# What Impulse Response Do Instrumental Variables Identify?

$\begin{array}[c]{ccccc}\text{Bonsoo Koo}\thanks{Monash University, Department
of Econometrics and Business Statistics,
<EMAIL_ADDRESS>Lee}\thanks{Seoul National University,
Department of Economics<EMAIL_ADDRESS>Hwan
Seo}\thanks{Seoul National University, Department of Economics,
<EMAIL_ADDRESS>Monash University, Department of
Econometrics and Business Statistics<EMAIL_ADDRESS>National
University, Department of Economics<EMAIL_ADDRESS>National
University, Department of Economics<EMAIL_ADDRESS>
(August 22, 2022111We acknowledge that this research is supported under the
Australian Research Council Discovery Project funding scheme (project number
DP210101440) and by the Seoul National University Research Grant in 2021.)

Instrumental variables (IV) are often used to provide exogenous variation in
the impulse response analysis but the heterogeneous effects the IV may
identify are rarely discussed. In microeconometrics, on the other hand, it is
well understood that an IV identifies the local average treatment effect
(Imbens and Angrist, 1994). Recognizing that macro shocks are often composites
(e.g., the government spending shock is the sum of sectoral spending shocks),
we show that the IV estimand can be written as a weighted average of
componentwise impulse responses. Negative weights can arise if the
correlations between the IV and the components of the macro shock have
different signs. This implies that the response of the variable of interest to
an IV shock may evolve in an arbitrary way regardless of the true underlying
componentwise impulse responses. We provide conditions under which the IV
estimand has a structural interpretation. When additional IVs or auxiliary
disaggregated data are available, we derive informative bounds on the
componentwise impulse responses. With multiple IVs and disaggregated data, the
impulse responses can be point-identified. To demonstrate the empirical
relevance of our theory, we revisit the IV estimation of the government
spending multiplier in the US. We show that the non-defense spending
multiplier larger than one can be obtained from the conventional IV estimates
of the aggregate spending multiplier, which are smaller than one.

Keywords: local projection, local average treatment effect, structural vector
moving average, instrumental variables, sectoral heterogeneity, impulse
response, government spending multiplier

## 1 Introduction

In line with recent advances in microeconometric identification, the
contemporary applied macroeconometrics literature has observed that an
external intervention by an instrumental variable (IV) generates exogenous
variation to the endogenous variable (e.g., government spending) enabling to
recover the impulse response of interest (e.g. government spending multiplier)
consistently. There has been growing attention to identification of various
policy multipliers by external IVs in the macroeconomics literature. Examples
include the proxy structural vector autoregressions (SVAR) (Stock and Watson,
2012; Mertens and Ravn, 2013) and local projections (LP) using instrumental
variables of Jordà (2005), Jordà, Schularick, and Taylor (2015, 2020), Stock
and Watson (2018), and Plagborg-Møller and Wolf (2021a). Yet, the current
literature on the identification builds on a strongly homogeneous structure.

Imbens and Angrist (1994) showed that a valid IV alone could only identify the
local average treatment effect (LATE) in the potential outcomes framework. In
the cross-sectional setting, it is natural to think of heterogeneity as
heterogeneous individual treatment effects. However, the notion of
heterogeneity in impulse response functions is not clearly established, and
little is known about what an IV identifies in the presence of heterogeneity
in the macroeconomic shock. We propose a formal theoretical framework that
allows heterogeneity in impulse responses, by establishing identifiability of
the IV estimator, and by providing conditions for identification of a LATE-
like IV estimand of impulse responses.

To this end, we adopt the non-invertible structural vector moving average
(SVMA) model to investigate identifiability of the structural parameters in
the model that determine the impulse responses and policy multipliers. It is
widely used to model the dynamics of a set of observable endogenous variables
in the macroeconomic system and impulse responses to macroeconomic shocks. It
proves to be particularly useful to compare different estimates for impulse
response functions from SVAR or LP in the recent literature; see Stock and
Watson (2018) and Plagborg-Møller and Wolf (2021a).

We introduce heterogeneity in the impulse response by noting that a one unit
macroeconomic shock to the endogenous variable can be generated by various
compositions of multiple structural shocks in the SVMA model. Therefore, the
composite macroeconomic shock can be viewed as a heterogeneous treatment. This
is possible due to the flexible nature of the SVMA model that allows for a
larger number of structural shocks than the observable endogenous variables.

To be specific, consider the response of GDP to a unit change in the
government spending shock, which is a composite shock consisting sectoral
spending shocks. A one unit exogenous change in government spending can
therefore be comprised of any combination of spending shocks from different
sectors. For example, it may consist entirely of defense spending, or some
combination of defense and non-defense spending. If GDP responds to defense
and non-defense spending shocks differently, then the response of GDP to the
government spending shock would depend on the structure of this composition.
It is this structure of the composition of how different sectoral shocks make
up the one unit government spending shock that we define as the heterogeneous
treatment. Then, the impulse response to the composite shock may be written as
a weighted sum of componentwise impulse responses, where the weights are
determined by the joint distribution of the component shocks.

Defining the composition of the macroeconomic shock as the treatment is a
distinctive feature of our approach from the potential outcomes framework in
Imbens and Angrist (1994), that is, the set of treatment is not a singleton or
finite but a continuum. Furthermore, the treatment is heterogeneous, but the
treatment effect (impulse response) is homogeneous conditional on the
composition of the shock. This is contrasted with the potential outcomes
framework where the treatment is homogeneous, but its effect is heterogeneous.
Angrist and Kuersteiner (2011) and Rambachan and Shephard (2021) extend the
cross-sectional potential outcomes framework to the time series setting.
Similarities and differences in the identified structural parameters and
identifying conditions between ours and the potential outcomes framework are
discussed in detail in Section 2.2.

An exogenous macroeconomic shock is not usually observed, so an external IV is
used. The IV is correlated with the macroeconomic shock in possibly different
degrees and signs of correlation with its components. In this sense, the IV
selects a particular composition (treatment). The treatment may be different
from that of another IV or from a policy rule. As a result, two valid IV’s may
identify completely different impulse responses. We refer to the impulse
response identified by an IV as the IV-impulse response.

Under treatment heterogeneity, we find that the IV-impulse response may not
identify a weighted average of componentwise impulse responses with non-
negative weights in general, contrary to common belief in the literature. For
instance, the government spending multiplier estimated by the military news
shock IV of Ramey and Zubairy (2018) may not be a proper weighted average of
the defense and non-defense spending multipliers, i.e., the weights can be
negative. Consequently, our result has an important implication that the
commonly used local projections-IV (LP-IV) estimate or more generally, the IV-
impulse response may not be an informative quantity and hence may well be
misleading to infer on the average of the componentwise impulse responses.

Only under the condition that the correlation between the instrument and each
of the shock components has the same sign the IV-impulse response identifies
an instrument-specific weighted average of structural componentwise impulse
responses of the macroeconomic variable of interest with non-negative weights.
This same-sign condition may be comparable to the so-called monotonicity
condition for identification of LATE by an IV. Unlike the monotonicity
assumption, we can sometimes test the same-sign condition when auxiliary
disaggregated variables are available. We also note that the same-sign
condition is not sensible for some IVs in the empirical literature.

Furthermore, we develop an identification-bounds analysis for the
componentwise impulse responses when additional IV’s or auxiliary
disaggregated data are available (but maybe not both). This helps obtain
informative bounds on the policy multipliers, in which the policy maker knows
the composition of the policy shock. With auxiliary sectoral data and some
structural restrictions on the SVMA model as well as multiple IV’s, we show
that componentwise impulse responses can be point-identified.

To demonstrate the empirical relevance of our theoretical results, we revisit
the IV estimation of the government spending multiplier in the US. Using the
military news shock of Ramey and Zubairy (2018) and the current defense
spending shock of Blanchard and Perotti (2002), we decompose the LP-IV
estimates of the government spending multipliers documented in Ramey and
Zubairy (2018) to the defense and non-defense spending multipliers. We find
that non-defense spending can have a prolonged effect, and its multiplier can
be larger than one. This finding is consistent with Auerbach and Gorodnichenko
(2012) and Nakamura and Steinsson (2014), where the former presents the
magnitude of the five-year cumulative non-defense spending multiplier to be
1.58 using the SVAR model and the latter estimates the open economy relative
multiplier of about 1.5 using regional variation.

## 2 Identification by IV-Impulse Response

###  2.1 Structural Vector Moving Average and Composite Shock

Let $\bm{Y}_{t}$ be an $n\times 1$ vector of observed endogenous variables and
let $\bm{\varepsilon}_{t}$ be an $m\times 1$ vector of unobserved structural
shocks. The endogenous variables $\bm{Y}_{t}$ is written as a linear
combination of current and past $\bm{\varepsilon}_{t}$:

$\bm{Y}_{t}=\bm{\Theta}(L)\bm{\varepsilon}_{t}$ (1)

where $L$ is the lag operator,
$\bm{\Theta}(L)=\bm{\Theta}_{0}+\bm{\Theta}_{1}L+\bm{\Theta}_{2}L^{2}+\cdots$,
and $\bm{\Theta}_{h}$ for $h=0,1,2,...$ is an $n\times m$ matrix of impulse
response coefficients. We assume that $E[\bm{\varepsilon}_{t}]=0$,
$E[\bm{\varepsilon}_{t}\bm{\varepsilon}_{t}^{\prime}]>0$, and the shocks are
mutually uncorrelated. A word on notation: we reserve the index ‘$s$’ to
indicate an element in the system (1), e.g., $\varepsilon_{s,t}$ then denotes
the $s$-th element of $\varepsilon_{t}$.

Let $y_{t}$ be the last element of $\bm{Y}_{t}$ without loss of generality. We
define the impulse response of $y_{t}$ at horizon $h$ to the shock
$\varepsilon_{s,t}$ as

$\theta_{h,ys}\equiv
E[y_{t+h}|\varepsilon_{s,t}=1]-E[y_{t+h}|\varepsilon_{s,t}=0].$

Here we are slightly abusing notation by letting $\theta_{h,ys}$ be the
$(n,s)$-th element of $\bm{\Theta}_{h}$. This is to emphasize that $y_{t}$ is
the main variable of interest.

The researcher is interested in the impulse response of $y_{t}$ at horizon $h$
to a macroeconomic shock $\xi_{t}$,
$E[y_{t+h}|\xi_{t}=1]-E[y_{t+h}|\xi_{t}=0]$. The shock $\xi_{t}$ may not be an
element of $\varepsilon_{t}$ but a composite shock that consists of multiple
structural shocks $\varepsilon_{s,t}$ for
$s\in\mathcal{S}_{\xi}\subseteq\\{1,2,...,m\\}$. Without loss of generality,
let the elements of $\mathcal{S}_{\xi}$ be the first $S$ elements of
$\bm{\varepsilon}_{t}$, so that we can write

$\xi_{t}=\sum_{s=1}^{S}\varepsilon_{s,t}.$

Conventionally in the literature, the shock $\xi_{t}$ has been often treated
as a single unit as if it consisted of identical structural shocks. Our
definition of $\xi_{t}$ is relevant for many empirical macro policy analyses
in the literature. Below we provide some examples of composite shocks.

