+1)+1)$ conditions in face of $(J+1)^{2}$
parameters to be estimated. The SPVAR is therefore under-identified
as there can be infinitely many solutions that satisfy the covariance
restrictions.
The IV procedure for the estimation of the structural matrix in SVARs
generally corresponds to either one or two separate steps, depending
on whether the economist is interested in a unit normalized shock
or a standard shock\footnote{See \citet{gertler2015monetary,Mertens2013,Mertens2014,StockWatson2018,Olea2021}
for a full description of the procedure.}. First, an IV is estimated with the following first and second stages
\[
\begin{aligned}\widetilde{W}_{i,t}=\delta Z_{i,t}+\eta_{i,t} & \text{ first stage}\\
\widetilde{Y}_{j,i,t}=\beta_{j}\widetilde{W}_{i,t}+\epsilon_{i,t} & \text{ second stage(s)}
\end{aligned}
\]
Then, if the economist is interested in a unit normalized shock, the
IV estimator simply becomes $\beta_{j}^{IV}=(\rho_{j}/\delta)$, where
$\rho_{j}$ comes from the regression $\widetilde{Y}_{j,i,t}=\rho_{j}Z_{i,t}+\nu_{i,t}$,
$R_{1,1}=\delta$ ,and $R_{1,j}=\beta_{j}^{IV}$. In the case in which
the economist is interested in a standardized shock, instead, the
IV estimator becomes $\beta_{j}^{IV}=c_{j}(\rho_{j}/\delta)$.\footnote{Following \citet{gertler2015monetary}, consider the partition of
the covariance matrix of the residuals $R=[R_{1}R_{2}]=\left[\begin{array}{cc}
r_{11} & r_{12}\\
r_{21} & r_{22}
\end{array}\right]$, $Q=\frac{r_{21}}{r_{11}}\Sigma_{11}\frac{r_{21}^{\prime}}{r_{11}}-(\Sigma_{21}\frac{r_{21}^{\prime}}{r_{11}}+\frac{r_{21}}{r_{11}}\Sigma_{21}^{\prime})+\Sigma_{22}$,
and $r_{12}r_{12}^{\prime}=(\Sigma_{21}-\frac{r_{21}}{r_{11}}\Sigma_{11})^{\prime}Q^{-1}(\Sigma_{21}-\frac{r_{21}}{r_{11}}\Sigma_{11}).$
Then, it follows that, to obtain the structural form, $c_{j}=\sqrt{r_{12}}$
and $R_{1,1}=\delta\cdot c_{1}$ and $R_{1,k}=\rho_{k}c_{k}$.} In this case, the estimator is plugged in the covariance matrix as
an affine transformation that depends on a normalization that allows
to move to the reduced form and fully identifies the first column,
so that $R_{1,1}=\delta c_{1}$ and $R_{1,j}=\beta_{j}c_{j}$. More
simply, in the first case the estimator reduces to $\beta_{j}^{IV}=(\rho_{j}/\delta)$;
in the second case the normalization provided by the vector $c_{j}$
returns the modified estimator $\beta_{j}^{IV}=\frac{1}{\sqrt{\sigma_{\widetilde{W}_{it}}}}(\rho_{j}/\delta)$.
I will focus on the first case as it provides an interpretation that
is akin to the one of a standard IV estimator.
Finally, the following normalizing assumptions will be considered
to hold across the rest of the paper.
\begin{assumption}
\label{assu:(Normalising-assumptions):(1)-}\textbf{(Normalizing assumptions)}:\\
(1) $\widetilde{x}_{it}$ and $Z_{it}$ are stationary,\\
(2) $\widetilde{x}_{it}\sim\mathcal{N}(\boldsymbol{0}_{J+1},\Sigma)$
and $Z_{it}\sim\mathcal{N}(\mu_{Z},\sigma_{Z})$.
