#  Coarsened Bayesian VARs
Correcting BVARs for Incorrect Specification††thanks:  Corresponding author:
Massimiliano Marcellino, Department of Economics, Bocconi University. Address:
Via Roentgen 1, 20122, Milano, Italy. Email:
<EMAIL_ADDRESS>Huber gratefully acknowledges financial
support from the Austrian Science Fund (FWF, grant no. ZK 35). We would like
to thank Luca Barbaglia, Tony Chernis, Todd Clark, Niko Hauzenberger, Gary
Koop, Dimitris Korobilis, Christian Matthes, Stuart McIntyre, Elmar Mertens,
Luca Onorante, Anthoulla Phella, Francesco Ravazollo, Cathy Yi-Hsuan Chen and
participants at research seminars at the University of Strathclyde and
University of Glasgow for helpful comments and suggestions.

Florian Huber University of Salzburg International Institute for Applied
Systems Analysis Massimiliano Marcellino Bocconi University, CEPR, IGIER,
BAFFI, BIDSA

Abstract

Model mis-specification in multivariate econometric models can strongly
influence estimates of quantities of interest such as structural parameters,
forecast distributions or responses to structural shocks, even more so if
higher-order forecasts or responses are considered, due to parameter
convolution. We propose a simple method for addressing these specification
issues in the context of Bayesian VARs. Our method, called coarsened Bayesian
VARs (cBVARs), replaces the exact likelihood with a coarsened likelihood that
takes into account that the model might be mis-specified along important but
unknown dimensions. Coupled with a conjugate prior, this results in a
computationally simple model. As opposed to more flexible specifications, our
approach avoids overfitting, is simple to implement and estimation is fast.
The resulting cBVARs perform well in simulations for several types of mis-
specification. Applied to US data, cBVARs improve point and density forecasts
compared to standard BVARs, and lead to milder but more persistent negative
effects of uncertainty shocks on output.

JEL Codes: C11, C32, C53

Keywords: Bayesian VARs, Conjugate priors, likelihood tempering, approximate
inference.

##  1 Introduction

Inference in multivariate econometric models is often plagued by issues such
as measurement errors, structural breaks, outliers, or non-Gaussian features,
such as asymmetric and fat tailed errors. Especially in recessionary episodes
or when large shocks hit the economy (e.g., the global financial crisis or the
Covid-19 pandemic), violations of the standard model assumptions become
prevalent. Moreover, selecting appropriate variables is necessary to strike a
balance between a large model, which might include many irrelevant predictors,
and a small tractable model, which could omit important information. These
issues can be partly addressed by making the model larger, more flexible, and
non-Gaussian, but this typically leads to substantial increases in
computational complexity and risks overfitting the data. For example, Cogley
and Sargent (2005); Primiceri (2005); Kalli and Griffin (2018); Huber, et al.
(2020); Clark, et al. (in-press); Goulet Coulombe, et al. (2022); Korobilis,
et al. (2021); Karlsson, Mazur, and Nguyen (2023) propose models that allow
for time-variation and non-linearities in the conditional mean, the
conditional variance or both. These methods often outperform their linear
counterparts in terms of estimation reliability and predictive accuracy.
However, they are difficult to tune and the computational burden increases
markedly with the size of the dataset. Similarly, to address the omitted
variable problem it is common to rely on factor models (e.g., Stock and
Watson, 2002; Bernanke, Boivin, and Eliasz, 2005; Kaufmann and Schumacher,
2019) or large Bayesian VARs (see, e.g., Bańbura, Giannone, and Reichlin,
2010; Koop, 2013; Chan, 2022; Gefang, Koop, and Poon, 2023). Factor models,
however, require the selection of the number of factors and call for filtering
techniques to estimate the latent factors. Large VARs, on the other hand, are
often specified to be linear and homoskedastic. Both assumptions are necessary
to retain conjugacy, and thus enable fast estimation, but could be
unrealistic. And one could introduce a separate stochastic process to capture
measurement errors (see, e.g., Cogley and Sargent, 2015), but doing so
requires knowledge about the nature of the measurement error and is thus not
easy to apply in general.

In summary, while there exist methods to handle specific types of mis-
specification of standard, simple, econometric models, these methods are not
commonly used in practice, due to their complexity and computational costs, in
addition to the fact that the type of mis-specification is often unknown.
Small, linear, Gaussian econometric models, such as vector autoregressions
(VARs), remain the workhorse of applied macroeconomists. Hence, our objective
in this paper is to modify standard (Bayesian) VARs (see, e.g., Kadiyala and
Karlsson, 1997; Sims and Zha, 1998; Carriero, Clark, and Marcellino, 2015;
Giannone, Lenza, and Primiceri, 2015b) to make them more robust to general and
unknown forms of mis-specification, without changing the simple model
specification and retaining computational simplicity and efficiency. We build
on a recent paper, Miller and Dunson (2018), and propose a robust version of a
conjugate VAR. We call this model coarsened Bayesian VAR (cBVAR). The cBVAR
replaces the exact likelihood with a coarsened likelihood that takes general
and unknown forms of model mis-specification into account. The key idea is
that instead of conditioning on the observed data $\bm{Y}$, one conditions on
the event that the difference in the sampling distribution of the observed
data $P_{\bm{Y}}$ and of the idealized data $P_{\bm{Y}^{*}}$ (with
$\bm{Y^{*}}$ denoting the idealized data) is smaller than a constant number
$c$, where the idealized data are such that all the model assumptions are
valid. The conditioning event effectively requires a distance metric and we
use relative entropy. Miller and Dunson (2018) provide a simple approximation
of the coarsened likelihood under relative entropy distance that reduces to
raising the standard likelihood to a fraction. This fraction can be
interpreted as a learning rate and controls how much the model learns from the
data. The smaller the extent of the mis-specification, the closer the fraction
is to one. Viceversa, for large mis-specification the fraction is close to
zero, making the posteriors similar to the priors, as the observed data are
considered as ”unreliable” (at least for learning via a standard BVAR).

Under a standard natural conjugate prior, we derive closed form posterior
distributions for the cBVAR, and provide a simple and scalable algorithm to
estimate the model. The resulting cBVAR relies on a single parameter (the
learning rate), which can be selected either by cross-validation or using an
information criterion, whereas the other hyperparameters are set using the
coarsened marginal likelihood. This constitutes our main technical
contribution.

To assess the empirical performance of the cBVAR, we first use simulated data,
generated from a variety of data generating processes (DGPs), characterized by
different types of mis-specification. Specifically, we compare the impulse
responses functions (IRFs) obtained from the cBVAR and from a standard BVAR
(both conditional on the true identification strategy) to the true IRFs
generated by DGPs featuring alternative combinations of non-Gaussianity,
stochastic volatility, random coefficients, and larger information sets than
those considered in the BVAR. It turns out that the standard conjugate BVAR
produces IRFs which are far off from the true ones, much more so than those
from the cBVAR (except for the case where the DGP coincides with the BVAR).
The precision of the cBVAR IRFs depends on the learning rate, and selecting
the latter with our proposed information criterion generally improves upon the
BVAR.

Next, we use a forecasting application to show that coarsening can often
lessen the effects of mis-specification for actual US data. Specifically, we
focus on point and density forecasting monthly US unemployment, inflation and
short-term interest rate, using three model sizes (small, medium, and large),
Our findings indicate that gains from coarsening the likelihood are largest if
models are small (i.e. the risk of omitted variable bias is particularly
pronounced), with relative gains decreasing if the model becomes larger, but
still present and systematic, in particular for longer forecast horizons. As
macroeconomists tend to prefer small models for empirical analysis, this
finding can be of substantial practical use. Moreover, the gains from
coarsening are generally larger when the Covid period is included in the
evaluation sample, in line with the increased mis-specification of the
standard BVAR during uncommon times. As with the simulated data, the
predictive accuracy strongly depends on the learning rate, and the information
criterion provides good guidance on how to set it.

Finally, as an illustration of the effects of coarsening the likelihood for
structural analysis, we compute IRFs to uncertainty shocks from a standard
BVAR and the associated cBVAR. It turns out that with the latter the
uncertainty shocks produce milder but more persistent negative responses of
output, the more so the smaller the learning rate.

The remainder of the paper is structured in the following way. The next
section provides an intuitive introduction to the coarsening idea of Miller
and Dunson (2018). This section mainly serves to set the stage for our model
developments that follow in Section 3 in which we derive the coarsened
likelihood for the BVAR and then back out the corresponding posterior
distributions. Section 4 provides synthetic data evidence while Section 5
includes our real-data forecasting exercise. Section 6 deals with the effects
of uncertainty shocks on real activity and Section 7 briefly discusses
possible model extensions. The final section offers a brief summary and
conclusions.

##  2 Coarsened posterior distributions

This section follows the discussion in Miller and Dunson (2018) and motivates
our use of the coarsened likelihood function to back out a coarsened posterior
distribution. To set the stage, the data we observe is defined as
$\bm{Y}=(\bm{y}_{1},\dots,\bm{y}_{T})^{\prime}$ which is a $T\times M$ matrix
with empirical sampling distribution
$P_{\bm{Y}}=\frac{1}{T}\sum_{t=1}^{T}\delta_{y_{t}}$. However, the unobserved
idealized dataset is given by $\bm{Y^{*}}$ with associated sampling
distribution $P_{\bm{Y}^{*}}$ defined analogously to $P_{\bm{Y}}$. Suppose
that $\bm{Y}$ arises from $\bm{Y}^{*}$ through a stochastic process which is,
unfortunately, unknown and let $d(P_{\bm{Y}},P_{\bm{Y}^{*}})$ denote a
distance function with $d(\cdot,\cdot)\geq 0$. We assume that
$d(P_{\bm{Y}},P_{\bm{Y}^{*}})<c$ for some threshold parameter $c\geq 0$.
Particular examples of such stochastic processes are simple measurement error
models that assume that $\bm{y}_{t}=\bm{y}_{t}^{*}+\bm{\varpi}$ with
$\bm{\varpi}$ denoting a random measurement error with a particular
distribution such as a Gaussian or multivariate student t or nonlinear models
that assume that $\bm{y}_{t}=g(\bm{y}_{t}^{*})$ for some nonlinear function
$g$.

