# Decision Synthesis in Monetary Policy

Tony Chernis
Bank of Canada
Gary Koop
University of Strathclyde
Emily Tallman
Duke University Mike West
Duke University

Abstract

The macroeconomy is a sophisticated dynamic system involving significant
uncertainties that complicate modelling. In response, decision makers consider
multiple models that provide different predictions and policy recommendations
which are then synthesized into a policy decision. In this setting, we
introduce and develop Bayesian predictive decision synthesis (BPDS) to
formalize monetary policy decision processes. BPDS draws on recent
developments in model combination and statistical decision theory that yield
new opportunities in combining multiple models, emphasizing the integration of
decision goals, expectations and outcomes into the model synthesis process.
Our case study concerns central bank policy decisions about target interest
rates with a focus on implications for multi-step macroeconomic forecasting.

Keywords: Bayesian predictive decision synthesis, dynamic model averaging,
interest rates, macroeconomic forecasting, model combination, path
forecasting, policy decisions, scenario analysis

##  Introduction

Monetary policy makers are tasked with simple, but hard to achieve,
objectives. A common objective is to target the future inflation rate, or
other macroeconomic outcomes, using the interest rates as the policy
instrument. These decisions are made based on uncertain information from many
sources. In this paper, these sources are econometric models that generate
predictive distributions for the macroeconomic outcomes and the policy
instruments over multiple time periods. For a single model, it is
straightforward to select an optimal policy instrument using decision analysis
and conditional forecasting. Applying standard methods– such as Bayesian model
averaging (BMA)– is one way to address the issue of model uncertainty, as
routine decision analysis can then be applied to the weighted average as a
single model. However, this traditional view ignores the reality that a set of
models may each individually recommend very different optimal policy
decisions. The question then arises as to how to synthesize this information
and, potentially, exploit it in the overall final decision process. This paper
addresses and answers this question.

There is an extensive Bayesian econometrics literature on model combination,
but discussion of the question that models are built for purposes– specific
prediction and decision goals– is very sparse. Traditional BMA analysis
weights models according to purely statistical model fit, and in time series
explicitly and only scores 1-step ahead forecast outcomes. Extensions and
alternatives have arisen to define model weightings based on aspects of past
forecast performance with respect to specific forecast goals. Martin et al.
(2023) survey Bayesian forecasting in economics and finance and overviews
various forecast combination approaches, including some that are more
explicitly concerned with goal-focused prediction (e.g. Mitchell and Hall,
2005; Geweke and Amisano, 2011; Conflitti et al., 2015; Kapetanios et al.,
2015; Loaiza-Maya et al., 2021; Chernis and Webley, 2022; Aastveit et al.,
2023; Bernaciak and Griffin, 2024). Lavine et al. (2021) provide additional
perspectives and put many of the earlier approaches in a foundational Bayesian
context, justifying model weights based on utilities in forecasting using
historical model-specific “scoring” of past forecast outcomes. The underlying
theoretical justifications come from Bayesian predictive synthesis (BPS) and
the specific class of “mixture BPS” models (McAlinn and West, 2019, section
2.2; Johnson and West, 2022). However, while ultimate decision goals may be
implicit in specific applications of model combination, they are rarely if
ever taken into account in the analysis and resulting decision-making. This
raises questions: A model that has fit or forecast specific outcomes well in
the past may be a good bet for use in resulting decision analysis– in our
settings, to define optimal decisions about values of policy instruments– but,
there is no guarantee that this will be so.

Our view is that models that have recommended optimal policy decisions that
turn out to be “good” should be more heavily weighted in looking ahead, just
as past statistical predictive performance is– and should be– generally
positively weighted. The catch, of course, is defining “good”; econometric
models were and are not explicitly used and scored in past policy decisions.
The challenge is then to operationalize the concept of “good decision”
performance. For example, a vector autoregression (VAR) model can be evaluated
on forecast performance using a pseudo real-time forecasting exercise. But, it
is not clear how to evaluate such a model when used to advise policy
decisions. We can, however, use the VAR model at present to inform near-term
decisions and explore how it would have advised on decisions in the past.
Evaluations can then compare such analysis to decisions actually made by
policy makers in the past (albeit recognizing that past decisions of policy
makers were not necessarily correct – rather, just the outcomes of the
amorphous reality of monetary policy-making).

Bayesian Predictive Decision Synthesis (BPDS–Tallman and West, 2023; Tallman,
2024) addresses these questions. As part of the theoretical framework of
Bayesian predictive synthesis (BPS–McAlinn and West, 2019; Johnson and West,
2022), BPDS explicitly allows and encourages scoring of models based on
decision analysis performance as well as statistical predictive accuracy. In
addition to reflecting historical outcomes of predictions and decisions, BPDS
critically also allows for differential model weighting based on expected
decision outcomes. This is a complete decision parallel to the proven use of
BPS models that incorporate outcome-dependent weights that modify BMA-like
mixtures to differentially favour models in different parts of the future
outcome space for pure forecasting. The latter concept was introduced by
Kapetanios et al. (2015), whose empirically inspired developments recognized,
for example, that one model may be better at predicting inflation when
inflation is high and rising, while another model may be better when inflation
is low and stable. BPS defines the conceptual and theoretical Bayesian bases
and a broader methodological framework for this. BPDS goes further by
integrating both historical and expected decision outcomes; here we develop,
extend, and exemplify BPDS in our central macroeconomic policy context.

BPDS applies the broader Bayesian mixture model approach of BPS using defined
utility– or “score”–functions that relate to explicit decision goals.
Importantly, this allows for multiple objectives, i.e., multi-attribute
decision analysis. For example, a purely predictive vector score function can
allow for multiple forecast horizons (e.g. to produce inflation near a target
for each of the next eight quarters) and/or multiple outcome criteria (e.g. to
separately reflect inflation targeting, interest rate smoothing, and stable
growth patterns over coming quarters), among others. For policy makers
juggling multiple objectives, this is a key feature; it is rather distinct
from conventional approaches that adopt single, scalar criteria for model
weighting. For example, a forecast combination approach might choose model
weights based on the $h-$step ahead predictive likelihood for a single choice
of $h$, with BMA simply focused on $h=1$, whereas BPDS can address multi-steps
ahead in parallel, along with scoring of outcomes of decision goals that
simultaneously target several macroeconomic outcomes.

The opportunities for exploring such practically relevant questions, and the
differences relative to traditional single-model and BMA-based analyses, are
showcased in our empirical studies. This involves an exploratory case study
using US macroeconomic data with multiple-objective score functions to define
BPDS model weights, exemplifying the use of BPDS in macroeconomic forecasting
and advisory decision-making.

##  BPDS Framework

We present and discuss the structure of BPDS at a particular point in time,
ignoring the time dependency and relevance in the notation for clarity in
communicating these essentials. Practical implementation in time series is of
course sequential, with models at time $t$ depending on all relevant
historical data and information.

###  Mixture BPDS and Decision Setting

At a given time point, let $\mathbf{y}$ denote the $q-$dimensional outcome
variable of interest (e.g. inflation in each of the next $q$ quarters) and
$\mathbf{x}$ the vector of control/decision variables (e.g., a target profile
of central bank interest/base rates over the next $q$ quarters). Each of a set
of $J$ models ${\mathcal{M}}_{j}$, $j=1{:}J,$ predicts the outcome
$\mathbf{y}$ via a predictive density
$p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})$ conditional on any considered
decision $\mathbf{x}.$ The policy-maker responsible for ultimate decisions
adopts a general BPDS approach that implies the overall conditional (on
$\mathbf{x}$) predictive pdf of the form

$f(\mathbf{y}|\mathbf{x})\propto\sum_{j=0{:}J}\pi_{j}(\mathbf{x})\alpha_{j}(\mathbf{y}|\mathbf{x})p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})$
(1)

with the following ingredients.

