# Subspace Shrinkage in Conjugate Bayesian Vector Autoregressions††thanks:  We
would like to thank Michael Pfarrhofer and Niko Hauzenberger for constructive
comments and suggestions. Huber gratefully acknowledges financial support from
the Austrian Science Fund (FWF, grant no. ZK 35) and the Jubiläumsfonds of the
Oesterreichische Nationalbank (OeNB, grant no. 18304).

Florian Huber University of Salzburg Gary Koop University of Strathclyde

Abstract

Macroeconomists using large datasets often face the choice of working with
either a large Vector Autoregression (VAR) or a factor model. In this paper,
we develop methods for combining the two using a subspace shrinkage prior.
Subspace priors shrink towards a class of functions rather than directly
forcing the parameters of a model towards some pre-specified location. We
develop a conjugate VAR prior which shrinks towards the subspace which is
defined by a factor model. Our approach allows for estimating the strength of
the shrinkage as well as the number of factors. After establishing the
theoretical properties of our proposed prior, we carry out simulations and
apply it to US macroeconomic data. Using simulations we show that our
framework successfully detects the number of factors. In a forecasting
exercise involving a large macroeconomic data set we find that combining VARs
with factor models using our prior can lead to forecast improvements.

JEL Codes: C11, C32, C53

Keywords: Subspace shrinkage, reduced-rank regression, Bayesian VAR

##  1 Introduction

Macroeconomists are increasingly working with multivariate time series models
involving large numbers of variables. Traditionally dynamic factor models
(DFMs) or Factor Augmented Vector Autoregressions (FAVARs) have been used (see
e.g. Geweke (1977), Stock and Watson (2002), Bernanke, Boivin, and Eliasz
(2005) and many others). These typically involve the use of principal
components methods to extract the information in the large number of variables
into a small number of factors thus avoiding over-parameterization concerns.
Starting with Banbura, Giannone, and Reichlin (2010) many researchers have
been simply including all the variables in a Vector Autoregression (VAR) and
using Bayesian shrinkage priors to avoid over-fitting (see, among many others,
Koop (2013), Giannone, Lenza, and Primiceri (2015), Jarocinski and Mackowiak
(2017), Carriero, Clark, and Marcellino (2019), Koop and Korobilis (2019),
Korobilis and Pettenuzzo (2019), Giannone, Lenza, and Primiceri (2019), Huber
and Feldkircher (2019), Chan (2020) and Hauzenberger, Huber, and Onorante
(2021)).

How should the researcher decide whether to use a factor model or a large
Bayesian VAR? This question can be answered through a comparison of their
predictive performance in a pseudo out of sample forecasting exercise.
Alternatively, marginal likelihoods can be used. But pseudo out of sample
forecasting evaluation can be time consuming and marginal likelihoods can be
sensitive to the prior used. In this paper, we develop an alternative method
for choosing between factor models and large VARs.

But why is there a need to choose between them when something in between might
lead to better forecast performance? This is another question addressed in
this paper. We propose a model which shrinks the VAR coefficients towards the
implied coefficients of a factor model leading to a model which combines the
two. We do so using a subspace shrinkage prior, see Shin, Bhattacharya, and
Johnson (2020). A conventional prior shrinks the posterior of a coefficient
towards its prior mean, which is typically zero. In contrast a subspace
shrinkage prior is a prior on function spaces that shrinks towards a class of
functions. In the present paper, we choose the class of functions to be a
factor model such as the FAVAR or DFM. We stay in the class of conjugate
priors (although we will discuss how other VAR priors can be accommodated)
and, thus, our methods are simple to implement. They do not require the use of
computationally-demanding Markov Chain Monte Carlo (MCMC) methods, implying
that these techniques are useful in very high dimensional models. We develop a
method for estimating the weight put on the DFM restrictions and the number of
factors involved in these restrictions. The result is a model which combines
the large VAR with a DFM in an optimal way. Alternatively, output from our
model can be used to select between the large VAR and the DFM and, if the
latter is selected, determine the number of factors.

We consider two versions of our subspace VAR prior. First, the subspace prior
can be combined with a conventional informative VAR prior such as the popular
Minnesota prior (see Doan, Litterman, and Sims (1984), Litterman (1986),
Kadiyala and Karlsson (1997), Sims and Zha (1998) and Banbura, Giannone, and
Reichlin (2010) for a natural conjugate implementation). We demonstrate that
results from such a model can be interpreted as a weighted average of the
Minnesota prior VAR and the factor model. Second, the subspace prior can be
combined with a non-informative VAR prior. The result is a new Bayesian VAR
prior. In contrast to conventional priors which shrink towards plausible
values for the VAR coefficients, our new prior shrinks towards the factor
model.

Our approach is illustrated using synthetic as well as real data. In
simulations, we show that our framework accurately detects the number of
factors if the true number of factors is small. This finding is independent of
the model size. In larger dimensions, and for a larger number of true factors,
our model slightly underestimates the true number of factors. To investigate
the merits of our approach we apply it to US macroeconomic data. In a
forecasting exercise, the different priors which shrink the VAR towards a
factor model improve upon a standard BVAR and the DFM. These improvements are
pronounced during the global financial crisis and the Covid-19 pandemic.

The remainder of the paper is structured as follows. Section 2 introduces the
econometric framework. After providing a brief overview on conjugate Bayesian
VARs and DFMs in Sub-section 2.1, we discuss our subspace shrinkage prior
which can be used to force the coefficients of the VAR towards the
restrictions implied by a DFM in Sub-sections 2.2 and 2.3. Sub-section 2.4
discusses how our approach can be used to estimate the number of factors
alongside the remaining model parameters and provides a brief overview on
posterior simulation. Section 3 applies our techniques to a big US
macroeconomic dataset and illustrates its favorable forecasting properties.
Section 4 discusses how alternative Bayesian VAR priors and extensions such as
stochastic volatility can be incorporated in our techniques. The final section
summarizes and concludes the paper.

##  2 Subspace Shrinkage in VARs

###  2.1 Conjugate Bayesian VARs and Dynamic Factor Models

Let $\\{\bm{Y}_{t}\\}_{t=1}^{T}$ denote an $M$-dimensional vector of
macroeconomic and financial quantities. The number of time series can be large
and, in addition, display substantial co-movements. One popular approach of
modeling this panel of time series is to assume $\bm{Y}_{t}$ to follow a
VAR($p$) process:

$\bm{Y}_{t}=\bm{A}_{1}\bm{Y}_{t-1}+\dots+\bm{A}_{p}\bm{Y}_{t-p}+\bm{\varepsilon}_{t},$
(1)

where $\bm{A}_{j}~{}(j=1,\dots,p)$ is an $M\times M$-dimensional coefficient
matrix and $\bm{\varepsilon}_{t}$ is a zero mean Gaussian shock with variance-
covariance matrix $\bm{\Sigma}$.111For brevity we exclude deterministic terms.
In our empirical work we include an intercept. Equation (1) can be written as
a multivariate regression model as follows:

$\bm{Y}_{t}=\bm{A}^{\prime}\bm{X}_{t}+\bm{\varepsilon}_{t},$

where $\bm{A}=(\bm{A}_{1},\dots,\bm{A}_{p})^{\prime}$ and
$\bm{X}_{t}=(\bm{Y}^{\prime}_{t-1},\dots,\bm{Y}^{\prime}_{t-p})^{\prime}$
denote $K(=Mp)\times M$ and $K\times 1$ matrices, respectively. Stacking
$\bm{Y}_{t},\bm{X}_{t}$ and $\bm{\varepsilon}_{t}$ allows us to recast the
model in full-data form:

$\bm{Y}=\bm{X}\bm{A}+\bm{\varepsilon},$ (2)

with typical $t^{th}$ row of $\bm{Y}$ given by $\bm{Y}^{\prime}_{t}$, of
$\bm{X}$ given by $\bm{X}^{\prime}_{t}$ and of $\bm{\varepsilon}$ a typical
row is $\bm{\varepsilon}^{\prime}_{t}$.

Notice that the number of VAR coefficients in $\bm{a}=\text{vec}(\bm{A})$ is
$k=(KM)$, which sharply increases with the number of endogenous variables
and/or the number of lags. Since $T$ is moderate for typical macroeconomic
datasets, shrinkage is necessary to obtain well behaved estimates and to rule
out implausible regions of the parameter space (e.g. regions which would imply
explosive roots of the VAR process).

Bayesian priors on $\bm{a}$ are often used to provide such shrinkage. If $M$
is large, natural conjugate priors are popular since they allow for fast
computation. This arises because they preserve a convenient Kronecker
structure for the posterior covariance matrix of $\bm{a}$, see Chan (2020).
The conjugate prior on $\bm{a}$ is specified conditionally on $\bm{\Sigma}$
and takes a Gaussian form:

$\bm{a}|\bm{\Sigma}\sim\mathcal{N}(\text{vec}(\underline{\bm{A}}),\bm{\Sigma}\otimes\underline{\bm{V}}).$
(3)

Here, we let $\underline{\bm{A}}$ denote a prior mean matrix of dimension
$K\times M$ and $\underline{\bm{V}}$ is a $K\times K$ matrix. The full
conditional posterior distribution of $\bm{a}$ is also Gaussian with:

$\displaystyle\bm{a}$
$\displaystyle|\bm{\Sigma},\bm{Y}\sim\mathcal{N}(\text{vec}(\overline{\bm{A}}),\bm{\Sigma}\otimes\overline{\bm{V}}),$
$\displaystyle\overline{\bm{V}}$
$\displaystyle=(\bm{X}^{\prime}\bm{X}+\underline{\bm{V}}^{-1})^{-1},\quad\overline{\bm{A}}=\overline{\bm{V}}(\bm{X}^{\prime}\bm{Y}+\underline{\bm{V}}^{-1}\underline{\bm{A}}).$

The prior on $\bm{\Sigma}$ is inverted Wishart with prior degrees of freedom
$\underline{\nu}$ and scaling matrix $\underline{\bm{S}}$ which, when combined
with the likelihood, yields a marginal posterior which also follows an
inverted Wishart distribution whose posterior moments take a standard form
(see, e.g., chapter 21 of Chan, et al. (2019)).

