# BVARs and Stochastic Volatility

Joshua C. C. Chan
Purdue University
ORCID: 0000-0003-3632-128X  This review is prepared for the Handbook of
Macroeconomic Forecasting edited by Mike Clements and Ana Galvao. I thank Todd
Clark, Frank Wu, Wei Zhang and the editors Mike Clements and Ana Galvao for
their helpful comments and suggestions.

(October 2023)

###### Abstract

Bayesian vector autoregressions (BVARs) are the workhorse in macroeconomic
forecasting. Research in the last decade has established the importance of
allowing time-varying volatility to capture both secular and cyclical
variations in macroeconomic uncertainty. This recognition, together with the
growing availability of large datasets, has propelled a surge in recent
research in building stochastic volatility models suitable for large BVARs.
Some of these new models are also equipped with additional features that are
especially desirable for large systems, such as order invariance—i.e.,
estimates are not dependent on how the variables are ordered in the BVAR—and
robustness against COVID-19 outliers. Estimation of these large, flexible
models is made possible by the recently developed equation-by-equation
approach that drastically reduces the computational cost of estimating large
systems. Despite these recent advances, there remains much ongoing work, such
as the development of parsimonious approaches for time-varying coefficients
and other types of nonlinearities in large BVARs.

Keywords: large Bayesian VAR, heavy tail, heteroskedasticity, forecasting,
order invariance, COVID-19

##  1 Introduction

Bayesian vector autoregressions (BVARs) are the workhorse in empirical
macroeconomics, especially for forecasting applications and structural
analysis. Since the pathbreaking work by Sims (1980) and Doan, Litterman, and
Sims (1984) in the early 1980s, there have been many important methodological
advances in the BVAR literature. The most notable are the following two
developments that motivate a large and expanding body of research in the last
decade or so.

First, there is a growing interest in exploiting richer information in
macroeconomic analysis, following the seminal paper by Bańbura, Giannone, and
Reichlin (2010) that demonstrates the benefits of including a large number of
variables in BVARs.111The earlier paper by Leeper, Sims, and Zha (1996)
develops various medium-sized structural VARs to study the effects of monetary
policy, including one with 18 variables. Koop (2013) compares the forecast
performance of a range of shrinkage priors for BVARs using a dataset
containing 168 variables, and finds that BVARs tend to forecast better than
factor models. Indeed, there are many situations in which it is necessary to
consider many variables simultaneously. For example, central banks and policy
institutions routinely monitor and forecast dozens of key macroeconomic
variables (Crump, Eusepi, Giannone, Qian, and Sbordone, 2021). In applications
involving regional time-series, where a region can be a country within an
economic union or a state/province within a country, it is often essential to
explicitly model the interactions among the regions (Canova and Ciccarelli,
2009; Koop and Korobilis, 2016; Koop, McIntyre, and Mitchell, 2020). For
nowcasting applications, it is important to incorporate the flow of data
releases in real time, which necessitates a data-rich approach (Giannone,
Reichlin, and Small, 2008; Bańbura, Giannone, Modugno, and Reichlin, 2013).

Many other applications also naturally call for the inclusion of a large
number of time-series, such as those involving data in multiple frequencies
(Schorfheide and Song, 2015; McCracken, Owyang, and Sekhposyan, 2021),
disaggregated data (Giannone, Lenza, Momferatou, and Onorante, 2014; Ellahie
and Ricco, 2017), firm-level data (Demirer, Diebold, Liu, and Yilmaz, 2018)
and financial data (Carriero, Kapetanios, and Marcellino, 2009). Finally, the
wide availability of large time-series datasets (McCracken and Ng, 2016, 2020;
Baumeister, Leiva-León, and Sims, 2022; Bokun, Jackson, Kliesen, and Owyang,
2023) further propels this development.

The second development is the increasing recognition of the need to allow for
time-varying volatility in modeling most macroeconomic datasets. Influential
papers by Cogley and Sargent (2005), Primiceri (2005) and Sims and Zha (2006)
underscore the secular variations in volatility—e.g., many key macroeconomic
variables were more volatile in the Great Inflation of the 1970s than the
following Great Moderation. At the business cycle frequency, a large body of
evidence has shown that macroeconomic volatility is strongly counter-cyclical
(see, e.g., Bloom, 2014; Jurado, Ludvigson, and Ng, 2015). Lastly, the
unexpected drastic movements in many macroeconomic variables at the onset of
the COVID-19 pandemic and the subsequent heightened volatility further
underline the need to allow for time-varying volatility.

Within the BVAR literature, papers such as Clark (2011), D’Agostino, Gambetti,
and Giannone (2013), Clark and Ravazzolo (2015), Cross and Poon (2016) and
Chan and Eisenstat (2018) highlight the empirical relevance of time-varying
volatility for model-fit and forecasting in small BVARs. For large BVARs,
Carriero, Clark, and Marcellino (2016) and Chan (2023a) present Bayesian model
comparison results that show overwhelming data support in favor of stochastic
volatility. Koop and Korobilis (2013) and Carriero, Clark, and Marcellino
(2019) demonstrate that BVARs with stochastic volatility forecast better than
their homoskedastic counterparts. Clark and Mertens (2023) provide a recent
review on the wide range of applications of BVARs with stochastic volatility.

The convergence of these two developments drives a surge in recent research in
building stochastic volatility models suitable for large BVARs and designing
efficient methods for estimating these models. We first introduce in Section 2
three classes of stochastic volatility models—the common stochastic
volatility, the Cholesky stochastic volatility and the factor stochastic
volatility—that are especially suitable for large BVARs. We then dive into
some recent research in developing BVARs with features that are particularly
desirable for large systems in Section 3. These include order invariance—i.e.,
estimation results that are not dependent on how the variables are ordered in
the BVAR—and robustness against COVID-19 outliers. Section 4 first reviews
Bayesian shrinkage priors, particularly the family of Minnesota priors. It
then outlines Markov chain Monte Carlo (MCMC) methods for estimating BVARs
with various types of stochastic volatility, followed by a discussion on
alternative Bayesian approaches and additional strategies to speed up
computations. Section 5 further explores BVARs with time-varying parameters
and other types of nonlinearities. Finally, Section 6 highlights a few
outstanding challenges and ongoing research directions.

##  2 Stochastic Volatility Models for VARs

In this section we discuss a few commonly-used stochastic volatility models in
empirical studies involving macroeconomic forecasting and structural analysis.
Since there is an increasing demand to incorporate a large number of time-
series, we pay special attention to the trade-off between model flexibility
and tractability in larger systems.