#### BPDS model probabilities

The decision dependent model probabilities $\pi_{j}(\mathbf{x})$ can
differentially weight models $j$ over the decision space of $\mathbf{x},$
incorporating any prior information relevant to model weighting based on past
predictive model fit and decision outcomes, and now explicitly allowing for
adjustments based on a currently considered decision $\mathbf{x}$. Dependence
of $\pi_{j}(\mathbf{x})$ on $\mathbf{x}$ is simply fundamental and critical in
our policy setting.

#### BPDS calibration functions

The $\alpha_{j}(\mathbf{y}|\mathbf{x})$ are calibration functions that define
outcome-dependence of model weights over the outcome space of $\mathbf{y}$ for
any chosen $\mathbf{x}$. This defines the opportunity to increase or decrease
the model weights differentially over the outcome $\mathbf{y}$ space to
address model-specific biases and preferences and address questions of model-
specific calibration more generally. The BPDS mixture of eqn. (1) has the
equivalent form

$f(\mathbf{y}|\mathbf{x})=\sum_{j=0{:}J}\tilde{\pi}_{j}(\mathbf{x})f_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})$
(2)

where

$f_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})=\alpha_{j}(\mathbf{y}|\mathbf{x})p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})/a_{j}(\mathbf{x})\quad\textrm{and}\quad\tilde{\pi}_{j}(\mathbf{x})=k(\mathbf{x})\pi_{j}(\mathbf{x})a_{j}(\mathbf{x})$
(3)

with normalizing terms $k(\mathbf{x})$ and $a_{j}(\mathbf{x})$ explicitly
dependent on $\mathbf{x}$. This form shows how the calibration functions
$\alpha_{j}(\cdot|\cdot)$ modify the initial mixture pdfs
$p_{j}(\cdot|\cdot)\to f_{j}(\cdot|\cdot)$ with corresponding changes of
mixture weights $\pi_{j}(\mathbf{x})\to\tilde{\pi}_{j}(\mathbf{x}).$

Note that the choice of relevant calibration functions
$\alpha_{j}(\mathbf{y}|\mathbf{x})$ will, in any given application, be partly
dependent on characteristics of the model pdfs
$p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})$. In particular, the
expectation of each $\alpha_{j}(\mathbf{y}|\mathbf{x})$ under
$p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})$ must be finite in order that
eqn. (1) defines a valid BPDS density $f(\mathbf{y}|\mathbf{x}).$ Unbounded
score functions may sometimes apply, but this point supports the use of
bounded scores in general.

#### Baseline mixture component

The model index $j=0$ explicitly allows for a baseline model component
${\mathcal{M}}_{0}$ in the mixture pdf $f(\cdot|\cdot)$ that can, among other
things, address the ever-present issue of “model set incompleteness” (Tallman
and West, 2023, section 2.2.3). ${\mathcal{M}}_{0}$ can be chosen to produce a
pdf $f_{0}(\cdot|\cdot)$ that is over-dispersed relative to the mixture of the
initial $J$ models, so supporting outcomes $\mathbf{y}$ that are unusual under
the $J$ models; the baseline is then a suitable “fall back” model for times
when the other models are forecasting poorly.

#### Initial mixture

The special case with each $\alpha_{j}(\mathbf{y}|\mathbf{x})=1$ defines the
initial mixture with no BPDS calibration. We use $p(\mathbf{y}|\mathbf{x})$ in
notion, i.e.,
$p(\mathbf{y}|\mathbf{x})=\sum_{j=1{:}J}\pi_{j}(\mathbf{x})p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})$

Special cases fix ideas. First, if $\pi_{j}(\mathbf{x})=\pi_{j}$ with
$\pi_{0}=0$ are model probabilities based on historical BMA analysis, and with
$\alpha_{j}(\mathbf{y}|\mathbf{x})=1,$ then eqn. (1) specialises to BMA. Thus
BMA analyses– with or without this decision dependence in model-specific
forecasts– are very special cases of BPDS. Second, again with
$\pi_{j}(\mathbf{x})=\pi_{j}$, $\pi_{0}=0$ and
$\alpha_{j}(\mathbf{y}|\mathbf{x})=1,$ the decision maker has the freedom to
specify the initial mixture probabilities $\pi_{j}$ in other ways than with
BMA. This includes using historical performance defined by scoring of past
forecast outcomes, justifying various approaches to goal-focused model
weighting (e.g. Lavine et al., 2021; Loaiza-Maya et al., 2021, and references
therein) as special cases of BPDS. Third, mixture BPS (McAlinn and West, 2019;
Johnson and West, 2022) is a special case in which models are combined with
outcome-dependent weights. In these settings, $\pi_{j}(\mathbf{x})=\pi_{j}$
depends on past predictive performance,
$p_{j}(\mathbf{y}|\mathbf{x})=p_{j}(\mathbf{y})$ and
$\alpha_{j}(\mathbf{y}|\mathbf{x})=\alpha_{j}(\mathbf{y})$ define outcome-
dependent modifications of model probabilities, but there is no decision
context so no $\mathbf{x}-$dependence. BPDS critically recognizes that the
foundational BPS theory allows explicit incorporation of decision goals–
admitting the conditioning on $\mathbf{x}$ throughout all components of eqn.
(1)– to extend the foregoing analyses.

With predictions of $(\mathbf{y}|\mathbf{x})$ based on eqn. (2), the Bayesian
decision maker acts to identify the optimal decision $\mathbf{x}$ based on a
chosen utility function $U(\mathbf{y},\mathbf{x}).$ This involves numerical
optimization to maximize the implied expected utility
$\bar{U}(\mathbf{x})=E_{f}[U(\mathbf{y},\mathbf{x})|\mathbf{x}]$ over the
decision space of $\mathbf{x}.$ The notation $E_{f}[\cdot|\cdot]$ here
explicitly represents expectation with respect to the BPDS distribution, and
we use $E_{p}[\cdot|\cdot]$ to denote expectation under the initial mixture.

###  Decision-dependent Scores for Calibration Functions

The key step in integrating decision outcomes into relative model weightings
is to address the question of how each ${\mathcal{M}}_{j}$ would inform
decisions if used alone. Given the predictive pdf
$p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})$ and a chosen, potentially
model-specific utility function $u_{j}(\mathbf{y},\mathbf{x}),$ acting based
only on ${\mathcal{M}}_{j}$ leads to the optimal decision $\mathbf{x}_{j}$
that maximizes $E_{p_{j}}[u_{j}(\mathbf{y},\mathbf{x})|\mathbf{x}]$ over
$\mathbf{x}.$ The decision maker has access to this set of model
recommendations and is interested in model combination to preferentially
weight “good decision models” as well as models that generate good
predictions. BPDS formalizes this with specified score functions
$\mathbf{s}_{j}(\mathbf{y},\mathbf{x}_{j}),$ each being a $k-$vector of
utilities that can be chosen to reflect both predictive and decision goals.
The use of multi-dimensional scores addresses multiple goals simultaneously.

The Bayesian decision-theoretic development of Tallman and West (2023)
generates the resulting functional forms of the BPDS calibration functions as

$\alpha_{j}(\mathbf{y},\mathbf{x})=\exp\\{{\bm{\tau}}(\mathbf{x})^{\prime}\mathbf{s}_{j}(\mathbf{y},\mathbf{x}_{j})\\},\,\,j=0{:}J,$
(4)

where ${\bm{\tau}}(\mathbf{x})$ is a $k-$vector with elements differentially
weighting the multiple utility dimensions of the score vector. The reasoning
and theory behind this key result is as follows.

