# Robust Impulse Responses using External Instruments: the Role of
Information††thanks: This paper has benefited from discussions with Mario
Forni and participants at the UEA Time Series Workshop, the Workshop in
Empirical and Theoretical Macroeconomics and various research seminars.

Davide Brignone Università di Roma Tor Vergata, email:
<EMAIL_ADDRESS>Alessandro Franconi Università di Pavia, email:
<EMAIL_ADDRESS>Marco Mazzali Università di Bologna, email:
<EMAIL_ADDRESS>
(This version: )

###### Abstract

External-instrument identification leads to biased responses when the shock is
not invertible and the measurement error is present. We propose to use this
identification strategy in a structural Dynamic Factor Model, which we call
Proxy DFM. In a simulation analysis, we show that the Proxy DFM always
successfully retrieves the true impulse responses, while the Proxy SVAR
systematically fails to do so when the model is either misspecified, does not
include all relevant information, or the measurement error is present. In an
application to US monetary policy, the Proxy DFM shows that a tightening shock
is unequivocally contractionary, with deteriorations in domestic demand,
labor, credit, housing, exchange, and financial markets. This holds true for
all raw instruments available in the literature. The variance decomposition
analysis highlights the importance of monetary policy shocks in explaining
economic fluctuations, albeit at different horizons.

Keywords: Proxy Dynamic Factor Model, Monetary Policy, Fundamentalness,
Impulse Response Functions, Variance Decomposition.
JEL codes: C32, C38, E52.

##  1 Introduction

The use of external instruments to identify structural shocks has now become a
widely used methodology in macroeconomics for estimating dynamic causal
effects. The main advantage of employing such identification strategy,
compared to traditional schemes, is its independence from any assumption on
sign or timing of the impulse responses (see Stock and Watson, 2018, for a
detailed review).

However, this identification strategy is not immune to problems affecting
standard vector autoregressive (VARs) models. As largely discussed by the
literature, the reliability of this class of models in estimating the
transmission of structural shocks may be undermined by two major limitations:
the curse of dimensionality and the presence of measurement errors. The first
is linked to the restricted amount of information available to the
econometrician, which is forced to use small- or, at most, medium-scale
systems due to the exponentially growing parameter space. Implicitly, this
procedure assumes that the information from the past and present observations
of the selected endogenous variables is sufficient to recover the structural
shock of interest, i.e., the model is informational sufficient (following the
definition of Forni and Gambetti, 2014). However, several studies demonstrated
that this assumption does not always hold true, as for instance, in the
presence of anticipated shocks, that is, shocks with a delayed effect on some
variables (see Leeper et al., 2013; Forni et al., 2014, among others).
Consequently, the issues of non-invertibility or non-fundamentalness may
arise.111While invertibility requires that the shocks can be inferred from
past and current values of the endogenous variables, fundamentalness implies
that the shocks only need to belong to the space spanned by those values. Note
that while the two properties are closely related, they are not exactly the
same thing, although they are often used as synonyms. For example, consider
the case where the vector moving average representation of a square system has
at least one root equal to one in modulus. The system would not be invertible,
but it would still be fundamental. See, inter alia, Hansen and Sargent (1991),
Lippi and Reichlin (1993, 1994) and the review by Alessi et al. (2011). A
second source of bias may be the presence of a non-negligible idiosyncratic
component of the series, which is usually interpreted as measurement error or
as sectoral disturbances. Indeed, despite often neglected in the literature,
this could further affect the estimates of the impulse response functions,
leading to inaccurate results - see, for instance, Giannone et al. (2006) and
Forni et al. (2020).

In the last decades, the literature attempted to deal with the curse of
dimensionality proposing econometric models that allow to include a larger
number of variables w.r.t. VARs. A non-exhaustive list of examples includes
Bayesian VARs (see, e.g., De Mol et al., 2008; Bańbura et al., 2010), Reduced-
Rank VARs (see,e.g., Carriero et al., 2011; Cubadda and Hecq, 2022), Factor
Augmented VARs (FAVAR, henceforth - see, e.g., Bernanke et al., 2005), and
Dynamic Factor Models (DFM, henceforth - see, e.g., Forni et al., 2000, 2009;
Stock and Watson, 2002). Although they manage to solve the missing information
problem, only the last approach is able to deal with measurement errors.

In this paper, we propose using external instruments in a Dynamic Factor Model
(Proxy DFM, henceforth). We describe how to apply the external instrument
identification in a structural DFM framework to estimate a unit-variance
shock. This allows the estimation of the variance decomposition and, possibly,
the historical decomposition. By means of the simple theoretical model with
perfect foresight studied in Leeper et al. (2013), we show the benefits of
applying the proposed identification strategy within a data-rich environment
compared to a small-scale (VAR) model. We compare the theoretical responses
with those obtained by estimating impulse responses using both a Proxy SVAR
and Proxy DFM from the simulated series of the model. We explore various model
specifications and highlight the role of measurement error and non-
fundamentalness in biasing the estimated results.

Our findings indicate that, when the information set is insufficient, Proxy
VAR delivers biased responses. Conversely, the DFM successfully estimates the
true IRFs. Moreover, informational deficiency may be compounded by other
sources of bias, such as the sensitivity of estimates to the chosen model
specification and the presence of measurement error. For example, even if the
VAR is correctly specified and the system is fundamental, the presence of
measurement error can bias the final estimates. In contrast, the DFM is not
affected by any distortion. Its robustness holds even when higher levels of
the idiosyncratic component are added to the simulated series.

We document the practical value of the proposed approach with an empirical
application, contributing to a never-ending debate for macroeconomists and
policymakers: the macroeconomic effect and the transmission of monetary
policy.

We select several instruments recently proposed in the literature,
specifically by Gertler and Karadi (2015) (GK), Romer and Romer (2004) (RR),
Miranda-Agrippino and Ricco (2021) (MAR), the raw high-frequency surprise
series of Jarociński and Karadi (2020) (JK) and Bauer and Swanson (2022) (BS)
and compare the results between the estimates obtained from different model
specifications using a Proxy SVAR with the unique set of responses estimated
using a Proxy DFM. The choice of instruments is not casual. In fact, we
consider also instruments that tend to capture the monetary policy shock along
with a news component coming from the central bank’s assessment of the
economic outlook, which makes the underlying shock also partially anticipated
(Ramey, 2016; Jarociński and Karadi, 2020; Miranda-Agrippino and Ricco, 2021).
This event potentially exposes the VAR estimates to the problems introduced
above. We find that the SVAR responses exhibit both output and price puzzles
in almost all the specifications analyzed and across all instruments except
the one proposed by MAR. On the other hand, the unique set of responses
estimated through the Proxy DFM does not suffer from either of these puzzles.
In other words, one can solve issues arising from the instrument by simply
enlarging the information set of the econometrician, mitigating the effect of
the news component of the monetary policy shock.

In the last part of our paper, we provide a thorough analysis of monetary
policy shocks transmission. We now exclusively focus on Proxy DFM results,
using GK as the baseline instrument. In addition, we compare the results with
all the instruments previously introduced. We study the various channels
through which a monetary policy shock propagates, analyzing both impulse
responses and the variance decomposition.

The first interesting result is that our proposed model estimate impulse
responses are robust across all instruments under analysis. Turning to the
analysis of the transmission, we find that a monetary policy tightening shock
is unequivocally contractionary for the economy. Real variables contract, but
only after some months, with the exception of consumption and retail sales,
which react at impact. All measures of prices decline, and their behavior
indicates the presence of price rigidities in the economy, as they do not
fully adjust at impact. The housing sector is deeply affected by the shock
and, by weakening household balance sheets, can explain the sharp contraction
in consumption (see Mian et al., 2013, for this mechanism). Its effect could
compound with other important channels. For instance, equity prices fall,
further negatively affecting household wealth; the exchange rate rises,
signaling an appreciation of the domestic currency that translates into a
reduction in net exports; and interest rates and spreads all point to an
overall contraction of the financial sector, which could amplify the negative
business cycle impact of the shock (Jordà et al., 2017). The variance
decomposition analysis indicates that monetary policy shocks are important in
explaining business cycle fluctuations of the economy, with results that
differ depending on the nature of the variable. For real variables, the shock
does not explain much variance at the very impact, but its importance
increases towards the third year. As expected, monetary policy shocks explain
a large part of the variance already at impact for financial variables,
exchange rates, and prices.

##### Related Literature.

Our study is linked to different strands of literature. First, we refer to the
literature related to Dynamic Factor Models. This model can include a large
number of variables in the estimation, and can therefore enlarge the
information set of the econometrician, a feature which makes it useful both in
forecasting and structural analysis (Forni et al., 2000; Stock and Watson,
2002). On the latter, the model can be particularly appealing, given that it
has been proved that the measurement error and non-fundamentalness issues are
solved by construction. Furthermore, the estimation of its Wold representation
- and reduced form shocks - does not depart too much from the standard
procedure which is applied in the context of VAR models (Forni et al., 2009).
For this reason, the DFM has been widely used in the context of structural
analysis in recent years, applied with different identification techniques -
see, for instance, Del Negro and Otrok (2007); Forni and Gambetti (2010);
Barigozzi et al. (2014); Luciani (2015); Bjørnland and Thorsrud (2019);
Kerssenfischer (2019); Brignone and Mazzali (2022). Among these, Forni and
Gambetti (2010) is particularly related to our work as it was the first to
explore the effects of U.S. monetary policy shocks in a DFM. In this paper we
departure by using external instruments to identify the monetary policy shock
rather than imposing timing restrictions of the variables in the model. Recent
contributions related to our work, who analyze monetary policy shocks in a
Proxy DFM are Alessi and Kerssenfischer (2019) and Corsetti et al. (2022). The
former uses it to estimate the response of asset prices to monetary policy
shocks in the U.S., while the latter to study the heterogeneity in the
transmission of shocks across euro area countries. Our work differs from
theirs in terms of both methodology and scope. In a simulation exercise, we
provide a comprehensive analysis of the advantages that a researcher can gain
by using the Proxy DFM relative to the Proxy SVAR. In the empirical
application, instead, we study the macroeconomic effects of monetary policy in
the U.S., along with the channels through which it propagates.222Focusing on
the issue of information and shock recoverability, our study is also related
to the recent literature that has proposed milder conditions based on the
concept of information sufficiency, see Forni et al. (2019); Chahrour and
Jurado (2022) Another major departure from them is our unit-variance shock
estimation, which allows us to perform variance and historical
decompositions.333In the paper, we only show the former decomposition.