  * Example 1 (Sectoral government spending)

Government spending can be categorized by different sectors, such as defense
spending and non-defense spending or, more granularly into education, social
security, social infrastructure, health care, and defense etc. $\xi_{t}$ can
be the aggregate government spending shock and $\varepsilon_{s,t}$ are
disaggregated sectoral spending shocks in this case. There exists an
increasing interest in the sectoral aspect of government spending. For
instance, Auerbach and Gorodnichenko (2012), Cox et al. (2020), Bouakez,
Rachedi and Santoro (2020), Woodford (2020), and Canova (2022) among many
others.

  * Example 2 (Forward guidance monetary policy)

Under the environment of the close-to-zero interest rates, forward guidance
such as public statements of monetary policy committee (MPC) meetings, as well
as lectures and seminars of MPC members could be used to substitute lower
rates at the zero bound. The literature classifies forward guidance into two
types: Odyssean forward guidance and Delphic forward guidance, and it is
believed that both types can bring about different monetary policy shocks (see
Campbell, Evans, Fisher, and Justiniano (2012) for more details). In this
case, $\xi_{t}$ is the aggregate forward guidance shock consisting of two
distinctive shocks ($\varepsilon_{1,t}$ and $\varepsilon_{2,t}$) from Odyssean
and Delphic forward guidance respectively.

When researchers consider the impulse response of $y_{t+h}$ to a composite
shock $\xi_{t}$, what they wish to identify is some weighted average of
$\theta_{h,ys}$ for $s=1,2,...,S$. To illustrate this, assume that the shocks
are discrete random variables. Further assuming the convention that
$E[y_{t+h}|\xi_{t}=0]$ corresponds to
$\bm{e}_{t}\equiv(\varepsilon_{1,t},\varepsilon_{2,t},\cdots,\varepsilon_{S,t})^{\prime}=\bm{0}$,
we can write

$\displaystyle E[y_{t+h}|\xi_{t}=1]-E[y_{t+h}|\xi_{t}=0]=$
$\displaystyle\sum_{\bm{a}\in\\{\xi_{t}=1\\}}\frac{P(\bm{e}_{t}=\bm{a})}{P(\\{\xi_{t}=1\\})}\left(E[y_{t+h}|\bm{e}_{t}=\bm{a}]-E[y_{t+h}|\bm{e}_{t}=\bm{0}]\right)$
(2) $\displaystyle=$
$\displaystyle\sum_{\bm{a}\in\\{\xi_{t}=1\\}}\frac{P(\bm{e}_{t}=\bm{a})}{P(\\{\xi_{t}=1\\})}\bm{a}^{\prime}\bm{\theta}_{h,y}$
$\displaystyle=$ $\displaystyle
E[\bm{e}_{t}|\xi_{t}=1]^{\prime}\bm{\theta}_{h,y}$ (3)

where $\bm{a}$ is an $S\times 1$ non-random vector, $\\{\xi_{t}=1\\}$ is a
collection of realizations of $\bm{e}_{t}$ such that $\xi_{t}=1$, and
$\bm{\theta}_{h,y}=(\theta_{h,y1},\theta_{h,ys},...,\theta_{h,yS})^{\prime}$.
Thus, the left-hand side (LHS) of (2) is a weighted average of componentwise
impulse responses and the weights are the mean of $\varepsilon_{s,t}$
conditional on $\xi_{t}=1$ which are non-negative for a large class of
reasonable distributions of $\bm{e}_{t}$. For example, if we assume
$\varepsilon_{s,t}$ for $s=1,...,S$ are i.i.d., then
$E[\varepsilon_{s,t}|\xi_{t}=1]=1/S$ and (3) becomes the equal-weighted
average.

The impulse response (3) would be identified if $\xi_{t}$ were observed, but
typically $\xi_{t}$ is unobserved and its scale is indeterminate. A common
solution to this problem is to measure the magnitude of $\xi_{t}$ by means of
an observable endogenous variable $x_{t}$ and then to use an instrument
$z_{t}$ correlated with $x_{t}$ to get an exogenous variation. As a result,
the response of $y_{t+h}$ to $z_{t}$ relative to the response of $x_{t}$ to
$z_{t}$ can be interpreted as the average response of $y_{t+h}$ to $\xi_{t}$
of a magnitude that corresponds to a unit change in $x_{t}$. Stock and Watson
(2018) provide an example that the causal effect of GDP growth ($y_{t}$) to a
monetary policy shock ($\xi_{t}$) can be identified by the ratio of the
impulse responses of GDP growth and the federal fund rate ($x_{t}$) to a
monetary policy announcement ($z_{t}$). Ramey and Zubairy (2018) calculate the
government spending multiplier as a ratio of the impulse responses of GDP
($y_{t}$) and total government spending ($x_{t}$) to the military news shock
($z_{t}$).

Let $x_{t}$ be the first element of $\bm{Y}_{t}$ without loss of generality.
Similar to $y_{t}$, we define the impulse response of $x_{t}$ at horizon $h$
to the shock $\varepsilon_{s,t}$ as $\theta_{h,xs}$ by slightly abusing
notation. To fix the scale of $\varepsilon_{s,t}$, we assume that
$\theta_{0,xs}=1$ for all $s=1,2,...,S$, which is referred to as the unit
effect normalization in the literature (Stock and Watson, 2018). In the
government spending shock example, the unit effect normalization means a unit
change in a sectoral spending shock changes the total spending by one unit.

###  2.2 IV-Impulse Response by Local Projections

Let $z_{t}$ be an IV that satisfies the following assumptions.

######  Assumption 1.

  1. (i)

$E[z_{t}\xi_{t}]\neq 0$ (relevance)

  2. (ii)

$E[z_{t}\varepsilon_{s,t}]=0$ for all $s=S+1,S+2,...,m$ (contemporaneous
exogeneity)

  3. (iii)

$E[z_{t}\bm{\varepsilon}_{t+j}]=0$ for $j\neq 0$ (lead-lag exogeneity)

Define $E[z_{t}\varepsilon_{s,t}]=\alpha_{s}$. Assumption 1(i) implies that
$\alpha_{s}\neq 0$ for some $s=1,2,...,S$. Assumption 1 is an extension of
Condition LP-IV of Stock and Watson (2018) allowing that more than one
structural shocks are correlated with the instrument $z_{t}$ and the
correlations are heterogeneous across the shocks.

For illustration purposes, let us assume that the instrument $z_{t}$ is binary
and $0<P(z_{t}=1)<1$. Write $P_{z}=P(z_{t}=1)$. Also assume that $S=2$ so that
$\xi_{t}$ is the sum of two structural shocks, $\varepsilon_{1,t}$ and
$\varepsilon_{2,t}$. A direct IV estimator of the impulse response is the LP-
IV estimator, whose population version is given by

$\beta_{h}^{IV}\equiv\frac{E[y_{t+h}|z_{t}=1]-E[y_{t+h}|z_{t}=0]}{E[x_{t}|z_{t}=1]-E[x_{t}|z_{t}=0]}.$

The LP-IV estimand $\beta_{h}^{IV}$ is the ratio of two IV-impulse responses
and it corresponds to the Wald estimand with a binary instrument in the
microeconometrics literature. Since $y_{t+h}$ is a linear combination of
$\bm{\varepsilon}_{t+h-j}$ for $j=0,1,2,...$ by (1), the numerator of
$\beta_{h}^{IV}$ can be written as

$E[y_{t+h}|z_{t}=1]-E[y_{t+h}|z_{t}=0]=\sum_{s=1}^{2}(E[\varepsilon_{s,t}|z_{t}=1]-E[\varepsilon_{s,t}|z_{t}=0])\theta_{h,ys}=\frac{\alpha_{1}\theta_{h,y1}+\alpha_{2}\theta_{h,y2}}{P_{z}(1-P_{z})}$

under Assumption 1. Likewise,

$E[x_{t}|z_{t}=1]-E[x_{t}|z_{t}=0]=\sum_{s=1}^{2}(E[\varepsilon_{s,t}|z_{t}=1]-E[\varepsilon_{s,t}|z_{t}=0])\theta_{0,xs}=\frac{\alpha_{1}+\alpha_{2}}{P_{z}(1-P_{z})},$

under the unit effect normalization. As a result, $\beta_{h}^{IV}$ can be
written as

$\frac{E[y_{t+h}|z_{t}=1]-E[y_{t+h}|z_{t}=0]}{E[x_{t}|z_{t}=1]-E[x_{t}|z_{t}=0]}=\frac{\alpha_{1}}{\alpha_{1}+\alpha_{2}}\theta_{h,y1}+\frac{\alpha_{2}}{\alpha_{1}+\alpha_{2}}\theta_{h,y2},$
(4)

which is an affine combination of the impulse responses $\theta_{h,y1}$ and
$\theta_{h,y2}$. Equation (4) clearly shows that without a restriction on
$\alpha_{s}$ we would not be able to interpret $\beta_{h}^{IV}$ as a
structural parameter. To see this, suppose that $\theta_{h,y1}=1$ and
$\theta_{h,y2}=2$ so the impulse responses are both positive. However, if
$\alpha_{1}=2$ and $\alpha_{2}=-1$, we have $\beta_{h}^{IV}=0$. This can
happen if the instrument $z_{t}$ is correlated with the two structural shocks,
$\varepsilon_{1,t}$ and $\varepsilon_{2,t}$, in the opposite direction.
Without a restriction on $\alpha_{s}$, $\beta_{h}^{IV}$ can be any real
number.

We do not have this problem when $S=1$. This special case corresponds to the
standard LP-IV setup of Stock and Watson (2018). When the composite shock
consists of one structural shock, $\xi_{t}=\varepsilon_{s,t}$ and the LP-IV
estimand $\beta_{h}^{IV}=\theta_{h,ys}$.

To interpret $\beta_{h}^{IV}$ as a meaningful average of the underlying
structural impulse responses, we require the following assumption.

######  Assumption 2.

For all $s=1,2,...,S$, either $\alpha_{s}\geq 0$ or $\alpha_{s}\leq 0$. (same
sign)

Assumption 2 restricts the way the instrument $z_{t}$ is correlated with the
endogenous variable $x_{t}$ via the relevant structural shocks
$\varepsilon_{s,t}$. Assumption 2 is analogous to the monotonicity assumption
for identification of LATE (Imbens and Angrist, 1994). We provide further
discussion of this assumption after Proposition 1.

The following proposition establishes that the LP-IV estimand with a general
non-binary IV is an affine combination of the structural impulse responses,
and with Assumption 2 it becomes a weighted average with non-negative weights
(i.e. convex combination).

######  Proposition 1.

If random variables $y_{t}$ and $x_{t}$ are elements of $\bm{Y}_{t}$ generated
according to (1) and a random variable $z_{t}$ satisfies Assumption 1, then
for $h=0,1,2,...$,

$\beta_{h}\equiv\frac{Cov(y_{t+h},z_{t})}{Cov(x_{t},z_{t})}=\sum_{s=1}^{S}w_{s}\theta_{h,ys},$
(5)

under the unit effect normalization, where

$w_{s}=\frac{\alpha_{s}}{\sum_{s^{\prime}=1}^{S}\alpha_{s^{\prime}}}.$

Furthermore, $w_{s}\geq 0$ for all $s=1,...,S$ if $z_{t}$ further satisfies
Assumption 2.

