# Random Subspace Local Projections††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS>and<EMAIL_ADDRESS>††thanks: We thank
the editor, three anonymous referees, Silvia Gonçalves, Eleonora Granziera,
Luke Hartigan, Òscar Jordà, Daniel Lewis, Adrian Pagan, Barbara Rossi and
participants at seminars at the BI Norwegian Business School, 2023 Econometric
Society of Australasia Meeting, 2023 International Conference in Computing in
Economics and Finance, the Advances in Local Projections and Empirical Methods
for Central Banking Conference, Workshop of the Australasia Macroeconomic
Society, OzMac Workshop, and the CAMA Workshop on Macroeconomic Challenges for
Open Economies for helpful comments and suggestions. We acknowledge funding
support from the Australian Research Council (DP190100202, DE200100693 and
DP240100970).

Viet Hoang Dinh  Monash University, Australia  Didier Nibbering  Monash
University, Australia  Benjamin Wong  Monash University, Australia

###### Abstract

We show how random subspace methods can be adapted to estimating local
projections with many controls. Random subspace methods have their roots in
the machine learning literature and are implemented by averaging over
regressions estimated over different combinations of subsets of these
controls. We document three key results: (i) Our approach can successfully
recover the impulse response functions across Monte Carlo experiments
representative of different macroeconomic settings and identification schemes.
(ii) Our results suggest that random subspace methods are more accurate than
other dimension reduction methods if the underlying large dataset has a factor
structure similar to typical macroeconomic datasets such as FRED-MD. (iii) Our
approach leads to differences in the estimated impulse response functions
relative to benchmark methods when applied to two widely studied empirical
applications.

Keywords: Local Projections, Random Subspace, Impulse Response Functions,
Large datasets

##  1 Introduction

Impulse response functions (IRFs) are a common tool in macroeconomics to study
the dynamics of variables in response to shocks. A recent development is to
estimate IRFs using local projections (LPs), developed by Jordà (2005) (e.g.,
see Tenreyro and Thwaites, 2016; Ramey and Zubairy, 2018; Swanson, 2021). LPs
are single-equation methods, and can be thought of as the direct forecast
counterpart of traditional multiple equation systems, such as a vector
autoregressions (VARs), which rely on iterating on a system of equations.

A parallel development in the macroeconometrics literature has been the use of
large datasets. The cost of putting together a large dataset for empirical
analysis, at least for U.S. macroeconomic data, is now becoming exceedingly
trivial, especially with datasets such as FRED-MD and FRED-QD (McCracken and
Ng, 2021). Like any regression model, the choice of control variables to use
in an LP forms part of the model specification. With the availability of large
datasets, the dimension of this control set has become potentially large.

The contribution of our paper is to introduce random subspace methods for the
estimation of IRFs using LPs. The method is simple to implement by three
steps: First, we include a random subset of controls in the LP and estimate
the IRF. Second, we repeat the first step many times. Finally, we take the
average IRF. Through averaging over random subsets of controls, where the
subsets are generated independently of the data, the variance of the IRF
estimate is reduced while maintaining most of the signal of the controls.

A practical implication of our approach is that it frees up the researcher to
focus on the IRFs, which is the key object of interest. Conditional on a base
set of controls ensuring identification of the IRF, the random subspace method
deals with the issue of appropriately including controls to improve estimation
efficiency. In case a large set of controls is required for identification,
the averaging across different subspaces mitigates the risk of identification
bias, while avoiding the signal of the shock of interest to get swamped by the
controls. This is an appealing practical feature because the coefficients on
the controls are almost never the object of interest, as long as the
appropriate controls are included in order to obtain accurate estimates of the
IRFs.

We highlight that random subspace methods apply more naturally within the LP
setting despite the now-known result that LPs and VARs recover the same IRF in
population (see Plagborg-Møller and Wolf, 2021). In a VAR setting, an
additional control variable implies an extra equation. In the LP setting,
extra variables imply just more controls within the single equation. With the
potential control variables numbering into the tens or hundreds, appropriately
dealing with extra, possibly extraneous, controls is arguably more manageable
than estimating tens or hundreds of extra, possibly irrelevant, equations.

Our key results are as follows. First, we show that random subspace methods
can recover the true IRFs under plausible factor structures and usual
identification techniques for the structural shock of interest. We base this
conclusion of two sets of Monte Carlo experiments, one using a real business
cycle model with fiscal foresight introduced by Leeper, Walker, and Yang
(2013), and one using the dynamic factor model of Stock and Watson (2016). The
proposed method is implemented with both SVAR identification (Plagborg-Møller
and Wolf, 2021), and instrumental variable (IV) identification (Stock and
Watson, 2018). Random subspace methods help accurately estimate IRFS using a
large set of controls that exhibits different factor structures resembling
macroeconomic datasets.

Second, the Monte Carlo experiments demonstrate that random subspace LPs are
competitive in terms of mean squared error to a wide range of methods,
including commonly used VAR methods and LPs with alternative dimension
reduction methods. More specifically, the use of random subspaces improves
upon the use of factors when the factor structure resembles that of U.S.
macroeconomic datasets such as FRED-MD. Random subspaces are also favoured
over variable selection methods, possibly in part due to the dense structure
of macroeconomic data (see Giannone, Lenza, and Primiceri, 2021), though
whether our method outperforms VAR based methods can depend on the specific
context.

Third, we document in two widely studied empirical applications (i.e. the
effect of the macroeconomy to an identified monetary policy and technology
shock) that random subspace LPs produce meaningful differences in the
estimated IRFs relative to common empirical strategies. Our results are
encouraging as it suggests that the random subspace LP is a simple procedure
or robustness check for practitioners using LPs who may be concerned that they
may have omitted relevant controls.

We link our work to two strands of the broader literature. First, although
random subspace methods have their roots in the machine learning literature,
researchers have recently applied them to improve forecast accuracy for
economic indicators based on a large number of possible predictors (Koop,
Korobilis, and Pettenuzzo, 2019; Kotchoni, Leroux, and Stevanovic, 2019; Boot
and Nibbering, 2019; Pick and Carpay, 2022). In general, they find strong
performance of random subspace methods across different macroeconomic
forecasting exercises. Since the datasets used in macroeconomic structural
analysis are similar, exploring random subspace methods to estimating IRFs
seems natural.

Second, various approaches have been proposed to improve the efficiency of the
LP estimator (e.g. Barnichon and Brownlees, 2019; Ferreira, Miranda-Agrippino,
and Ricco, forthcoming; Lusompa, 2023; Ho, Lubik, and Matthes, forthcoming),
which is known to have a lower bias but higher variance than VARs (see, e.g.
Li, Plagborg-Møller, and Wolf, forthcoming). One can view our approach in a
similar vein. Given that proliferation of controls leads to inefficiency,
researchers naturally economize on the number of controls. Random subspace
methods are a form of regularization, in which averaging across subsets of
controls targets the bias-variance trade-off between omitting potentially
relevant information and including all available data as controls.

The remainder of the paper is organised as follows: Section 2 provides a
detailed discussion of our framework. Section 3 summarizes how to implement
our procedure in a step-by-step manner. Section 4 uses a Monte Carlo exercise
to understand how random subspace methods help in appropriately estimating
IRFs. Section 5 applies our proposed approach to two widely studied empirical
applications. Section 6 provides some concluding remarks.

##  2 Method

Consider the standard LP regressions one specifies to estimate the IRF to the
variable $y$ from an exogenous one unit impulse on $x_{t}$:

${y}_{t+h}=\mu_{h}+\beta_{h}x_{t}+{\Phi}_{h}^{\prime}{W}_{t}+\xi_{t+h},\quad
h=0,1,\dots,H,$ (1)

where $\mu_{h},\beta_{h}$, and ${\Phi}_{h}$ are projection coefficients,
$\xi_{t+h}$ is the projection error, and ${W}_{t}$ is a vector of controls. We
are interested in the response of $y_{t+h}$ with respect to an exogenous one
unit impulse to $x_{t}$, which equals $\beta_{h}$. The problem we investigate
in this paper is how one deals with the control set ${W}_{t}$. While one is
almost never interested in the coefficients ${\Phi}_{h}$, the specification of
the control set can matter to obtaining accurate estimates of $\beta_{h}$.

We briefly highlight two roles the controls play in the estimation of
$\beta_{h}$. First, the set of control variables accounts for relevant
determinants which are correlated with $x_{t}$. In this setting, failure to
include the relevant controls results in omitted variable bias, and so biased
estimates of the IRF. Second, even if $x_{t}$ is strictly exogenous, or we
possess a strictly exogenous instrument, including additional controls in the
projection may reduce the variation in forecasting $y_{t+h}$, which may result
in a more precise estimate for $\beta_{h}$. However, in this setting, the
effect is less obvious. If the included set of controls is too large, the
increase in parameter uncertainty reverses possible efficiency gains, and may
then increase the variance of the estimate for $\beta_{h}$.

With large macroeconomic datasets like FRED-MD, empirical work in
macroeconomics now has access to over a hundred variables that may serve as
potential controls. This leads to the familiar trade-off for practitioners
where omitting relevant controls results in bias, but including irrelevant
variables leads to an increase in variance. In cases where a variable is known
to be irrelevant, such a variable should be omitted, although in most
empirical settings it is not obvious whether a variable is irrelevant or not.
This is the setting that we propose solving through random subspace local
projections (RSLP): we alleviate the issue of specifying ${W}_{t}$, mindful
that a practitioner is often only interested in $\beta_{h}$.

###  2.1 Random Subspace Local Projections (RSLP)

Instead of a generic set of controls ${W}_{t}$, we make a distinction between
the $p_{V}\times 1$ vector ${V}_{t}$ of variables which are considered
essential controls, and the $p_{G}\times 1$ vector ${G}_{t}$ of possibly
relevant controls. Rewriting (1) in terms of these essential and possible
controls results in

$\displaystyle{y}_{t+h}$
$\displaystyle=\mu_{h}+\beta_{h}x_{t}+{\Theta}_{h}{V}_{t}+{\Psi}_{h}{G}_{t}+\xi_{t+h},$
(2)

where ${\Theta}_{h}$ and ${\Psi}_{h}$ are the projection coefficients of
${V}_{t}$ and ${G}_{t}$, respectively. While nothing precludes specifying
${V}_{t}$ as an empty set, most macroeconomic applications would a priori
treat some variables as essential. For example, the usual monetary policy
shock application would include at least some measure of real activity,
inflation, and the interest rate. The identification strategy for $\beta_{h}$
may also inform which variables have to be included in ${V}_{t}$, a matter
which we discuss in detail in Section 2.2.111Even in cases where one possesses
a strictly exogenous and valid instrument –and no controls are required for
the LP estimator to be consistent– the inclusion of controls known to be
relevant in ${V}_{t}$ may improve the accuracy of the estimate for
$\beta_{h}$.

Some applications may only provide essential control categories, but leave the
precise variable unspecified. For example, even if the model should feature an
interest rate variable, there are many such variables in datasets like FRED-
MD. Our empirical examples select one of the interest variables to be included
in $V_{t}$ in this case, and include the rest in ${G}_{t}$. Alternatively,
Appendix A.1 extends our approach to using prior knowledge on which categories
are essential, without having to select a specific variable from those
categories.

The number of possibly relevant controls in ${G}_{t}$ is potentially large as
previously motivated. Instead of estimating $\beta_{h}$ in (2), a form of
dimension reduction is usually applied to ${G}_{t}$. Consider the linear
projection from (2) with a $k\times p_{G}$ compression matrix ${R}^{(j)}$
indexed by $j$, where $k\leq p_{G}$:

$\displaystyle{y}_{t+h}=\mu_{h}^{(j)}+\beta_{h}^{(j)}x_{t}+{\Theta}_{h}^{(j)}{V}_{t}+{\Gamma}_{h}^{(j)}{R}^{(j)}{G}_{t}+\xi_{t+h}^{(j)},$
(3)

where ${\Gamma}_{h}^{(j)}$ is a $k$-dimensional vector of projection
coefficients, instead of the $p_{G}$-dimensional vector of projection
coefficients ${\Psi}_{h}$ in (2). The construction of the compression matrix
can be data-driven. For instance, variable selection methods use data to
estimate a selection matrix for ${R}^{(j)}$. Factor-augmented models take
${R}^{(j)}$ as the matrix of the principal component loadings corresponding to
the $k$ largest eigenvalues from the sample covariance matrix of ${G}_{t}$.