Second, our study focuses on the strand of literature which identifies
structural shocks of interest in a VAR framework using external instruments
available (see Stock and Watson, 2018, for a survey). This approach typically
involves a two-step strategy. Firstly, reduced form shocks are estimated,
usually from a VAR model. Then, the structural shock of interest is identified
by projecting a single selected reduced-form shock on the available external
instrument. Its first appearance in macro-econometrics dates back to Stock
(2008), reaching broader consensus some years later. Among the many
contributions using Proxy SVARs we find Stock and Watson (2012), which examine
the macroeconomic dynamics of the Great Recession in the U.S. and the
subsequent slow recovery, Mertens and Ravn (2013), which provide evidence on
the U.S. personal and corporate income tax changes. More closely related
papers are the ones using a Proxy SVAR to estimate the macroeconomic effects
of monetary policy shocks. Among those, we find Gertler and Karadi (2015),
Jarociński and Karadi (2020), Miranda-Agrippino and Ricco (2021), and Bauer
and Swanson (2022). This literature has also been favoured by the development
of the high-frequency identification (HFI) strategy, which allows using
information from “outside” the VAR to identify the shock of interest.444The
narrative approach, instead, has been less involved in the shock
identification of Proxy SVAR. One potential reason is that it involves
constructing a series for the shock from related historical documents,
exposing this methodology to the problem of subjective choices. The HFI
exploits high-frequency asset price changes around policy meetings to quantify
exogenous changes in monetary policy actions (Kuttner, 2001; Gürkaynak et al.,
2004, among others). The identifying assumption underlying the HFI approach is
that unexpected changes in interest rates in a short window surrounding policy
meetings are only due to monetary policy actions.

In between the two bodies of literature mentioned above, our work is also
related to Miescu and Mumtaz (2019); Bruns (2021); De Nora (2023), which use
external instruments within a FAVAR approach. As we show in the simulations,
FAVAR is able to address the missing variable problem, but its estimates are
still affected by measurement error and by model specification.

Finally, we also refer to the literature which directly focuses on the role of
information in the proxy SVAR identification (see the recent works by Forni et
al., 2022; Plagborg-Møller and Wolf, 2022; Bruns, 2021; Miescu and Mumtaz,
2019), and generally to the literature which has proposed milder conditions
based on the concepts of informational sufficiency and recoverability of the
shocks555Recoverability requires that the structural shock is a linear
combination of present and future values of VAR residuals. For this,
fundamentalness implies recoverability, but not vice versa.  \- see Forni et
al. (2019, 2022); Chahrour and Jurado (2022). The Proxy DFM is invertible by
construction and, thus, the shocks are always recoverable, as the
invertibility condition is stricter than the latter.

The remainder of the paper is organised as follows. Section 2 describes the
econometrics behind the DFM and the methodology we follow to apply the
external instrument identification within this framework. Section 3 is devoted
to the comparison between the Proxy SVAR and Proxy DFM results using a
theoretical model with perfect foresight. Section 4 covers the empirical
application to monetary policy. First, we compare impulse responses between
the VAR and DFM identified with multiple instruments, then we use a Proxy DFM
to shed light on the transmission mechanism of a monetary policy shock.
Section 5 concludes.

##  2 Econometric framework

In this section, we present the model and the identification procedure.
Regarding the DFM, we present two specifications. First, we show the
stationary case, which is used in the simulation, which builds on Forni et al.
(2009). Second, we present the methodology proposed by Barigozzi et al. (2021)
to handle non-stationary variables. The latter is used in the empirical
application.

###  2.1 Dynamic Factor Model

####  2.1.1 The stationary I(0) specification

Consider a $N$-vector $x_{t}$ of weakly stationary time series. As standard in
DFM literature, we assume each variable $x_{it}$, $i=1,...,N$, can be
rewritten as the sum of an idiosyncratic component, $\xi_{it}$, and a common
component, $\chi_{it}$. The former represents the source of variation
affecting a specific variable and thus interpreted as measurement error or
sectoral shocks. It is assumed that the $\xi_{it}$ are poorly correlated in
the cross-sectional dimension, a milder and more realistic assumption than
uncorrelation. This assumption is crucial for ascribing this model to the
class of approximate factor models, rather than the more traditional exact
factor models à la Sargent and Sims (1977) and Geweke (1977). On the other
hand, the common components, $\chi_{it}$, are assumed to permeate the entire
dataset and to be function of $q$ common shocks
$u_{t}=(u_{1t},u_{2t},...,u_{qt})^{\prime}$, with $q<<N$, such that666In order
to be consistent with the literature, throughout the paper, we will refer to
$u_{t}$ interchangeably as common shocks or dynamic factors.

$\displaystyle\chi_{it}$
$\displaystyle=b_{i1}(L)u_{1t}+b_{i2}(L)u_{2t}+...+b_{iq}(L)u_{qt}$

Defining $\chi_{t}=(\chi_{1t}\dots\chi_{Nt})^{\prime}$ and
$\xi_{t}=(\xi_{1t}\dots\xi_{Nt})^{\prime}$ we can rewrite the model in vector
notation as

$\displaystyle\chi_{t}$ $\displaystyle=B_{\chi}(L){u_{t}}$ (1)

where $B_{\chi}(L)$ is a $N\times q$ matrix, whose $(i,j)$-th entry is
$b_{ij}(L)$, and $u_{t}$ is an orthonormal white noise vector such that
$u_{t}\perp\xi_{t}$. Since the vector $u_{t}$ is orthogonal to $\xi_{t}$, the
latter is also orthogonal to $\chi_{t}$. Moreover, being the vector $\chi_{t}$
singular, the dynamic factors are fundamental.

One can further rewrite the expression above in terms of static factors as

$\displaystyle\chi_{t}$ $\displaystyle=\Lambda F_{t}$ (2)

where $F_{t}$ is a vector of $r>q$ static factors, still orthogonal to
$\xi_{t}$, and $\Lambda$ is a $N\times r$ matrix of factor loadings. Notice
that the static factors are only loaded contemporaneously, whereas we consider
present and past values of the dynamic factors. It is further assumed that the
vector $F_{t}$ follows a VAR of order p, and that $F_{t}$ and $u_{t}$ are
linked as follows

$D(L)F_{t}=\varepsilon_{t},\quad\text{with }\varepsilon_{t}=Ru_{t}$ (3)

where $D(L)$ is a $r\times r$ polynomial matrix of coefficients,
$\varepsilon_{t}$ is the vector of VAR errors of the static factors, R is a
$r\times q$ matrix resulting from a spectral decomposition of the errors
$\varepsilon_{t}$. Given that $r>q$, the stochastic vector $F_{t}$ is still
singular.

By inverting the matrix of coefficient $D(L)$ we obtain the moving average
representation

$F_{t}=D(L)^{-1}Ru_{t}=B_{F}(L)u_{t}$ (4)

and the following relation

$\displaystyle\chi_{t}$ $\displaystyle=\Lambda F_{t}=\Lambda
B_{F}(L)u_{t}=B_{\chi}(L)u_{t}$ (5)

where $B_{F}(L)$ is a $r\times q$ polynomial matrix of coefficients. Equations
(3) - (5) uncover the link between dynamic and static factors.

####  2.1.2 The non-stationary I(1) specification

In case the data is $I(1)$, we need a few adjustments, as suggested by the
recent work of Barigozzi et al. (2021). Let us suppose we have a $N$-vector
$x_{t}$ of non-stationary time series. Considering a deterministic trend, one
can still describe those as the sum of two orthogonal unobservable components,
namely a common and idiosyncratic component, as

$\displaystyle x_{t}$ $\displaystyle=\alpha+\beta t+\chi_{t}+\xi_{t}$ (6)

where $\alpha$ is a vector of constants and $\beta t$ is the linear trend. In
this framework, moreover, the factors are assumed to be $I(1)$ and the
idiosyncratic components are either $I(0)$ or $I(1)$.

The procedure in this case is as follows: (i) we estimate the number of static
factors $r$ and dynamic factors $q$ on the $I(0)$ transformed data; (ii) we
estimate the loadings $\Lambda$ from the differenciated data by means of the
principal component; (iii) given $\Lambda$, we retrieve $F_{t}$ from the non-
stationary dataset; (iv) we estimate $\alpha$ and the coefficient $\beta$
associated to the trend and we project
$\hat{\tilde{x}}_{t}=x_{t}-\hat{\alpha}-\hat{\beta}t$ on the previously
estimated loadings; (v) having $F_{t}$, we derive the MA representation of
$\chi_{t}$, analogously to equation (5), either applying a VAR-in-level
specification or a VECM. The latter case requires the estimation of the number
of cointegration relationships $d$ .

###  2.2 External Instrument Identification in a DFM environment

To identify the structural shocks from the estimated reduced form shock in
(3), we follow the external instrument procedure developed by Stock and Watson
(2012) and Mertens and Ravn (2013). We start by describing the standard
procedure in a quadratic system such as in VAR models, given that most of the
assumptions also apply in a DFM framework.

Let us assume we observe a variable $z_{t}$ satisfying the following
conditions:

$\displaystyle\mathbb{E}_{t}(z_{t}\eta_{it})=\alpha$ (7)
$\displaystyle\mathbb{E}_{t}(z_{t}\eta_{-it})=0$ (8)

with $\eta_{it}$ representing the structural shock we want to identiy and
$\eta_{-it}$ the remainig structural shocks. The two above conditions state
that the variable $z_{t}$ needs a) to be correlated with the structural shock
$\eta_{it}$ we want to estimate and b) to be orthogonal to all the remaining
structural shocks.

If the two condition apply, one can retrieve the structural shock $\eta_{it}$
from the estimated reduced-form shock $u_{it}$ by regressing $u_{it}$ on the
instrument $z_{t}$ and then by a proper rescaling of the coefficient coming
from the regression. This, however, makes automatically impossible to retrieve
a unit-variance shock and to perform variance and historical decomposition.
Moreover, this way of proceeding would not fit in a DFM framework. Indeed, the
methodology would require the choice of the estimated reduced-form residual
which is directly associated to the policy variable. This is always possible
in a VAR setting, given that it follows a non-singular quadratic $N\times N$
model, whereas the DFM is characterised by a singular $N\times q$
representation, which makes a clear one-to-one link between variables and
shocks impossible.

A viable solution is offered by Forni et al. (2022), where they describe an
alternative way to estimate a unit-variance shock using an external instrument
identification in a non-singular quadratic system as the VAR.777A similar
procedure is employed by Alessi and Kerssenfischer (2019), which, however, is
unable to retrieve the unit-variance shock and, therefore, to perform a
variance and historical decomposition analysis. We borrow from them and
consider the projection of the instrument, $z_{t}$, on the vector of
residuals, $u_{t}$, as follows

$z_{t}=\delta^{\prime}u_{t}+v_{t}$ (9)

The unit-variance shock and the respective IRFs can be estimated as

$\displaystyle\hat{\eta}_{it}=$
$\displaystyle\frac{\hat{\delta}^{\prime}\hat{u}_{t}}{\sqrt{\hat{\delta}^{\prime}\widehat{\Sigma}_{u}\hat{\delta}}}$
(10) $\displaystyle C_{\chi,i}(L)=$ $\displaystyle
B_{\chi}(L)\widehat{\Sigma}_{u}\frac{\hat{\delta}}{\sqrt{\hat{\delta}^{\prime}\widehat{\Sigma}_{u}\hat{\delta}}}$
(11)

where $\Sigma_{u}$ is the variance-covariance matrix of $u_{t}$, which is
equal to the identity matrix in our framework.888Additional calculations are
available in Section A of the Appendix. We obtain that the instrument $z_{t}$
is correlated to the structural shock of interest $\eta_{it}$, spanned
throughout all the $u_{t}$, and that it is orthogonal to all the remaining
structural shocks $\eta_{-it}$. As a direct consequence, we do not need to
choose a specific variable to instrument, as is the case in standard VARs - a
choice that has often left room for discussion in the literature, providing an
additional source of sensitivity that could reduce the external validity of
the results.999One recent example is Bauer and Swanson (2022), which argue
that the two-year Treasury yield is a better measure of the monetary policy
stance than the one-year Treasury yield used by Gertler and Karadi (2015).