Proof. Write $y_{t+h}$ and $x_{t+h}$ as

$\displaystyle y_{t+h}=$
$\displaystyle\sum_{j=0}^{\infty}\left(\sum_{s=1}^{S}\theta_{j,ys}\varepsilon_{s,t+h-j}+\sum_{r=S+1}^{m}\theta_{j,yr}\varepsilon_{r,t+h-j}\right),$
$\displaystyle x_{t+h}=$
$\displaystyle\sum_{j=0}^{\infty}\left(\sum_{s=1}^{S}\theta_{j,xs}\varepsilon_{s,t+h-j}+\sum_{r=S+1}^{m}\theta_{j,xr}\varepsilon_{r,t+h-j}\right).$

By Assumption 1 and the unit effect normalization, we obtain

$\displaystyle Cov(y_{t+h},z_{t})=$ $\displaystyle
E[y_{t+h}z_{t}]=\sum_{s=1}^{S}E[z_{t}\varepsilon_{s,t}]\theta_{h,ys}=\sum_{s=1}^{S}\alpha_{s}\theta_{h,ys},$
$\displaystyle Cov(x_{t},z_{t})=$ $\displaystyle
E[x_{t}z_{t}]=\sum_{s=1}^{S}E[z_{t}\varepsilon_{s,t}]\theta_{0,xs}=\sum_{s=1}^{S}\alpha_{s},$

as desired. ∎

When there is only one component in $\xi_{t}$ ($S=1$), (5) simplifies to
Equation (8) of Stock and Watson (2018). Thus, Proposition 1 generalizes the
identification result of Stock and Watson (2018) to cases where the shock
$\xi_{t}$ is the sum of the structural shocks in the SVMA model (1).

Comparison with LATE Imbens and Angrist (1994) and Angrist, Imbens, and Rubin
(1996) show that the local average treatment effect (LATE) is identified by an
instrument under treatment effect heterogeneity. Let $Y_{i}$ be the observed
outcome and $D_{i}$ be an indicator of treatment for individual $i$. Their
analysis is based on the potential outcomes framework: the potential outcomes
$Y_{i}(1)$ and $Y_{i}(0)$ are defined as the outcome with treatment
($D_{i}=1$) and without treatment ($D_{i}=0$), respectively. The observed
outcome is written as $Y_{i}=Y_{i}(0)+(Y_{i}(1)-Y_{i}(0))D_{i}$. The
individual treatment effect is $Y_{i}(1)-Y_{i}(0)$, which is assumed to be
heterogeneous. Since the potential outcomes for an individual are not observed
at the same time, the individual treatment effect is not identified.
Therefore, the goal is to identify the average treatment effect (ATE),
$E[Y_{i}(1)-Y_{i}(0)]$, or a version of ATE. Here, the treatment is
homogeneous within the treated and non-treated groups, but the treatment
effects are heterogeneous.

In comparison, the composite shock $\xi_{t}$ is the (unobserved) treatment.
The ATE corresponds to the impulse response of $y_{t+h}$ to $\xi_{t}$:
$E[y_{t+h}|\xi_{t}=1]-E[y_{t+h}|\xi_{t}=0]$, which is a weighted average of
componentwise impulse responses as shown in (3). Unlike the ATE, heterogeneity
arises in the composition of $\xi_{t}$, so that the treatment is
heterogeneous. Since the componentwise impulse responses are assumed constant,
the treatment effect of a particular composition in $\xi_{t}$ is homogeneous.

What is the role of the IV? In both cases, it provides exogenous variation in
the endogenous variable. However, due to the different nature of the
endogeneity (selection vs simultaneity), how the instrument provides
identification of economically meaningful parameters in the two frameworks are
different. This is illustrated in Figure 1.

Figure 1: Monotonicity and Same-Sign

First, consider a binary instrument $Z_{i}$ in the LATE case (left panel in
Figure 1). $D_{i}$ is endogenous because it is not randomly assigned. For each
value of $Z_{i}$, define the potential treatment status $D_{i}(1)$ and
$D_{i}(0)$, which corresponds to $Z_{i}=1$ and $Z_{i}=0$, respectively.
Angrist, Imbens, and Rubin (1996) define four subpopulations depending on the
potential treatment status: always-takers ($D_{i}(1)=D_{i}(0)=1$), never-
takers ($D_{i}(1)=D_{i}(0)=0$), compliers ($D_{i}(1)=1$, $D_{i}(0)=0$), and
defiers ($D_{i}(1)=0$, $D_{i}(0)=1$).222Or equivalently, compliers and defiers
can be defined as $D_{i}(1)=0$ and $D_{i}(0)=1$, and $D_{i}(1)=1$,
$D_{i}(0)=0$, respectively. Imbens and Angrist (1994) show that the IV
estimand identifies the ATE of the compliers (thus the local average), who
would receive the treatment if $Z_{i}=1$ but not otherwise. The key condition
for identification of LATE is the monotonicity condition (Condition 2 of
Imbens and Angrist, 1994) that there is no defiers (who behave in the opposite
way to the compliers) in the population.

Now consider our framework (right panel in Figure 1). Since $\xi_{t}$ is not
observed and its scale is indeterminate, we need to measure the response of
$y_{t+h}$ to $\xi_{t}$ relative to the response of another observable
endogenous variable $x_{t}$ to $\xi_{t}$. $x_{t}$ is endogenous due to
simultaneity because the shock $\xi_{t}$ affects $y_{t+h}$ and $x_{t}$
simultaneously. To ensure structural interpretations of the LP-IV estimand, we
require that the correlation between the instrument and the shock components
in $\xi_{t}$ have the same sign. This same-sign condition plays an analogous
role to the monotonicity condition. Unlike the monotonicity condition that
restricts the individual behavior, the same-sign condition restricts the
average relationship between the instrument and the shocks.

Potential outcomes framework for time series Angrist and Kuersteiner (2011)
develop semiparametric tests for conditional independence, also known as the
unconfoundedness condition. Analogous to the cross-sectional case, the
potential outcomes with and without the treatment at time $t$ are not observed
simultaneously and the time-specific treatment effects are heterogeneous.
Their framework is different from ours because (i) the policy variable
(treatment) is observed, (ii) the policy variable is independent of potential
outcomes after conditioning on observables, and as a result, (iii) the role of
IV is not discussed.

Rambachan and Shephard (2021) provide a more general potential outcomes
framework allowing for unobserved treatment and the use of IV. Their IV
identification result relies on a time-series version of the monotonicity
condition. Since our same-sign condition plays a similar role to their
monotonicity condition, it is worth comparing the two conditions. Consider a
binary instrument $Z_{t}$ and two components in $\xi_{t}$ so that
$\xi_{t}=\varepsilon_{1,t}+\varepsilon_{2,t}$. Following the framework of
Rambachan and Shephard, we define the potential shock (“assignment” in their
term) with and without the instrument as $\varepsilon_{s,t}(1)$ and
$\varepsilon_{s,t}(0)$, respectively, for $s=1,2$. Also assume that
$(\varepsilon_{s,t}(1),\varepsilon_{s,t}(0))^{\prime}\sim_{iid}N((\delta,-\delta)^{\prime},I_{2})$
for some constant $\delta>0$ and $Z_{t}$ is independently generated with
$E[Z_{t}]=0.5$. Then $E[\varepsilon_{s,t}]=0$, $s=1,2$ because
$\varepsilon_{s,t}=\varepsilon_{s,t}(1)Z_{t}+\varepsilon_{s,t}(0)(1-Z_{t})$
and $\varepsilon_{1,t}$ and $\varepsilon_{2,t}$ are uncorrelated. By some
algebra we can also show that the same-sign condition is strictly satisfied
because $E[Z_{t}\varepsilon_{s,t}]=\delta/2>0$ for $s=1,2$. However, the
monotonicity condition (Assumption iv of Corollary 1 in Rambachan and
Shephard), which states
$\xi_{t}(0)\equiv\varepsilon_{1,t}(0)+\varepsilon_{2,t}(0)\leq\varepsilon_{1,t}(1)+\varepsilon_{2,t}(1)\equiv\xi_{t}(1)$
with probability one, is not satisfied for this basic distribution because
$\xi_{t}(0)\sim N(-2\delta,2)$ and $\xi_{t}(1)\sim N(2\delta,2)$. This example
highlights that our approach can provide causal interpretations of the IV
estimand under weaker distributional assumptions and is complementary to the
existing potential outcomes framework for time series setting.

Since $E[\bm{Y}_{t}]=0$, $\bm{Y}_{t}$ is typically specified in differences or
changes, but researchers may be interested in the level of $\bm{Y}_{t}$. The
difference in the levels between $t$ and $t+h$ can be written as the
cumulative changes over the $h$ periods: $\sum_{j=0}^{h}\bm{Y}_{t+j}$. Write
$\widetilde{y}_{t+h}=\sum_{j=0}^{h}y_{t+j}$ and
$\widetilde{x}_{t+h}=\sum_{j=0}^{h}x_{t+j}$. Define the cumulative impulse
responses as $\widetilde{\theta}_{h,ys}=\sum_{j=0}^{h}\theta_{j,ys}$ and
$\widetilde{\theta}_{h,xs}=\sum_{j=0}^{h}\theta_{j,xs}$, respectively.

######  Corollary 1.

If random variables $y_{t}$ and $x_{t}$ are elements of $\bm{Y}_{t}$ generated
according to (1), a random variable $z_{t}$ satisfies Assumption 1, and
$\widetilde{\theta}_{h,xs}\neq 0$ for all $s=1,2,...,S$, then for
$h=0,1,2,...$,

$\widetilde{\beta}_{h}\equiv\frac{Cov\left(\widetilde{y}_{t+h},z_{t}\right)}{Cov\left(\widetilde{x}_{t+h},z_{t}\right)}=\sum_{s=1}^{S}\left(\frac{\alpha_{s}\widetilde{\theta}_{h,xs}}{\sum_{s^{\prime}=1}^{S}\alpha_{s^{\prime}}\widetilde{\theta}_{h,xs^{\prime}}}\right)\frac{\widetilde{\theta}_{h,ys}}{\widetilde{\theta}_{h,xs}}.$
(6)

Proof. The proof is similar to the proof of Proposition 1, and thus omitted. ∎

Corollary 1 is relevant for identification and estimation of the government
spending multiplier using external instruments. Ramey and Zubairy (2018)
estimate the multiplier by LP-IV using the military news shock as an
instrument. Their LP-IV estimand takes the form of the LHS of (6) after
controlling for the lagged variables, where $y_{t}$ is the GDP and $x_{t}$ is
government spending, both relative to the trend GDP. Assume that the
government spending shock consists of non-defense ($s=1$) and defense ($s=2$)
spending shocks: $\xi_{t}=\varepsilon_{1,t}$ \+ $\varepsilon_{2,t}$. By
Corollary 1,

$\widetilde{\beta}_{h}=\left(\frac{\alpha_{1}\widetilde{\theta}_{h,x1}}{\alpha_{1}\widetilde{\theta}_{h,x1}+\alpha_{2}\widetilde{\theta}_{h,x2}}\right)\frac{\widetilde{\theta}_{h,y1}}{\widetilde{\theta}_{h,x1}}+\left(\frac{\alpha_{2}\widetilde{\theta}_{h,x2}}{\alpha_{1}\widetilde{\theta}_{h,x1}+\alpha_{2}\widetilde{\theta}_{h,x2}}\right)\frac{\widetilde{\theta}_{h,y2}}{\widetilde{\theta}_{h,x2}}.$
(7)

Thus, the LP-IV estimand is an affine combination of the cumulative non-
defense and defense spending multipliers,
$\widetilde{\theta}_{h,ys}/\widetilde{\theta}_{h,xs}$, $s=1,2$.

