# Conditional Forecasts in Large Bayesian VARs with Multiple Equality and
Inequality Constraints††thanks: We thank Dario Caldara, Dimitris Korobilis,
Gary Koop, Giulia Mantoan, Michael McCracken, Hashem Pesaran, Ivan Petrella
and Mike West for many constructive comments and useful suggestions.

Joshua C. C. Chan
Purdue University  Davide Pettenuzzo
Brandeis University  Aubrey Poon
Orebro University  Dan Zhu
Monash University

###### Abstract

Conditional forecasts, i.e. projections of a set of variables of interest on
the future paths of some other variables, are used routinely by empirical
macroeconomists in a number of applied settings. In spite of this, the
existing algorithms used to generate conditional forecasts tend to be very
computationally intensive, especially when working with large Vector
Autoregressions or when multiple linear equality and inequality constraints
are imposed at once. We introduce a novel precision-based sampler that is
fast, scales well, and yields conditional forecasts from linear equality and
inequality constraints. We show in a simulation study that the proposed method
produces forecasts that are identical to those from the existing algorithms
but in a fraction of the time. We then illustrate the performance of our
method in a large Bayesian Vector Autoregression where we simultaneously
impose a mix of linear equality and inequality constraints on the future
trajectories of key US macroeconomic indicators over the 2020–2022 period.

Keywords: precision-based method, conditional forecast, vector autoregression

JEL classification: C11, C32, C51

##  1 Introduction

Conditional forecasts are projections of a set of variables of interest on the
future paths of some other variables. This is in contrast to unconditional
forecasts, where no knowledge of the future paths of any variables is assumed.
Since the seminal work of Waggoner and Zha (1999), conditional forecasts have
become a popular tool for forecasters and policymakers within empirical
macroeconomics. They are often paired with Vector Autoregressions (VAR) and
are used routinely to project the future path of a set of macroeconomic
variables after conditioning on a particular policy instrument or an important
macroeconomic indicator, such as the Fed fund rate or real GDP.111To produce
conditional forecasts, one could alternatively treat the variable that is
conditioned on as an exogenous deterministic process over the forecast period.
When proceeding in this way, all existing methods, both frequentistic and
Bayesian, can be applied. See, for example, the method of Sims and Zha (1998)
for statistical inferences on point forecasts. However, this approach has some
limitations, as it is often advisable to include the variable conditioned on
as endogenous in a VAR setting.

A first important consideration when generating conditional forecasts is
whether one wants to compute classic (reduced-form) conditional forecasts or
instead is interested in scenario analysis. In the former case, it is
sufficient to rely on the empirical correlations between the variables in the
system and not take a stand on the underlying causal mechanisms behind the
results. This is equivalent to assuming that the conditions are generated by
all the structural shocks of the model, and it is the most commonly adopted
approach in the literature. Examples along this line of work include
Andersson, Palmqvist, and Waggoner (2010), Giannone, Lenza, Pill, and Reichlin
(2012); Giannone, Lenza, Momferatou, and Onorante (2014), Altavilla, Giannone,
and Lenzaa (2016), Aastveit, Carriero, Clark, and Marcellino (2017), Giannone,
Lenza, and Reichlin (2019), and Tallman and Zaman (2020). In practice,
assuming that the conditions are generated by all structural shocks of the
model may be undesirable and economically irrelevant, and there are,
therefore, reasons to prefer a more structural approach, where the conditional
forecasts are obtained from a sequence of specific shocks derived from a
(point or set) identified structural VAR. See, for example, the discussion in
Dieppe, Legrand, and Van Roye (2016). An important contribution in this regard
is the recent work of Antolín-Díaz, Petrella, and Rubio-Ramírez (2021), who
introduced a unified framework for conditional forecasts and structural
scenario analysis with VAR models.222Antolín-Díaz, Petrella, and Rubio-Ramírez
(2021) define a structural scenario as a combination of a path for one or more
variables in the system and a restriction that only a subset of structural
shocks can deviate from their unconditional distribution. Their approach works
by first specifying a set of conditions on observable variables and then
pairing these conditions with the subset of structural shocks that are
believed to be driving the forecasts.333Relative to the existing methods in
the literature (i.e., the seminal work of Waggoner and Zha (1999), the Kalman
filter based approach of Clarida and Coyle (1984) and Bańbura, Giannone, and
Lenza (2015), and the “forecast scenario” method of Baumeister and Kilian
(2014)), the approach developed in Antolín-Díaz, Petrella, and Rubio-Ramírez
(2021) is closed-form and valid for an arbitrary number of
conditions.,444Antolín-Díaz, Petrella, and Rubio-Ramírez (2021) also show that
in the Gaussian case, their approach to conditional forecasting is equivalent
to the entropic forecast tilting method of Robertson, Tallman, and Whiteman
(2005).

A second and equally important consideration when producing conditional
forecasts is whether one wants to fix the future paths of the conditioned
variables at specific values (i.e., hard or equality conditions) or instead
prefers to allow the future values of the conditioned variables to lie within
a certain range (i.e., inequality conditions). The equality constraint case is
the most commonly employed approach in the literature (see again Giannone,
Lenza, Pill, and Reichlin (2012); Giannone, Lenza, Momferatou, and Onorante
(2014) as well as Jarocinski and Smets (2008) and Lenza, Pill, and Reichlin
(2010)), while to date there has been only limited work with inequality
constrained conditional forecasts (in addition to Waggoner and Zha (1999), see
also Andersson, Palmqvist, and Waggoner (2010)). This is largely due to the
computational challenges that the inequality constraints entail. For example,
the algorithm of Waggoner and Zha (1999) for the inequality constraints is
computationally very heavy, even when using the more efficient version of the
algorithm proposed by Jarociński (2010). This is because the approach relies
on a simple accept-reject algorithm that obtains candidates from the original,
unconstrained distribution. Consequently, it requires a large number of
candidate draws to obtain one that satisfies all the constraints.

The limited use of inequality constraint applications is unfortunate, as there
are many good reasons why inequality constraints should appeal to forecasters
and policymakers. For one, it is often the case that one does not know the
actual future realization path of the constrained endogenous variables, and in
these situations, it is much simpler to impose that the future values of the
variables conditioned on will be within a range or an interval instead of an
exact path. In addition, inequality constraints allow the forecaster to
acknowledge the uncertainty surrounding the future realization path of the
constrained endogenous variables. For example, Andersson, Palmqvist, and
Waggoner (2010) show that ignoring the uncertainty about the conditions can
lead to density forecasts of the unrestricted variables that are too narrow.

In view of these considerations, we introduce a novel approach to conditional
forecasting that generalizes and extends the existing methods available in the
literature in a number of ways. First, much like Antolín-Díaz, Petrella, and
Rubio-Ramírez (2021), our approach is closed-form and can be used for both
conditional forecast and structural scenario analysis. Second, thanks to the
way we derive the conditional forecasts’ distribution, our approach is
significantly more efficient and better suited to handling large dimensional
VARs as well as situations in which we have a large number of conditioning
variables and long forecast horizons. We accomplish this by building on the
intuition of Bańbura, Giannone, and Lenza (2015), who shows that the
conditional forecasts can be considered as time series with missing data.
This, in turn, allows us to build on the efficient sampling approach developed
in CPZ23—designed for handling complex missing data patterns in linear
Gaussian state space models—to our settings, where the missing data is subject
to inequality constraints. Third, our approach can be used to produce
conditional forecasts and structural scenario analysis under both equality and
inequality conditions without compromising the computational efficiency and
scalability of the algorithm. This is in contrast to the traditional methods
available in the literature, such as Waggoner and Zha (1999) and Andersson,
Palmqvist, and Waggoner (2010).555Andersson, Palmqvist, and Waggoner (2010)
show how to implement the inequality restrictions using a Gibbs sampler
instead of the original accept-reject method introduced in Waggoner and Zha
(1999). They rely on the Gibbs sampler proposed by Geweke (1996) and Genz
(1992) to sample from a number of univariate truncated Gaussian distributions.
However, their method is only suitable for a small-dimension VAR model with
only one inequality constraint. To accomplish this, we pair the use of the
precision sampler of Chan and Jeliazkov (2009) to the exponential minimax
tilting method of Botev (2017), which allows us to draw the inequality
constraint conditional forecasts in a fast and efficient manner within the
precision sampler. More specifically, Botev (2017) provides an efficient
method to generate random draws from high-dimensional Gaussian distributions
under linear restrictions. Even though the algorithm is based on an accept-
reject sampler, it carefully constructs a suitable proposal distribution that
satisfies all the constraints via exponential tilting. As such, it has good
theoretical properties, and the acceptance probability of the candidate draws
is typically high.

We conduct a simulation study to compare the computation time and accuracy of
our proposed precision-based conditional forecast sampler against the existing
methods in the literature. Specifically, we compare our precision-based
equality constraint conditional forecasts against the Waggoner and Zha (1999),
Bańbura, Giannone, and Lenza (2015) and Antolín-Díaz, Petrella, and Rubio-
Ramírez (2021) approaches. We show that our approach generates exactly the
same conditional forecasts and credible sets as the three existing methods but
is substantially less demanding computationally. Next, we investigate the
accuracy and computational efficiency of our inequality constraint precision-
based approach algorithm by comparing it against the Waggoner and Zha (1999),
and Andersson, Palmqvist, and Waggoner (2010) accept-reject methods (both
Bańbura, Giannone, and Lenza (2015) and Antolín-Díaz, Petrella, and Rubio-
Ramírez (2021) are not designed to generate conditional forecasts in the
presence of inequality constraints) and once again find that our inequality
constraint algorithm is significantly faster than the existing alternatives.
For example, using the Waggoner and Zha (1999) algorithm, it takes about 80
and 459 minutes to generate the in- and out-of-sample conditional forecasts
from a small dimension VAR with only one inequality constraint. In contrast,
our proposed approach only takes 1-2 minutes to generate the corresponding
conditional forecasts from the same VAR model.

We next move to illustrate the benefits of our proposed precision-based
conditional forecast sampler with an empirical application where we follow
Crump, Eusepi, Giannone, Qian, and Sbordone (2021) and estimate a large BVAR
including 31 quarterly US macro and financial variables from 1976 to the end
of 2019. We use this setting to investigate the effect of simultaneously
imposing equality and inequality constraints on the trajectories of CPI
inflation, unemployment rate, and the 10-year Treasury rate over the following
13 quarters, setting our constraints to mimic the baseline and severe
scenarios prepared by the Federal Reserve Board for their 2020 stress test
analysis. We find that when we condition on the Fed’s baseline projections for
CPI inflation, unemployment and the 10-year Treasury rate, meant to
approximate a moderate expansion over the next 13 quarters, we observe small
temporary increases in real GDP, industrial production, housing starts and the
S&P 500 followed by a gradual convergence back to their original levels
sometime around the end of 2022. On the contrary, the story that emerges from
conditioning our forecasts on the adverse scenario, which the Fed describes as
a severe global recession accompanied by a period of heightened stress in
commercial real estate and corporate debt markets, is quite different. In this
case, we observe a sharp decrease in real GDP, industrial production, and
housing starts, bottoming up around the second half of 2021 before a gradual
recovery begins. Similarly, we see a significant drop in housing starts and
the S&P 500 index and a sudden increase in the market volatility, as measured
by the VIX. Interestingly, with the exception of the effect on the S&P 500
index, the initial responses we see in our conditional forecasts are in the
same direction and, in most cases, of a magnitude similar to what the US
economy experienced after the large and unexpected COVID-19 shocks that
occurred in the first half of 2020. However, the sudden and unprecedented
nature of the COVID-19 shock and the unfolding of events that followed the
initial impact were such that the US economy’s initial reaction was much more
immediate and the recovery that followed much faster.