##  3  Non-fundamentalness and Proxy identification

This section aims at showing some of the possible advantages offered by the
proxy identification strategy in the DFM framework, as described in section
2.2, compared to a standard VAR model. We conduct the first part of the
analysis by deploying a theoretical macroeconomic model. Specifically, we
borrow the simple Real Business Cycle (RBC) model of Leeper et al. (2013).
Such model is characterized by log preferences, inelastic labor supply, full
capital depreciation and fiscal foresight. The latter, in particular, is what
matters in our case. In fact, in the presence of fiscal foresight, agents are
able to foresee future values of the tax rate and behave according to such
information.

The evolution of capital through time is dictated by the following equation

$k_{t}=\alpha
k_{t-1}+a_{t}-\kappa\sum_{i=0}^{\infty}\theta^{i}E_{t}\tau_{t+i+1}$ (12)

where $0<\alpha<1,|\theta|<1,\kappa=(1-\theta)\tau/(1-\tau)$, $\tau$ is the
steady state tax rate, $k_{t}$ is capital, $a_{t}$ is technology and
$\tau_{t}$ is tax rate. All the variables are considered as log deviations
from the their steady state value.

For the sake of simplicity, technology and tax are assumed to follow two
i.i.d. processes, $u_{a,t}$ and $u_{\tau,t}$, respectively. In order to allow
for fiscal foresight, it is assumed that

$\tau_{t}=u_{\tau,t-h}$

Depending on the value of $h$, we face different scenarios. When $h=0$,
equation (12) reduces to $k_{t}=\alpha k_{t-1}+a_{t}$, $i.e.$ the
representative agent does not have information about the future. As a
consequence capital accumulation does not depend on the tax shock. In this
situation, the information set of the econometrician is aligned with that of
the agent. A completely different case, instead, is when $h>0$. Since the
information set at time $t$ of the agent includes present and past
observations of $u_{a,t}$ and $u_{\tau,t}$, she has knowledge about future
values of $\tau_{t}$, according to which capital will be adjusted. Such
knowledge generates misalignment between the two information sets, causing
non-fundamentalness of the shocks.

In what follows, we consider the case with $h=2$, that is characterized by the
following equations

$\displaystyle a_{t}$ $\displaystyle=u_{a,t}$ $\displaystyle k_{t}$
$\displaystyle=\alpha k_{t-1}+a_{t}-\kappa(u_{\tau,t-1}+\theta u_{\tau,t})$
$\displaystyle\tau_{t}$ $\displaystyle=u_{\tau,t-2}$

which implies the MA representation

$\left(\begin{array}[]{l}a_{t}\\\ k_{t}\\\
\tau_{t}\end{array}\right)=\left(\begin{array}[]{cc}0&1\\\
\frac{-\kappa(L+\theta)}{1-\alpha L}&\frac{1}{1-\alpha L}\\\
L^{2}&0\end{array}\right)\left(\begin{array}[]{c}u_{\tau,t}\\\
u_{a,t}\end{array}\right)=B(L)u_{t}.$ (13)

As pointed out by Forni et al. (2020), $u_{t}$ is fundamental for none of the
square subsystems generated by the possible pairs of
$y_{t}=(a_{t},k_{t},\tau_{t})^{\prime}$. On the other hand, considering all
the three variables is sufficient to recover $u_{t}$.101010The fact that
adding information helps in recovering the true structural shocks is not new
in the literature. See inter alia Giannone and Reichlin (2006).

###  3.1 Simulation

We propose a simulation exercise to quantitatively assess to what extent non-
funtamentalness threatens the results coming from the empirical analysis. The
exercise is very close in spirit to the one performed in Forni et al. (2020)
and Miescu and Mumtaz (2019).

We consider a static factor model representation of the form

$\chi_{t}=\Lambda F_{t}$ (14)

where $\chi_{t}$ is a $n$-dimensional vector of economic variables, $\Lambda$
is a $n\times r$ matrix of loadings and $F_{t}$ is a vector of static factors
of dimension $r$.

Let us define
$F_{t}=(k_{t},u_{a,t},u_{\tau,t}u_{\tau,t-1},u_{\tau,t-2})^{\prime}$, which is
assumed to have the following VAR(1) dynamics

$F_{t}=AF_{t-1}+Bu_{t}$ (15)

where

$A=\left(\begin{array}[]{ccccc}\alpha&0&-\kappa&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&1&0&0\\\ 0&0&0&1&0\end{array}\right)\quad\quad
B=\left(\begin{array}[]{cc}1&-\kappa\theta\\\ 1&0\\\ 0&1\\\ 0&0\\\
0&0\end{array}\right)\quad\quad u_{t}=\left(\begin{array}[]{c}u_{a,t}\\\
u_{\tau,t}\end{array}\right)$

Having the factors $F_{t}$, one can easily find that
$y_{t}=\bar{\Lambda}^{x}F_{t}$, where $y_{t}=(a_{t},k_{t},\tau_{t})^{\prime}$
and

$\bar{\Lambda}^{x}=\left(\begin{array}[]{ccccc}0&1&0&0&0\\\ 1&0&0&0&0\\\
0&0&0&0&1\end{array}\right)$

We assume the econometrician observes
$\chi_{t}=(x_{t}^{\prime},x_{t}^{*}{{}^{\prime}})^{\prime}$ where
$x_{t}=(\tau_{t},k_{t})^{\prime}$ is a non-fundamental subsystem of $y_{t}$
and $x^{*}_{t}$ is a set of $n$ survey series generated artificially. The
cross-sectional dimension of the entire sample is thus $N=n+2$. The series in
$x^{*}_{t}$ are generated by a linear transformation of the factors, given by
$\Lambda^{*}F_{t}$, where the entries of $\Lambda^{*}$ are drawn from
independent $N(0,1)$.

Considering $\Lambda^{x}$ as the third and second rows of $\bar{\Lambda}^{x}$
in such order, we can rewrite equation (14) as

$\left(\begin{array}[]{c}x_{t}\\\
x^{*}_{t}\end{array}\right)=\left(\begin{array}[]{c}\Lambda^{x}\\\
\Lambda^{*}\end{array}\right)F_{t}$ (16)

Finally, we assume that the varibles are observed with a measurement error

$Y_{t}=\chi_{t}+\xi_{t}=\Lambda F_{t}+\xi_{t}$ (17)

where $\xi_{it}\sim N(0,\sigma_{i})$ and $\sigma_{i}\sim U(0,\nu)$, with
$\nu\in[0.5,2,5]$. To higher values of $\nu$ corresponds an higher share of
measurement errors.

We simulate $1000$ different datasets of $T=200$ observations from the model
presented above, using the parametrization of Leeper et al. (2013):
$\alpha=0.36,\theta=0.2673,\tau=0.25$ and $u_{t}\sim N(0,I)$. We arbitrarily
fix $n=100$, so to have a sufficiently large cross-section.

For each dataset, we estimate a standard VAR, a DFM and, in addition, the
Factor Augmented VAR (FAVAR) of Bernanke et al. (2005).111111 The inclusion of
the FAVAR is motivated by the work of Miescu and Mumtaz (2019), that proposed
a Proxy FAVAR, which is clearly related to our methodology. Differently, their
simulation is based on an enlarged version of the theoretical model of Leeper
et al. (2013), but the results are coherent to ours. Our choice of a
restricted number of variables is due to the sake of simplicity. Unless
otherwise stated, in each case we set the lag order equal to $2$. 121212 We
repeated the estimation considering more lags and the findings are virtually
identical. Results are available from the authors upon request.

While DFM and FAVAR estimation involves the whole sample, the VAR analysis
includes only few key variables, specifically capital and tax rate, though we
add also technology in some exercises. Moreover, the DFM is estimated with
parameters $r=5$ and $q=2$, whereas the FAVAR is based on a bivariate VAR
augmented with three additional factors.131313In this way we have that the
space generated is the same as the one generated by the $F_{t}$ of the DFM.

To identify the shock from the simulated series for all the three models, we
apply the external instrument procedure described in Section 2.2. As
instrument, we consider the structural shock itself, i.e. $u_{\tau,t}$, which
is the best instrument one can possibly have.141414We also consider
instruments of different quality. Since the results do not show substantially
difference with those presented here, we show them as robustness in Appendix
B. Furthermore, we repeat each exercise with different and increasing values
of $\nu$, so that we are also able to observe the role of the measurement
error.

Figure 1: Comparison of VAR specifications (a) Notes: black dotted lines are
the theoretical IRF, the red lines are the responses obtained estmating a VAR
with different specifications along with their $68\%$ confidence bands
interval in grey. The empirical IRF are computed as the sample average of the
resposes obtained across simulations. Going form the upper panel to the lower,
we see the results for a bivariate VAR(3) on capital and tax rate observed
without measurement errors, for a trivariate VAR(3) on capital, tax rate and
technology without measurement error and finally the same trivare VAR(3)
obeserve with small measuremnt error , specifically $\nu=0.5$.

###  3.2 Simulation results

In the next section we show the possible issues that may affect the IRFs
estimated with a proxy identification within a VAR framework.

Figure 1(a) plots the responses of capital and tax rate to a tax shock, for
three different VAR specifications. In each case, the series are simulated
with two-periods fiscal foresight and the IRFs are computed as the average
across the model simulations. The black dotted lines in the figure represents
the model true responses. The agents know two periods in advance how and when
a tax shock will hit, therefore they adjust capital accordingly. However, when
the same responses are estimated from the data simulated from the same model,
the econometrician is not always able to recover the true responses. In fact,
the results depend on two different aspects: first, on the information set at
the econometrician’s disposal and, second, on whether what is observed is
contaminated by the measurement error.

The results reported in the upper panels of Figure 1(a) are derived from a
bivariate VAR on capital and tax rate, where no measurement error is added.
Clearly, the VAR model is unable to recover the structural shock, even without
measurement error contaminating the estimation, thus providing misleading
impulse responses. Indeed, as explained in the previous section and more in
detail in Leeper et al. (2013), considering only capital and tax rate would be
insufficient to solve the non-fundamentalness issue, leading to
misspecifications. In the middle panel we show that adding a third variable,
namely technology, is crucial for the reliability of VAR estimates. The
additional variable allows VAR empirical responses to retrace almost perfectly
their theoretical counterparts. Nevertheless, in reality, variables are often
observed with measurement errors. We then augment the same trivariate system
with a small measurement error (obtained setting $\nu=0.5$), and proceed in
identical manner. The results, shown in the last panel of Figure 1(a),
highlight that the overall estimates of the true IRFs can still be distorted
by measurement errors, despite the system itself is fundamental.