For the estimand (7) to have structural interpretation, a version of
Assumption 2 is required to ensure non-negative weights. We can check if the
assumption is plausible. First, it is reasonable to assume that positive
sectoral spending shocks have positive impacts on the cumulative government
spending: $\widetilde{\theta}_{h,xs}>0$ for $s=1,2$. Next, it is reasonable to
assume that the military news shock is positively correlated with the defense
spending shock ($\alpha_{2}>0$). Thus, the weight for the non-defense spending
multiplier would be non-negative if the military news shock is non-negatively
correlated with the non-defense spending shock ($\alpha_{1}\geq 0$). If this
is justified, then the estimand is a weighted average of the cumulative
sectoral spending multiplier. On the other hand, if $\alpha_{1}<0$, the
estimand may not provide meaningful information about the government spending
multiplier because the estimand can be close to zero or even negative while
the defense and non-defense spending multipliers are strictly greater than
one. We find empirical evidence supporting $\alpha_{1}<0$ in Section 6.

## 3 Identification of Componentwise Impulse Responses

Equation (5) in Proposition 1 shows that the LP-IV estimand is an affine
combination of componentwise impulse responses (Assumption 2 ensures that it
is a convex combination). In this section, we provide a set of conditions
under which the componentwise impulse responses in the LP-IV estimand are
point-identified.

The first condition is to have componentwise observations $x_{s,t}$
corresponding to $\varepsilon_{s,t}$ for $s=1,2,...,S$ such that
$x_{t}=\sum_{s=1}^{S}x_{s,t}$. The $x_{s,t}$’s are components of the aggregate
variable $x_{t}$. To avoid unnecessary confusion, we call $s$ as sectors and
$x_{s,t}$ as sectoral variables, although they do not necessarily have to be.
The SVMA model (1) is augmented with these auxiliary sectoral data333By pre-
multiplying
$\bm{B}=\left[\begin{array}[]{cccccccc}1&1&\cdots&1&0&0&\cdots&0\\\
0&0&\cdots&0&1&0&\cdots&0\\\ &&&\vdots&&&&\\\ 0&0&\cdots&0&0&0&\cdots&1\\\
\end{array}\right]$ both sides of (8), we can obtain (1) and
$\bm{\Theta}(L)=\bm{B}\bm{\Psi}(L)$. Without any further restrictions on the
impulse response matrices, (8) is more general than (1).:

$\left(\begin{array}[]{c}x_{1,t}\\\ x_{2,t}\\\ \vdots\\\ x_{S,t}\\\ \vdots\\\
y_{t}\end{array}\right)=\bm{\Psi}(L)\left(\begin{array}[]{c}\varepsilon_{1,t}\\\
\varepsilon_{2,t}\\\ \vdots\\\ \varepsilon_{S,t}\\\ \vdots\\\
\varepsilon_{m,t}\end{array}\right)$ (8)

where $\bm{\Psi}(L)=\bm{\Psi}_{0}+\bm{\Psi}_{1}L+\bm{\Psi}_{2}L^{2}+\cdots$,
and $\bm{\Psi}_{h}$ for $h=0,1,2,...,$ is an $(n+S-1)\times m$ matrix of
impulse responses. Note that the last $n-1$ rows of $\bm{\Psi}(L)$ are
identical to the last $n-1$ rows of $\bm{\Theta}(L)$. Thus, we maintain
$\theta_{h,ys}$ for the impulse response of $y_{t+h}$ to $\varepsilon_{s,t}$
as a component of $\bm{\Psi}_{h}$.

When it is further restricted to have $n+S-1=m$, the number of endogenous
variables is the same with the number of structural shocks. This is a standard
assumption in the structural VAR model because it is a necessary condition for
the so-called invertibility. Similar to the SVMA model (1), the structural
shocks on the augmented SVMA model (8) are assumed to be mean zero, have non-
zero variances, and are mutually uncorrelated.

Let $\psi_{h,rs}$ be the response of sectoral variable $x_{r,t+h}$ to the
shock $\varepsilon_{s,t}$. This impulse response is the $(r,s)$-th element of
$\bm{\Psi}_{h}$. With the unit effect normalization for each sectoral shock
($\psi_{0,ss}=1$ for $s=1,2,...,S$), $\varepsilon_{s,t}$ is scaled so that one
unit change in $\varepsilon_{s,t}$ corresponds to the unit change in
$x_{s,t}$. For example, if $x_{1,t}$ and $x_{2,t}$ are non-defense and defense
spending and $\varepsilon_{1,t}$ and $\varepsilon_{2,t}$ are the corresponding
sectoral spending shocks, then a positive unit sectoral spending shock
increases the corresponding sectoral spending by one unit.

Since $x_{t}=\sum_{s=1}^{S}x_{s,t}$, we write
$\sum_{r=1}^{S}\psi_{h,rs}=\theta_{h,xs}$ for $s=1,...,S$. This connects the
impulse response of the augmented model (8) and the baseline model (1).

The second condition for identification of the componentwise impulse responses
is a restriction on $\bm{\Psi}_{0}$ that there is no contemporaneous inter-
sectoral causal effects, i.e., for all $r,s=1,2,...,S$ and $r\neq s$,

$0=E[x_{r,t}|\varepsilon_{s,t}=1]-E[x_{r,t}|\varepsilon_{s,t}=0].$ (9)

In other words, the $s$-th sectoral shock $\varepsilon_{s,t}$ does not enter
the moving average representation of the $r$-th sectoral observation $x_{r,t}$
where $r\neq s$. Consider again the government spending multiplier example.
The condition (9) states that the defense spending shock does not affect non-
defense spending and vice versa. The condition is formally stated as follows:

######  Assumption 3.

For all $r\neq s$ and $r,s\in\mathcal{S}_{\xi}$, $\psi_{0,rs}=0$ (no
contemporaneous inter-sectoral causal effects)

Assumption 3 may be tested if we have enough number of instruments. A formal
testing procedure is given in Section 3.1. Note that
$\psi_{0,ss}=\theta_{0,xs}=1$ under Assumption 3 and the unit effect
normalization.

Using the sectoral data, Assumptions 1 and 3, and the unit effect
normalization, the weights are identified. Since

$Cov(x_{s,t},z_{t})=\alpha_{s},$

for $s=1,2,...,S$, the LP-IV estimand can be written as

$\frac{Cov(y_{t+h},z_{t})}{Cov(x_{t},z_{t})}=\sum_{s=1}^{S}\frac{Cov(x_{s,t},z_{t})}{Cov(x_{t},z_{t})}\theta_{h,ys}.$
(10)

Since $Cov(y_{t+h},z_{t})$ and $Cov(x_{s,t},z_{t})$ are the moments of
observables, (10) shows that the $S$ unknown parameters $\theta_{h,ys}$ can be
identified from the simultaneous equations if we have $S$ or more instruments.

Let $\bm{Z}_{t}$ be an $l\times 1$ vector of instruments and
$\bm{X}_{t}=(x_{1,t},x_{2,t},\cdots,x_{S,t})^{\prime}$ be the $S\times 1$
vector of sectoral observations. We require that the instruments are jointly
relevant, which is Assumption 4.

######  Assumption 4.

$rank(E[\bm{Z}_{t}\bm{X}_{t}^{\prime}])=S$.

A necessary condition for Assumption 4 is $l\geq S$.

Let $\bm{\theta}_{h,y}\equiv(\theta_{h,y1},...,\theta_{h,yS})^{\prime}$ be the
$S\times 1$ vector of sectoral impulse responses of $y_{t+h}$ to the shocks
$(\varepsilon_{1,t},\varepsilon_{2,t},...,\varepsilon_{S,t})^{\prime}$. The
following proposition establishes identification of $\bm{\theta}_{h,y}$.

######  Proposition 2.

Suppose that random variables $y_{t}$, $x_{s,t}$ for $s=1,2,...,S$, are
generated according to (8) under Assumption 3, each element of $\bm{Z}_{t}$
satisfies Assumptions 1 (with possibly different instrument-specific
$\alpha_{s}$), and $\bm{Z}_{t}$ satisfies Assumption 4. Assuming the unit
effect normalization ($\psi_{0,ss}=1$ for $s=1,2,...,S$), for $h=0,1,2,...,$

$Cov\left(\bm{Z}_{t},\bm{X}_{t}^{\prime}\right)^{-1}Cov(y_{t+h},\bm{Z}_{t})=\bm{\theta}_{h,y},$

if $l=S$, and

$\left(Cov\left(\bm{X}_{t},\bm{Z}_{t}^{\prime}\right)\bm{W}Cov\left(\bm{Z}_{t},\bm{X}_{t}^{\prime}\right)\right)^{-1}Cov\left(\bm{X}_{t},\bm{Z}_{t}^{\prime}\right)\bm{W}Cov(y_{t+h},\bm{Z}_{t})=\bm{\theta}_{h,y},$

if $l>S$, where $\bm{W}$ is a non-random $l\times l$ positive definite matrix.

Proof. By Assumption 3, each element of $\bm{X}_{t}$ can be written as

$x_{r,t}=\sum_{j=0}^{\infty}\left(\sum_{s=1}^{S}\psi_{j,rs}\varepsilon_{s,t-j}+\sum_{s^{\prime}=S+1}^{n}\psi_{j,rs^{\prime}}\varepsilon_{s^{\prime},t-j}\right)\\\
=\sum_{j=0}^{\infty}\psi_{j,rr}\varepsilon_{r,t-j}.$

For each $\varepsilon_{s,t}$, define the $l\times 1$ vector
$\bm{\lambda}_{s}=E[\bm{Z}_{t}\varepsilon_{s,t}]$ to allow heterogeneous
correlation between the instruments and the sectoral shocks.