Let $\mathbf{y}_{t}=(y_{1,t},\ldots,y_{n,t})^{\prime}$ denote the $n$-vector
of endogenous variables for $t=1,\ldots,T$. Consider the following VAR with a
generic time-varying error covariance matrix $\boldsymbol{\Sigma}_{t}$:

$\mathbf{y}_{t}=\mathbf{a}+\mathbf{A}_{1}\mathbf{y}_{t-1}+\cdots+\mathbf{A}_{p}\mathbf{y}_{t-p}+\boldsymbol{\varepsilon}_{t},\quad\boldsymbol{\varepsilon}_{t}\sim\mathcal{N}(\mathbf{0}_{n},\boldsymbol{\Sigma}_{t}),$
(1)

where $\mathbf{a}$ is an $n\times 1$ vector of intercepts,
$\mathbf{A}_{1},\ldots,\mathbf{A}_{p}$ are $n\times n$ coefficient matrices
and $\mathbf{0}_{n}$ denotes an $n$-vector of zeros. To model volatility
clustering—e.g., the empirical observation that large changes tend to be
followed by large changes, and small changes followed by small changes—it is
important to specify a persistent law of motion for $\boldsymbol{\Sigma}_{t}$.
While there are many ways to construct persistent processes, the key challenge
here is that for any time $t$, $\boldsymbol{\Sigma}_{t}$ needs to be a
positive-definite covariance matrix. In what follows, we discuss various
models for $\boldsymbol{\Sigma}_{t}$ that satisfy these requirements.

###  2.1 Common Stochastic Volatility

We first consider the common stochastic volatility model introduced in
Carriero, Clark, and Marcellino (2016), which is the very first stochastic
volatility model designed for large BVARs. The model is motivated by the
observation that the estimated time-varying volatilities of many US
macroeconomic variables have broadly similar low-frequency movements: they
tend to have high volatility in the 1970s, a drastic decrease in volatility
starting in the early 1980s that mark the beginning of the Great Moderation,
followed by a new surge in volatility at the onset of the Great Recession of
2007-2009. In particular, Carriero, Clark, and Marcellino (2016) obtain the
estimated volatilities of 14 major US macroeconomic time-series from
univariate autoregressive models with stochastic volatility, and find that the
first principal component explains about 66% of the variation in the
individual volatility time-series.

Motivated by this common component in volatilities, Carriero, Clark, and
Marcellino (2016) propose the common stochastic volatility model in which the
error covariance matrix is scaled by a common, time-varying factor that may be
interpreted as the overall macroeconomic volatility:

$\boldsymbol{\Sigma}_{t}=\text{e}^{h_{t}}\boldsymbol{\Sigma},$ (2)

where $\boldsymbol{\Sigma}$ is a time-invariant covariance matrix. The log-
volatility $h_{t}$ in turn follows a stationary AR(1) process:

$h_{t}=\phi h_{t-1}+u_{t}^{h},\quad u_{t}^{h}\sim\mathcal{N}(0,\sigma^{2}),$
(3)

for $t=2,\ldots,T$, where $|\phi|<1$ and the initial condition is specified as
$h_{1}\sim\mathcal{N}(0,\sigma^{2}/(1-\phi^{2}))$. Note that for
identification purposes, the unconditional mean of the AR(1) process is
assumed to be zero.

For large systems, estimation of the generic VAR with stochastic volatility in
(1) is computationally intensive because of the large number of VAR
coefficients
$\mathbf{A}=(\mathbf{a}_{0},\mathbf{A}_{1},\ldots,\mathbf{A}_{p})^{\prime}$:
it grows quadratically in the number of endogenous variables $n$. The key
advantage of the common stochastic volatility model is that it leads to many
useful analytical results that make estimation fast, with computational
complexity of the order $\mathcal{O}(n^{3})$—provided that the natural
conjugate prior on $(\mathbf{A},\boldsymbol{\Sigma})$ is used (see, e.g., Koop
and Korobilis, 2010; Karlsson, 2013, and Chapter 2 of this volume by
Hauzenberger, Huber and Koop). Consequently, estimation of BVARs with dozens
endogenous variables can be done in minutes instead of hours for other
multivariate stochastic volatility models.

The main drawback of the common stochastic volatility model is that it appears
to be restrictive along two dimensions. First, fast estimation with
computational complexity $\mathcal{O}(n^{3})$ is only possible when the
natural conjugate prior is used, but the natural conjugate prior can be
restrictive as it rules out cross-variable shrinkage (see, e.g. Carriero,
Clark, and Marcellino, 2015; Chan, 2022). Second, the common stochastic
volatility model assumes in particular that all variances and covariances are
scaled by a single factor. Consequently, they are always proportional to each
other. Nevertheless, as discussed above, there is empirical evidence that the
error variances of many US macroeconomic variables share broadly similar low-
frequency movements, and a common stochastic volatility is a parsimonious way
to model this empirical feature.

The common stochastic volatility model in (2)–(3) can be extended in various
ways. In particular, Chan (2020) considers a more general family of BVARs that
can accommodate non-Gaussian, heteroskedastic and serially dependent errors.
In addition, fast estimation of complexity $\mathcal{O}(n^{3})$ can be
attained for this wide family of BVARs.

Recent empirical applications using this common stochastic volatility model
and its extensions include Mumtaz (2016), Mumtaz and Theodoridis (2017), Götz
and Hauzenberger (2018), Poon (2018), Louzis (2019), LeSage and Hendrikz
(2019), Hartwig (2021), Chan, Poon, and Zhu (2023) and Hou, Nguyen, and Zhang
(2023).

###  2.2 Cholesky Stochastic Volatility

Instead of scaling the error covariance matrix by a common volatility factor,
a more flexible approach for modeling time-varying volatilities and
correlations is to incorporate multiple stochastic volatility processes. In
particular, Cogley and Sargent (2005) build a multivariate stochastic
volatility model based on the modified Cholesky decomposition. More
specifically, consider again the VAR in (1), but now the error covariance
matrix is constructed via

$\boldsymbol{\Sigma}_{t}=\mathbf{B}_{0}^{-1}\mathbf{D}_{t}(\mathbf{B}_{0}^{-1})^{\prime},$
(4)

where $\mathbf{B}_{0}$ is an $n\times n$ lower triangular matrix with ones on
the diagonal and
$\mathbf{D}_{t}=\text{diag}(\text{e}^{h_{1,t}},\ldots,\text{e}^{h_{n,t}})$.
The law of motion for each element of
$\mathbf{h}_{t}=(h_{1,t},\ldots,h_{n,t})^{\prime}$ is specified as an
independent autoregressive process:

$h_{i,t}=\mu_{i}+\phi_{i}(h_{i,t-1}-\mu_{i})+u_{i,t}^{h},\quad
u_{i,t}^{h}\sim\mathcal{N}(0,\sigma_{i}^{2})$ (5)

for $t=2,\ldots,T$, where the initial condition is specified as
$h_{i,1}\sim\mathcal{N}(\mu_{i},\sigma_{i}^{2}/(1-\phi_{i}^{2}))$. Due to its
construction using the modified Cholesky decomposition, this stochastic
volatility model is sometimes called the Cholesky stochastic volatility.

Comparing to the common stochastic volatility model in (2), it is clear that
the Cholesky stochastic volatility is more flexible—it contains $n$ stochastic
volatility processes—and can accommodate more complex co-volatility patterns.
In particular, the Cholesky stochastic volatility can in principle recover the
common stochastic volatility specification if we reparameterize the model in
(4) so that $\mathbf{B}_{0}$ is the lower Cholesky factor of
$\boldsymbol{\Sigma}^{-1}$ and
$\mathbf{D}_{t}=\text{e}^{h_{t}}\mathbf{I}_{n}$, where $\mathbf{I}_{n}$ is the
$n$-dimensional identity matrix.