\end{assumption}
Notice that assumption \ref{assu:(Normalising-assumptions):(1)-}
is required for several different reasons. Part (1) allows to consider
the estimator of first stage and second stage(s) without violations
of the Wold theorem and mean that $R^{-1}$ is invertible and part (2) allows the derivative interpretation
of $\delta$ and $\rho_{j}$.
\begin{rem}
\textbf{(Why normality?)} While normality is not a necessary condition,
it has some important properties that are convenient when discussing
the estimators. In fact, alternative estimators that make different
assumptions about the distribution of $Z_{it}$ and $\widetilde{x}_{it}$
may be considered. For example, in the case in which $Z_{it}$ and
$\widetilde{W}_{it}$ are treatment dummy indicators, the theory goes
back to the traditional case of \citet{angrist1996identification},
and in the case in which $Z_{it}$ and $\widetilde{W}_{it}$ are multi-valued,
the theory goes back to the cases analyzed by \citet{vytlacil2002independence,heckman2001policy}.
\end{rem}
Under the assumptions laid out in section \ref{sec:Assumptions} it
can be shown that
\begin{thm}
\label{thm:PVAR-LATE}\textbf{(PSVAR-IV estimates a ratio of derivatives)}.
Under assumptions \ref{assu:(Independence)};\ref{assu:(Exclusion)};\ref{assu:(Monotonicity)-For-all};
\ref{assu:(Normalising-assumptions):(1)-};$\beta^{IV}$ estimates
\[
\beta_{j}^{IV}=\frac{\delta\mathbb{E}[\widetilde{Y}_{j}(z^{\circ})]/\delta z^{\circ}}{\delta\mathbb{E}[\widetilde{W}(z^{\circ})]/\delta z^{\circ}}.
\]
\end{thm}
Then, according to theorem \ref{thm:PVAR-LATE}, $\beta_{j}^{IV}$
simply captures the ratio of the effect of moving along different
values of $z^{\circ}$ on $\widetilde{Y}_{j}$ and on $\widetilde{W}$.
Therefore, a useful property of impulse response function can be established
according to the following theorem.
\begin{thm}
\label{thm:(Interpretation-of-the-IRF)}\textbf{(Interpretation of
the impulse response functions)}. The immediate impulse response function
of a shock in $\widetilde{W}$ captures
\[
\hat{\text{IRF}}_{j}=\mu-\text{LATE}_{j}=\mathbb{E}[\widetilde{Y}_{j}(\widetilde{w})-\widetilde{Y}_{j}(\widetilde{w}^{\prime})|\widetilde{W}(z)=\widetilde{w}]
\]
representing the difference between the shock being equal to $\widetilde{w}$
and $\widetilde{w}^{\prime}$.
\end{thm}
Notice that theorem \ref{thm:(Interpretation-of-the-IRF)} implies
that, considering two different impulse response functions, such as
the difference between a $1\%$ and a $0\%$ shock, results in the
$\mu-\text{LATE}$ that captures the difference of GDP growth for
those units that complies with the national spending growth. Hence,
the impulse response captures $\mathbb{E}[\widetilde{Y}_{j}(1\%)|\widetilde{W}(z)=1\%]$,
the impact of a one percent deviation in regional military spending
growth on GDP growth for those units that observed a spending increase.
\section{Inference \label{sec:Inference}}
Instrumental variables are useful only as far as they satisfy the
relevance condition (assumption \ref{assu:(Relevance).}). In fact,
it is easy to see that, being the IV estimator
\[
\beta_{j}^{IV}=(\rho_{j}/\delta),
\]
weak identification could be tested by the means of a null hypothesis
$H_{0}:\delta=0$. Hence, for $\delta\rightarrow0$, it must be that
either $\beta_{j}^{IV}\rightarrow\infty$ or $\beta_{j}^{IV}\rightarrow-\infty$
depending on the sign of $\rho_{j}$. For some time the general consensus
has been to carry two different inferential procedures: one in the
first stage, by the means of the Cragg-Donald statistic, frequently
defined as first-stage F-statsistic (\citet{StockStaiger1997}); and
one, separately, in the second stage, by the means of standard confidence
intervals or bootstrap. Such procedure heavily relied on the idea
that standard confidence intervals possess asymptotically good coverage
properties under the alternative hypothesis ($H_{1}:\delta\neq0$).