A conventional Bayesian VAR prior such as the Minnesota prior would make
particular choices for $\underline{\bm{A}}$ and $\underline{\bm{V}}$. An
alternative would be to exploit the fact that the data might feature a factor
structure. That is, the information in $\bm{X}$ might be characterized by a
small number of $q~{}(q\ll K)$ latent factors. These can be estimated using
principal components (PCs) which can be implemented through a singular value
decomposition (SVD). The SVD allows to decompose
$\bm{X}=\bm{F}_{q}\bm{L}_{q}^{\prime}$ in terms of a $T\times q$ matrix
$\bm{F}_{q}$, which are the estimated factors, and a $K\times q$ matrix
$\bm{L}_{q}$, which is a matrix of factor loadings. If the matrix $\bm{X}$ is
of rank $q$, this equation is exact. In general, if the rank of $\bm{X}$
exceeds $q$, $\bm{F}_{q}\bm{L}_{q}^{\prime}$ approximates $\bm{X}$. Replacing
$\bm{X}$ with $\bm{F}_{q}\bm{L}_{q}^{\prime}$ in Eq. (missing)2 shows that the
corresponding matrix of regression coefficients
$\bm{B}=\bm{L}^{\prime}_{q}\bm{A}$ is of dimension $q\times M$, a substantial
reduction in the dimension of the state space. Using the Moore-Penrose inverse
of $\bm{L}_{q}$, $\bm{L}_{q}^{\dagger}$, allows us to express $\bm{A}$ in
terms of $\bm{B}$ and the estimated loadings:

$\bm{A}=(\bm{L}_{q}^{\dagger})\bm{B}.$

This equation enables us to think about a DFM in terms of an otherwise
unrestricted VAR with specific restrictions (which are driven by $\bm{L}_{q}$)
on the VAR coefficients. In a conventional DFM, these restrictions are always
dogmatically imposed. In this paper, our goal is to introduce a shrinkage
prior which softly pushes the elements in $\bm{A}$ towards the implied
restrictions of the PC regression model.

###  2.2 Shrinking the flat prior VAR towards a factor model

Shrinking the regression model towards a subspace spanned by, e.g., the
principal components can be done in several ways. For instance, Oman (1982)
show how shrinkage estimators can be used to force an unrestricted regression
model towards a projection on a subspace (such as the one spanned by the PCs)
as opposed to the origin. This approach uses the eigenvalues of
$\bm{X}^{\prime}\bm{X}$ to shrink coefficients towards the space spanned by
the first $q$ eigenvectors. Our approach is similar but relies on a modified
variant of the functional Horseshoe prior stipulated in Shin, Bhattacharya,
and Johnson (2020). This is achieved by setting
$\underline{\bm{A}}=\bm{0}_{K\times M}$ and $\underline{\bm{V}}$ as follows:

$\underline{\bm{V}}=\left(\frac{\omega}{1-\omega}\bm{X}^{\prime}(\bm{I}_{T}-\bm{\Phi}_{0})\bm{X}\right)^{-1}.$

Here, $\omega\in[0,1]$ is a tightness parameter and the $T\times T$ matrix
$\bm{\Phi}_{0}=\bm{F}_{q}(\bm{F}_{q}^{\prime}\bm{F}_{q})^{-1}\bm{F}_{q}^{\prime}$
is the projection of $\bm{F}_{q}$. Recall that we obtain $\bm{F}_{q}$ from the
SVD of $\bm{X}$.222Note that if we were to set $\bm{\Phi}_{0}=\bm{0}_{T\times
T}$ then the prior would reduce to a standard $g$-prior with hyperparameter
$\omega/(1-\omega)$. We let $\omega$ be an unknown parameter and estimate it
in a data-based fashion as described below. The posterior is given in the
preceding sub-section with these particular choices of $\underline{\bm{A}}$
and $\underline{\bm{V}}$ inserted.

To see how $\omega$ shrinks the VAR towards the factor model, it is convenient
to exploit the fact that if the rank of $\bm{X}$ is $K$, the matrix $\bm{X}$
and the matrix $\bm{F}_{K}$ (i.e., the first $K$ principal components of
$\bm{X}$) span the same column space $C$. Moreover, notice that
$\bm{F}_{K}=(\bm{F}_{q},\bm{F}_{(q+1):K})$ with $\bm{F}_{(q+1):K}$ storing the
final $K-q$ principal components of $\bm{X}$. Using these definitions and the
result that $C(\bm{X})=C(\bm{F}_{K})$, the corresponding projection matrices
coincide:

$\displaystyle\bm{X}\left(\bm{X}^{\prime}\bm{X}+\frac{\omega}{1-\omega}\bm{X}^{\prime}(\bm{I}_{T}-\bm{\Phi}_{0})\bm{X}\right)^{-1}\bm{X}^{\prime}$
(4)
$\displaystyle=\bm{F}_{K}\left(\bm{F}_{K}^{\prime}\bm{F}_{K}+\frac{\omega}{1-\omega}\bm{F}^{\prime}_{K}(\bm{I}_{T}-\bm{\Phi}_{0})\bm{F}_{K}\right)^{-1}\bm{F}_{K}^{\prime}.$
(5)

Notice that conditional on a standard normalization, we have that
$\bm{F}^{\prime}_{K}\bm{F}_{K}=\bm{I}_{K}$ and
$\bm{F}_{q}^{\prime}\bm{F}_{(q+1):K}=\bm{0}$. This allows us to rewrite Eq.
(missing)5 as:

$\displaystyle\bm{F}_{K}\left(\bm{I}_{K}+\frac{\omega}{1-\omega}\left[\bm{I}_{K}-\begin{pmatrix}\bm{I}_{q}&\bm{0}\\\
\bm{0}&\bm{0}\end{pmatrix}\right]\right)^{-1}\bm{F}_{K}^{\prime}$ (6)
$\displaystyle=(\bm{F}_{q},~{}\bm{F}_{q+1:K})\begin{pmatrix}\bm{I}_{q}&\bm{0}\\\
\bm{0}&\bm{(}1-\omega)\bm{I}_{K-q}\end{pmatrix}\begin{pmatrix}\bm{F}^{\prime}_{q}\\\
\bm{F}^{\prime}_{q+1:K}\end{pmatrix}$ (7)
$\displaystyle=\bm{\Phi}_{0}+(1-\omega)\bm{\Phi}_{1},$ (8)

with
$\bm{\Phi}_{1}=\bm{F}_{q+1:K}(\bm{F}^{\prime}_{q+1:K}\bm{F}_{q+1:K})^{-1}\bm{F}^{\prime}_{q+1:K}$.
It is straightforward to show that

$\bm{\Phi}=\bm{\Phi}_{0}+\bm{\Phi}_{1},$

where $\bm{\Phi}=\bm{X}(\bm{X}^{\prime}\bm{X})^{-1}\bm{X}^{\prime}$ is the
projection matrix of $\bm{X}$. Thus, we can substitute
$\bm{\Phi}_{1}=\bm{\Phi}-\bm{\Phi}_{0}$ in Eq. (missing)8 and multiply from
the right with $\bm{Y}$ to arrive at:

$\mathbb{E}(\bm{X}\bm{A}|\bm{Y},\omega)=\omega\bm{\Phi}_{0}\bm{Y}+(1-\omega)\bm{\Phi}\bm{Y},$
(9)

which shows that the posterior mean of the regression function is a convex
combination of the VAR fit, $\bm{\Phi}\bm{Y}$, and the fit of the PCA
regression, $\bm{\Phi}_{0}\bm{Y}$. This result can be used to show that the
resulting predictive distribution (or impulse responses) are weighted averages
of the ones obtained from estimating an unrestricted VAR and a PC regression,
both estimated using OLS. Larger values of $\omega$ imply estimates which are
closer to the ones obtained from estimating a PC regression while values of
$\omega$ closer to zero yield estimates closer to those of a non-informative
prior Bayesian VAR.

Note that in the preceding material we are not incorporating any conventional
Bayesian VAR prior such as the Minnesota prior. The prior defined above is a
new one which can be used if the researcher wishes to use a prior which only
shrinks towards the factor model. We will use the acronym subVAR-Flat to
denote this prior which combines the subspace prior with a flat prior for the
VAR coefficients. The fact that $\omega=0$ yields a flat prior VAR illustrates
an important aspect and potential shortcoming of this prior. Flat prior VARs
tend to over-fit unless $M$ is very small and if $K>T$, as commonly occurs
with large VARs, the OLS estimator will not be defined. Adding subspace
shrinkage will ensure the posterior is proper, but small values of $\omega$
can potentially lead to over-fitting. As we will document in our empirical
results, using a non-informative prior for $\omega$ can lead to poor forecast
performance in large VARs. Hence, the need for a suitable prior for $\omega$.
This will be provided below.

###  2.3 Shrinking the Minnesota prior VAR towards a factor model

Since our prior is conjugate, we can easily add additional VAR priors to
complement our subspace prior. In this sub-section, we show how this can be
done for the natural conjugate Minnesota prior as implemented in Banbura,
Giannone, and Reichlin (2010).

Let $\overline{\bm{Y}}=(\bm{Y}^{\prime},\underline{\bm{Y}}^{\prime})^{\prime}$
and $\overline{\bm{X}}=(\bm{X}^{\prime},\underline{\bm{X}}^{\prime})^{\prime}$
denote dummy-augmented data matrices. The dummies $\underline{\bm{Y}}$ and
$\underline{\bm{X}}$ can be specified to match features of the different
priors in the Minnesota tradition. We assume that these dummies are
parameterized by a hyperparameter $\vartheta$, with values of $\vartheta$
close to zero implying strong shrinkage towards the prior mean. In our
empirical work we set the dummies as follows:

$\underline{\bm{Y}}=\begin{pmatrix}\text{diag}(\underline{a}_{1}\hat{\sigma}_{1},\dots,\underline{a}_{M}\hat{\sigma}_{M})/\vartheta\\\
\bm{0}_{M(p-1)\times M}\\\
\text{diag}(\hat{\sigma}_{1},\dots,\hat{\sigma}_{M})\\\ \bm{0}_{1\times
M}\end{pmatrix},\quad\underline{\bm{X}}=\begin{pmatrix}\bm{J}_{p}\otimes\text{diag}(\hat{\sigma}_{1},\dots,\hat{\sigma}_{M})/\vartheta&\bm{0}_{K\times
1}\\\ \bm{0}_{M\times K}&\bm{0}_{M\times 1}\\\ \bm{0}_{1\times
K}&\kappa\end{pmatrix},$

with $\hat{\sigma_{j}}~{}(j=1,\dots,M)$ denoting the OLS residual standard
deviation of an AR($p$) model for $y_{jt}$, the $j^{th}$ variable in
$\bm{Y}_{t}$, $\underline{a}_{j}$ is the $j^{th}$ diagonal element of
$\underline{\bm{A}}$, and $\bm{J}_{p}=\text{diag}(1,\dots,p)$. Notice that
this set of dummies includes the prior for the intercept which depends on the
hyperparameter $\kappa$. $\kappa$ is set to a very small number (in our
empirical work it equals $0.001$), leading to a weakly informative prior for
the intercepts.