The remainder of the paper is organized as follows. Section 2 introduces the
proposed precision-based conditional forecast sampler and illustrates how the
approach can be used with equality and inequality constraints on observables
as well as for scenario analysis and entropic tilting. Next, Section 3
presents the results of our simulation study, where we compare our proposed
precision-based conditional forecast sampler against the existing methods
available in the literature. Section 4 focuses on the empirical application,
where we show how our approach can be used to generate conditional forecasts
in a large Bayesian VAR when multiple inequality and equality constraints are
at play simultaneously. Finally, Section 5 provides some concluding remarks.

##  2 Methodology

In this section, we introduce our approach to generating unconditional and
conditional forecasts. Our starting point is a structural VAR (SVAR), a very
flexible and general approach used routinely by empirical macroeconomists to
produce forecasts and impulse response analysis. We follow the setups
considered in both Waggoner and Zha (1999) and Antolín-Díaz, Petrella, and
Rubio-Ramírez (2021), but at the same time we highlight that our framework is
more general as we are able to accommodate within the same general methodology
both equality and inequality constraints on observables, as well as
constraints on structural shocks and scenario analysis. Most importantly,
thanks to the use of precision-based sampling methods and the ability to
exploit fast band matrix algorithms, we are able to achieve this generality
without sacrificing computational efficiency, and this, in turn, makes our
methods particularly suitable for handling large VARs, long forecast horizons,
and multiple conditions.

We begin in subsection 2.1 with derivations of both unconditional and
conditional predictive densities given the observed data and the model
parameters. Next, we show in subsubsection 2.2.1 and subsubsection 2.2.2 how
to apply this setup to producing equality- and inequality-constrained
conditional forecasts. Finally, in subsection 2.3 we show how the approach can
be extended to accommodate situations where the conditional forecasts depend
only on a subset of structural shocks or to carry out structural scenario
analysis.

###  2.1 Unconditional Forecasts

As our starting point, consider an $n\times 1$ vector of variables
$\mathbf{y}_{t}=(y_{1,t},\ldots,y_{n,t})^{\prime}$, and write the following
SVAR with $p$ lags:

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

where $\mathbf{a}$ is an $n\times 1$ vector of intercepts,
$\mathbf{A}_{1},\ldots,\mathbf{A}_{p}$ are the $n\times n$ VAR coefficient
matrices, $\mathbf{A}_{0}$ is a full-rank contemporaneous impact matrix,
$\mathbf{0}_{n}$ is an $n\times 1$ vector of zeros and $\mathbf{I}_{n}$ is the
$n$-dimensional identity matrix.

Given the whole history of observations
$\mathbf{y}^{T}=(\mathbf{y}_{1-p}^{\prime},\ldots,\mathbf{y}_{T}^{\prime})^{\prime}$,
the unconditional forecast of our $n$ variables for the next $h$ periods,
$\mathbf{y}_{T+1,T+h}=(\mathbf{y}_{T+1}^{\prime},\ldots,\mathbf{y}_{T+h}^{\prime})^{\prime}$,
can be written as

$\mathbf{H}\mathbf{y}_{T+1,T+h}=\mathbf{c}+\boldsymbol{\varepsilon}_{T+1,T+h},\quad\boldsymbol{\varepsilon}_{T+1,T+h}\sim\mathcal{N}(\mathbf{0}_{nh},\mathbf{I}_{nh}),$
(2)

where

$\mathbf{c}=\begin{bmatrix}\mathbf{a}+\sum_{j=1}^{p}\mathbf{A}_{j}\mathbf{y}_{T+1-j}\\\
\mathbf{a}+\sum_{j=2}^{p}\mathbf{A}_{j}\mathbf{y}_{T+2-j}\\\
\mathbf{a}+\sum_{j=3}^{p}\mathbf{A}_{j}\mathbf{y}_{T+3-j}\\\ \vdots\\\
\mathbf{a}+\mathbf{A}_{p}\mathbf{y}_{T}\\\ \mathbf{a}\\\ \vdots\\\
\mathbf{a}\end{bmatrix},\;\mathbf{H}=\begin{bmatrix}\mathbf{A}_{0}&\mathbf{0}_{n\times
n}&\cdots&\cdots&\cdots&\cdots&\cdots&\mathbf{0}_{n\times n}\\\
-\mathbf{A}_{1}&\mathbf{A}_{0}&\mathbf{0}_{n\times
n}&\cdots&\cdots&\cdots&\cdots&\mathbf{0}_{n\times n}\\\
-\mathbf{A}_{2}&-\mathbf{A}_{1}&\mathbf{A}_{0}&\mathbf{0}_{n\times
n}&\cdots&&&\mathbf{0}_{n\times n}\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&&\vdots\\\
-\mathbf{A}_{p-1}&\cdots&&-\mathbf{A}_{1}&\mathbf{A}_{0}&\mathbf{0}_{n\times
n}&&\vdots\\\ \mathbf{0}_{n\times n}&&&&\ddots&\ddots&\ddots&\vdots\\\
\vdots&&\ddots&&\ddots&\ddots&\ddots&\vdots\\\ \mathbf{0}_{n\times
n}&\cdots&\mathbf{0}_{n\times
n}&-\mathbf{A}_{p}&\cdots&-\mathbf{A}_{2}&-\mathbf{A}_{1}&\mathbf{A}_{0}\end{bmatrix}$

(3)

and $\mathbf{0}_{n\times n}$ denotes the $n\times n$ zero matrix. Since
$\mathbf{A}_{0}$ is of full-rank and the determinant of $\mathbf{H}$ is
$|\mathbf{A}_{0}|^{h}\neq 0$, the inverse $\mathbf{H}^{-1}$ exists. It follows
from (2) that

$\mathbf{y}_{T+1,T+h}\sim\mathcal{N}\left(\mathbf{H}^{-1}\mathbf{c},(\mathbf{H}^{\prime}\mathbf{H})^{-1}\right).$
(4)

It’s worth noting that since $\mathbf{H}$ is an $nh\times nh$ band matrix with
band width $np$, the precision-based sampling approach of Chan and Jeliazkov
(2009) can be used to efficiently draw from the unconditional distribution in
(4), and this becomes particularly convenient when either (or both) $n$ and
$h$ are large.666More specifically, to obtain a draw from the Gaussian
distribution in (4), we first obtain
$\mathbf{u}=(u_{1},\ldots,u_{nh})^{\prime}$, where
$u_{i}\sim\mathcal{N}(0,1),i=1,\ldots,nh$. Then, we solve the linear system
$\mathbf{H}\mathbf{z}=\mathbf{c}+\mathbf{u}$ for $\mathbf{z}$ by, e.g.,
Gaussian elimination. The last operation can be done very quickly as the
matrix $\mathbf{H}$ is banded. It can be easily verified that
$\mathbf{z}\sim\mathcal{N}\left(\mathbf{H}^{-1}\mathbf{c},(\mathbf{H}^{\prime}\mathbf{H})^{-1}\right)$.
We will rely heavily on this result as we expand our methods below, as this
will allow us to introduce a fast and scalable algorithm to generate
conditional forecasts in large BVARs.

###  2.2 Conditional Forecasts

With this in mind, we now move to describe our approach to conditional
forecasting. As in Andersson, Palmqvist, and Waggoner (2010) and Antolín-Díaz,
Petrella, and Rubio-Ramírez (2021), we write the conditional forecasts as a
set of linear restrictions on the path of future observables
${\mathbf{y}}_{T+1,T+h}$, i.e.

$\mathbf{R}{{\mathbf{y}}_{T+1,T+h}}\sim\mathcal{N}(\mathbf{r},\boldsymbol{\Omega}),$
(5)

where $\mathbf{R}$ is a $r\times nh$ constant matrix with full row rank (so
that there are no redundant restrictions), while $\mathbf{r}$ and
$\boldsymbol{\Omega}$ are $r\times 1$ and $r\times r$ matrices representing
the mean and covariance of the restrictions. As Antolín-Díaz, Petrella, and
Rubio-Ramírez (2021) note, the setup in (5) is very general and can
accommodate both the classic “hard” or linear equality constraint case as
defined in Waggoner and Zha (1999) (simply setting
$\boldsymbol{\Omega}=\mathbf{0}_{r\times r}$) and the more general density
forecast case as defined by Andersson, Palmqvist, and Waggoner (2010). It also
replicates entropic tilting for the Gaussian case, the approach introduced and
popularized by Robertson, Tallman, and Whiteman (2005), which looks for the
density forecast distribution that meets the constraints in (5) while
minimizing the relative entropy with the unconditional forecast distribution
in (4) (see Antolín-Díaz, Petrella, and Rubio-Ramírez (2021) for a formal
proof of this equivalence in the Gaussian case). Next, combining (2) and (5),
we obtain

$\mathbf{R}{\mathbf{y}}_{T+1,T+h}=\mathbf{R}\mathbf{H}^{-1}\mathbf{c}+\mathbf{R}\mathbf{H}^{-1}{\boldsymbol{\varepsilon}}_{T+1,T+h}\sim\mathcal{N}(\mathbf{r},\boldsymbol{\Omega}).$
(6)

In what follows, we first derive the set of restrictions on the future shocks
implied by (5) and (6). In particular, following Antolín-Díaz, Petrella, and
Rubio-Ramírez (2021), we let
$\left.{\boldsymbol{\varepsilon}}_{T+1,T+h}\right|\mathbf{R},\mathbf{r},\boldsymbol{\Omega}$
denote the restricted future shocks with the distribution

$\left.{\boldsymbol{\varepsilon}}_{T+1,T+h}\right|\mathbf{R},\mathbf{r},\boldsymbol{\Omega}\sim\mathcal{N}(\boldsymbol{\mu}_{\boldsymbol{\varepsilon}},\mathbf{I}_{nh}+\boldsymbol{\Psi}_{\boldsymbol{\varepsilon}}),$
(7)

where $\boldsymbol{\mu}_{\boldsymbol{\varepsilon}}$ and
$\boldsymbol{\Psi}_{\boldsymbol{\varepsilon}}$ are, respectively, the
deviations of the mean vector and covariance matrix of the restricted future
shocks from their unconditional counterparts in (2). In turn, equations (6)
and (7) combined imply the following restrictions on
$\boldsymbol{\mu}_{\boldsymbol{\varepsilon}}$ and
$\boldsymbol{\Psi}_{\boldsymbol{\varepsilon}}$:

$\begin{split}\mathbf{R}\mathbf{H}^{-1}(\mathbf{c}+\boldsymbol{\mu}_{\boldsymbol{\varepsilon}})&=\mathbf{r}\\\
\mathbf{R}\mathbf{H}^{-1}(\mathbf{I}_{nh}+\boldsymbol{\Psi}_{\boldsymbol{\varepsilon}})\mathbf{H}^{-1\prime}\mathbf{R}^{\prime}&=\boldsymbol{\Omega}.\end{split}$
(8)

In typical situations where $r<nh$, the system in (8) is underdetermined and
has multiple solutions.777If $r=nh$, the system is just determined and has a
unique solution; if $r>nh$, the system is inconsistent and there are no
solutions. Following Antolín-Díaz, Petrella, and Rubio-Ramírez (2021), we
choose a solution that can be expressed in terms of the Moore-Penrose inverse
of $\mathbf{R}\mathbf{H}^{-1}$, which we denote as
$\left(\mathbf{R}\mathbf{H}^{-1}\right)^{+}$:

$\begin{split}\boldsymbol{\mu}_{\boldsymbol{\varepsilon}}&=\left(\mathbf{R}\mathbf{H}^{-1}\right)^{+}\left(\mathbf{r}-\mathbf{R}\mathbf{H}^{-1}\mathbf{c}\right)\\\
\boldsymbol{\Psi}_{\boldsymbol{\varepsilon}}&=\left(\mathbf{R}\mathbf{H}^{-1}\right)^{+}\left(\boldsymbol{\Omega}-\mathbf{R}(\mathbf{H}^{\prime}\mathbf{H})^{-1}\mathbf{R}^{\prime}\right)\left(\mathbf{R}\mathbf{H}^{-1}\right)^{+\prime}.\end{split}$
(9)

This solution minimizes the sum of the Frobenius norms of
$\boldsymbol{\mu}_{\boldsymbol{\varepsilon}}$ and
$\boldsymbol{\Psi}_{\boldsymbol{\varepsilon}}$. In other words, this solution
represents the smallest deviations of the mean vector and covariance matrix of
the conditional future shocks from the unconditional ones. Finally, we can map
the constraints on the future shocks implied by (5) and (6) to the
corresponding constraints on the forecasts. If we denote the conditional
forecast distribution with

$\left.{\mathbf{y}}_{T+1,T+h}\right|\mathbf{R},\mathbf{r},\boldsymbol{\Omega}\sim\mathcal{N}(\boldsymbol{\mu}_{\mathbf{y}},\boldsymbol{\Sigma}_{\mathbf{y}}),$
(10)

then, (2) and (9) imply that

$\begin{split}\boldsymbol{\mu}_{\mathbf{y}}&=\mathbf{H}^{-1}\left[\mathbf{c}+\left(\mathbf{R}\mathbf{H}^{-1}\right)^{+}\left(\mathbf{r}-\mathbf{R}\mathbf{H}^{-1}\mathbf{c}\right)\right]\\\
\boldsymbol{\Sigma}_{\mathbf{y}}&=\mathbf{H}^{-1}\left[\mathbf{I}_{nh}+\left(\mathbf{R}\mathbf{H}^{-1}\right)^{+}\left(\boldsymbol{\Omega}-\mathbf{R}(\mathbf{H}^{\prime}\mathbf{H})^{-1}\mathbf{R}^{\prime}\right)\left(\mathbf{R}\mathbf{H}^{-1}\right)^{+\prime}\right]\mathbf{H}^{-1\prime}.\end{split}$
(11)

This result is extremely general and in fact encompasses a number of useful
and popular applications of conditional forecasting. For example, if one
wished to impose restrictions on the mean of the conditional forecasts
$\boldsymbol{\mu}_{\mathbf{y}}$ while preserving the variance of the
unconditional forecasts, this could be easily accomplished by setting
$\boldsymbol{\Omega}=\mathbf{R}(\mathbf{H}^{\prime}\mathbf{H})^{-1}\mathbf{R}^{\prime}$.
In that case, it is easy to show that
$\boldsymbol{\Sigma}_{\mathbf{y}}=(\mathbf{H}^{\prime}\mathbf{H})^{-1},$ i.e.
the covariance matrix of the unconditional forecasts. Similarly, if one were
to consider imposing restrictions on the second moment of the conditional
forecasts while preserving the mean of the unconditional forecasts, this could
be accomplished by setting $\mathbf{r}=\mathbf{R}\mathbf{H}^{-1}\mathbf{c}$.
Also note, to conclude, that our result is equivalent to what is reported in
Antolín-Díaz, Petrella, and Rubio-Ramírez (2021). The key difference is that
they parameterize the unconditional and conditional forecast distributions in
terms of the covariance matrix, while we work with the precision matrix (i.e.,
the inverse covariance matrix), and this, in turn, allows us to exploit fast
band matrix algorithms such as those introduced by Chan and Jeliazkov (2009)
to substantially speed up computations, especially as the dimension of the
SVAR $n$ or the dimension of the forecast horizon $h$ (or both) growth. We
will return to this point in the simulation section, where we will focus on
the computational gains of the approach.

In many applications, the constraints in (5) may be considered too strong, or
one may not have enough information to elicit both the mean and covariance of
the restrictions, especially as the forecast horizon increases. Similarly, in
other settings, one may question the choice of the normal distribution to
express the constraint. As a concrete example of this, consider a situation
where there is significant disagreement on the future path of some variable(s)
we would like to condition our forecasts on, and this disagreement is best
described via a bi-modal distribution. Examples where this may be the case are
not hard to find, and this was in fact the case with the Survey of
Professional Forecasters (SPF) for a number of macroeconomic variables right
after the first wave of the Covid-19 pandemic in the second half of
2020.888See for example the projections of GDP and unemployment in the 2020Q2
SPF, released on May 15, 2020 and available here:
https://www.philadelphiafed.org/-/media/frbp/assets/surveys-and-data/survey-
of-professional-forecasters/2020/spfq220.pdf?la=en. In these situations, it
may be easier and more plausible for the forecaster to specify a range for the
future path of the constrained endogenous variables. It is trivial to extend
the setup above to accommodate the case of inequality conditioning, where one
allows the future values of the conditioned variables to lie within a certain
range. Let the inequality constraints be expressed as

$\underline{\mathbf{c}}<\mathbf{S}\mathbf{y}_{T+1,T+h}<\bar{\mathbf{c}}$ (12)

where $\mathbf{S}$ is a $s\times nh$ pre-specified full-rank constant matrix,
$\underline{\mathbf{c}}$ and $\bar{\mathbf{c}}$ are $s\times 1$ vectors of
constants (with generic elements $\underline{c}_{i}$ and $\bar{c}_{i}$ in
$\mathbb{R}\cup\\{\pm\infty\\}$) and the inequalities hold component-wise. In
typical applications, $\mathbf{S}$ would be a selection matrix, but our
framework allows for inequality restrictions on any linear combinations of the
variables. Combining (4) with (12) leads to a truncated multivariate normal
distribution for $\mathbf{y}_{T+1,T+h}$,

$\mathbf{y}_{T+1,T+h}\,|\,\underline{\mathbf{c}}<\mathbf{S}\mathbf{y}_{T+1,T+h}<\bar{\mathbf{c}}\sim\mathcal{N}\left(\mathbf{H}^{-1}\mathbf{c},(\mathbf{H}^{\prime}\mathbf{H})^{-1}\right)\mathbb{1}(\underline{\mathbf{c}}<\mathbf{S}\mathbf{y}_{T+1,T+h}<\bar{\mathbf{c}}),$
(13)

where $\mathbb{1}(\cdot)$ is the indicator function. We note that this is a
slightly more general formulation of inequality constraints than that in
Waggoner and Zha (1999) and it reduces to the latter when setting
$\mathbf{S}=\mathbf{I}_{nh}$. We also note that this approach is closely
related to Andersson, Palmqvist, and Waggoner (2010), but again since we are
parameterizing the forecast distribution using the precision matrix
$\mathbf{H}^{\prime}\mathbf{H}$, our approach allows us to exploit the
computational efficiency of the precision-based sampling approach of Chan and
Jeliazkov (2009) and it will therefore lead to large computational gains.

It is also straightforward to combine the two types of constraints in (5) and
(13). In particular, suppose we impose restrictions on
$\boldsymbol{\mu}_{\mathbf{y}}$ while preserving the variance of the
unconditional forecasts by setting
$\boldsymbol{\Omega}=\mathbf{R}(\mathbf{H}^{\prime}\mathbf{H})^{-1}\mathbf{R}^{\prime}$.
Then, using the same logic we applied above, we obtain

$\mathbf{y}_{T+1,T+h}\,|\,\mathbf{R},\mathbf{r},\boldsymbol{\Omega},\underline{\mathbf{c}}<\mathbf{S}\mathbf{y}_{T+1,T+h}<\bar{\mathbf{c}}\sim\mathcal{N}(\boldsymbol{\mu}_{\mathbf{y}},(\mathbf{H}^{\prime}\mathbf{H})^{-1})\mathbb{1}(\underline{\mathbf{c}}<\mathbf{S}\mathbf{y}_{T+1,T+h}<\bar{\mathbf{c}})$
(14)

where the definition of $\boldsymbol{\mu}_{\mathbf{y}}$ is given in (11).999In
the special case without inequality constraints, i.e. when
$\underline{c}_{i}=\infty$ and $\bar{c}_{i}=\infty,i=1,\ldots,nh$, we revert
back to the results presented in (5).

Next, we showcase the generality of the approach with a broad set of examples.
We begin with the classic “hard” or equality conditional forecast case
introduced by Doan, Litterman, and Sims (1984) and popularized by Waggoner and
Zha (1999). Next, we move to the inequality constraint case, also discussed in
Waggoner and Zha (1999). In both of these two cases, the conditional forecasts
are in reduced form, i.e. rely on all shocks. While this is useful, it is
often very relevant to conditions on a subset of structural shocks. We review
that last.

####  2.2.1 Equality Constraints on Future Observables

The first special case is the classic conditional forecast setup introduced by
Doan, Litterman, and Sims (1984) and popularized by Waggoner and Zha (1999).
It is used to produce forecasts of the variables of interest given the future
path of a subset of other variables. As an example, a policymaker might be
interested in the future path of GDP, inflation and unemployment that is
conditioned on the scenario that future policy rate follows a fixed path
across the forecast horizon.

This type of restriction can be represented as

$\mathbf{R}_{o}\mathbf{y}_{T+1,T+h}=\mathbf{r}_{o},$ (15)

where $\mathbf{R}_{o}$ is a $r_{o}\times nh$ pre-specified full-rank selection
matrix—a matrix in which each row has exactly one element that is 1 and all
other elements are 0—and $\mathbf{r}_{o}$ is a $r_{o}\times 1$ vector of
constants. The setup in (15) can be cast within our general framework in (5)
by setting $\mathbf{R}=\mathbf{R}_{o}$, $\mathbf{r}=\mathbf{r}_{o}$ and
$\boldsymbol{\Omega}=\mathbf{0}_{r_{o}\times r_{o}}$ and is sometimes referred
to as conditional forecasting under hard or equality constraints.

In this case, some of the computations described in the general framework
become unnecessary and the algorithm can be streamlined. In particular, the
efficient sampling approach of CPZ23 designed for missing data with linear
equality constraints can be directly applied. More specifically, we partition
the $nh\times 1$ vector $\mathbf{y}_{T+1,T+h}$ into $\mathbf{y}_{T+1,T+h}^{o}$
and $\mathbf{y}_{T+1,T+h}^{u}$, where the former is a $r_{o}\times 1$ vector
including the equality-constrained endogenous variables—the set of variables
that are selected by $\mathbf{R}_{o}$—and the latter is a $(nh-r_{o})\times 1$
vector of free or unconstrained variables. For example, if we are interested
in the conditional forecasts of GDP, inflation and unemployment given that the
policy rate follows a fixed path over the next $h$ periods, then
$\mathbf{y}_{T+1,T+h}^{o}$ is the $h\times 1$ vector of policy rates
($r_{0}=h$) over the forecast horizons $t=T+1,\ldots,T+h$ and
$\mathbf{y}_{T+1,T+h}^{u}$ is the $3h\times 1$ vector consisting of GDP,
inflation and unemployment. With this in mind, let $\mathbf{R}_{o}^{-}$ denote
the associated $(nh-r_{o})\times nh$ selection matrix that selects
$\mathbf{y}_{T+1,T+h}^{u}$. Then, we can write $\mathbf{y}_{T+1,T+h}$ as
follows:

$\mathbf{y}_{T+1,T+h}=\mathbf{M}_{u}\mathbf{y}_{T+1,T+h}^{u}+\mathbf{M}_{o}\mathbf{y}_{T+1,T+h}^{o},$
(16)

where $\mathbf{M}_{u}=(\mathbf{R}_{o}^{-})^{\prime}$ and
$\mathbf{M}_{o}=\mathbf{R}_{o}^{\prime}$. Note that both $\mathbf{M}_{u}$ and
$\mathbf{M}_{o}$ have full column rank and are sparse with only, respectively,
$nh-r_{o}$ and $r_{o}$ non-zero elements.