Therefore, the estimated responses following a proxy identification procedure
can in principle be affected by both non-fundamentalness and the measurement
error issues. As a solution, we propose to apply the same identification
procedure into a DFM frameowork.The comparison is given in Figure 2(a), which
complements the results displayed in the previous chart. In this case, we
still have a model with a two-period fiscal foresight, with the true impulse
responses of the tax rate and capital to a tax shock which are depicted in
black dotted line. We add different levels of measurement error to the
variables - specifically considering $\nu=0.5,2$ and $5$, and we then re-
estimate the shock following the same proxy identification technique for each
level of $\nu$. The estimated IRFs are reported in blue, red and yellow for
each level of the added measurement error.

Moreover, in figure 2(a) we compare the results coming from three different
models: bivariate VAR (top panel), FAVAR (middle panel) and DFM (bottom
panel). Not surprisingly, the higher the measurement error, the more distorted
are the estimated responses of the VAR. Conversely, both the FAVAR and DFM
successfully recover the true responses, although we observe an increase in
the estimation-bias in the former as the error increases. These results
underline the usefulness of expanding the information set at econometrician’s
disposal also in the proxy identification procedure: not only it helps solving
the non-fundamentalness issue by construction, but one can also better deal
with the measurement error problem, which usually affects macroeconomic time
series. Specifically on this matter, the DFM behaves better then the FAVAR
because in the latter some of the variables are observed with the measurement
error, a feature which can lead to a contamination of the estimated IRFs, as
we will observe later in this section.

Figure 2: IRF of tax rate and capital to a tax shock, with a two-period fiscal
foresight. (a) Notes: Black dotted lines are the ”true” responses. Responses
obtained in case $\nu=0.5$ are displayed in blue, along with the $68\%$
confidence bands in grey. The red lines are the response obtained in case
$\nu=2$. The yellow lines are the response obtained in case $\nu=5$. In each
case the external intrument is the real structural shock $u_{\tau,t}$. The
empirical IRF are computed as the sample average of the resposes obtained
across simulations. Going form the upper panels to the lower, we see the
results of the bivariate VAR(2) on capital and tax rate, of the FAVAR(2) on
capital, tax rate and three factors and finally of the DFM(2).

Figure 8(a) complements the above picture, though focusing this time on the
underlying shock which is estimated by the different models. The chart is
obtained following the same procedure explained above for the IRFs, and it
depicts the true shock simulated from the model (black dotted line) along with
the unit-variance shocks obtained from a VAR (top panel), FAVAR (middle panel)
and DFM (bottom panel) which are estimated each time with different level of
measurement errors.151515As for the IRFs, the chart plots the mean of the
distribution of shocks which are estimated for each model simulation As
evident, the non-fundamental VAR is never able to recover the true structural
shock and higher measurement errors contribute to further distorting the
estimated shock. On the other hand, FAVAR and DFM perform largely better,
especially the latter which shows great robustness.

In Table 2(a), we also report the Frobenius Norm computed between the true
model IRFs and structural shocks and the estimated counterparts. Results
corroborate what seen so far. Using VAR as benchmark, we show that the
additional information exploited by FAVAR and DFM is effective. Let us focus
on the specific case when the model is simulated with fiscal foresight: here
both DFM and FAVAR largely overcome VAR. As $\nu$ increases their results tend
to converge to those of VAR, but at different rates: DFM converges at a lower
rate with respect to the FAVAR, suggesting a higher reliability of DFM result.

For a matter of completeness, we also briefly comment the results obtained
following the same procedure, but with a model with no fiscal foresight. In
this model specification, the econometrician can estimate a bivariate VAR from
of the simulated series of tax rate and capital to have a sufficient
information set, i.e. the VAR is fundamental.161616See Forni and Gambetti
(2014) for a discussion. However, although results are quantitatively
different from the case studied in the previous paragraphs, they still lead to
analogous conclusions. We spend only few words describing the results of
figure 9(a), as it is not crucial in our analysis. Here we observe that the
the VAR estimated on the simulated series with small measurement error (blu
line) is now able to estimate reliable responses, which are almost identical
to those of the other models. However, as measurement error increases (red and
yellow lines), the VAR estimation deteriorates quickly, whereas the other two
models are consistent. We refer again to table 2(a), which help us complete
the overall picture. By looking at the Frobenius norm, it is evident that with
the measurement error increasing, both the DFM and FAVAR perform better with
respect to the VAR. 171717 This result seems in contrast with what observed
before under the case of fiscal foresight, where an increase of measurement
error was pushing the factor models closer to VAR in terms of distance. A
possible explanation lies upon the different rates of deterioration. If
variables are only affected by measurement error, VAR deterioration rate is
higher than those of the other two models. This is because the latter models
are able to better handle the issue and manage to better recover theoretical
results. Thus, when this error increases, DFM and FAVAR perform relatively
better than VAR. Conversely, when there is fiscal foresight, VAR is
informationally deficient and, consequently, is already largely distorted by
non-fundamentalness. The distortion due to the increasing measurement error
add up to the pre-existing one, but the sum deteriorate at a slower rate wrt
DFM and FAVAR, reducing the performance gap. The conclusion we draw from the
table is: VAR is largely affected by measurement error, but even more by
informational deficiency. In other words, ift the former is of small size -
and the shock to be estimated is fundamental - VAR may be comparable to DFM
and FAVAR and its results are reliable. Conversely, if the shock is no-
fundamental, but we do not observe any measurement error contaminating the
series, the estimation will always be distorted.

As a final note, we stress another point which is also important in the
estimation of the IRFs, represented by the variables choice and the overall
model specification. We do it through an exercise which differs from the ones
we showed above, where we show that not only the VAR, but also the FAVAR is
highly dependent on such choice. We proceed as follows: we estimate a
trivariate VAR and a FAVAR having three variables observed with measurement
errors plus two factors. We fix the first two observed variables in both the
VAR and the FAVAR to be capital and tax rate, whereas the third variable
changes for each iteration. The measurement error, instead, is always kept
equal to $\nu=0.5$. Figure 7(a) compares the $n$ different estimates of the
two models, with the VAR shown in the top panel, while FAVAR in the middle
one. As evident, it is sufficient to vary only one variable at the time to
obtain responses that greatly differ across specifications. In some cases, the
results may end up being very misleading. This happens because each observed
variable brings in the model a different type of information, along with extra
measurement error, which can therefore contaminate the results. Conversely,
the DFM does not have this weakness by construction: the econometrician
already has all the information set available at his disposal, and the
variables are already cleaned by the measurement error.

##  4 Empirical application

In this section, we provide a detailed account of our empirical analysis by
describing the data and the model specification used. We present our findings
in two sub-sections. Firstly, we compare impulse responses obtained using a
Proxy VAR versus a Proxy DFM. For the analysis, we identify the monetary
policy shock with a broad range of external instruments available in the
literature. Then, we explore the transmission of monetary policy shocks in the
United States, using the Proxy DFM identified with all the instruments
previously analysed. Finally, we present the variance decomposition obtained
using the new “unit-variance shock” methodology, which we have described in
section 2.2.

###  4.1 Data, specifications and procedure

We use data from the FRED monthly dataset, which is described in McCracken and
Ng (2016), covering the period from January 1963 to December 2018. The dataset
contains $N=100$ macroeconomic and financial variables. For the VAR model, we
specify a core subset of variables, which include the industrial production
index, the unemployment rate, the consumer price index (CPI), and the policy
rate. We use the one-year Treasury yield as the policy variable, as it is
common in the literature. In contrast, the DFM includes all available
variables in the dataset. The variables are left in levels or log-levels and
are not transformed to reach stationarity.181818It is important to note that
we do not perform a VECM estimation in either the VAR or DFM cases. Sims
(1980) and Sims et al. (1990) show that the cointegration relationship is
correctly taken into account within a standard VAR in levels, at least at
short horizons. Similarly, as shown by Barigozzi et al. (2021), the VAR
estimated on $I(0)$ static factors produces IRFs that, at a short horizon, are
equal to the VECM specification, without the need to explicitly estimate the
number of cointegration relationships.

We estimate a VAR with $p=8$ lags, chosen based on both the Akaike and Schwarz
information criteria. These criteria suggest $p=11$ and $p=6$, respectively.
The estimation of the DFM in based on Section 2.1.2, and we determine the
number of static and dynamic factors using the tests provided by Bai and Ng
(2002) and Hallin and Liška (2007), respectively. Based on these tests, we set
the number of static factors to $\hat{r}=9$ and the number of dynamic factors
to $\hat{q}=4$, which represent our baseline parameter specification of the
model. To maintain consistency with the VAR, we set the number of lags to
$p=8$. We conduct a robustness analysis, as presented in Appendix E, and find
that the results are virtually identical when varying $p$, which holds true
for both the VAR and DFM models.

After having estimated the models, we identify the structural shock of
interest exploiting the external information provided by the proxy (see
Section 2.2). The baseline analysis is based on GK, which covers a period
stretching from January 1990 to June 2012. However, we also offer a comparison
with instruments developed by RR, MAR, JK, and BS.191919Regarding RR, we take
the version extended by Miranda-Agrippino and Rey (2020). As mentioned in
Section 2.2, our new procedure has the advantage of being agnostic regarding
the choice of the policy variable to instrument. Indeed, we do not need to
choose between the one- and two-year Treasury yield to capture the monetary
policy stance, as long as the information of the structural shock we want to
estimate (in our case, the monetary policy shock) is spread throughout the $q$
estimated reduced-form innovations.

###  4.2 A VAR-DFM comparison across instruments

The aim of this section is to compare the estimated impulse responses from the
core specification of the VAR identified with the most common instruments in
the literature, namely GK, RR, MAR, JK, and BS, with those of the DFM obtained
with the same instruments. The comparison is shown in Figure 3(a). Three main
points can be made here: firstly, there are significant differences between
the impulse responses estimated with the two models; secondly, in most cases,
the VAR model produces results that are at odd with standard macroeconomic
theory, with both price and output puzzles, whereas the DFM model does not;
lastly, the VAR estimates show large variability across instruments, while by
construction the DFM is unaffected by this issue.

Overall, the VAR estimates show that a contractionary shock raises industrial
production and prices and lowers the unemployment rate. This issue is related
to the “information effect,” i.e., the instrument not only contains
information about the underlying monetary policy shock, but at the same time
it also carries information, or news, on the future macroeconomic outlook
which is implicitly comunicated by the central banks when during the policy
announcements. Therefore, the instrument is not exogenous and can be predicted
with any publicly available information at the time of the FOMC announcement.
The puzzles are evident for the responses obtained for all the instruments
except with MAR. This is because the authors have developed an
informationally-robust instrument that combines the high-frequency approach
with the central bank’s information set (Greenbook forecasts) to control for
the macroeconomic information revealed by the central bank with its policy
change.202020The main assumption of this paper is that they assume that if the
econometrician can correctly identify the monetary policy shock, then private
agents should respond to it as a true monetary policy shock. The problem,
however, is that private agents do not have the Greenbook forecasts in real
time, as they are released to the public with a five-year lag. This is not the
only attempt in the literature, and others have addressed the same issue in
different ways. Jarociński and Karadi (2020) take a high-frequency approach
and distinguish a true monetary policy shock from an information shock by
looking at the comovement of interest rates and stock prices around policy
announcements. Bauer and Swanson (2022) obtain the series of high-frequency
monetary policy surprises and then orthogonalize them using a set of
predictors that are closely related to the Fed’s monetary policy rule. Given
our interest in exploring the role of our model in eliminating any kind of
puzzle in the estimation of impulse responses without subjectively choosing
how to clean the instrument, we use the raw high-frequency surprise series for
JK and BS.212121The cleaned version for MAR is simply GK.