Standard Bayesian practice would specify a likelihood
$p(\bm{Y}|\bm{\vartheta})$, with $\bm{\vartheta}$ denoting a vector of
parameters, and a prior on $\bm{\vartheta}\sim p(\bm{\vartheta})$, both of
which are then used to back out the posterior distribution:

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

This procedure, however, neglects the fact that $\bm{Y}$ is a corrupted
version of $\bm{Y}^{*}$. An alternative would be to set up an auxiliary model
$p(\bm{Y}|\bm{Y}^{*})$. However, this is not feasible since, in applications
with actual data, the process giving rise to $\bm{Y}$ from $\bm{Y}^{*}$ is not
known (or might render the resulting model computationally involved). Miller
and Dunson (2018) propose a simple alternative. Instead of conditioning on
$\bm{Y}$ when forming the posterior, one could condition on
$d(P_{\bm{Y}},P_{\bm{Y}^{*}})<c$ instead. Since $c$ is typically unknown to
the researcher one could specify a prior on it, i.e. $C\sim\pi$. Doing so
leads to the coarsened posterior:

$p(\bm{\vartheta}|\hat{d}(P_{\bm{Y}},P_{\bm{Y}^{*}})<C)\propto\mathbb{P}(\hat{d}(P_{\bm{Y}},P_{\bm{Y}^{*}})<C|\bm{\vartheta})~{}p(\bm{\vartheta}),$

where the probability
$\mathbb{P}(\hat{d}(P_{\bm{Y}},P_{\bm{Y}^{*}})<C|\bm{\vartheta})$ can be
interpreted as a likelihood function. Notice that this coarsened likelihood is
generally not a probability distribution of $\bm{Y}$ given $\bm{\vartheta}$.

Different alternatives for the distance function $d$ can be used. The choice
of the discrepancy function can be based on the expected (or more feared) type
of mis-specification. For example, as noted by Miller and Dunson (2018),
robustness to outliers requires a discrepancy function that is little
sensitive to movements of small amounts of probability mass to the outlying
region (e.g., the first Wasserstein distance). Among different distance
functions $d$, one particularly appealing variant stands out, which considers
differences in the entire likelihood function: the relative entropy. Moreover,
using the relative entropy between $P_{\bm{Y}}$ and $P_{\bm{Y}^{*}}$ and an
exponential prior on $C\sim Exp(\alpha)$ leads to a particularly simple and
accurate approximation of the coarsened posterior. In this case, it can be
approximated as follows:

$p(\bm{\vartheta}|\hat{d}(P_{\bm{Y}},P_{\bm{Y}^{*}})<C)\mathrel{\raisebox{3.01385pt}{\mbox{$\underset{\vbox{\hbox{$\sim$}}}{\vbox{\hbox{$\propto$}}}$}}}p(\bm{\vartheta})~{}\prod_{t=1}^{T}p(\bm{y}_{t}|\bm{\vartheta},\bm{y}_{1},\dots,\bm{y}_{t-1})^{\zeta_{T}},$
(1)

with
$\mathrel{\raisebox{3.01385pt}{\mbox{$\underset{\vbox{\hbox{$\sim$}}}{\vbox{\hbox{$\propto$}}}$}}}$
denoting approximately proportional to. Equation (1) implies that the
coarsened posterior is simply equal to the prior times a tempered
likelihood.111This particular form of the coarsened posterior resembles the
intermediate approximating distribution of sequential Monte Carlo (SMC)
methods. The main difference is that the learning rate in SMC grows from $0$
(i.e. the likelihood plays no role) to $1$ (i.e. one obtains the uncoarsened
posterior). If one would fix the learning rate in SMC to $\zeta_{T}$, the
result would be a sequential coarsened posterior.

The standard likelihood is raised by the learning rate $\zeta_{T}$ which is
given by:

$\zeta_{T}=\frac{\alpha}{\alpha+T}.$

To understand the role of the coarsening parameter $\alpha$, it is useful to
note that the resulting posterior corresponds to shrinking the sample size
from $T$ to $\zeta_{T}T$. Hence, when $\alpha$ is small and coarsening is
relevant, the sample size is substantially reduced and the posterior is much
less concentrated. This is sensible, as substantial coarsening should be
associated to a larger extent of mis-specification, and hence uncertainty
should indeed be larger. Yet, if $\alpha$ is set to a small number when the
model is instead approximately correct, posterior credible sets will be too
large and model complexity likely under-estimated. Therefore, the choice of
the parameter $\alpha$ is very important for coarsening to be helpful.

It is worth stressing that this form of the coarsened posterior does not
require the computation of the relative entropy term since it is absorbed in
the constant of proportionality (and thus independent of $\bm{\vartheta}$). In
macroeconomic time series it might make sense that $\alpha$ is a function of
$T$ as well. This is because longer time series have the potential to be
corrupted by extraordinary events that imply appreciable departures from a
linear and homoskedastic model. In this regard, Miller and Dunson (2018)
suggest that as a simple starting point one could set $\alpha=T$, implying
that you need to observe at least $T$ additional observations to be robust to
a presumed perturbation. Alternatively, $\alpha$ could be set as a decreasing
function in $T$, effectively capturing the notion that longer series tend to
feature more outliers, structural breaks and other non-linear features.

An interesting question is whether the choice of a specific prior, in
particular a robust one, could make coarsening less relevant. As discussed by
Miller and Dunson (2018), this is not the case since a robust prior makes the
results less sensitive to the choice of the prior but the importance of mis-
specification of the likelihood remains the same, and the likelihood dominates
the prior when the sample size grows (even though the coarsened posterior does
not concentrate as $T$ diverges just because of coarsening). A related issue
is whether the choice of the prior matters more in the coarsened than in the
standard case, since the ”weight” of the likelihood decreases, particularly so
when $\alpha$ is low. Intuitively, this makes sense. Conditional on the prior,
if we believe that the information contained in the likelihood is severely
corrupted (either through wrongly specifying the likelihood or measurement
errors, both of which fall into our definition of mis-specification) a good
choice would be to downweight this piece of information.

It is also worth discussing the relationship of coarsening with robust control
theory and with entropic tilting, to clarify they they are quite different. In
robust control theory (see e.g. Hansen, et al. (2006), Hansen and Sargent
(2008)) the decision maker has one reference model but she evaluates a
decision rule under a set of alternative models that are perturbed versions of
the reference model. Hansen, et al. (2006) measure the difference between the
reference and perturbed models using relative entropy, taking the maximum
value of the difference as a parameter that measures the set of perturbations
against which the decision maker seeks robustness (and restricts the extent of
model misspeciﬁcation). They also provide conditions that permit to consider
the perturbed models as the multiple priors that appear in the max–min
expected utility theory of Gilboa and Schmeidler (1989). Instead, in our
context, the prior is unique, we condition on the fact that the reference
model can be at a certain (entropic) distance from the true model, and we
contaminate the likelihood to take that into account when forming the
posterior distribution of the model parameters.

Entropic tilting (see e.g. Robertson, Tallman, and Whiteman (2005) instead
minimizes the Kullback-Leibler distance between the predictive density under
restrictions and the unconditional one. Coarsening does not aim to reduce the
distance between the model in use and the true one, rather it takes it as
given and assumes that it is smaller than a certain value, and tries to
robustify estimation and inference for the model in use.

To sum up the discussion so far, the coarsened posterior under a relative
entropy distance metric can be approximated through a simple power posterior
with learning rate $\zeta_{T}$. In principle, the value of $\alpha$ controls
the amount of mis-specification the researcher expects from a given dataset.
Hence, choosing $\alpha$ wisely is key for robust statistical inference and in
the next sections we offer some guidance on how to select it. But before doing
so, we discuss a simple yet practically relevant example.

######  Example 2.1.

To illustrate the concept of the coarsened posterior, we now focus on a simple
multivariate example. In our example, we set up a DGP that matches the
dynamics of US inflation, unemployment and short-term interest rates well.
These three focus variables are often analyzed, closely monitored by policy
makers and form the basis of the small New Keynesian model (Clarida, Gali, and
Gertler, 1999).

Suppose that the data is generated by a VAR(2) model as follows:

$\bm{y}_{t}=\bm{A}_{1}\bm{y}_{t-1}+\bm{A}_{2}\bm{y}_{t-2}+\bm{Q}\bm{\varepsilon}_{t},\quad\varepsilon_{jt}\sim{t}_{3}(0,1)\text{
for }j=1,2.$

where the VAR coefficients and the lower Cholesky factor of the error
covariances, labeled $\bm{Q}$, are given by:

$\bm{A}_{1}=\begin{pmatrix}1.60&0.09&0.32\\\ -0.16&1.54&-0.49\\\
0.02&0.00&1.01\end{pmatrix},\quad\bm{A}_{2}=\begin{pmatrix}-0.61&-0.09&-0.22\\\
0.16&-0.57&0.53\\\
-0.02&0.04&-0.12\end{pmatrix},\quad\bm{Q}=\begin{pmatrix}0.30&0.00&0.00\\\
0.00&0.28&0.00\\\ 0.17&-0.28&0.65\end{pmatrix}.$

These values are obtained by estimating a VAR(2) model on US unemployment,
inflation and the short-term interest rate using OLS and monthly data. We
initialize the model by setting $\bm{y}_{0}=\bm{y}_{1}=\bm{0}_{M}$. Notice
that the structural shocks in $\bm{\varepsilon}_{t}$ follow independent
t-distributions with three degrees of freedom.

In a VAR, interest often centers on highly nonlinear functions of the
parameters such as impulse response functions (IRFs) or multi-step-ahead
forecasts. In typical conjugate VARs such as the ones proposed in Sims and Zha
(1998) or Giannone, Lenza, and Primiceri (2015b), shocks are assumed to be
Gaussian. However, in light of our simulation design this assumption would
imply a mis-specified model and estimates of the IRFs or forecast
distributions might be biased. We illustrate this and show how the model
proposed in Section 3 can be used to substantially reduce the bias of the
estimated IRFs to a one standard deviation shock to $\varepsilon_{2t}$.

Notes: The orange lines refer to the 5/95 percentiles of the posterior
distribution of the IRFs associated with the BVAR while the gray shaded area
refers to the 5/95 percentiles of the posterior distribution of the cBVAR. The
solid black line is the posterior median of the impulse responses of the
cBVAR. The dark blue dots are the true IRFs. Figure 1: Impulse responses to a
one standard deviation shock in $\varepsilon_{2t}$

Figure 1 shows the posterior quantiles ($5^{th}$, $95^{th}$) of the IRFs
associated with the standard BVAR (solid, orange lines) and of the cBVAR (gray
shaded areas). Both models feature 12 lags. The median of the cBVAR IRFs is in
solid black and the true values of the IRFs are the dark blue dots. When the
one-step-ahead forecast errors are heavy tailed, the standard BVAR yields
impulse responses that do not include the true IRFs and indicate much stronger
effects. This effect is driven by the fact that the BVAR (even with a
shrinkage prior) estimates larger coefficients to control for model mis-
specification. By contrast, the IRFs arising from cBVAR are much closer to the
truth, with credible sets including the true outcomes almost entirely over the
impulse response horizon. Notice, in addition, that the credible sets of the
cBVAR IRFs are slightly wider. This effect can be traced back to the fact that
we are tempering the likelihood and the posterior becomes more dispersed (see
the discussion towards the end of Sub-section 3.2).