We can add these dummies to $\bm{Y}$ and $\bm{X}$ and then combine it with our
subspace shrinkage prior. The posterior covariance matrix of the VAR
coefficients then becomes:

$\overline{\bm{V}}=\left(\overline{\bm{X}}^{\prime}\overline{\bm{X}}+\underbrace{\frac{\omega}{1-\omega}\overline{\bm{X}}^{\prime}\left[\begin{pmatrix}\bm{I}_{T}&\bm{0}\\\
\bm{0}&\bm{0}\end{pmatrix}-\begin{pmatrix}\bm{\Psi}_{0}&\bm{0}\\\
\bm{0}&\bm{0}\end{pmatrix}\right]\overline{\bm{X}}}_{\underline{\bm{V}}^{-1}_{\Psi_{0}}}\right)^{-1}.$

This equation implies that
$\underline{\bm{V}}=\left(\underline{\bm{X}}^{\prime}\underline{\bm{X}}+\underline{\bm{V}}^{-1}_{\Psi_{0}}\right)^{-1}$.
Adding the Minnesota prior means that the result in (9) no longer holds
exactly. Intuitively speaking, if $\vartheta$ is set too tight, the Minnesota-
type prior overrules the subspace shrinkage prior. However, for reasonable
values of $\vartheta$ the following will hold approximately:

$\mathbb{E}(\bm{X}\bm{A}|\bm{Y},\vartheta,\omega)\approx\omega\bm{\Phi}_{0}\bm{Y}+(1-\omega)\bm{X}(\overline{\bm{X}}^{\prime}\overline{\bm{X}})^{-1}\overline{\bm{X}}^{\prime}\overline{\bm{Y}}.$

This result states that the posterior mean of the regression function is a
convex combination of the (OLS) fit of a PCA regression and the posterior mean
based on a Minnesota prior VAR.

To investigate the accuracy of this approximation for different values of
values of $\vartheta$ we can compute the average squared approximation error:

$\Xi(\theta,\omega)=\frac{1}{M}\sum_{j=1}^{M}T^{-1}||\bm{X}\overline{\bm{A}}_{j}-\omega\bm{\Phi}_{0}\bm{Y}_{j}-(1-\omega)\bm{X}(\overline{\bm{X}}^{\prime}\overline{\bm{X}})^{-1}\overline{\bm{X}}^{\prime}\overline{\bm{Y}}_{j}||^{2},$
(10)

with $\overline{\bm{A}}_{j},\bm{Y}_{j}$ and $\overline{\bm{Y}}_{j}$ denoting
the $j^{th}$ column of the corresponding matrix and $||\bullet||$ denotes the
Euclidean norm of a vector. This approximation error quickly approaches zero
if $\vartheta$ becomes moderately large. If $\omega\approx 0$, the
approximation error also vanishes since then we obtain the Minnesota prior
BVAR estimate. The interaction between $\vartheta$ and $\omega$ in determining
$\Xi(\vartheta,\omega)$ is highly non-linear. The key point to take away is
that if $\vartheta$ approaches zero faster than $\omega$ approaches one, the
standard Minnesota prior dominates the subspace prior.

These points are illustrated in Figure 1 which plots the approximation error
for different values of $\vartheta$ and $\omega$ using data sets simulated
from different data generating processes (DGPs) for different values of $M$
and $T$. The DGP is a dynamic factor model with $q=3$ and the factors evolving
according to a multivariate random walk with a full error variance-covariance
matrix.333For more details on the DGP, see Section 2.5.

Figure 1 suggests that $\omega$ does not have a large effect on the
approximation, but that $\vartheta$ does. In particular, for values of
$\vartheta>0.1$, the log approximation error is less than $-8$ for all the
different values of $\omega$. In the next section we will specify a prior on
$\vartheta$ which allocates substantial mass to this region. It is worth
stressing that even if $\vartheta$ is smaller than this, our prior is still a
valid prior combining the Minnesota prior with the subspace prior, it is just
that the posterior mean that results will deviate more from being a linear
combination of a posterior mean using the Minnesota prior and a PC regression.

$M=30$

$M=120$

$T=250$

$T=350$

$T=500$

Figure 1: Log Squared Approximation Error for different DGPs and $q=3$

###  2.4 Selecting the number of factors and estimating the hyperparameters

Our prior depends on a choice for the number of factors ($q$), the weight
attached to the VAR relative to the PC regression ($\omega$) and the degree of
shrinkage in the Minnesota prior ($\vartheta$). The posterior for these is:

$p(q,\omega,\vartheta|\bm{Y})\propto
p(\bm{Y}|q,\omega,\vartheta)~{}p(q,\omega,\vartheta).$

Estimation is straightforward since the natural conjugate prior leads to an
analytical form for the marginal likelihood:

$p(\bm{Y}|q,\omega,\vartheta)\propto\left(|\underline{\bm{V}}|~{}|\overline{\bm{V}}|\right)^{-\frac{M}{2}}~{}|\overline{\bm{S}}|^{-\frac{T+M+\underline{v}}{2}},$
(11)

where
$\overline{\bm{S}}=\underline{\bm{S}}+\bm{Y}^{\prime}\bm{Y}+\underline{\bm{A}}^{\prime}\underline{\bm{V}}^{-1}\underline{\bm{A}}-\overline{\bm{A}}^{\prime}\overline{\bm{V}}^{-1}\overline{\bm{A}}$
is the posterior scaling matrix of the inverse Wishart posterior of
$\bm{\Sigma}$. This can be multiplied by the prior to produce the posterior.
We define discrete grids for $\omega$, $\vartheta$ and $q$ and evaluate the
posterior at points in the grids. We can then do Monte Carlo integration by
sampling from the Multinomial distributions that arise. Hence, our predictive
densities reflect uncertainty in these parameters. This is similar to a
strategy suggested in Giannone, Lenza, and Primiceri (2015) for the Minnesota
prior VAR but, as detailed below, avoids carrying out complex matrix
operations during posterior simulation and thus offers substantial
computational gains (at the cost of approximating a continuous posterior
distribution using a discrete one).

It remains to specify the priors. For $\vartheta$ we follow suggestions in
Giannone, Lenza, and Primiceri (2015) and use a Gamma prior which we set to
have mode $0.2$ and standard deviation $0.4$. This value implies that the
approximation error in (1) is extremely small and the posterior mean of the
model fit can be safely interpreted as a convex combination between the BVAR
and the DFM fit.

For $\omega$ we use a Beta prior: $\mathcal{B}(c_{0},c_{1})$. In our empirical
work we consider two ways of specifying the hyperparameters $c_{0}$ and
$c_{1}$. The first sets $c_{0}=c_{1}=1$, yielding a non-informative uniform
prior on $\omega$. The second prior sets $c_{0}=\mathfrak{c}_{0}\times M$ and
$c_{1}=\mathfrak{c}_{1}\times M$, with $\mathfrak{c}_{0},\mathfrak{c}_{1}$
being scalars greater than zero. This choice implies that the prior mean on
$\omega$ is equal to $\mathfrak{c}_{0}/(\mathfrak{c}0+\mathfrak{c}_{1})$ and
the prior variance equals
$(\mathfrak{c}_{0}~{}\mathfrak{c}_{1})/((\mathfrak{c}_{0}+\mathfrak{c}_{1})^{2}~{}(M(\mathfrak{c}_{0}+\mathfrak{c}_{1})+1)$.
In our empirical work we set $\mathfrak{c}_{0}=8$ and $\mathfrak{c}_{1}=6$,
yielding a prior mean on $\omega$ of around $0.6$ and thus placing
considerable mass on the factor model restrictions while the prior variance
decreases in $M$. In large dimensions, this choice increasingly forces the
model towards the factor restrictions but still provides sufficient
flexibility for individual time series to exhibit VAR dynamics.

We assume a discrete uniform prior on $q$:

$q\sim\mathcal{U}(1,q_{0}),$

which implies that all values up to $q_{0}$ (which denotes some integer
smaller than $K$ set by the researcher) are a-priori equally likely. Other
choices which utilize sample information (such as the eigenvalues of $\bm{X}$)
are in principle possible.

Two hyperparameters remain to be chosen. If we use a flat prior in combination
with a subspace prior, we set $\underline{v}=M+2$ and
$\underline{\bm{S}}=\frac{1}{100}\bm{I}_{M}$. If we use a Minnesota prior we
set $\underline{v}$ equal to the number of rows of $\underline{\bm{Y}}$ and
$\underline{\bm{S}}=(\underline{\bm{Y}}-\underline{\bm{X}}\underline{\bm{A}})^{\prime}(\underline{\bm{Y}}-\underline{\bm{X}}\underline{\bm{A}})$
(see Kadiyala and Karlsson (1997)).

It is worth noting that, in Bayesian factor analysis, selecting the number of
latent factors is a difficult task. Bayesian solutions include using
reversible jump MCMC algorithms which treat the number of factors as an
unknown quantity (see, e.g., Lopes and West (2004) and Frühwirth-Schnatter and
Lopes (2018)). Another strand of the literature estimates an overfitting
factor model and applies Bayesian shrinkage priors to force the columns of the
factor loadings matrix associated with irrelevant factors to zero, see
Bhattacharya and Dunson (2011). However, the following sub-section
investigates, using simulated data, the simple and computationally efficient
approach given here. These simulations show that, even under a uniform prior,
our approach selects the true number of factors successfully.

It is also worth noting that, conditional on $\omega$ and $\vartheta$, all
quantities used in the Monte Carlo sampling of $q$ can be pre-computed and
thus estimation of huge models (i.e., with $M>100$) is feasible. This requires
specifying a grid for $\omega$, $\theta$ and $q$. In all our empirical work we
set the grid for $q\in\\{1,\dots,\text{min}(10,\mathcal{L}^{*})\\}$ with
$\mathcal{L}^{*}$ denoting the Ledermann bound.444The Ledermann bound is
defined as the solution $\mathcal{L}^{*}$ to the equation
$(M-\mathcal{L}^{*})^{2}\geq M+\mathcal{L}^{*}$. The grid on $\omega$ is
specified to go from $0.01$ to $0.99$ with a step-size of $0.05$. Finally, the
grid on $\vartheta$ is
$\\{0.001,0.01,0.025,0.05,0.10,0.20,0.3,0.4,0.5,2,3,4,5\\}$.

Evaluating marginal likelihoods can be challenging in very large models and
they depend on the prior. Accordingly, in our empirical work (which involves
forecasting three variables of interest), we also investigate an alternative
way of choosing $q$, $\omega$ and $\vartheta$. This is to use the Bayesian
Information Criterion (BIC) for the three variables of interest to choose
them.