On the other side, performing the same identification strategy in a DFM
environment implictly solves the information for those instruments which were
affected. This is linked to the the large dimension of data which is at
econometrician’s disposal: including variables which also carry information
about the future helps purging the responses from the news component.
Therefore, the estimated effect of monetary policy using the Proxy DFM yields
puzzle-free impulse responses across all instruments: a contractionary shock
that raises the one-year government bond yield lowers industrial production
and prices (with some lag for some instruments), while raising the
unemployment rate. Clearly, the use of a model that incorporates a large
amount of information is critical to recover plausible monetary policy
responses, no matter the instrument.

The problem of a limited information set available in small- and medium-scale
models also determines a large variability in the estimates when including
additional variables (see Forni et al., 2020). We proceed along their lines
and show the impulse response for the core variables when one additional
variable is added to the model at a time, with a total of 95 different
specifications. We also test for each model specification whether the shock is
invertible. To do this, we use the test recently proposed by Forni et al.
(2022), where the proxy of interest is projected on the current value and the
first $r$ leads of the Wold residuals $v_{t}$ as follows:

$z_{t}=\sum_{k=0}^{r}\hat{\gamma}_{k}^{{}^{\prime}}\hat{v}_{t+k}+\hat{\xi}_{r,t}$
(18)

where $z_{t}$ is the proxy, and $v_{t}$ are the VAR reduced-form residuals. We
test for invertibility using the F-test for the significance of the
regressors, with the null hypothesis being
$H_{0}:\gamma_{1}=\gamma_{2}=\dots=\gamma_{r}=0$ against the alternative that
at least one of the coefficients is non-zero.222222The regression does not
include a constant as $E(z_{t}|x=0)=0$, where
$x=\sum_{k=0}^{r}\hat{\gamma}_{k}^{{}^{\prime}}\hat{v}_{t+k}$. We estimate the
regression in equation (18) with $k=8$ and a 5% confidence level. We choose
this calibration because too many leads would introduce significant noise in
the regression and too few would be insufficient, both of which would
undermine the validity of the test. If the estimated VAR specification has a
shock invertible, then the corresponding impulse response is colored in
yellow. Grey lines, instead, represent models with non invertible shocks.

Figure 3: VAR vs DFM: proxy comparison (a) Notes: Comparison between impulse
responses identified with all instruments for monetary policy shocks included
in the analysis. The VAR specification includes our core variables: the one-
year rate, industrial production, unemployment and CPI.

The results are plotted in Figure 4(a) and indicate that the additional
variable added at each iteration to the core specification is sufficient to
include additional information, along with measurement error, that can
significantly affect the estimated impulse responses. However, it is often not
sufficient to overcome the puzzles in the estimates, and even when the impulse
responses are estimated with a cleaned instrument (MAR), some specifications
estimate price puzzles. Overall, the high sensitivity of these estimates
underscores the importance of carefully selecting the variables to be included
in the model.232323This problem affects not only the impulse responses but
also the underlying structural shocks. An additional issue highlighted in the
figure is that in most of the cases, the shock is not invertible (gray line).
Conversely, the DFM specification does not suffer from the above problems
because it is able to deal with a large number of variables simultaneously,
which is not possible in the VAR framework due to the curse of dimensionality.
Lastly, as described in section 2.1.1, the DFM by construction cleans the data
from the measurement error, thereby eliminating the corresponding bias in the
estimated responses.

Figure 4: VAR vs DFM with GK identification: IRFs comparison (a) Notes: The
Figure shows impulse responses to a monetary policy shock estimated from the
95 different VAR specifications (grey lines=non invertible shock; yellow
lines=invertible shock) and the DFM (red line). The identification is
performed using all external instruments used in this paper. The invertibility
test in equation (18) includes 8 leads and uses 5% as confidence level.

###  4.3 Propagation Channels of the Monetary Policy Shock

This section further investigates the various channels of monetary policy (see
Mishkin, 1995, for a survey) and, at the same time, shows the power of the
Proxy DFM which succesfully estimates robust puzzle-free impulse responses
across all the analysed external instruments. Compared to VAR models, a
distinct advantage of our methodology relies in the possibility to study a
wider range of variables at the same time, allowing for a broader
understanding of the monetary policy transmission mechanism. Using the newly
developed “unit-variance shock”, we are also able to estimate the variance
decomposition, which was not previously feasible within the external
instrument approach (see section 2.2).

Figure 5(a) plots the impulse responses of a representative sample of
variables selected to explore the transmission channels of monetary policy.
Specifically, we look at measures of economic activity, labor, housing,
financial and labor markets, exchange rates, and uncertainty. The figure
reports the median impulse responses obtained from the GK instrument, along
with the 68 and 95 percent confidence bands. However, the median responses
using all of the other instruments analysed in the previous section are also
reported. Notice that, for comparability reason, all the results are
normalized at 100 basis point increase in the one-year Treasury yield at
impact, a fairly common normalization in the literature.

#### Real Economy and Labor Market

Figure 5(a) shows that a contractionary monetary policy has a negative effect
on the economy. Industrial production reacts with a delay of two months and
builds up gradually over time to reach its peak impact after 10 months at -2%
for GK before returning to its original level, while the other instruments
depict slighly higher magnitudes, but still within the confidence
bands.242424This is in line with standard macro textbook (Mankiw, 2010) and
what central banks believe today. In the United States, the November 2022 FOMC
statement indicates that the Committee will consider lags as it determines the
pace of future increases in the federal funds rate. This is also true for
other central banks. For instance, Philip R. Lane, a Member of the Executive
Board of the ECB, in a recent speech at the Conference EU and US Perspectives:
New Directions for Economic Policy, has commented on the lagged effect of
monetary policy transmission to the real economy. All the other real variables
follow a similar path, even though no restrictions on the shape of the impulse
responses are imposed. This suggests that monetary policy shocks are
transmitted to the real economy with a lag of a few months, in line with what
monetary policymakers believe. Indeed, both real consumption, real income, and
capacity utilization decline in a hump-shaped manner following the shock,
reaching the maximum impact just before the end of the first year before
returning to their trend, with a peak magnitude ranging between -1% and -3%.
Real consumption, on the other hand, seems to react more quickly to the shock,
which can be explained by the sharp drop in house prices, a mechanism well
documented in the literature (e.g., Mian et al., 2013; Slacalek et al., 2020),
a result we will comment on later. At the same time, business sales, business
inventories, and new orders for durable goods all contract in line with
industrial production. Turning to the labor market, the unemployment rate
shows a sluggish increase, with no response at impact. The increase starts
only from the second month onwards and it reaches its peak around 0.5% after
about ten months (see e.g., Christiano et al., 1999). The average real
earnings do not seem to react significantly to a monetary shock at impact, but
progressively decrease over time, confirming the rather sluggish nature of
real wages and possibly suggesting the presence of frictions in the economy,
an idea that is now found in almost all the standard macro-theoretical models.

Figure 5: The transmission of monetary policy (a) Notes: Impulse response
functions of a monetary policy shock identified using GK, MAR, RR, JK, and BS
as instruments in a DFM framework. The impact of the shock is normalised such
that the one-year Treasury yield increases by 100 basis points at impact. The
reported confidence bands are related to GK only and correspond to the 84th
(dark grey) and the 95th percentile (light grey) respectively, and are
estimated via wild bootstrap.

#### Housing Market

The analysis of both housing market and financial variables can further help
us in better understanding which other channels are at play in the propagation
of the monetary policy shock. Housing investments reduce at a large magnitude
following a monetary policy tightening, with both housing starts and new
housing permits which shrink at impact by around 10%, reaching almost -20% at
peak within six months, though they also exhibit a rather short-living
response. Coherently, house prices decrease by 0.5% at impact for GK, MAR and
BS, and all the instruments point to a more long-lasting effect compared to
the other housing sector indicators. Overall, the high sensitivity of the
housing market to monetary policy shocks may directly impact household balance
sheets, as also confirmed by the sharp reduction in consumption expenditure.
As underlined by recent studies, this channel might be more relevant than
previously believed. Indeed, when combining the large percentage of real
estate over total assets in the portfolio of households with low level of
wealth (Franconi and Rella, 2023) along with the large marginal propensity to
consume (MPC) of hand-to-mouth households (e.g. see Kaplan and Violante, 2018)
and the larger MPC in response to a negative income shock (Christelis et al.,
2019), the household balance sheets channel takes on a greater relevance for
the transmission of monetary policy shocks (see for instance Slacalek et al.,
2020).

#### Financial Market

The negative wealth effect coming from the housing market can potentially
compoud with that coming from the financial market, further contributing to
the amplification of the monetary policy shock. In the literature, these
mechanisms are referred to as the financial accelerator and the credit channel
(Bernanke and Gertler, 1995; Bernanke et al., 1999). Asset prices, proxied by
the S&P 500, show a sudden and large repricing for the Gk isntrument,
experiencing a decline of more than 10% already in the second month.
Interesting, the other instruments also show a significant drop, equal to 5%,
which is however half compared to what implied by GK. More generally, all
financial variables included, such as the Moody’s Corporate Bond Spread (BAA-
AAA) and all interest rates across different maturities, react at impact.
Interest rates rise across all maturities, albeit to different degrees, with
the ten-year Treasury yield which rises less compared to the federal funds
rate and the one-year Treasury yieldṪhis is reflected in an inversion of the
yield curve and a negative reaction of the term spread, which reduces by
around 50bp in the first months. Turning to the credit channel, the tightening
of financial conditions drives an increase in reserves and the corresponding
decline in credit throughout the economy. Business loans, real estate loans,
and nonrevolving consumer credit fall sluggishly, reaching a trough of roughly
2 to 4 per cent the former and 1 to 3 percent the latter about 20 months after
the shock.

#### Exchange Rates

In terms of exchange rates, the US dollar appreciates following a monetary
policy tightening, as shown by the GBP/USD, CAD/USD, CHF/USD and JPY/USD
exchange rates, which increase by roughly 4% (2% for JPY) in the first two
months. This implies higher prices for those countries that import goods
produced in the US, and thus a decrease in US exports, which can further
negatively weigh on the overall impact of the monetary policy shock on the
domestic economy. It is also worth noting that both exchange rates react
quickly, peaking within the second month before gradually declining. Thus,
they do not exhibit the delayed overshooting puzzle that was instead presented
by the VAR analysis of Eichenbaum and Evans (1995), a result that confirms
what was also found by Forni and Gambetti (2010).

#### Prices

Finally, a monetary policy tightening unequivocally reduces prices. All of the
different price measures share a similar pattern: the CPI, the PCE deflator,
the PPI, house prices and oil prices all react at impact for almost all the
instruments, a feature that is at odds with the classical recursive
identification scheme, which assumes zero contemporaneous effects at impact.
Although the contraction is immediate, prices do not fully adjust at impact,
but continue to fall for several months. The large fall in oil prices observed
with GK, MAR and BS instruments could potentially be explained by a fall in
consumption and investment following the negative monetary policy shock, which
then affects oil prices through the demand channel by directly reducing the
demand for oil.