With this setup in hand, we can now derive the joint conditional distribution
of $\mathbf{y}_{T+1,T+h}^{u}$ given $\mathbf{y}_{T+1,T+h}^{o}$ and the model
parameters $\mathbf{A}_{0}$ and
$\mathbf{A}=(\mathbf{a},\mathbf{A}_{1},\ldots,\mathbf{A}_{p})^{\prime}$. To
this end, substitute (16) into (2) to obtain

$\mathbf{H}(\mathbf{M}_{u}\mathbf{y}_{T+1,T+h}^{u}+\mathbf{M}_{o}\mathbf{y}_{T+1,T+h}^{o})=\mathbf{c}+\boldsymbol{\varepsilon}_{T+1,T+h},\quad\boldsymbol{\varepsilon}_{T+1,T+h}\sim\mathcal{N}(\mathbf{0}_{n},\mathbf{I}_{nh}).$

Next, the conditional density of $\mathbf{y}_{T+1,T+h}^{u}$ given
$\mathbf{y}_{T+1,T+h}^{o}$ and the model parameters can be expressed as
(recall $\mathbf{H}$ and $\mathbf{c}$ are functions of the model parameters
$\mathbf{A}_{0}$ and $\mathbf{A}$):

$\displaystyle p($
$\displaystyle\mathbf{y}_{T+1,T+h}^{u}\,|\,\mathbf{y}_{T+1,T+h}^{o},\mathbf{A}_{0},\mathbf{A})$
$\displaystyle\propto\exp\left\\{-\frac{1}{2}(\mathbf{H}(\mathbf{M}_{u}\mathbf{y}_{T+1,T+h}^{u}+\mathbf{M}_{o}\mathbf{y}_{T+1,T+h}^{o})-\mathbf{c})^{\prime}(\mathbf{H}(\mathbf{M}_{u}\mathbf{y}_{T+1,T+h}^{u}+\mathbf{M}_{o}\mathbf{y}_{T+1,T+h}^{o})-\mathbf{c})\right\\}$
$\displaystyle\propto\exp\left\\{-\frac{1}{2}\left(\mathbf{y}_{T+1,T+h}^{u^{\prime}}\mathbf{M}_{u}^{{}^{\prime}}\mathbf{H}^{{}^{\prime}}\mathbf{H}\mathbf{M}_{u}\mathbf{y}_{T+1,T+h}^{u}-2\mathbf{y}_{T+1,T+h}^{u^{\prime}}\mathbf{M}_{u}^{{}^{\prime}}\mathbf{H}^{{}^{\prime}}(\mathbf{c}-\mathbf{H}\mathbf{M}_{o}\mathbf{y}_{T+1,T+h}^{o})\right)\right\\}$
$\displaystyle\propto\exp\left\\{-\frac{1}{2}(\mathbf{y}_{T+1,T+h}^{u}-\boldsymbol{\mu}_{u})^{\prime}\mathbf{K}_{u}(\mathbf{y}_{T+1,T+h}^{u}-\boldsymbol{\mu}_{u})\right\\},$

where
$\mathbf{K}_{u}=\mathbf{M}_{u}^{{}^{\prime}}\mathbf{H}^{\prime}\mathbf{H}\mathbf{M}_{u}$
and
$\boldsymbol{\mu}_{u}=\mathbf{K}_{u}^{-1}\mathbf{M}_{u}^{{}^{\prime}}\mathbf{H}^{\prime}\mathbf{H}(\mathbf{H}^{-1}\mathbf{c}-\mathbf{M}_{o}\mathbf{y}_{T+1,T+h}^{o})$.
That is,

$\mathbf{y}_{T+1,T+h}^{u}\,|\,\mathbf{y}_{T+1,T+h}^{o},\mathbf{A}_{0},\mathbf{A}\sim\mathcal{N}(\boldsymbol{\mu}_{u},\mathbf{K}_{u}^{-1}).$

Since $\mathbf{H}$ and $\mathbf{M}_{u}$ are band matrices, so is the precision
matrix $\mathbf{K}_{u}$. Therefore, we can again use the precision sampler of
Chan and Jeliazkov (2009) to draw $\mathbf{y}_{T+1,T+h}^{u}$ efficiently.

####  2.2.2 Inequality Constraints on Future Observables

We now move to discussing the case of inequality constraints on observables.
Continuing with the previous example, instead of conditioning on a fixed path
of the future policy rate, we could be interested in restricting the future
path of the policy rate to be, say, between 1% and 2% for the next 8 quarters
and between 1.5% and 2.5% afterward. In terms of our general framework, this
type of inequality constraints on the observables can be formulated via (13).
In particular, this case can be specified simply by setting
$\mathbf{S}=\mathbf{I}_{nh}$ and the appropriate elements in
$\underline{\mathbf{c}}$ and $\bar{\mathbf{c}}$ to the desired values.

We note, however, that for large VARs with many inequality constraints,
sampling from the truncated multivariate Gaussian distribution in (13) can be
a computationally daunting task. For example, Waggoner and Zha (1999)
implement a simple accept-reject algorithm where the proposal is the
unrestricted multivariate Gaussian distribution. In their empirical
application with a small VAR, they note that out of 185,000 simulated draws
only 8,000 draws satisfy the inequality constraints. For high-dimensional
problems with many endogenous variables and inequality constraints, it is
clear that this approach is computationally infeasible. To circumvent this
challenge, Andersson, Palmqvist, and Waggoner (2010) instead use the approach
in Geweke (1996) to draw from the truncated multivariate Gaussian distribution
implied by the inequality constraints. Specifically, it relies on a Gibbs
sampler that iteratively samples from each univariate conditional
distribution, which in this case is a univariate truncated Gaussian
distribution. Note that being a Gibbs sampler, in high-dimensional settings
where the components are highly correlated this approach would tend to
generate highly auto-correlated MCMC draws.

Our proposed solution to this problem is to rely on the minimax tilting method
of Botev (2017), a general algorithm that directly samples from a potentially
high-dimensional Gaussian distribution under linear inequality restrictions.
The key idea is to locate a proposal distribution by tilting the mean vector
of the original unrestricted Gaussian distribution. This is done optimally by
solving a minimax problem: it minimizes the worst-case behavior of the
likelihood ratio, defined as the ratio of the original density divided by the
proposal density. Due to the log-concavity of the Gaussian distribution, this
optimization problem can be solved efficiently. The solution to this minimax
optimization problem then provides a proposal distribution for the accept-
reject algorithm with strong efficiency properties. As opposed to the Gibbs
sampler of Geweke (1996), this approach directly samples from the target
truncated multivariate Gaussian distribution.

While the minimax tilting method of Botev (2017) can be directly applied to
draw from the truncated multivariate Gaussian distribution given in (13), we
can further improve sampling efficiency by exploiting the special structure of
our simulation problem. More specifically, suppose a subset of the future
observables $\mathbf{y}_{T+1,T+h}$ is subjected to the inequality constraints
such that $\mathbf{S}$ is a $s_{o}\times nh$ selection matrix. Then, as in the
equality-constraints-on-observables case, we can partition
$\mathbf{y}_{T+1,T+h}$ into $\mathbf{y}_{T+1,T+h}^{o}$ and
$\mathbf{y}_{T+1,T+h}^{u}$, where the former is a $s_{o}\times 1$ vector of
the inequality-constrained endogenous variables such that
$\mathbf{y}_{T+1,T+h}^{o}=\mathbf{S}\mathbf{y}_{T+1,T+h}$ and the latter is a
$(nh-s_{o})\times 1$ vector of unconstrained variables. Using the
representation in (16) (with $\mathbf{M}_{o}=\mathbf{S}^{\prime}$ in this
case) and a similar derivation as before, we can decompose the joint
distribution of $\mathbf{y}_{T+1,T+h}$ into the product of the marginal
distribution of $\mathbf{y}_{T+1,T+h}^{o}$ and the conditional distribution of
$\mathbf{y}_{T+1,T+h}^{u}$ given $\mathbf{y}_{T+1,T+h}^{o}$. In fact, the
former is a truncated $s_{o}$-variate Gaussian distribution and the latter is
a $(nh-s_{o})$-variate Gaussian distribution (with no restrictions):

$\displaystyle\mathbf{y}_{T+1,T+h}^{o}\,|\,\mathbf{A}_{0},\mathbf{A}$
$\displaystyle\sim\mathcal{N}(\boldsymbol{\mu}_{o},\mathbf{K}_{o}^{-1})\mathbb{1}(\underline{\mathbf{c}}<\mathbf{y}_{T+1,T+h}^{o}<\bar{\mathbf{c}})$
$\displaystyle\mathbf{y}_{T+1,T+h}^{u}\,|\,\mathbf{y}_{T+1,T+h}^{o},\mathbf{A}_{0},\mathbf{A}$
$\displaystyle\sim\mathcal{N}(\boldsymbol{\mu}_{u},\mathbf{K}_{u}^{-1}),$

where
$\mathbf{K}_{u}=\mathbf{M}_{u}^{{}^{\prime}}\mathbf{H}^{\prime}\mathbf{H}\mathbf{M}_{u}$,
$\boldsymbol{\mu}_{u}=\mathbf{K}_{u}^{-1}\mathbf{M}_{u}^{{}^{\prime}}\mathbf{H}^{\prime}\mathbf{H}(\mathbf{H}^{-1}\mathbf{c}-\mathbf{M}_{o}\mathbf{y}_{T+1,T+h}^{o})$,
$\mathbf{K}_{o}=(\mathbf{M}_{o}^{{}^{\prime}}(\mathbf{H}^{\prime}\mathbf{H})^{-1}\mathbf{M}_{o})^{-1},$
and
$\boldsymbol{\mu}_{o}=\mathbf{M}_{o}^{\prime}\mathbf{H}^{-1}\mathbf{c}$.101010Note
that in general, a marginal distribution of a truncated Gaussian is not a
truncated Gaussian. Here the marginal distribution of
$\mathbf{y}_{T+1,T+h}^{o}$ is a truncated Gaussian because only
$\mathbf{y}_{T+1,T+h}^{o}$ is restricted and $\mathbf{y}_{T+1,T+h}^{u}$ is
not. Hence, to obtain a draw for $\mathbf{y}_{T+1,T+h}$, we can first sample
$\mathbf{y}_{T+1,T+h}^{o}$ marginally from its truncated $s_{o}$-variate
Gaussian distribution using the algorithm of Botev (2017). Given a draw for
$\mathbf{y}_{T+1,T+h}^{o}$, we can then sample $\mathbf{y}_{T+1,T+h}^{u}$ from
its Gaussian conditional distribution using the precision sampler of Chan and
Jeliazkov (2009), which can be done very quickly and the computational cost
increases only linearly in the dimension. In typical applications where
$s_{o}$ is much smaller than $nh$, this approach based on the marginal-
conditional decomposition is substantially more efficient, as it reduces the
dimension of the more computationally intensive sampling step from the
truncated Gaussian from $nh$ to $s_{o}$.