This modeling flexibility, however, comes at a cost of more computationally
intensive posterior simulations. In particular, the conventional approach of
drawing all VAR coefficients jointly, which has $\mathcal{O}(n^{6})$
computational complexity, becomes excessively time-consuming when $n$ is
large. To tackle this computational problem, Carriero, Clark, and Marcellino
(2019) introduce a blocking scheme that makes it possible to estimate the
model equation by equation that drastically reduces the computational
complexity to $\mathcal{O}(n^{4})$; see Section 4 for more details.
Nevertheless, the estimation of the Cholesky stochastic volatility model is
still an order of magnitude slower than that of the common stochastic
volatility (when the natural conjugate prior is used).

There are various useful extensions of the basic setup in (4)-(5). First,
instead of the independent autoregressive processes specified in (5),
$\mathbf{h}_{t}$ can be modeled as a VAR(1) to allow for potential
correlations among the volatility processes. In the extreme, Chan, Eisenstat,
and Strachan (2020) consider a singular VAR for $\mathbf{h}_{t}$ to capture
the idea that only a few common shocks drive the evolution of the $n$
volatility processes. Second, the modified Cholesky factor $\mathbf{B}_{0}$ in
(4) can be made time-varying, as introduced in Primiceri (2005). In fact, the
multivariate stochastic volatility model of Primiceri (2005) is perhaps the
most widely used specification in empirical macroeconomics, at least for small
systems.

Recent applications using large BVARs with the Cholesky stochastic volatility
include Bańbura and van Vlodrop (2018), Bianchi, Guidolin, and Ravazzolo
(2018), Huber and Feldkircher (2019), Cross, Hou, and Poon (2019), Koop,
McIntyre, Mitchell, and Poon (2020), Tallman and Zaman (2020), Zens, Böck, and
Zörner (2020), Baumeister, Korobilis, and Lee (2022), Fischer, Hauzenberger,
Huber, and Pfarrhofer (2023) and Gianfreda, Ravazzolo, and Rossini (2023).

###  2.3 Factor Stochastic Volatility

Another approach to model the law of motion of the error covariance matrix in
(1) is through the class of factor stochastic volatility models (Pitt and
Shephard, 1999; Aguilar and West, 2000; Chib, Nardari, and Shephard, 2006). A
conventional factor model with homoskedastic factors and idiosyncratic errors
is typically used for dimensionality reduction—to model linear relationships
among a large number of variables in a very parsimonious way. For instance, an
unrestricted $n\times n$ covariance matrix $\boldsymbol{\Sigma}$ has
$n(n+1)/2$ free elements; the number of parameters thus grows quadratically in
$n$. To reduce the number of parameters to be estimated, a factor model in
essence specifies $\boldsymbol{\Sigma}$ using simpler matrices
$\mathbf{G},\mathbf{D}$ and $\mathbf{L}$ via
$\boldsymbol{\Sigma}=\mathbf{L}\mathbf{G}\mathbf{L}^{\prime}+\mathbf{D}$,
where $\mathbf{G}$ and $\mathbf{D}$ are $r\times r$ and $n\times n$ diagonal
matrices, respectively, and $\mathbf{L}$ is an $n\times r$ matrix, typically
with $r\ll n$. Even for an unrestricted $\mathbf{L}$, the number of parameters
in a factor model is only $nr+n+r$; it grows linearly in $n$. Consequently,
for high-dimensional settings with a large $n$ but a small $r$, a factor model
can substantially reduce the number of parameters.

Factor stochastic volatility models extend the classic factor model by
introducing time-varying volatility. An example is to specify the error
covariance matrix $\boldsymbol{\Sigma}_{t}$ as

$\boldsymbol{\Sigma}_{t}=\mathbf{L}\mathbf{G}_{t}\mathbf{L}^{\prime}+\mathbf{D}_{t},$
(6)

where
$\mathbf{D}_{t}=\text{diag}(\text{e}^{h_{1,t}},\ldots,\text{e}^{h_{n,t}})$ and
$\mathbf{G}_{t}=\text{diag}(\text{e}^{h_{n+1,t}},\ldots,\text{e}^{h_{n+r,t}})$,
and $h_{i,t},i=1,\ldots,n+r,$ follows the AR(1) process in (5). To facilitate
estimation, data augmentation is typically used to represent the model in
terms of latent factors. More specifically, consider the same VAR in (1), but
now the innovation is constructed via

$\boldsymbol{\varepsilon}_{t}=\mathbf{L}\mathbf{f}_{t}+\mathbf{u}_{t},$ (7)

where $\mathbf{f}_{t}=(f_{1,t},\ldots,f_{r,t})^{\prime}$ is an $r\times 1$
vector of latent factors and $\mathbf{L}$ is the associated $n\times r$ factor
loadings matrix. The errors $\mathbf{u}_{t}$ and the latent factors
$\mathbf{f}_{t}$ are assumed to be independent at all leads and lags and
jointly Gaussian:

$\begin{pmatrix}\mathbf{u}_{t}\\\
\mathbf{f}_{t}\end{pmatrix}\sim\mathcal{N}\left(\begin{pmatrix}\mathbf{0}\\\
\mathbf{0}\end{pmatrix},\begin{pmatrix}\mathbf{D}_{t}&\mathbf{0}\\\
\mathbf{0}&\mathbf{G}_{t}\end{pmatrix}\right),$ (8)

where
$\mathbf{D}_{t}=\text{diag}(\text{e}^{h_{1,t}},\ldots,\text{e}^{h_{n,t}})$ and
$\mathbf{G}_{t}=\text{diag}(\text{e}^{h_{n+1,t}},\ldots,\text{e}^{h_{n+r,t}})$.
Under the representation in (7)-(8), it is clear that the correlations among
the elements of $\boldsymbol{\varepsilon}_{t}$ are induced solely by the
latent factors.

For identification purposes, it is common to assume $\mathbf{L}$ to be lower
triangular with ones on the main diagonal. However, Chan, Eisenstat, and Yu
(2022) show that in the presence of stochastic volatility and
$r\leqslant(n-1)/2$, the factor loadings matrix $\mathbf{L}$ is identified up
to permutations and sign switches (with the restriction that
$\mu_{i},i=n+1,\ldots,n+r$, the unconditional means of the log-volatilities
associated with the factors, are set to 0).222To handle the so-called label
switching problem (the latent factors are permutations invariant), one can
postprocess the posterior draws to sort them into the correct categories
using, e.g., the approach in Kaufmann and Schumacher (2019).

In practice one also needs to determine the number of factors $r$. A common
approach is to first fix the number of factors $r$ in the estimation, and then
select $r$ using the marginal likelihood or some information criterion. An
alternative approach is to include a large number of factors, and then use a
shrinkage prior on the factor loadings so that they are increasingly shrunk to
0 as the column index increases, as considered in, e.g., Bhattacharya and
Dunson (2011) and Kastner (2019). This latter approach circumvents the need to
select the number of factors, but it makes the interpretation of the factors
more difficult.