The initial mixture
$p(\mathbf{y}|\mathbf{x})=\sum_{j=1{:}J}\pi_{j}(\mathbf{x})p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})$
is the $\mathbf{y}-$margin of the joint distribution
$p(\mathbf{y},{\mathcal{M}}_{j}|\mathbf{x})=\pi_{j}(\mathbf{x})p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j}),$
$(j=1{:}J)$. Under this initial distribution for any candidate decision
$\mathbf{x}$, and with score vectors
$\mathbf{s}_{j}(\mathbf{y},\mathbf{x}_{j})$ defined and evaluated at model-
specific optimal decisions $\mathbf{x}_{j},$ the decision maker has initial
expected score
$\mathbf{m}_{p}(\mathbf{x})=\sum_{j=0{:}J}\pi_{j}(\mathbf{x})\mathbf{m}_{jp}(\mathbf{x})$
where
$\mathbf{m}_{jp}(\mathbf{x})=\int_{\mathbf{y}}\mathbf{s}_{j}(\mathbf{y},\mathbf{x}_{j})p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})d\mathbf{y}.$
Treating $\mathbf{m}_{p}(\mathbf{x})$ as a benchmark to improve on in
expectation, the BPDS theory enquires about distributions
$f(\mathbf{y},{\mathcal{M}}_{j}|\mathbf{x})$ that yield expected scores
$\mathbf{m}_{f}(\mathbf{x})\geq\mathbf{m}_{p}(\mathbf{x})+{\bm{\epsilon}}(\mathbf{x})$
for some non-negative $k-$vector (with at least one positive entry)
${\bm{\epsilon}}(\mathbf{x});$ this may be chosen to depend on $\mathbf{x},$
or may be a specified constant “decision score improvement”. Given
$\mathbf{m}_{f}(\mathbf{x}),$ the BPDS theory identifies a unique
$f(\cdot,\cdot|\mathbf{x})$ that minimizes the Küllback-Leibler (KL)
divergence of $p(\cdot,\cdot|\mathbf{x})$ from $f(\cdot,\cdot|\mathbf{x})$ and
has expected score exactly $\mathbf{m}_{f}(\mathbf{x}).$ The theory is that of
relaxed entropy tilting (Tallman and West, 2022, 2023; West, 2023) and yields
$f(\mathbf{y},{\mathcal{M}}_{j}|\mathbf{x})\propto\pi_{j}(\mathbf{x})\alpha_{j}(\mathbf{y},\mathbf{x})p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})$
with calibration function precisely as in eqn. (4). The tilting vector
${\bm{\tau}}(x)$ is implicitly defined by the vector of $k$ target score
constraints
$E_{f}[\mathbf{s}_{j}(\mathbf{y},\mathbf{x}_{j})|\mathbf{x}]=\mathbf{m}_{f}(\mathbf{x}).$

BPDS takes the view that the initial mixture is based on past performance, and
additional small changes in models and the way they are weighted based on
their expected performance may lead to better future decisions. It asks the
question as to whether there are perturbations of the mixture based on the
initial model probabilities that can lead to improved scores. Consider a
stylized example with $J=2$ models which in the past have forecast equally
well. Traditional model averaging methods focused only on past forecasting
experience– and BMA in particular– would confer equal weights in the
combination. If, however, the models have different expected scores
$\mathbf{m}_{jp}(\mathbf{x})$, conferring slightly more weight on the model
expected to lead to a higher score makes sense. The entropic (or exponential)
tilting theory is general. It is, of course, possible to tilt the initial
joint distribution to most targets, so long as they are technically achievable
under the initial distribution; but an overly ambitious target score will
result in a tilted joint distribution that is empirically unreasonable. Hence
we emphasize the importance of selecting $\mathbf{m}_{f}(\mathbf{x})$ that
represents a “small ” improvement over the initial benchmark score
$\mathbf{m}_{p}(\mathbf{x}).$ This is bolstered by the assumption that the
initial model probabilities reflect the empirical plausibility of models as
well as any available information about historical predictive and decision
performance. Further, as we exemplify in the case study later, aspects of the
computational methodology for model fitting in the sequential time series
setting naturally inform on, and allow monitoring of, relevant choices of
target expected scores.

###  BPDS Summary

This section has outlined the main ideas underlying BPDS and the key
ingredients of the theory and resulting technical machinery. Specifications of
score and utility functions, initial model probabilities, and target scores
are all required for implementation and are, of course, application specific.
The following section develops full details in the context of the
macroeconomic decision-making application. In terms of computation, BPDS
requires the use of posterior simulation methods (i.e. draws from conditional
predictive densities from each model are required) as well as numerical
optimization methods (i.e. to find $\mathbf{x}_{j}$ or the overall optimal
decision $\mathbf{x}$ under the final BPDS analysis).

##  BPDS for Optimal Monetary Policy Decisions

The choice of data and models is inspired by Furlanetto et al. (2019). We use
quarterly macroeconomic and financial variables from 1973:Q1 to 2022:Q2 from
the FRED-QD database maintained by the Federal Reserve Bank of St. Louis. The
data set includes GDP (the log of real GDP), prices (the log of the GDP
deflator), the interest rate (the shadow rate111This is the Federal Funds rate
when the latter is positive but can go negative when it is at the zero lower
bound, taking into account unconventional monetary policy; see Wu and Xia
(2016). which we treat as the policy rate), investment (the ratio of real
gross private domestic investment to GDP), stock prices (log of the S&P500)
and the spread (the spread between BAA bonds and the Fed funds rate). Models
are run over multiple years, and at the end of each quarter produce forecasts–
full predictive distributions in terms of Monte Carlo samples– of outcomes of
interest over the following $k=8$ quarters, conditional on candidate settings
of the decision vector which is taken as the trajectory of interest (shadow)
rates over those quarters. Within-model decision analysis then delivers model-
specific optimal decisions about these rates.

###  Models, Forecasts and Model-Specific Decisions

We consider $J=2$ models: ${\mathcal{M}}_{1}$ is a three-variable monetary
policy VAR involving GDP, prices, and the interest rate; ${\mathcal{M}}_{2}$
is the model of Furlanetto et al. (2019), a VAR with the same variables as
${\mathcal{M}}_{1}$ plus investment, stock prices, and the spread. Following
the latter, we include 5 lags in the VARs. The two structural VARs are
identified using the sign restrictions from Table 1 of Furlanetto et al.
(2019). In ${\mathcal{M}}_{1}$ these restrictions define supply, demand and
monetary policy shocks. In ${\mathcal{M}}_{2}$ investment and financial shocks
are additionally identified. We condition on a given value of the policy rate
and set the monetary policy to be the driving shock. We do this by imposing
restrictions on the set of structural shocks underlying the conditional
forecasts. Structural shocks other than the monetary policy shock have zero
means. We use the asymmetric conjugate prior of Chan (2022), with the
advantage that the marginal likelihoods for each can be easily calculated;
prior hyperparameter choices are made to maximize the marginal likelihood as
in this referenced paper. At each quarter, multi-step ahead predictions are
based on simulations using the precision-based sampler of Chan et al. (2023).
Details on the conditional forecast computations are summarized in Appendix B.
In short, this generates $p_{j}(\mathbf{y},|\mathbf{x})$ with zero-mean
constraints on all shocks apart from the monetary policy variable. We do not
restrict the variance (i.e., ‘soft’ restrictions) such that we also have
uncertainty around the path of $\mathbf{x}$. This can be thought of as
conditional commitment– we allow the possibility of $\mathbf{x}$ deviating
from the proposed policy path with deviations informed by historical
uncertainty around forecasts of the interest rate outcomes.

This is a model setting in which the decision variable is also modeled – it is
an outcome variable. In the current quarter,
$\mathbf{x}=(x_{1},\ldots,x_{k})^{\prime}$ is the $k-$vector of interest rate
values over the next $k$ quarters, and
$\mathbf{y}=(y_{1},\ldots,y_{k})^{\prime}$ the corresponding $k-$vector of
inflation rates. Whatever other variables are in the VAR model
${\mathcal{M}}_{j},$ our interest focuses on the implied
$p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})$ required for BPDS eqn. (1).
This conditional predictive is used in decision analysis with the same utility
function for each model, namely
$u_{j}(\mathbf{y},\mathbf{x})=U(\mathbf{y},\mathbf{x})$ given by

$U(\mathbf{y},\mathbf{x})=-\sum_{h=1{:}k}\\{\rho^{k-h}(y_{h}-y^{*})^{2}+(x_{h}-x_{h-1})^{2}\\}$
(5)

where $\rho\in(0,1)$ is a discount factor. This is a conventional quadratic
function that reflects the dual goals of inflation rate targeting and interest
rate smoothing over the next $k=8$ periods. The $\mathbf{y}$ terms relates an
inflation targeting mandate of $y^{*}=2\%$ over the longer run, while the
$\mathbf{x}$ terms encourage relatively constrained changes in quarter-to-
quarter interest rates (the latter being a “don’t rock the boat”
consideration, as large swings in interest rates can/will have otherwise
unduly effects on the macro-economic system). The terms involving the discount
factor $\rho$ represent the fact that monetary policy works with a lag, so it
is desirable to less heavily penalize deviations from the target at shorter
horizons $h$. With our two-year horizon $(k=8$ quarters) our example analysis
below adopts $\rho=0.95$. The model-specific optimal decision vector
$\mathbf{x}_{j}$ then maximizes
$E_{p_{j}}[U(\mathbf{y},\mathbf{x})|\mathbf{x}]$ over $\mathbf{x}.$ Again,
this analysis is repeated each quarter over the period of time series of
interest, producing rolling updates of the “currently optimal” projections for
interest rates over the coming 8 quarters.