Since all the elements of $\bm{Z}_{t}$ satisfy Assumption 1, we can write

$\displaystyle Cov(y_{t+h},\bm{Z}_{t})=$
$\displaystyle\sum_{s=1}^{S}\bm{\lambda}_{s}\theta_{h,ys},$ (11)
$\displaystyle Cov(x_{s,t},\bm{Z}_{t})=$ $\displaystyle\bm{\lambda}_{s},$ (12)

where (12) holds by the unit effect normalization. By replacing (12) into
(11), we have

$Cov(y_{t+h},\bm{Z}_{t})=Cov(\bm{Z}_{t},\bm{X}_{t}^{\prime})\bm{\theta}_{h,y}.$

When $l=S$, $Cov(\bm{Z}_{t},\bm{X}_{t}^{\prime})$ is invertible under
Assumption 4. When $l>S$, consider an $l\times l$ non-random positive definite
matrix $\bm{W}$ and let
$\bm{A}=Cov\left(\bm{X}_{t},\bm{Z}_{t}^{\prime}\right)\bm{W}Cov\left(\bm{Z}_{t},\bm{X}_{t}^{\prime}\right)$.
By Assumption 4, $\bm{A}$ is invertible. Thus, the desired conclusion follows.
∎

Next we present an analogous identification result to Corollary 1 for the
relative cumulative impulse responses of $y_{t}$ and $x_{t}$ over the $h$
period. Write $\widetilde{x}_{s,t+h}=\sum_{j=0}^{h}x_{s,t+j}$ and
$\widetilde{\bm{X}}_{t+h}=\sum_{j=0}^{h}\bm{X}_{t+j}$. Define the cumulative
impulse response as $\widetilde{\psi}_{h,ss}=\sum_{j=0}^{h}\psi_{j,ss}$ for
$s=1,...,S$ and the $S\times 1$ vector

$\widetilde{\bm{\theta}}_{h,y}\equiv(\widetilde{\theta}_{h,y1}/\widetilde{\psi}_{h,11},...,\widetilde{\theta}_{h,yS}/\widetilde{\psi}_{h,SS})^{\prime}.$

Note that each element of $\widetilde{\bm{\theta}}_{h,y}$ is normalized with
respect to the cumulative change in the corresponding sector.

######  Corollary 2.

Suppose that the assumptions of Proposition 2 hold, Assumption 4 holds for
$\widetilde{\bm{X}}_{t}$ in place of $\bm{X}_{t}$, and
$\widetilde{\psi}_{h,ss}\neq 0$ for $s=1,2,...,S$. Then for $h=0,1,2,...,$

$Cov\left(\bm{Z}_{t},\widetilde{\bm{X}}_{t+h}^{\prime}\right)^{-1}Cov\left(\widetilde{y}_{t+h},\bm{Z}_{t}\right)=\widetilde{\bm{\theta}}_{h,y},$

if $l=S$, and

$\left(Cov\left(\widetilde{\bm{X}}_{t+h},\bm{Z}_{t}^{\prime}\right)\bm{W}Cov\left(\bm{Z}_{t},\widetilde{\bm{X}}_{t+h}^{\prime}\right)\right)^{-1}Cov\left(\widetilde{\bm{X}}_{t+h},\bm{Z}_{t}^{\prime}\right)\bm{W}Cov\left(\widetilde{y}_{t+h},\bm{Z}_{t}\right)=\widetilde{\bm{\theta}}_{h,y},$

if $l>S$, where $\bm{W}$ is a non-random $l\times l$ positive definite matrix.

Proof. The proof is similar to the proof of Proposition 2, and thus omitted. ∎

###  3.1 Testing No Inter-Sectoral Causal Effects

Assumption 3 can be tested by the generalized method of moments (GMM). Let
$\bm{e}_{t}\equiv(\varepsilon_{1,t},\varepsilon_{2,t},...,\varepsilon_{S,t})^{\prime}$
and $\bm{\Lambda}\equiv E[\bm{Z}_{t}\bm{e}_{t}^{\prime}]$. Also let
$\bm{\Psi}_{0,X}$ be the $S\times S$ upper left submatrix of $\bm{\Psi}_{0}$
in the augmented SVMA model (8). Assuming the unit effect normalization
without Assumption 3, $\bm{\Psi}_{0,X}$ is given by

$\bm{\Psi}_{0,X}=\left(\begin{array}[]{cccc}1&\psi_{0,12}&\cdots&\psi_{0,1S}\\\
\psi_{0,21}&1&\cdots&\psi_{0,2S}\\\ \vdots&\vdots&\vdots&\vdots\\\
\psi_{0,S1}&\psi_{0,S2}&\cdots&1\end{array}\right).$

Since
$E[\bm{Z}_{t}x_{s,t}]=\bm{\Lambda}\times(\psi_{0,s1},\psi_{0,s2},...,\psi_{0,sS})^{\prime}$,

$Cov(\bm{Z}_{t},\bm{X}_{t}^{\prime})=E[\bm{Z}_{t}\bm{X}_{t}^{\prime}]=\bm{\Lambda}\bm{\Psi}_{0,X}^{\prime}.$
(13)

Assume that $\bm{\Psi}_{0,X}$ is invertible.444For example, consider a two-
sector model ($S=2$). In this case,
$\bm{\Psi}_{0,X}=\left(\begin{array}[]{cc}1&\psi_{0,12}\\\ \psi_{0,21}&1\\\
\end{array}\right)$ is invertible if and only if $1-\psi_{0,12}\psi_{0,21}\neq
0$. Plugging
$E[\bm{Z}_{t}\bm{X}_{t}^{\prime}]\bm{\Psi}_{0,X}^{-1^{\prime}}=\bm{\Lambda}$
from (13) into $E[\bm{Z}_{t}y_{t+h}]=\bm{\Lambda}\bm{\theta}_{h,y}$, we have
the moment condition

$E[\bm{Z}_{t}(y_{t+h}-\bm{X}_{t}^{\prime}\bm{\Psi}_{0,X}^{-1^{\prime}}\bm{\theta}_{h,y})]=0.$

Since the number of unknown parameters is $S^{2}$, i.e., $S^{2}-S$ unknown
elements in $\bm{\Psi}_{0,X}$ plus $S$ unknown elements in
$\bm{\theta}_{h,y}$, if at least $l=S^{2}$ instruments are available then a
necessary condition for identification is satisfied. Since the moment
condition is nonlinear in the parameters, having as many instruments as the
unknown parameters does not necessarily imply global identification. In
general, finding primitive conditions for identification of parameters in a
nonlinear model is difficult (Newey and McFadden, 1994). Nevertheless, by
assuming identification, Assumption 3 can be written as the null hypothesis:

$H_{0}:~{}\bm{\Psi}_{0,X}=I_{S}$

where $I_{S}$ is the $S\times S$ identity matrix. We can use the GMM based
tests such as the Wald test or the Lagrange multiplier test. When the
instruments are weak, which might be relevant in practice, one may use the
weak-instrument robust GMM Lagrange multiplier statistic of Kleibergen (2005).

## 4 Bounds on Componentwise Impulse Responses and Counterfactual Analysis

The previous section provided the conditions for point identification of the
componentwise impulse responses, which require both auxiliary sectoral data
and additional instruments. In practice, both may not be available at the same
time. In this section, we show that the bounds for the componentwise impulse
responses can be obtained without the sectoral data or additional instruments.

For simplicity, we assume $S=2$ (two sectors). The shock of interest is
$\xi_{t}=\varepsilon_{1,t}+\varepsilon_{2,t}$. The LP-IV estimand can be
written as

$\beta_{h}=w_{1}\theta_{h,y1}+(1-w_{1})\theta_{h,y2},$ (14)

where $w_{1}=\alpha_{1}/(\alpha_{1}+\alpha_{2})$. To investigate what can be
said about the componentwise impulse responses $\theta_{h,y1}$ and
$\theta_{h,y2}$ when some of the identification assumptions are not satisfied,
we consider two cases when the auxiliary sectoral data are not available (Case
I) and when only one IV is available (Case II).

###  4.1 Case I: Aggregate Data and Additional Instruments

The first case is when we have the aggregate data $(y_{t},x_{t})$, but the
sectoral data is not available. Without the sectoral data, $w_{1}$ cannot be
point identified. Instead, we combine the additional information from
additional instruments and suitable restrictions on the sign of the weights to
obtain reasonable bounds for $\theta_{h,y1}$ and $\theta_{h,y2}$.

By solving (14) for $w_{1}$, we have

$w_{1}=\frac{\beta_{h}-\theta_{h,y2}}{\theta_{h,y1}-\theta_{h,y2}},$ (15)

provided that $\theta_{h,y1}\neq\theta_{h,y2}$. Note that $w_{2}=1-w_{1}$.
Since $w_{s}=E[z_{t}\varepsilon_{s,t}]/E[z_{t}x_{t}]$ and $E[z_{t}x_{t}]$ is
observable, the sign of $w_{s}$ is determined by assuming the sign of
$E[z_{t}\varepsilon_{s,t}]$. For example, one may assume that the military
news shock ($z_{t}$) is positively correlated with the defense spending shock
($\varepsilon_{2,t}$) to have $w_{2}>0$, if $E[z_{t}x_{t}]>0$ (which can be
justified by the data). The identification strategy here is to use the signs
of $w_{1}$ and $w_{2}$ to derive some estimable bounds for $\theta_{h,y1}$ and
$\theta_{h,y2}$.

For each sign restriction on the weight, we can derive the following sets from
(15):

$\displaystyle w_{1}>0:~{}$
$\displaystyle\\{(\theta_{h,y1},\theta_{h,y2})|\theta_{h,y1}>\theta_{h,y2},~{}\beta_{h}>\theta_{h,y2}\\}\text{~{}or~{}}\\{(\theta_{h,y1},\theta_{h,y2})|\theta_{h,y1}<\theta_{h,y2},~{}\beta_{h}<\theta_{h,y2}\\}$
(16) $\displaystyle w_{1}<0:~{}$
$\displaystyle\\{(\theta_{h,y1},\theta_{h,y2})|\theta_{h,y1}>\theta_{h,y2},~{}\beta_{h}<\theta_{h,y2}\\}\text{~{}or~{}}\\{(\theta_{h,y1},\theta_{h,y2})|\theta_{h,y1}<\theta_{h,y2},~{}\beta_{h}>\theta_{h,y2}\\}$
(17) $\displaystyle w_{2}>0:~{}$
$\displaystyle\\{(\theta_{h,y1},\theta_{h,y2})|\theta_{h,y1}>\theta_{h,y2},~{}\beta_{h}<\theta_{h,y1}\\}\text{~{}or~{}}\\{(\theta_{h,y1},\theta_{h,y2})|\theta_{h,y1}<\theta_{h,y2},~{}\beta_{h}>\theta_{h,y1}\\}$
(18) $\displaystyle w_{2}<0:~{}$
$\displaystyle\\{(\theta_{h,y1},\theta_{h,y2})|\theta_{h,y1}>\theta_{h,y2},~{}\beta_{h}>\theta_{h,y1}\\}\text{~{}or~{}}\\{(\theta_{h,y1},\theta_{h,y2})|\theta_{h,y1}<\theta_{h,y2},~{}\beta_{h}<\theta_{h,y1}\\}$
(19)

Trivially, if $w_{1}=0$ then $\beta_{h}=\theta_{h,y2}$ and if $w_{1}=1$, then
$\beta_{h}=\theta_{h,y1}$. The identified sets are illustrated in Figure 2. In
each panel, the sets that correspond those in (16)-(19) are shown as grey
areas (we set $\beta_{h}>0$). For instance,
$\mathcal{C}_{1}=\\{(\theta_{h,y1},\theta_{h,y2})|\theta_{h,y1}>\theta_{h,y2},~{}\beta_{h}>\theta_{h,y2}\\}$
and
$\mathcal{C}_{2}=\\{(\theta_{h,y1},\theta_{h,y2})|\theta_{h,y1}<\theta_{h,y2},~{}\beta_{h}<\theta_{h,y2}\\}$
in panel (a), and so on.