###  2.3 Constraints on Structural Shocks and Structural Scenario Analysis

While the discussion above has focused on the classic (or reduced-form)
conditional forecast case, where one assumes that the equality or inequality
conditions are generated by all the structural shocks of the model, it is
worthwhile showing that our framework can be straightforwardly extended to
accommodate the situation in which it is of interest to produce conditional
forecasts by restricting the path of a subset of structural shocks over the
forecast horizon. This specific case is discussed for example in Baumeister
and Kilian (2014) and Antolín-Díaz, Petrella, and Rubio-Ramírez (2021). Using
our notation, this type of restriction on structural shocks can be formulated
as

$\mathbf{W}\boldsymbol{\varepsilon}_{T+1,T+h}\sim\mathcal{N}(\mathbf{w},\boldsymbol{\Psi}),$
(17)

where $\mathbf{W}$ is a $w\times nh$ full-rank selection matrix, $\mathbf{w}$
is a $w\times 1$ vector of constants and $\boldsymbol{\Psi}$ is a $w\times w$
covariance matrix. To see how this case is also nested within our general
framework, note that we can use (2) to write the structural shocks
$\boldsymbol{\varepsilon}_{T+1,T+h}$ in terms of $\mathbf{y}_{T+1,T+h}$. Then,
the restrictions on the structural shocks can be rewritten as restrictions on
the observables:

$\displaystyle\begin{split}\mathbf{W}\boldsymbol{\varepsilon}_{T+1,T+h}&=\mathbf{W}\left(\mathbf{H}\mathbf{y}_{T+1,T+h}-\mathbf{c}\right)\\\
&=\mathbf{W}\mathbf{H}\mathbf{y}_{T+1,T+h}-\mathbf{W}\mathbf{c}\sim\mathcal{N}(\mathbf{w},\boldsymbol{\Psi})\end{split}$
(18)

Comparing this expression with (5), we see that we can implement this type of
restriction by setting $\mathbf{R}=\mathbf{W}\mathbf{H}$,
$\mathbf{r}=\mathbf{W}\mathbf{c}+\mathbf{w}$ and
$\boldsymbol{\Omega}=\boldsymbol{\Psi}$. And, as with the constraints-on-
observables case discussed in subsubsection 2.2.1, simulating the conditional
forecasts conditional on the structural shocks can often be streamlined by
exploiting the special structure of the problem (e.g., in the case of equality
constraints with $\boldsymbol{\Psi}=\mathbf{0}_{w\times w}$).

Finally, we consider the structural scenario case discussed in Antolín-Díaz,
Petrella, and Rubio-Ramírez (2021). A structural scenario combines
restrictions on the path of future observations with the restriction that only
a subset of the structural shocks—what Antolín-Díaz, Petrella, and Rubio-
Ramírez (2021) call the driving shocks—can deviate from their unconditional
distribution over the forecast horizon, while the remaining structural
shocks—the non-driving shocks—are restricted to retain their unconditional
distribution. As such, this setup is more flexible and plausible than
conditioning on a particular future path of structural shocks, as structural
shocks are unobserved and it is typically difficult to elicit restrictions on
their future path. It is also more appealing than restricting only the future
path of observables since the user in this case can specify which structural
shocks deviate from their unconditional distribution to drive the future
observables.

Our setup can be easily extended to accommodate this particular case, as a
structural scenario can be formulated by combining restrictions on observables
and restrictions on structural shocks in a straightforward way. First, we can
restrict the path of future observables using (15). We can then augment these
restrictions by appending (17), where $\mathbf{W}$ now denotes the $w\times
nh$ full-rank selection matrix flagging the non-driving shocks. Over the
forecasting horizon, these non-driving shocks, as Antolín-Díaz, Petrella, and
Rubio-Ramírez (2021) discussed, should retain their unconditional
distribution, i.e.
$\mathbf{W}\boldsymbol{\varepsilon}_{T+1,T+h}\sim\mathcal{N}(\mathbf{0}_{w},\mathbf{I}_{w})$.
This, in turn, implies

$\mathbf{W}\mathbf{H}\mathbf{y}_{T+1,T+h}\sim\mathcal{N}(\mathbf{W}\mathbf{c},\mathbf{I}_{w}).$
(19)

Combining (15) with (19) yields

$\underbrace{\begin{bmatrix}\mathbf{R}_{o}\\\
\mathbf{W}\mathbf{H}\end{bmatrix}}_{\widetilde{\mathbf{R}}}\mathbf{y}_{T+1,T+h}\sim\mathcal{N}\left(\underbrace{\begin{bmatrix}\mathbf{r}_{o}\\\
\mathbf{W}\mathbf{c}\end{bmatrix}}_{\widetilde{\mathbf{r}}},\underbrace{\begin{bmatrix}\boldsymbol{\Omega}_{o}&\mathbf{0}_{r_{0}\times
w}\\\ \mathbf{0}_{w\times
r_{0}}&\mathbf{I}_{w}\end{bmatrix}}_{\widetilde{\boldsymbol{\Omega}}}\right).$
(20)

Again, comparing this expression with (5), we see that this setting can be
nested within our general framework by setting
$\mathbf{R}=\widetilde{\mathbf{R}}$, $\mathbf{r}=\widetilde{\mathbf{r}}$ and
$\boldsymbol{\Omega}=\widetilde{\boldsymbol{\Omega}}$.

It is worth noting that while the cases reviewed in this section follow
closely the discussion in Antolín-Díaz, Petrella, and Rubio-Ramírez (2021), by
casting these restrictions within our general setup we are able to generalize
and improve upon their approach. First, as noted earlier, both equality or
inequality constraints can be imposed on the observables, and this carries
over to the structural scenario restrictions. For example, inequality
constraints on observables can be easily added by simply switching from (4) to
(13). Second, the precision-based methods described in earlier sections can be
used in both the structural shock restriction and the structural scenario
cases to speed up the sampling of $\mathbf{y}_{T+1,T+h}$ and seamlessly apply
these methods in large dimensional VARs or situations with many restrictions
being imposed.

It is also worth pointing out that while the exposition in this section
centers around the structural-form representation, it is straightforward to
apply the methodology in reduced-form settings for simple conditional
forecasting. This can be done, for example, by setting the impact matrix
$\mathbf{A}_{0}$ as the inverse Cholesky factor of the reduced-from error
covariance matrix, and computing the implied structural-form coefficient
matrices $\mathbf{A}_{1},\ldots,\mathbf{A}_{p}$ from the reduced-form
parameters.

##  3 A Simulation Study

To illustrate the performance of our precision-based conditional forecast
samplers, we conduct a series of simulations in which we compare our proposed
algorithms against existing conditional forecast algorithms in the literature.
Our benchmarks in this exercise are the approaches of Waggoner and Zha (1999),
Andersson, Palmqvist, and Waggoner (2010), Bańbura, Giannone, and Lenza (2015)
and Antolín-Díaz, Petrella, and Rubio-Ramírez (2021). We will focus on
reporting both accuracy (i.e. how similar our conditional forecasts are to
those produced by the various benchmarks) and speed (i.e. what is the
computational cost of implementing the various methods). For all calculations
in this section, we implement the algorithms using MATLAB and a standard
desktop with an Intel Xeon W-2224 @3.60GHz processor and 16GB memory.

In most of our simulations, we consider a data-generating process (DGP) that
follows an $n$-variable VAR with $p=2$ lags:

$\mathbf{y}_{t}=\mathbf{b}+\mathbf{B}_{1}\mathbf{y}_{t-1}+\mathbf{B}_{2}\mathbf{y}_{t-2}+\boldsymbol{\varepsilon}_{t},\quad\boldsymbol{\varepsilon}_{t}\sim\mathcal{N}(\mathbf{0}_{n},\boldsymbol{\Sigma}).$
(21)

but as a robustness we will also investigate the case of $p=4$. We set $T=300$
and $\mathbf{b}=0.01\times\mathbf{1}_{n}$, where $\mathbf{1}_{n}$ is an
$n\times 1$ column of ones. We then generate the diagonal elements of the
first lag coefficient matrix from $\mathcal{U}(0,0.5)$ and the off-diagonal
elements from $\mathcal{U}(-0.2,0.2)$. All the other elements of the higher
VAR coefficient matrix are generated independently from
$\mathcal{N}(0,0.05^{2}/p^{2})$. Finally, we generate the covariance matrix
$\boldsymbol{\Sigma}$ from the inverse-Wishart distribution
$\mathcal{IW}(n+10,0.07\mathbf{I}_{n}+0.03\mathbf{1}_{n}\mathbf{1}_{n}^{{}^{\prime}})$.

We estimate the model above by eliciting a normal prior for the VAR
coefficients and an inverse-Wishart prior for the covariance matrix. More
specifically, let
$\boldsymbol{\beta}=\text{vec}\left(\left[\mathbf{b}_{0},\mathbf{B}_{1},\mathbf{B}_{2}\right]^{\prime}\right)$
denote the $k\times 1$ vector of VAR coefficients stacked by rows. We then
specify uninformative independent priors for $\boldsymbol{\beta}$ and
$\boldsymbol{\Sigma}$, i.e.
$\boldsymbol{\beta}\sim\mathcal{N}(\mathbf{0}_{k},\mathbf{I}_{k})$ and
$\boldsymbol{\Sigma}\sim\mathcal{IW}(3+{n},\mathbf{I}_{n})$.

###  3.1 Equality Constraints on Observables

In the first set of simulations, we work with equality constraints on
observables. We specify a forecast horizon of length $h$ and constrain the
entire out-of-sample paths of $n_{o}$ endogenous variables while leaving those
of the remaining $n-n_{o}$ variables unconstrained. That is, we set the
forecasts of the $n_{o}$ variables to be the actual simulated data
$\mathbf{y}_{T:T+h}^{o}$. We consider VARs of different dimensions: a medium
size VAR with a short forecast horizon ($n=8$, $h=5$), a large VAR with a
longer forecast horizon ($n=15$, $h=20$), and an extra-large VAR with an even
longer forecast horizon ($n=40$, $h=30$). In all three cases, we let the
number of constrained endogenous variables, $n_{o}$, range from one to five
and retain 25,000 MCMC draws, after discarding the first 10,000 draws.

We compare the proposed algorithms against the three conditional forecast
algorithms in the literature: Waggoner and Zha (1999), Bańbura, Giannone, and
Lenza (2015) and Antolín-Díaz, Petrella, and Rubio-Ramírez (2021). We note
that although these Kalman filter based methods are implemented differently,
they all aim to draw from the same Gaussian distribution. Consequently, the
resulting samplers have the same mixing properties as ours, and it suffices to
compare the computational speed. Focusing on accuracy first, Figure 1 and
Figure 2 plot the conditional forecasts of the first four variables of both
the medium and large VAR that we obtained from our method as well as the three
benchmarks. We show both the median estimates (solid lines) as well as the
associated 68 per cent credible intervals (dashed lines). As we can see from
both figures, our proposed precision-based method produces virtually identical
posterior conditional forecasts and credible sets as those of the other three
benchmarks.