However, such approach does not cover situations in which the instrument
is weak but satisfies independence. Indeed, an approach that generates
confidence intervals on $\beta_{j}^{IV}$ on the basis of the strength
of the instrument, even when the first stage coefficient is near zero,
may be preferred to one that may end up discarding interesting research
hypothesis on the basis of a weak - but independent - instrument.
On the basis of such wisdom, the two step approach may be sub optimal
compared to the Anderson-Rubin statistic approach (\citet{Anderson1949,Olea2021,AndrewsStockSun2019,stock2002testing,mikusheva2006tests,mikusheva2010robust}).
\citet{mikusheva2010robust} introduces a procedure for generating
confidence sets for the second stage that have good coverage properties
even in the null hypothesis case.\footnote{Hence, for $\delta=0$, the confidence set would be distributed between
minus and plus infinity.} \citet{Olea2021} extend the result by introducing a confidence set
for the Impulse Response function generated by a SVAR-IV by using
the AR statistic.
While it is known that the Anderson-Rubin confidence sets are optimal
in the case of one instrument, there is yet to form a consensus about
which approach may be preferred in the case of two or more instruments
(some recent advancements include the CLR test of \citet{andrews2006optimal}).
Appendix \ref{sec:Appendix-B:-Monte} extends the conventional wisdom
present in the SVAR-IV field to the PSVAR-IV case using rotations
of the Anderson-Rubin statistic and demonstrates the good coverage
properties of the AR statistic.
\section{Estimation of the dynamic fiscal multiplier \label{sec:Fiscal-multiplier}}
The fiscal multiplier is generally defined as the coefficient a regression
with gdp as the dependent variable and government spending as the independent variable. Such measure is of great
interest because of its policy relevance: a relatively large fiscal
multiplier is often times evoked by governments as the reason to increase
spending\footnote{For example, \citet{Nakamura2014} observes that the American Recovery
and Reinvestment Act (ARRA) was justified on the basis of large estimates
of the fiscal multiplier.}.
The aggregate fiscal multiplier is generally computed using vector autoregressions (\citet{romer2010macroeconomic,Blanchard2002}) or local projections (\citet{Ramey2018}). Normally, the aggregate fiscal multiplier was found to be rarely above one.
A local fiscal multiplier could be preferred to the aggregate fiscal multiplier for several reasons. First, the assumptions required to obtain an unbiased estimand are less restrictive than their aggregate counterpart. Indeed, the computation of a national aggregate fiscal multiplier often poses some credibility issues due to the unreliability
of the underlying assumptions. The fiscal policy literature has therefore
explored the quantification of the impact of a fiscal expansion on
GDP by using more granular and localized data, either at the state
or regional-level. Second, the open economy multiplier can be potentially more interesting to central bankers because, by using state heterogeneity, it is essentially independent of monetary policy, as overnight rates are fixed for all states.
The recent emergence of this literature has generated
different relevant contributions that seem to indicate a regional
fiscal multiplier of about 1.5 (\citet{farhi2016fiscal,Nakamura2014,shoag2010impact,chodorow2019geographic}).
To the best of my knowledge, there is currently little work done in
estimating the dynamic \emph{regional} fiscal multiplier with the
same type of external instrument approach that has characterized the
aggregate data literature. Perhaps closer to the main idea of this
paper is \citet{dupor2023regional}, which estimates the dynamic regional
fiscal multiplier using a model to frame the impact of the ARRA. Yet,
the ARRA is representative of a particular context of the US economy
of low inflation and low interest rates, and may not be representative
of different state dependencies.