###  BPDS Model Specification

The BPDS framework requires specification by the decision maker of a relevant
class of baseline pdfs $p_{0}(\mathbf{y}|\mathbf{x}),$ the model-specific
vector score functions $\mathbf{s}_{j}(\mathbf{y},\mathbf{x}),$ the initial
BPDS model probabilities $\pi_{j}(\mathbf{x})$ as functions of candidate
decisions $\mathbf{x},$ and the target expected scores
$\mathbf{m}_{f}(\mathbf{x})$ at any $\mathbf{x}$. These are discussed in turn.
In addition to customizing BPDS to the specific application, this section
highlights a number of methodological developments relevant to other
applications, particularly in: (a) the linkages of the $\pi_{j}(\mathbf{x})$
to $\mathbf{x}$ that are relevant more generally when $\mathbf{x}$ is both an
outcome to be forecast as well as a putative decision variable (highlighted in
Section 3.2.3 below); and (b) the relevance of dependence structure among the
elements of the vector score under the initial distribution
$p(\mathbf{y},{\mathcal{M}}_{j})$ (highlighted in Section 3.2.4 below).

####  Baseline Distribution

To complete the main BPDS pdf in eqn. (1) requires the baseline
$p_{0}(\mathbf{y}|\mathbf{x}).$ This is taken as a multivariate T distribution
with 10 degrees of freedom, using the location from the initial mixture
$p(\mathbf{y}|\mathbf{x})$ ignoring the baseline (i.e., with
$\pi_{0}(\mathbf{x})=0)$ and corresponding variance of that mixture inflated
by 4. This defines a relevant, tractable ${\mathcal{M}}_{0}$ that can capture
outcomes $\mathbf{y}$ that the two VAR models are not predicting well for any
$\mathbf{x}$ under consideration, and signal that to the decision maker.

####  BPDS Score Functions

The considerations of inflation targeting and interest rate smoothing
reflected in the model-specific decision analysis in Section 3.1 are relevant
to the choices of BPDS score functions. Our example takes
$\mathbf{s}_{j}(\mathbf{y},\mathbf{x})=[s_{j1}(y_{1},x_{1}),\ldots,s_{jk}(y_{k},x_{k})]^{\prime}$
with elements

$s_{jh}(y_{h},x_{h})=\exp{\\{-(y_{h}-y^{*})^{2}/(2z_{y}^{2})}\\}+\exp{\\{-(x_{h}-x_{h-1})^{2}/(2z_{x}^{2})}\\},\quad
h=1{:}k,$ (6)

where $y^{*}=2\%$ is the inflation target and $z_{y}$ and $z_{x}$ are score
bandwidth parameters. This defines a class of bounded score functions, always
relevant in decision analysis and here ensuring that the entropically tilted
BPDS pdf of eqn. (3) is always integrable. The score bandwidths are set so
that a certain deviation $d_{y}=(y-y^{*})^{2}$ has a score of
${\bm{\varepsilon}}$; given a choice of ${\bm{\varepsilon}}$ we set
$z_{y}=d_{y}/\sqrt{-2\log({\bm{\varepsilon}})}$. Similar considerations apply
to choosing $z_{x}$. Our analyses use ${\bm{\varepsilon}}=0.4$, $d_{y}=2$, and
$d_{x}=1$ to ensure the score function is dispersed enough to accommodate
modest changes in the Federal Funds rate while is more lenient in deviations
from the inflation target. Obvious modifications could incorporate horizon
$h-$specific inflation targets and differentially weight the two exponential
terms, but this form suffices for our main goals in this paper. Note also
that, if inflation deviations from target and interest rate changes are
“small”, then $s_{jh}(y_{h},x_{h})$ is approximately quadratic in
$|y_{h}-y^{*}|$ and $|x_{h}-x_{h-1}|$ for all $h,$ a perhaps more familiar
utility form.

####  Initial Model Probabilities

For clarity in the presentation in this section, we now make explicit the
dependency on time, so that the ingredients of the full BPDS predictive pdf in
eqn. (1)– with the exponential form of the calibration function of eqn. (4)–
are now indexed by current time $t$, i.e.,

$f_{t}(\mathbf{y}_{t}|\mathbf{x}_{t})\propto\sum_{j=0{:}J}\pi_{tj}(\mathbf{x}_{t}){\textrm{e}}^{{\bm{\tau}}_{t}(\mathbf{x}_{t})^{\prime}\mathbf{s}_{tj}(\mathbf{y}_{t},\mathbf{x}_{tj})}p_{tj}(\mathbf{y}_{t}|\mathbf{x}_{t},{\mathcal{M}}_{j}).$

Bayesian model weighting based on historical predictive performance with
respect to define forecast goals, as developed in (Lavine et al., 2021),
provides the starting point for specification of the
$\pi_{tj}(\mathbf{x}_{t}).$ The general form adopted is

$\pi_{tj}(\mathbf{x}_{t})\propto\pi_{tj}p_{tj}(\mathbf{x}_{t}|{\mathcal{M}}_{j}),\qquad
j=0{:}J,$ (7)

subject to summing to 1 over $j=0{:}J$ and with ingredients as follows.

#### Initial model probabilities

Traditional Bayesian analysis (e.g. West and Harrison, 1997, chapter 12)
defines the starting point. Here the time $t$ initial model probabilities are
based on standard sequential Bayesian updating from those at $t-1$: That is,
$\pi_{tj}\propto\pi_{t-1,j}p_{tj}(\mathbf{z}_{t-1,j}|{\mathcal{M}}_{j})$ where
the “marginal model likelihood” term
$p_{tj}(\mathbf{z}_{t-1,j}|{\mathcal{M}}_{j})$ is the value of the 1-step
ahead predictive pdf under ${\mathcal{M}}_{j}$ at the observed values of the
last period outcomes $\mathbf{z}_{t-1,j}$ under that model. In our applied
setting, this $\mathbf{z}_{t-1,j}$ includes time $t-1$ outcome values of
inflation $(y)$, interest rate $(x)$, and other economic indicators modelled
and forecast in ${\mathcal{M}}_{j}$ in our setting. In general, these can
differ across models, but in consideration for the initial weights we restrict
to variables common across models.

Then, BPDS allows the decision maker freedom to make alternative choices of
the $\pi_{tj},$ and the goal and decision focus recommends modification of the
standard BMA choice. BMA, after all, only reweights models based on 1-step
ahead predictive accuracy. Hence we adopt two modifications, based on recent
literature consonant with the goal foci.

First, we use simple power discounting of historically accrued support across
models, in which the time $t-1$ to time $t$ evolution is reflected in
$\pi_{tj}\propto\pi_{t-1,j}^{\gamma}p_{tj}(\mathbf{z}_{t-1,j}|{\mathcal{M}}_{j})$
where $\gamma$ is a discount factor in $(0,1]$, closer to 1 for most
applications. This acts to discount historically accrued support for model $j$
at a per-time unit discount rate $\gamma$ prior to updating by the time $t-1$
information. Going back at least to Smith (1979) and then, in a formal dynamic
model uncertainty context, West and Harrison (1989, chapter 12, p.445), power-
discounting has been shown to be of value in empirical studies in implicitly
allowing for time-variation in the predictive relevance of different models
(e.g. Raftery et al., 2010; Koop and Korobilis, 2013; Zhao et al., 2016). Our
case study below uses $\gamma=0.95$.