###  2.5 Simulated data exercise on selecting the number of factors

In this sub-section we investigate whether our approach successfully detects
the correct number of factors by means of synthetic data. To analyze how
estimation accuracy changes across different datasets, we consider DGPs that
vary along the number of variables ($M$) as well as the number of factors. The
DGP is a dynamic factor model given by:

$\displaystyle\bm{y}_{t}=\bm{\Lambda}\bm{f}_{t-1}+\bm{\epsilon}_{t},\quad\bm{\epsilon}_{t}\sim\mathcal{N}(\bm{0}_{M},\bm{\Sigma}),$

with $\bm{f}_{t}$ evolving according to a multivariate random walk with full
state-innovation variance $\bm{\Omega}$, an initial state
$\bm{f}_{0}=\bm{0}_{q}$ and
$\bm{\Sigma}=\bm{\Lambda}\bm{\Omega}\bm{\Lambda}^{\prime}+\bm{W}$ being a full
matrix with $\bm{W}$ denoting a diagonal matrix of measurement error
variances. Notice that this is a standard dynamic factor model which is
rewritten by plugging the random walk state equation into an observation
equation which typically includes the contemporaneous factors and uncorrelated
measurement errors.

In all our simulations we assume that $\lambda_{ij}$, the $(i,j)^{th}$ element
of $\bm{\Lambda}$, is drawn from a normal distribution with zero mean and
variance $0.1^{2}$ if $i\neq j$ or set equal to unity if $i=j$. Instead of
specifying $\bm{\Omega}$ we obtain the lower Cholesky factor of $\bm{\Sigma}$,
$\bm{A}^{-1}_{0}$, by simulating the off-diagonal elements from a Gaussian
distribution with zero mean and variance $0.1^{2}$ and the main diagonal
elements are set equal to $0.1$.

We simulate $T=500$ observations from small ($M=10$), moderate ($M=60$) and
large ($M=120$) datasets. For each of these, we vary the number of factors
$q\in\\{1,3,6,8\\}$. All simulations are repeated $100$ times and, in Table 1,
we report averages of posterior medians across these replications.

It can be seen that all of the versions of our subVAR prior are doing a good
job of choosing the correct number of factors. It is mainly in the least
parsimonious cases (i.e. DGPs with $M=120$ and $q=8$) where it is slightly
underestimating the number of factors. But this is due to the large VAR
providing some of the fit, leaving less for the DFM to explain. In the context
of these very large models, slight over-shrinkage is better than the over-
fitting which would have occurred if the prior had failed to shrink enough.

Table 1: Simulation results for differing values of $q$ and $M$. Averages
across 100 replications from the DGP

subVAR-Minn0 | subVAR-Minn1 | subVAR-Flat0 | subVAR-Flat1 |
---|---|---|---|---
$q=$ | 1 | 3 | 6 | 8 | 1 | 3 | 6 | 8 | 1 | 3 | 6 | 8 | 1 | 3 | 6 | 8
Posterior mean of $q$ |
$M=10$ | 1.32 | 3.03 | 5.87 | 2.54 | 1.29 | 3.11 | 6.00 | 8.00 | 1.32 | 3.04 | 5.90 | 2.53 | 1.28 | 3.06 | 6.00 | 8.00
$M=60$ | 1.00 | 2.96 | 5.23 | 6.01 | 1.03 | 2.95 | 5.34 | 6.24 | 1.00 | 2.95 | 5.26 | 6.01 | 1.02 | 2.99 | 5.28 | 6.22
$M=120$ | 1.00 | 2.58 | 3.92 | 4.54 | 1.00 | 2.67 | 3.92 | 4.59 | 1.00 | 2.57 | 3.84 | 4.50 | 1.00 | 2.63 | 3.95 | 4.61

Notes: subVAR denotes the VAR coupled with the subspace shrinkage prior, Minn
is the combination between subspace and Minnesota shrinkage while flat is the
subspace shrinkage prior without additional shrinkage. The $0$ and $1$
attached to the respective label indicate a flat ($0$) or informative ($1$)
prior on $\omega$. Each number is based on computing the mean of posterior
medians across 100 replications from the respective DGPs. For $q$, we use the
posterior median as our point estimate while for $\omega$ we use the posterior
mean.

##  3 Forecasting Using US Macroeconomic Data

###  3.1 Data

We use a large set of $166$ quarterly macroeconomic variables taken from the
St. Louis Fed’s FRED data base (fred.stlouisfed.org) and discussed in
McCracken and Ng (2020). These are listed in the appendix in Table 3.
Variables are transformed to stationarity following recommendations there. Our
forecasting results focus on three variables of interest: GDP growth (based on
real GDP growth, GDPC1), the Fed Funds rate (FEDFUNDS) and inflation (based on
the consumer price index, CPIAUCSL).

The data runs from 1960:Q1 to 2020:Q3 and in our forecasting exercise, the
evaluation period is from 1990:Q3-2020:Q3. We adopt a recursive forecasting
design. We use the initial estimation period (1960:Q1 to 1990:Q2) to produce
one- and iterated four-quarter-ahead forecast distributions for 1990:Q3 and
1991:Q2, respectively. After obtaining these, we expand the initial estimation
period by one observation until we reach the end of the sample.

###  3.2 Models

We present results for four models involving subspace priors (acronym subVAR).
These involve two priors for the VAR coefficients: the non-informative one
(flat) and the Minnesota prior (Minn). There are also two priors for $\omega$:
one non-informative (flat) and one informative. These are indicated by adding
a $0$ (flat) and $1$ (tight) to the relevant labels of the VAR coefficient
prior.

For each of these four models, we present results for data sets of four
different sizes: small (S, $12$ variables), medium (M, $22$ variables), large
(L, $78$ variables) and extra large (XL, $166$ variables). Table 3 lists which
variable belongs in which category.

For comparison we also present results for Minnesota prior VARs (implemented
by setting $\omega=0$ in the subVAR-Minn) and factor models (labeled DFM). The
factor model is a FAVAR for the three variables of interest and the factors
estimated by extracting the PCs from the remaining time series within a given
model size. The number of PCs is chosen by retaining the PCs with standard
deviations greater than unity. It is estimated using a relatively non-
informative Minnesota prior. The lag length in all models is set to two.

###  3.3 Summary of forecasting results

We begin by summarizing the results of our pseudo-out-of-sample forecasting
exercise in Table 2. This table contains Root Mean Squares Forecast Errors
(RMSFEs) and averages (over time) of log predictive likelihoods (LPLs) for our
three variables of interest and for two different forecast horizons. The
RMSFEs are ratios between the RMSFEs of a given model and the Minnesota VAR
while the LPLs are differences between a given model and the Minnesota VAR
(both for a given model size).

Before discussing our subVAR models, consider the comparison between the
Minnesota prior VAR and the factor model. For some variables, forecast
horizons and model sizes, the VAR yields more precise forecasts. That is, for
the interest rate it consistently forecasts better and for inflation and GDP
growth for larger models at longer horizons, its forecasts tend to be more
precise than the ones of the DFM. But for other cases the factor model
outperforms the Bayesian VAR. This result raises the possibility that an
approach such as ours, which combines the two, could lead to better overall
forecast performance than either the BVAR or DFM individually and, with
several exceptions discussed below, this is what we find.

Consider first the most informative subVAR model which uses the Minnesota
prior on the VAR coefficients and the informative prior on $\omega$ (Minn1).
With some exceptions, this model is yielding forecasts which are better than
the predictions produced by the BVAR and are often the most precise ones. The
main exceptions are the one-year-ahead GDP growth predictions which are
marginally worse than the the BVAR benchmark. However, this is a case where
the DFM and some of the less informative subVAR approaches are forecasting
substantially worse than the BVAR.

Consider now the second most informative subVAR approach (Minn0) which retains
the Minnesota prior for the VAR coefficients, but uses a non-informative prior
on $\omega$. Its forecasts are comparable to those of Minn1, but overall are
slightly worse. But clearly results are robust to the prior on $\omega$. Both
of these Minnesota prior subVAR approaches are forecasting well most of the
time and even the few exceptions reveal only slight deterioration in forecast
performance relative to the BVAR benchmark.

Using a non-informative prior for the VAR coefficients, however, goes wrong in
some cases in larger models. In one sense, this is unsurprising. Non-
informative priors work poorly in large VARs since they suffer from severe
over-fitting problems. It might have been possible that these would have been
corrected by adding the subspace prior shrinking towards the factor model. But
clearly this effect is not strong enough in the L and XL models to counter-
balance the over-fitting problem. One might have hoped that estimates of
$\omega$ would have been pulled towards $1$ in these cases, leading to results
similar to the DFM but (unless we use an extremely dogmatic prior on $\omega$)
this is not happening in the larger models. Similar to the results based on
synthetic data, this is because the larger (unrestricted) VARs explain the
majority of variation in the data and leave little variation to explain for
the factor model, yielding posterior estimates of $\omega$ close to zero.
However, the subVAR-Flat models are performing well for our small and medium-
sized models and for the one-quarter-ahead forecast horizon for the larger
models. And it is only the iterated one-year-ahead forecasts that are
deteriorating. This suggests that this approach might be found useful by
researchers working with VARs up to a dimension of approximately $20$ who wish
to avoid the use of standard BVAR priors such as the Minnesota prior,
particularly if the focus is on short-term forecasts. Such a researcher may
also wish to avoid using marginal likelihoods since they are prior-dependent.
For them, it is also interesting to note that using the BIC to estimate the
prior hyperparameters also works roughly as well as using marginal
likelihoods.