Compared to the Cholesky stochastic volatility, the factor stochastic
volatility contains $r$ additional volatility processes and is apparently more
flexible. However, as discussed earlier, and is clear in the decomposition in
(6), the latter inherits the dimensionality reduction property of the classic
factor model. As such, whether it fits macroeconomic time-series better than
the two other stochastic volatility models is an empirical question. In terms
of estimation, the factor stochastic volatility is more computationally
intensive to estimate relative to the common stochastic volatility, but it can
still be fitted reasonably quickly even when $n$ is large. In particular,
given the latent factors, the VAR becomes $n$ unrelated regressions.
Consequently, the $n$-equation system can be estimated equation by equation.

While a variety of factor stochastic volatility models are commonly employed
in financial applications, with recent examples in Jin, Maheu, and Yang
(2019), Hautsch and Voigt (2019), Kastner (2019), Loaiza-Maya and Smith (2020)
and McCausland, Miller, and Pelletier (2020), they are not yet widely used in
the context of large BVARs. Notable exceptions are Kastner and Huber (2020)
and Hauzenberger, Huber, Koop, and Mitchell (2022).

##  3 Order-Invariant and Outlier-Augmented Models

In this section we introduce a variety of recently proposed stochastic
volatility models with features that are especially relevant for large systems
and for fitting post COVID-19 data.

###  3.1 Order-Invariant Stochastic Volatility Models

As outlined in Section 2, the law of motion for $\boldsymbol{\Sigma}_{t}$ is
typically constructed by combining multiple volatility processes using some
version of matrix decomposition or factorization. This construction process
often leads to order dependence—estimates might depend on the order of
variables arranged in the vector $\mathbf{y}_{t}$. For instance, the Cholesky
stochastic volatility, which is based on a lower triangular parameterization,
is not order invariant. In particular, as explained in Carriero, Clark, and
Marcellino (2019), since the model is constructed using a lower triangular
impact matrix $\mathbf{B}_{0}$ and the priors are independently elicited on
$\mathbf{B}_{0}$ and the volatility processes, the implied prior on
$\boldsymbol{\Sigma}_{t}$ is not order invariant.

Even though this ordering issue is well-known and is explicitly discussed in
Cogley and Sargent (2005) and Primiceri (2005), its empirical relevance has
only been appreciated relatively recently. For example, Arias, Rubio-Ramirez,
and Shin (2022) use the model of Primiceri (2005) to produce point and density
forecasts of 4 macroeconomic variables. They show that while the point
forecasts are essentially the same across all the permutations of the 4
variables, the density forecasts can differ substantially across different
variable orderings. Since the number of permutations grows exponentially as
the number of variables increases, this ordering issue is expected to be more
acute in large BVARs involving dozens of dependent variables. In fact, Chan,
Doucet, León-González, and Strachan (2018) find evidence that estimates of
reduced-form error variances based on the Cholesky stochastic volatility can
drastically change across different variable orderings in large systems.333The
structural analysis in both Cogley and Sargent (2005) and Primiceri (2005) are
based on a recursive identification scheme. As such, the order of the
variables is part of the identification scheme and is therefore predetermined.
However, many later studies (e.g., Benati, 2008; Baumeister and Peersman,
2013) use the BVARs in Cogley and Sargent (2005) and Primiceri (2005) as
reduced-form models and implement other, non-recursive identification schemes.
In those cases, the reduced-form models do not have a natural ordering. One
such example is given in Bognanni (2018), who demonstrates that responses to
structural shocks identified with sign restrictions—using the reduced-form
BVAR of Primiceri (2005)—can be sensitive to how the variables are ordered;
see also Hartwig (2020) for another example.

Given the sensitivity of the estimates and forecasts on different variable
orderings, it is therefore desirable to employ order-invariant stochastic
volatility models in empirical applications. Which of the stochastic
volatility models discussed in Section 2 are order invariant? It is clear that
the common stochastic volatility is order invariant, whereas, as discussed
earlier, the Cholesky stochastic volatility is not. However, the latter can be
extended to be order invariant by a slight modification: avoid the use of the
lower triangular parameterization of $\mathbf{B}_{0}$ in (4). Based on this
observation, Chan, Koop, and Yu (2023) extend the Cholesky stochastic
volatility model by allowing the impact matrix $\mathbf{B}_{0}$ to be any
dense, non-degenerate matrix. They prove that the model is order invariant,
and based on the results in Bertsche and Braun (2022), the impact matrix
$\mathbf{B}_{0}$ is identified up to permutations and sign switches. Moreover,
estimation can be done similarly as in the Cholesky stochastic volatility
model, particularly that the VAR coefficients can be sampled equation by
equation. Chan, Koop, and Yu (2023) also consider a version of the model with
a time-varying impact matrix $\mathbf{B}_{0}$, thus extending the model of
Primiceri (2005). An alternative approach based on a different factorization
of $\boldsymbol{\Sigma}_{t}$ is considered in Wu and Koop (2022). More
specifically, they construct $\boldsymbol{\Sigma}_{t}$ using the
eigendecomposition, where the $n$ log-eigenvalues evolve as random walks. They
develop a fast posterior sampler to estimate the model and show that it is
applicable to large datasets.

Finally, whether a factor stochastic volatility model is order invariant
depends on the type of restrictions imposed on the factor loadings. As noted
in Kastner (2019), if one imposes the usual identification restrictions that
the factor loadings matrix is triangular, then the model is not order
invariant. By contrast, Chan, Eisenstat, and Yu (2022) show that, under some
mild conditions discussed in the previous section, a factor stochastic
volatility model is order invariant and the factor loadings are identified up
to permutations and sign switches.

There are other order-invariant modeling approaches, though they are typically
designed for small systems. One popular approach is to construct multivariate
stochastic volatility models using Wishart or inverse-Wishart processes;
examples include Philipov and Glickman (2006) and Asai and McAleer (2009). A
few papers extend these multivariate stochastic volatility models to BVARs,
including Chan, Doucet, León-González, and Strachan (2018), Shin and Zhong
(2020) and Arias, Rubio-Ramirez, and Shin (2022). More recently, Arias, Rubio-
Ramirez, and Shin (2022) introduce a new order-invariant approach that builds
on the new parameterization of correlation matrices proposed in Archakov and
Hansen (2021). More specifically, they construct a time-varying correlation
matrix based on random walk processes under the new parameterization. However,
estimation of these stochastic volatility models is generally computationally
intensive, as it often involves drawing from high-dimensional, non-standard
distributions. Consequently, these stochastic volatility models do not scale
well to large datasets.

Another order-invariant approach is based on the discounted Wishart process,
which admits efficient filtering and smoothing algorithms for estimation.
Stochastic volatility models constructed under this approach are considered in
Uhlig (1997), West and Harrison (2006), Prado and West (2010), Bognanni (2018)
and Arias, Rubio-Ramirez, and Shin (2022). However, the discounted Wishart
process appears to be too tightly parameterized for typical macroeconomic
data, and it tends to underperform in terms of both point and density
forecasts relative to standard stochastic volatility models such as Cogley and
Sargent (2005) and Primiceri (2005), as demonstrated in a forecasting exercise
in Arias, Rubio-Ramirez, and Shin (2022).