Second, reflecting the foci on specific predictive and decision goals, initial
model probabilities should also generally be modified based on the recent
relative performance of models with respect to the defined goals. This is the
premise underlying the specific variants of BPS in the setting of adaptive
variable selection, or BPS-AVS, in Lavine et al. (2021), and related
developments in Loaiza-Maya et al. (2021), for example. This leads to the
immediate BPDS extension of these prior approaches in which the above
reasoning is extended to define

$\pi_{tj}\propto\pi_{t-1,j}^{\gamma}p_{tj}(\mathbf{z}_{t-1,j}|{\mathcal{M}}_{j})\textrm{e}^{{\bm{\tau}}_{t-1}(\mathbf{x}_{t-1})^{\prime}\mathbf{s}_{t-1,j}(\mathbf{y}_{t-1},\mathbf{x}_{t-1,j})}.$

Here the discounted Bayesian model probabilities are further updated with AVS-
style weights based on the the realized BPDS calibration function based on
relative model scores based on the actual decision outcomes at the last time
period. As a result, models are initially and naturally reweighted based on
both predictive and decision outcome performance at the last time period.

In our applied setting this means that, as a result, models achieving “good”
recent trajectories of interest rate smoothness, as well as relatively
accurate forecasting performance of realized inflation outcomes, will be
rewarded with higher initial BPDS model probabilities in looking forward to
the next time point. And we note that the specification here can cut back to
define special cases including BPS-AVS (by setting
${\bm{\tau}}_{t-1}(\mathbf{x}_{t-1})=\mathbf{0})$, and within that to
traditional BMA (by setting $\gamma=1$) for comparisons.

Finally, the inclusion in BPDS of the baseline model and its forecast
densities leads to a modification of these initial model probabilities to
provide a non-zero value $\pi_{t0}$ for the baseline. We choose a fixed
probability– in our analysis $\pi_{t0}=0.1$ at each time point $t$– and simply
renormalize the $\pi_{tj}$ above over $j=1{:}J$ accordingly.

#### Informative Conditioning on $\mathbf{x}_{t}$

As noted earlier, in our setting the future values of decision variables are
also considered outcomes predicted under the models. The models each forecast
the future evolution of interest rates as part of the complex, dynamic
macroeconomic system, whereas for decisions we must condition on
$\mathbf{x}_{t}$. This is reflected in the conditional (on $\mathbf{x}_{t}$)
distributions $p_{tj}(\mathbf{y}_{t}|\mathbf{x}_{t},{\mathcal{M}}_{j})$ in
BPDS where $\mathbf{x}_{t}$ is treated as known. The theoretical implication
for the BPDS model probabilities is the term
$p_{tj}(\mathbf{x}_{t}|{\mathcal{M}}_{j})$ in eqn. (7)– this is the value of
the current marginal predictive pdf of the vector $\mathbf{x}_{t}$ under
${\mathcal{M}}_{j}.$ Assuming the prior (to time $t$) probabilities $\pi_{tj}$
are specified, this form arises directly via Bayes’ theorem; the act of
conditioning on $\mathbf{x}_{t}$ is informative, and the implied update is,
simply by Bayes’ theorem, that in eqn. (7). Critically, this implies that
candidate decision values that are not well-supported under the joint
distribution of a model are down-weighted. Conversely, at any candidate
decision vector $\mathbf{x}_{t},$ models that are more predictively supportive
of the decision $\mathbf{x}_{t}$ will be relatively rewarded with higher
values of resulting $\pi_{tj}(\mathbf{x}_{t}).$

In other applications of BPDS the decision variables may be exogenous, i.e.,
control variables that are to be chosen by the decision maker but that are not
forecast jointly with $\mathbf{y}_{t}$ in the set of models. In such cases, it
will be common to assume that the external choice of $\mathbf{x}_{t}$ is not
informative, and then eqn. (7) results in decision-independent BPDS
probabilities $\pi_{tj}(\mathbf{x}_{t})=\pi_{tj}$ based only on historical
data and information.

####  BPDS Target Scores

The BPDS target expected score
$\mathbf{m}_{f}(\mathbf{x})=E_{f}[\mathbf{s}(\mathbf{y},\mathbf{x})]$
represents a desired improvement over the initial expected score
$\mathbf{m}_{p}(\mathbf{x})=E_{p}[\mathbf{s}(\mathbf{y},\mathbf{x})]$. In the
multi-objective case the resulting ${\bm{\tau}}(\mathbf{x})$ that defines
$f(\mathbf{y}|\mathbf{x})$ to satisfy this target expectation is sensitive to
both the relative scales and dependence of elements of
$\mathbf{s}(\mathbf{y},\mathbf{x})$ under the initial mixture $\mathbf{y}\sim
p(\mathbf{y}|\mathbf{x})$ at any candidate decision $\mathbf{x}.$ As functions
of $\mathbf{y}$ the elements of the random score vector
$\mathbf{s}(\mathbf{y},\mathbf{x})$ can be strongly correlated, leading to
challenges in specifying relevant targets. This can also complicate the
calculation of the implied BPDS tilting vector ${\bm{\tau}}(\mathbf{x})$– the
vector that is needed to satisfy
$\mathbf{m}_{f}(\mathbf{x})=E_{f}[\mathbf{s}(\mathbf{y},\mathbf{x})]$ under
the BPDS density of eqn. (1). We address this by explicitly recognizing score
dependencies and defining an approach that explicitly incorporates dependence.

Some theoretical intuition is gained by considered cases of “small
perturbations” in which
$\mathbf{m}_{f}(\mathbf{x})-\mathbf{m}_{p}(\mathbf{x})$ has small elements. In
this setting, entropic tilting theory in Tallman and West (2022) yields the
second-order approximation
${\bm{\tau}}(\mathbf{x})\approx\mathbf{V}_{p}(\mathbf{x})^{-1}(\mathbf{m}_{f}(\mathbf{x})-\mathbf{m}_{p}(\mathbf{x}))$
where $\mathbf{V}_{p}(\mathbf{x})$ is the variance matrix of
$\mathbf{s}(\mathbf{y},\mathbf{x})$ under the initial mixture
$p(\mathbf{y}|\mathbf{x}).$ This shows that the implied tilting vector will be
very sensitive to the initial score scales and dependencies as reflected in
$\mathbf{V}_{p}(\mathbf{x}),$ and suggests a prime focus on a standardized
score scale. That is, define $\mathbf{C}_{p}(\mathbf{x})$ as the scaled
eigenvector matrix such that
$\mathbf{V}_{p}(\mathbf{x})=\mathbf{C}_{p}(\mathbf{x})\mathbf{C}_{p}(\mathbf{x})^{\prime}$
and set the target score using
$\mathbf{m}_{f}(\mathbf{x})=\mathbf{m}_{p}(\mathbf{x})+\mathbf{C}_{p}(\mathbf{x}){\bm{\epsilon}}(\mathbf{x})$
for a specified standardized expected score vector
${\bm{\epsilon}}(\mathbf{x})$. The usual convention is taken in which the
eigenvector columns of $\mathbf{C}_{p}(\mathbf{x})$ are ordered according to
decreasingly values of the corresponding eigenvalues, so that the first column
is “dominant”, and so forth. This provides insights into how to practically
define target scores related to the absolute standardized scale. As examples
of the two extremes, taking
${\bm{\epsilon}}(\mathbf{x})=\epsilon(\mathbf{x})\mathbf{1}$ for some scalar
$\epsilon(\mathbf{x})$ represents targets deviating from the initial expected
score in equal amounts of $\epsilon(\mathbf{x})$ along each of the
standardized eigendimensions. At the other extreme, and most relevant when
there are strong score dependencies, taking
${\bm{\epsilon}}(\mathbf{x})=(\epsilon(\mathbf{x}),0,\ldots,0)^{\prime}$
defines the resulting target $\mathbf{m}_{f}(\mathbf{x})$ based on the major,
dominant eigen-dimension alone. The latter is a starting point in general and
is taken to define our BPDS case study below. In that setting, we choose
$\epsilon(\mathbf{x})$ such that
$\min\\{(\mathbf{m}_{f}(\mathbf{x})/\mathbf{m}_{p}(\mathbf{x}))\\}=0.75$ to
define the maximum expected improved score in any dimension.222Additionally,
due to the arbitrariness of signs of eigenvectors, we apply a $\pm 1$
multiplier to the first column of $\mathbf{C}_{p}(\mathbf{x})$ so that the sum
of elements are positive, ensuring the target score improves upon
$\mathbf{m}_{p}(\mathbf{x})$. It is obviously straightforward to extend this
methodology to define target scores impacted by higher eigen-dimensions,
though that is left for future applications.