#### Global Spillovers

A US monetary policy shock plays also as an important role in terms of global
spillovers, thus contributing in shaping the global outlook. As shown by many
studies, for instance Ca’Zorzi et al. (2020), many of the channels which we
analysed in the above paragraph, such as the demand channel, or the exchange
channel, contribute to the propagatation of negative spillovers to other
countries’ financial markets and real activity sectors. Though representing an
interesting topic, our analysis is not strictly related to the global
dimension of US monetary policy, which may be explored in future works.

Table 1: Forecast Error Variance Decomposition: Proxy DFM (GK) Variables |  | h=0 | h=6 | h=12 | h=18 | h=24 | h=30
---|---|---|---|---|---|---|---
Industrial production |  | 0.37 | 10.46 | 22.07 | 25.02 | 26.18 | 26.10
Real Consumption |  | 9.07 | 30.65 | 32.65 | 31.53 | 30.20 | 29.23
Real Income ex.Trans. |  | 7.01 | 10.32 | 21.42 | 23.41 | 22.74 | 21.03
Business Sales |  | 3.68 | 29.11 | 35.67 | 35.61 | 34.67 | 33.54
New Orders: Durables |  | 4.76 | 27.16 | 33.35 | 34.41 | 34.71 | 34.27
Business Inventories |  | 8.72 | 0.95 | 5.89 | 10.38 | 13.34 | 14.62
Cap. Util. Manuf. |  | 0.04 | 13.03 | 24.49 | 26.80 | 27.24 | 26.56
Business Sentiment |  | 1.37 | 37.49 | 38.57 | 31.22 | 26.39 | 24.32
Unemployment rate |  | 2.53 | 16.61 | 26.65 | 29.82 | 30.33 | 28.84
Avg. Hours manuf. |  | 1.21 | 17.70 | 27.35 | 27.13 | 26.35 | 26.70
Avg. Earnings manuf. |  | 23.54 | 5.44 | 9.79 | 9.68 | 9.46 | 9.24
CPI Headline |  | 15.28 | 7.92 | 4.30 | 3.06 | 2.97 | 3.42
PCE |  | 14.74 | 9.19 | 5.26 | 3.79 | 3.60 | 3.96
PPI |  | 18.04 | 13.23 | 8.21 | 6.24 | 5.63 | 5.60
Oil Price |  | 22.58 | 18.25 | 12.29 | 9.75 | 8.66 | 8.20
House Price |  | 20.15 | 22.16 | 19.09 | 18.04 | 17.26 | 15.99
Housing Starts |  | 30.40 | 61.10 | 46.70 | 33.01 | 25.14 | 21.88
New Housing Permits |  | 29.35 | 60.44 | 45.46 | 32.62 | 26.14 | 24.12
Total Reserves |  | 45.03 | 38.35 | 31.00 | 27.80 | 26.27 | 25.41
Business Loans |  | 72.01 | 17.90 | 12.38 | 16.66 | 19.47 | 20.08
Real Estate Loans |  | 53.88 | 10.97 | 15.51 | 19.63 | 21.29 | 20.99
One-year rate |  | 39.21 | 18.39 | 14.20 | 12.18 | 11.68 | 11.50
Ten-year rate |  | 28.27 | 11.64 | 8.81 | 7.27 | 6.38 | 5.89
BAA-AAA |  | 35.12 | 16.17 | 12.41 | 10.14 | 8.99 | 8.38
S&P 500 |  | 51.54 | 53.53 | 40.85 | 34.98 | 31.86 | 29.93
Exch. GBP/USD |  | 71.65 | 67.14 | 60.74 | 59.67 | 58.08 | 54.28
Exch. CAD/USD |  | 60.88 | 61.73 | 51.05 | 46.80 | 44.83 | 43.93
Exch. CHF/USD |  | 54.56 | 58.47 | 53.40 | 47.51 | 41.80 | 36.11
Exch. JPY/USD |  | 17.84 | 22.43 | 23.14 | 21.19 | 18.74 | 16.32
(a) Notes: Forecast Error Variance Decomposition of the contractionary
monetary policy shock for a selection of varibales, at different horizons. The
shock in the Proxy DFM is identified with the instrument GK. Values in red
represents the peak relevance.

#### Variance Decomposition

Table 1(a) presents the variance decomposition analysis obtained from the GK
instrument and confirms that a monetary policy shock has an important role in
explaining the cyclical fluctuations of the economy. For real variables and
indicators of economic activity, the shock explains a small fraction of the
variance at impact, ranging from 1% for industrial production to 9% for real
income. This is consistent with the impulse response analysis, where a
monetary policy shock has a limited effect on most of the real variables at
the very impact, with the shock taking a few months to be transmit to the
economy. However, its importance increases at longer frequencies, generally
peaking during the course of the second year at roughly 30%. Similar results
are shared by labor market variables, specifically for the unemployment rate
and hours worked. The variance explained of the average earnings seems,
however, to remain rather low, suggesting what already observed in the IRFs.
Turning to the nominal variables, this analysis shows a larger share of
variance explained at the beginning of the forecast horizon, from 14.7% of PCE
to 22.5% of oil prices, with the importance of the monetary policy shock
dissipating at longer horizons. On the housing and financial market,
conversely, a monetary shock explains a large fraction of the total variance
of these variables within the first six months following the shock. The peak
in variance explained by the monetary policy shock for the housing sector is
around 60%, for the credit market in the range 53% to 72%, for interest rates
between 28% to 35%, for S&P 500 index above 50% and quite persistent, for
exchange rates in the range 54% to 71%, with the exception of the $JPY/USD$
which amounts to almost 18%.252525The Bank of Japan conducts interventions in
the foreign exchange rate market under the input of the Ministry of Finance.
This may explain why US monetary policy shocks are comparatively less relevant
for changes in the JPY/USD exchange rate.

##  5 Conclusion

External instruments identification procedure is not safe from issues
affecting traditional SVARs. Even if the instrument is perfect, the estimates
can be biased.

By means of a theoretical model with perfect foresight, we show that, if the
underlying shock is non-fundamental or the variables are observed with a
measurement error, the SVAR consistently fails to estimate the true impulse
responses. Moreover, subjective choices about the variables included in the
model can further increase the uncertainty in both the magnitude and the sign
of the estimated responses. The latter is a problem that also affects FAVARs.
As a solution, we propose using external instruments in a DFM, which is able
to address all mentioned issues at once and to estimate the correct IRFs.

In the empirical exercise, we focus on an application to monetary policy and
consider the most well-known monetary policy instruments in the literature.
Results show that, unlike SVAR, the information included in the DFM is enough
to estimate puzzle-free responses in line with economic theory. Interestingly,
results and consistent regardless the instrument considered, suggesting that
the larger information set is able to deal with the distorting effect of
monetary policy news.

Moreover, DFM proves invaluable in examining the behavior of a large set of
variables, simultaneously. A tool of great value, especially for central
banks, which allows for an internally consistent examination of the
transmission channels of monetary policy. Our analysis shows that a monetary
policy tightening shock has a clear contractionary effect on the economy,
leading to a decline in both economic activity and prices. Multiple channels
come into play, with both the financial and housing sectors deteriorating and
directly affecting private consumption. Finally, the variance decomposition
analysis shows that the monetary policy shock explains a significant portion
of the variance of both real and nominal variables, albeit at different
horizons, further highlighting the role of monetary policy in influencing
business cycle fluctuations.