###  3.2 Outlier-Augmented Stochastic Volatility Models

The COVID-19 pandemic has caused extreme movements in many macroeconomic and
financial time-series. One consequence is that impulse response functions and
forecasts from homoskedastic BVARs are heavily distorted by these extreme
observations, as demonstrated in Schorfheide and Song (2021) and Lenza and
Primiceri (2022) using US data and Bobeica and Hartwig (2023) using euro area
data. This problem can be ameliorated by using BVARs with stochastic
volatility, which downweight extreme observations. However, as pointed out by
Carriero, Clark, Marcellino, and Mertens (2022), extreme observations, by
definition, should reflect transitory spikes, not permanent increases, in
volatility. But in a typical stochastic volatility, changes in volatility are
assumed to be highly persistent. It is therefore useful to augment standard
stochastic volatility models to include an explicit component to model
infrequent volatility spikes.

There is a general approach to directly model infrequent volatility spikes,
and it can be incorporated into the many multivariate stochastic volatility
models discussed earlier. The key idea is to replace the normal distribution
in (1) with a more robust distribution that puts more mass on extreme events.
A common example is a continuous heavy-tailed distribution, such as the
Student-$t$ distribution. As long as the robust distribution can be
represented as a scale, finite or an infinite mixture of normals, estimation
is straightforward thanks to data augmentation and the modular nature of MCMC
methods.

As an example, take the Cholesky stochastic volatility model with a
generalized $t$ distribution considered in Clark and Ravazzolo (2015), Cross
and Poon (2016) and Chiu, Mumtaz, and Pinter (2017), which is constructed from
$n$ univariate Student-$t$ distributions with different degree of freedom
parameters.444A similar heavy-tailed stochastic volatility model is introduced
by Cúrdia, Del Negro, and Greenwald (2014) to construct a dynamic stochastic
general equilibrium model that can accommodate extreme events during the Great
Recession. It can be represented as a conditionally Gaussian VAR specified in
(1) with $\boldsymbol{\Sigma}_{t}$ constructed as

$\boldsymbol{\Sigma}_{t}=\mathbf{B}_{0}^{-1}\boldsymbol{\Lambda}_{t}\mathbf{D}_{t}(\mathbf{B}_{0}^{-1})^{\prime}.$
(9)

Compared to the standard Cholesky stochastic volatility in (4), the new
addition is the diagonal matrix of latent variables
$\boldsymbol{\Lambda}_{t}=\text{diag}(\lambda_{1,t},\ldots,\lambda_{n,t})$,
where $\lambda_{i,t}\sim\mathcal{IG}(\nu_{i}/2,\nu_{i}/2)$ and
$\mathcal{IG}(a,b)$ denotes the inverse-gamma distribution with mean $b/(a-1)$
when $a>1$.555For the special case with
$\lambda_{1,t}=\cdots=\lambda_{n,t}=\lambda_{t}\sim\mathcal{IG}(\nu/2,\nu/2)$,
the marginal distribution of the innovation $\boldsymbol{\varepsilon}_{t}$
(unconditional on the latent variable $\lambda_{t}$) is a Student-$t$
distribution with degree of freedom parameter $\nu$ (see, e.g., Geweke, 1993;
Ni and Sun, 2005).

Another example is the outlier-augmented stochastic volatility model proposed
by Carriero, Clark, Marcellino, and Mertens (2022), which extends the Cholesky
stochastic volatility by incorporating a discrete mixture representation first
introduced in Stock and Watson (2016) for handling outliers in unobserved
components models. More specifically, it has the same conditionally Gaussian
VAR representation with $\boldsymbol{\Sigma}_{t}$ specified in (9), but the
latent variable $\lambda_{i,t}$ is specified as $\lambda_{i,t}=o_{i,t}^{2}$,
where $o_{i,t}$ follows a 2-part distribution with a point mass at 1 and a
uniform distribution on the interval $(2,20)$. The first part of the
distribution represents ‘regular’ observations with scale normalized to 1,
whereas the second part captures ‘outliers’ that have 2-20 times larger
standard deviations relative to regular observations.666In the original setup
in Stock and Watson (2016), the range of the uniform distribution is from 2 to
10. As noted in Carriero, Clark, Marcellino, and Mertens (2022), many
macroeconomic variables exhibit extreme variability at the onset of the
COVID-19 pandemic, and the upper range is accordingly increased to 20.

Other distributional assumptions for $\lambda_{i,t}$ are possible and they
would induce different marginal distributions on the innovations. Examples
include the multivariate Laplace distribution (Eltoft, Kim, and Lee, 2006), a
finite mixture of scale mixtures of normals (Karlsson and Mazur, 2020) and an
infinite mixture of normals (Braun, 2021). These non-Gaussian distributions
can also be incorporated in other stochastic volatility models, as is done in
Chan (2020) and Hartwig (2021).

##  4 Shrinkage Priors and Posterior Simulations

What makes BVARs Bayesian is the use of informative priors on the large number
of VAR coefficients. Below we first outline a family of priors generally
referred to as the Minnesota priors. We then discuss Bayesian estimation of
BVARs with various types of stochastic volatility introduced earlier,
particularly the posterior simulation of the log-volatilities and the VAR
coefficients.

###  4.1 Shrinkage Priors on the VAR Coefficients

We first discuss a family of priors on the VAR coefficients that can be traced
back to the work in the 1980s by Doan, Litterman, and Sims (1984) and
Litterman (1986). Due to their affiliations at that time, this family of
priors is commonly known as the Minnesota priors. In the original formulation
for homoskedastic VARs, the error covariance matrix is fixed at some estimate,
and a normal prior is elicited on the VAR coefficients
$\boldsymbol{\alpha}=\text{vec}(\mathbf{A}^{\prime})$. Later Kadiyala and
Karlsson (1993, 1997) extend the original version to a joint prior on
$\boldsymbol{\alpha}$ and the time-invariant error covariance matrix. Since
then many other more flexible variants have been developed, for both
homoskedastic and heteroskedastic VARs, and they can be written as a
conditionally normal prior given some latent variables or hyperparameters
$\boldsymbol{\xi}$:

$(\boldsymbol{\alpha}\,|\,\boldsymbol{\xi})\sim\mathcal{N}(\boldsymbol{\alpha}_{0},\mathbf{V}),$
(10)

where the prior covariance matrix $\mathbf{V}$ may depend on
$\boldsymbol{\xi}$. The prior mean $\boldsymbol{\alpha}_{0}$ is typically set
to zero for growth rates data, such as GDP growth. This reflects the prior
belief that growth rates data are generally not very persistent, and the VAR
coefficients are thus shrunk to 0. For levels data such as private investment
or inflation indices, $\boldsymbol{\alpha}_{0}$ is set to be zero except for
the coefficients associated with the first own lag, which are set to be one.
This reflects the prior belief that levels data are highly persistent—in
particular, it expresses the preference for a random walk specification.

In early versions of the Minnesota priors there are no latent variables and
the prior hyperparameters are fixed at some subjective values. The prior
covariance matrix $\mathbf{V}$ is typically assumed to be diagonal, and the
diagonal elements are elicited to reflect a range of subjective beliefs, such
as cross-variable shrinkage—i.e., the idea that coefficients on other
variables’ lags are on average smaller than coefficients on own lags. This is
often implemented by introducing two hyperparameters, say, $\xi_{1}$ and
$\xi_{2}$, where $\xi_{1}$ controls the overall prior variance for
coefficients on own lags, whereas $\xi_{2}$ for coefficients on other lags.
Then, $\xi_{1}$ and $\xi_{2}$ are carefully calibrated to incorporate the
belief that $\xi_{2}$ is much smaller than $\xi_{1}$.