###  BPDS Implementation and Optimal Decisions

The final step in BPDS is to couple the decision maker’s utility function with
the BPDS predictive equations (1,2) to define the optimal decisions from the
model synthesis. The decision maker can adopt any utility function but an
initial neutral analysis will be based on using the same form as usual in the
model-specific decisions, the function $U(\mathbf{y},\mathbf{x})$ of eqn. (5).
This is used in the example analysis to follow, with the aim of computing
$\mathbf{x}$ to maximize the implied expected utility function
$\bar{U}(\mathbf{x}).$ In the case study analysis we compare decisions
recommended by BPDS to those from each of the models and to a traditional BMA-
based analysis. On the latter, the BMA mixture uses model weights proportional
to the marginal likelihoods of the data which is common to all of the models
(which includes inflation, interest rate, and GDP) under each
${\mathcal{M}}_{j}$. The BMA mixture naturally involves only the
$p_{j}(\mathbf{y}|\mathbf{x})$ with no BPS/BPDS outcome-dependence, no notion
of a baseline model to address model-set incompleteness, and no regard for the
decision-focused use of the models. The foundational BPDS framework
theoretically allows for these critical considerations as fundamental to the
broader subjective Bayesian decision-analytic and goal-focused approach. Then,
technically, BMA arises as a special case of the BPDS analysis as earlier
discussed throughout Section 3.

The computation of BPDS involves two key components. First, is the overall
optimization over $\mathbf{x}$ which explores potential BPDS decisions and
finds the optimizing vector $\mathbf{x}$. This requires an “outer loop”
numerical optimization to explore $\mathbf{x}$ space. In our study, this is
performed using a trust region method, namely Powell’s Derivative Free
Optimization Solvers (PDFO–Ragonneau and Zhang, 2023). Due to the possibility
of multi-modality in $\bar{U}(\mathbf{x})$, the optimization is run repeatedly
(in parallel) from multiple starting values. In some periods over time, we do
find evidence of multi-modality, so repeat starts of the optimization routine
are mandated. Second, within each evaluation of a potential BPDS decision, it
is necessary to compute the tilting vector ${\bm{\tau}}(\mathbf{x})$ given the
constraint
$E_{f}[\mathbf{s}(\mathbf{y},\mathbf{x})]=\mathbf{m}_{f}(\mathbf{x})$ for a
target expected score $\mathbf{m}_{f}(\mathbf{x}).$. The theoretical basis of
this is an implicit equation that is solved via standard, generic numerical
optimization methods. Relevant details follow Tallman and West (2023, section
4.4) and are summarized in our Appendix A.

The BPDS forecast distributions are evaluated using importance sampling. At
any given $\mathbf{x},$ the BPDS predictive distribution in eqn. (1) is
simulated by sampling from the
$p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})$ in proportions defined by the
BPDS probabilities $\pi_{j}(\mathbf{x})$; then the resulting importance
sampling weights are simply proportional to the realized values of
$\alpha_{j}(\mathbf{y}|\mathbf{x}).$ This provides for efficient computation
as well as access to traditional methods and metrics– such as importance
sampling effective sample sizes (ESS, e.g. Gruber and West, 2016, 2017, in
related contexts)– to monitor and evaluate the quality of the resulting Monte
Carlo approximations to resulting predictive expectations. Note that this can
deliver such metrics to assess “concordance” between the initial densities
$p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})$ and their corresponding BPDS-
tilted versions $f_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})$ in eqn. (2),
as well as that of the initial mixture $p(\mathbf{y}|\mathbf{x})$ and the
resulting $f(\mathbf{y}|\mathbf{x}).$ More aggressive BPDS target scores will
generally lead to lower concordance, and choices can be partly guided by such
empirical evaluations.

##  Case Study

###  Overview

The analysis and all empirical results proceed sequentially on an expanding
window of data beginning in 1992Q2. Our summaries begin with a comparison of
the decisions recommended by BPDS to those suggested by BMA. This is followed
by a discussion of the individual models and how they are combined by BPDS and
BMA. Additional discussion highlights some operational BPDS details to provide
further insights into the resulting decision outcomes.

###  Optimal Decisions

Figure 1 shows the actual policy rate each quarter along with the $1{-}8$
quarters ahead policy recommendations that would have been made by BPDS and
BMA. In using the shadow rate, the zero lower bound is not in effect and
negative values for the policy rate are possible. Recommendations for negative
values for the policy rate are not to be taken literally as advising cuts to a
negative Federal Funds rate, but rather a suggestion to undertake other forms
of monetary easing that would be expected to proxy such cuts.

Since 2014 the optimal policy paths recommended by the two approaches are
generally similar, though there are notable differences prior to that time.
Some specific periods of interest are now highlighted.

#### 2014 to the Present

During this period, BMA and BPDS provide similar recommendations that are
often quite different from the actual policy rate. For almost all of these
times, the policy recommendations are to cut interest rates whereas (apart
from 2019-2021) the actual policy rate increased. Some differences do arise
between BMA and BPDS, for example, during the post-COVID inflation BPDS
recommends a higher rate path. BPDS is closer to the decision actually made by
the Federal Reserve, although according to BPDS rate cuts should have begun
coming down by now.

#### The Financial Crisis and Subsequent Recession

It is during this period that the differences between BPDS and BMA are most
acute. The actual policy rate fell slowly during this period. BPDS recommends
rate cuts as well, initially at a more rapid rate than what actually occurred,
but as of 2010, its recommendations are similar to the ones the policy makers
actually made. In contrast, BMA recommends huge cuts to the policy rate right
at the start of the financial crisis but subsequently consistently argues for
rate increases.

#### The First Years of the 21st Century

In the period from 2003 through the beginning of the financial crisis, the
actual policy rate was gradually increasing. In this period, BMA consistently
recommended rapid rate increases. In contrast, BPDS recommendations were
generally similar to what actually transpired, apart from at the beginning of
this period where the advice was to raise the policy rate more slowly than
occurred.

#### The 1990s

During this period, the pattern is more mixed. Optimal policy recommendations
generated under each of BMA and BPDS often differed from actual decisions,
with no consistent pattern; at times the recommended rates were higher than
the actual policy rate, and at other times they were lower.

Figure 1: Recursively calculated policy decisions

A general pattern, one that occurs throughout the sample period, is that BMA
and BPDS typically recommend larger changes in policy rates than were actually
implemented by policy makers. Part of this is presumably due to differences
between the policy makers utility function and those used in our analyses.
Other possible explanations are that policy makers can affect expectations
through their communications which is a channel not captured in the model or
their models use a much steeper Phillips Curve. Also, we focus only on
inflation up to 2 years ahead, without considering the possibility of an over-
or under-shoot of inflation after 8 quarters while policy makers would
generally desire longer-term targets sustainably.