The random subspace approach to dimension reduction is to generate the
elements of ${R}^{(j)}$ from a probability distribution that is independent of
the data. More precisely, $R^{(j)}$ is sampled from a uniform distribution
across all combinations of the $p_{G}$ available predictors of size $k$. It
follows that ${R}^{(j)}$ is a random subset selection matrix which randomly
selects a subset of $k$ predictors out of the $p_{G}$ predictors. For
instance, if there are 5 possible predictors and we wanted to choose 3,
$p_{G}=5$ and $k=3$, and one possible draw $j$ could be the following:

$\displaystyle{R}^{(j)}=\begin{bmatrix}0&1&0&0&0\\\ 0&0&1&0&0\\\
0&0&0&0&1\end{bmatrix}.$

Conditional on a draw ${R}^{(j)}$, $\beta_{h}^{(j)}$ in (3) can be estimated
using least squares. The random subspace estimate for $\beta_{h}^{(j)}$ is
constructed by averaging over the least squares estimates
$\hat{\beta}_{h}^{(j)}$ corresponding to different draws ${R}^{(j)}$, with
$j=1,\dots,n_{R}$:

$\displaystyle\hat{\beta}_{h}=\frac{1}{n_{R}}\sum_{j=1}^{n_{R}}\hat{\beta}_{h}^{(j)}.$
(4)

Through averaging, random subspace methods reduce the variance of the LP
estimate, while it maintains most of the signal of the controls. We note that
one could combine these subspaces using weights reflecting the performance of
each subspace, instead of equal weights. However, estimation of these optimal
weights introduces additional variance, and hence a simple average is
generally hard to beat, as shown by Elliott and Timmermann (2013).222In
Appendix A.2, we use a weighting scheme based on the Bayesian information
criterion as discussed by Elliott, Gargano, and Timmermann (2013) in our
empirical examples, which indeed results in little differences relative to
equal weighting.

Implementing the random subspace LPs requires a choice for the subspace
dimension $k$. We find that setting $k=50$ is a reasonable choice in our Monte
Carlo experiments and empirical applications. While one could use an
information criterion to choose the dimension, as we discuss in Appendix A.3,
this may not work well in macroeconomic settings with a large number of
controls and a small sample size. Sensitivity analyses reported in Appendix B
suggest that the optimal subspace dimension is often in the 40-60 range.
Hence, we proceed with an empirical strategy that uses $k=50$, and
subsequently check for the robustness of the results.

Random subspace methods have shown to improve forecast accuracy for economic
indicators based on a large number of possible predictors (Koop, Korobilis,
and Pettenuzzo, 2019; Kotchoni, Leroux, and Stevanovic, 2019; Boot and
Nibbering, 2019; Pick and Carpay, 2022). The relative effectiveness of the
random subspace approach may be explained by three features of macroeconomic
data. First, macroeconomic variables are often highly correlated. Variables
that are irrelevant when all relevant controls are included, are often
correlated with omitted relevant controls in a random subset. Hence, the use
of random subsets results in a small bias while it reduces the variance.
Second, most macroeconomic data is characterized by a low frequency and low
signal. As a result, the selection of the controls in ${G}_{t}$, or the
derived factors, may be subject to substantial uncertainty. Third, Giannone,
Lenza, and Primiceri (2021), among others, find that often many variables are
relevant in explaining macroeconomic quantities of interest. This makes it
even more challenging to find an accurate low-dimensional representation of
the data by variable selection methods. Boot and Nibbering (2020) discuss
random subspace methods and its theoretical justification in the context of
macroeconomic forecasting in more detail.

###  2.2 Structural identification within RSLP

The exposition so far discusses how one can use random subspace methods to
estimate IRFs in an LP setting. We now connect our approach to recovering the
object of interest, namely the IRF from an underlying structural model.

Define ${w}_{t}$ as an $n\times 1$ vector of macroeconomic variables, in which
both the variable of interest $y_{t}$ and the variable $x_{t}$ to which we are
introducing an exogenous impulse in (1) are included. The variables in
${w}_{t}$ are driven by a $m\times 1$ vector of independent and identically
distributed (i.i.d.) shocks ${\epsilon}_{t}$:

${w}_{t}={\Theta}(L){\epsilon}_{t},\qquad{\epsilon}_{t}\sim
i.i.d.(0,{I_{m}}),$ (5)

with lag polynomial ${\Theta}(L)\equiv\sum_{j=0}^{\infty}{\Theta}_{j}L^{j}$,
and $n\times m$ coefficient matrices ${\Theta}_{j}$.

Suppose, without loss of generality, that we are interested in the effect of
the first shock $\epsilon_{1t}$, and $y_{t}$ and $x_{t}$ are respectively the
$i^{th}$ and $j^{th}$ variable in the vector ${w}_{t}$. After normalizing
$\epsilon_{1t}$ to imply a unit increase in $x_{t}$,333This normalization is
without loss of generality since shocks are unobserved, so the variance and
sign of shocks are ultimately normalized. Our choice of normalization
simplifies the exposition from needing to introduce a parameter to link
$\epsilon_{1t}$ to $x_{t}$. we can write $x_{t}$ as a linear combination of
shocks:

$\displaystyle
x_{t}=\epsilon_{1t}+f(\epsilon_{2:m,t},\epsilon_{t-1},\epsilon_{t-2},...),$
(6)

where $\epsilon_{2:m,t}\equiv(\epsilon_{2t},...,\epsilon_{mt})^{\prime}$ and
$f(.)$ is a linear combination of its argument. Leading the $i^{th}$ equation
of (5), which describes the dynamics of $y_{t}$, by $h$ periods, and
subsequently rearranging (6) to substitute out for $\epsilon_{1t}$, we obtain

$y_{t+h}=\theta^{i1}_{h}x_{t}+f(\epsilon_{t+h}...,\epsilon_{t+1},\epsilon_{2:m,t},\epsilon_{t-1},\epsilon_{t-2},...),$
(7)

where the argument in $f(\cdot)$ now subsumes $y_{t+h}$ being a linear
function of past and future shocks and
$f(\epsilon_{2:m,t},\epsilon_{t-1},\epsilon_{t-2},...)$ from (6). The object
of interest is $\theta^{i1}_{h}$, as it is the IRF from (5). Based on the form
in which (7) is written it would be analogous to $\beta_{h}$ in (1). Due to
endogeneity, (7) cannot be consistently estimated since both $x_{t}$ and
$y_{t+h}$ are a function of all past and current shocks. More precisely, in
our context, $x_{t}$ is not only correlated with $\epsilon_{1t}$ but also with
other shocks that affect $y_{t+h}$.

####  2.2.1 Identification through instruments or external variation

The IRF can be estimated using two-stage least squares and an instrument
$z_{t}$ that satisfies the following conditions, termed Condition LP-IV by
Stock and Watson (2018):

  1. 1.

$\mathbb{E}\left[\epsilon_{1t}z_{t}\right]\neq 0$ (relevance),

  2. 2.

$\mathbb{E}\left[\epsilon_{2:m,t}z_{t}\right]={0}$ (contemporaneous
exogeneity),

  3. 3.

$\mathbb{E}\left[\epsilon_{t+j}z_{t}\right]={0},\,j\neq 0$ (lead-lag
exogeneity).

With a strictly exogenous instrument satisfying the LP-IV conditions, the
instrument and the shocks are assumed to be uncorrelated contemporaneously and
at all lags, and hence the IRF can be estimated consistently. Directly
observing the shock is equivalent to observing a $z_{t}$ which is perfectly
correlated with $\epsilon_{1t}$. In this case, Condition LP-IV allows one to
directly regress the shock on $y_{t+h}$, similar as in approaches such as
Romer and Romer (2004). In these settings, the inclusion of controls is to the
extent that they may lead to efficiency in finite samples, but they are not
required for the identification of the IRF.

Condition LP-IV is a strong assumption but can be relaxed by including
controls:

$\displaystyle\mathbb{E}\left[\epsilon_{2:m_{\epsilon},t}^{\perp}z_{t}^{\perp}\right]$
$\displaystyle=$ $\displaystyle{0},\,j\neq 0\,\text{(conditional
contemporaneous exogeneity)},$ (8)
$\displaystyle\mathbb{E}\left[\epsilon_{t+j}^{\perp}z_{t}^{\perp}\right]$
$\displaystyle=$ $\displaystyle{0},\,j\neq 0\,\text{(conditional lead-lag
exogeneity)},$ (9)

where $u_{t}^{\perp}=u_{t}-Proj(u_{t}\mid{W}_{t})$ for some variable $u$. This
is a more relevant setting empirically, given widely used instruments for
monetary policy shocks have been shown to be either contaminated by other
shocks (see Miranda-Agrippino and Ricco, 2021, for a prominent example), or
possibly forecastable by other macro variables (see Ramey, 2016). These
suggest violations of strict exogeneity. Controls that account for this
information can aid in satisfying conditional exogeneity, and so help
constructing a valid instrument $z_{t}^{\perp}$.444This argument is analogous
to what Lloyd and Manuel (2024) label as the one-step approach, where one uses
controls in the first stage to render the instrument conditionally exogenous.
If there are particular controls that one knows to help the instrument to
satisfy conditional exogeneity, they should be treated as essential controls.
In addition, RSLP may efficiently account for the information in the possibly
relevant controls so that the conditions in (8) and (9) are satisfied.

We thus extend random subspace methods to the two-stage least squares settings
where the first and second stage regressions are

$\displaystyle{x}_{t}$
$\displaystyle=\alpha^{(j)}+\rho^{(j)}z_{t}+{\Lambda}^{(j)}{V}_{t}+{\Upsilon}^{(j)}{R}^{(j)}{G}_{t}+\eta_{t}^{(j)},$
(10) $\displaystyle y_{t+h}$
$\displaystyle=\mu_{h}^{(j)}+\beta_{h}^{(j)}\hat{x}_{t}^{(j)}+{\Theta}_{h}^{(j)}{V}_{t}+{\Psi}_{h}^{(j)}{R}^{(j)}{G}_{t}+\xi_{t+h}^{(j)},$
(11)

where ${\Lambda}^{(j)}$ and ${\Upsilon}^{(j)}$ are the projection coefficients
from the first stage regression, $\hat{x}_{t}$ is the fitted value from (10),
and the estimates for $\beta_{h}^{(j)}$ in (11) are used in (4) to average
across random draws for selection matrices.

####  2.2.2 Implementing SVAR identification

Plagborg-Møller and Wolf (2021) show that SVAR or system-based identification
(i.e. recursive identification, long-run restrictions, and sign restrictions)
can be implemented using LPs. These identification schemes require SVAR
invertibility, which allows one to map a VAR in $w_{t}$ to the form in (5) by
inverting the VAR lag polynomials. Under invertibility there is a linear
combination of $w_{t}$ and its lags which accounts for
$f(\epsilon_{2:m,t},\epsilon_{t-1},\epsilon_{t-2},...)$ in (6), so that what
gets used in place of $x_{t}$ in (7) identifies the shock of interest
$\epsilon_{1t}$. If not all relevant information is included, identification
fails due to not satisfying invertibility. The literature has also long
recognized the link between invertibility and the inclusion of all relevant
information (see, e.g. Hansen and Sargent, 1991; Fernández-Villaverde et al.,
2007; Stock and Watson, 2018).

Where a standard LP has a VAR representation estimating the same IRF in
population, estimating an IRF by RSLP is analogous to averaging over the IRFs
from VARs with different sets of variables, but the same identification
strategy.555Note that we will be applying the same sign, long, or short run
restriction across different subspaces. Therefore, all that is changing is the
information set, or the set of controls, one is using to apply the
identification scheme. In this regard, the approach has connections to the
approach of Ho, Lubik, and Matthes (forthcoming), although they allow the
identifying restrictions to differ across the different models that they
average over. In cases where the essential controls in $V_{t}$ are sufficient
for invertibility, RSLP may help reducing the variance in the IRF estimates by
regularizing over the information from the rest of the dataset. In cases where
the possibly relevant controls in $G_{t}$ are required to satisfy
invertibility, and thus play a role with identification, the averaging over
different subspaces of $G_{t}$ implicitly takes uncertainty about the exact
identifying assumptions into account. The averaging over many different
information sets is also related to the estimation of IRFs by factor models or
large Bayesian VARs, such as respectively those by Bernanke, Boivin, and
Eliasz (2005) and Bańbura, Giannone, and Reichlin (2010). These methods may be
viewed as alternative approaches to fulfill the invertibility condition by
applying dimension reduction to large information sets.