##  3 Coarsened Bayesian VAR

In this section we develop the cBVAR model. We start by discussing the
coarsened likelihood of the VAR and then briefly sketch the conjugate Gaussian
prior in Sub-section 3.1. We then provide details on the full conditional
posterior distributions and on how we set $\alpha$ in, respectively, Sub-
sections 3.2 and 3.3.

###  3.1 Coarsened likelihood

We consider linear and homoskedastic VAR($p$) models given by:

$\bm{y}_{t}=\bm{a}_{0}+\bm{A}_{1}\bm{y}_{t-1}+\dots+\bm{A}_{p}\bm{y}_{t-p}+\bm{\varepsilon}_{t},$
(2)

with $\bm{A}_{j}~{}(j=1,\dots,M)$ denoting $M\times M$ matrices of
autoregressive coefficients and
$\bm{\varepsilon}_{t}\sim\mathcal{N}(\bm{0},\bm{\Sigma})$ a Gaussian vector
white noise process with zero mean and $M\times M$-dimensional variance-
covariance matrix $\bm{\Sigma}$. The model in Eq. (missing)2 contains
$k=(pM+1)M$ regression coefficients and $v=M(M+1)/2$ free elements in the
error variance-covariance matrix. If $M$ and $p$ are moderately large, the
number of coefficients quickly exceeds the available number of observations
$T$ and the VAR is prone to overfitting.

In full matrix form, the VAR can be written as:

$\bm{Y}=\bm{X}\bm{A}+\bm{E},$ (3)

where $\bm{X}$ is a $T\times K(=pM+1)$ matrix with $t^{th}$ row given by
$\bm{x}^{\prime}_{t}=(1,\bm{y}^{\prime}_{t-1},\dots,\bm{y}^{\prime}_{t-p})$,
$\bm{A}=(\bm{a}_{0},\bm{A}_{1},\dots,\bm{A}_{p})$, and $\bm{E}$ is a $T\times
M$ matrix with typical row given by $\bm{\varepsilon}^{\prime}_{t}$. We can
use Eq. (missing)3 to derive the coarsened likelihood:

$\tilde{p}(\bm{Y}|\bm{A},\bm{\Sigma},\text{Data})=\left(\prod_{t=1}^{T}p(\bm{y}_{t}|\bm{A},\bm{\Sigma})^{\zeta_{T}}\right).$

With a few simple algebraic manipulations, the coarsened likelihood reads:

$\displaystyle\begin{split}\tilde{p}(\bm{Y}|\bm{A},\bm{\Sigma},\text{Data})=|\bm{\Sigma}|^{-\frac{{\zeta_{T}}T}{2}}&\exp\left(-\frac{1}{2}\text{tr}\left(\bm{\Sigma}^{-1}(\bm{A}-\hat{\bm{A}})^{\prime}\underbrace{\zeta_{T}^{1/2}\bm{X}^{\prime}}_{\tilde{\bm{X}}^{\prime}}\underbrace{\bm{X}\zeta_{T}^{1/2}}_{\tilde{\bm{X}}}(\bm{A}-\hat{\bm{A}})\right)\right)\\\
&\times\exp\left(-\frac{1}{2}\text{tr}\left(\bm{\Sigma}^{-1}\underbrace{\bm{U}^{\prime}\zeta_{T}^{1/2}}_{\tilde{\bm{U}}^{\prime}}\underbrace{\bm{U}\zeta_{T}^{1/2}}_{\tilde{\bm{U}}}\right)\right),\end{split}$
(4)

where $|\bullet|$ and $\text{tr}(\bullet)$ denote the determinant and trace of
a matrix, respectively, and $\bm{U}=(\bm{Y}-\bm{X}\hat{\bm{A}})$ is the matrix
of residuals with $\hat{\bm{A}}$ denoting the maximum likelihood/OLS estimate
of $\bm{A}$.

For large values of $M$ and $p$, the number of parameters can quickly exceed
the number of available observations and the OLS estimator ceases to exist.
Hence, regularization is necessary and Bayesian approaches that rely on
specifying priors on $\bm{A}$ and $\bm{\Sigma}$ are commonly employed. In the
next sub-section we will derive the coarsened posterior under a natural
conjugate prior.

###  3.2 Coarsened Bayesian Analysis of the VAR

The likelihood in Eq. (missing)4 can be combined with a natural conjugate
Gaussian - Wishart prior on $\bm{a}=\text{vec}(\bm{A})$ and
$\bm{\Sigma}^{-1}$. The natural conjugate prior is given by (see, e.g.,
Kadiyala and Karlsson, 1997; Koop, 2013; Carriero, Clark, and Marcellino,
2015):

$\bm{a}|\bm{\Sigma}\sim\mathcal{N}(\underline{\bm{a}},\bm{\Sigma}\otimes\overline{\bm{V}})$

with $\underline{\bm{a}}=\text{vec}(\underline{\bm{A}})$ denoting a $k\times
1$ prior mean vector that is obtained by vectorizing a $K\times M$ prior mean
matrix $\underline{\bm{A}}$ and $\overline{\bm{V}}$ is a $K\times K$ prior
covariance matrix. Notice that this prior is specified conditionally on
$\bm{\Sigma}$ and that the prior covariance features a Kronecker structure. On
$\bm{\Sigma}$ we use an inverted Wishart prior:

$\bm{\Sigma}\sim\mathcal{W}^{-1}(\underline{v},\underline{\bm{S}}),$ (5)

where $\underline{v}$ denote the prior degrees of freedom and
$\underline{\bm{S}}$ is a $M\times M$ prior scaling matrix. The prior
hyperparameters
$\underline{\bm{a}},\underline{v},\underline{\bm{V}},\underline{\bm{S}}$ can
be set in various ways. The most popular approach is the one proposed in Sims
and Zha (1998) and Giannone, Lenza, and Primiceri (2015a). Alternatives are
based on using structural models to inform the parameter estimates of the
model (Ingram and Whiteman, 1994; Del Negro and Schorfheide, 2004; De Luigi
and Huber, 2018; Loria, Matthes, and Wang, 2022) or use information on the
long-run behavior of the time series under scrutiny (Giannone, Lenza, and
Primiceri, 2019). In this paper, we follow the approach set forth in Sims and
Zha (1998). We set $\bm{a}_{0}$ so as to capture the notion that the elements
in $\bm{y}_{t}$ evolve according to a random walk a priori. This implies that
$\bm{A}_{1}$ is centered on $\bm{I}_{M}$ and $\bm{A}_{j}~{}(j>2)$ on
$\bm{0}_{M\times M}$. The prior covariance matrix is parameterized by three
parameters $\lambda,\kappa,\xi$ where $\lambda$ control the tightness on the
Minnesota, $\kappa$ on the so-called Sum-of-Coefficients and $\xi$ on the
Single Unit Root prior. We follow Bańbura, Giannone, and Reichlin (2010) and
set $\kappa=\xi=10\times\lambda$ to render the latter two priors mildly
informative.

The joint prior $p(\bm{a},\bm{\Sigma})=p(\bm{a}|\bm{\Sigma})\times
p(\bm{\Sigma})$ can be combined with Eq. (missing)4 to obtain the joint
coarsened posterior distribution:

$\displaystyle\begin{split}\tilde{p}(\bm{a},\bm{\Sigma}|\text{Data})\propto|\bm{\Sigma}|^{-\frac{\underline{v}+M+K+1+\zeta_{T}T}{2}}\times\exp\left(-\frac{1}{2}\text{tr}\left(\bm{\Sigma}^{-1}(\bm{A}-\overline{\bm{A}})^{\prime}\overline{\bm{V}}^{-1}(\bm{A}-\overline{\bm{A}})\right)\right)\\\
\times\exp\left(-\frac{1}{2}\bm{\Sigma}^{-1}\underbrace{\left(\zeta_{T}\bm{Y}^{\prime}\bm{Y}+\underline{\bm{A}}^{\prime}\underline{\bm{V}}^{-1}\underline{\bm{A}}-\overline{\bm{A}}^{\prime}\overline{\bm{V}}^{-1}\overline{\bm{A}}+\underline{\bm{S}}\right)}_{\overline{\bm{S}}}\right)\end{split}$

which implies that

$\displaystyle\bm{a}|\bm{\Sigma},\text{Data}$
$\displaystyle\sim\mathcal{N}(\text{vec}(\overline{\bm{A}}),\bm{\Sigma}\otimes\overline{\bm{V}}),$
$\displaystyle\bm{\Sigma}|\text{Data}$
$\displaystyle\sim\mathcal{W}^{-1}(\underline{v}+\zeta_{T}T,\overline{\bm{S}}).$

The posterior variance and mean of $\bm{a}$ are given by, respectively:

$\displaystyle\overline{\bm{V}}=(\zeta_{T}\bm{X}^{\prime}\bm{X}+\underline{\bm{V}}^{-1})^{-1},\quad\overline{\bm{A}}=\overline{\bm{V}}(\zeta_{T}\bm{X}^{\prime}\bm{Y}+\underline{\bm{V}}^{-1}\underline{\bm{a}}).$

Notice that we end up with the original posterior in the case that
$\alpha\to\infty$ and thus $\zeta_{T}=1$.

The shrinkage parameter $\lambda$ governing the tightness of the prior is set
as follows. The coarsened marginal likelihood (ML) is given by:

$\tilde{p}(\bm{Y}|\lambda)=\int\int\tilde{p}(\bm{Y}|\bm{A},\bm{\Sigma},\lambda)p(\bm{A},\bm{\Sigma})d\bm{A}d\bm{\Sigma}\propto\left(\frac{|\overline{\bm{V}}|}{|\underline{\bm{V}}|}\right)^{M/2}|\overline{\bm{S}}|^{-\frac{\zeta_{T}T+\underline{v}}{2}}.$

We then maximize the coarsened ML with respect to $\lambda$:

$\lambda^{*}=\operatorname*{arg\,max}_{\lambda}\tilde{p}(\bm{Y}|\lambda).$

This choice neglects the fact that $\lambda$ itself might be a random
variable. However, notice that using this plug-in estimate implies that we do
not have to recompute possibly high dimensional quantities such as the
posterior covariance matrix and its Cholesky factor and/or inverse. The fact
that the shrinkage parameter is selected based on the coarsened likelihood
immediately implies that the amount of coarsening impacts the prior tightness
as well.

The coarsened posterior can be interpreted as a posterior that arises from
adjusting the sample size from $T$ to $\zeta_{T}T$ (Miller and Dunson, 2018).
Hence, if the data is corrupted, coarsening the likelihood by $\zeta_{T}$
(that depends on $\alpha$) makes the model tolerant for values of $\bm{A}$ and
$\bm{\Sigma}$ that might have been learned from a sample of size $\alpha$ from
the idealized sampling density $P_{\bm{Y}^{*}}$ but effectively comes from
$P_{\bm{Y}}$ (with sample size $T$). Put differently, if the amount of mis-
specification is large (implying that $\alpha$ should be set small) less
weight is put on the likelihood function and hence more weight is attributed
to the prior. For a given tightness of the prior, this immediately implies
more uncertainty surrounding parameter estimates and functions thereof. This
makes sense since any inferences made from the mis-specified model should be
considered with some caution.