Naturally, one set of fixed values for $\xi_{1}$ and $\xi_{2}$ is not expected
to be optimal for all datasets with widely different time-series and sample
periods. Indeed, Giannone, Lenza, and Primiceri (2015) demonstrate that
substantial benefits can be obtained—e.g., better out-of-sample forecast
performance using real data and more accurate impulse response functions using
simulated data—simply by estimating these prior hyperparameters from the data
instead of fixing them at some commonly-used subjective values. In a Bayesian
model comparison exercise, Chan (2023a) finds strong evidence in favor of
estimating these prior hyperparameters across BVARs with a variety of
stochastic volatility specifications.777Earlier papers such as Del Negro and
Schorfheide (2004) and Carriero, Kapetanios, and Marcellino (2012) have
adopted an empirical Bayes approach of selecting these prior hyperparameters
by maximizing the marginal likelihood of the homoskedastic BVAR, which is
available analytically. For time-varying parameter models, Amir-Ahmadi,
Matthes, and Wang (2020) show similar benefits in treating the prior
hyperparameters as unknown parameters to be estimated.

There is a surge of interest in the statistics literature to develop
hierarchical shrinkage priors, such as the Bayesian Lasso (Park and Casella,
2008), the normal-gamma prior (Griffin and Brown, 2010), the horseshoe prior
(Carvalho, Polson, and Scott, 2010) and the Dirichlet-Laplace prior
(Bhattacharya, Pati, Pillai, and Dunson, 2015). These shrinkage priors are
originally designed for the setting of a linear regression with a large number
of arbitrary predictors. More recently, they have been introduced in BVAR
settings. Applications include Huber and Feldkircher (2019), Follett and Yu
(2019), Kastner and Huber (2020), Korobilis and Pettenuzzo (2019) and Gefang,
Koop, and Poon (2023). In contrast to Minnesota priors that tend to shrink all
VAR coefficients, these adaptive hierarchical priors have the desirable
theoretical property of only shrinking ‘small’ coefficients strongly to zero,
while leaving ‘large’ coefficients mostly intact.

In practice, however, these new hierarchical priors do not seem to forecast
better than some sophisticated Minnesota priors, as demonstrated in Cross,
Hou, and Poon (2019). One reason for this surprising result could be because
the new hierarchical priors typically treat all VAR coefficients identically,
and do not take into account the many plausible prior beliefs, such as cross-
variable shrinkage and that variables of higher lags are less important,
incorporated into the Minnesota priors. There are a few papers that aim to
adapt the new adaptive hierarchical priors so that they are more suitable for
BVARs. For instance, Huber and Feldkircher (2019) develop a normal-gamma prior
in which additional lag-specific shrinkage parameters are introduced with the
goal of shrinking higher-order lags more strongly to zero. Korobilis and
Pettenuzzo (2019) consider a class of hierarchical shrinkage priors that
incorporate some features of the Minnesota priors. Chan (2021) develops a
framework that explicitly nests both the Minnesota priors and the new adaptive
hierarchical priors. These Minnesota-type adaptive hierarchical priors can be
written as conditionally Gaussian priors with additional latent variables
$\boldsymbol{\xi}$ given in (10). For example, under the Minnesota-type
normal-gamma prior $\boldsymbol{\xi}$ becomes a vector of gamma random
variables; for the Minnesota-type horseshoe prior, elements of
$\boldsymbol{\xi}$ have a half-Cauchy distribution. Chan (2021) shows that
these Minnesota-type adaptive hierarchical priors forecast better than both
the conventional Minnesota priors and the standard adaptive hierarchical
priors.

The literature on shrinkage priors on VAR coefficients is vast and expanding.
Here we have focused on the family of Minnesota priors. More discussions on
other shrinkage priors and approaches can be found in Chapter 2 of this volume
by Hauzenberger, Huber and Koop and excellent reviews by Koop and Korobilis
(2010) and Karlsson (2013).

###  4.2 Bayesian Estimation

Estimating BVARs with stochastic volatility tends to be computationally
intensive, especially for large systems. But a lot of progress has been made
in the last decade, and it is now feasible to fit large BVARs with a variety
of stochastic volatility specifications using MCMC methods. Below we discuss
the posterior simulation of the two most computationally intensive parts:
sampling the log-volatilities and the VAR coefficients.

####  4.2.1 Posterior Simulation of the Log-Volatilities

Conditional on the data and other model parameters, the VAR in (1) defines the
observation equation for the log-volatilities. But since the log-volatilities
enter the model via the time-varying covariance matrix, the state space
representation is nonlinear and high-dimensional, which makes estimation very
challenging in general. However, many multivariate stochastic volatility
models, such as common, Cholesky and factor stochastic volatility, are
constructed in such a way that they can be transformed into a linear (though
non-Gaussian) state space model, for which the auxiliary mixture sampler of
Kim, Shephard, and Chib (1998) can be applied to sample the log-volatilities
efficiently.

More specifically, after transforming the original nonlinear Gaussian
representation into a linear non-Gaussian state space model, Kim, Shephard,
and Chib (1998) observe that the non-Gaussian errors in the new observation
equation can be well approximated using a 7-component mixture of normals.
Then, given the mixture indicators, the approximate model becomes a linear
Gaussian state space model for which sampling the latent states can be done
using standard algorithms, such as the Kalman-filter based smoothers of Carter
and Kohn (1994) and Durbin and Koopman (2002) or the precision-based samplers
of Chan and Jeliazkov (2009) and McCausland, Miller, and Pelletier (2011).
Estimation details of the common, Cholesky and factor stochastic volatility
models can be found in Chan (2023a). Other stochastic volatility models with
additional features can also be estimated using the auxiliary mixture sampler
with minor modifications; see, for example, Carriero, Clark, Marcellino, and
Mertens (2022) and Chan, Koop, and Yu (2023).

One important point to emphasize is that, as pointed out in Del Negro and
Primiceri (2015), when the auxiliary mixture sampler is used as part of a
larger posterior simulator, the latter becomes a collapsed Gibbs sampler,
where the augmented mixture indicators and all other model parameters should
be sampled jointly. This joint sampling can be done by first drawing the model
parameters given the log-volatilities but marginal of the mixture indicators,
followed by sampling the mixture indicators given the log-volatilities and
other model parameters. In contrast to a standard Gibbs sampler, here the
sampling order is important.888It is also feasible to apply a standard Gibbs
sampler by conditioning on the mixture indicators in all the Gibbs steps. This
approach is typically more cumbersome as, e.g., the VAR in (1) is no longer
Gaussian given the mixture indicators. See Stock and Watson (2016) for an
example of this alternative approach. In practice, the main implication is
that the log-volatilities should be sampled immediately after the mixture
indicators.999For stochastic volatility models with multiple time-series of
log-volatilities, where each series has an associated vector of mixture
indicators, it is often more convenient to sample each pair of mixture
indicators and log-volatilities separately. This approach is valid if the
time-series of log-volatilities are conditionally independent given the model
parameters—an example is the Cholesky stochastic volatility model specified in
(4)-(5). If they are not conditionally independent, then all the mixture
indicators should be sampled first before the log-volatilities.