Figure 2: Identified Sets for Sign Restrictions on the Weight

A few observations can be made about the sign restrictions and the sets.
First, $w_{2}<0$ if and only if $w_{1}>1$, which implies $w_{1}>0$. In terms
of the sets, $\mathcal{F}_{1}\subset\mathcal{C}_{1}$ and
$\mathcal{F}_{2}\subset\mathcal{C}_{2}$. By the same argument, $w_{1}<0$
implies $w_{2}>0$, and $\mathcal{D}_{1}\subset\mathcal{E}_{1}$ and
$\mathcal{D}_{2}\subset\mathcal{E}_{2}$. This also implies that negative sign
restriction is more informative. Second, $w_{2}>0$ if and only if $w_{1}<1$.

Either using the sets in (16)-(19), or using additional sets generated by
other instruments, the intersection of the sets corresponding to the available
sign restrictions is the identified set for $(\theta_{h,y1},\theta_{h,y2})$.
From the identified sets, we may find the lower or upper bound. Therefore, the
more IVs with sign restrictions, the tighter bounds for
$(\theta_{h,y1},\theta_{h,y2})$. Furthermore, we may present the identified
sets for general cases other than the sign restrictions.

For example, consider two instruments, $z_{t}^{A}$ and $z_{t}^{B}$. We write
the LP-IV estimand and weights based on $z_{t}^{A}$ ($z_{t}^{B}$) with the
superscript $A$ ($B$). Suppose that $w_{1}^{A}>0$ and and $w_{1}^{B}<0$. There
are three possibilities: If $\beta_{h}^{A}=\beta_{h}^{B}$ then the identified
set is empty. If $\beta_{h}^{A}<\beta_{h}^{B}$, then the identified set is
$\mathcal{C}_{1}\cap\mathcal{D}_{1}$ in Figure 3(a) because
$\mathcal{C}_{2}\cap\mathcal{D}_{2}=\phi$. In this case, the bounds are
$\theta_{h,y1}>\theta_{h,y2}$ and $\beta_{h}^{B}<\theta_{h,y2}<\beta_{h}^{A}$.
Lastly, if $\beta_{h}^{A}>\beta_{h}^{B}$, then the identified set is
$\mathcal{C}_{2}\cap\mathcal{D}_{2}$ in Figure 3(b) because
$\mathcal{C}_{1}\cap\mathcal{D}_{1}=\phi$. The bounds are
$\theta_{h,y1}<\theta_{h,y2}$ and $\beta_{h}^{A}<\theta_{h,y2}<\beta_{h}^{B}$.
Figure 3(c) shows the identified set for another example with only one
instrument $z_{t}^{A}$ and $w_{1}^{A}>0$ and $w_{2}^{A}>0$. The bounds are
$\theta_{h,y2}<\beta_{h}^{A}<\theta_{h,y1}$ or
$\theta_{h,y1}<\beta_{h}^{A}<\theta_{h,y2}$.

Figure 3: Intersection of Identified Sets

###  4.2 Case II: Auxiliary Sectoral Data and One Instrument

The next case is when we have the auxiliary sectoral data
$(y_{t},x_{1,t},x_{2,t})$ so that we can use the augmented SVMA model (8), but
only one instrument $z_{t}$ is available. Since we do not have enough number
of IVs (two sectors and one IV), we cannot use the point identification result
of componentwise impulse responses in Proposition 2. Nevertheless, the
auxiliary sectoral data can be used to derive the informative bounds for the
componentwise impulse responses.

Using the augmented SVMA model (8) and Assumption 3, the weight $w_{1}$ is
identified as

$w_{1}=\frac{Cov(x_{1,t},z_{t})}{Cov(x_{t},z_{t})}.$ (20)

By plugging (20) into (14), we have

$Cov(y_{t+h},z_{t})=Cov(x_{1,t},z_{t})\theta_{h,y1}+Cov(x_{2,t},z_{t})\theta_{h,y2}.$
(21)

Since $Cov(y_{t+h},z_{t})$, $Cov(x_{1,t},z_{t})$, and $Cov(x_{2,t},z_{t})$ are
the moments of the observables, $\theta_{h,1}$ and $\theta_{h,2}$ are set-
identified from (21). Note that if $Cov(x_{s,t},z_{t})>0$ for $s=1,2$, then
the identified set for $(\theta_{h,y1},\theta_{h,y2})$ coincides with the one
in Figure 3(c).

For example, suppose that non-defense spending ($x_{1,t}$) and defense
spending ($x_{2,t}$) data are available. The instrument $z_{t}$ is a military
news shock. By appropriately normalizing the data to satisfy the zero mean
assumption of the SVMA model, we can obtain the sample estimates of
$Cov(y_{t+h},z_{t})$, $Cov(x_{1,t},z_{t})$, and $Cov(x_{2,t},z_{t})$. Using
these values, we can estimate the identified set of
$(\theta_{h,1},\theta_{h,2})$ that satisfy (21). Furthermore, if the impulse
response to defense spending shock $\theta_{h,y2}$ is calibrated then the
corresponding value of $\theta_{h,y1}$ can be computed. By varying the
calibrated value of $\theta_{h,y2}$, we can obtain different counterfactual
values of $\theta_{h,y1}$. We illustrate this counterfactual analysis in
Section 6.

## 5 Estimation

In this section, we provide estimation procedures based on the identification
results in the previous sections. Estimation of LP-IV estimand and the
componentwise (sectoral) impulse responses are straightforward by replacing
the population moments with the sample moments.

The LP-IV model is a single equation linear IV model given by

$y_{t+h}=x_{t}\beta_{h}+\bm{R}_{t}^{\prime}\bm{\gamma}_{h}+u_{t+h},$ (22)

for $t=1,2,...,T$ and $h=0,1,2,...,H$, where the control variables
$\bm{R}_{t}$ include a constant and the lagged values of the endogenous
variable and the instrument. There may be several benefits of including
control variables in the LP-IV model. Stock and Watson (2018) argue that
adding control variables can strengthen the plausibility of Assumption 1
(their Condition LP-IV⟂) and reduce the sampling variance of the IV estimator.
Montiel Olea and Plagborg-Møller (2021) show that the LP estimation using
lagged variables as controls can be applicable to broader data-generating
processes and the standard errors need not be corrected for serial
correlation.

The parameters $(\beta_{h},\bm{\gamma}_{h}^{\prime})$ can be estimated by the
IV regression using $(z_{t},\bm{R}_{t}^{\prime})$ as the IV for
$(x_{t},\bm{R}_{t}^{\prime})$. The LP-IV estimator is

$\left(\begin{array}[]{c}\widehat{\beta}_{h}\\\
\widehat{\bm{\gamma}}_{h}\end{array}\right)=\left(\sum_{t=1}^{T-h}\left[\begin{array}[]{cc}z_{t}x_{t}&z_{t}\bm{R}_{t}^{\prime}\\\
\bm{R}_{t}x_{t}&\bm{R}_{t}\bm{R}_{t}^{\prime}\\\
\end{array}\right]\right)^{-1}\sum_{t=1}^{T-h}\left(\begin{array}[]{c}z_{t}y_{t+h}\\\
\bm{R}_{t}y_{t+h}\end{array}\right).$

The standard errors can be obtained by using the heteroskedasticity and
autocorrelation robust (HAR) covariance matrix estimator under the usual
assumptions such as weak stationarity. Since inference is not the main focus
of this paper, we leave the LP-IV inference as future research.

Estimation of the sectoral impulse response $\bm{\theta}_{h,y}$ in Proposition
2 requires attention when control variables are included in the IV regression.
It is a common practice to include the lagged values of the instrument when
the LP-IV model (22) is estimated, e.g., Ramey and Zubairy (2018) and
Plagborg-Møller and Wolf (2022). This implies that the set of control
variables in (22) changes when multiple instruments are used one at a time to
estimate $\beta_{h}$. Define $v_{t}^{\perp(j)}$ be the residual of $v_{t}$
after regressing it on $\bm{R}_{t}^{(j)}$. Let $\otimes$ denote the Kronecker
product and $\circ$ denote the element-wise multiplication (Hadamard product).
Also let $1_{S}$ be the $S\times 1$ vector of ones.

  1. 1.

Obtain the residuals $y_{t+h}^{\perp(j)}$, $x_{s,t}^{\perp(j)}$, and
$z_{i,t}^{\perp(j)}$ for $s=1,2,...,S$ and $i=1,2,...,l$.

  2. 2.

Construct the $l\times 1$ vectors
$\bm{Z}_{t}^{\perp}=(z_{1,t}^{\perp(1)},z_{2,t}^{\perp(2)},...,z_{l,t}^{\perp(l)})^{\prime}$
and
$\bm{\Upsilon}_{t+h}=(y_{t+h}^{\perp(1)},y_{t+h}^{\perp(2)},...,y_{t+h}^{\perp(l)})^{\prime}$,
and the $l\times S$ matrix

$\overline{\bm{X}}_{t}=\left(\begin{array}[]{cccc}x_{1,t}^{\perp(1)}&x_{2,t}^{\perp(1)}&\cdots&x_{S,t}^{\perp(1)}\\\
x_{1,t}^{\perp(2)}&x_{2,t}^{\perp(2)}&\cdots&x_{S,t}^{\perp(2)}\\\
\vdots&\vdots&&\vdots\\\
x_{1,t}^{\perp(l)}&x_{2,t}^{\perp(l)}&\cdots&x_{S,t}^{\perp(l)}\\\
\end{array}\right).$

  3. 3.

Calculate the $l\times S$ matrix $\widehat{\bm{C}}_{ZX}$ and the $l\times 1$
vector $\widehat{\bm{C}}_{Zy}$ as follows:

$\displaystyle\widehat{\bm{C}}_{ZX}=$
$\displaystyle\frac{1}{T}\sum_{t=1}^{T-h}(\bm{Z}_{t}^{\perp}\otimes\bm{1}_{S}^{\prime})\circ\overline{\bm{X}}_{t},$
$\displaystyle\widehat{\bm{C}}_{Zy}=$
$\displaystyle\frac{1}{T}\sum_{t=1}^{T-h}(\bm{Z}_{t}^{\perp}\circ\bm{\Upsilon}_{t+h}).$

  4. 4.

The sectoral impulse response estimator is

$\widehat{\bm{\theta}}_{h,y}=\widehat{\bm{C}}_{ZX}^{-1}\widehat{\bm{C}}_{Zy}$

when $l=S$ (just-identified) and

$\widehat{\bm{\theta}}_{h,y}=\left(\widehat{\bm{C}}_{ZX}^{\prime}\widehat{\bm{W}}\widehat{\bm{C}}_{ZX}\right)^{-1}\widehat{\bm{C}}_{ZX}^{\prime}\widehat{\bm{W}}\widehat{\bm{C}}_{Zy}$

when $l>S$ (over-identified) where $\widehat{\bm{W}}$ is a positive-definite
matrix.