###  3.3 Setting the learning rate

Apart from the hyperparameters that control the tightness of the prior, we
need to decide on the learning rate $\alpha$. Miller and Dunson (2018) propose
several ways of choosing $\alpha$. The first one would be to use prior
information on the amount of mis-specification. However, for possibly large-
dimensional macroeconomic datasets this turns out to be unfeasible. A second
approach relies on cross-validation to decide on $\alpha$. This essentially
implies to recompute quantities of interest (such as forecasts or impulse
responses) for different values of $\alpha$. Third, a principled approach to
deciding on $\alpha$ is to introduce a measure of model fit and a loss
function that takes model complexity into account.

We briefly discuss the third approach here. A natural measure of model fit
would be the likelihood evaluated at the posterior mean of $\bm{A}$ and
$\bm{\Sigma}$. This is given by
$\text{MF}(\alpha)=p(\bm{Y}|\overline{\bm{A}},\overline{\bm{\Sigma}})$ with
$\overline{\bm{\Sigma}}=\overline{\bm{S}}/(\underline{v}+\zeta_{T}T-M-1)$.
Measuring model complexity is slightly more complicated. Since we use a
continuous shrinkage prior the number of non-zero elements in
$\overline{\bm{A}}$ is always $k$. However, under the coarsened posterior the
corresponding prior implies stronger shrinkage (keeping $\lambda$ fixed).
Hence, we simply measure complexity by counting the number of effective
coefficients as follows:

$\text{MC}(\alpha)=\sum_{j=1}^{k}\mathbb{I}(|\overline{a}_{j}|<\tau),$ (6)

with $\mathbb{I}(\bullet)$ denoting the indicator function that equals one if
its argument is true and $\overline{a}_{j}$ is the $j^{th}$ element of
$\overline{\bm{a}}=\text{vec}(\overline{a})$. As a standard choice for the
macro datasets we consider in this paper we use $\tau=0.01$.

We compute $\text{MF}(\alpha)$ and $\text{MC}(\alpha)$ over a grid
$\alpha\in\\{25,50,75,100,125,250,350,500,1000,\infty\\}$. This yields a curve
in the $(\text{MF}(\alpha),\text{MC}(\alpha))$-space. The optimal value of
$\alpha,\alpha^{*}$, is then selected by searching the $\alpha$ that yields
the most favorable fit/penalty combination. This is achieved by computing a
straight line from the first point $(\text{MF}(25),\text{MC}(25))$ to the last
point $(\text{MF}(\infty),\text{MC}(\infty))$. The value $\alpha^{*}$ is then
the one that has the greatest distance to the line. In the following, we will
refer to this method of selecting $\alpha^{*}$ as BIC, due to the similarity
to the BIC criterion used in time series analysis for model specification.

##  4 Simulation-based evidence on the effect of coarsening

The empirical illustration in section 2 suggests that coarsening has the
potential of substantially improving inference about impulse responses if the
DGP is non-Gaussian and features few but substantial outliers. In this
section, we provide more simulation evidence on how model mis-specification
negatively impacts inference about IRFs and how the coarsened BVAR can be used
to improve estimation accuracy of the IRFs.

To do so, we generate data from a variety of models, all using the matrices
$\bm{A}_{1},\bm{A}_{2}$ and $\bm{Q}$ defined in Example 1. Based on this, we
let our models differ along two dimensions. First, we consider three different
types of shocks. Specifically, we consider Gaussian shocks, t-distributed
shocks with three degrees of freedom and shocks that are Gaussian but feature
stochastic volatility (SV).222The true (log) volatilites, $h_{t}$, evolve
according to an AR(1) process: $w_{t}=0.8w_{t-1}+s_{t}$ with
$s_{t}\sim\mathcal{N}(0,1)$ being a Gaussian shock.

The second dimension relates to how we specify the conditional mean. Again, we
focus on three important cases. The first assumption is that the conditional
mean of the model is the same as in Example 1 (i.e. a VAR(2) specification).
The second assumes that the model is a random coefficient (RC) model with
$\text{vec}(\bm{A}_{1t})=\text{vec}(\bm{A}_{1})+\bm{\xi}_{1t}$ and
$\text{vec}(\bm{A}_{2t})=\text{vec}(\bm{A}_{2})+\bm{\xi}_{2t}$ with
$\bm{\xi}_{jt}\sim\mathcal{N}(\bm{0},(0.035)^{2}\times\bm{I}_{M^{2}})$. The
third assumes that instead of a standard VAR(2) model the DGP includes $30$
additional covariates that are simulated from Gaussian distributions with zero
mean and variance $4$.

These 9 different DGPs give rise to a great variety of possible forms of model
mis-specification. For instance, we can have situations where we ignore
structural breaks in the conditional mean relations and the error variances or
situations where the DGP is linear and Gaussian without any form of time-
variation. Notice that in only one of the 9 DPGs we have a correctly specified
model.

Our aim is to recover the true impulse responses
$\frac{\partial\bm{y}_{t+h}}{\partial\varepsilon_{jt}}$ for $h=1,\dots,13$ and
$j=1,2,3$. This is achieved by comparing the posterior median of the IRFs of
the cBVAR and the standard BVAR to the true IRFs. Let $\bm{\Psi}(\alpha)_{j}$
denote a $12\times M$ matrix that contains the impulse responses to a shock to
the $j^{th}$ variable from the cBVAR with coarsening parameter $\alpha$.
Stacking over the columns of the different variables $j$ yields a $12\times
3M$ matrix $\bm{\Psi}(\alpha)$. The true $12\times 3M$ matrix of IRFs is
denoted by $\bm{\Psi}^{*}$. We then compute the mean absolute errors (MAEs)
between $\bm{\Psi}(\alpha)$ and $\bm{\Psi}^{*}$.

To make sure that results are not driven by particular draws from the DGP, we
repeat our simulation study $50$ times for each DGP. Relative MAEs between
cBVAR and the BVAR, averaged across these 50 replications, are shown in Table
1. In the columns, we show how estimation accuracy changes with $\alpha$
whereas the rows include the 9 different DGPs. Numbers smaller than unity
indicate that a particular model (i.e. cBVAR with some specific $\alpha$)
improves upon the BVAR, whereas the opposite is the case if a given number
exceeds one.

Table 1: Simulation performance: Impulse responses functions | $\alpha\rightarrow$ | 25 | 50 | 75 | 100 | 125 | 150 | 250 | 350 | 500 | 1000 | $\infty$ | BIC
---|---|---|---|---|---|---|---|---|---|---|---|---|---
VAR | Gauss | 3.04 | 2.32 | 1.82 | 1.64 | 1.48 | 1.41 | 1.24 | 1.17 | 1.13 | 1.08 | 1.00 | 1.66
| t-dist | 1.21 | 0.79 | 0.48 | 0.50 | 0.59 | 0.62 | 0.72 | 0.79 | 0.84 | 0.92 | 1.00 | 0.53
| SV | 0.80 | 0.55 | 0.50 | 0.57 | 0.61 | 0.67 | 0.77 | 0.82 | 0.87 | 0.93 | 1.00 | 0.57
RC | Gauss | 0.89 | 1.11 | 0.70 | 0.67 | 0.71 | 0.75 | 0.81 | 0.84 | 0.91 | 0.95 | 1.00 | 0.69
| t-dist | 0.57 | 0.59 | 0.58 | 0.64 | 0.67 | 0.72 | 0.81 | 0.85 | 0.88 | 0.94 | 1.00 | 0.63
| SV | 0.54 | 0.54 | 0.60 | 0.67 | 0.69 | 0.73 | 0.80 | 0.84 | 0.92 | 0.96 | 1.00 | 0.66
Exo | Gauss | 0.49 | 0.63 | 0.71 | 0.76 | 0.78 | 0.81 | 0.87 | 0.91 | 0.94 | 0.97 | 1.00 | 0.76
| t-dist | 0.50 | 0.63 | 0.71 | 0.76 | 0.79 | 0.81 | 0.87 | 0.92 | 0.94 | 0.97 | 1.00 | 0.77
| SV | 0.49 | 0.63 | 0.71 | 0.75 | 0.80 | 0.81 | 0.88 | 0.91 | 0.94 | 0.97 | 1.00 | 0.76
Notes: VAR is a standard VAR model, RC is a VAR with random coefficients, Exo
is a VAR model with exogenous factors included as well. Gauss are Gaussian
shocks with a constant error variance, t-dist are t-distributed shocks with 3
degrees of freedom and SV is a stochastic volatility model in the shocks.
Reported values are MAEs for the IRFs of each model with respect to the true
ones, relative to those of the standard VAR, averaged over 50 replications
from the DGP.

At a very general level, Table 1 shows that for 8 out of 9 DGPs, using
coarsening leads to improvements in estimation accuracy that depend on the
specific level of $\alpha$. This is not surprising given the fact that the
only DGP where coarsening reduces estimation accuracy is the one that implies
a correctly specified model (at least in terms of having a linear conditional
mean and Gaussian shocks). The gains reach up to 50 percent in MAE terms (for,
e.g., the model that includes exogenous factors). In the least favorable
scenario (i.e., a correctly specified DGP and a coarsening parameter smaller
than 100), coarsening leads to a substantial increase in errors made when
estimating impulse responses. In this case, the likelihood introduces little
information and the posterior IRFs will be close to the prior IRFs.

Zooming into the cases where coarsening helps, we find that in the presence of
a VAR conditional mean and t-distributed shocks, setting a coarsening
parameter between 75 to 250 yields gains that range from 52 to 28 percent,
respectively. If the DGP is heteroskedastic, the gains are similar and
improvements are largest for values of $\alpha$ between $50$ and $150$.

Turning to the DGP that has time-varying parameters and homoskedastic Gaussian
shocks we find that, in principle, coarsening helps and values of $\alpha$
between 75 and 250 lead to sizable improvements. When we use a DGP that
features time-varying parameters and flexible error distributions the gains
become even more pronounced. Irrespective of whether t-distributed shocks or
Gaussian shocks with SV are used, gains are sizable and tend to be largest
when $\alpha$ is set rather small (i.e. below $150$). This makes sense given
the fact that both cases indicate that the linear Gaussian VAR is strongly
mis-specified and the non-coarsening case would fail in recovering impulse
responses that are close to the actual ones.

If we focus on the DGP that features exogenous covariates (which we omit when
estimating the VAR) we find the largest improvements across all
specifications. Even in the case of Gaussian shocks the improvements can be as
large as 50 percent. Interestingly, for this DGP it does not matter whether
shock distributions are more flexible; the relative MAEs are very close to
each other and indicate that ignoring a large portion of the information set
dominates the effects of a mis-specified likelihood function.