Table 2: Forecasting results across focus variables, models and forecast
horizons

| FEDFUNDS | CPIAUCSL | GDPC1
---|---|---|---
| SubVAR | DFM | BVAR | SubVAR | DFM | Minn | SubVAR | DFM | Minn
| Minn0 | Minn1 | Flat0 | Flat1 |  |  | Minn0 | Minn1 | Flat0 | Flat1 |  |  | Minn0 | Minn1 | Flat0 | Flat1 |  |
Marginal Likelihood to estimate $q$, $\omega$ and $\vartheta$
Relative RMSEs
One-quarter-ahead
S | 1.02 | 0.95 | 1.03 | 0.96 | 1.07 | 0.58 | 0.99 | 0.98 | 0.99 | 0.98 | 1.01 | 1.21 | 0.99 | 0.99 | 1.00 | 0.99 | 1.00 | 1.15
M | 1.04 | 0.93 | 1.12 | 1.05 | 1.18 | 0.64 | 0.99 | 0.97 | 1.01 | 1.00 | 0.98 | 1.26 | 1.00 | 1.00 | 1.04 | 1.02 | 1.02 | 1.14
L | 0.96 | 0.94 | 1.89 | 1.79 | 1.37 | 0.74 | 0.98 | 0.98 | 1.33 | 1.30 | 0.96 | 1.32 | 0.99 | 0.99 | 1.41 | 1.38 | 0.92 | 1.19
XL | 0.96 | 0.91 | 1.48 | 1.29 | 1.37 | 0.74 | 0.98 | 0.96 | 1.08 | 1.04 | 1.05 | 1.35 | 0.99 | 0.97 | 1.11 | 1.07 | 0.93 | 1.21
One-year-ahead
S | 1.00 | 1.01 | 1.03 | 1.01 | 1.03 | 0.49 | 1.01 | 1.01 | 1.00 | 1.00 | 1.03 | 1.14 | 1.00 | 1.00 | 0.99 | 1.00 | 1.00 | 1.12
M | 1.04 | 0.95 | 1.19 | 1.11 | 0.95 | 0.53 | 1.01 | 1.00 | 1.02 | 1.01 | 1.06 | 1.14 | 1.00 | 1.00 | 1.01 | 1.00 | 1.01 | 1.12
L | 0.90 | 0.88 | 6.16 | 9.47 | 1.48 | 0.69 | 0.99 | 0.98 | 2.72 | 3.74 | 1.03 | 1.18 | 1.00 | 1.00 | 3.10 | 5.54 | 1.07 | 1.12
XL | 0.87 | 0.71 | $>10$ | $>10$ | 1.42 | 0.85 | 0.99 | 0.97 | $>10$ | $>10$ | 1.06 | 1.19 | 1.00 | 0.97 | $>10$ | $>10$ | 1.03 | 1.13
Average LPLs
One-quarter-ahead
S | -0.01 | 0.05 | 0.01 | 0.06 | -0.52 | -0.91 | 0.05 | 0.11 | 0.05 | 0.13 | 0.32 | -2.00 | 0.02 | 0.06 | 0.05 | 0.05 | 0.00 | -1.67
M | -0.01 | 0.05 | -0.05 | -0.01 | -0.63 | -0.87 | 0.01 | 0.20 | 0.07 | 0.18 | 0.75 | -2.36 | 0.01 | 0.01 | -0.03 | -0.02 | 0.10 | -1.66
L | 0.11 | 0.14 | -0.63 | -0.46 | -0.67 | -1.13 | 0.26 | 0.39 | 1.37 | 1.53 | 2.06 | -3.76 | -0.04 | 0.05 | -0.47 | -0.45 | 0.10 | -1.69
XL | 0.84 | 1.36 | 0.84 | 0.56 | 0.62 | -2.49 | 2.12 | 3.28 | 4.61 | 4.29 | 4.65 | -6.44 | 0.28 | 0.36 | 0.15 | -0.21 | 0.71 | -2.36
One-year-ahead
S | -0.07 | 0.05 | -0.26 | -0.16 | -0.47 | -1.08 | 0.02 | -0.00 | -0.04 | 0.00 | 0.04 | -1.64 | -0.02 | 0.02 | -0.07 | -0.05 | 0.02 | -1.63
M | -0.07 | 0.04 | -0.49 | -0.40 | -0.62 | -1.07 | 0.06 | 0.01 | 0.05 | 0.05 | 0.05 | -1.75 | -0.06 | -0.05 | -0.20 | -0.15 | -0.05 | -1.60
L | 0.03 | 0.06 | -3.64 | -3.93 | -2.14 | -1.57 | 0.03 | 0.03 | -3.45 | -3.73 | -1.59 | -1.58 | -0.02 | -0.03 | -3.74 | -4.04 | -1.29 | -1.66
XL | -0.12 | -0.31 | -6.61 | -8.18 | -2.01 | -1.45 | -0.09 | -0.28 | -6.60 | -8.23 | -1.39 | -1.56 | -0.19 | -0.49 | -7.03 | -8.63 | -1.25 | -1.55
BIC for the three focus variables used to estimate $q$, $\omega$ and
$\vartheta$
Relative RMSEs
One-quarter-ahead
S | 0.96 | 0.91 | 0.97 | 0.88 | 0.96 | 0.66 | 0.99 | 0.95 | 0.99 | 0.95 | 0.98 | 1.24 | 1.00 | 0.99 | 1.00 | 0.99 | 0.99 | 1.15
M | 0.92 | 0.74 | 0.94 | 0.74 | 0.88 | 0.85 | 0.97 | 0.89 | 0.98 | 0.89 | 0.90 | 1.37 | 0.99 | 0.93 | 1.00 | 0.95 | 0.97 | 1.21
L | 0.93 | 0.82 | 1.81 | 1.15 | 1.47 | 0.71 | 0.98 | 0.92 | 1.30 | 1.07 | 0.97 | 1.30 | 0.98 | 0.96 | 1.35 | 1.09 | 0.92 | 1.18
XL | 0.94 | 0.84 | 1.43 | 1.03 | 1.45 | 0.71 | 0.97 | 0.91 | 1.05 | 0.96 | 1.06 | 1.34 | 0.99 | 0.94 | 1.08 | 0.99 | 0.94 | 1.20
One-year-ahead
S | 0.98 | 0.94 | 1.00 | 0.95 | 0.93 | 0.52 | 1.01 | 0.99 | 1.00 | 0.99 | 1.01 | 1.16 | 1.01 | 1.01 | 1.01 | 1.02 | 1.01 | 1.11
M | 0.84 | 0.56 | 0.89 | 0.56 | 0.56 | 0.94 | 0.97 | 0.92 | 0.97 | 0.93 | 0.97 | 1.25 | 0.97 | 0.94 | 0.97 | 0.93 | 0.95 | 1.20
L | 0.87 | 0.75 | 8.15 | $>10$ | 1.63 | 0.64 | 0.99 | 0.99 | 3.77 | $>10$ | 1.06 | 1.15 | 0.99 | 0.99 | 5.31 | $>10$ | 1.09 | 1.12
XL | 0.84 | 0.65 | $>10$ | $>10$ | 1.66 | 0.76 | 1.00 | 0.98 | $>10$ | $>10$ | 1.07 | 1.17 | 0.98 | 0.98 | $>10$ | $>10$ | 1.04 | 1.11
Average LPLs
One-quarter-ahead
S | 0.05 | 0.09 | 0.05 | 0.12 | -0.42 | -1.01 | 0.05 | 0.19 | 0.09 | 0.22 | 0.43 | -2.09 | 0.03 | 0.06 | 0.02 | 0.05 | 0.07 | -1.67
M | 0.07 | 0.20 | 0.06 | 0.21 | -0.45 | -1.05 | 0.18 | 0.63 | 0.13 | 0.60 | 1.23 | -2.83 | -0.01 | 0.05 | 0.01 | 0.07 | 0.15 | -1.70
L | 0.14 | 0.22 | -0.48 | -0.31 | -0.76 | -1.04 | 0.39 | 0.81 | 1.20 | 1.63 | 1.69 | -3.39 | 0.13 | 0.17 | -0.33 | -0.27 | 0.17 | -1.78
XL | 0.63 | 1.02 | -0.06 | -0.55 | -0.06 | -1.81 | 1.53 | 2.45 | 2.91 | 2.47 | 3.19 | -4.99 | 0.16 | 0.31 | -0.34 | -0.78 | 0.48 | -2.13
One-year-ahead
S | 0.06 | 0.29 | 0.00 | 0.26 | -0.14 | -1.41 | 0.02 | 0.06 | 0.01 | 0.06 | 0.11 | -1.69 | -0.01 | 0.06 | -0.04 | 0.00 | 0.07 | -1.67
M | 0.09 | 0.57 | 0.06 | 0.54 | 0.03 | -1.72 | 0.04 | -0.02 | -0.01 | 0.04 | 0.03 | -1.73 | 0.05 | 0.20 | 0.04 | 0.19 | 0.20 | -1.83
L | 0.06 | 0.25 | -4.22 | -5.04 | -2.21 | -1.50 | -0.00 | -0.03 | -3.98 | -4.77 | -1.63 | -1.55 | -0.02 | -0.05 | -4.29 | -5.15 | -1.32 | -1.64
XL | -0.20 | -0.73 | -8.11 | -9.95 | -1.98 | -1.45 | -0.20 | -0.74 | -8.15 | -9.97 | -1.37 | -1.57 | -0.27 | -0.96 | -8.44 | -7.59 | -1.13 | -1.65

Notes: subVAR denotes the VAR coupled with the subspace shrinkage prior, Minn
is the combination between subspace and Minnesota shrinkage while flat is the
subspace shrinkage prior without additional shrinkage. The $0$ and $1$
attached to the respective label indicate a flat ($0$) or informative ($1$)
prior on $\omega$. DFM is a VAR in the three focus variables augmented with
principal components and BVAR refers to a Minnesota VAR. The upper part of the
table shows relative root mean squared forecast errors (RMSEs) between a given
model and the Minnesota VAR while the lower part of the table shows
differences in average log predictive likelihoods (LPLs) to the Minnesota VAR.
The numbers in the BVAR columns include the actual RMSEs and LPLs of the
Minnesota VAR.

###  3.4 A deeper examination of forecast performance of subspace VAR methods

To examine more deeply the properties of our subVAR prior, in this sub-section
we provide plots over time of predictive Bayes factors against the Minnesota
prior VAR. Moreover, we investigate how the estimates of the prior
hyperparameters $q$, $\omega$ and $\vartheta$ evolve over the hold-out period.

Figures 2 and 3 plot the log predictive Bayes factors for the three variables
being forecast for the four subspace VAR priors. Figure 2 uses the marginal
likelihood for all the variables in the model to estimate the prior
hyperparameters and Figure 3 uses the BIC for the three variables of interest.
The overall best performance of the priors which combine subspace shrinkage
with the Minnesota prior can be seen in both figures. An examination of the
main exception to this pattern is informative. This occurs for inflation
forecasts where the combination of the non-informative prior VAR with the
subspace shrinkage prior forecasts well. But this result holds only for the
one-quarter-ahead horizon. The iterated one-year-ahead forecasts are very poor
(see Table 2).

Another pattern worth noting is that substantial changes tend to occur during
either the financial crisis (around 2009) or the pandemic (2020). The tendency
at these times is for the subVAR-Flat models to do better. Results for GDP
growth from larger VARs are particularly striking. The forecast performance of
these models was extremely poor up until the pandemic when the subVAR-Flat
models almost caught up to the subVAR-Minn models. The stronger prior
information in the latter is a great benefit in normal times, but in the
pandemic this makes it less able to adjust to the extreme observations which
arise. This is because the corresponding predictive density is narrow which
helps in tranquil periods while in turbulent times (such as during the
pandemic) the variance is too low, rendering outliers less likely under the
posterior predictive distribution.