To illustrate how SVAR identification may be implemented within an RSLP, we
consider a recursive identification scheme from the seminal work by
Christiano, Eichenbaum, and Evans (1999). They identify a monetary policy
shock by restricting the contemporaneous effect of “fast moving” variables on
the Federal funds rate to be zero, and the contemporaneous effect of monetary
policy shocks on “slow moving” variables to be zero. This suggests that
contemporanous values of the “slow moving” variables, such as real activity
and price variables, have to be included in the LP. The number of available
real activity and price variables in modern macroeconomic datasets is large,
and the identification scheme does not provide guidance on which specific
variables to select.666For example, it is known that researchers have used
industrial production, unemployment, employment, or real GDP, amongst others,
to reflect real activity. While the motivation is to include at least one real
activity variable, there is little guidance of which one to include. From this
perspective, one could argue that researchers aim to control for the same
information even if they disagree on which precise variable reflects that
information.  By including a small number of variables in $V_{t}$, such as
industrial production and inflation, and the remaining available “slow moving”
variables in $G_{t}$, one can view RSLP as a procedure to balance the risk of
non-invertibility with overfitting.

###  2.3 Error bands with RSLP

We briefly discuss how to construct error bands for the IRF estimated by RSLP.
A key challenge is that the random subspace literature, like most machine
learning methods, provides little guidance on how to quantify estimation
uncertainty.

The standard error of a standard LP can be estimated with a block bootstrap to
account for serially correlated errors (see, e.g. Kilian and Kim, 2011; Inoue,
Jordà, and Kuersteiner, 2023).777We note that if one were certain that the
lags of the correct variables are included as essential controls, it may be
possible to obviate the need to account for serial correlation (see Montiel
Olea and Plagborg-Møller, 2021) and just consider the cross correlation across
the regressions. Since RSLP is constructed by averaging across LP regressions,
the bootstrap should not only preserve correlation across residuals in each
regression, but also the correlation across the regressions. Therefore, a
natural choice would be a block bootstrap, using the same block of residuals
across regressions to also preserve the correlation across subspace estimates.
Although estimating a single RSLP requires $n_{R}$ least squares regressions,
which would at most take a few seconds in a modern macro context, the
bootstrap may amount to substantial computational costs. Given each horizon is
bootstrapped separately, an IRF for one variable would require bootstrapping
$n_{R}\times(H+1)$ regressions. The computation time of this procedure can be
alleviated by parallelizing the computation across subspaces.

Since the block bootstrap for RSLP is computationally expensive, we also
discuss an alternative approach derived from Buckland, Burnham, and Augustin
(1997). This approach relies on the assumption that there is perfect
correlation across the IRF estimates from different subspaces. Although the
correlation across subspaces is high in practice, this assumption is likely
too strict, and therefore the bands will be conservative. The error bands of
standard LP bands are already known to be wide, although RSLP may alleviate
this issue by efficiently incorporating controls that help explain variation
in the variable of interest. The implementation details of both the bootstrap
and the Buckland, Burnham, and Augustin (1997) approach are deferred to
Appendix C.

##  3 Implementing RSLP

We briefly summarize the preceding discussion to outline how RSLP can be
implemented:

  1. 1.

Make a distinction between essential controls, $V_{t}$, and the controls that
the random subspace method will be applied to, $G_{t}$. This distinction
specifies the number of possible covariates that each random subspace matrix
will select from, $p_{G}$.

  2. 2.

Choose a subspace dimension, $k$. The subspace dimension dictates how many
covariates are included in each subspace regression. In practice, we find a
subspace dimension of 50 to work well with U.S. macroeconomic data.

  3. 3.

Randomly draw from the $p_{G}$ possible controls, assigning equal probability
that any of these $p_{G}$ controls can be chosen. Create a matrix indexed by
$j$, $R^{(j)}$ of dimension $k\times P_{G}$, which acts as a selector matrix
that sets the appropriate element to 1 if a control is chosen, and zero
otherwise.

  4. 4.

For each horizon $h\in\\{0,1,\ldots,H\\}$, estimate
${y}_{t+h}=\mu_{h}^{(j)}+\beta_{h}^{(j)}x_{t}+{\Theta}_{h}^{(j)}{V}_{t}+{\Gamma}_{h}^{(j)}{R}^{(j)}{G}_{t}+\xi_{t+h}^{(j)}$.
Depending on the identification strategy, $x_{t}$ may need to be constructed
with a first stage regression. This regression may be similar to (10) if one
used an external instrument, or is specified using system-based identification
restrictions such as long-run or sign restrictions.888See Plagborg-Møller and
Wolf (2021) for a general discussion on how to implement system-based
identification in an LP setting. Alpanda, Granziera, and Zubairy (2021)
discuss sign restrictions, and Appendix D long-run restrictions in the spirit
of Blanchard and Quah (1989) in LPs. Because the first stage regression also
requires choosing controls, the $R^{(j)}$’s will also be used in the first
stage.

  5. 5.

Repeat steps 3 and 4 a large number of times. In practice, we find a number of
subspace draws $n_{R}=1000$ to be sufficiently large that results are stable.
The estimated IRF for variable $y$ at horizon $h$ to the identified shock of
interest, can be obtained via averaging over the $n_{R}$ draws:
$\beta_{h}=\frac{1}{n_{R}}\sum_{j=1}^{n_{R}}\beta_{h}^{(j)}$.

##  4 Monte Carlo Experiments

As an illustration of the RSLP procedure, we consider Monte Carlo experiments
based on the real business cycle model with fiscal foresight by Leeper,
Walker, and Yang (2013). The phenomenon of fiscal foresight occurs when at
time $t$ agents know the tax rate they will face at time $t+h$. The model
includes income taxes, inelastic labor supply, and full capital depreciation.
The log-linearized equilibrium condition for capital is given by999See Leeper,
Walker, and Yang (2013), page 1118, Equation (4).

$\displaystyle k_{t}=\alpha
k_{t-1}+u_{at}-(1-\theta)\frac{\tau}{1-\tau}\sum_{k=0}^{\infty}\theta^{k}E_{t}\hat{\tau}_{t+k+1},$
(12)

where $k_{t}$ and $\hat{\tau}_{t}$ are respectively the percentage deviations
from the steady state capital and tax rate, $\tau$ is the steady state value
of the tax rate, $\theta$ and $\alpha$ are parameters satisfying the
inequalities $0<\theta<1$ and $0<\alpha<1$, and $u_{at}$ is an i.i.d.
technology shock. The tax rule is $\hat{\tau}_{t+h}=u_{\tau t}$, where
$u_{\tau t}$ is an i.i.d. tax shock. Allowing $h=2$ (a two-period foresight),
(12) becomes

$\displaystyle k_{t}=\alpha k_{t-1}+u_{at}-\kappa(\theta u_{\tau t}+u_{\tau
t-1}),$ (13)

where $\kappa=\tau(1-\theta)/(1-\tau)$. The structural moving average
representation equals

$\displaystyle\begin{bmatrix}\hat{\tau}_{t}\\\
k_{t}\end{bmatrix}=\begin{bmatrix}L^{2}&0\\\ -\frac{\kappa(L+\theta)}{1-\alpha
L}&\frac{1}{1-\alpha L}\end{bmatrix}\begin{bmatrix}u_{\tau t}\\\
u_{at}\end{bmatrix}.$ (14)

To parametrize the above, we follow Leeper, Walker, and Yang (2013) and Forni
and Gambetti (2014) by setting $\theta=0.2673$, $\tau=0.25$, $\alpha=0.36$,
and $u_{\tau t}$ and $u_{at}$ to be from independent standard Normal
distributions.

###  4.1 Information sets and identification schemes

The econometrician’s objective is to estimate the IRFs of both taxes and
capital to a tax shock. Due to fiscal foresight, there is insufficient
information to recover the IRFs by just observing capital and the tax rate at
time $t$. The Monte Carlo experiments consider different ways of extending the
information set of the econometrician.

##### Strictly exogenous instrument

First, we allow the econometrician to observe a strictly exogenous instrument
together with additional series that can aid in recovering the IRFs. The
instrument $z_{t}$ for the tax shock $u_{\tau t}$ is generated as

$\displaystyle z_{t}=0.7u_{\tau
t}+\nu_{1t-1}+\nu_{2t-1}+\epsilon_{z},\quad\epsilon_{z}\sim N(0,0.01),$ (15)

where the shocks $\nu_{1,t}$ and $\nu_{2,t}$ are both generated from a
$N(0,4)$ independent from the structural model. The instrument identifies the
IRF without controls. However, being able to account for variation in $z_{t}$
due to $\nu_{1t-1}$ and $\nu_{2t-1}$ can lead to efficiency gains. To this
end, 100 informational variables $y_{it}^{*}$ are generated as

$\displaystyle
y_{it}^{*}=b_{i}\nu_{1t}+(1-b_{i})\nu_{2t}+\xi_{it},\quad\xi_{it}\sim
N(0,\sigma_{i}^{2}),\quad i=1,\dots,100,$ (16)

where $b_{i}$ is a Bernoulli random variable assuming value 1 with probability
0.1.

##### Conditionally exogenous instrument

Second, we allow the econometrician to observe an instrument for the tax shock
which is only exogenous conditional on a set of informational variables. The
instrument $z_{t}$ is generated as

$\displaystyle z_{t}=0.7u_{\tau t}+u_{at-1}+u_{\tau
t-1}+\epsilon_{z},\quad\epsilon_{z}\sim N(0,0.01),$ (17)

where $u_{at-1}$ and $u_{\tau t-1}$ are structural shocks. Failure to control
for these lagged shocks will lead to a violation of lead-lag exogeneity. To be
able to recover the IRFs with this instrument, we generate 100 informational
variables $y_{it}^{*}$ as

$\displaystyle y_{it}^{*}=b_{i}u_{\tau
t-1}+(1-b_{i})u_{at-1}+\xi_{it},\quad\xi_{it}\sim N(0,\sigma_{i}^{2}),\quad
i=1,\dots,100.$ (18)

The settings with strictly and conditionally exogenous instruments will help
to understand the extent to which RSLP can achieve either reduction in
estimation variance or identification bias.

##### SVAR identification

Third, we assume that the econometrician knows that the technology shock has
no cumulative effect on the tax rate, and has access to the informational
series from (18). As shown by Leeper, Walker, and Yang (2013), the model
implied by (14) does not admit an invertible SVAR representation in taxes and
capital as the model is missing the two period ahead tax rate (or the current
tax shock) in the information set. Since the informational variables contain
this information, including these variables admits an invertible VAR
representation. It follows from Plagborg-Møller and Wolf (2021) that
subsequently applying SVAR identification with this additional information in
an LP can recover the IRF.101010We note that Forni and Gambetti (2014) already
demonstrate that one can use SVAR identification to recover the IRF in this
setting by using a FAVAR to account for this information. We impose the SVAR
identification restriction in an LP setting by using the procedure presented
by Plagborg-Møller and Wolf (2021). We leave the implementation details to
Appendix D.

##### Controls with a strong and weakened factor structure

To examine the role of the strength of the factor structure of the information
series in recovering the IRFs, we consider two settings that differentiate the
factor structure of the informational variables in (16) and (18):

Strong case: $\displaystyle\sigma_{i}\sim U(0,1),\quad\text{Weak case:}$
$\displaystyle\sigma_{i}\sim U(0,4).$ (19)

The strong case mimics a structure that one would probably encounter in a
panel where these individual series have a high level of comovement, such as
cross country asset prices and interest rates (e.g. Miranda-Agrippino and Rey,
2020), or commodity prices (e.g. West and Wong, 2014; Alquist, Bhattarai, and
Coibion, 2020). The weak case is akin to the sort one would expect to
encounter when using U.S. macroeconomic data: Appendix E shows that the first
two factors of both the FRED-MD and FRED-QD dataset explain a similar amount
of variation as in the weak case.

We run each of the three identification schemes with both the strong and weak
cases. This amounts to six Monte Carlo experiments, each simulating 1000
artificial datasets of 200 observations for capital and the tax rate according
to (14). The experiments corresponding to different information sets and
identification schemes accompany each of these artificial datasets with: an
exogenous instrument and informational variables generated from (15) and (16);
a conditionally exogenous instrument and a set of informational variables from
(17) and (18); informational variables from (18) within SVAR identification.

###  4.2 RSLP can recover the true impulse response functions

To understand if RSLP can recover the IRF from the DGP, we study the
expectation of the estimated IRFs. For each artifical dataset, RSLP constructs
the IRF following the procedure in Section 2.1 with 1000 draws of the
selection matrix ${R}^{(j)}$ and ${G}_{t}$ equal to the first lag of
$y^{*}_{t}$. To investigate whether RSLP is a valid approach to accounting for
the omitted information, we estimate LPs without ${G}_{t}$, which we label LP
with a base set of controls. Both specifications include the two lags of tax
rate and capital in ${V}_{t}$, together with two lags of the instrument when
using IV identification. Appendix D elaborates on the specification of the LPs
when using SVAR identification.