Finally, the key question that typically arises is how $\alpha$ should be set.
Since our DGPs are patterned so as to fit the time series behavior of the US
unemployment rate, inflation and short-term interest rates, our simulations
indicate that if the researcher believes the DGP to be heteroskedastic, a
conservative choice would be to set $\alpha$ between $125$ and $350$. In this
case, even if the DGP would be linear and Gaussian, losses would not be too
high, whereas in the case of mis-specification accuracy gains remain large.
Using our BIC-inspired approach to decide on $\alpha$ yields MAEs that are
never best but also never worst. In fact, in most cases we find that choosing
$\alpha$ using this heuristic is a good and simple way on deciding on the
appropriate amount of correction for the mis-specification.

##  5 Forecasting with Coarsened Bayesian VARs

In the previous section we have shown that, if the model is mis-specified, the
empirical performance in terms of IRFs can be drastically improved. We now
investigate whether this carries over to a standard US macroeconomic dataset
and consider the forecasting performance of the different coarsened VARs vis-
á-vis the standard (uncoarsened) BVAR.

###  5.1 Data and design of the forecasting exercise

We use the monthly version of the McCracken and Ng (2016) dataset and include
time series that span from $1974$:M$01$ to $2021$:M$07$. To analyze how
coarsening interacts with model size, we focus on three different model sizes
(with FRED abbreviations in parentheses). A small model that includes the
unemployment rate (UNRATE), CPI inflation (CPIAUCSL) and the federal funds
rate (FEDFUNDS), a medium-sized model that, in addition to these three
variables, includes the logarihm of non-borrowed reserves (NONBORRES), real M2
(M2REAL and total reserves (TOTRESNS). This medium-sized dataset resembles the
information set of the monetary VAR in Christiano, Eichenbaum, and Evans
(1999). Then we consider a larger model that adds the following additional
time series to the medium-sized dataset: log industrial production (INDPRO),
log real personal income (RPI), the log S&P 500 stock market index (S.P.500),
capacity utilization in manufacturing (CUMFNS), the 10 year government bond
spread over the federal funds rate (T10YFFM), average weekly hours (AWHMAN),
log M1 (M1SL), the log GBP/USD exchange rate (EXUSUKx) and log housing starts
(HOUST). Our largest model thus includes 15 endogenous variables. In
principle, for most of the variables our transformations imply non-
stationarity. All models include 13 lags of the endogenous variables.

Since our model is fully conjugate, we could estimate much larger models with
hundreds of endogenous variables. However, our main goal in this section is to
show in a common setting that coarsening can dramatically improve the
empirical forecasting performance of a conjugate VAR.

Our forecasting design is a recursive one. This implies that we start
estimation using the period from $1974$M$01$ to $2001$M$06$, and we compute
$h$-step-ahead forecast densities (for $h\in\\{1,3,12\\})$. Next, we expand
the initial estimation sample by one month and compute a new set of predictive
densities. This procedure is repeated until we reach the end of the sample
($2021$M$07$). Hence, we have a rather long evaluation period, 24 years or 288
months.

To compare across different model sizes, we focus on forecasts for the
unemployment rate $u_{t}$, inflation $\pi_{t}$ and the short-term interest
rate ($i_{t}$), as these variables are included in all the models. The
forecasts are evaluated using MAEs and log predictive likelihoods (LPLs),
benchmarked against those for the large BVAR without coarsening.

###  5.2 Overall forecasting performance

We start our discussion by considering how coarsening impacts point forecasts
of the three focus variables. Table 2 presents the MAEs of the various models
relative to that of the large BVAR (where the results for the BVARs are in
rows indicated by $\alpha=\infty$), so that values smaller than one indicate
gains from the cBVAR.

Table 2: Point forecasting performance relative to the large BVAR: $2001$M$07$
to $2021$M$07$

| One-month-ahead | One-quarter-ahead | One-year-ahead
---|---|---|---
$\alpha\downarrow$ | $u_{t}$ | $\pi_{t}$ |  $i_{t}$ | $u_{t}$ | $\pi_{t}$ |  $i_{t}$ | $u_{t}$ | $\pi_{t}$ | $i_{t}$
Small
25 | 0.75 | 1.02 | 0.23 | 0.73 | 1.22 | 0.22 | 0.22 | 0.66 | 0.28
50 | 0.75 | 1.02 | 0.23 | 0.73 | 1.22 | 0.21 | 0.22 | 0.65 | 0.28
75 | 0.75 | 1.03 | 0.23 | 0.73 | 1.21 | 0.21 | 0.22 | 0.65 | 0.28
100 | 0.75 | 1.02 | 0.24 | 0.73 | 1.21 | 0.22 | 0.22 | 0.66 | 0.28
125 | 0.76 | 0.96 | 0.24 | 0.74 | 1.07 | 0.24 | 0.22 | 0.55 | 0.30
150 | 0.76 | 0.89 | 0.35 | 0.74 | 0.89 | 0.35 | 0.22 | 0.42 | 0.42
250 | 0.80 | 0.88 | 0.42 | 0.79 | 0.87 | 0.42 | 0.23 | 0.43 | 0.44
350 | 0.80 | 0.88 | 0.44 | 0.79 | 0.87 | 0.44 | 0.23 | 0.43 | 0.44
500 | 0.81 | 0.88 | 0.45 | 0.81 | 0.88 | 0.45 | 0.23 | 0.43 | 0.44
1000 | 0.83 | 0.88 | 0.48 | 0.82 | 0.88 | 0.48 | 0.23 | 0.43 | 0.45
BIC | 0.75 | 1.02 | 0.24 | 0.73 | 1.21 | 0.22 | 0.21 | 0.65 | 0.28
$\infty$ | 0.86 | 0.90 | 0.54 | 0.87 | 0.91 | 0.57 | 0.24 | 0.44 | 0.48
Medium
25 | 0.77 | 0.93 | 0.32 | 0.75 | 0.98 | 0.31 | 0.24 | 0.50 | 0.38
50 | 0.77 | 0.91 | 0.36 | 0.76 | 0.95 | 0.35 | 0.24 | 0.51 | 0.41
75 | 0.78 | 0.91 | 0.39 | 0.77 | 0.95 | 0.38 | 0.24 | 0.51 | 0.42
100 | 0.78 | 0.92 | 0.41 | 0.77 | 0.94 | 0.40 | 0.24 | 0.52 | 0.43
125 | 0.79 | 0.91 | 0.43 | 0.78 | 0.94 | 0.42 | 0.25 | 0.52 | 0.43
150 | 0.79 | 0.92 | 0.43 | 0.79 | 0.94 | 0.42 | 0.25 | 0.52 | 0.43
250 | 0.80 | 0.92 | 0.48 | 0.81 | 0.93 | 0.47 | 0.25 | 0.52 | 0.45
350 | 0.81 | 0.92 | 0.51 | 0.82 | 0.93 | 0.49 | 0.25 | 0.51 | 0.46
500 | 0.81 | 0.93 | 0.56 | 0.83 | 0.92 | 0.56 | 0.25 | 0.50 | 0.45
1000 | 0.89 | 0.97 | 0.68 | 0.90 | 0.96 | 0.69 | 0.27 | 0.49 | 0.50
BIC | 0.78 | 0.91 | 0.40 | 0.78 | 0.94 | 0.40 | 0.25 | 0.52 | 0.43
$\infty$ | 0.95 | 1.00 | 0.74 | 0.97 | 0.96 | 0.74 | 0.27 | 0.49 | 0.52
Large
25 | 0.70 | 0.92 | 0.51 | 0.66 | 0.94 | 0.49 | 0.21 | 0.49 | 0.47
50 | 0.70 | 0.89 | 0.56 | 0.66 | 0.91 | 0.55 | 0.21 | 0.49 | 0.49
75 | 0.71 | 0.88 | 0.60 | 0.67 | 0.91 | 0.59 | 0.21 | 0.50 | 0.52
100 | 0.72 | 0.88 | 0.62 | 0.68 | 0.91 | 0.61 | 0.23 | 0.51 | 0.53
125 | 0.72 | 0.88 | 0.64 | 0.69 | 0.91 | 0.63 | 0.23 | 0.51 | 0.55
150 | 0.73 | 0.88 | 0.66 | 0.70 | 0.92 | 0.65 | 0.24 | 0.52 | 0.55
250 | 0.75 | 0.88 | 0.70 | 0.73 | 0.92 | 0.68 | 0.28 | 0.53 | 0.59
350 | 0.77 | 0.88 | 0.72 | 0.75 | 0.93 | 0.71 | 0.34 | 0.55 | 0.61
500 | 0.80 | 0.89 | 0.74 | 0.78 | 0.94 | 0.75 | 0.38 | 0.58 | 0.66
1000 | 0.85 | 0.90 | 0.79 | 0.84 | 0.97 | 0.81 | 0.52 | 0.65 | 0.71
BIC | 0.72 | 0.88 | 0.62 | 0.68 | 0.91 | 0.61 | 0.23 | 0.50 | 0.53
$\infty$ | 0.15 | 0.66 | 0.09 | 0.32 | 0.79 | 0.26 | 2.61 | 1.49 | 0.74

For unemployment, the improvements over the large BVAR are substantial for all
model sizes and forecast horizons and range from 10 to 30 percent one-month-
ahead, 10 to 34 percent one-quarter- ahead, and even 50 to 80 percent one-
year-ahead, with in general larger gains for smaller values of $\alpha$ and
larger model sizes. Yet, the small BVAR is generally best for unemployment,
and if we take that as the benchmark the gains from the cBVAR shrink somewhat,
though they systematically remain. This result is surprising since evidence
in, e.g., Bańbura, Giannone, and Reichlin (2010) suggests that employment
predictions should profit from larger information sets. In our case, the
opposite is the case. However, as we show in Table A1 that includes point
forecasting results through $2019$:M$12$, this result is driven by the
pandemic period, when the large BVAR produces quite inaccurate unemployment
forecasts, likely due to the parameter instability caused by the substantial
outliers in monthly unemployment rates. Indeed, once we exclude the pandemic
observations, results change: unemployment rate forecasts from bigger model
are more accurate, but coarsening, particularly for larger models and the one-
step-ahead horizon, still improves predictive accuracy by up to 16 percent.

For inflation, the gains with respect to the large BVAR are smaller but
systematic at shorter horizons (i.e., one-month and one-quarter-ahead) . This
is particularly pronounced for medium and large model sizes, reaching 10 to 12
percent, often for rather large values of $\alpha$. Turning to one-year-ahead
forecasts reveals much larger gains from using coarsening. For this horizon,
the gains are in the range of 34 to 58 percent, with the largest gains arising
from the small model for larger values of $\alpha$. Yet, also for this
variable, if we use the small BVAR as benchmark, the gains from coarsening
decrease. This finding again points towards possible instabilities that arise
during the pandemic period. Coarsening again handles this issue particularly
well. When we only consider the results before the pandemic, we find that the
small cBVAR for values of $\alpha$ between 150 and 250 improves upon both the
large BVAR and the small BVAR.