For other stochastic volatility models for which the auxiliary mixture sampler
cannot be applied, there are two alternatives. While they are both very
general and can be applied to a wide range of stochastic volatility models,
they are more computationally demanding. The first is to use sequential Monte
Carlo (SMC) or particle filters. There is a large and rapidly expanding
literature on sequential Monte Carlo methods; Doucet, De Freitas, and Gordon
(2001) and Creal (2012) provide excellent reviews on this large family of
algorithms. The most widely used member is perhaps the bootstrap particle
filter, where the state equation is used to generate particle values from time
$t$ to values at time $t+1$. While the bootstrap particle filter is easy to
implement, its empirical performance is often poor. The main reason is that
the generated particle values from the state equation are unlikely to be
consistent with the data in typical settings. There are a number of more
sophisticated filters that use information from the observation equation to
construct better proposal distributions. A recent example is the tempered
particle filter proposed in Herbst and Schorfheide (2019) that adaptively
constructs the proposal distribution through a sequence of tempering steps.

The second general approach is to implement a Metropolis-Hastings step to
sample the log-volatilities. This can be done by using either a random-walk
Metropolis-Hastings algorithm as in Asai and McAleer (2009), or an
independence-chain Metropolis-Hastings algorithm with a tailored proposal
distribution. The first option is straightforward to implement, but is often
very inefficient in terms of highly autocorrelated posterior draws, especially
in high-dimensional settings. The second option requires a good proposal
distribution that well approximates the conditional distribution of the log-
volatilities. As such, it demands more analysis and is therefore more
difficult to implement. However, there has been a lot of progress in
developing good proposal distributions. For example, McCausland (2012)
introduces the HESSIAN method based on a fifth-order Taylor approximation that
works remarkably well for a range of univariate stochastic volatility models.
Chan and Eisenstat (2018) and Cross, Hou, Koop, and Poon (2023) instead
consider quadratic (hence, Gaussian) approximations of the target
distributions in the context of BVARs with stochastic volatility, which are
easier to obtain and work well for their applications.101010This approach of
using Gaussian approximations is first introduced to estimate univariate
stochastic volatility models in the seminal papers by Durbin and Koopman
(1997) and Shephard and Pitt (1997). They formulate the approximating
algorithm via a linear Gaussian state space model, which implies a Gaussian
density for the states. Chan and Eisenstat (2018) and Cross, Hou, Koop, and
Poon (2023) improve upon this approach by directly computing the Gaussian
approximating density using Newton-Raphson method based on fast band matrix
routines. In addition, sequential Monte Carlo methods can also be used to
construct good proposal distributions; see, e.g., Andrieu, Doucet, and
Holenstein (2010).

####  4.2.2 Posterior Simulation of the VAR Coefficients

Given the conditionally Gaussian VAR in (1) and the conditionally Gaussian
prior on the VAR coefficients $\boldsymbol{\alpha}$ in (10), the conditional
posterior distribution of $\boldsymbol{\alpha}$ is also Gaussian by standard
linear regression results. Hence, in principle sampling $\boldsymbol{\alpha}$
is straightforward. In practice, however, when the number of endogenous
variables in the VAR is large, conventional ways of drawing all VAR
coefficients jointly—e.g., by first obtaining the Cholesky factor of the
covariance matrix or the precision matrix—become excessively computationally
intensive. In particular, for an $n$-variable VAR with $p$ lags, the number of
VAR coefficients (ignoring intercepts) is $n^{2}p$. Since the covariance
matrix of the joint distribution is of dimension $n^{2}p\times n^{2}p$,
obtaining its Cholesky factor has computational complexity of the order
$\mathcal{O}(n^{6}p^{3})$.111111Sampling the VAR coefficients using the
Cholesky factor of the precision matrix is faster, as it avoids explicitly
inverting the precision matrix to obtain the covariance matrix. But since the
precision matrix is dense in this case, its computational complexity is of the
same order of $\mathcal{O}(n^{6}p^{3})$.

To tackle this computational problem, Carriero, Clark, and Marcellino (2019)
consider an alternative sampling scheme of drawing the VAR coefficients
equation by equation that can drastically reduce the computational complexity
by two orders of magnitude to $\mathcal{O}(n^{4}p^{3})$.121212As noted in
Bognanni (2022), the algorithm in Carriero, Clark, and Marcellino (2019) can
only be viewed as an approximation. Carriero, Chan, Clark, and Marcellino
(2022) provide an exact algorithm to draw the VAR coefficients equation by
equation that has the same order of computational complexity. The key reason
for the speed-up is as follows. Since the number of VAR coefficients in each
equation is $np$, drawing these coefficients conditional on coefficients in
other equations can be done in $\mathcal{O}(n^{3}p^{3})$ operations. Iterating
this step over $n$ equations, the total computational complexity is therefore
$\mathcal{O}(n^{4}p^{3})$. This equation-by-equation estimation algorithm is
formulated for the Cholesky stochastic volatility model, but it can be used
for other types of stochastic volatility or homoskedastic VARs with minor
modifications.131313For some special cases, other faster algorithms are
available. For example, under the natural conjugate prior, the common
stochastic volatility model and its variants can be estimated using algorithms
with complexity $\mathcal{O}(n^{3}p^{3})$; see Carriero, Clark, and Marcellino
(2016) and Chan (2020). Carriero, Chan, Clark, and Marcellino (2022) report a
10-50 times speed-up of the equation-by-equation approach for $n$ up to 40
compared to sampling all coefficients jointly.

Another venue to improve sampling speed is to find faster ways to simulate
from high-dimensional Gaussian distributions. For a linear regression with $r$
observations and $m$ covariates, the conventional, Cholesky-factor based
sampling approach to simulate the $m$-vector of coefficients requires
$\mathcal{O}(m^{3})$ operations. Bhattacharya, Chakraborty, and Mallick (2016)
propose an alternative way to sample the coefficients based on the Woodbury
formula, which has computational complexity of $\mathcal{O}(r^{2}m)$. As such,
this new algorithm is expected to be faster in cases where there are far more
covariates than observations. For our BVAR setting with $r=Tn$ and $m=np$
(sampling the coefficients in one equation), the new algorithm has
computational complexity of $\mathcal{O}(T^{2}n^{3}p)$ compared to the
conventional sampling approach of $\mathcal{O}(n^{3}p^{3})$. Both have the
same order of complexity in $n$, but they have different complexity in other
dimensions. For instance, when $T$ is small and $p$ is large, the algorithm of
Bhattacharya, Chakraborty, and Mallick (2016) is expected to perform better.

####  4.2.3 Alternative Bayesian Approaches

So far we have focused on various MCMC methods to estimate the reduced-form
VAR given in (1). There is a range of additional strategies to speed up
computations so that one can fit larger BVARs with more flexible features.