Figure 1 presents the IRFs of both capital and tax rate to a tax shock from
both the IV and SVAR identification schemes, under both the strong and weak
case. We also plot IRFs estimated via LP without the additional controls from
the generated informational series, and the true IRFs. The plotted estimated
IRFs are taken as the average across the Monte Carlo replications, so
deviations from the true IRF represent bias.

Figure 1: Impulse response functions in the Monte Carlo experiments

(a) Strictly exogenous instrument

(b) Conditionally exogenous instrument

(c) SVAR identification

_Note: This figure shows IRFs estimated by LP with a base set of controls
(orange, dot-dashed), RSLP using controls from the strong (blue, solid) and
weak case (blue, dotted) in ( 19), and the true responses (black, dashed). The
panels correspond to different identification schemes. The shock is normalized
to a 100 basis point increase in the tax rate. The results are the average
IRFs across 1000 Monte Carlo replications._

Since the tax shock occurs in period 2, the true IRF of the tax rate sees a
one off spike in period $t+2$. Capital falls for two periods, then adjusts
back in a monotonic fashion. Panel (a) in Figure 1 shows that with a strictly
exogenous instrument the true IRFs are recovered. This result holds even
without information on the controls, although the dot-dashed orange line shows
some more variation around the true IRF. Now suppose we estimate the IRF
without controls in Panel (b) and (c). The dot-dashed orange line shows that
we will miss the timing of the tax rate, and as a result, also miss the
response of capital. LP with a base set of controls and a conditionally
exogenous instrument in Panel (b) is biased given the instrument is invalid
due to not being exogenous to the lagged shocks. Panel (c) is an illustration
of a fiscal foresight issue, where information on only the tax rate and
capital is insufficient to recover the true IRF, which has been previously
documented (e.g. Leeper, Walker, and Yang, 2013; Forni and Gambetti, 2014).

Our proposed approach is able to largely recover the true IRF. In particular,
RSLP is able to estimate the tax rate spiking in period 2, and recover the
dynamics of the response to capital. While our proposed method is close to
being unbiased in the strong case, the bias is still minimal under the weak
case with either IV or SVAR identification.

###  4.3 Comparisons relative to other methods

This section compares RSLP relative to widely used existing methods for
constructing IRFs, and to LPs employing alternative dimension reduction
methods. More specifically, we consider FAVARs and factor augmented local
projections (FALPs), small and medium Bayesian VARs (BVAR),111111The small
BVAR only includes the base set of variables; taxes and capital. The medium
BVAR considers 20 variables, with another 18 informational variables in
addition to the base set. Extant work suggests that with the standard
Minnesota type shrinkage and natural conjugate priors, a 20 variable BVAR
often produces similar results relative to large BVARs of over hundred
variables (see Bańbura, Giannone, and Reichlin, 2010; Morley and Wong, 2020).
In our setting, a 20 variable BVAR suffices to make our point. the Bayesian
local projection (BLP) of Ferreira, Miranda-Agrippino, and Ricco
(forthcoming), and LPs exploiting sparsity with either a LASSO penalty or the
variable selection method OCMT proposed by Chudik, Kapetanios, and Pesaran
(2018). Finally, we consider the LP with a base set of controls (Base).
Implementation details of these methods are deferred to Appendix F.

Table 1 shows the root mean squared error (RMSE) of the IRFs estimated by each
method relative to the RMSE of the IRFs estimated by RSLP.121212Note that
there are seven horizons ($h=0,\dots,6$) in the IRFs, across which we average
to calculate the RMSE:
$\sqrt{\frac{1}{7}\sum_{h=0}^{6}\frac{1}{1000}\sum_{i=1}^{1000}(\hat{\beta}_{hi}-\beta_{h,\text{true}})^{2}}$,
where $\hat{\beta}_{hi}$ is the estimated IRF at horizon $h$ in the $i$-th
replication and $\beta_{h,\text{true}}$ is the true IRF at horizon $h$. Values
above one favour RSLP. These values are reported for the six experiments with
the real business cycle model with fiscal foresight, which we refer to as the
baseline experiments. We also compare the methods in experiments with a
dynamic factor model (DFM) in the spirit of Stock and Watson (2016). Appendix
G discusses the DFM experiments in detail. Here we summarize broad
observations from our baseline experiments, noting that the DFM experiments
have similar results unless otherwise stated.

Table 1: RMSE relative to RSLP in the baseline Monte Carlo experiments

| Strict Instrument | Conditional Instrument | SVAR
---|---|---|---
| Strong | Weak | Strong | Weak | Strong | Weak
| Tax | Cap | Tax | Cap | Tax | Cap | Tax | Cap | Tax | Cap | Tax | Cap
| LP methods
FALP | 0.955 | 0.948 | 1.046 | 1.039 | 0.964 | 0.954 | 1.259 | 1.031 | 0.886 | 1.042 | 1.156 | 1.100
LASSO | 4.523 | 4.298 | 8.167 | 15.219 | 6.664 | 1.789 | 3.931 | 1.666 | 1.867 | 1.012 | 2.002 | 1.001
OCMT | 1.340 | 1.341 | 1.279 | 1.301 | 0.974 | 1.002 | 0.737 | 0.926 | 1.194 | 1.183 | 0.748 | 1.069
BLP | 7.093 | 4.380 | 6.125 | 3.761 | 8.280 | 4.284 | 4.603 | 3.824 | 7.164 | 1.683 | 5.121 | 1.633
Base | 9.634 | 17.815 | 8.319 | 15.295 | 7.283 | 1.886 | 4.049 | 1.683 | 7.336 | 1.744 | 5.243 | 1.693
| VAR methods
FAVAR | 5.779 | 3.361 | 5.075 | 2.790 | 0.871 | 0.783 | 2.340 | 1.281 | 0.824 | 1.035 | 1.197 | 0.998
Small | 7.042 | 4.314 | 6.081 | 3.704 | 8.219 | 4.215 | 4.570 | 3.762 | 7.142 | 1.564 | 5.105 | 1.518
Medium | 4.964 | 1.557 | 4.385 | 1.241 | 4.996 | 1.303 | 2.535 | 1.210 | 2.377 | 1.284 | 2.639 | 1.497

_Note: Root mean squared error (RMSE) relative to RSLP. Strict and conditional
instruments refer to instruments generated by ( 15) and (17), respectively.
SVAR refers to using an SVAR identification scheme. Strong and weak refers to
the cases in (19). FALP refers to a factor augmented LP. LASSO and OCMT (see
Chudik, Kapetanios, and Pesaran, 2018) refer to these procedures applied to
estimating the LP. BLP refers to the Bayesian LP by Ferreira, Miranda-
Agrippino, and Ricco (forthcoming). Base refers to an LP with the base set of
controls; taxes and capital. FAVAR refers to a factor augmented VAR. Small and
medium refers to respectively a bivariate BVAR of tax and capital and a 20
variable BVAR with tax, capital and 18 informational variables. See text and
Appendix F for further details._

##### RSLP improves upon factor models when the factor structure is weakened

We focus our initial comparisons relative to factor models given its
established tradition in the macroeconomics literature. We first discuss the
FAVAR, since this model is shown to be able to recover the IRF in the baseline
DGP with SVAR identification (see Forni and Gambetti, 2014).131313The FAVAR
augments a bivariate VAR system containing taxes and capital with the first
two principal components of the informational variables. The IV identification
case uses the instrument as per the approach by Mertens and Ravn (2013) and
Gertler and Karadi (2015). The SVAR identification case identifies the shock
by assuming there is no cumulative effect of the technology shock on the tax
shock. Table 1 shows that in the strong factor case in the conditionally
exogenous instrument or SVAR identification schemes, the FAVAR models have a
lower or similar RMSE relative to our RSLP approach.141414Note that we focus
the comparison of the FAVAR to the RSLP to these two cases as the FAVAR is
invertible. The FAVAR is non-invertible in the strictly exogenous instrument
case in our baseline DGP as the controls capture noise in the instrument,
explaining its underperformance. However, the conclusions flip when we
consider the weak cases, where RSLP outperforms the FAVAR, at times by
substantial margins.

The comparison between FAVAR with RSLP may be driven by differences between a
VAR and LP, or between using factors or random subspaces. Comparing RSLP with
FALP isolates the latter. Although the margin of improvement decreases, RSLP
still outperforms FALP in the weak case. This result suggests that the
outperformance of the RSLP relative to FAVAR is both driven by the VAR
structure and possible issues related to the factors not being able to account
for all relevant information. We conclude that with a strong factor structure,
such as cross-country data on asset prices and interest rates, factor models
do well. RSLP becomes useful with a weaker factor structure, resembling that
of a macroeconomic dataset such as the FRED-MD. The DFM experiments also show
results in favour of RSLP over FALP. However, since the DGP in those
experiments is a closer approximation to a FAVAR, FAVAR is outperforming both.

##### RSLP does well compared to other local projection methods

RSLP is competitive compared to applying other dimension reduction methods to
LPs. Together with FALP, RSLP often performs best in terms of RMSE. Hence,
depending on the strength of the underlying factor structure in the data, FALP
also appears to be a reasonable method. Variable selection with LASSO does not
appear to outperform the FALP or RSLP. OCMT performs well with a conditionally
exogenous instrument or SVAR identification, but is less accurate when the
instrument is strictly exogenous. Note that, without any guidance from the
literature on how to implement variable selection methods within LPs,
improvement may be made in selecting the tuning parameters in these methods.

Comparing the base LP with the LPs using dimension reduction in the
experiments with a strictly exogenous instrument shows that using a large set
of controls often leads to efficiency gains. All the LP methods that use large
information sets do similarly or better than an LP with a standard set of
controls. Since the controls are not required to identify the shock with a
strictly exogenous instrument, any improvements in RMSE are driven by
efficiency gains from inclusion of the potentially relevant controls.

In the DFM experiments, both variable selection methods are outperformed by
FALP and RSLP. Although BLP is outperformed in the baseline experiments, it
performs well in the DFM experiments. The BLP is shrinking towards a BVAR.
While a BVAR is a poor approximation in the baseline DGP due to information
insufficiency, a BVAR is a reasonable approximation to the DFM DGP, and so
correspondingly, the BLP does much better in the latter setting. Therefore,
while the BLP is a valuable addition to the empirical toolkit, our exercise
also shows that how well the BLP does in practice depends on the underlying
BVAR that one chooses to shrink towards.

##### RSLP may compare favourably relative to VARs, depending on the DGP

Relative to VAR methods other than FAVAR, RSLP performs well across the
baseline experiments. Within these DGPs, the small BVAR suffers from
information insufficiency. The medium BVAR is technically information
sufficient, and hence the loss of accuracy relative to RSLP is driven by two
features. First, the VARs include equations for all included variables and are
therefore vastly overparametrized. Second, to prevent the overparametrization
to lead to substantial estimation uncertainty, the VAR methods apply shrinkage
to all coefficients. This shrinkage does not discriminate between essential
and potentially relevant controls, which may induce a bias.

Note that the results in Table 1 are based on mean squared errors averaged
across horizons. It is well known that the relative accuracy of VARs improves
when the horizons increase. Appendix H shows that this is also the case in our
baseline experiments. For instance, the relative performance of RSLP and FALP
stays relatively constant across horizons, while the relative accuracy of
FAVAR to RSLP improves with larger horizons.

While the baseline experiments show that RSLP can be a serious competitor
relative to VAR methods, the DFM experiments show that this is not a general
result. In the DFM experiments, the BVAR produces accurate IRFs for most
variables considered, and also BLP outperforms the LP methods for most
variables. The medium BVAR performs better than RSLP across all variables.
These results suggest that VARs are a better approximation to the DFM than to
the baseline DGP. The DFM DGP generates the data from factors without
distinguishing essential variables, making the application of dimension
reduction to all variables appropriate. This may benefit VAR models by
reducing both the variance due to potential overparametrization and bias due
to the shrinkage priors.

###  4.4 Take-aways

We summarize two key take-aways from our Monte Carlo exercise. First, RSLP is
capable of recovering the true IRF, which at least provides prima facie
evidence that it is a viable approach for applied work. In particular, RSLP is
close to being unbiased across a range of different Monte Carlo experiments.
Second, relative to a set of alternative benchmarks RSLP at least performs
competitively, suggesting it to be an appropriate addition to the standard
toolkit for applied macroeconomists.

##  5 Empirical applications

We use RSLP to estimate the dynamic responses to technology and monetary
policy shocks. Both applications can be traced to a broad empirical
literature, where there are suggestions that baseline specifications with
minimal controls are insufficient. Hence, these applications are natural
settings to understand whether random subspace is a useful method to
incorporate information from a large set of controls.