The largest improvements in terms of point forecasting performance are found
for the final focus variable, the federal funds rate. In this case,
improvements approach 77 percent for the one-month and one-quarter-ahead
horizon and the small BVAR with low values of $\alpha$. For one-year-ahead
predictions, the gains slightly decline but still remain very large, with
improvements of just over 70 percent. It is worth noting that for short-term
rates, even the small model is outperformed drastically by the coarsened
models. Inspection of the predictive densities (not shown) suggests that
nonlinearities during the zero lower bound period are well captured by a model
that places little weight on the likelihood and thus the prior dominates,
implying that the predictions are tightly centered around the random walk
forecasts. And precisely this behavior improves predictive performance
throughout our hold-out sample.

Table 3: Density forecasting performance relative to the large BVAR:
$2001$M$07$ to $2021$M$07$

| One-month-ahead | One-quarter-ahead | One-year-ahead
---|---|---|---
$\alpha\downarrow$ | $u_{t}$ | $\pi_{t}$ |  $i_{t}$ | $u_{t}$ | $\pi_{t}$ |  $i_{t}$ | $u_{t}$ | $\pi_{t}$ | $i_{t}$
Small
25 | -4.45 | -0.53 | 0.43 | -2.19 | -0.31 | 0.84 | -2.72 | -0.04 | 0.62
50 | -0.72 | -0.21 | 0.29 | 0.48 | -0.16 | 0.71 | -1.39 | -0.11 | 0.55
75 | 0.30 | -0.12 | 0.22 | 1.46 | -0.13 | 0.66 | -1.00 | -0.15 | 0.52
100 | 0.75 | -0.09 | 0.18 | 1.95 | -0.11 | 0.62 | -0.70 | -0.17 | 0.49
125 | 0.93 | -0.03 | 0.17 | 1.97 | -0.09 | 0.57 | -0.47 | -0.08 | 0.44
150 | 1.73 | 0.12 | 0.15 | 2.95 | 0.13 | 0.49 | 0.33 | 0.37 | 0.32
250 | 2.01 | 0.15 | 0.11 | 3.47 | 0.16 | 0.41 | 0.56 | 0.41 | 0.32
350 | 2.13 | 0.13 | 0.10 | 3.70 | 0.16 | 0.39 | 0.57 | 0.41 | 0.32
500 | 2.32 | 0.14 | 0.09 | 3.70 | 0.15 | 0.37 | 0.56 | 0.42 | 0.33
1000 | 2.15 | 0.14 | 0.08 | 3.44 | 0.14 | 0.34 | 0.56 | 0.38 | 0.32
BIC | 0.82 | -0.08 | 0.18 | 1.73 | -0.11 | 0.62 | -0.73 | -0.17 | 0.49
$\infty$ | 2.41 | 0.08 | 0.05 | 3.71 | 0.04 | 0.26 | 0.56 | 0.36 | 0.31
Medium
25 | -2.82 | -0.24 | 0.47 | 0.67 | 0.03 | 0.70 | 0.97 | 0.34 | 0.33
50 | -0.43 | 0.00 | 0.36 | 1.43 | 0.08 | 0.62 | 0.82 | 0.35 | 0.34
75 | -0.34 | 0.08 | 0.29 | 2.07 | 0.11 | 0.57 | 0.83 | 0.35 | 0.33
100 | 0.66 | 0.11 | 0.25 | 2.34 | 0.11 | 0.53 | 0.86 | 0.35 | 0.34
125 | 0.89 | 0.14 | 0.22 | 2.40 | 0.13 | 0.51 | 0.83 | 0.34 | 0.34
150 | 1.34 | 0.14 | 0.20 | 2.62 | 0.13 | 0.50 | 0.84 | 0.33 | 0.35
250 | 1.42 | 0.16 | 0.16 | 2.81 | 0.13 | 0.44 | 0.86 | 0.35 | 0.36
350 | 1.41 | 0.17 | 0.13 | 2.75 | 0.12 | 0.42 | 0.80 | 0.34 | 0.36
500 | 1.83 | 0.18 | 0.11 | 2.82 | 0.14 | 0.38 | 0.84 | 0.31 | 0.39
1000 | 1.73 | 0.15 | 0.08 | 2.59 | 0.13 | 0.31 | 0.67 | 0.33 | 0.38
BIC | 1.05 | 0.12 | 0.25 | 2.45 | 0.11 | 0.53 | 0.86 | 0.34 | 0.34
$\infty$ | 1.82 | 0.13 | 0.02 | 2.60 | 0.14 | 0.25 | 0.71 | 0.31 | 0.39
Large
25 | -6.51 | -1.05 | 0.56 | -5.68 | -0.22 | 0.68 | 0.39 | 0.32 | 0.46
50 | -3.20 | -0.43 | 0.45 | -3.47 | -0.14 | 0.56 | 0.41 | 0.21 | 0.41
75 | -1.91 | -0.22 | 0.37 | -2.28 | -0.09 | 0.48 | 0.41 | 0.17 | 0.36
100 | -0.82 | -0.12 | 0.33 | -1.59 | -0.08 | 0.44 | 0.40 | 0.15 | 0.34
125 | -0.36 | -0.05 | 0.29 | -1.42 | -0.06 | 0.41 | 0.41 | 0.14 | 0.32
150 | -0.43 | -0.03 | 0.27 | -1.24 | -0.07 | 0.38 | 0.35 | 0.12 | 0.30
250 | 0.11 | 0.04 | 0.21 | -0.52 | -0.05 | 0.31 | 0.29 | 0.10 | 0.24
350 | -0.20 | 0.07 | 0.18 | -0.28 | -0.02 | 0.27 | 0.29 | 0.10 | 0.21
500 | 0.51 | 0.09 | 0.15 | -0.11 | -0.02 | 0.22 | 0.23 | 0.06 | 0.16
1000 | 0.37 | 0.08 | 0.12 | -0.15 | -0.02 | 0.19 | 0.08 | 0.03 | 0.14
BIC | -1.16 | -0.11 | 0.32 | -1.83 | -0.08 | 0.44 | 0.34 | 0.15 | 0.35
$\infty$ | -6.32 | -1.43 | 0.84 | -6.81 | -1.63 | -0.21 | -2.58 | -1.84 | -0.94

Table 3 presents the same model comparison but instead of using MAE ratios it
shows differences in the average log predictive likelihood to assess the
density forecasting performance. Positive values indicate gains from the cBVAR
whereas negative values indicate that coarsening hurts density forecasting
performance.

For unemployment, at the one-month and one-quarter horizons there are gains
with respect to the large BVAR. Notice, however, that the small BVAR without
coarsening is difficult to beat. Again, the poor performance of the large BVAR
for unemployment is mostly driven by the pandemic. Table A2 shows that if we
exclude the pandemic, large cBVARs have the potential to improve upon the
standard BVAR. But the gains appear to be muted. Moving to one-year-ahead
forecasts we find that the medium-sized cBVAR produces the largest
improvements, in particular for small values of $\alpha$.

For inflation, the small BVAR is better than the large one, but now the cBVAR
is even better, at all horizons, with gains that are not very large but quite
systematic and arise for rather large values of $\alpha$. For the federal
funds rate, the overall picture is even more favorable. We find pronounced
gains for all forecast horizons, with small BVARs with $\alpha$ between $25$
and $100$ producing the highest LPLs.

One interesting difference between the point and density forecasting
performance is that changing $\alpha$ has a much stronger effect on LPLs. This
finding is driven by the fact that coarsening impacts point forecasts but also
higher-order moments of the predictive distribution. Hence, whereas MAEs are
only affected through changes in the first moment, LPLs are impacted by
changes in both the predictive mean and variance.

###  5.3 Forecasting performance over time

The previous discussion was concerned with the average forecasting performance
throughout the hold-out period. However, in this discussion we highlighted how
the pandemic changed the relative ordering of the models we consider. In this
sub-section, our aim is to investigate in more details how the forecasting
performance changes over time.

Figures 2 to 4 report for, respectively, unemployment, inflation and the
interest rate, the evolution of the cumulative one-step-ahead LPL of the cBVAR
(for different values of $\alpha$ and model sizes) relative to that of the
large BVAR over time, so that positive values indicate a better performance of
the cBVAR. The larger left panels cover the period through $2019$M$12$, the
smaller right panels focus on the Covid-19 period (2020M01 to 2021M07), when
the scale gets very different.

(a) Small

(b) Medium

(c) Large

Figure 2: Relative cumulative log predictive scores to the large BVAR over
time: UNRATE

Considering unemployment (see Figure 2), the pattern of temporal evolution in
the pre-Covid period depends on the model size, with more variability for the
smaller models and larger gains from the large cBVAR. This is particularly the
case if we set $\alpha$ in the range $50$ to $100$. Yet, even for the smaller
models the cBVAR does generally better than the corresponding BVAR. In the
Covid-19 period, as mentioned, the ranking changes. In this case, smaller
models perform better than larger ones. The small BVAR exhibits the best
overall performance, though with little differences with respect to the
corresponding cBVAR with $\alpha$ set to large values.

(a) Small

(b) Medium

(c) Large

Figure 3: Relative cumulative log predictive scores to the large BVAR over
time: CPIAUCSL

When we turn to inflation forecasting (see Figure 3), smaller models are often
slightly better than larger ones, with the standard BVAR displaying the best
performance before the global financial crisis. After that episode, the cBVAR
with larger values of $\alpha$ turns out as the best performing model and it
remains so also during the Covid-19 pandemic. Changes over time are relatively
small and often do not have a discernible effect on model rankings.

(a) Small

(b) Medium

(c) Large

Figure 4: Relative cumulative log predictive scores to the large BVAR over
time: FEDFUNDS

For the interest rate (see Figure 4), the dominance of the cBVAR over the BVAR
is even more clear-cut and holds throughout the sample period. The gains
become larger after the financial crisis and during the Covid-19 period,
likely because the ensuing unconventional monetary policy can be hardly
captured by the standard BVAR (see, e.g., Carriero, et al., 2021), so that the
mis-specification of the interest rate equation in the BVAR is substantial,
and in fact very small values of $\alpha$ are the best choice for the cBVAR,
in line with the findings in Table 3, but larger values also lead to gains.