The role of the prior on $\omega$ can to be seen to be relatively unimportant
in most cases. Although it is interesting to note that in smaller models it
can be beneficial to use the informative prior for $\omega$ (see, in
particular, the forecasts of inflation for the small and medium VARs).

(a) FEDFUNDS

(b) CPIAUCSL

(c) GDPC1

Notes: Solid lines refer to the model which includes both subspace and
Minnesota shrinkage whereas the dashed lines show the variable-specific log-
predictive Bayes factor (to the Minnesota prior VAR) of the subspace shrinkage
VAR without the Minnesota prior. Black lines refer to models with an
uninformative prior on $\omega$. Red lines denote models with an informative
prior on $\omega$.

Figure 2: Evolution of the log predictive Bayes factor between subVAR and the
Minnesota VAR across focus variables when the marginal likelihood is used to
select $q,\vartheta$ and $\omega$

(a) FEDFUNDS

(b) CPIAUCSL

(c) GDPC1

Notes: Solid lines refer to the model which includes both subspace and
Minnesota shrinkage whereas the dashed lines show the variable-specific log-
predictive Bayes factor (to the Minnesota prior VAR) of the subspace shrinkage
VAR without the Minnesota prior. Black lines refer to models with an
uninformative prior on $\omega$. Red lines denote models with an informative
prior on $\omega$.

Figure 3: Evolution of the log predictive Bayes factor between subVAR and the
Minnesota VAR across focus variables when the BIC over the three focus
variables is used to select $q,\vartheta$ and $\omega$

Figures 4 and 5 plot the posterior mean of $q$ over time for the marginal
likelihood-based and BIC-based methods, respectively. The figures illustrate
some interesting differences between these two methods. In particular, use of
the BIC allows for more time variation in the parameter estimates for the XL
model suggesting it allows for quicker adjustment to new information. Consider
the best-performing approach which is the subVAR-Minn prior with an
informative prior on $\omega$. For this case, using the marginal likelihood
leads to a choice of $10$ factors for all time periods for the XL model. But
using the BIC, there is more variation over time. For the XL model in
particular the number of factors increases gradually from $7$ to $9$ before
quickly collapsing down to a posterior mean near $6$ when the financial crisis
hits. In general, for the XL model the marginal likelihood is consistently
leading to large estimates for $q$ which vary little over time. This pattern
does not recur for the lower dimensional models where the marginal likelihood-
based estimates of $q$ tend to be lower. For instance, in the smallest model
the sub-VAR flat model chooses $q=1$ for all periods, which contrasts with
much larger BIC-based estimates. Marginal likelihood calculation in large VARs
can be unstable and sensitive to the prior and our results suggest that in
larger models at least it may be safer to use BIC-based estimates.

(a) SubVAR-Minn (flat prior on $\omega$)

(b) SubVAR-Minn (informative prior on $\omega$)

(c) SubVAR-Flat (flat prior on $\omega$)

(d) SubVAR-Flat (informative prior on $\omega$)

Figure 4: Evolution of the posterior mean of $q$ over the hold-out period when
the marginal likelihood is used to select $q,\vartheta$ and $\omega$

(a) SubVAR-Minn (flat prior on $\omega$)

(b) SubVAR-Minn (informative prior on $\omega$)

(c) SubVAR-Flat (flat prior on $\omega$)

(d) SubVAR-Flat (informative prior on $\omega$)

Figure 5: Evolution of the posterior mean of $q$ over the hold-out period when
the BIC over the three focus variables is used to select $q,\vartheta$ and
$\omega$

Figures 6 and 7 present evidence on the estimation of $\omega$. For the XL and
L models, we are finding striking differences between the BIC and marginal-
likelihood based estimates. Note that for subVAR-Minn model with the
informative prior on $\omega$ we are finding the posterior mean of $\omega$ to
be approximately $0.6$ when estimated using BIC whereas the marginal
likelihood based estimates are much lower at approximately $0.25$/$0.35$ for
the XL/L models. Hence, the former model is shrinking much more closely to the
factor model than the latter.

We can also see the role that the prior for $\omega$ has in that estimates
using the non-informative prior for $\omega$ tend to be substantially lower
than those produced using the informative prior. In fact, with rare
exceptions, using the non-informative prior never leads to estimates of
$\omega$ above $0.2$. At least in this data set, it is necessary to use an
informative prior for $\omega$ to achieve substantial shrinkage towards the
factor model. It is interesting to note that, when we do so, we are
consistently finding $\omega$ to be in the region $[0.25,0.60]$ being far from
the region where one would feel confident selecting either the Minnesota prior
VAR ($\omega=0$) or the factor model ($\omega=1$) thus indicating again the
potential benefits of our approach which combines the two.

(a) SubVAR-Minn (flat prior on $\omega$)

(b) SubVAR-Minn (informative prior on $\omega$)

(c) SubVAR-Flat (flat prior on $\omega$)

(d) SubVAR-Flat (informative prior on $\omega$)

Figure 6: Evolution of the posterior mean of $\omega$ over the hold-out period
when the marginal likelihood is used to select $q,\vartheta$ and $\omega$

(a) SubVAR-Minn (flat prior on $\omega$)

(b) SubVAR-Minn (informative prior on $\omega$)

(c) SubVAR-Flat (flat prior on $\omega$)

(d) SubVAR-Flat (informative prior on $\omega$)

Figure 7: Evolution of the posterior mean of $\omega$ over the hold-out period
when the BIC over the three focus variables is used to select $q,\vartheta$
and $\omega$

Finally we turn to main shrinkage parameter of the Minnesota prior,
$\vartheta$. This hyperparameter only appears in the approaches involving the
Minnesota prior. Note that smaller values of $\vartheta$ imply stronger
shrinkage. Posterior means are plotted in Figure 8.

The most striking pattern here is that using the marginal likelihood leads to
much lower estimates of this hyperparameter than using the BIC, especially for
the small and medium models. In general, and consistent with Giannone, Lenza,
and Primiceri (2015), we find that larger models generally feature smaller
values of $\vartheta$ (and thus more shrinkage). If we combine this with the
fact that the marginal likelihood-based estimates of $\omega$ are lower for
these models we have the interesting finding that it is choosing to put more
weight on a Minnesota prior VAR with more shrinkage. In contrast, the BIC
based weights are closer to be a combination of a factor model with a
Minnesota prior that is implemented rather loosely.

Another interesting finding is that $\vartheta$ tends to sharply increase
during the pandemic. This is especially pronounced for small and medium-sized
models. Since the variance of the predictive distribution is positively
related to $\vartheta$, larger values of $\vartheta$ are (all else being
equal) accompanied by wider predictive intervals. This explains why some of
the models improve appreciable against the benchmark in 2020.

(a) SubVAR-Minn (flat prior on $\omega$, ML)

(b) SubVAR-Minn (flat prior on $\omega$, BIC)

(c) SubVAR-Minn (informative prior on $\omega$, ML)

(d) SubVAR-Minn (informative prior on $\omega$, BIC)

Figure 8: Evolution of the posterior mean of $\vartheta$ over the hold-out
period

##  4 Further Discussion

In this paper, we have worked with two popular models (i.e., the conjugate
version of the Minnesota prior VAR with a single shrinkage hyperparameter and
the factor model) both of which are homoskedastic. We did this to draw out all
the theoretical insights in a clear and simple way and because, in many
empirical contexts, simple approaches such as these have been found to work
well (see, e.g., Banbura, Giannone, and Reichlin (2010) and Carriero, Clark,
and Marcellino (2015)). Furthermore, computation is vastly simplified since
analytical results are available and we can avoid the use of MCMC methods.

However, many recent Bayesian VAR papers have used richer econometric
structures. These can be classified in two main categories: other priors and
other forms for the error covariance matrix. Here we discuss using the subVAR
prior in the context of such extensions.

One restrictive feature of our Minnesota prior is that it involves a single
shrinkage parameter. Allowing for each equation to have its own shrinkage
parameter could be a useful extension of our approach. This raises two issues:
i) the resulting priors would no longer be conjugate and ii) the number of
prior hyperparameters would become larger leading to a higher dimensional grid
of values to evaluate in the Monte Carlo step. These issues could partly be
surmounted by decomposing
$\bm{\Sigma}=\bm{A}_{0}^{-1}(\bm{A}_{0}^{-1})^{\prime}$ and writing the VAR in
structural form (i.e. with $\bm{A}_{0}\bm{Y}_{t}$ on the left hand side and
the error covariance matrix being diagonal). Conditional on $\bm{A}_{0}$, the
subspace shrinkage prior could be applied in an equation-specific manner and
estimation could proceed one equation at a time (which would help reduce the
computational burden, although the computational cost of having $M$ different
sets of prior hyperparameters would still be large). $\bm{A}_{0}$ could be
drawn using MCMC methods. In essence, it would be straightforward to develop
an MCMC algorithm for drawing $\bm{A}_{0}$, $\omega_{j}$, $q_{j}$ and
$\vartheta_{j}$ for $j=1,..,M$. Being an MCMC algorithm it would be inherently
much more computationally demanding than the methods developed in this paper,
but could be useful at least in medium-sized VARs.

There have also been many global-local shrinkage priors (e.g., the Horseshoe
and Lasso priors) which are conditionally Gaussian (i.e., conditional on some
new parameters in the prior they are Gaussian). Estimation proceeds by adding
blocks to the MCMC algorithm for drawing these new parameters. Since our
Minnesota prior is Gaussian, it is trivial to replace it with any
conditionally Gaussian prior. The theory developed in this paper would hold,
conditional on the new parameters. Estimation would proceed through an MCMC
algorithm which drew these new parameters and then conditional on each draw
exploited the subVAR methods developed in this paper.

In a similar fashion, the assumption of homoskedasticity could be relaxed to
allow for stochastic volatility. This would lead to an MCMC algorithm which
involved drawing the volatilities and, conditional on each draw, the results
for the subspace VAR prior developed in this paper could be used. Several
forms for stochastic volatility in VARs have been proposed in the literature,
see for instance Carriero, Clark, and Marcellino (2019) for a particularly
popular form, and the general strategy outlined here would work with any of
them.

In sum, many extensions of the conjugate subVAR approach developed in this
paper are possible. However, they would require the use of MCMC methods.
Provided the likelihood and prior remain Gaussian conditional on some new
parameters, the theory derived in Section 2 would hold, conditional on these
new parameters.