In the over-identified case, the estimator $\widehat{\bm{\theta}}_{h,y}$ can
be interpreted as a GMM estimator. The corresponding moment condition is
$E[\bm{Z}_{t}^{\perp}\circ(\bm{\Upsilon}_{t+h}-\overline{\bm{X}}_{t}\bm{\theta}_{h,y})]=0$.555This
is slight abuse of notation because the residuals in the moment condition are
the population versions. The weight matrix can be set
$\widehat{\bm{W}}=(\sum_{t=1}^{T-h}\bm{Z}_{t}^{\perp}\bm{Z}_{t}^{\perp^{\prime}})^{-1}$
or take the form of the efficient weight matrix.

The standard errors are calculated as

$s(\widehat{\bm{\theta}}_{h,y})=\sqrt{\frac{\text{diag}(\widehat{\bm{V}})}{T}},$

where

$\widehat{\bm{V}}=\widehat{\bm{C}}_{ZX}^{-1}\widehat{\bm{\Omega}}\widehat{\bm{C}}_{ZX}^{-1^{\prime}}$

when $l=S$,

$\widehat{\bm{V}}=\left(\widehat{\bm{C}}_{ZX}^{\prime}\widehat{\bm{W}}\widehat{\bm{C}}_{ZX}\right)^{-1}\widehat{\bm{C}}_{ZX}^{\prime}\widehat{\bm{W}}\widehat{\bm{\Omega}}\widehat{\bm{W}}\widehat{\bm{C}}_{ZX}\left(\widehat{\bm{C}}_{ZX}^{\prime}\widehat{\bm{W}}\widehat{\bm{C}}_{ZX}\right)^{-1}$

when $l>S$, and $\widehat{\bm{\Omega}}$ is a consistent estimator of the long-
run covariance matrix

$\bm{\Omega}=\lim_{T\rightarrow\infty}Var\left[\frac{1}{\sqrt{T}}\sum_{t=1}^{T-h}\left(\bm{Z}_{t}^{\perp}\bm{Z}_{t}^{\perp^{\prime}}\right)\circ\left((\bm{\Upsilon}_{t+h}-\overline{X}_{t}\widehat{\bm{\theta}}_{h,y})(\bm{\Upsilon}_{t+h}-\overline{\bm{X}}_{t}\widehat{\bm{\theta}}_{h,y})^{\prime}\right)\right].$

## 6 Application: Government Spending Multiplier in the U.S.

The government spending multiplier is the ratio of the change in the GDP to
the change in government spending. The magnitude of the multiplier is the key
to making fiscal policies but there still is debate in the literature whether
it is larger than one, e.g., Blanchard and Perotti (2002), Barro and Redlick
(2011), Ramey (2011), Auerbach and Gorodnichenko (2012), Nakamura and
Steinsson (2014), and Ramey and Zubairy (2018), among many important others.
Most studies focus on the estimation of the aggregate or defense spending
multipliers using the variation in the defense spending. This is because non-
defense spending varies less than defense-spending, and more importantly, non-
defense spending is likely to be endogenous with respect to GDP (Barro and
Redlick, 2011).

In this section, we estimate the non-defense spending multiplier in the United
States in the post WWII period. By using the identification results in the
previous sections, we decompose the aggregate spending multipliers estimated
by external instruments (LP-IV estimates) into the sectoral spending
multipliers. We focus on the post-WWII period because our identification
strategy requires defense spending data, which are only available after WWII.

The LP-IV model is

$\sum_{j=0}^{h}y_{t+j}=\mu_{h}+\beta_{h}\sum_{j=0}^{h}x_{t+j}+\bm{\phi}_{h}(L)^{\prime}\bm{R}_{t-1}+u_{t+h},$

where $y_{t}$ is GDP, $x_{t}$ is government spending, $\bm{R}_{t}$ is a set of
control variables, and $\bm{\phi}_{h}(L)$ is a coefficient vector of
polynomial in the lag operator of order 4. Since $\sum_{j=0}^{h}y_{t+j}$ is
the sum of GDP over $h$ periods and $\sum_{j=0}^{h}x_{t+j}$ is the sum of
government spending over $h$ periods, the parameter $\beta_{h}$ is the
cumulative government spending multiplier. The instrument is $z_{t}$. The
control variables include lagged values of $y_{t}$, $x_{t}$, and $z_{t}$.

We use two instruments: the military news shock of Ramey and Zubairy (2018),
labeled as ‘RZ news shock’ and the current defense spending shock of Blanchard
and Perotti (2002), labeled as ‘BP defense shock’. The BP defense shock is
current defense spending.666We conducted the weak instrument test of Montiel
Olea and Pflueger (2013) to accommodate possible serial correlation in the
errors. We did not find that statistical evidence of weak instruments for both
instruments. When the BP defense shock is used as an IV, the control variables
include the lagged values of GDP, government (aggregate) spending, and defense
spending. This is analogous to the construction of the Blanchard-Perotti
(2002) shock in Ramey and Zubairy (2018) using current government
spending.777Following Ramey and Zubairy (2018), the identification of the BP
defense shock is equivalent to the SVAR defense spending shock of Blanchard
and Perotti (2002). The SVAR system in Section IX. B. of Blanchard and Perotti
(2002) includes four variables, taxes, defense spending, non-defense spending,
and the GDP. Our main specification does not include taxes, but we obtained a
similar result when we included taxes in the control variables.

Figure 4: Cumulative Multipliers Across Different Horizons with 90% Confidence
Bands. Left: aggregate spending multipliers identified by RZ news shock
(solid), BP defense shock (dash-dotted). Right: non-defense spending
multiplier (x), defense spending multiplier (circle)

Figure 4 shows the cumulative government spending multipliers for each horizon
from two quarters to five years out. The bands are the pointwise 90%
confidence bands using Newey-West (1987) standard errors. The left panel shows
the LP-IV estimates of the cumulative government spending multiplier using the
RZ news shock (solid line) and the BP defense shock (dash-dotted) one at a
time. These estimates correspond to the LHS of (6) in Corollary 1. Consistent
with Ramey and Zubairy (2018), the estimated multipliers using either of the
RZ and BP shocks are below one after one year. When the BP defense shock is
used, the initial multiplier is smaller than the RZ shock but shows more
prolonged effects after three years.

The right panel of Figure 4 shows the cumulative sectoral spending multipliers
decomposed from the two LP-IV estimates of the government spending multipliers
on the left panel. The magnitude of the defense spending multiplier (circle
markers) is similar to that of the aggregate spending multiplier using the BP
defense shock.

The cumulative non-defense spending multiplier (x markers) shows interesting
trajectories. It is around 0.7 after two years but keeps increasing past 1
after four years. This suggests a prolonged effect of non-defense spending on
GDP. Furthermore, the non-defense spending multiplier can be larger than one
even when the government spending multipliers estimated using the RZ news
shock and the BP defense shock are below one. It turns out that this is due to
the negative weight for the non-defense spending multiplier in the
decomposition of the government spending multiplier.

To get a better understanding of the role of weights in the government
spending multiplier, consider the decomposition of the eighteen-quarters
($h=18$) cumulative government spending multiplier. Let $s=1$ denote non-
defense spending and $s=2$ denote defense spending. The government spending
multiplier estimated by the RZ news shock, is then decomposed as

$\begin{array}[]{ccccccccc}\widehat{\beta}^{RZ}_{h}&=&\widehat{w}_{h,1}^{RZ}&\times&\widehat{\frac{\widetilde{\theta}_{h,y1}}{\widetilde{\theta}_{h,x1}}}&+&\widehat{w}_{h,2}^{RZ}&\times&\widehat{\frac{\widetilde{\theta}_{h,y2}}{\widetilde{\theta}_{h,x2}}},\\\
0.37&&-0.87&&1.02&&1.87&&0.68\\\ \end{array}$ (23)

and the multiplier estimated by the BP defense shock is decomposed as

$\begin{array}[]{ccccccccc}\widehat{\beta}^{BP}_{h}&=&\widehat{w}_{h,1}^{BP}&\times&\widehat{\frac{\widetilde{\theta}_{h,y1}}{\widetilde{\theta}_{h,x1}}}&+&\widehat{w}_{h,2}^{BP}&\times&\widehat{\frac{\widetilde{\theta}_{h,y2}}{\widetilde{\theta}_{h,x2}}}.\\\
0.69&&0.03&&1.02&&0.97&&0.68\\\ \end{array}$ (24)

Using sectoral spending data, the weights are estimated by

$\widehat{w}_{h,s}^{(j)}=\frac{\sum_{t=1}^{T-h}z_{t}^{\perp(j)}\left(\sum_{j=0}^{h}x_{s,t+j}^{\perp(j)}\right)}{\sum_{t=1}^{T-h}z_{t}^{\perp(j)}\left(\sum_{j=0}^{h}x_{t+j}^{\perp(j)}\right)},$

where $j$ denotes the RZ news shock or the BP defense shock and
$v_{t}^{\perp(j)}$ denotes the residual of $v_{t}$ after regressing it on the
set of control variables including the lagged values of the instrument $j$.
The estimated weights are quite similar across $h$ in terms of the magnitude
and the sign for both of the instruments.

The decomposition (23)-(24) show that the government spending multiplier
estimated by the RZ news shock is smaller than both of the sectoral spending
multipliers due to the negative weight for the non-defense spending multiplier
in (23). This suggests a violation of the same-sign condition (Assumption 2)
for the RZ news shock as an IV; a positive military news shock has a positive
impact on defense spending, but at the same time has a negative impact on non-
defense spending, which in turn has a negative impact on GDP through a
positive non-defense spending multiplier. As a result, the government spending
multiplier estimated by the RZ news shock underestimates the sectoral spending
multipliers and may not be structurally interpreted.

For the government spending multiplier estimated using the BP defense shock,
the sign of the weights are both positive but the weight for the non-defense
spending multiplier is very small (0.03). As a result, the government spending
multiplier estimated by the BP defense shock is similar to the defense
spending multiplier.

Figure 5: Impulse Response of Sectoral Spending to the Instruments with 90%
Confidence Bands: Defense spending (solid), Non-defense spending (circle)

Figure 5 shows the impulse response of defense spending (solid line) and non-
defense spending (circle marker) to the RZ news shock (left panel) and to the
BP defense shock (right panel). Note that the dependent variables are sectoral
spending ($x_{s,t+h}$), rather than the cumulative sectoral spending
($\sum_{j=0}^{h}x_{s,t+j}$). The RZ news shock has a relatively small but
statistically significant negative effect on non-defense spending. In
contrast, the response of non-defense spending to the BP defense shock is
statistically insignificant for all $h$. Since the decomposition (24) suggests
that the BP defense shock is quite close to the structural defense spending
shock, the insignificant response of non-defense spending to the BP shock
supports the no inter-sectoral causal effects condition (a cumulative version
of Assumption 3).

###  6.1 Bound Estimates for Sectoral Multipliers

In this subsection, we apply the bound analysis in Section 4 by considering
the cases when the sectoral multipliers are not point identified.

Figure 6: Identified Set for Sectoral Spending Multipliers using Two IVs

First consider the case when the aggregate data and two instruments are
available (Case I in Section 4). We use GDP, total government spending, and
the two instruments.888Although the BP defense shock instrument cannot be
constructed using the aggregate data, we use this instrument for illustration
purposes. We impose the following restrictions:

  1. (i)

Both of defense and non-defense spending shocks have positive impacts on the
cumulative government spending over the $h$ periods
($\widetilde{\theta}_{h,xs}\geq 0$ for $s=1,2$).