###  5.1 Technology shock application

Since Gali (1999), a strand of the SVAR literature has investigated the impact
of technology shocks on a variety of macroeconomic variables, with a focus on
labor market variables (e.g., Francis and Ramey, 2005; Forni and Gambetti,
2014; Barnichon, 2010). Keeping in the spirit of Gali (1999), the technology
shock is identified as being the only shock that has a long-run impact on
labor productivity.151515For settings that we are working with LPs instead of
SVARs, we implement the long-run identification restrictions as suggested by
Plagborg-Møller and Wolf (2021). Implementation requires one to nominate a
horizon at which all short-run effects of the shock are expected to dissipate.
We set this horizon to 3 years. Appendix D discusses further implementation
details.

Mirroring the baseline bivariate VAR in the broader literature, our baseline
set of controls includes four lags of both the growth rate of labor
productivity and unemployment in ${V}_{t}$ as we use quarterly data. The set
of possible controls in ${G}_{t}$ includes 127 macroeconomic time series
specified at the quarterly frequency, which includes 117 series from the FRED-
QD database (see McCracken and Ng, 2021), 6 total factor productivity series
from Fernald’s website,161616https://www.johnfernald.net/TFP and 4 consumer
confidence indicators from the Michigan Survey. We consider the first lags for
these variables, which gives us a set of 127 possible control variables. Our
sample is from 1960Q1 to 2019Q4.

###  5.2 Monetary policy shock application

This application features a baseline set of controls that echoes the external-
VAR of Gertler and Karadi (2015). As the model is monthly, our baseline set of
controls includes 12 lags of the log difference of CPI, log difference of
industrial production (IP), the excess bond premium by Gilchrist and Zakrajšek
(2012), and the 1-year government bond rate in ${V}_{t}$.

Our possible control set ${G}_{t}$ includes 111 FRED-MD series, and we
consider the first lag of these variables. The sample spans January 1990 to
June 2012 to match up with the time-span of the instrument. We use the high
frequency surprises around announcements by Gertler and Karadi (2015) as an
instrument to identify the effect of monetary policy shocks.

###  5.3 Results

Figure 2 presents the estimated IRFs from our two applications.171717Note that
the IRFs are on the level of labor productivity, CPI and industrial production
index. Hence, we follow the usual practice to specify the left-hand side
variable in the LP as $y_{t+h}-y_{t-1}$ (see Stock and Watson, 2018, Section
1.5). The RSLPs are based on 1000 draws of the selection matrix, a subspace
dimension equal to 50,181818Appendix B investigates the robustness of the
results with respect to the subspace dimension. and accompanied by an
approximate 90% bootstrap interval constructed as in Appendix C. The IRFs are
also estimated by the benchmark methods discussed in Section 4.3. The
implementation details of these methods in both applications are deferred to
Appendix I.

Figure 2: Impulse response functions in the empirical applications

(a) Response to a technology shock

(b) Response to a monetary policy shock

_Note: Panel (a) presents estimated IRFs of labor productivity and
unemployment rate to a technology shock which increases labor productivity by
0.25 $\%$ on impact. Panel (b) presents estimated IRFs of CPI inflation and
industrial production to a monetary policy shock which increases the one year
bond rate by 100 basis points. The blue-shaded areas indicate the 90% error
bands of RSLP. See text and note to Table 1 for details of models estimated. _

For the technology shock application, we present the response of labor
productivity and the unemployment rate to a technology shock which raises
labour productivity by $0.25\%$, which is equivalent to a one standard
deviation shock in the FAVAR model. Using RSLP, labor productivity increases
and the unemployment rate decreases in response to an expansionary technology
shock. These results are consistent with the predictions of the real business
cycle model and those shown by, for example, Christiano, Eichenbaum, and
Vigfusson (2003).

For the monetary policy shock application, we present the response of CPI and
industrial production to a contractionary monetary policy shock which
increases the 1-year government bond rate by a 100 basis points. Using RSLP,
both CPI and industrial production fall in response to a contractionary
monetary policy shock, consistent with what one expects from a standard
monetary policy shock.191919The information effect does not seem to produce
puzzles here. We nonetheless note that our approach may be well-suited for
dealing with informational effects, as variables representing central bank
information can be used as additional controls. We experimented with using the
27 variables representing forecasts and forecast revisions constructed by
Miranda-Agrippino and Ricco (2021) as part of a larger set of possible
controls for our monetary policy application, and obtained almost identical
results, which suggests that our large information set already accounted for
possible informational effects.

Comparing relative to other methods, we find that while the direction of the
response of the variables is largely consistent across all the different
methods, our proposed RSLP also leads to noticeable differences relative to
the other approaches. These differences can be qualitatively meaningful, as
the estimated responses from the different methods can be outside the
bootstrapped 90% intervals for RSLP.

More specifically, we find that the relative differences across LP methods
vary across the two applications. FALP does not differ much from the base LP
in the technology application, but produces IRFs different from all other
methods with the monetary policy shock. The IRFs of the LPs with variable
selection are similar to the ones from RSLP in the technology application, but
do not pick up any information from the controls in the monetary policy shock
example. From the comparison with the VAR methods, we find that FAVAR and the
medium BVAR are closer to RSLP than BLP and the small BVAR across both
applications, indicating that they pick up some signal from the controls. The
BLP can be quite similar to the estimated response from the small BVAR, which
highlights the finding from our Monte Carlo experiments that the performance
of BLP can be driven by the choice of the BVAR one chooses to shrink towards.

Overall, our empirical exercise suggests that RSLP does lead to reasonable
empirical results, especially when compared with other methods and the broader
literature. Consistent with our Monte Carlo experiments, we are encouraged
that our approach can be a useful addition to the applied macroeconomist’s
toolkit.

##  6 Conclusion

We show how one can apply a dimension reduction technique traditionally used
in machine learning to LPs in order to estimate IRFs with many controls. This
random subspace method is simple to implement and it basically contains 3
steps: Step 1: take a random draw; Step 2: do it many times; and Step 3:
average. We have shown that our approach is a plausible addition to the
toolkit as it can recover the true IRFs in settings encountered in
macroeconomics.

It is worth stressing that while our proposed approach is a plausible
empirical strategy, it does not compete to supplant any of the recent
innovations in the broader development of LP estimation. To highlight some
important developments in the LP literature, Barnichon and Brownlees (2019)
consider smoothing LP, and Ferreira, Miranda-Agrippino, and Ricco
(forthcoming) combine LPs with additional prior information. Both approaches
can be applied to RSLPs instead of standard LPs with a base set of controls,
so providing a more appropriate starting point for their procedures. Since
RSLP averages across estimates from standard LP regressions, each regression
can in principle be corrected for finite-sample bias as proposed by Herbst and
Johannsen (2024), or estimated by generalized least squares as discussed by
Lusompa (2023).

We also note that while we show that our method can outperform factor models
in some empirical plausible settings, we do not argue that one has to choose
between either factor or random subspace methods. For instance, there is
nothing to stop one from using factors in conjunction with subspace methods.
One possibility is the application of subspace methods to a set of estimated
factors instead of the original set of controls. Therefore, one should not
necessarily view our work as a replacement for existing methods. Instead, we
are keen to stress the potential for future work to combine our insights with
existing developments in the LP literature in attempts to further improve the
properties of these LP estimators.

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## ONLINE APPENDIX

##  Appendix A Selecting and weighting random subspaces

This appendix discusses three different extensions to the RSLP method
discussed in Section 2.1: (1) the random subset selection matrices take
predictor categories into account; (2) the subspaces are weighted by Bayesian
information criterions (BICs); and (3) the subspace dimension is selected
using BIC.

###  A.1 Randomly selecting subspaces using predictor categories

Most macroeconomic applications include datasets in which the variables can be
categorized. For instance, in the FRED-MD database, the variables have clear
categories, e.g., prices, employment, production, etc. In cases that the
researcher has prior knowledge that these categories –or a subset of them– are
essential to the LP, every random subset selection matrix should at least
include one predictor from each category. Prior knowledge on which categories
are essential may be more common than prior knowledge on which specific
predictors are essential. In the latter case the essential predictors should
be included in $V_{t}$.

Suppose the vector of possibly relevant controls $G_{t}$ is characterized by
$C$ mutually exclusive categories. Without loss of generality, $G_{t}$ can now
be written as
$G_{t}=(G_{1t}^{\prime},G_{2t}^{\prime},\dots,G_{Ct}^{\prime})^{\prime}$, with
$p_{G,c}\times 1$ vectors $G_{ct}$ for each category $c=1,\dots,C$. To ensure
at least one predictor is selected per category by each random subset
selection matrix, the matrix $R^{(j)}$ is constructed as follows. First, draw
a $k_{c}\times p_{G,c}$ subset selection matrix $R_{c}^{(j)}$ for each
category $c$. Second, construct the $k\times p_{G}$ matrix
$R^{(j)}=\text{blockdiag}(R_{1}^{(j)},R_{2}^{(j)},\dots,R_{C}^{(j)})$, where
$\text{blockdiag}(A,B)$ is a block diagonal matrix with the matrices $A$ and
$B$ on its diagonal. The subspace dimension for category $c$ is $1\leq
k_{c}\leq p_{G,c}$ and $k=\sum_{c=1}^{C}k_{c}$ with $C\leq k_{c}\leq p_{G}$.
We set $k_{c}$ to the nearest integer such that it reflects the distribution
of number of predictors across the categories, and set $k=50$. Alternatively,
categories which are a priori considered to be more important can be assigned
a relatively large $k_{c}$.

###  A.2 Weighting random subspaces using the BIC

Elliott, Gargano, and Timmermann (2013) and Elliott, Gargano, and Timmermann
(2015) have experimented with alternative weighting schemes to the equal
weights in (4) in a forecasting context. They show that weights proportional
to the exponential of the BIC value of a subspace do not result in substantial
improvements over equal weighting schemes. This result is consistent with
findings in the general forecast combinations literature, usually explained by
the additional variance introduced by estimating the weights. Due to the small
sample sizes in macroeconomic applications and the relatively large subspace
dimension of 50, estimating optimal weights in our setting is therefore also
not expected to work well.

Since most identification schemes require the LP to be estimated in two-
stages, as is also the case for our empirical applications, the computation of
the BIC is not as straightforward as in a predictive regression. Omitting
information required for identification in the first stage result in an
identification bias. Therefore we estimate the weights based on the BIC from
the first stage which does not depend on the horizon $h$.

Suppose the first stage equation corresponding to draw $R^{(j)}$ is

$\displaystyle{x}_{t}$
$\displaystyle=\alpha^{(j)}+\rho^{(j)}z_{t}+{\Lambda}^{(j)}{V}_{t}+{\Upsilon}^{(j)}{R}^{(j)}{G}_{t}+\eta_{t}^{(j)},$
(A1)

which is (10) in the main paper. The BIC corresponding to draw $R^{(j)}$ now
equals

$\displaystyle\text{BIC}_{j}=T\ln\left(\frac{1}{T}\left({x}_{t}-\left(\hat{\alpha}^{(j)}+\hat{\rho}^{(j)}z_{t}+\hat{\Lambda}^{(j)}{V}_{t}+\hat{\Upsilon}^{(j)}{R}^{(j)}{G}_{t}\right)\right)^{2}\right)+\ln(T)(2+p_{V}+k),$
(A2)

where $T$ is the number of observations. The weighted IRF is constructed as

$\displaystyle\hat{\beta}_{h}=\sum_{j=1}^{n_{R}}\frac{\exp(-\text{BIC}_{j})}{\sum_{j=1}^{n_{R}}\exp(-\text{BIC}_{j})}\hat{\beta}_{h}^{(j)}.$
(A3)

The weight of each subspace is proportional to the exponential of its BIC,
which results in larger weights for subspaces with a high likelihood value.

###  A.3 Selecting the subspace dimension by BIC

The accuracy of RSLP relies on an appropriate choice of subspace dimension.
Papers proposing random subspace methods in forecasting, have selected the
subspace dimension recursively based on past predictive performance (Elliott,
Gargano, and Timmermann, 2013; Boot and Nibbering, 2019). Since the IRFs are
unobserved, this approach does not work in our setting. Elliott, Gargano, and
Timmermann (2015) also select the subspace dimension recursively, but use the
BIC instead. In principle, the BIC can be computed for the RSLP, and
subsequently be used to select a subspace dimension as well. However, as
Elliott, Gargano, and Timmermann (2015) also note, selection based on the BIC
relies on estimates of the residual variance, and hence is not expected to
work well when the subspace dimension is large relative to the sample size.
Therefore, selection by BIC might not work well in most macroeconomic
applications of the RSLP.