Finally, it is worth noting that the error covariance structure proposed in
Chan (2020) maintains many of the benefits of the conjugate form. It assumes
the VAR error covariance matrix is $\bm{\Sigma}\otimes\bm{\Omega}$ where
$\bm{\Omega}$ can be any positive definite matrix. This nests many possible
specifications, including a common stochastic volatility model, moving average
errors and non-Gaussian errors. This Kronecker structure in the likelihood
matches up with the Kronecker structure in the conjugate prior leading to
derivations which are similar to those in Section 2 of this paper. Roughly
speaking, whereas the derivations in Sub-section 2.2 show how the subspace
prior leads to a posterior mean which is a combination of the OLS estimate of
the VAR with a PC regression, using the model of Chan (2020) leads to a
combination of a GLS estimate with a PC regression. Chan (2020) develops a
computationally efficient MCMC algorithm for models with this error covariance
structure.

##  5 Conclusions

Macroeconomic researchers with large data sets have traditionally been forced
to make a choice between a large VAR or a dynamic factor model. In this paper,
we have shown how to combine the two. We have developed a subspace prior for
the VAR which shrinks towards a dynamic factor model. A parameter, $\omega$
controls the degree of shrinkage and we have developed methods for estimating
it from the data. Thus, we have developed a Bayesian methodology for averaging
a large VAR with a factor model or choosing between them.

We illustrate our approach using synthetic and real data. In simulations, we
show that our approach accurately detects the number of factors if the true
DGP suggests relatively few factors (irrespective of the model size). If the
DGP features a large number of factors, our approach slightly underestimates
the true number of factors. In a forecasting exercise involving a large number
of macroeconomic variables, we demonstrate the benefits of combining the two
model classes using our subspace VAR approach. Using subspace shrinkage in
combination with a Minnesota prior often yields more precise forecasts than
the ones obtained from either the factor model or the VAR.

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##  Appendix A Data Appendix

Table 3: Description of the Dataset

ID | FRED Code | Description | Transformation Codes | S | M | L | XL
---|---|---|---|---|---|---|---
1 | GDPC1 | Real Gross Domestic Product | 5 | X | X | X | X
2 | PCECC96 | Real Personal Consumption Expenditures | 5 |  | X | X | X
3 | PCDGx | Real personal consumption expenditures: Durable goods | 5 |  |  |  | X
4 | PCESVx | Real Personal Consumption Expenditures: Services | 5 |  |  | X | X
5 | PCNDx | Real Personal Consumption Expenditures: Nondurable Goods | 5 |  |  | X | X
6 | GPDIC1 | Real Gross Private Domestic Investment | 5 |  |  | X | X
7 | FPIx | Real private fixed investment | 5 |  | X | X | X
8 | Y033RC1Q027SBEAx | Real Gross Private Domestic Investment: Fixed Investment: Nonresidential Equipment | 5 |  |  | X | X
9 | PNFIx | Real private fixed investment: Nonresidential | 5 |  |  | X | X
10 | PRFIx | Real private fixed investment: Residential | 5 |  |  | X | X
11 | A014RE1Q156NBEA | Shares of gross domestic product: Gross private domestic investment: Change in private inventories | 1 |  |  | X | X
12 | GCEC1 | Real Government Consumption Expenditures and Gross Investment | 5 |  | X | X | X
13 | A823RL1Q225SBEA | Real Government Consumption Expenditures and Gross Investment: Federal | 1 |  |  |  | X
14 | FGRECPTx | Real Federal Government Current Receipts | 5 |  |  |  | X
15 | SLCEx | Real government state and local consumption expenditures | 5 |  |  |  | X
16 | EXPGSC1 | Real Exports of Goods and Services | 5 |  |  | X | X
17 | IMPGSC1 | Real Imports of Goods and Services | 5 |  |  | X | X
18 | DPIC96 | Real Disposable Personal Income | 5 |  |  | X | X
19 | OUTNFB | Nonfarm Business Sector: Real Output | 5 |  |  |  | X
20 | OUTBS | Business Sector: Real Output | 5 |  |  |  | X
21 | INDPRO | IP:Total index Industrial Production Index (Index 2012=100) | 5 |  | X | X | X
22 | IPFINAL | IP:Final products Industrial Production: Final Products (Market Group) (Index 2012=100) | 5 |  |  | X | X
23 | IPCONGD | IP:Consumer goods Industrial Production: Consumer Goods (Index 2012=100) | 5 |  |  | X | X
24 | IPMAT | Materials (Index 2012=100) | 5 |  |  |  | X
25 | IPDMAT | Durable Materials (Index 2012=100) | 5 |  |  |  | X
26 | IPNMAT | Nondurable Materials (Index 2012=100) | 5 |  |  |  | X
27 | IPDCONGD | Durable Consumer Goods (Index 2012=100) | 5 |  |  |  | X
28 | IPB51110SQ | Durable Goods: Automotive products (Index 2012=100) | 5 |  |  |  | X
29 | IPNCONGD | Nondurable Consumer Goods (Index 2012=100) | 5 |  |  |  | X
30 | IPBUSEQ | Business Equipment (Index 2012=100) | 5 |  |  |  | X
31 | IPB51220SQ | Consumer energy products (Index 2012=100) | 5 |  |  |  | X
32 | CUMFNS | Capacity Utilization: Manufacturing (SIC) (Percent of Capacity) | 1 |  |  |  | X
33 | IPMANSICS | Industrial Production: Manufacturing (SIC) (Index 2012=100) | 5 |  |  |  | X
34 | IPB51222S | Industrial Production: Residential Utilities (Index 2012=100) | 5 |  |  |  | X
35 | IPFUELS | Industrial Production: Fuels (Index 2012=100) | 5 |  |  |  | X
36 | PAYEMS | Emp:Nonfarm All Employees: Total nonfarm (Thousands of Persons) | 5 |  |  | X | X
37 | USPRIV | All Employees: Total Private Industries (Thousands of Persons) | 5 |  |  |  | X
38 | MANEMP | All Employees: Manufacturing (Thousands of Persons) | 5 |  |  |  | X
39 | SRVPRD | All Employees: Service-Providing Industries (Thousands of Persons) | 5 |  |  |  | X
40 | USGOOD | All Employees: Goods-Producing Industries (Thousands of Persons) | 5 |  |  |  | X
41 | DMANEMP | All Employees: Durable goods (Thousands of Persons) | 5 |  |  |  | X
42 | NDMANEMP | All Employees: Nondurable goods (Thousands of Persons) | 5 |  |  |  | X
43 | USCONS | All Employees: Construction (Thousands of Persons) | 5 |  |  |  | X
44 | USEHS | All Employees: Education & Health Services (Thousands of Persons) | 5 |  |  |  | X
45 | USFIRE | All Employees: Financial Activities (Thousands of Persons) | 5 |  |  |  | X
46 | USINFO | All Employees: Information Services (Thousands of Persons) | 5 |  |  |  | X
47 | USPBS | All Employees: Professional & Business Services (Thousands of Persons) | 5 |  |  |  | X
48 | USLAH | All Employees: Leisure & Hospitality (Thousands of Persons) | 5 |  |  |  | X
49 | USSERV | All Employees: Other Services (Thousands of Persons) | 5 |  |  |  | X
50 | USMINE | All Employees: Mining and logging (Thousands of Persons) | 5 |  |  |  | X
51 | USTPU | All Employees: Trade, Transportation & Utilities (Thousands of Persons) | 5 |  |  |  | X
52 | USGOVT | All Employees: Government (Thousands of Persons) | 5 |  |  |  | X
53 | USTRADE | All Employees: Retail Trade (Thousands of Persons) | 5 |  |  |  | X
54 | USWTRADE | All Employees: Wholesale Trade (Thousands of Persons) | 5 |  |  |  | X
55 | CES9091000001 | All Employees: Government: Federal (Thousands of Persons) | 5 |  |  |  | X
56 | CES9092000001 | All Employees: Government: State Government (Thousands of Persons) | 5 |  |  |  | X
57 | CES9093000001 | All Employees: Government: Local Government (Thousands of Persons) | 5 |  |  |  | X
58 | CE16OV | Civilian Employment (Thousands of Persons) | 5 | X | X | X | X
59 | CIVPART | Civilian Labor Force Participation Rate (Percent) | 2 |  |  |  | X
60 | UNRATE | Civilian Unemployment Rate (Percent) | 2 | X | X | X | X
61 | UNRATESTx | Unemployment Rate less than 27 weeks (Percent) | 2 |  |  |  | X
62 | UNRATELTx | Unemployment Rate for more than 27 weeks (Percent) | 2 |  |  | X | X
63 | LNS14000012 | Unemployment Rate - 16 to 19 years (Percent) | 2 |  |  |  | X
64 | LNS14000025 | Unemployment Rate - 20 years and over, Men (Percent) | 2 |  |  |  | X
65 | LNS14000026 | Unemployment Rate - 20 years and over, Women (Percent) | 2 |  |  |  | X
66 | UEMPLT5 | Number of Civilians Unemployed - Less Than 5 Weeks (Thousands of Persons) | 5 |  |  |  | X
67 | UEMP5TO14 | Number of Civilians Unemployed for 5 to 14 Weeks (Thousands of Persons) | 5 |  |  |  | X
68 | UEMP15T26 | Number of Civilians Unemployed for 15 to 26 Weeks (Thousands of Persons) | 5 |  |  |  | X
69 | UEMP27OV | Number of Civilians Unemployed for 27 Weeks and Over (Thousands of Persons) | 5 |  |  |  | X
70 | AWHMAN | Average Weekly Hours of Production and Nonsupervisory Employees: Manufacturing (Hours) | 1 |  |  | X | X
71 | AWOTMAN | Average Weekly Overtime Hours of Production and Nonsupervisory Employees: Manufacturing (Hours) | 2 |  |  | X | X
72 | HWIx | Help-Wanted Index | 1 |  |  |  | X
73 | CES0600000007 | Average Weekly Hours of Production and Nonsupervisory Employees: Goods-Producing | 2 | X | X | X | X
74 | CLAIMSx | Initial Claims | 5 |  |  | X | X
75 | HOUST | Housing Starts: Total: New Privately Owned Housing Units Started | 5 |  | X | X | X
76 | HOUST5F | Privately Owned Housing Starts: 5-Unit Structures or More | 5 |  |  |  | X
77 | HOUSTMW | Housing Starts in Midwest Census Region (Thousands of Units) | 5 |  |  |  | X
78 | HOUSTNE | Housing Starts in Northeast Census Region (Thousands of Units) | 5 |  |  |  | X
79 | HOUSTS | Housing Starts in South Census Region (Thousands of Units) | 5 |  |  |  | X
80 | HOUSTW | Housing Starts in West Census Region (Thousands of Units) | 5 |  |  |  | X
81 | RSAFSx | Real Retail and Food Services Sales (Millions of Chained 2012 Dollars) | 5 |  |  | X | X
82 | AMDMNOx | Real Manufacturers New Orders: Durable Goods (Millions of 2012 Dollars) | 5 |  |  |  | X
83 | AMDMUOx | Real Value of Manufacturers Unfilled Orders for Durable Goods Industries | 5 |  |  |  | X
84 | BUSINVx | Total Business Inventories (Millions of Dollars) | 5 |  |  |  | X
85 | ISRATIOx | Total Business: Inventories to Sales Ratio | 2 |  |  |  | X
86 | PCECTPI | Personal Consumption Expenditures: Chain-type Price Index | 6 | X | X | X | X
87 | PCEPILFE | Personal Consumption Expenditures Excluding Food and Energy | 6 | X | X | X | X
88 | GDPCTPI | Gross Domestic Product: Chain-type Price Index | 6 | X | X | X | X
89 | GPDICTPI | Gross Private Domestic Investment: Chain-type Price Index | 6 |  |  | X | X
90 | IPDBS | Business Sector: Implicit Price Deflator (Index 2012=100) | 6 |  |  | X | X