Motivated by the lack of empirical evidence at the intersection of
the two streams of literature, I estimate a regional fiscal multiplier
for the US using aggregate national military spending as an instrument
for the innovations of regional military spending. Two cautions are
however invited to the reader. First, the fiscal multiplier identified
using military spending data is particularly useful, but may not be
representative of a generic spending multiplier. It is useful as it
is inherently a measure of direct spending of the US government (\citet{Nakamura2014});
but it is not representative as it does not include \emph{all} the
government spending (\citet{koo2023impulseresponseinstrumentalvariables}.
In the case of panel vector autoregressions, the dynamic regional
fiscal multiplier can be estimated by simply by running a PVAR estimation
on the vector
\[
x_{it}=(\frac{\text{exp}_{it}-\text{exp}_{it-1}}{\text{gdp}_{it-1}}^{\prime},\frac{\text{gdp}_{it}-\text{gdp}_{it-1}}{\text{gdp}_{it-1}}^{\prime})^{\prime}.
\]
Here, the issue of endogeneity arises because the contemporaneous
innovations of fiscal expanses growth may not be thought as exogenous
with respect to the contemporaneous innovations of GDP growth.
The leading assumption for the case of PVAR therefore is that the
United States do not embark on military buildups because states that
receive a disproportionate amount of military spending are doing\emph{
more poorly than before} relative to other states. To exploit this
assumption, I use data from the US extracted from the electronic database
of DD-350 military procurement forms available from the US Department
of Defense by \citet{Nakamura2014}, which includes military spending
for equipment of 10000\$ or more in the period 1966-1984 and above
100000\$ in the period until 2006.\footnote{Unfortunately, the data is not updated any further.}
The rest of the analysis follows \citet{Nakamura2014} fairly closely:
the data is at a yearly frequency, and region
refers to the aggregation of different states that are close and not
densely populated, resulting in 10 different macro-regions and 39
different years. Differently from the original paper, however, the
main estimation makes use of the variable's growth with respect to
the previous year, rather than the previous two years; and the Bartik/shift-share
instrument is given a preference over using 10 different instruments
(one for each state). The reason I made such choice is that conventionally
time series regressions are framed in terms of growth with respect
to the previous period, and a one dimensional instrument is known
to have an optimal confidence set, whereas the case of multiple instruments
may provide less reliable confidence sets.\footnote{In the appendix, I show that alternative formulations using growth
with respect to two previous periods may change the results slightly.
While the quantities tend to be similar in the impulse response, the
mechanism by which fiscal expansions tend to be associated with an
increase in output in the following year is by the means of a large
output autocorrelation, rather than a direct correlation between output
and past expanses.}
The model is estimated as follows. First, I use a one-lag model as
suggested by the MAIC, MBIC, MHIQ from table \ref{tab:MAIC,-BBIC,-MHIQ}.
\begin{table}
\begin{centering}
\begin{tabular}{cccc}
& & & \tabularnewline
\hline
\hline
& MBIC & MAIC & MQIC\tabularnewline
$p=1$ & -24.08 & 7.00 & -5.35\tabularnewline
$p=2$ & -23.02 & -7.48 & -13.66\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\caption{MAIC, BBIC, MHIQ tests. \label{tab:MAIC,-BBIC,-MHIQ}}
\end{table}
The residuals plotted in Figure \ref{fig:Regression-of-the-residuals-and-lags}.
\begin{figure}
\centering{}\includegraphics[scale=0.6]{scatter_residuals_IV.pdf}\caption{Regression of the residuals and their lags. \label{fig:Regression-of-the-residuals-and-lags}}
\end{figure}
From the figure, there appears to be no indication of residual autocorrelation.