###  Trajectories of BPDS and BMA Predictive Densities

Figures 2–4 shed light on these patterns. These images represent the time
trajectories of predictive densities of inflation at each relevant horizon,
using BPDS (Fig. 2) and BMA (Fig. 3), as well as their differences (Fig. 4).
These indicate that the BPDS mixtures are less dispersed than the BMA mixture
for much of the sample period, i.e., BMA predictive distributions are
relatively more heavy-tailed, especially at longer horizons. This is partly
due to the BPDS score function emphasizing that the policy maker wants to
avoid extreme inflation outcomes, and also accounts for why BMA often tells
policy makers to make larger changes to the policy rate than BPDS as discussed
in Section 4.2. The differences between BPDS and BMA become larger at longer
forecast horizons. Medium and long-term macroeconomic forecasting is difficult
which leads to standard methods such as BMA producing fairly dispersed
predictive densities at longer horizons. BPDS, on the other hand, is reducing
this effect, which dampens the BPDS optimal decisions and reduces predictive
uncertainty relative to BMA. Then, differences between BPDS and BMA forecast
densities are reduced after the financial crisis, which helps account for why
their policy recommendations are similar in the last decade of the sample.

Figure 2: BPDS forecast densities of inflation. The frames represent 1–8
quarter ahead forecasts, reading along the rows from top-left to bottom-right.
The colors represent probabilities with the shading from blue, being lower
probability, to red being higher probability  Figure 3: BMA forecast densities
of inflation. The frames represent 1–8 quarter ahead forecasts, reading along
the rows from top-left to bottom-right. The colors represent probabilities
with the shading from blue, being lower probability, to red being higher
probability  Figure 4: Difference between BPDS and BMA forecast densities of
inflation. Red-shaded regions have higher probability under BPDS than under
BMA, with blue shading indicating the reverse, and white shading equal
probability

###  Model Probabilities

Figure 5 shows the trajectories of model probabilities under BPDS and BMA.
These are the discounted AVS prior model probabilities $\pi_{tj}$ over time,
the implied initial decision-dependent probabilities
$\pi_{tj}(\mathbf{x}_{t})$ evaluated at the BPDS-optimal decision
$\mathbf{x}_{t}$ at each time, the resulting BPDS probabilities
$\tilde{\pi}_{tj}(\mathbf{x}_{t})$ of eqn. (3) at each time, and the standard
BMA probabilities over time.

Figure 5: Time trajectories of model probabilities. (a) Prior BPDS
probabilities $\pi_{tj}$ based on discounted AVS with a fixed baseline
$\pi_{t0}=0.1;$ (b) BPDS decision-dependent initial probabilities
$\pi_{tj}(\mathbf{x}_{t});$ (c) Implied BPDS weights
$\tilde{\pi}_{tj}(\mathbf{x}_{t});$ (d) BMA probabilities.

Under traditional BMA, the two model probabilities are appreciable until the
financial crisis. After the start of the crisis, the less parsimonious
${\mathcal{M}}_{2}$– that includes additional financial variables– receives
virtually all the weight. In contrast, BPDS weights vary more over time,
allocating most of the weight to the parsimonious ${\mathcal{M}}_{1}$ for much
of the period (i.e. 1997 through 2017), though ${\mathcal{M}}_{2}$ plays more
of a role at both the beginning and end of the sample period. That BPDS
generally favors the more parsimonious ${\mathcal{M}}_{1}$, with less
dispersed forecast distributions, partially accounts for why BPDS often
dampens extreme recommendations made when using BMA.

BPDS probabilities on the over-dispersed ${\mathcal{M}}_{0}$ are generally
small, though with notable increases at two critical periods: the start of the
financial crisis and the start of the COVID-19 pandemic. In such extreme
times, when neither ${\mathcal{M}}_{1}$ or ${\mathcal{M}}_{2}$ forecasts well,
the increased probability on the fall-back ${\mathcal{M}}_{0}$– though small–
provides an indicator of this.

The BPDS prior model probabilities $\pi_{tj}$ based on discounted AVS differ
noticeably from BMA probabilities (except at the very start of the time
period). A big impact then arises from the conditioning on information
generated in the decision space to map these $\pi_{tj}$ to the decision-
dependent weights $\pi_{tj}(\mathbf{x}_{t})$ at the BPDS optimal decisions
$\mathbf{x}_{t}$ at each time. Recall that this mapping theoretically properly
takes into account the likelihood of future, as yet unobserved, interest rate
outcomes; this relevant information is not accounted for in the prior weights
$\pi_{tj}$ and is, of course, absent under BMA. The subsequent map from
initial probabilities $\pi_{tj}(\mathbf{x}_{t})$ to the BPDS weights
$\tilde{\pi}_{tj}(\mathbf{x}_{t})$ is wholly based on the impact of the
entropic tilting towards “more favorable” decisions, in expectation. We see
that the impact is rather small over time, and this is to be expected: the
BPDS analysis uses “small” perturbations of the initial mixture based on
target expected scores that are only modest increases over those under the
initial mixture. We expect to see slight tilting towards models that are
expected to do well, but not large changes relative to the initial
probabilities.

###  Additional Insights from BPDS Results

Time trajectories of the evaluated tilting vectors
${\bm{\tau}}_{t}(\mathbf{x}_{t})$, evaluated at the optimal decisions
$\mathbf{x}_{t}$, are shown in Figure 7. The values generally tend to increase
with horizon $h$, thus attaching more weight to longer forecasting horizons.
This is partly to be expected due to the higher uncertainties at longer
forecast horizons.

Figure 6: Trajectories of the 8 elements of the evaluated BPDS tilting vector
${\bm{\tau}}_{t}(\mathbf{x}_{t})$ at the optimized $\mathbf{x}_{t}$ at each
quarter

Figure 7: Trajectories of the effective sample size (ESS) metrics for
individual models and for the BPDS initial mixture

Figure 7 plots trajectories of several effective sample size (ESS) measures
arising from the importance sampling to simulate BPDS predictive distributions
as discussed in Section 3.3. This provides a read-out of the extent of tilting
the initial mixture $p(\mathbf{y}|\mathbf{x})$ to the BPDS mixture
$f(\mathbf{y}|\mathbf{x})$ as well as that for tilting each of the individual
model pdfs from $p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})$ to
$f_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})$ (again, time-indexed and
updated throughout the time series). Until the COVID recession, the ESS of the
initial mixture is stable between 90-95% suggesting only a small amount of
tilting, as desired. The COVID recession is a period of rapid change as
expected, as we see large changes in the initial weights and larger values of
${\bm{\tau}}$ required to achieve the desired target. The low value of ESS
indicates that at that point the target expected scores are unrealistic given
the then current state of the economy. However, the resulting decisions during
this time period appear to be rather sensible, meaning we do not need to be
too concerned about the low ESS which can, in any case, be redressed by simply
increasing the overall Monte Carlo sample size accordingly. The ESS values of
individual models are generally lower than that of the overall mixture and
somewhat more volatile. One nice point is that, even when one of the models
seems to suffer a low ESS, the BPDS mixture ESS is generally maintained at
higher values. This indicates that BPDS is able to strike a balance in
weighting expected versus historical performance of models on both predictive
and decision outcomes.

Finally, Figure 8 compares the realized trajectories of expected utilities
under BPDS and BMA. Each uses the same utility function to define the final
optimal policy path decision, so these are directly comparable, and the
comparison is relevant in terms of the setting of forward, sequential
decisions where a change to much lower values at any time point should signal
concern to the decision-maker. BPDS is designed to target an expected utility
higher than that of the initial mixture, but whether it achieves a higher
expected utility than BMA– which has different initial probabilities and lacks
outcome-dependent weighting– is a question for empirical study. In this
example, as illustrated in the figure, BPDS utility does exceed that of BMA in
virtually every period. After the financial crisis, the two are similar,
consistent with the earlier finding that they typically produced similar
decisions during this time period. However, before the financial crisis, there
were several periods during which the BPDS expected utilities were
substantially higher than those of BMA. These correspond to times where we see
more differences between optimal policy path recommendations, and within which
there are some periods of greater concordance between BPDS and actual policy
decisions as well as more constrained (i.e., less extreme) recommended
decisions under BPDS relative to BMA.

Figure 8: Trajectories of expected utilities comparing BPDS with BMA

##  Summary Comments

BPDS is the formal, foundational Bayesian framework that extends traditional
Bayesian model uncertainty analysis to address explicit use of model-specific
decision outcomes as well as purely predictive performance in model comparison
and combination. This paper has adapted the BPDS foundations to define implied
methodology in formulating macro-economic decision-making when faced with
multiple objectives and multiple outcomes of interest in the monetary policy
setting.