Similar to weighting subspaces with BIC, we construct the BIC corresponding to
the first stage to minimise the risk that we omit variables key to the
identification of the IRF:

$\displaystyle\text{BIC}_{k}=T\ln\left(\frac{1}{T}\left({x}_{t}-\frac{1}{n_{R}}\sum_{j=1}^{n_{R}}\left(\hat{\alpha}^{(j)}+\hat{\rho}^{(j)}z_{t}+\hat{\Lambda}^{(j)}{V}_{t}+\hat{\Upsilon}^{(j)}{R}^{(j)}{G}_{t}\right)\right)^{2}\right)+\ln(T)(2+p_{V}+k).$
(A4)

The BIC is considered for a grid $k\in\\{0,10,20,30,40,50,60\\}$, and the $k$
corresponding to the smallest BIC is selected. Since the first stage usually
does not depend on $h$, the selected subspace dimension is the same across all
horizons.

###  A.4 Empirical results

We estimate the IRFs in the empirical applications using the three extensions
to the RSLP method, and compare the estimates to the estimated IRFs by RSLP.

First, consider the random selection of subspaces using predictor categories.
The technology shock application has 127 controls in $G_{t}$, from which 117
are from FRED-QD, 6 from the total factor productivity series at Fernald’s
website, and 4 from the Michigan Survey consumer confidence indicators. These
127 controls can be categorized into 12 categories: NIPA (1), Industrial
Production (15), Employment and Unemployment(28), Housing (11), Prices (22),
Interest Rates (18), Money and Credit (10), Exchange Rates (4) Stock Market
(5), FRED’s Other Series (3), Total Factor Productivity (6), and Michigan’s
Consumer Index (4), where the number of controls in each category is included
in parentheses. The monetary policy shock application has 111 controls from
FRED-MD in $G_{t}$. These controls can be categorized into 8 categories:
Output and Income (14), Labor Market (30), Consumption and Orders (10), Orders
and Inventories (9), Money and Credit (13), Interest Rates and Exchange Rates
(19), Prices (12), and Stock Market (4). Figure A1 shows that the estimated
IRFs from randomly selecting subspaces using these categories and without the
categories are almost identical.

Second, Figure A1 shows the estimated IRFs from RSLP with equal weights and a
fixed subspace dimension of 50, together with RSLP with weights based on the
BIC and RSLP with a subspace dimension selected by BIC. The estimated IRFs
with equal weights and BIC weights are very similar in our empirical
applications. The estimated IRFs from RSLP with a subspace dimension selected
by BIC is either close to RSLP with a fixed subspace dimension of 50 or in
between this RSLP and the LP with a base set of controls, due to the BIC often
selecting small subspace dimensions.

Figure A1: Estimated IRFs by RSLP and its extensions in the empirical
applications

(a) Response to a technology shock

(b) Response to a monetary policy shock

_Note: This figure shows the estimated LP with only a base set of controls
(orange, dot-dashed), RSLP (blue, solid), RSLP with subspaces weighted by BIC
(blue, dotted), RSLP with subspace dimension selected by BIC (blue, dashed),
and RSLP using predictor categories (blue, dot-dashed). The upper panel
presents estimated IRFs of labor productivity and unemployment rate to a
technology shock normalized to increase labor productivity by 0.25 $\%$ on
impact. The lower panel presents estimated IRFs of CPI and IP to a monetary
policy shock increasing the one year bond rate by 100 basis points. The blue-
shaded areas indicate the 90% error bands of RSLP._

##  Appendix B Subspace dimension

###  B.1 Subspace dimension in the Monte Carlo experiments

We explore the size of the subspace dimension in our Monte Carlo experiments.
For both variables and the different identification schemes, we calculate the
root mean squared error (RMSE) of the RSLP across horizons 0-6 for each
subspace dimension. Figure A2 reports these RMSEs relative to the RMSE of the
LP with the base set of controls.

Figure A2: Subspace dimension RSLP in the Monte Carlo experiments

(a) Strictly exogenous instrument

(b) Conditionally exogenous instrument

(c) SVAR identification

_Note: This figure shows the root mean squared error of RSLP relative to LP
with a base set of controls for various subspace dimensions, in the strong
(blue, solid) and the weak case (blue, dotted) in ( 19), respectively. The
panels correspond to different identification schemes._

While we do not find a unique subspace dimension that minimizes the RMSEs in
each setting under consideration, we highlight two findings from this
exercise. First, varying the subspace dimension reveals a “U-shaped” pattern,
suggesting that when there are a few controls in each subspace regression,
additional controls improve the estimates. This is, however, a trade-off where
for some subspace dimensions the variance in the estimates exceeds the
reduction in bias. Second, we find that the minimum is often achieved with a
subspace dimension within the 40-60 range, and the subspace dimension that
minimizes the RMSEs is smaller in the strong cases. Intuitively, one needs
fewer variables in the subspace regressions to filter out the relevant signal
with stronger signals from the informational variables.

In general, these findings are consistent across horizons as well, except for
some smaller horizons for tax rate where larger subspace dimensions are
optimal. Since the optimal subspace dimension is not known a priori, and
Appendix A.3 discusses that selecting this dimension does not necessarily
improves the results, it seems reasonable to set $k=50$.

Although this is a simulated data example, we stress that the Monte Carlo
experiments are useful in designing empirical strategies on how to set the
subspace dimension for applied work. First, the suggestion from the Monte
Carlo is that for a dataset with a factor structure like the FRED-MD, a
subspace dimension in the range of 40 to 60 is probably a good starting point.
Moreover, Boot and Nibbering (2020) also find an optimal subspace dimension of
40-60 with the FRED-MD, albeit in a forecasting context. Second, given
datasets are different, exploring some form of robustness of the subspace
dimension is probably warranted for applied work. Hence, we proceed with an
empirical strategy that uses a subspace dimension in the 40-60 range, and
subsequently check for the robustness of the results.

###  B.2 Subspace dimension in the empirical applications

In the two empirical applications, we set the subspace dimension equal to 50
as follows from the discussion above. However, it remains an open question
whether this choice of subspace dimension is appropriate in our applications.
To investigate this choice, Figure A3 presents the estimated IRFs by RSLP for
our applications, but now with varying subspace dimensions. The blue solid-
line corresponds to a subspace dimension of 50, which is displayed alongside
the estimates corresponding to the subspace dimensions of 40 and 60. We find
that these estimated IRFs are very similar to that considered in our baseline.
Moreover, they are within the 90% error bands of the baseline choice of 50.
This indicates that our choice of subspace dimension is appropriate.

Figure A3: Subspace dimension RSLP in the empirical applications

(a) Response to a technology shock

(b) Response to a monetary policy shock

_Note: The upper panel presents the estimated IRFs of labor productivity and
unemployment rate to a technology shock normalized to increase labor
productivity by 0.25 $\%$ on impact. The lower panel presents estimated IRFs
of CPI and IP to a monetary policy shock increasing the one year bond rate by
100 basis points. The blue-shaded areas indicate the 90% error bands of RSLP
with a subspace dimension of 50._

##  Appendix C Standard deviation of impulse response functions

###  C.1 Bootstrapping

To construct error bands via bootstrapping, we first recognize that the
estimated IRF $\hat{\beta}_{h}$ is the average of estimated IRFs
$\hat{\beta}_{h}^{(j)}$ from $n_{R}$ different regressions. It follows that
one can construct the standard error of $\hat{\beta}_{h}$ as long as one has
the variances and covariances associated with each of the
$\hat{\beta}_{h}^{(j)}$’s.

One natural approach is to construct the covariances between the
$\hat{\beta}_{h}^{(j)}$’s via bootstrapping. When designing the bootstrap, one
should allow for possible serial correlation up to an order of $h$ in the
residuals, as commonly done when one estimates LPs, and also allow for
possible correlation across the $\hat{\beta}_{h}^{(j)}$’s. We use a moving
block bootstrap of block size $h$ in order to allow for serial correlation up
to an order of $h$. To allow for cross correlation, we draw the same
corresponding block across the $n_{R}$ regressions to preserve any possible
cross correlation.

From a given horizon, $h\in\\{0,1,2,\ldots,H\\}$, with a sample of $T$
observations to estimate the IRF, we first define $T-h$ moving blocks of
residuals $\hat{\xi}_{t+h}^{(j)}$, where $j$ denotes the index of the
$j^{\text{th}}$ draw of the random subspace algorithm. We sample with
replacement $(T/h)+1$ blocks from the $T-h$ moving blocks. We only draw this
once for each bootstrap replication (as opposed to $n_{R}$ times), then use
the same corresponding blocks across all the $n_{R}$ regressions for the same
bootstrap replication. Using these blocks, we obtain bootstrapped samples for
each of the $\hat{\beta}_{h}^{j}$. These bootstrapped samples are used to form
an $n_{R}\times n_{R}$ covariance matrix for all $\hat{\beta}_{h}^{j}$’s, and
use these quantities to estimate the standard error of $\hat{\beta}_{h}$.
Using the critical values of a standard normal distribution, we construct an
approximate 90% interval.

###  C.2 Analytical Expression

Recall that our impulse response estimates are constructed as

$\displaystyle\hat{\beta}_{h}=\frac{1}{n_{R}}\sum_{j=1}^{n_{R}}\hat{\beta}_{h}^{(j)}.$
(A5)

Buckland, Burnham, and Augustin (1997) derive an expression for the variance
of $\hat{\beta}_{h}$ under two assumptions. First, assume that the expectation
across all possible draws $R^{(j)}$ of the misspecification bias in estimating
$\beta_{h}^{(j)}$ in the model corresponding to $R^{(j)}$ is zero. That is,
$\text{E}[\beta_{h}^{(j)}]=\beta_{h}$. Second, assume that $\beta_{h}^{(j)}$
and $\beta_{h}^{(l)}$ are perfectly correlated, for all $l\neq j$. Although
this is a strong assumption, each $\beta_{h}^{(j)}$ is estimated on the same
underlying dataset and these correlations are therefore indeed expected to be
high.

It now follows that

$\displaystyle\text{SD}[\hat{\beta}_{h}]=\frac{1}{n_{R}}\sum_{j=1}^{n_{R}}\sqrt{\text{var}(\hat{\beta}_{h}^{(j)}|\text{model
$(j)$ is correct})+({\beta}_{h}^{(j)}-{\bar{\beta}}_{h})^{2}},$ (A6)

which may be estimated by replacing $({\beta}_{h}^{(j)}-\bar{{\beta}}_{h})$ by
$(\hat{\beta}_{h}^{(j)}-\hat{{\beta}}_{h})$, and
$\text{var}(\hat{\beta}_{h}^{(j)}|\text{model $(j)$ is correct})$ by the
squared Newey-West standard error for the ordinary least squares estimate
$\hat{\beta}_{h}^{(j)}$ in model $(j)$ to account for the serial correlation
in the error terms (see Jordà, 2005). As discussed in the text, it is likely
that the assumption of perfect correlation is too strong in practice, and so
the bootstrap may be preferred.

##  Appendix D Implementing SVAR identification in RSLP

###  D.1 Monte Carlo experiments

Equation (14) shows that all the variation in the tax rate is from the tax
shock (and not the technology shock). Hence, the tax shock can be identified
by assuming that the cumulative impact of the tax rate on a technology shock
is zero. In practice, this approach corresponds to how the long-run
restrictions such as by Blanchard and Quah (1989) are implemented since they
are based on restrictions on the cumulative effect of shocks.

However, there is an important difference between the SVAR identification we
implement here and how long-run restrictions are usually implemented. In long-
run restrictions, one has an I(1) variable in which all the variation is
driven by the permanent shock. Since the I(1) variable enters the VAR or the
LP in differenced form, this implies a restriction where the cumulative impact
on the changes from the transitory shocks to the I(1) variable is equivalent
to zero, and thus has zero long-run impact on the level. In contrast, there is
no long-run in our setting since neither the tax rate nor capital is I(1), but
we can still identify the technology shock since the long-run cumulative
impact of this shock on the tax rate (whether differenced or in levels) is
zero.