Having established that our approach produces conditional forecasts that are
virtually identical to those obtained with the available methods in the
literature, we turn to compare the computation time of the proposed precision-
based conditional forecast sampler for the VAR models against the three other
methods.111111For the Waggoner and Zha (1999) conditional forecasts, we use
the approach of Jarociński (2010), who developed a more efficient algorithm to
simulate the conditional forecast with one equality constraint compared to the
original algorithm of Waggoner and Zha (1999). We present these results in
Table 1. The computation time reported is based on the computation cost of
solely drawing the conditional forecasts across the four methods. We excluded
the computation time of drawing the VAR parameters since these are the same
across the four methods.121212For proper Bayesian inference in conditional
forecasting, the usual Gibbs sampler for the VAR needs to be adjusted so that
the conditions are taken into account when drawing the model parameters. While
this point has been emphasized by Waggoner and Zha (1999), the bulk of the
literature instead implements a two-step procedure: first estimate the model
parameters using a standard Gibbs sampler; then, compute the conditional
forecasts given the parameter estimates. We follow the latter approach in this
paper and side-step the estimation issue, since our main contribution lies in
improving the computation of conditional forecasts.

Figure 1: Conditional forecasts from a medium VAR with three equality
constraints. The thick black line is the posterior median estimate of the
conditional forecast using our precision-based method. The thick red line is
the posterior median estimates of the conditional forecast using the Waggoner
and Zha (1999) approach. The thick dark blue line is the posterior median
estimate of the conditional forecast from the approach of Bańbura, Giannone,
and Lenza (2015) using the Durbin_Koopman_2002 smoother. The thick light blue
line is the posterior median estimate of the conditional forecast using the
Antolín-Díaz, Petrella, and Rubio-Ramírez (2021). The dashed lines are the
corresponding 68% credible intervals for all four methods.

Uniformly across all cases considered, our proposed precision-based method is
computationally more efficient in producing conditional forecasts than the
other three methods. We can also see that as the number of imposed equality
constraints increases in the VAR, the proposed precision-based method becomes
relatively more efficient in generating the conditional forecasts than the
three other methods.

Figure 2: Conditional forecasts from a large VAR with three equality
constraints. The thick black line is the posterior median estimate of the
conditional forecast using our precision-based method. The thick red line is
the posterior median estimates of the conditional forecast using the Waggoner
and Zha (1999) approach. The thick dark blue line is the posterior median
estimate of the conditional forecast from the approach of Bańbura, Giannone,
and Lenza (2015) using the Durbin_Koopman_2002 smoother. The thick light blue
line is the posterior median estimate of the conditional forecast using the
Antolín-Díaz, Petrella, and Rubio-Ramírez (2021). The dashed lines are the
corresponding 68% credible intervals for all four methods.

This implies that the proposed precision-based method is scalable in terms of
the VAR dimensions and the number of imposed equality constraints. On the
other hand, the conditional forecasting method proposed by Antolín-Díaz,
Petrella, and Rubio-Ramírez (2021) does not appear to scale well, becoming
computationally very intensive when the dimension of the VAR or the number of
equality constraints increase.131313As discussed in Section 2.2, the proposed
method parameterizes the conditional forecast distribution in terms of its
precision matrix as opposed to the dense covariance matrix as in Antolín-Díaz,
Petrella, and Rubio-Ramírez (2021). Furthermore, under the proposed method,
simulations of the conditional forecasts given the equality constraints
involve only primitive matrices such as $\mathbf{M}_{u}$ and $\mathbf{H}$,
which are banded. By contrast, Antolín-Díaz, Petrella, and Rubio-Ramírez
(2021) simulate the conditional forecasts using the MATLAB built-in function
mvnrnd, which requires additional computations such as computing the Cholesky
decomposition or the singular value decomposition of the covariance matrix.

Dimension |  |  | Precision-based | WZ | DK | APR
---|---|---|---|---|---|---
|  |  |  Model with $p=2$ lags
Medium | $h=5$ | $n_{o}=1$ | 3 | 6 | 9 | 10
$n_{o}=3$ | 3 | 6 | 9 | 11
$n_{o}=5$ | 2 | 7 | 9 | 12
Large | $h=20$ | $n_{o}=1$ | 24 | 71 | 50 | 141
$n_{o}=3$ | 23 | 73 | 50 | 147
$n_{o}=5$ | 22 | 88 | 51 | 174
Extra Large | $h=30$ | $n_{o}=1$ | 338 | 1554 | 410 | -
$n_{o}=3$ | 332 | 1865 | 405 | -
$n_{o}=5$ | 291 | 1873 | 419 | -
|  |  |  Model with $p=4$ lags
Medium | $h=5$ | $n_{o}=1$ | 4 | 7 | 10 | 15
$n_{o}=3$ | 3 | 8 | 11 | 15
$n_{o}=5$ | 3 | 9 | 10 | 17
Large | $h=20$ | $n_{o}=1$ | 44 | 104 | 68 | 238
$n_{o}=3$ | 39 | 118 | 70 | 254
$n_{o}=5$ | 34 | 122 | 66 | 466
Extra Large | $h=30$ | $n_{o}=1$ | 937 | 3066 | 1375 | -
$n_{o}=3$ | 834 | 3340 | 1341 | -
$n_{o}=5$ | 799 | 3340 | 1260 | -
Table 1: Computation time for the equality constraints case. This table
reports the computation time (in seconds) required to generate conditional
Forecasts from the precision-based sampler, the Waggoner and Zha (1999) (WZ),
Bańbura, Giannone, and Lenza (2015) implemented using the smoother of
Durbin_Koopman_2002 (DK) and Antolín-Díaz, Petrella, and Rubio-Ramírez (2021)
(APR) methods. The computation times are based on 25,000 MCMC draws with a
10,000 burn-in period.

###  3.2 Inequality Constraints on Observables

Next, we move to the inequality-constraint case. We follow the same simulation
structure as described above in the equality constraint case, except that we
can only compare our conditional forecasts to those obtained using the
Waggoner and Zha (1999) and Andersson, Palmqvist, and Waggoner (2010)
algorithms since both the approaches of Bańbura, Giannone, and Lenza (2015)
and Antolín-Díaz, Petrella, and Rubio-Ramírez (2021) are not designed to
handle inequality constraints. In terms of sampling efficiency, the proposed
method has the same mixing properties as the approach in Waggoner and Zha
(1999), as both aim to directly draw from the same multivariate truncated
distribution. It is in principle more efficient than the algorithm in
Andersson, Palmqvist, and Waggoner (2010), since they implement a Gibbs
sampling step to draw from the multivariate truncated distribution that
induces additional autocorrelation in the MCMC draws. However, we find that
the inefficiency factors of the posterior draws of the conditional forecasts
for each of the three methods are very similar, and we therefore focus on
comparing the computational speed. We set
$\bar{\mathbf{y}}_{T-h:T}^{o}-0.1<\mathbf{y}_{T:T+h}^{o}<\bar{\mathbf{y}}_{T-h:T}^{o}+0.1$,
where
$\bar{\mathbf{y}}_{T-h:T}^{o}=\frac{1}{h}\sum_{t=T-h+1}^{T}\mathbf{y}_{t,}^{o}$
is the average of the actual simulated data over the periods $t=T-h,\ldots,T.$
In this simulation exercise, we focus on the medium- and large-sized VARs,
where $n=8$ and $=n=15$, and a long forecast horizon $h=20$. As in the
equality constraint case, we left the number of constrained variables,
$n_{o}$, range from one to five. As before, we estimate the medium VAR using
25,000 MCMC draws with a burn-in period of 10,000 draws and implement the same
priors as in the equality constraint exercise.

Figure 3 plots the conditional forecasts for the first four variables from the
medium-sized VAR with one inequality constraint. As before, our proposed
precision-based method produces virtually identical posterior conditional
forecast estimates and credible sets as both the Waggoner and Zha (1999) and
Andersson, Palmqvist, and Waggoner (2010) (denoted in the figure as APW)
algorithms. However, as mentioned in the previous section, the Waggoner and
Zha (1999) inequality constraint algorithm is very computationally intensive.
For example, when working with a medium VAR and one inequality constraint,
their accept-reject algorithm took approximately 2,542 minutes to compute the
conditional forecasts. In contrast, our proposed precision-based method and
the Andersson, Palmqvist, and Waggoner (2010) take about 62 and 70 seconds to
obtain the corresponding conditional forecasts, respectively.

Figure 3: Conditional forecast from a medium VAR with one inequality
constraint. The thick black line is the posterior median estimate of the
conditional forecast using the proposed precision-based method. The thick red
line is the posterior median estimates of the conditional forecast using the
Waggoner and Zha (1999) method. The thick blue line is the posterior median
estimates of the conditional forecasting using the Andersson, Palmqvist, and
Waggoner (2010) (APW) method. The dash lines are the corresponding 68 per cent
credible intervals for all three methods.

Note that while the computation costs of our precision-based and the
Andersson, Palmqvist, and Waggoner (2010) methods are relatively similar when
focusing on a single inequality constraint, Table 2 shows that as the number
of inequality constraints increase our precision-based method becomes more
computationally efficient than the Andersson, Palmqvist, and Waggoner (2010).
This is true for both a VAR with $p=2$ (top half of the table) and $p=4$ lags
(bottom half of the table).

| No. of inequality constraints
---|---
| $n_{o}=1$ | $n_{o}=3$ | $n_{o}=5$
Medium VAR with $p=2$
Precision-based | 2 | 5 | 8
APW | 3 | 9 | 21
Large VAR with $p=2$
Precision-based | 4 | 7 | 9
APW | 4 | 9 | 22
Medium VAR with $p=4$
Precision-based | 3 | 5 | 9
APW | 4 | 9 | 21
Large VAR with $p=4$
Precision-based | 7 | 9 | 11
APW | 6 | 12 | 23
Table 2: Computation time for inequality constraints case. This table report
the computation time (in seconds) required to simulate 1,000 draws for the
precision-based sampler and the Andersson, Palmqvist, and Waggoner (2010)
(APW) method in the case of a medium VAR and a $h=20$ forecast horizon.

##  4 Empirical Application

To illustrate the performance of the proposed precision-based conditional
forecast method in an empirical setting, we next setup a large BVAR with 31
quarterly variables to model the dynamic evolution over time of the US
macroeconomic and financial sectors. The complete list of the variables used
along with their transformations is available in Appendix A and our sample
goes from 1971Q1 to 2022Q3. Our analysis is inspired by the recent work of
Crump, Eusepi, Giannone, Qian, and Sbordone (2021), who estimate a quarterly
BVAR with variables extracted from those included in the Federal Reserve Board
of Governor’s Tealbook A Greensheets and selected to provide a broad picture
the US macroeconomic and financial conditions. They then consider a number of
conditional forecasting and tilting exercises to produce counterfactual
scenarios and evaluate their impact on the US economy both retrospectively and
prospectively.141414The Federal Reserve Tealbook A, officially subtitled
“Economic and Financial Conditions: Current Situation and Outlook”, is
produced by the staff of the Board of Governors and provides in-depth analysis
and forecasts of the U.S. and international economy. See https://www.
federalreserve.gov/monetarypolicy for additional details on the variables
being tracked. We exploit the computational efficiency and generality of the
methods presented in the previous sections to expand on the analysis of Crump,
Eusepi, Giannone, Qian, and Sbordone (2021) and investigate the macroeconomic
impact of a combination of multiple inequality and equality constraints at
once. To the best of our knowledge, this is the first study within the
literature that considers conditional forecasting in a large VAR setting with
multiple equality and inequality constraints.151515Crump, Eusepi, Giannone,
Qian, and Sbordone (2021) employ a filtering-based method to generate their
conditional forecasts, simulating the conditional forecasts one period at a
time. In contrast, our proposed precision-based method allows us to simulate
all the conditional forecasts in one step and as we showed in Section 3 leads
to large computational savings.