Overall, the cBVAR seems capable of improving the forecasting performance for
these three key economic variables with respect to a standard BVAR, with gains
that are particularly large and persistent over time for the interest rate,
but that are also present for inflation and unemployment in the pre-Covid
period. In terms of the best values of $\alpha$, they depend on the extent of
mis-specification, with large values preferred for unemployment and inflation
and smaller ones for the interest rate, where mis-specification is likely
larger. It is worth stressing that choosing $\alpha$ using our BIC-inspired
approach also works: it never produces the most accurate forecasts but also
never the least accurate ones, and generally beats the benchmark, in line with
the results with simulated data. Hence, if the researcher wishes to use an
automatic approach to selecting $\alpha$, applying the BIC-inspired approach
would be a reasonable choice. Another possible way of choosing $\alpha$ that
is easy to implement would be by using the first fraction of the hold-out
period to learn $\alpha$ and then fix it (i.e., pick the one that produces the
best forecasts over a certain time frame). While our discussion showed that
there is some time heterogeneity with respect to the best performing models
(defined by the value of $\alpha$), it is also rarely the case that a model
that does well over the first part of the hold-out fails to perform well in
later periods of the hold-out.

##  6 Structural analysis with Coarsened BVARs: The effects of macroeconomic
uncertainty

In this section we apply the cBVAR to a topical structural application and
show that coarsening has empirically relevant effects on estimated impulse
responses for real data as well. We focus on how uncertainty shocks impact the
economy and pay particular attention on how real activity reacts to unexpected
variations in uncertainty. The study of the effects of uncertainty shocks has
attracted considerable attention in the literature, in particular after the
seminal paper by Bloom (2009), see also the review in Bloom (2014). One common
finding in the uncertainty literature is that real activity, after declining
initially, tends to overshoot its initial level after some time. However, most
of the papers that find this real activity overshoot phenomenon use observed
proxies of economic uncertainty. Since uncertainty is a latent quantity, using
proxies ignores possible measurement errors and this could impact IRF
estimates (Carriero, Clark, and Marcellino, 2018) so that the real activity
overshoot vanishes. However, models that estimate uncertainty alongside the
remaining model parameters and allow for heteroskedasticity and time-varying
parameters are more difficult to estimate and tune. Hence, a simpler model
that is robust with respect to measurement errors in uncertainty would be
preferable and we use the cBVAR to investigate whether coarsening helps in
reducing the real activity overshoot in response to increases in economic
uncertainty.

While there is no consensus on the definition of uncertainty, the forecast-
based measure suggested by Jurado, Ludvigson, and Ng (2015, JLN) is quite
popular. We now add this JLN uncertainty measure to our large-scale VAR,
ordering it first, and compute IRFs to the resulting uncertainty shock. Our
main focus is on understanding whether and what changes in the IRFs are
implied by different values of $\alpha$. Since the forecasting results pointed
towards important effects of the pandemic, we recompute the IRFs for the full
sample (i.e., including the pandemic observations) and for a shorter sample
that ends in 2019M12.

Figure 5: Output reactions to a one standard error uncertainty shock

(a) Data through 2021M06

(b) Data through 2019M12

Notes: The orange lines refer to the 5/95 percentiles of the posterior
distribution of the IRFs associated with the BVAR while the gray shaded area
refers to the 5/95 percentiles of the posterior distribution of the cBVAR. The
solid black line is the posterior median of the impulse responses of the
cBVAR.

We start by discussing the responses of industrial production to uncertainty
shocks. Figure 5 shows the corresponding IRFs (and associated 68 percent
posterior credible intervals) for different values of $\alpha$. 68 percent
posterior intervals of IRFs arising from the standard BVAR are shown in
orange. The top panel (panel (a)) shows the results for the full sample (i.e.,
including pandemic observations) while panel (b) shows the results based on
stopping the sample in 2019M12. Starting with panel (a) reveals that the
standard BVAR produces responses that are consistent with the literature on
uncertainty shocks using observed shock measures. With a lag of one month,
increases in economic uncertainty trigger a decline in output of around one
percent in the short run. Medium-run reactions (i.e. between 20 and 30 months)
indicate a pronounced and statistically significant overshoot in real
activity. If we follow the more recent uncertainty literature that treats
uncertainty as latent and leverages more sophisticated models, this overshoot
is an artifact arising from mis-specification.

If we now consider the output responses produced by the cBVAR for different
values of $\alpha$ we find that in cases where $\alpha$ is below $1000$, there
is no evidence of an overshoot in real activity. Interestingly, the specific
choice of $\alpha$ plays only a minor role as long as it does not exceed
$500$. The main features are that output declines, reaching a trough after
around half a year. If $\alpha$ is between 25 and 125, output gradually
recovers and posterior intervals include zero for longer-run responses in most
cases. When we increase $\alpha$, short-run reactions are very similar in
terms of shape but appear to be much stronger. In fact, for the first year the
IRFs of the cBVAR for $\alpha\in\\{250,350,500\\}$ closely resemble the ones
of the BVAR. It is only for impulse horizons above 1.5 years that IRFs
increasingly disagree, with the BVAR producing the real activity overshoot and
the cBVAR (for these values of $\alpha$) producing a rather persistent decline
in industrial production. The fact that short-run responses quickly converge
to the ones of the uncoarsened model while longer-run responses can disagree
for sizable values of $\alpha$ squares well with what we have learned from the
forecasting exercise, where we found that in general longer-run forecasts gain
more from coarsening. Once a certain threshold for $\alpha$ is crossed, the
IRFs of the cBVAR approach the ones of the BVAR and also produce a real
activity overshoot.

Figure 6: Unemployment reactions to a one standard error uncertainty shock

(a) Data through 2021M06

(b) Data through 2019M12

Notes: The orange lines refer to the 5/95 percentiles of the posterior
distribution of the IRFs associated with the BVAR while the gray shaded area
refers to the 5/95 percentiles of the posterior distribution of the cBVAR. The
solid black line is the posterior median of the impulse responses of the
cBVAR.

Turning to labor market reactions, still with data through 2021M6 (see Figure
6, panel (a)) reveals similar insights to the ones obtained from considering
output reactions. The standard BVAR indicates that unemployment first
increases but then, after around 30 months, puts substantial posterior mass on
negative values. This, again, points to an overshoot in real activity. Once we
apply coarsening to the BVAR, this rebound in the labor market vanishes.
Unless $\alpha$ is set to rather large values (i.e. between 500 and 1,000),
unemployment increases and then slowly moves back towards the steady state.

The lower panel of Figure 6 indicates that, after excluding the pandemic, the
standard BVAR produces IRFs more in line with the ones obtained from the
different cBVAR versions include pandemic information. However, notice that
for longer-run responses (i.e., above 80 months) unemployment declines. This
points towards a delayed rebound. If we apply coarsening, this effect is
substantially reduced and long-run responses, albeit having some posterior
mass below zero, appear to be insignificant and centered on zero.

These results suggest that coarsening is relevant also for a proper assessment
of the effects of economic shocks. Interestingly, values of $\alpha$ close to
the threshold that implies no real activity overshoot (both in terms of
industrial production and the unemployment rate) are also those that produce
favorable forecasts (in particular for the one-year-ahead forecast horizon
that roughly matches the timing of the output overshoot).

Finally, in the discussion so far we have focused on the IRFs conditional on
specific values of $\alpha$. Using our BIC-inspired approach to selecting
$\alpha$ leads to an optimal value of $\alpha=100$ for both samples, and hence
to impulse responses which do not display an overshoot in real activity (both
in terms of industrial production and unemployment rates).

##  7 Extensions

In this paper we apply the general idea of coarsening to multivariate time
series models with a particular, conjugate, prior structure. This conjugate
structure implies that the likelihood and prior have the same Kronecker
structure, which simplifies computations but can be restrictive in some cases.
For example, it could be (and this is to a certain degree evidenced by our
forecasting results) that some variables are more subject to model mis-
specification than others. For instance, during the first few months of the
pandemic the unemployment rate exhibited substantial fluctuations, outside the
range of past movements. By contrast, inflation remained rather stable. Hence,
using separate $\alpha$’s for unemployment and inflation might be preferable
over using a single $\alpha$ (and thus a common learning rate $\zeta_{T}$).

If the researcher is willing to trade computational simplicity for
flexibility, she could use any equation-by-equation estimation algorithm for
VARs (such as the one proposed in Carriero, Clark, and Marcellino (2019)), and
then use equation-specific $\alpha$’s. These can be set using the BIC-inspired
criterion outlined above. Another alternative, which would remain conjugate
(and thus permit order-invariant inference) and allow for equation-specific
$\alpha$’s, is to use the large asymmetric conjugate VAR of Chan (2022). But
both options are much more involved to implement than the cBVAR, and thus face
the risk of being less used in practice.

Another potential shortcoming of our approach is that the parameter $\alpha$
is time-invariant. For example, it could be that time series models feature
more mis-specification in turbulent times (such as recessions). To capture
this freature, one needs to take a stance about which regimes require
different values of $\alpha$ and how to group observations into these regimes.
This is feasible, but it would substantially complicate inference and require
a law of motion for $\alpha$, so that we leave it aside.

Finally, our approach could be applied to other popular models in
macroeconomics. For instance, dynamic stochastic general equilibrium (DSGE)
models such as the ones developed by Smets and Wouters (2007) rely on Bayesian
updating to obtain estimates of the structural parameters. However, they are
likely to be mis-specified and Canova and Matthes (2021) discuss this issue
and provide a solution based on model combination. Another straightforward way
of handling mis-specification in DSGE models is to apply coarsening by
obtaining an estimate of the likelihood through filtering and then simply
modifying this likelihood estimate.

##  8 Conclusions

Common practice in empirical macro is to estimate simple and interpretable
models to derive stylized facts and inform decision making. However, these
simple models often suffer from mis-specification of various, and unknown,
types, which has a deleterious effect on estimation, inference, and predictive
performance. In this paper, we offer a simple solution to get more robust
results. Our proposal is to modify standard Bayesian multivariate econometric
models. Instead of conditioning on the observed data we propose to condition
on the event that there is a distance between the sampling distribution of the
observed data and a hypothetical idealized sampling distribution. Using
relative entropy as loss metric gives rise to a simple approximation that
amounts to tempering the likelihood. We use this result and apply it to the
conjugate VAR case, deriving analytically coarsened posterior distributions
that can be used for robust forecasting and structural analysis.

In simulations, we show that standard BVARs in the presence of mis-
specification of different forms produce estimates of impulse responses that
are substantially biased. This bias can be reduced through coarsening, the
gains in terms of MAE can be up to almost 50 percent.

Using actual US monthly data, we show that the cBVAR produces more accurate
point and density forecasts than the standard BVAR, with larger gains at
longer horizons and when using small models, and it does not suffer much from
the substantial outliers observed during the pandemic. This suggests that our
approach can be also used to obtain more robust inference in the presence of
large outliers.

Finally, we show that a coarsened model can help also for more structural
applications. When we consider the effects of uncertainty shocks, we find that
the cBVAR produces no evidence of a real activity overshoot, contrary to the
literature that relies on observed proxies of uncertainty, with negative real
activity responses that are milder but more persistent than with a standard
BVAR.