The estimation of the cumulative identification in an LP setting is shown by
Plagborg-Møller and Wolf (2021). This method constructs a new tax variable by
accumulating our tax rate variable in Equation (14):

$\displaystyle\hat{\tau}^{\text{accum}}_{1}=\hat{\tau}_{1},\quad\quad\hat{\tau}^{\text{accum}}_{t}=\hat{\tau}_{t}+\hat{\tau}^{\text{accum}}_{t-1},\quad\text{for}\;t=2,3,...T.$
(A7)

Define
$\upsilon_{t}=\hat{\tau}^{\text{accum}}_{t+2}-\hat{\tau}^{\text{accum}}_{t-1}$.
The IRFs can now be estimated by two-stage least squares in the first and
second stage regressions

$\displaystyle\upsilon_{t}$
$\displaystyle=\alpha^{(j)}+{\Lambda}^{(j)}{V}^{1}_{t}+{\Upsilon}^{(j)}{R}^{(j)}{G}^{1}_{t}+\eta_{t}^{(j)},$
(A8) $\displaystyle y_{t+h}$
$\displaystyle=\mu_{h}^{(j)}+\beta_{h}^{(j)}\hat{\upsilon}_{t}^{(j)}+{\Theta}_{h}^{(j)}{V}^{2}_{t}+{\Psi}_{h}^{(j)}{R}^{(j)}{G}^{2}_{t}+\xi_{t+h}^{(j)},$
(A9)

where ${V}^{1}_{t}$ contains the contemporaneous values of the tax rate and
capital, ${G}^{1}_{t}$ contains contemporaneous values of the informational
series, ${V}^{2}_{t}$ consists of the two lags of tax rate and capital, and
${G}^{2}_{t}$ consists of the first lag of the informational series. The first
stage uncovers the linear combinations of the data that explain the two-period
ahead movement of the tax rate. Therefore the fitted values of the first stage
recover the tax shock. Once the tax shock is recovered, one can estimate the
impulse responses from the second stage.

###  D.2 Empirical application with a technology shock

The procedure outlined above also applies to our empirical application to a
technology shock, with minor adjustments. First, $\upsilon_{t}$ now equals the
long-run movement of labor productivity in levels:
$\upsilon_{t}=\text{labor}^{\text{level}}_{t+12}-\text{labor}^{\text{level}}_{t-1}$.
Second, the controls in both stages are now as follows: The first stage
controls in ${V}^{1}_{t}$ are the first-differences of contemporaneous values
of labor productivity and the unemployment rate, and ${G}^{1}_{t}$ contains
contemporaneous values of 127 FRED-QD series. The second stage controls in
${V}^{2}_{t}$ are the first four lags of the first-differences of labor
productivity and the unemployment rate, and ${G}^{2}_{t}$ consists of the
first lag of the 127 FRED-QD series.

##  Appendix E FRED data structure

Figure A4 shows the factor structure of the FRED-QD and FRED-MD datasets,
together with the factor structure of the simulated informational series in
(18) in Section 4. More precisely, we present the cumulative variance being
accounted for as we sequentially add an additional principal component. This
is a standard metric for understanding the strength of the factor structure.
The figure shows that the first two factors of FRED-QD and FRED-MD account for
25$\%$ and 20$\%$ of the variation, respectively. Similarly, our simulation
data under the weak case also requires two factors to account for 20$\%$ of
the variation in the controls.

Figure A4: Factor structure

_Note: This figure shows the factor structure of FRED-QD, FRED-MD, and our
simulated informational series in ( 18) in the strong and weak case._

Note that in Figure A4, we calculate the factor structure of the simulated
data in population. Let $n_{y^{*}}=100$ be the number of informational series,
which have mean zero and unit variance. Given the setting in Section 4.1, the
expected number of informational series containing the tax shock equals 10,
and the expected number of informational series containing the capital shock
equals 90. The population covariance matrix of the informational series is
therefore equal to $\Sigma_{y^{*}}=H+\sigma^{2}I_{n_{y^{*}}}$, where $H$ is a
$100\times 100$ blockmatrix with a $10\times 10$ and $90\times 90$ blockmatrix
with all elements equal to 1 on its diagonal, $\sigma^{2}=0.25$ in the strong
case and $\sigma^{2}=4$ in the weak case, and $I_{n_{y^{*}}}$ is a $100\times
100$ identity matrix.

Then the factor structure of the dataset can be calculated by solving the
eigenvalue-eigenvector decomposition of the correlation matrix $\rho_{y^{*}}$
of the covariance matrix $\Sigma_{y^{*}}$: $\rho_{y^{*}}\nu=\Lambda\nu$, where
$\nu$ is a matrix whose columns are the eigenvectors and $\Lambda$ is a
diagonal matrix with the eigenvalues $\lambda_{j}$ on its diagonal. The
cumulative variance explained by the first $i$ principal components can be
calculated as $\frac{1}{100}\sum_{j=1}^{i}\lambda_{j}$, where $\lambda_{j}$ is
the $j^{th}$ largest eigenvalue of $\lambda$.

##  Appendix F Benchmark methods in the Monte Carlo experiments

All LP methods estimate (2) with two lags of tax rate and capital in
${V}_{t}$, together with two lags of the instrument when using IV
identification. FALP defines $G_{t}$ as the first lag of the two principal
components that explain the most variation in $\\{y^{*}_{s}\\}_{s=1}^{t}$.

The identification schemes in the experiments require the LP to be estimated
in two-stages. Since omitting information required for identification in the
first stage results in an identification bias, we apply variable selection
methods to the first stage. We use both the LASSO as implemented by Friedman,
Tibshirani, and Hastie (2010) and OCMT by Chudik, Kapetanios, and Pesaran
(2018) to find a subset of $\\{y^{*}_{s}\\}_{s=1}^{t}$ to be included in
$G_{t}$. Both the first and second stage regressions are subsequently
estimated with the selected variables in $G_{t}$.

Conditional on the instrument and $V_{t}$, LASSO includes $y^{*}_{t}$ in
$G_{t}$ in the first stage and adds a LASSO penalty to the corresponding
coefficients. All variables with a nonzero coefficient estimate are selected.
The penalty parameter is selected from the grid $\exp(-j/2)$ with
$j=0,1,\dots,20$, by minimizing the one-observation-ahead out-of-sample errors
of the final hundred observations. To construct each error, the model is
estimated using an expanding estimation window. OCMT takes $y^{*}_{t}$ as the
set of potential signals for the first stage, and includes the instrument and
$V_{t}$ as preselected variables. We take the default settings $p=0.05$,
$c=1$, $\delta=1$, $\delta^{*}=2$.

The VAR methods include tax rate and capital, the instrument is accounted for
by the external-IV method (see Mertens and Ravn, 2013; Gertler and Karadi,
2015), and two lags are considered. The BVAR is implemented according to
Giannone, Lenza, and Primiceri (2015) and the BLP as proposed in Ferreira,
Miranda-Agrippino, and Ricco (forthcoming). The BVAR uses a natural conjugate
prior with a standard Minnesota type structure, with each VAR equation shrunk
towards a random walk (see Bańbura, Giannone, and Reichlin, 2010). The degree
of shrinkage is then specified as a single hyperparameter, where the Giannone,
Lenza, and Primiceri (2015) approach calibrates this hyperparameter by
maximizing the marginal likelihood of the model. The BLP approach takes the
posterior mode of the BVAR coefficients estimated using the Giannone, Lenza,
and Primiceri (2015) approach as given, and iterates on these coefficients in
order to specify a prior on the coefficients of the LP regression using a
natural conjugate prior. The degree of shrinkage towards the BVAR coefficients
is then estimated by maximizing the marginal likelihood. We shrink the BLP
towards the small BVAR in our example as the small BVAR would be considered a
“standard” specification in each application. We consider two BVARs. The small
BVAR includes only the essential variables (i.e. tax rate and capital),
whereas the medium BVAR includes 20 variables, by adding 18 information series
from $y^{*}_{s}$ in addition to the 2 essential variables. The FAVAR includes
the two principal components that explain the most variation in
$\\{y^{*}_{s}\\}_{s=1}^{t}$ in addition to the essential variables and is
estimated as in Bernanke, Boivin, and Eliasz (2005).

##  Appendix G Dynamic factor model experiments

This appendix conducts additional Monte Carlo experiments based on a the
dynamic factor model (DFM) discussed by Stock and Watson (2016). This model
assumes the following process for the $n_{X}\times 1$ vector of macroeconomic
variables $X_{t}$:

$\displaystyle X_{t}=\Lambda f_{t}+v_{t},\quad
v_{i,t}=\Delta_{i}(L)v_{i,t-1}+\Xi_{i}\xi_{i,t},\quad\xi_{i,t}\sim
iid\;N(0,1),$ (A10)

where the $n_{f}\times 1$ vector of latent factors follows a stationary VAR
model

$\displaystyle f_{t}=\Phi(L)f_{t-1}+\vartheta_{t},\quad\vartheta_{t}\sim
N(0,\Sigma_{\vartheta}).$ (A11)

Following Stock and Watson (2012), we set $n_{f}=6$ and the number of lags in
both (A10) and (A11) equal to 4. The parameters in $\Lambda$, $\Delta_{i}(L)$,
$\Xi_{i}$, $\Phi(L)$, and $\Sigma_{\vartheta}$ are set to the values estimated
on the Stock and Watson (2016) dataset consisting of 222 quarterly
observations from 1959Q3 to 2014Q4 on $n_{X}=207$ macroeconomic
series.202020We follow Stock and Watson (2012, 2016) with all the data
handling steps and transformations, including the adjustments of outliers, the
treatment of missing observations, and detrending using a biweight filter with
a bandwidth of 100 quarters. We empirically calibrate the monetary policy
shock, $\epsilon_{t}^{\text{MP}}$ by estimating a regression of the monetary
policy shock instrument by Romer and Romer (2004) on each of the empirically
estimated residuals $\hat{\vartheta_{t}}$.

###  G.1 Information sets and identification schemes

The objective is to estimate the IRFs to a monetary policy shock. We consider
two experiments. First, the econometrician observes a strictly exogenous
instrument. Second, the econometrician observes a conditionally exogenous
instrument. In both experiments, we also assume that 105 of the 207
macroeconomic series in $X_{t}$ are observed, mimicking the size of the
information set when working with datasets like FRED-QD or FRED-MD. These 105
series are selected so that all thirteen variable groups in the dataset are
represented.

##### Strictly exogenous instrument

First, the instrument $z_{t}$ for the monetary policy shock
$\epsilon^{\text{MP}}_{t}$ is generated as

$\displaystyle
z_{t}=\sqrt{0.5}\epsilon^{\text{MP}}_{t}+\nu_{t},\quad\nu_{t}\sim N(0,1).$
(A12)

The instrument identifies the IRF without controls. However, being able to
account for variation in the variable of interest due to movements in the
other observed variables from $X_{t}$ can lead to efficiency gains. Note that
this is a slightly different setting compared to the strictly exogenous
instrument experiment in Section 4. In that experiment, the controls address
the variation in the instrument $z_{t}$ in the first stage, where the controls
in the experiment here target variation in the variable of interest $y_{t}$ in
the second stage.

##### Conditionally exogenous instrument

Second, the instrument $z_{t}$ for the monetary policy shock
$\epsilon^{\text{MP}}_{t}$ is generated as

$\displaystyle
z_{t}=\epsilon^{\text{MP}}_{t}+\epsilon^{\text{MP}}_{t-1}+\eta^{\text{Tax}}_{t-1}+0.9\eta^{\text{Tech}}_{t-1}+\nu_{t},\quad\nu_{t}\sim
N(0,1),$ (A13)

where $\eta^{\text{Tax}}_{t-1}$ and $\eta^{\text{Tech}}_{t-1}$ are
respectively tax and technology components of the reduced form forecast
errors, $\vartheta_{t-1}$. The crucial point here is because these components
are a function of lagged shocks, the instrument is now only conditionally
exogenous since it fails lead-lag exogeneity. Using the instrument without
controlling for the information in $\eta^{\text{Tax}}_{t-1}$ and
$\eta^{\text{Tech}}_{t-1}$ will thus render the instrument invalid. Since the
observed variables from $X_{t}$ are a function of the lagged shocks, they may
help recover the IRFs by making the instrument valid conditionally, as long as
they control for the information contained in the other shocks in
$\eta^{\text{Tax}}_{t-1}$ and $\eta^{\text{Tech}}_{t-1}$.

We empirically calibrate $\eta^{\text{Tax}}_{t}$ and $\eta^{\text{Tech}}_{t}$
using the tax instrument by Romer and Romer (2010) and the technology
instrument by Fernald (2014), respectively. We first obtain the coefficients,
$h^{\text{Tax}}$, from a regression of the tax instrument by Romer and Romer
(2010) on the residuals of the monetary policy shock regression which we used
to obtain $\epsilon_{t}^{\text{MP}}$, and subsequently get the coefficients
$h^{\text{Tech}}$ by estimating a regression of the technology instrument by
Fernald (2014) on the residuals from the tax instrument regression. We then
form $\eta^{\text{Tax}}_{t}$ and $\eta^{\text{Tech}}_{t}$ as linear
combinations of the forecast errors where each of the forecast errors loads on
each component. We construct the unnormalized loadings by taking the inverse
of the coefficients $h^{\text{Tax}}$ and $h^{\text{Tech}}$. We then normalize
by ensuring that the loadings of each component sum to unity. Note that
$\eta^{\text{Tax}}_{t}$ and $\eta^{\text{Tech}}_{t}$ are linear combinations
of the forecast errors which are functions of other shocks in the model. Hence
the instrument fails lead-lag exogeneity without any controls, whether
$\eta^{\text{Tax}}_{t}$ and $\eta^{\text{Tech}}_{t}$ are viewed as structural
shocks or not.212121The only structural shock which we interpret in the model
is the monetary policy shock $\epsilon^{\text{MP}}_{t}$, and deliberately
labelled $\eta^{\text{Tax}}_{t}$ and $\eta^{\text{Tech}}_{t}$ as components to
make the distinction clear. This simplifies the analysis, as the samples of
the different instruments are not aligned and instruments are also not
orthogonal in practice (see Stock and Watson, 2012, for documentation of and
discussion of the latter).