Our estimation strategy for the VAR parameters follows closely Chan (2022),
who proposes a novel asymmetric natural conjugate Minnesota-type prior for
large VARs. Unlike the more traditional natural conjugate prior, which does
not allow for asymmetric shrinkage of own lags and lags of other variables,
this new prior is more flexible and can be used, for example, to enforce
stronger shrinkage on the other variables’ lags. At the same time, this prior
maintains many useful analytical results of the traditional conjugate prior,
such as an extremely fast and scalable posterior simulator and a closed-form
expression of the marginal likelihood. As in subsection 2.1, we start with a
VAR($p$) in structural form,

$\mathbf{A}_{0}\mathbf{y}_{t}=\mathbf{a}+\mathbf{A}_{1}\mathbf{y}_{t-1}+\cdots+\mathbf{A}_{p}\mathbf{y}_{t-p}+\boldsymbol{\varepsilon}_{t},\quad\boldsymbol{\varepsilon}_{t}\sim\mathcal{N}(\mathbf{0}_{n},\mathbf{\Sigma}),$
(22)

where $\mathbf{A}_{0}$ is a lower triangular matrix with ones on its main
diagonal and
$\mathbf{\Sigma}=diag\left(\sigma^{2}_{1},...,\sigma^{2}_{n}\right)$ is
diagonal. This, in turn, allows us to estimate the $n$-dimensional VAR above
in a recursive manner, one equation at a time. For this purpose, let $a_{i}$
denote the $i$-th element of $\mathbf{a}$ and let $\mathbf{a}_{j,i}$ represent
the $i$-th row of $\mathbf{A}_{j}$. In addition, let $\boldsymbol{\alpha}_{i}$
denote the $i-1$ free elements from the $i$-th row of $\mathbf{A}_{0}$, and
define
$\boldsymbol{\beta}_{i}=\left(a_{i},\mathbf{a}_{1,i},...,\mathbf{a}_{p,i}\right)^{\prime}$.
With this notation in hand, we can rewrite the $i$-th equation of (22) as
follows:

$\displaystyle\begin{split}y_{it}&=\mathbf{w}_{it}\boldsymbol{\alpha}_{i}+\mathbf{z}_{it}\boldsymbol{\beta}_{i}+\varepsilon_{it}\end{split}$
(23)

where $\mathbf{w}_{it}=\left(-y_{1t},...,-y_{i-1,t}\right)$
$\mathbf{z}_{it}=\left(1,\mathbf{y}_{t-1}^{\prime},...,\mathbf{y}_{t-p}^{\prime}\right)$
and $\varepsilon_{it}\sim\mathcal{N}\left(0,\sigma^{2}_{i}\right)$. We pair
the model in (23) with the following normal-inverse-gamma prior for
$\left(\boldsymbol{\alpha}_{i},\boldsymbol{\beta}_{i},\sigma^{2}_{i}\right)$,

$\displaystyle\begin{split}\left.\boldsymbol{\alpha}_{i}\right|\sigma^{2}_{i}&\sim\mathcal{N}\left(\mathbf{m}_{i}^{\alpha},\sigma^{2}_{i}\mathbf{V}_{i}^{\alpha}\right)\\\
\left.\boldsymbol{\beta}_{i}\right|\sigma^{2}_{i}&\sim\mathcal{N}\left(\mathbf{m}_{i}^{\beta},\sigma^{2}_{i}\mathbf{V}_{i}^{\beta}\right)\end{split}$
(24)

and

$\sigma^{2}_{i}\sim\mathcal{IG}\left(\frac{v_{0}+i-n}{2},\frac{s_{i}}{2}\right).$
(25)

where $s_{i}^{2}$ denotes the sample variance of the residuals from an AR($p$)
model estimated on variable $i$, $i=1,...,n$. We specify the prior
hyperparameters by mirroring the choices in Chan (2022). In particular, we set
$v_{0}=n+2$, $\mathbf{m}_{i}^{\alpha}=\mathbf{0}_{i-1}$ and
$\mathbf{V}_{i}^{\alpha}=diag\left(1/s^{2}_{1},...,1/s^{2}_{i-1}\right)$.
Next, we set all elements of $\mathbf{m}_{i}^{\beta}$ to zero, except for the
coefficient associated with the first own lag, which we set to one. As for
$\mathbf{V}_{i}^{\beta}$, we specify it to be a diagonal matrix with the
$k$-th diagonal element, $(\mathbf{V}_{i}^{\beta})_{k}$, inspired by the
Minnesota prior,

$(\mathbf{V}_{i}^{\beta})_{k}=\left\\{\begin{matrix}[l]\frac{\kappa_{1}}{l^{2}s_{i}^{2}}&\text{for
the coefficient on the }l\text{th lag of variable }i\\\
\frac{\kappa_{2}}{l^{2}s_{j}^{2}}&\text{for the coefficient on the }l\text{th
lag of variable }j,~{}~{}j\neq i\\\ 100&\text{for the
intercept}\end{matrix}\right.$ (26)

and where the hyperparameters $\kappa_{1}$ and $\kappa_{2}$ control the
overall shrinkage intensity for the coefficients on the own lags and other
variables’ lags, respectively. As in Chan (2022), we estimate the optimal
degrees of shrinkage from the data.

As a first step, we estimate the model in (22) on data ranging from 1976Q3 to
2019Q3 (this is the same sample Crump, Eusepi, Giannone, Qian, and Sbordone
(2021) used to explore a number of their scenarios), after setting the lag
length of the VAR to $p=4$.161616As in Crump, Eusepi, Giannone, Qian, and
Sbordone (2021), rates enter the VAR in levels, while most other variables
enter in log-levels. See Appendix A for the full list of all variable
transformations. Before turning to the conditional forecasts, we quickly
review the estimation results and in particular we focus on whether the
asymmetric conjugate priors we employ allow to fit the data better than the
traditional natural conjugate priors (the latter is the prior adopted by
Crump, Eusepi, Giannone, Qian, and Sbordone (2021)). Table 3 reports the
results of this comparison, showing both the posterior means of the
$\kappa_{1}$ and $\kappa_{2}$ hyperparameters (in the case of the natural
conjugate prior, with symmetric shrinkage on the own and other variables’
lags, $\kappa_{1}=\kappa_{2}$), as well as the marginal likelihoods. Under the
symmetric prior, the optimal hyperparameter value is 0.0045, while when we
allow the two hyperparameters to differ the results are quite different, with
the data clearly pointing to a much stronger shrinkage on the coefficients of
the other variables’ lags. In addition, allowing for asymmetric shrinkage
increases the marginal log-likelihood by about 43 points. Taken all together,
these results overwhelmingly point to the data favoring the asymmetric prior
specification.

| Symmetric Prior | Asymmetric Prior
---|---|---
${\kappa}_{1}$ | 0.0045 | 0.083
${\kappa}_{2}$ | 0.0045 | 0.0024
Log-ML | -6121.4 | -6078.4
Table 3: Optimal shrinkage. This table reports the values of the
hyperparameters under the symmetric natural conjugate prior and the asymmetric
conjugate prior.

Next, we move to studying the model’s forecasts, both conditional and
unconditional. We consider two exercises. In the first one, we replicate one
of the analysis in Crump, Eusepi, Giannone, Qian, and Sbordone (2021), where
we estimate the BVAR in (22) using data from 1976Q4 to 2019Q3 and increase the
one-period ahead forecasts for real GDP by one percent, while leaving all
other variables unconstrained. We then compute, for each of the 31 variables
in the VAR, the difference between the conditional and unconditional forecasts
over the next three years. This, as Crump, Eusepi, Giannone, Qian, and
Sbordone (2021) note, is equivalent to computing an unorthogonalized impulse
response function (IRF). Figure 4 displays the responses of 16 selected
macroeconomic and financial variables to a one percent increase in real GDP.
Along with the posterior mean of the IRFs, shown in the figure with solid
lines, we report the 68% credibile intervals. As in Crump, Eusepi, Giannone,
Qian, and Sbordone (2021), a one percent increase in the level of real GDP
produces higher real GDP, higher PCE inflation, higher 1-year Treasury yield,
and higher exports and imports. On the other end, the unemployment rate falls
at first, to then revert back to the level of the unconditional forecasts by
the beginning of 2022.

In the second exercise, we leverage the generality of our approach and
condition the VAR forecasts on a combination of equality and inequality
constraints. In particular, after updating the BVAR estimates to include data
from 2019Q4, we set a number of equality constraints on the future paths of
unemployment rate and the 10-year Treasury rate by relying on the projections
reported by the Federal Reserve Board in their 2020 stress tests. These tests
are known as the Dodd-Frank Act stress test (DFAST) and the Comprehensive
Capital Analysis and Review (CCAR) and are used on an annual basis to ensure
that large bank holding companies operating in the United States will be able
to lend to households and businesses even in a severe recession.171717The
Federal Reserve Board annual stress tests from 2013 onward can be found at
https://www.federalreserve.gov/publications/dodd-frank-act-stress-test-
publications.htm. We consider two scenarios, namely (i) baseline and (ii)
severely adverse and as in the previous exercise restrict our attention to the
13 quarter period following the end of the estimation sample, i.e. 2020Q1
through 2023Q1. As noted in the Fed publications, the 2020 baseline scenario
is a moderate economic expansion over the 13-quarter stress-test period and
mimic closely the January 2020 consensus projections from Blue Chip Economic
Indicators. The 2020 severely adverse scenario is instead characterized by a
severe global recession accompanied by a period of heightened stress in
commercial real estate and corporate debt markets, and is designed to assess
the strength of banking organizations and their resilience to unfavorable
economic conditions. We complement the 26 equality constraints described above
with 13 more inequality constraints on the path of CPI inflation over the same
out-of-sample period. To design these constraints, we once again start from
the baseline and severely adverse scenarios provided by the Federal Reserve
Board stress tests and match these with a measure of disagreement that we
extract from the Survey of Professional Forecasters (SPF). The Federal Reserve
of Philadelphia reports the cross-sectional dispersion, defined as the
difference between the 75th percentile and the 25th percentile of the SPF
projections, for almost all variables included in their surveys for each
quarterly vintage, going back to the beginning of 1968.181818The cross-
sectional dispersion from the survey of professional forecasters is available
at https://www.philadelphiafed.org/surveys-and-data/real-time-data-
research/dispersion-forecasts. More specifically, for each quarter past the
end of our estimation sample, we use the difference between the P75 and P25 of
the SPF forecasts for CPI inflation to create an offset to the stress test
projections, such that our inequality constraints for CPI inflation are
centered on the FED stress test numbers and have a P25-P75 range that reflects
the disagreement we see in the SPF forecasts. Two clarifications are in order.
First, since the SPF does not provide projections under different scenarios,
we use the same measure of disagreement in the baseline and severely adverse
scenarios. Second, since the SPF cross-sectional dispersion is only available
for the first five quarters following the end of the sample, we fix the
inequality constraint intervals from the sixth quarter (i.e., from 2021Q2)
onward to match the forecast dispersion we observed in 2021Q1.191919As a
sensitivity analysis, we also experimented with the idea of gradually widening
the inequality constraint intervals around the corresponding FED stress test
numbers from 2021Q2 onward as we progress through the forecast horizon, as a
way of modeling the higher degree of uncertainty that could potentially arise
on future inflation. The results from this alternative scenario are very
similar to the ones we have reported. Table 4 summarizes the inequality and
equality constraints for the baseline and adversely severe scenarios.