ID | FRED Code | Description | Transformation Codes | S | M | L | XL
---|---|---|---|---|---|---|---
91 | DGDSRG3Q086SBEA | Personal consumption expenditures: Goods | 6 |  |  | X | X
92 | DDURRG3Q086SBEA | Personal consumption expenditures: Durable goods | 6 |  |  | X | X
93 | DSERRG3Q086SBEA | Personal consumption expenditures: Services | 6 |  |  | X | X
94 | DNDGRG3Q086SBEA | Personal consumption expenditures: Nondurable goods | 6 |  |  | X | X
95 | DHCERG3Q086SBEA | Personal consumption expenditures: Services: Household consumption expenditures | 6 |  |  |  | X
96 | DMOTRG3Q086SBEA | Personal consumption expenditures: Durable goods: Motor vehicles and parts | 6 |  |  |  | X
97 | DFDHRG3Q086SBEA | Personal consumption expenditures: Durable goods: Furnishings and durable household equipment | 6 |  |  |  | X
98 | DREQRG3Q086SBEA | Personal consumption expenditures: Durable goods: Recreational goods and vehicles | 6 |  |  |  | X
99 | DODGRG3Q086SBEA | Personal consumption expenditures: Durable goods: Other durable goods | 6 |  |  |  | X
100 | DFXARG3Q086SBEA | Personal consumption expenditures: Nondurable goods: Food and beverages purchased for off-premises consumption | 6 |  |  |  | X
101 | DCLORG3Q086SBEA | Personal consumption expenditures: Nondurable goods: Clothing and footwear | 6 |  |  |  | X
102 | DGOERG3Q086SBEA | Personal consumption expenditures: Nondurable goods: Gasoline and other energy goods | 6 |  |  |  | X
103 | DONGRG3Q086SBEA | Personal consumption expenditures: Nondurable goods: Other nondurable goods | 6 |  |  |  | X
104 | DHUTRG3Q086SBEA | Personal consumption expenditures: Services: Housing and utilities | 6 |  |  |  | X
105 | DHLCRG3Q086SBEA | Personal consumption expenditures: Services: Health care | 6 |  |  |  | X
106 | DTRSRG3Q086SBEA | Personal consumption expenditures: Transportation services | 6 |  |  |  | X
107 | DRCARG3Q086SBEA | Personal consumption expenditures: Recreation services | 6 |  |  |  | X
108 | DFSARG3Q086SBEA | Personal consumption expenditures: Services: Food services and accomodations | 6 |  |  |  | X
109 | DIFSRG3Q086SBEA | Personal consumption expenditures: Financial services and insurance | 6 |  |  |  | X
110 | DOTSRG3Q086SBEA | Personal consumption expenditures: Other services | 6 |  |  |  | X
111 | CPIAUCSL | Consumer Price Index for All Urban Consumers: All Items | 6 | X | X | X | X
112 | CPILFESL | Consumer Price Index for All Urban Consumers: All Items Less Food & Energy | 6 | X | X | X | X
113 | WPSFD49207 | Producer Price Index by Commodity for Finished Goods | 6 |  |  | X | X
114 | PPIACO | Producer Price Index for All Commodities | 6 |  |  | X | X
115 | WPSFD49502 | Producer Price Index by Commodity for Finished Consumer Goods | 6 |  |  |  | X
116 | WPSFD4111 | Producer Price Index by Commodity for Finished Consumer Foods | 6 |  |  |  | X
117 | PPIIDC | Producer Price Index by Commodity Industrial Commodities | 6 |  |  |  | X
118 | WPSID61 | Producer Price Index by Commodity Intermediate Materials: Supplies & Components | 6 |  |  |  | X
119 | WPU0561 | Producer Price Index by Commodity for Fuels and Related Products and Power | 5 |  |  | X | X
120 | OILPRICEx | Real Crude Oil Prices: West Texas Intermediate (WTI) - Cushing, Oklahoma | 5 |  |  | X | X
121 | WPSID62 | Producer Price Index: Crude Materials for Further Processing | 6 |  |  |  | X
122 | PPICMM | Producer Price Index: Commodities: Metals and metal products: Primary nonferrous metals | 6 |  |  |  | X
123 | CPIAPPSL | Consumer Price Index for All Urban Consumers: Apparel | 6 |  |  | X | X
124 | CPITRNSL | Consumer Price Index for All Urban Consumers: Transportation | 6 |  |  | X | X
125 | CPIMEDSL | Consumer Price Index for All Urban Consumers: Medical Care | 6 |  |  | X | X
126 | CUSR0000SAC | Consumer Price Index for All Urban Consumers: Commodities | 6 |  |  | X | X
127 | CES2000000008x | Real Average Hourly Earnings of Production and Nonsupervisory Employees: Construction | 5 |  |  | X | X
128 | CES3000000008x | Real Average Hourly Earnings of Production and Nonsupervisory Employees: Manufacturing | 5 |  |  | X | X
129 | COMPRNFB | Nonfarm Business Sector: Real Compensation Per Hour (Index 2012=100) | 5 |  |  | X | X
130 | CES0600000008 | Average Hourly Earnings of Production and Nonsupervisory Employees: | 6 | X | X | X | X
131 | FEDFUNDS | Effective Federal Funds Rate (Percent) | 2 | X | X | X | X
132 | TB3MS | 3-Month Treasury Bill: Secondary Market Rate (Percent) | 2 |  |  | X | X
133 | TB6MS | 6-Month Treasury Bill: Secondary Market Rate (Percent) | 2 |  |  | X | X
134 | GS1 | 1-Year Treasury Constant Maturity Rate (Percent) | 2 |  | X | X | X
135 | GS10 | 10-Year Treasury Constant Maturity Rate (Percent) | 2 | X | X | X | X
136 | AAA | Moody’s Seasoned Aaa Corporate Bond Yield (Percent) | 2 |  |  |  | X
137 | BAA | Moody’s Seasoned Baa Corporate Bond Yield (Percent) | 2 |  |  | X | X
138 | BAA10YM | Moody’s Seasoned Baa Corporate Bond Yield Relative to Yield on 10-Year Treasury | 1 |  |  |  | X
139 | TB6M3Mx | 6-Month Treasury Bill Minus 3-Month Treasury Bill, secondary market (Percent) | 1 |  |  | X | X
140 | GS1TB3Mx | 1-Year Treasury Constant Maturity Minus 3-Month Treasury Bill, secondary market | 1 |  |  | X | X
141 | GS10TB3Mx | 10-Year Treasury Constant Maturity Minus 3-Month Treasury Bill, secondary market | 1 |  |  | X | X
142 | CPF3MTB3Mx | 3-Month Commercial Paper Minus 3-Month Treasury Bill, secondary market | 1 |  |  | X | X
143 | GS5 | 5-Year Treasury Constant Maturity Rate | 2 |  |  | X | X
144 | TB3SMFFM | 3-Month Treasury Constant Maturity Minus Federal Funds Rate | 1 |  |  | X | X
145 | T5YFFM | 5-Year Treasury Constant Maturity Minus Federal Funds Rate | 1 |  |  | X | X
146 | AAAFFM | Moody’s Seasoned Aaa Corporate Bond Minus Federal Funds Rate | 1 |  |  | X | X
147 | M1REAL | Real M1 Money Stock | 5 |  |  | X | X
148 | M2REAL | Real M2 Money Stock | 5 |  | X | X | X
149 | BUSLOANSx | Real Commercial and Industrial Loans, All Commercial Banks | 5 |  |  | X | X
150 | CONSUMERx | Real Consumer Loans at All Commercial Banks | 5 |  |  | X | X
151 | NONREVSLx | Total Real Nonrevolving Credit Owned and Securitized, Outstanding | 5 |  |  | X | X
152 | REALLNx | Real Real Estate Loans, All Commercial Banks | 5 |  |  | X | X
153 | TOTALSLx | Total Consumer Credit Outstanding | 5 |  |  | X | X
154 | TOTRESNS | Total Reserves of Depository Institutions | 6 |  | X | X | X
155 | NONBORRES | Reserves Of Depository Institutions, Nonborrowed | 7 |  | X | X | X
156 | DTCOLNVHFNM | Consumer Motor Vehicle Loans Outstanding Owned by Finance Companies | 6 |  |  |  | X
157 | DTCTHFNM | Total Consumer Loans and Leases Outstanding Owned and Securitized by Finance Companies | 6 |  |  |  | X
158 | INVEST | Securities in Bank Credit at All Commercial Banks | 6 |  |  |  | X
159 | TABSHNOx | Real Total Assets of Households and Nonprofit Organizations | 5 |  |  |  | X
160 | EXSZUSx | Switzerland / U.S. Foreign Exchange Rate | 5 |  |  | X | X
161 | EXJPUSx | Japan / U.S. Foreign Exchange Rate | 5 |  |  | X | X
162 | EXUSUKx | U.S. / U.K. Foreign Exchange Rate | 5 |  |  | X | X
163 | EXCAUSx | Canada / U.S. Foreign Exchange Rate | 5 |  |  | X | X
164 | S.P.500 | S&P’s Common Stock Price Index: Composite | 5 |  | X | X | X
165 | S.P..indust | S&P’sCommon Stock Price Index: Industrials | 5 |  |  |  | X
166 | S.P.div.yield | S&P’s Composite Common Stock: Dividend Yield | 2 |  |  |  | X

Notes: This table provides an overview of the dataset employed. The
transformation codes are applied to each time series $\bm{Y}_{j}$ and
described in McCracken and Ng (2020): (1) no transformation; (2) $\Delta
y_{jt}$; (3) $\Delta^{2}y_{jt}$; (4) $\log(y_{jt})$; (5) $\Delta\log(y_{jt})$;
(6) $\Delta^{2}\log(y_{jt})$; (7) $\Delta(y_{jt}/y_{jt-1}-1)$. ’X’ marks the
inclusion of a variable into one of the datasets.