  2. (ii)

The RZ news shock is positively correlated with defense spending
($E[z_{t}^{RZ}\varepsilon_{2,t}]>0$).

  3. (iii)

The BP defense shock is positively correlated with defense spending
($E[z_{t}^{BP}\varepsilon_{2,t}]>0$).

  4. (iv)

The RZ news shock is negatively correlated with non-defense spending
($E[z_{t}^{RZ}\varepsilon_{1,t}]<0$).

  5. (v)

The correlation between the BP defense shock and non-defense spending is close
to zero, regardless of the sign ($E[z_{t}^{BP}\varepsilon_{1,t}]\approx 0$).

The restrictions (i)-(iii) are straightforward. (iv) is based on the
government budget constraint argument. (v) is a structural restriction similar
to the ordering of endogenous variables in the SVAR model. Recall that

$\displaystyle\beta_{h}^{RZ}=$ $\displaystyle
w_{h,1}^{RZ}\times\frac{\widetilde{\theta}_{h,y1}}{\widetilde{\theta}_{h,x1}}+(1-w_{h,1}^{RZ})\times\frac{\widetilde{\theta}_{h,y2}}{\widetilde{\theta}_{h,x2}},$
(25) $\displaystyle\beta_{h}^{BP}=$ $\displaystyle
w_{h,1}^{BP}\times\frac{\widetilde{\theta}_{h,y1}}{\widetilde{\theta}_{h,x1}}+(1-w_{h,1}^{BP})\times\frac{\widetilde{\theta}_{h,y2}}{\widetilde{\theta}_{h,x2}},$

where

$w_{h,1}^{j}=\frac{E[z_{t}^{j}\varepsilon_{1,t}]\widetilde{\theta}_{h,x1}}{E[z_{t}^{j}\widetilde{x}_{t+h}]}$

for $j=RZ,BP$. In the post WWII sample, the estimates of
$E[z_{t}^{j}\widetilde{x}_{t+h}]$ are positive for $h\geq 1$ for both
instruments. Thus, the sign of $w_{h,1}^{j}$ is determined by the sign of the
numerator, which are given by (i)-(iv). By (i) and (ii), $1-w_{h,1}^{RZ}>0$.
By (i) and (iii), $1-w_{h,1}^{BP}>0$. By (i) and (iv), $w_{h,1}^{RZ}<0$.
Finally, by (i) and (v), $w_{h,1}^{RZ}<w_{h,1}^{BP}$.

Using (16)-(19) and the estimates of $\beta_{h}^{RZ}$ and $\beta_{h}^{BP}$, we
can estimate the identified set for the sectoral spending multipliers. The
sets are shown as grey areas in Figure 6. For the cumulative non-defense
spending multiplier, it is bounded above by the LP-IV estimates using the RZ
news shock until around two years, when it is then bounded below by the LP-IV
estimates using the BP defense shock. The identified set for the cumulative
defense spending multiplier is identical to that of the non-defense spending
multiplier until the two LP-IV estimates cross around two years out, but the
set is bounded below by the LP-IV estimates using the RZ news shock which is
wider.

Note that the point estimates of the sectoral spending multipliers in Figure 4
are in the identified set in Figure 6. This may not be surprising as the
restrictions (i)-(v) are consistent with those estimated weights in (23) and
(24). The illustrated bound analysis shows that even when the sectoral data
are not available we can derive the bounds for the componentwise impulse
responses by using reasonable structural assumptions.

Next consider the case when the sectoral data and one instrument are available
(Case II in Section 4). We use GDP, total government spending, defense and
non-defense spending, and the RZ news shock. With the RZ news shock as the
instrument, the LP-IV estimand can be decomposed according to (25). Under
Assumptions 1 and 3, the weight is identified as

$w_{h,1}^{RZ}=\frac{E[z_{t}^{RZ}\widetilde{x}_{1,t+h}]}{E[z_{tt}^{RZ}\widetilde{x}_{t+h}]}.$

Since we can estimate $\beta_{h}^{RZ}$ and $w_{h,1}^{RZ}$, we have the set of
pairs
$(\widetilde{\theta}_{h,y1}/\widetilde{\theta}_{h,x1},\widetilde{\theta}_{h,y2}/\widetilde{\theta}_{h,x2})$
which satisfy the estimated decomposition in (25).

Figure 7: Identified Set for Sectoral Spending Multipliers using Auxiliary
Sectoral Data

Figure 7 shows the sets for $h=0,...,20$ (blue solid lines). When $h=0$, the
line has a steeper slope with a larger y-intercept and the slope and the
y-intercept become smaller in general as $h$ increases. This implies that for
a given magnitude of the defense spending multiplier (y-axis) the
corresponding magnitude of the non-defense spending multiplier is increasing
with $h$. Specifically, to have the non-defense spending multiplier larger
than one, the magnitude of the defense spending multiplier needs to be larger
than 1.5 when $h=0$ but 0.67 when $h=20$ (five years out).

The identified set in Figure 7 can also be used for counterfactual analyses.
We calibrate the defense spending multiplier and find the corresponding non-
defense spending multiplier for each $h$. Our justification of calibrating the
defense spending multiplier is that there is a general consensus in the
literature about the magnitude of the defense spending multiplier, especially
compared with the magnitude of the non-defense spending multiplier. Figure 7
shows two trajectories of the sectoral spending multipliers (with the circle
and the x markers). The transitory999Named to be clearly contrasted to the
persistent defense spending multiplier. defense spending multiplier is
obtained by using the result of Auerbach and Gorodnichenko (2012), who used
the smooth transition VAR model to estimate the sectoral spending multipliers.
The persistent defense spending multiplier is calibrated using the finding of
Barro and Redlick (2011) that the defense spending multiplier is higher by
0.1-0.2 if the shock is permanent. The defense spending multiplier in Auerbach
and Gorodnichenko (2012) is 0.67 and the calibrated persistent multiplier is
0.85 after two years.

Figure 8: Sectoral Spending Multipliers using Calibration

The two trajectories in Figure 7 and Figure 8 reveal that a small difference
in the magnitude of the calibrated defense spending multiplier can lead to a
big difference in the non-defense spending multiplier. In particular, if the
magnitude of the defense spending multiplier is persistently around 0.7-0.8
after two years, then the non-defense spending multiplier can be larger than
one even when the government (aggregate) and defense spending multipliers are
below one.

## 7 Conclusion

In response to the increasing popularity of using external instruments in
identifying and estimating the impulse response functions of macroeconomic
shocks, we investigate what the IV estimand for a policy multiplier actually
identifies. We show that the IV estimand may be uninformative and misleading
in the presence of heterogeneous componentwise impulse responses to a policy
shock. Therefore, when interpreting an IV estimate for a policy multiplier,
appropriate caution should be exercised, such as testing the same-sign
condition and analyzing identification bounds.

## References

  * [1] Angrist, J. D., Imbens, G. W., & Rubin, D. B. (1996). Identification of causal effects using instrumental variables. Journal of the American Statistical Association, 91(434), 444-455.
  * [2] Angrist, J. D., & Kuersteiner, G. M. (2011). Causal effects of monetary shocks: Semiparametric conditional independence tests with a multinomial propensity score. Review of Economics and Statistics, 93(3), 725-747.
  * [3] Auerbach, A. J., & Gorodnichenko, Y. (2012). Measuring the output responses to fiscal policy. American Economic Journal: Economic Policy, 4(2), 1-27.
  * [4] Barro, R. J., & Redlick, C. J. (2011). Macroeconomic effects from government purchases and taxes. The Quarterly Journal of Economics, 126(1), 51-102.
  * [5] Blanchard, O., & Perotti, R. (2002). An empirical characterization of the dynamic effects of changes in government spending and taxes on output. The Quarterly Journal of Economics, 117(4), 1329-1368.
  * [6] Bouakez, H., Rachedi, O., & Santoro, E. (2020). The sectoral origins of the spending multiplier. Working paper.
  * [7] Campbell, J. R., Evans, C. L., Fisher, J. D., Justiniano. (2012). Macroeconomic effects of federal reserve forward guidance. Brookings papers on economic activity, 1-80.
  * [8] Canova, F. (2022) Should we trust cross sectional multiplier estimates? forthcoming, Journal of Applied Econometrics.
  * [9] Cox, L., Müller, G., Pasten, E., Schoenle, R., & Weber, M. (2020). Big G (No. w27034). National Bureau of Economic Research.
  * [10] Imbens, G. W., & Angrist, J. D. (1994). Identification and Estimation of Local Average Treatment Effects. Econometrica, 62(2), 467-475.
  * [11] Jordà, Ò. (2005). Estimation and inference of impulse responses by local projections. American Economic Review, 95(1), 161-182.
  * [12] Jordà, Ò., Schularick, M., & Taylor, A. M. (2015). Betting the house. Journal of International Economics, 96, S2-S18.
  * [13] Jordà, Ò., Schularick, M., & Taylor, A. M. (2020). The effects of quasi-random monetary experiments. Journal of Monetary Economics, 112, 22-40.
  * [14] Kleibergen, F. (2005). Testing parameters in GMM without assuming that they are identified. Econometrica, 73(4), 1103-1123.
  * [15] Mertens, K., & Ravn, M. O. (2013). The dynamic effects of personal and corporate income tax changes in the United States. American Economic Review, 103(4), 1212-47.
  * [16] Montiel Olea, J. L., & Plagborg‐Møller, M. (2021). Local projection inference is simpler and more robust than you think. Econometrica, 89(4), 1789-1823.
  * [17] Nakamura, E., & Steinsson, J. (2014). Fiscal stimulus in a monetary union: Evidence from US regions. American Economic Review, 104(3), 753-92.
  * [18] Newey, W. K., & McFadden, D. (1994). Large sample estimation and hypothesis testing. Handbook of econometrics, 4, 2111-2245.
  * [19] Newey, W. K., & West, K. D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55(3), 703-708.
  * [20] Plagborg-Møller, M., & Wolf, C. K. (2021a). Local projections and VARs estimate the same impulse responses. Econometrica, 89(2), 955-980.
  * [21] Plagborg-Møller, M., & Wolf, C. K. (2022). Instrumental variable identification of dynamic variance decompositions. Journal of Political Economy, 130(8), 2164-2202.
  * [22] Ramey, V. A. (2011). Identifying government spending shocks: It’s all in the timing. The Quarterly Journal of Economics, 126(1), 1-50.
  * [23] Rambachan, A., & Shepherd, N. (2021). When do common time series estimands have nonparametric causal meaning?. Working paper.
  * [24] Ramey, V. A., & Zubairy, S. (2018). Government spending multipliers in good times and in bad: evidence from US historical data. Journal of Political Economy, 126(2), 850-901.
  * [25] Stock, J. H., & Watson, M. W. (2012). Disentangling the Channels of the 2007-2009 Recession (No. w18094). National Bureau of Economic Research.
  * [26] Stock, J. H., & Watson, M. W. (2018). Identification and estimation of dynamic causal effects in macroeconomics using external instruments. The Economic Journal, 128(610), 917-948.
  * [27] Woodford, M. (2020). Effective demand failures and the limits of monetary stabilization policy (No. w27768). National Bureau of Economic Research.