This is confirmed by the regression coefficients obtained by regressing
the residuals against their lags. The coefficients, being not statistically
different from zero, do not seem to suggest to reject the null hypothesis
of a statistically significant relationship between the residuals
and their lags. Finally, I turn to the assumption of normality of
the model, discussed in assumption \ref{assu:(Normalising-assumptions):(1)-}.
In fact, violations of assumption \ref{assu:(Normalising-assumptions):(1)-}(iii)
would suggest that non-parametric estimators could be preferred over
the ones implied by the 2sls utilized for the IV regression because
of the assumption that are required from parametric estimators. The
histograms displaying the error's distribution in figure \ref{fig:Distribution-of-residuals}
seem to indicate that the residuals may be normally distributed and
therefore continuously differentiable with appropriate weights, indicating
the adequateness of the normality assumption.\footnote{Several statistical tests (such as \citet{shapiro1965analysis,shapiro1972approximate})
with the null hypothesis of normality fail to reject the null, indicating
that the residuals may indeed be normally distributed.}
\begin{figure}
\begin{centering}
\includegraphics[scale=0.6]{Histograms_normal.pdf}
\par\end{centering}
\caption{Distribution of the residuals and of the national military spending
instruments. \label{fig:Distribution-of-residuals}}
\end{figure}
The results from the IV regression are instead displayed in Table
\ref{tab:Table-IV}. The coefficient from the first stage is statistically
significantly different from zero, and the first stage F-statistic
is above the commonly advocated threshold level of 10 (see \citet{StockStaiger1997}).
Moreover, the AR statistic is above the $\chi_{1,1-\alpha}$ critical
value for $\alpha=.05$, suggesting that the results may be statistically
significantly different from zero.
\begin{table}
\begin{centering}
\begin{tabular}{cc|ccc}
\hline
\multicolumn{2}{|c|}{\textsc{first stage}} & \multicolumn{3}{c|}{\textsc{Second stage}}\tabularnewline
\hline
\hline
& \textsl{State spending} & & & \textsl{GDP growth}\tabularnewline
\textsl{Total spending} & 0.46 & & \textsl{State spending} & 1.74\tabularnewline
\textsl{Standard CI} & {[}0.37 0.54{]} & & \textsl{Anderson-Rubin CS} & {[}0.53 3.05{]}\tabularnewline
\textsl{Fist stage F statistic} & 120.86 & & \textsl{Anderson-Rubin statistic} & 92.63\tabularnewline
\hline
\hline
& \multicolumn{1}{c}{} & & & \tabularnewline
\end{tabular}
\par\end{centering}
\caption{The first column of this table reports the result of the regression
of state spending innovations on total spending, with standard confidence
intervals and the first stage F-statistic. The second column of the
table instead reports the results of the 2sls of GDP growth innovations
on the instrumented state spending innovation, with the confidence
sets built using the Anderson-Rubin statistic and the Anderson-Rubin
statistic itself. In both cases, the confidence interval and the confidence
set are at the 95\% level.\label{tab:Table-IV}}
\end{table}
Finally, consider figure \ref{fig:Impulse-response-functions}, which
displays the impulse response functions of a 1\% shock in fiscal spending
growth. The results are similar to the ones of the literature, suggesting
a value of the fiscal multiplier of approximately $\sim1.7$ in the
first period\footnote{Notice that, by definition, the IRF on impact is the second stage
regression in table \ref{tab:Table-IV}.}. However, the dynamic fiscal multiplier displays an interesting feature,
as it appears that the impact of a change in the fiscal spending in
year $t$ results in a corresponding increase in output growth by
$\sim1.5$. To better highlight the mechanism by which such response
happens, table \ref{tab:Autoregressive-coefficients-of-PVAR-main}
displays the AR coefficients. The high correlation between GDP growth
and fiscal policy in the previous period is the main mechanism by
which the fiscal multiplier can result in a GDP growth that may last
for more than one year. Moreover, fiscal policy tends to not be particularly
correlated with past fiscal policy or output, resulting in a response
close to zero in the second horizon.