## References

  * (1)
  * Alessi and Kerssenfischer (2019) Alessi, Lucia and Mark Kerssenfischer, “The response of asset prices to monetary policy shocks: stronger than thought,” Journal of Applied Econometrics, 2019, 34 (5), 661–672.
  * Alessi et al. (2011) , Matteo Barigozzi, and Marco Capasso, “Non-fundamentalness in structural econometric models: A review,” International Statistical Review, 2011, 79 (1), 16–47.
  * Bai and Ng (2002) Bai, Jushan and Serena Ng, “Determining the number of factors in approximate factor models,” Econometrica, 2002, 70 (1), 191–221.
  * Bańbura et al. (2010) Bańbura, Marta, Domenico Giannone, and Lucrezia Reichlin, “Large Bayesian vector auto regressions,” Journal of applied Econometrics, 2010, 25 (1), 71–92.
  * Barigozzi et al. (2014) Barigozzi, Matteo, Antonio M Conti, and Matteo Luciani, “Do euro area countries respond asymmetrically to the common monetary policy?,” Oxford bulletin of economics and statistics, 2014, 76 (5), 693–714.
  * Barigozzi et al. (2021) , Marco Lippi, and Matteo Luciani, “Large-dimensional Dynamic Factor Models: Estimation of Impulse–Response Functions with I (1) cointegrated factors,” Journal of Econometrics, 2021, 221 (2), 455–482.
  * Bauer and Swanson (2022) Bauer, Michael D and Eric T Swanson, “A Reassessment of Monetary Policy Surprises and High-Frequency Identification,” NBER Macroeconomics Annual, 2022.
  * Bernanke and Gertler (1995) Bernanke, Ben S and Mark Gertler, “Inside the black box: the credit channel of monetary policy transmission,” Journal of Economic perspectives, 1995, 9 (4), 27–48.
  * Bernanke et al. (2005) , Jean Boivin, and Piotr Eliasz, “Measuring the effects of monetary policy: a factor-augmented vector autoregressive (FAVAR) approach,” The Quarterly journal of economics, 2005, 120 (1), 387–422.
  * Bernanke et al. (1999) , Mark Gertler, and Simon Gilchrist, “The financial accelerator in a quantitative business cycle framework,” Handbook of macroeconomics, 1999, 1, 1341–1393.
  * Bjørnland and Thorsrud (2019) Bjørnland, Hilde C and Leif Anders Thorsrud, “Commodity prices and fiscal policy design: Procyclical despite a rule,” Journal of Applied Econometrics, 2019, 34 (2), 161–180.
  * Brignone and Mazzali (2022) Brignone, Davide and Marco Mazzali, “Evidence on the Confounding Nature of the Main Business Cycle Driver,” Working Paper, 2022.
  * Bruns (2021) Bruns, Martin, “Proxy vector autoregressions in a data-rich environment,” Journal of Economic Dynamics and Control, 2021, 123, 104046.
  * Carriero et al. (2011) Carriero, Andrea, George Kapetanios, and Massimiliano Marcellino, “Forecasting large datasets with Bayesian reduced rank multivariate models,” Journal of Applied Econometrics, 2011, 26 (5), 735–761.
  * Ca’Zorzi et al. (2020) Ca’Zorzi, Michele, Luca Dedola, Georgios Georgiadis, Marek Jarocinski, Livio Stracca, and Georg Strasser, “Monetary policy and its transmission in a globalised world,” 2020.
  * Chahrour and Jurado (2022) Chahrour, Ryan and Kyle Jurado, “Recoverability and expectations-driven fluctuations,” The Review of Economic Studies, 2022, 89 (1), 214–239.
  * Christelis et al. (2019) Christelis, Dimitris, Dimitris Georgarakos, Tullio Jappelli, Luigi Pistaferri, and Maarten Van Rooij, “Asymmetric consumption effects of transitory income shocks,” The Economic Journal, 2019, 129 (622), 2322–2341.
  * Christiano et al. (1999) Christiano, Lawrence J, Martin Eichenbaum, and Charles L Evans, “Monetary policy shocks: What have we learned and to what end?,” Handbook of macroeconomics, 1999, 1, 65–148.
  * Corsetti et al. (2022) Corsetti, Giancarlo, Joao B Duarte, and Samuel Mann, “One money, many markets,” Journal of the European Economic Association, 2022, 20 (1), 513–548.
  * Cubadda and Hecq (2022) Cubadda, Giangluca and Alain Hecq, “Reduced Rank Regression Models in Economics and Finance,” in “Oxford Research Encyclopedia of Economics and Finance,” Oxford University Press, 2022.
  * Eichenbaum and Evans (1995) Eichenbaum, Martin and Charles L Evans, “Some empirical evidence on the effects of shocks to monetary policy on exchange rates,” The Quarterly Journal of Economics, 1995, 110 (4), 975–1009.
  * Forni and Gambetti (2010) Forni, Mario and Luca Gambetti, “The dynamic effects of monetary policy: A structural factor model approach,” Journal of Monetary Economics, 2010, 57 (2), 203–216.
  * Forni and Gambetti (2014) and  , “Sufficient information in structural VARs,” Journal of Monetary Economics, 2014, 66, 124–136.
  * Forni et al. (2009) , Domenico Giannone, Marco Lippi, and Lucrezia Reichlin, “Opening the black box: Structural factor models with large cross sections,” Econometric Theory, 2009, 25 (5), 1319–1347.
  * Forni et al. (2022) , Luca Gambetti, and Giovanni Ricco, “External Instrument SVAR Analysis for Noninvertible Shocks,” Technical Report 2022.
  * Forni et al. (2014) ,  , and Luca Sala, “No news in business cycles,” The Economic Journal, 2014, 124 (581), 1168–1191.
  * Forni et al. (2019) ,  , and  , “Structural VARs and noninvertible macroeconomic models,” Journal of Applied Econometrics, 2019, 34 (2), 221–246.
  * Forni et al. (2020) ,  , Marco Lippi, and Luca Sala, “Common component structural VARs,” Available at SSRN 3753929, 2020.
  * Forni et al. (2000) , Marc Hallin, Marco Lippi, and Lucrezia Reichlin, “The generalized dynamic-factor model: Identification and estimation,” Review of Economics and statistics, 2000, 82 (4), 540–554.
  * Franconi and Rella (2023) Franconi, Alessandro and Giacomo Rella, “Monetary policy across the wealth distribution,” Working Paper, 2023.
  * Gertler and Karadi (2015) Gertler, Mark and Peter Karadi, “Monetary policy surprises, credit costs, and economic activity,” American Economic Journal: Macroeconomics, 2015, 7 (1), 44–76.
  * Geweke (1977) Geweke, John, “The dynamic factor analysis of economic time series,” Latent variables in socio-economic models, 1977.
  * Giannone and Reichlin (2006) Giannone, Domenico and Lucrezia Reichlin, “Does information help recovering structural shocks from past observations?,” Journal of the European Economic Association, 2006, 4 (2-3), 455–465.
  * Giannone et al. (2006) ,  , and Luca Sala, “VARs, common factors and the empirical validation of equilibrium business cycle models,” Journal of Econometrics, 2006, 132 (1), 257–279.
  * Gürkaynak et al. (2004) Gürkaynak, Refet S, Brian P Sack, and Eric T Swanson, “Do actions speak louder than words? The response of asset prices to monetary policy actions and statements,” The Response of Asset Prices to Monetary Policy Actions and Statements (November 2004), 2004.
  * Hallin and Liška (2007) Hallin, Marc and Roman Liška, “Determining the number of factors in the general dynamic factor model,” Journal of the American Statistical Association, 2007, 102 (478), 603–617.
  * Hansen and Sargent (1991) Hansen, Lars Peter and Thomas J Sargent, “Two difficulties in interpreting vector autoregressions,” Rational expectations econometrics, 1991, 1, 77–119.
  * Jarociński and Karadi (2020) Jarociński, Marek and Peter Karadi, “Deconstructing monetary policy surprises—the role of information shocks,” American Economic Journal: Macroeconomics, 2020, 12 (2), 1–43.
  * Jordà et al. (2017) Jordà, Òscar, Moritz Schularick, and Alan M Taylor, “Macrofinancial history and the new business cycle facts,” NBER macroeconomics annual, 2017, 31 (1), 213–263.
  * Kaplan and Violante (2018) Kaplan, Greg and Giovanni L Violante, “Microeconomic heterogeneity and macroeconomic shocks,” Journal of Economic Perspectives, 2018, 32 (3), 167–94.
  * Kerssenfischer (2019) Kerssenfischer, Mark, “The puzzling effects of monetary policy in VARs: Invalid identification or missing information?,” Journal of Applied Econometrics, 2019, 34 (1), 18–25.
  * Kuttner (2001) Kuttner, Kenneth N, “Monetary policy surprises and interest rates: Evidence from the Fed funds futures market,” Journal of monetary economics, 2001, 47 (3), 523–544.
  * Leeper et al. (2013) Leeper, Eric M, Todd B Walker, and Shu-Chun Susan Yang, “Fiscal foresight and information flows,” Econometrica, 2013, 81 (3), 1115–1145.
  * Lippi and Reichlin (1993) Lippi, Marco and Lucrezia Reichlin, “The dynamic effects of aggregate demand and supply disturbances: Comment,” The American Economic Review, 1993, 83 (3), 644–652.
  * Lippi and Reichlin (1994) and  , “VAR analysis, nonfundamental representations, Blaschke matrices,” Journal of Econometrics, 1994, 63 (1), 307–325.
  * Luciani (2015) Luciani, Matteo, “Monetary policy and the housing market: A structural factor analysis,” Journal of applied econometrics, 2015, 30 (2), 199–218.
  * Mankiw (2010) Mankiw, N.G., Macroeconomics, 7th Edition, Worth Publishers, 2010.  The Intermediate-Level TextbookAlso available as Macroeconomics and the Financial System, with Laurence Ball.
  * McCracken and Ng (2016) McCracken, Michael W and Serena Ng, “FRED-MD: A monthly database for macroeconomic research,” Journal of Business & Economic Statistics, 2016, 34 (4), 574–589.
  * Mertens and Ravn (2013) Mertens, Karel and Morten O Ravn, “The dynamic effects of personal and corporate income tax changes in the United States,” American economic review, 2013, 103 (4), 1212–47.
  * Mian et al. (2013) Mian, Atif, Kamalesh Rao, and Amir Sufi, “Household balance sheets, consumption, and the economic slump,” The Quarterly Journal of Economics, 2013, 128 (4), 1687–1726.
  * Miescu and Mumtaz (2019) Miescu, Mirela S and Haroon Mumtaz, “Proxy structural vector autoregressions, informational sufficiency and the role of monetary policy,” Technical Report, Working Paper 2019.
  * Miranda-Agrippino and Ricco (2021) Miranda-Agrippino, Silvia and Giovanni Ricco, “The transmission of monetary policy shocks,” American Economic Journal: Macroeconomics, 2021, 13 (3), 74–107.
  * Miranda-Agrippino and Rey (2020) and Hélene Rey, “US monetary policy and the global financial cycle,” The Review of Economic Studies, 2020, 87 (6), 2754–2776.
  * Mishkin (1995) Mishkin, Frederic S, “Symposium on the monetary transmission mechanism,” Journal of Economic perspectives, 1995, 9 (4), 3–10.
  * De Mol et al. (2008) Mol, Christine De, Domenico Giannone, and Lucrezia Reichlin, “Forecasting using a large number of predictors: Is Bayesian shrinkage a valid alternative to principal components?,” Journal of Econometrics, 2008, 146 (2), 318–328.
  * Del Negro and Otrok (2007) Negro, Marco Del and Christopher Otrok, “99 Luftballons: Monetary policy and the house price boom across US states,” Journal of Monetary Economics, 2007, 54 (7), 1962–1985.
  * De Nora (2023) Nora, Giorgia De, “Factor-Augmented Vector Autoregression with narrative identification. An application to monetary policy in the US,” Economics Letters, 2023, p. 111201.
  * Plagborg-Møller and Wolf (2022) Plagborg-Møller, Mikkel and Christian K Wolf, “Instrumental variable identification of dynamic variance decompositions,” Journal of Political Economy, 2022, 130 (8), 2164–2202.
  * Ramey (2016) Ramey, Valerie A, “Macroeconomic shocks and their propagation,” Handbook of macroeconomics, 2016, 2, 71–162.
  * Romer and Romer (2004) Romer, Christina D and David H Romer, “A new measure of monetary shocks: Derivation and implications,” American Economic Review, 2004, 94 (4), 1055–1084.
  * Sargent and Sims (1977) Sargent, Thomas J and Christopher A Sims, “Business cycle modeling without pretending to have too much a priori economic theory,” New methods in business cycle research, 1977, 1, 145–168.
  * Sims (1980) Sims, Christopher A, “Macroeconomics and reality,” Econometrica: journal of the Econometric Society, 1980, pp. 1–48.
  * Sims et al. (1990) , James H Stock, and Mark W Watson, “Inference in linear time series models with some unit roots,” Econometrica: Journal of the Econometric Society, 1990, pp. 113–144.
  * Slacalek et al. (2020) Slacalek, Jiri, Oreste Tristani, and Giovanni L Violante, “Household balance sheet channels of monetary policy: A back of the envelope calculation for the euro area,” Journal of Economic Dynamics and Control, 2020, 115, 103879.
  * Stock (2008) Stock, James, “What’s New in Econometrics-Time Series, Lecture 7: Structural VARs,. Minicourse 2008,” 2008.
  * Stock and Watson (2002) Stock, James H and Mark W Watson, “Forecasting using principal components from a large number of predictors,” Journal of the American statistical association, 2002, 97 (460), 1167–1179.
  * Stock and Watson (2012) and  , “Disentangling the channels of the 2007-09 recession,” Brookings Papers on Economic Activity, 2012, pp. 120–157.
  * Stock and Watson (2018) and  , “Identification and estimation of dynamic causal effects in macroeconomics using external instruments,” The Economic Journal, 2018, 128 (610), 917–948.