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##  Appendix A Empirical appendix

###  A.1 Forecasting results excluding the pandemic

Table A1: Point forecasting performance relative to the large BVAR:
$2001$M$07$ to $2019$M$12$

| One-month-ahead | One-quarter-ahead | One-year-ahead
---|---|---|---
$\alpha\downarrow$ | $u_{t}$ | $\pi_{t}$ |  $i_{t}$ | $u_{t}$ | $\pi_{t}$ |  $i_{t}$ | $u_{t}$ | $\pi_{t}$ | $i_{t}$
Small
25 | 0.97 | 1.04 | 0.25 | 0.99 | 1.27 | 0.23 | 1.08 | 1.27 | 0.39
50 | 0.97 | 1.04 | 0.25 | 0.99 | 1.27 | 0.23 | 1.06 | 1.27 | 0.39
75 | 0.97 | 1.05 | 0.25 | 0.99 | 1.27 | 0.23 | 1.05 | 1.26 | 0.39
100 | 0.97 | 1.05 | 0.26 | 0.99 | 1.27 | 0.24 | 1.04 | 1.27 | 0.39
125 | 1.00 | 0.97 | 0.26 | 1.02 | 1.10 | 0.26 | 1.08 | 1.05 | 0.43
150 | 1.00 | 0.90 | 0.39 | 1.02 | 0.92 | 0.39 | 1.07 | 0.80 | 0.60
250 | 1.01 | 0.88 | 0.41 | 1.04 | 0.89 | 0.41 | 1.10 | 0.80 | 0.61
350 | 1.01 | 0.88 | 0.42 | 1.04 | 0.89 | 0.42 | 1.10 | 0.81 | 0.61
500 | 1.01 | 0.88 | 0.43 | 1.04 | 0.90 | 0.43 | 1.10 | 0.81 | 0.61
1000 | 1.02 | 0.88 | 0.45 | 1.04 | 0.89 | 0.45 | 1.10 | 0.81 | 0.61
BIC | 0.97 | 1.04 | 0.25 | 0.99 | 1.26 | 0.23 | 1.04 | 1.26 | 0.39
$\infty$ | 1.01 | 0.89 | 0.50 | 1.06 | 0.91 | 0.52 | 1.12 | 0.84 | 0.63
Medium
25 | 1.01 | 0.95 | 0.35 | 1.05 | 1.01 | 0.34 | 1.16 | 0.94 | 0.54
50 | 1.01 | 0.93 | 0.39 | 1.06 | 0.99 | 0.38 | 1.18 | 0.96 | 0.59
75 | 1.00 | 0.92 | 0.42 | 1.06 | 0.98 | 0.41 | 1.19 | 0.98 | 0.60
100 | 1.01 | 0.93 | 0.43 | 1.05 | 0.97 | 0.43 | 1.19 | 0.98 | 0.61
125 | 1.01 | 0.92 | 0.46 | 1.05 | 0.97 | 0.44 | 1.19 | 0.99 | 0.61
150 | 1.01 | 0.93 | 0.46 | 1.06 | 0.97 | 0.45 | 1.19 | 0.99 | 0.61
250 | 1.02 | 0.93 | 0.50 | 1.06 | 0.96 | 0.50 | 1.18 | 0.99 | 0.63
350 | 1.03 | 0.94 | 0.53 | 1.08 | 0.96 | 0.51 | 1.20 | 0.98 | 0.65
500 | 1.01 | 0.94 | 0.58 | 1.07 | 0.95 | 0.58 | 1.20 | 0.96 | 0.63
1000 | 1.04 | 0.98 | 0.64 | 1.07 | 0.98 | 0.64 | 1.23 | 0.94 | 0.66
BIC | 1.01 | 0.92 | 0.43 | 1.05 | 0.97 | 0.43 | 1.20 | 0.98 | 0.60
$\infty$ | 1.05 | 1.00 | 0.69 | 1.08 | 0.98 | 0.67 | 1.23 | 0.94 | 0.66
Large
25 | 0.86 | 0.93 | 0.54 | 0.82 | 0.96 | 0.53 | 1.00 | 0.94 | 0.66
50 | 0.84 | 0.90 | 0.59 | 0.80 | 0.94 | 0.59 | 0.98 | 0.96 | 0.70
75 | 0.85 | 0.89 | 0.64 | 0.80 | 0.93 | 0.64 | 0.96 | 0.97 | 0.73
100 | 0.85 | 0.89 | 0.67 | 0.80 | 0.93 | 0.66 | 0.95 | 0.97 | 0.74
125 | 0.85 | 0.89 | 0.69 | 0.81 | 0.94 | 0.69 | 0.95 | 0.98 | 0.75
150 | 0.86 | 0.89 | 0.71 | 0.81 | 0.94 | 0.70 | 0.94 | 0.98 | 0.77
250 | 0.86 | 0.90 | 0.75 | 0.82 | 0.94 | 0.74 | 0.93 | 0.99 | 0.80
350 | 0.87 | 0.90 | 0.78 | 0.83 | 0.95 | 0.77 | 0.92 | 0.99 | 0.82
500 | 0.90 | 0.90 | 0.81 | 0.86 | 0.95 | 0.81 | 0.94 | 1.02 | 0.85
1000 | 0.96 | 0.92 | 0.86 | 0.90 | 0.97 | 0.88 | 0.93 | 1.01 | 0.89
BIC | 0.85 | 0.89 | 0.66 | 0.81 | 0.93 | 0.66 | 0.95 | 0.97 | 0.74
$\infty$ | 0.08 | 0.66 | 0.08 | 0.14 | 0.75 | 0.24 | 0.49 | 0.76 | 0.52

Table A2: Density forecasting performance relative to the large BVAR:
$2001$M$07$ to $2019$M$12$

| One-month-ahead | One-quarter-ahead | One-year-ahead
---|---|---|---
$\alpha\downarrow$ | $u_{t}$ | $\pi_{t}$ |  $i_{t}$ | $u_{t}$ | $\pi_{t}$ |  $i_{t}$ | $u_{t}$ | $\pi_{t}$ | $i_{t}$
Small
25 | -0.09 | -0.61 | 0.38 | -0.40 | -0.36 | 0.78 | -2.56 | -0.07 | 0.60
50 | -0.02 | -0.27 | 0.23 | -0.23 | -0.20 | 0.65 | -1.31 | -0.15 | 0.53
75 | -0.02 | -0.17 | 0.17 | -0.20 | -0.16 | 0.60 | -0.95 | -0.18 | 0.50
100 | -0.02 | -0.13 | 0.12 | -0.19 | -0.14 | 0.56 | -0.66 | -0.20 | 0.47
125 | -0.03 | -0.07 | 0.12 | -0.20 | -0.13 | 0.51 | -0.43 | -0.11 | 0.42
150 | -0.01 | 0.09 | 0.09 | -0.14 | 0.11 | 0.42 | 0.28 | 0.34 | 0.29
250 | -0.01 | 0.13 | 0.08 | -0.17 | 0.16 | 0.38 | 0.44 | 0.39 | 0.31
350 | -0.02 | 0.12 | 0.07 | -0.17 | 0.16 | 0.37 | 0.45 | 0.40 | 0.31
500 | -0.03 | 0.12 | 0.06 | -0.18 | 0.15 | 0.35 | 0.44 | 0.40 | 0.32
1000 | -0.03 | 0.13 | 0.05 | -0.18 | 0.14 | 0.33 | 0.44 | 0.37 | 0.32
BIC | -0.02 | -0.12 | 0.13 | -0.18 | -0.14 | 0.56 | -0.69 | -0.20 | 0.47
$\infty$ | -0.04 | 0.07 | 0.05 | -0.20 | 0.03 | 0.27 | 0.45 | 0.35 | 0.31
Medium
25 | -0.06 | -0.30 | 0.44 | -0.02 | 0.02 | 0.66 | 0.85 | 0.31 | 0.32
50 | 0.01 | -0.03 | 0.32 | -0.03 | 0.07 | 0.58 | 0.69 | 0.33 | 0.32
75 | 0.03 | 0.06 | 0.26 | -0.03 | 0.11 | 0.52 | 0.69 | 0.33 | 0.32
100 | 0.03 | 0.09 | 0.21 | -0.03 | 0.11 | 0.49 | 0.72 | 0.33 | 0.32
125 | 0.03 | 0.13 | 0.19 | -0.04 | 0.13 | 0.47 | 0.69 | 0.32 | 0.33
150 | 0.03 | 0.13 | 0.17 | -0.03 | 0.12 | 0.45 | 0.70 | 0.32 | 0.33
250 | 0.02 | 0.15 | 0.12 | -0.05 | 0.12 | 0.40 | 0.73 | 0.33 | 0.34
350 | 0.00 | 0.16 | 0.09 | -0.07 | 0.12 | 0.38 | 0.67 | 0.33 | 0.34
500 | 0.00 | 0.17 | 0.08 | -0.07 | 0.14 | 0.34 | 0.71 | 0.29 | 0.37
1000 | -0.03 | 0.14 | 0.08 | -0.09 | 0.13 | 0.31 | 0.54 | 0.31 | 0.37
BIC | 0.03 | 0.10 | 0.22 | -0.03 | 0.11 | 0.49 | 0.73 | 0.32 | 0.32
$\infty$ | -0.05 | 0.13 | 0.04 | -0.10 | 0.13 | 0.28 | 0.58 | 0.29 | 0.38
Large
25 | -0.23 | -1.15 | 0.55 | 0.20 | -0.24 | 0.66 | 0.36 | 0.30 | 0.45
50 | 0.01 | -0.48 | 0.43 | 0.26 | -0.16 | 0.54 | 0.39 | 0.20 | 0.41
75 | 0.07 | -0.26 | 0.35 | 0.26 | -0.11 | 0.46 | 0.38 | 0.16 | 0.36
100 | 0.11 | -0.16 | 0.30 | 0.26 | -0.09 | 0.42 | 0.38 | 0.14 | 0.34
125 | 0.12 | -0.08 | 0.27 | 0.25 | -0.07 | 0.38 | 0.39 | 0.13 | 0.31
150 | 0.13 | -0.05 | 0.24 | 0.25 | -0.08 | 0.35 | 0.33 | 0.11 | 0.30
250 | 0.14 | 0.02 | 0.18 | 0.23 | -0.06 | 0.28 | 0.27 | 0.10 | 0.24
350 | 0.14 | 0.06 | 0.15 | 0.21 | -0.03 | 0.24 | 0.28 | 0.10 | 0.21
500 | 0.12 | 0.08 | 0.12 | 0.18 | -0.02 | 0.19 | 0.21 | 0.06 | 0.16
1000 | 0.06 | 0.07 | 0.09 | 0.13 | -0.03 | 0.15 | 0.07 | 0.03 | 0.14
BIC | 0.11 | -0.14 | 0.30 | 0.25 | -0.09 | 0.41 | 0.31 | 0.14 | 0.34
$\infty$ | 0.94 | -1.43 | 0.89 | 0.34 | -1.60 | -0.15 | -2.40 | -1.80 | -0.92