\begin{table}
\begin{centering}
\begin{tabular}{|c|cc|}
\multicolumn{1}{c}{} & \multicolumn{2}{c}{}\tabularnewline
\cline{2-3} \cline{3-3}
\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\widetilde{\frac{\text{exp}_{it-1}-\text{exp}_{it-2}}{\text{gdp}_{it-2}}}$} & $\widetilde{\frac{\text{gdp}_{it-1}-\text{gdp}_{it-2}}{\text{gdp}_{it-2}}}$\tabularnewline
\hline
$\widetilde{\frac{\text{exp}_{it}-\text{exp}_{it-1}}{\text{gdp}_{it-1}}}$ & $-0.10$ & $-0.03^{\ast\ast\ast}$\tabularnewline
\multicolumn{1}{|c}{} & $[-0.28,0.06]$ & $[-0.05,-0.01]$\tabularnewline
\hline
$\widetilde{\frac{\text{gdp}_{it}-\text{gdp}_{it-1}}{\text{gdp}_{it-1}}}$ & $0.80^{\ast\ast\ast}$ & $0.34^{\ast\ast\ast}$\tabularnewline
\multicolumn{1}{|c}{} & $[0.20,1.41]$ & $[0.21,0.46]$\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\caption{Autoregressive coefficients of the PVAR. The confidence interval represent
are set at the $95\%$ level.\label{tab:Autoregressive-coefficients-of-PVAR-main} }
\end{table}
\begin{figure}
\begin{centering}
\includegraphics[scale=0.6]{IRF.pdf}
\par\end{centering}
\caption{\label{fig:Impulse-response-functions}Impulse response functions
of regional military spending and regional GDP growth to a 1\% shock
in regional military spending. The confidence sets are built using
the Anderson-Rubin test statistic developed in section \ref{sec:Inference}
at the 95\% level.}
\end{figure}
\section{Conclusions\label{sec:Conclusions}}
This paper discussed the causal interpretation of panel vector autoregressions
identified by the means of external instruments. The IRF generated
by a PVAR can estimate a LATE representing the difference between
the outcome variable under a treatment and no treatment status for
the compilers. However, such LATE needs to be read differently from
the panel linear regression literature, as it refers to the residuals
and emerges as a counterfactual assignment of different predictions,
such as a 1\% shock versus a 0\% shock. I have discussed under which
assumptions the LATE may be captured: independence, exclusion, and
monotonicity. Some drawbacks of the proposed identification scheme
include the severity of the parametric linear nature of the monotonicity
assumption.
Moreover, I discussed the best approaches to conduct inference in
a PVAR identified using external instrument. In appendix \ref{sec:Appendix-B:-Monte}
I showcase the good small sample properties of the AR confidence sets
calibrating a simulation on the basis of the dataset from the application.
Finally, I have applied these tools to the estimation of a dynamic
regional fiscal multiplier for the United States, a quantity that
has been rarely targeted by the literature. My empirical findings
suggest that the dynamic regional fiscal multiplier may be above one
in the second period, indicating some possibly longer term effects
of fiscal expansions on GDP growth.
Future researchers are invited to develop two points. First, the $\mu-\text{LATE}$
interpretation of the PSVAR-IV relies on an underlying linearity assumption.
Yet, non-parametric estimators, which could potentially alleviate
the linearity assumption, are never utilized in the SVAR nor the PSVAR
literature. If the data utilised is sufficiently large, such methods
could be further explored. Second, because the IV literature is still
uncertain about which statistics to use when dealing with multiple
instruments, the inference issue of overidentification naturally carry
to SVAR-IV and PSVAR-IV. Hence, future researchers should properly
discuss the unreliability of confidence sets in such cases and possibly
implement novel methodologies with better coverage properties.
\newpage
\bibliographystyle{apalike}
\bibliography{bibliography}
\newpage