Figure 5 to Figure 8 present the results of this exercise for six key
variables, namely (i) Real GDP, (ii) Industrial Production, (iii) Hourly
compensation in the business sector, (iv) Housing starts, (v) the S&P 500
index, and (vi) the CBOE Volatility index. Starting with the baseline
scenario, Figure 5 displays the unconditional forecasts (blue bands and solid
line) and conditional forecasts (red bands and solid line), where the bands
correspond to 68% coverage intervals and the solid lines correspond to the
posterior means.

Figure 4: Responses to a one percent increase in the level of real GDP one
quarter ahead. Within each panel, the solid line(s) shows the posterior mean
while the shaded area depicts the 68% coverage intervals.

| CPI Inflation (Inequality Constraint) | Equality Constraints
---|---|---
Date | Lower Bound | Fed’s projection | Upper Bound | UNRATE | GS10
Baseline Scenario
2020Q1 | 1.69 | 2.20 | 2.71 | 3.60 | 1.80
2020Q2 | 1.55 | 2.10 | 2.65 | 3.60 | 1.90
2020Q3 | 1.58 | 2.00 | 2.42 | 3.60 | 1.90
2020Q4 | 1.47 | 1.90 | 2.33 | 3.70 | 2.00
2021Q1 | 1.57 | 2.10 | 2.63 | 3.70 | 2.00
2021Q2 | 1.57 | 2.10 | 2.63 | 3.70 | 2.10
2021Q3 | 1.57 | 2.10 | 2.63 | 3.80 | 2.10
2021Q4 | 1.57 | 2.10 | 2.63 | 3.80 | 2.20
2022Q1 | 1.77 | 2.30 | 2.83 | 3.90 | 2.20
2022Q2 | 1.67 | 2.20 | 2.73 | 3.90 | 2.40
2022Q3 | 1.67 | 2.20 | 2.73 | 3.90 | 2.50
2022Q4 | 1.67 | 2.20 | 2.73 | 3.90 | 2.60
2023Q1 | 1.67 | 2.20 | 2.73 | 3.90 | 2.70
Adverse Scenario
2020Q1 | 1.19 | 1.70 | 2.21 | 4.50 | 0.70
2020Q2 | 0.55 | 1.10 | 1.65 | 6.10 | 0.90
2020Q3 | 0.58 | 1.00 | 1.42 | 7.40 | 1.00
2020Q4 | 0.67 | 1.10 | 1.53 | 8.40 | 1.10
2021Q1 | 0.77 | 1.30 | 1.83 | 9.20 | 1.20
2021Q2 | 0.87 | 1.40 | 1.93 | 9.70 | 1.30
2021Q3 | 0.97 | 1.50 | 2.03 | 10.00 | 1.40
2021Q4 | 1.17 | 1.70 | 2.23 | 9.90 | 1.50
2022Q1 | 1.27 | 1.80 | 2.33 | 9.70 | 1.60
2022Q2 | 1.27 | 1.80 | 2.33 | 9.50 | 1.80
2022Q3 | 1.27 | 1.80 | 2.33 | 9.20 | 1.90
2022Q4 | 1.27 | 1.80 | 2.33 | 8.80 | 2.10
2023Q1 | 1.17 | 1.70 | 2.23 | 8.50 | 2.20

Table 4: Summary of equality and inequality constraints. This table reports
the full set of inequality and equality constraints over the 2021Q1–2023Q1
time period that we use to generate the BVAR conditional forecasts. We
consider two different scenarios from the Federal Reserve Board stress tests,
namely (i) baseline and (ii) adverse.

Within each panel we also overlay, with black solid and dotted lines, the
actual realization of each series, both in-sample and out-of-sample. The
latter provides a reference point for how the economy actually evolved over
the particularly tumultuous period that followed the initial COVID-19 shock
and its unfolding.

Next, Figure 6 plots the differences between the unconditional and conditional
forecasts, once again showing the posterior means and the 68% coverage
intervals. As the figure shows, relative to the unconditional forecasts,
conditioning on the Fed’s baseline projections for CPI inflation, unemployment
and the 10-year Treasury rate leads to temporary increases in most of the
variables, with the most notable case being the industrial production which
increases as much as 5% over the first half of 2021, before gradually
returning back to its original level by the end of 2022.

Figure 5: Conditional and unconditional forecasts when CPI inflation,
unemployment, and the 10-year Treasury rate in 2021Q1–2023Q1 match the Fed’s
stress test baseline projections. The shaded bands correspond to the 68%
coverage intervals while the solid and dotted black lines denote the in-sample
and out-of-sample values. Figure 6: Difference between the conditional and
unconditional forecasts when CPI inflation, unemployment, and the 10-year
Treasury rate in 2021Q1–2023Q1 match the Fed’s baseline projections. Bands
denote to the 68% coverage intervals. The black dashed line shows the
difference between the actual realizations and the posterior mean of the
unconditional forecasts.

The story that emerges from the Fed’s adverse scenario, shown in Figure 7 and
Figure 8 is instead quite different. Conditioning on the Fed’s adverse
scenario projections for CPI inflation, unemployment and the 10-year Treasury
rate leads to a large decrease in real GDP and industrial production,
bottoming up around the second half of 2021 before a gradual recovery begins.
Similarly, we see a significant drop in housing starts and the S&P 500 and a
spike in the market volatility, as picked up the by VIX measure.
Interestingly, with the exception of the effect on the S&P 500 index, the
initial responses we see in our conditional forecasts are in the same
direction and in most cases of a magnitude similar to what the US economy
experienced after the large and unexpected COVID-19 shocks that occurred
during the first half of 2020. However, the sudden and unprecedented nature of
the COVID-19 shock, as well as the unfolding of events that followed the
initial impact, were such that the US economy’s initial reaction was much more
immediate and the recovery that followed much faster.

Figure 7: Conditional and unconditional forecasts when CPI inflation,
unemployment, and the 10-year Treasury rate in 2021Q1–2023Q1 match the Fed’s
adverse projections. The shaded bands correspond to the 68% coverage intervals
while the solid and dotted black lines denote the in-sample and out-of-sample
values. Figure 8: Difference between the conditional and unconditional
forecasts when CPI inflation, unemployment, and the 10-year Treasury rate in
2021Q1–2023Q1 match the Fed’s stress test adverse projections. Bands denote to
the 68% coverage intervals. The black dashed line denotes the difference
between the actual realizations and the posterior mean of the unconditional
forecasts.

##  5 Conclusions

We introduced a novel precision-based approach that can be used for
conditional forecasting, scenario analysis and entropic tilting and can handle
both equality and inequality constraints. Thanks to the way we have derived
the conditional forecasts’ distribution, our approach is computationally very
efficient and particularly well suited to handle large dimensional VARs as
well as situations in which we have a large number of conditioning variables
and long forecast horizons. We have showed in a simulation study that the
proposed approach generates exactly the same conditional forecasts and
credible sets as those from Waggoner and Zha (1999), Bańbura, Giannone, and
Lenza (2015), and Antolín-Díaz, Petrella, and Rubio-Ramírez (2021), but is
substantially less demanding computationally. Finally, we conducted an
empirical exercise where we estimated a Bayesian VAR featuring 31 quarterly
macroeconomic and financial series, and we used our approach to investigate
the effect of simultaneously imposing a number of inequality and equality
constraints on the trajectories of CPI inflation, the unemployment rate, and
the 10-year Treasury rate over the 2020–2022 period. In the future, we believe
it would be useful to generalize the current approach to also work with
Bayesian VARs with time-varying parameters and stochastic volatility, and it
would also be interesting to extend the precision-based sampler to produce
conditional forecasts for binary variables, along the lines of McCracken,
McGillicuddy, and Owyang (2021).

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Appendices

##  Appendix A Data and Transformations

Table A.1: This table lists all the variables we included in our empirical
application. For each series, we report the FRED-QD mnemonic and the
transformation we adopted. We extract all variables from the FRED-QD dataset,
available at https://research.stlouisfed.org/econ/mccracken/fred-databases/.
The list of variables and transformations follow closely the choices from
Crump, Eusepi, Giannone, Qian, and Sbordone (2021).

Variable | Mnemonic | Transformation
---|---|---
Real Gross Domestic Product | GDPC1 | $100\text{ln}(x_{t})$
Real Personal Consumption Expenditures | PCECC96 | $100\text{ln}(x_{t})$
Real Private Fixed Investment: Residential | PRFIx | $100\text{ln}(x_{t})$
Real Private Fixed Investment: Non-Residential | PNFIx | $100\text{ln}(x_{t})$
Real Exports of Goods and Services | EXPGSC1 | $100\text{ln}(x_{t})$
Real Imports of Goods and Services | IMPGSC1 | $100\text{ln}(x_{t})$
Real Government Consumption Expenditures and Gross Investment | GCEC1 | $100\text{ln}(x_{t})$
Real Government Consumption Expenditures and Gross Investment: |  |
Federal (Chain-Type Quantity Index) | B823RA3Q086SBEA | $100\text{ln}(x_{t})$
Gross Domestic Product: Chain-Type Price Index | GDPCTPI | $100\text{ln}(x_{t})$
Producer Price Index by Commodity: All Commodities | PPIACO | $100\text{ln}(x_{t})$
Personal Consumption Expenditures Excluding Food and Energy |  |
(Chain-Type Price Index) | PCEPILFE | $100\text{ln}(x_{t})$
Consumer Price Index | CPIAUCSL | $100\text{ln}(x_{t})$
Consumer Price Index for All Urban Consumers: |  |
All Items Less Food and Energy in U.S. City Average | CPILFESL | $100\text{ln}(x_{t})$
Business Sector: Real Hourly Compensation for All Employed Persons | RCPHBS | $100\text{ln}(x_{t})$
All Employees, Total Nonfarm | PAYEMS | $100\text{ln}(x_{t})$
Unemployment Rate | UNRATE | Level
Industrial Production: Total Index | INDPRO | $100\text{ln}(x_{t})$
Capacity Utilization: Manufacturing | CUMFNS | $100\text{ln}(x_{t})$
New Privately-Owned Housing Units Started: Total Units | HOUST | $100\text{ln}(x_{t})$
Real Disposable Personal Income | DPIC96 | $100\text{ln}(x_{t})$
University of Michigan: Consumer Sentiment | UMCSENTx | Level
Market Yield on U.S. Treasury Securities at 1-Year Constant Maturity | GS1 | Level
Market Yield on U.S. Treasury Securities at 10-Year Constant Maturity | GS10 | Level
Moody’s Seasoned Aaa Corporate Bond Yield | AAA | Level
Moody’s Seasoned Baa Corporate Bond Yield | BAA | Level
Trade Weighted U.S. Dollar Index: Advanced Foreign Currencies | TWEXAFEGSMTHx | $100\text{ln}(x_{t})$
S&P 500 | S&P 500 | $100\text{ln}(x_{t})$
CBOE Volatility Index: VIX | VIXCLSx | Level
Personal Consumption Expenditures: Chain-type Price Index | PCECTPI | $100\text{ln}(x_{t})$
Real Crude Oil Prices: West Texas Intermediate (WTI) | OILPRICEx | $100\text{ln}(x_{t})$
Federal Funds Effective Rate | FEDFUNDS | Level