###  G.2 Methods

Each experiment simulates 1000 artificial datasets of 200 observations
according to (A10) and (A11), from which only 105 variables are retained. Each
artificial dataset is accompanied by an instrument generated from either (A12)
or (A13). Since the Monte Carlo experiments are parametrized by empirically
estimating the model on data, the artificial data generated from the Monte
Carlo thus represent particular variables of interest as they will mimic time
series properties of these variables. We therefore refer to these artificial
data according to the variables that they represent in the dataset which we
used to parameterize the DFM.

LP with a base set of controls includes four lags of the Federal funds rate,
industrial production, consumer price index, unemployment rate, and excess
bond premium in $V_{t}$, together with four lags of the instrument. RSLP uses
the same specification for $V_{t}$, and includes the first lag of the
remaining 100 observed macroeconomic series in $G_{t}$.

The benchmark methods are implemented as outlined in Appendix F, where the
macroeconomic time series included in $G_{t}$ for RSLP are considered as
information series $\\{y^{*}_{s}\\}_{s=1}^{t}$. We include six principal
components in FALP and FAVAR, which is the correct number of factors in the
data generating process. The small BVAR includes the same variables as in
$V_{t}$ and considers four lags. The medium BVAR includes 20 variables, and
hence we select in addition to $V_{t}$ the following 15 variables: (1) Real
Gross Domestic Product; (2) Real Personal Consumption Expenditures, (3)
Industrial Production: Final Products (Market Group), (4) IP: Consumer goods,
(5) Total Nonfarm Payrolls: All Employees, (6) Housing Starts: Total: New
Privately Owned Housing Units Started, (7) Mfrs new orders durable goods
industries, (8) Gross Domestic Product: Chain-type Price Index, (9) Average
Hourly Earnings: Total Private Industries Defl by PCE(LFE) Def, (10) 3-Month
Treasury Bill: Secondary Market Rate, (11) St. Louis Adjusted Monetary Base,
(12) S$\&$P 500, (13) Common Stock Prices: Dow Jones Industrial Average, (14)
Consumer expectations NSA, and (15) World Oil Production.

###  G.3 Results

Table A1 shows the RMSE of the IRFs estimated by each method relative to the
RMSE of the IRFs estimated by RSLP. Values above one favour RSLP. These values
are reported for the experiments with the strictly and conditionally exogenous
instruments. Section 4.3 discusses these results in more detail.

Table A1: RMSE relative to RSLP in the DFM Monte Carlo experiments

| Strict Instrument | Conditional Instrument
---|---|---
| Fed | IP | UNE | CPI | EBP | Fed | IP | UNE | CPI | EBP
| LP methods
FALP | 1.079 | 1.049 | 1.091 | 0.960 | 1.241 | 1.850 | 1.534 | 1.539 | 0.962 | 1.229
LASSO | 1.360 | 1.248 | 1.307 | 1.109 | 1.522 | 2.509 | 2.685 | 2.937 | 1.756 | 1.863
OCMT | 3.771 | 3.268 | 3.424 | 2.641 | 4.429 | 3.327 | 3.700 | 4.359 | 3.018 | 2.504
BLP | 0.453 | 0.753 | 0.915 | 0.436 | 0.569 | 0.870 | 1.689 | 2.007 | 0.587 | 0.976
Base | 3.770 | 3.268 | 3.424 | 2.641 | 4.429 | 3.452 | 3.909 | 4.580 | 3.134 | 2.589
| VAR methods
FAVAR | 0.501 | 0.653 | 0.686 | 0.499 | 0.546 | 0.464 | 0.608 | 0.625 | 0.485 | 0.498
Small | 0.458 | 0.781 | 0.949 | 0.429 | 0.578 | 0.872 | 1.700 | 2.023 | 0.574 | 0.978
Medium | 0.484 | 0.738 | 0.831 | 0.518 | 0.480 | 0.445 | 0.773 | 0.841 | 0.501 | 0.517

_Note: This table reports the root mean squared error (RMSE) relative to RSLP
for the DFM experiments with a strictly and conditionally exogenous instrument
(( A12) and (A13), respectively). See text and note to Table 1 for further
details of models estimated. The DGP is parameterized by estimating the DFM on
U.S. macroeconomic data. The relative RMSEs are reported for the following,
representing variables which we generated from the DGP: Fed - Federal funds
rate, IP - Industrial production, Unemp - Unemployment, CPI - CPI inflation,
EBP - excess bond premium as per Gilchrist and Zakrajšek (2012)._

##  Appendix H Additional results baseline Monte Carlo experiments

Table A2: RMSE of FAVAR relative to RSLP in the baseline Monte Carlo
experiments

|  | Strict Instrument | Conditional Instrument | SVAR
---|---|---|---|---
|  | Strong | Weak | Strong | Weak | Strong | Weak
h | method | Tax | Cap | Tax | Cap | Tax | Cap | Tax | Cap | Tax | Cap | Tax | Cap
0 | Base | 13.919 | 16.408 | 12.175 | 14.143 | 38.399 | 3.432 | 5.509 | 2.757 | 130.860 | 2.961 | 11.985 | 2.248
0 | FALP | 0.956 | 0.952 | 1.048 | 1.048 | 1.794 | 0.961 | 1.534 | 1.104 | 5.578 | 1.054 | 2.998 | 1.282
0 | FAVAR | 8.469 | 9.182 | 7.384 | 7.629 | 1.048 | 0.983 | 1.875 | 2.507 | 2.509 | 1.811 | 1.959 | 4.649
1 | Base | 13.764 | 12.699 | 11.641 | 10.796 | 22.921 | 1.902 | 4.644 | 1.673 | 4.525 | 1.822 | 0.703 | 1.641
1 | FALP | 0.973 | 0.942 | 1.055 | 1.035 | 1.018 | 0.968 | 1.315 | 1.048 | 0.647 | 0.979 | 0.716 | 1.082
1 | FAVAR | 2.752 | 2.187 | 2.303 | 1.826 | 0.665 | 0.941 | 2.944 | 1.473 | 1.164 | 0.956 | 7.302 | 1.865
2 | Base |  | 24.520 |  | 20.548 |  | 1.608 |  | 1.455 |  | 1.160 |  | 1.082
2 | FALP |  | 0.940 |  | 1.034 |  | 0.972 |  | 1.050 |  | 0.933 |  | 1.073
2 | FAVAR |  | 1.314 |  | 1.141 |  | 0.920 |  | 1.124 |  | 0.912 |  | 1.078
3 | Base | 6.298 | 16.246 | 5.452 | 13.893 | 1.498 | 1.476 | 1.391 | 1.350 | 0.946 | 0.967 | 0.887 | 0.890
3 | FALP | 0.944 | 0.968 | 1.047 | 1.048 | 0.955 | 0.943 | 1.013 | 0.992 | 0.935 | 0.923 | 1.126 | 1.009
3 | FAVAR | 0.527 | 0.670 | 0.448 | 0.577 | 0.904 | 0.898 | 1.124 | 0.968 | 0.907 | 0.878 | 1.056 | 0.984
4 | Base | 10.000 | 15.213 | 8.536 | 13.057 | 1.500 | 1.492 | 1.415 | 1.367 | 0.883 | 0.922 | 0.830 | 0.849
4 | FALP | 0.955 | 0.942 | 1.039 | 1.026 | 0.950 | 0.951 | 1.017 | 1.009 | 0.934 | 0.934 | 1.068 | 1.059
4 | FAVAR | 0.291 | 0.372 | 0.238 | 0.308 | 0.900 | 0.874 | 0.951 | 0.899 | 0.901 | 0.862 | 0.965 | 0.874
5 | Base | 4.982 | 12.747 | 4.350 | 11.135 | 1.442 | 1.505 | 1.357 | 1.364 | 0.871 | 0.934 | 0.813 | 0.865
5 | FALP | 0.956 | 0.947 | 1.052 | 1.048 | 0.954 | 0.936 | 1.021 | 0.988 | 0.923 | 0.917 | 1.045 | 1.029
5 | FAVAR | 0.123 | 0.184 | 0.103 | 0.157 | 0.874 | 0.400 | 0.841 | 0.417 | 0.883 | 0.400 | 0.823 | 0.417
6 | Base | 6.114 | 22.655 | 5.302 | 19.762 | 1.440 | 1.490 | 1.361 | 1.363 | 0.968 | 0.927 | 0.905 | 0.858
6 | FALP | 0.948 | 0.949 | 1.038 | 1.036 | 0.934 | 0.950 | 0.999 | 1.027 | 0.956 | 0.915 | 1.111 | 1.010
6 | FAVAR | 0.072 | 0.099 | 0.059 | 0.085 | 0.852 | 0.197 | 0.750 | 0.208 | 0.860 | 0.199 | 0.756 | 0.211

_Note: This table reports the root mean squared error (RMSE) of FAVAR relative
to RSLP for the IV and SVAR identification under both the strong and weak
factor structure settings. Strict and conditional instruments refer to
instruments generated by ( 15) and (17), respectively. Since the IRF of the
tax rate is scaled to a unit tax shock at horizon $h=2$, the corresponding
RMSEs are zero and the relative RMSE is undefined. _

##  Appendix I Implementation details empirical applications

The methods are implemented as outlined in Appendix F, where the macroeconomic
time series included in $G_{t}$ for RSLP are considered as information series
$\\{y^{*}_{s}\\}_{s=1}^{t}$. We include three principal components in FALP and
FAVAR in the technology shock application and four in the monetary policy
shock application, as suggested by the Granger causality test with the
variables in $V_{t}$ proposed by Forni and Gambetti (2014). The BVAR includes
the same variables as in $V_{t}$ and considers four lags in the technology
shock application and twelve in the monetary policy shock application.

The medium BVAR includes in total 20 variables, which means that in addition
to $V_{t}$ another 18 variables in the technology shock application and 16 in
the monetary policy shock application have to be selected. For the technology
shock application, the following variables are selected from FRED-QD (FRED
mnemonics in parentheses): (1) Consumer Price Index for All Urban Consumers:
All Items (CPI), (2) Industrial Production Index (IP), (3) Effective Federal
Funds Rate (FEDFUNDS), (4) Help-Wanted Index (HWIx), (5) Producer Price Index:
Commodities: Metals and metal products (PPICMM), (6) S$\&$P’s Common Stock
Price Index: Composite (S$\&$P 500), (7) S$\&$P’s Common Stock Price Index:
Industrials (S$\&$P:indust), (8) Housing Starts: Total (HOUST), (9) New
Private Housing Units Authorized by Building Permits (PERMIT), (10) 1-Year
Treasury Constant Maturity Rate (GS1), (11) Real M1 Money Stock (M1REAL), (12)
Real M2 Money Stock (M2REAL), (13) CP/GDPDEF, (14) GDPDEF, (15) dtfp, (16)
dtfp-util, (17) Business Condition next year, (18) Expected Index.

For the monetary policy shock application, the following variables are
selected from FRED-MD: (1) Civilian Unemployment Rate (UNRATE), (2) IP: Final
Products (Market Group) (IPFINAL), (3) IP: Consumer Goods (IPCONDGD), (4)
Initial Claims (CLAIMSx), (5) All Employees: Total nonfarm (PAYEMS), (6) Total
Business Inventories (BUSINVx), (7) Total Business: Inventories to Sales Ratio
(ISRATIOx), (8) CPI : Apparel (CPIAPPSL), (9) Avg Hourly Earnings:
Construction (CES2000000008), (10) Consumer Sentiment Index (UMCSENTx), (11)
Housing Starts: Total New Privately Owned (HOUST), (12)New Private Housing
Permits (PERMIT), (13) Nonrevolving consumer credit to Personal Income
(CONSPI), (14) S$\&$P’s Common Stock Price Index: Composite (S$\&$P 500), (15)
3-Month Treasury Bill (TB3MS), (16) Japan / U.S. Foreign Exchange Rate
(EXJPUSx).