##  Appendix A Absolute shock and IRFs

If we have fundamentalness, the reduced-form residuals, say $u_{t}$, can be
written as a linear combination of the structural shocks, say $\eta_{t}$.
Formally:

$\displaystyle u_{t}=$ $\displaystyle Q\eta_{t}$ (19) The external instrument
identification allows to obtain covariance restrictions from proxies for the
latent structural shock of interest in line with conditions (7) and (8). We
proceed as usual with the first-stage regression
$z_{t}=\delta^{{}^{\prime}}u_{t}+\nu_{t}$. Then, we have that for the shock
$i$: $\displaystyle\delta=$
$\displaystyle\frac{\mathbb{E}[z_{t}u_{t}]}{\Sigma_{u}}=q_{i}\frac{\mathbb{E}[z_{t}\eta_{it}]}{\Sigma_{u}}=q_{i}\frac{\alpha}{\Sigma_{u}}$
(20) where $q_{i}$ is the $i$-th column of the matrix $Q$, which corresponds
to the structural shock of interest $i$ and is then equal to
$q_{i}=\frac{\Sigma_{u}}{\alpha}\delta$, and $\Sigma_{u}$ is the variance-
covariance matrix of the reduced-form innovations (or common shocks) which is
equal to the identity matrix in a DFM framework. Forni et al. (2022) show that
if the shock is fundamental then we can estimate consistently $\alpha$ as
follows: $\displaystyle\hat{\alpha}=$
$\displaystyle\sqrt{\hat{\delta}^{{}^{\prime}}\hat{\Sigma}_{u}\hat{\delta}}$
(21) Hence, the absolute (unit-variance) structural shock $i$ is:
$\displaystyle\eta_{it}=$ $\displaystyle
q_{i}^{{}^{\prime}}\Sigma_{u}^{-1}u_{t}=\frac{\delta
u_{t}}{\sqrt{\delta^{{}^{\prime}}\Sigma_{u}\delta}}$ (22) and the
corresponding impulse response functions to the structural shock $i$ are:
$\displaystyle c_{i}(L)=$ $\displaystyle
B_{\chi}(L)\Sigma_{u}q_{i}=B_{\chi}(L)\Sigma_{u}\frac{\delta}{\sqrt{\delta^{{}^{\prime}}\Sigma_{u}\delta}}$
(23)

##  Appendix B On the Instrument

In the previous sections we adopted a rather strong condition and we assumed
the instrument available to the econometrician to be perfect, i.e. to be the
true structural shock itself. However, as originally pointed out by Mertens
and Ravn (2013), the instrument is very likely to be measured with an error,
which, moreover, could also compound with other sources of distortion.
Therefore, in what follows we relax some of the conditions assumed so far,
studying how different models behave in a more generic framework.

Following Plagborg-Møller and Wolf (2022) and Forni et al. (2022), we
generalize the data generating process of the instrument, which is now
represented by the equation (see also the discussion in Stock and Watson
(2018)):

$z_{t}=\phi(L)z_{t-1}+\lambda(L)y_{t-1}+\alpha\varepsilon_{1,t}+\epsilon_{t}$
(24)

where $\epsilon_{t}\sim N(0,\sigma_{\epsilon})$, $\alpha\geq 0$ is a scalar
measuring (along with $\sigma_{\epsilon}$) the overall strength of $z_{t}$ as
IV and $y_{t-1}$ are past observations of the endogenous variables. $\phi(L)$
and $\lambda(L)$ are rational polynomials in the lag operator, $L$.

One can immediately notice how using the shock of interest itself as a direct
instrument represent a special case of equation (24). In other words, in the
previous sections we were implicitly assuming the following assumptions to be
satisfied:

  1. A.1

The IV does not depend on past values, i.e. $\phi(L)=\lambda(L)=0$

  2. A.2

The IV is perfectly related to the shock of interest, i.e. $\alpha=1$

  3. A.3

The IV is measured without error, i.e. $\sigma_{\epsilon}=0$.

Let us briefly discuss their implications. Assumption A.1 is what Stock and
Watson (2018) define as ”lag exogeneity assumption”. Although rather stringent
in an LP-IV setting, it is not a strong requirement for the SVAR-IV
methodology. Nevertheless, we will also consider the case in which this
assumption does not hold. To understand the implications of A.2-A.3, we
instead focus on the signal-to-noise ratio. Without loss fo generality, let us
consider a situation in which only A.1 holds, so that the ratio is
$\alpha^{2}/\sigma_{\epsilon}^{2}$, and the value of the constant $\alpha$
measures the correlation between the instrument and the structural shock.
Clearly, when $\alpha=0$ the ratio is zero and, thus, the instrument is of
small interest because it is not correlated with the observable variable. On
the other hand, the higher $\alpha$, the greater the importance of the shock
(ceteris paribus). Analogously, as $\sigma_{\epsilon}$ tends to zero the
information that the instrument conveys is perfect up to the scale $\alpha$,
as the ratio goes to infinity.

To better understanding the importance of A.1-A.2-A.3 and their implications
in the estimation of structural shocks, we perform a simulation exercises in
which we compare estimates obtained with four instruments of different
quality. The four different instruments are obtained from equation 24 by
switching on-and-off some of the assumptions described in the previous
paragraph, specifically:

$\displaystyle I1:$ $\displaystyle z_{t}=\tilde{\alpha}^{2}\varepsilon_{1,t}$
$\displaystyle I2:$ $\displaystyle
z_{t}=\tilde{\alpha}^{2}\varepsilon_{1,t}+\epsilon_{t}$ $\displaystyle I3:$
$\displaystyle z_{t}=\tilde{\alpha}^{2}\varepsilon_{1,t}+\phi z_{t-1}$
$\displaystyle I4:$ $\displaystyle
z_{t}=\tilde{\alpha}^{2}\varepsilon_{1,t}+\tilde{\delta}^{\prime}y_{t-1}$

with $\tilde{\alpha}\sim N(0,1)$ and $\sigma_{\epsilon}\sim U(0,0.5)$, whereas
the other parameters are deterministic and arbitrarily calibrated following
Forni et al. (2022) as $\phi=0.5$ and $\tilde{\delta}=[-0.6\quad 0.4\quad
0]^{\prime}$.

The instruments are then used in estimating the underlying structural shocks
in two models, which consist in a trivariate (and fundamental) VAR augmented
with measurement error and a DFM on the entire dataset generated with a
measuremnt error. For the construction of both the models we follow the
procedure described in the previous section. Results are reported in Figure
6(c), and confirm what already found in previously. Indeed, the overall
picture does not change with respect to the perfect instrument case and
results are robust also over different specification of the instrument
genereting process. Not surprisinly, the trivariate VAR (Panel a)
systematicallty fails to estimate the structural shock in neither the cases
even if fundamental, with the overall results suffering from the presence of
the measurement error. Conversely, the DFM (Panel b) is able to recover the
true responses almost perfectly: results do not exhibit any bias and are
robust across all the four cases studied, thus further confirming the goodness
of this model and its good match with the proxy identification technique.

Figure 6: Comparison of weak instruments

(a) VAR (b) DFM

(c) Notes: the two panels show the responses of tax rate and capital to a tax
shock. Panel (a) is obtained with a fundamental trivariate VAR with tax reate,
capitla and technology but augemnted with a small measurement error. Panel (b)
is obtained with the Proxy DFM on the entire dataset generated with a
measuremnt error. In both cases, the measurement is generated with $\nu=0.5$.
In each panel, the black dashed lines are the true response, in magenta the
reposnes for $I1$, in green the responses for $I2$, in blue the responses for
$I3$ and in yellow for $I4$.

##  Appendix C Tables

Table 2: Frobenius Norm between theoretical and empirical IRF [Shocks] |  | Fiscal Foresight | No Fiscal Foresight
---|---|---|---
$\nu$ | Model | Shock | IRF | Shock | IRF
$0.5$ | VAR | 100.00 | 100.00 | 100.00 | 100.00
| FAVAR | 20.72 | 16.56 | 99.34 | 130.72
| DFM | 17.95 | 15.07 | 102.48 | 130.61
$2$ | VAR | 100.00 | 100.00 | 100.00 | 100.00
| FAVAR | 31.19 | 20.94 | 60.10 | 77.43
| DFM | 19.86 | 16.04 | 57.08 | 73.40
$5$ | VAR | 100.00 | 100.00 | 100.00 | 100.00
| FAVAR | 39.85 | 23.47 | 53.99 | 53.66
| DFM | 23.90 | 17.94 | 44.52 | 47.71
(a) Table 3: Fundamentalness Test |  | Leads
---|---|---
Model | $\alpha$ | 1 | 2 | 3 | 4 | 5
VAR | 1% | 64.29 | 85.71 | 85.71 | 89.80 | 88.78
| 5% | 75.51 | 87.76 | 88.78 | 89.80 | 89.80
| 10% | 76.53 | 89.80 | 91.84 | 91.84 | 91.84
FAVAR | 1% | 64.29 | 59.18 | 54.08 | 53.06 | 52.04
| 5% | 67.35 | 63.27 | 57.14 | 55.10 | 54.08
| 10% | 68.37 | 64.29 | 60.20 | 57.14 | 56.12
DFM | 1% | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
| 5% | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
| 10% | 0.00 | 0.00 | 0.00 | 0.00 | 0.00

##  Appendix D Figures

Figure 7: Comparison models specifications (a) Notes: black dotted lined are
the true responses, the red lines the empirical responses obtained with
different specificatio of the models. The resposnes are computed as point
estimates of the IRFs obtained with different specification of the model.
Going from the upper panels to the lower, we see results obtained with VAR(2)
on capital, tax rate and a third variable, with FAVAR(2) on capital, tax rate,
a third variable and two factors and with a DFM(2). The third variable for
VAR(2) and FAVAR(2) varies at each iteration. DFM(2) include all the variables
in the estimation.

Figure 8: Comparison of the shocks across models and $\nu$ (a) Notes:
Comparison between the mean of unit-variance shocks across models and
different values of $\nu$. Black dotted lines are the ”true” responses.
Responses obtained in case $\nu=0.5$ are displayed in blue, along with their
$68\%$ confidence bands in grey. The red lines are the response obtained in
case $\nu=2$. The yellow lines are the response obtained in case $\nu=5$. The
series are computed as sample average of the shocks obtained across
simulations.

Figure 9: IRF of tax rate and capital to a tax shock, with no fiscal
foresight. (a) Note: Black dotted lines are the ”true” responses. Responses
obtained in case $\nu=0.5$ are displayed in blue, along with the $68\%$
confidence bands in grey. The red lines are the response obtained in case
$\nu=2$. The yellow lines are the response obtained in case $\nu=5$. In each
case the external intrument is the real structural shock $u_{\tau,t}$.

Figure 10: VAR vs DFM with GK identification: shock comparisonn (a) Notes:
Comparison between the structural monetary policy shocks estimated from the 95
different VAR specifications (grey lines) and the DFM (red line). The
structural shock has been recovered via external instrument taken by Gertler
and Karadi.

##  Appendix E Robustness

Figure 11: Robustness on the number lags $p$ (a) Notes: Comparison of
responses to the shock identified using GK instrument with different lag
lengths $p$. Dark and light grey shaded areas are respectively the $84\%$ and
$90\%$ confidence bands for the baseline (i.e. $p=8$). Bands are obtained with
bootstrap techniques. Figure 12: Robustness on the number of static factors
$r$ (a) Notes: Comparison of responses to the shock identified using GK
instrument with different number of static factors $r$. Dark and light grey
shaded areas are respectively the $84\%$ and $90\%$ confidence bands for the
baseline (i.e. $r=9$). Bands are obtained with bootstrap techniques. Figure
13: Robustness on the number of dynamic factors $q$ (a) Notes: Comparison of
responses to the shock identified using GK instrument with different number of
dynamic factors $q$. Dark and light grey shaded areas are respectively the
$84\%$ and $90\%$ confidence bands for the baseline (i.e. $q=4$). Bands are
obtained with bootstrap techniques.