One approach is to reparameterize the standard reduced-form VAR in the
structural form with a recursive system (see, e.g., Chan, 2021; Gefang, Koop,
and Poon, 2023). The structural-form parameter estimates can then be
transformed to obtain the corresponding reduced-form estimates. The main
advantage of this approach is that the $n$-variable VAR can be directly
written as a system of $n$ unrelated univariate regressions, which can be
estimated separately. This reparameterization can offer up to 10 times speed
up compared to estimation of the reduced form, depending on the size of the
system. Moreover, further speed-up can be achieved via parallelization. The
key drawback of this approach is that to apply standard conjugacy results, one
needs to use normal priors on the structural-form VAR coefficients, which
often imply order-dependent priors on the reduced-form parameters. One way to
ameliorate this ordering issue is to first elicit prior beliefs on the
reduced-form parameters, and then work out the implied prior beliefs on the
structural-form parameters; see, e.g., Chan (2022) for an example.

There are other Bayesian estimation approaches that offer substantial
reduction in computation time compared to conventional MCMC methods. For
example, Bognanni and Zito (2020) introduce a sequential Monte Carlo (SMC)
algorithm for estimating BVARs with stochastic volatility that leverages
parallelization. In addition, the algorithm is especially suitable for online
learning tasks, such as out-of-sample forecasting, where new data can be
incorporated to update the posterior distribution without completely
reestimating the model.

While MCMC and SMC are exact methods in the sense that they can approximate
the posterior distribution arbitrarily well by increasing the simulation size,
there are approximate methods available in settings where neither MCMC nor SMC
are computationally feasible. One prominent example is the fast growing
collection of variational Bayesian methods, which are deterministic algorithms
for approximating the posterior distribution using a more tractable
distribution. More specifically, given a family of tractable densities, the
variational Bayesian approach locates the optimal density within this family
by minimizing the Kullback-Leibler divergence of the approximating density to
the posterior density. Key references on variational Bayesian methods can be
found in Jordan, Ghahramani, Jaakkola, and Saul (1999), Bishop (2006) and
Ormerod and Wand (2010).

Hajargasht and Woźniak (2018) appear to be the first to apply the variational
Bayesian approach to the estimation of a 7-variable homoskedastic VAR. Chan
and Yu (2022) and Gefang, Koop, and Poon (2023) develop variational Bayesian
methods for estimating large BVARs with stochastic volatility. In particular,
Chan and Yu (2022) use a 96-variable BVAR to measure global bank network
connectedness, whereas Gefang, Koop, and Poon (2023) conduct an out-of-sample
forecasting exercise using a large data set of 100 variables. While both Chan
and Yu (2022) and Gefang, Koop, and Poon (2023) parameterize their BVARs in
the structural form, Bernardi, Bianchi, and Bianco (2023) develop a
variational Bayes algorithm to estimate a reduced-form BVAR with stochastic
volatility. Hierarchical shrinkage priors on the reduced-form VAR coefficients
seem to perform better than those elicited on the structural form, but
estimation under the reduced-form parameterization is slightly more costly.

##  5 Time-Varying Parameter and Nonlinear VARs

Since the pioneering works of Cogley and Sargent (2001, 2005) and Primiceri
(2005), BVARs with time-varying VAR coefficients have been widely used to
model time-series with structural changes. In these time-varying parameter
(TVP) models, it is crucial to also allow for time-varying volatility—as
highlighted in Sims (2001), a failure to account for heteroskedasticity in a
TVP model with constant error variances might give an appearance of time
variation in the VAR coefficients. In fact, Sims and Zha (2006) compare
various regime-switching BVARs and find that the best model is the one that
allows for time variation in error variances only. Chan and Eisenstat (2018)
report similar findings in BVARs with TVP and stochastic volatility.

Of course, TVP models are but one nonlinear approach to model the conditional
mean in BVARs. For applications that study the impact of macroeconomic or
financial uncertainty on key macroeconomic variables, BVARs with stochastic
volatility in mean are an especially convenient framework. For example, Mumtaz
and Zanetti (2013) develop a TVP-VAR with stochastic volatility where the log-
volatilities and their lags enter the conditional mean as regressors. Using
this new model, they then study the impact of monetary shocks volatility. An
alternative approach based on a large BVAR with a multiplicative factor
stochastic volatility model is proposed in Carriero, Clark, and Marcellino
(2018). In their model the log-volatilities are driven by common factors
representing macroeconomic and financial uncertainty. The two common factors
and their lags then enter the conditional mean in the BVAR as regressors.
Estimation of these stochastic volatility in mean models is typically more
computationally intensive, as the auxiliary mixture sampler does not apply. In
view of this difficulty, Cross, Hou, Koop, and Poon (2023) develop an
efficient precision-based sampler for posterior and predictive inference in
large stochastic volatility in mean BVARs. They report substantial
computational and statistical efficiency gains over a standard particle
filter.

There are also a few nonparametric approaches designed for BVARs. For example,
Hauzenberger, Huber, Marcellino, and Petz (2021) develop a nonparametric BVAR
using a Gaussian process prior to model the relationship of each endogenous
variable on its own lags and lags of all other variables. The model also
features a multivariate stochastic volatility specification similar to the
Cholesky stochastic volatility. An alternative nonparametric approach based on
Bayesian additive regression trees is considered in Hauzenberger, Huber, Koop,
and Mitchell (2022) and Clark, Huber, Koop, Marcellino, and Pfarrhofer (2022).
Specifically, a TVP-VAR with stochastic volatility is first represented in the
structural form, and in each equation both the time-varying conditional mean
and the conditional variance are modeled using Bayesian additive regression
trees. Both find forecast accuracy gains over linear BVARs. For more detailed
discussions on Bayesian nonparametric methods, see Chapter 5 of this volume by
Marcellino and Pfarrhofer.

##  6 The Road Ahead

Over the last decade substantial progress has been made in developing a wide
range of stochastic volatility models and efficient estimation approaches for
large datasets. There remains, however, a lot of ongoing work in building more
flexible time-varying BVARs. One direction tackles the problem of modeling
time variation in large systems, since standard TVP models of Cogley and
Sargent (2005) and Primiceri (2005) cannot be directly applied to those
settings due to computational and overparameterization concerns. In addition,
allowing all VAR coefficients to be time-varying appears to be unnecessary.
For example, Chan (2023b) finds evidence that in a large BVAR, the
coefficients in some, but not all, equations are time-varying, and forcing all
coefficients to be time-varying worsens forecast performance. Papers such as
Koop and Korobilis (2013, 2018) propose fast estimation methods to approximate
the posterior distributions of large TVP models. Huber, Koop, and Onorante
(2019) develop an algorithm that first shrinks the time-varying coefficients,
followed by setting the small values to zero. The nonparametric approach based
on Bayesian additive regression trees developed in Hauzenberger, Huber, Koop,
and Mitchell (2022) can also accommodate time-varying coefficients.

Another research direction involves developing more general VAR models able to
accommodate the augmentation of typical macroeconomic datasets with additional
types of data. A promising recent approach is the functional VAR developed in
Chang, Chen, and Schorfheide (2021) that jointly models standard macroeconomic
time-series and individual-level cross-sectional data. This framework is
especially suitable for studying, for example, the distributional effects of
various structural macroeconomic shocks.

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