Earlier applications of BPDS have focused mainly on financial portfolio
forecasting and decisions (Tallman and West, 2023, section 6; Tallman and
West, 2024), a setting in which forecasting models do not (generally) depend
on the decisions of interest, while utility functions may and often do depend
on the models and their predictions. In contrast, the setting of monetary
policy analysis is one in which the dependence of models and their forecasts
on the decision variables (policy instruments) is simply fundamental. It is
also a setting in which the decision variables are treated simultaneously as
outcomes. The future paths of central bank interest rates, for example, are
modelled as time series outcomes along with other economic and financial
indicators in VAR models. This then leads to conditioning on decision
variables to define predictions of other indicators, with consequent
implications for relative model weights in the model uncertainty setting. This
latter point is critical as it then leads to relatively up- or down-weighting
a model based on how well-supported a particular candidate decision is under
its predictions; to our knowledge, this is the first time this central
question has been formally, statistically addressed. These central features of
predictive decision-making in monetary policy contexts are addressed with
extensions and customization of the existing theory of BPDS.

The BPDS perspective– of integrating historical and expected decision outcomes
with focused aspects of statistical predictive performance into relative model
weightings– is new to the policy arena. We argue for this perspective since
policy makers are primarily interested in utilizing sets of models for the
eventual policy decisions. Pure forecasting exercises– and evaluation and
combinations of models for prediction per se– is, of course, of parallel
interest and importance. We emphasize that BPDS also involves addressing
predictive performance on specific, defined outcomes of interest. But most
importantly, by putting the spotlight on decision-making, we gain additional
insights into policy making that are not possible in exercises that focus
solely on predictive performance.

In a recursive real-time decision-making exercise, we find substantial
differences at various periods of time between the policy recommendations of
BPDS and the traditional Bayesian model averaging approach, though good
concordance at other times. When recommended policy decisions differ between
the approaches, in most cases the BPDS policy paths are more intuitively
sensible and less extreme than under BMA, and more consistent with the actual
decisions made by the policy makers at the time. The case study presented
investigates and interprets aspects of BPDS– in terms of differential model
weights based on historical information alone and then updated based on
identified optimal decisions, with consequent insights into how the
differences relative to standard BMA arise and are exploited. This case study
is a first step towards broader development and evaluation of BPDS in a
setting with larger numbers of econometric models. The parallel next steps
will naturally include BPDS for scenario forecasting– analyses in
collaboration with policy-makers that explore and aim to understand the
sensitivity of model-based recommendations relative to chosen potential
economic scenarios.

## Acknowledgements

Research of Emily Tallman was partially supported by the U.S. National Science
Foundation through NSF Graduate Research Fellowship Program grant DGE 2139754.
Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the authors and do not necessarily reflect the views of
the National Science Foundation. Further, the views expressed in this paper
are solely those of the authors and may differ from the official views of the
Bank of Canada; no responsibility for the views expressed in this paper should
be attributed to the Bank of Canada.

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## Appendix

### Appendix A: Evaluation of Tilting Vectors

Evaluation of tilting vectors ${\bm{\tau}}(\mathbf{x})$ involves Monte Carlo
integration and numerical optimization, briefly summarized here. This follows
and customizes the general results in Tallman and West (2023, section 4.4).

The vector ${\bm{\tau}}(\mathbf{x})$ is implicitly defined to satisfy
$E_{f}[\mathbf{s}_{j}(\mathbf{y},\mathbf{x}_{j})|\mathbf{x}]=\mathbf{m}_{f}$
where $\mathbf{m}_{f}$ is the chosen target score and expectation is with
respect to

$f(\mathbf{y},{\mathcal{M}}_{j})=\tilde{\pi}_{j}(\mathbf{x})f_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})=k(\mathbf{x})\pi_{j}(\mathbf{x})\alpha_{j}(\mathbf{y}|\mathbf{x})p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j}),\quad
j=0{:}J,$

with components given in eqn. (3). With $\alpha_{j}(\mathbf{y}|\mathbf{x})$ of
the exponential form in eqn. (4), numerical optimization aims to solve

$k(\mathbf{x})\sum_{j=0{:}J}\int_{\mathbf{y}}\pi_{j}(\mathbf{x})\textrm{e}^{{\bm{\tau}}(\mathbf{x})^{\prime}\mathbf{s}_{j}(\mathbf{y},\mathbf{x}_{j})}p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})d\mathbf{y}-\mathbf{m}_{f}=\mathbf{0}$
(8)

for ${\bm{\tau}}(\mathbf{x}).$ The normalizing term $k(\mathbf{x})$ is defined
via $k(\mathbf{x})^{-1}=\sum_{j=0{:}J}\pi_{j}(\mathbf{x})a_{j}(\mathbf{x})$
where

$a_{j}(\mathbf{x})=\int_{\mathbf{y}}\textrm{e}^{{\bm{\tau}}(\mathbf{x})^{\prime}\mathbf{s}_{j}(\mathbf{y},\mathbf{x}_{j})}p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})d\mathbf{y},\quad
j=0{:}J.$ (9)

At any given $\mathbf{x},$ the integrals in equations (8,9) are approximated
using Monte Carlo integration based on random samples from each of the
$p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j}).$ This gives the ingredients
for numerical search over ${\bm{\tau}}(\mathbf{x})$ using derivative-free
optimization methods such as those used in the current paper.

For completeness, we note that the same approach applies where gradient-based
optimization is considered. Newton-Raphson and allied methods will require the
vector derivative $\dot{a}_{j}(\mathbf{x})$ of $a_{j}(\mathbf{x})$ with
respect to ${\bm{\tau}}(\mathbf{x}),$ and may also require the matrix of
second derivatives $\ddot{a}_{j}(\mathbf{x}).$ These are given by

$\displaystyle\dot{a}_{j}(\mathbf{x})$
$\displaystyle=\int_{\mathbf{y}}\mathbf{s}_{j}(\mathbf{y},\mathbf{x}_{j})\textrm{e}^{{\bm{\tau}}(\mathbf{x})^{\prime}\mathbf{s}_{j}(\mathbf{y},\mathbf{x}_{j})}p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})d\mathbf{y},$
$\displaystyle\ddot{a}_{j}(\mathbf{x})$
$\displaystyle=\int_{\mathbf{y}}\mathbf{s}_{j}(\mathbf{y},\mathbf{x}_{j})\mathbf{s}_{j}(\mathbf{y},\mathbf{x}_{j})^{\prime}\textrm{e}^{{\bm{\tau}}(\mathbf{x})^{\prime}\mathbf{s}_{j}(\mathbf{y},\mathbf{x}_{j})}p_{j}(\mathbf{y}|\mathbf{x},{\mathcal{M}}_{j})d\mathbf{y}.$

Hence the Monte Carlo integration approach used to evaluate the
$a_{j}(\mathbf{y})$ also delivers approximations to these derivatives.

### Appendix B: Conditional Forecasts

This section discusses the construction of conditional forecasting in VAR
models, following Chan et al. (2023) and discussion in Antolin-Diaz et al.
(2021).

#### Observational Constraints

The general theory for constraining on exact or uncertain linear constraints
in multivariate normal forecast distributions (e.g. West and Harrison, 1997,
sect. 16.3.2) underlies these results used in that paper and here in the
specific case of VAR models.

The $n\times 1$ vector time series
$\mathbf{y}_{t}=(y_{t1},\dots,y_{tn})^{\prime}$ follows a VAR$(p)$ in
structural form

$\mathbf{A}_{0}\mathbf{y}_{t}=\textrm{a}+\mathbf{A}_{1}y_{t-1}+\dots+\mathbf{A}_{p}y_{t-p}+{\bm{\varepsilon}}_{t},\quad{\bm{\varepsilon}}_{t}\sim
N(\mathbf{0}_{n},\mathbf{I}_{n})$ (10)

with the following terms: a is a $n\times 1$ vector of intercepts,
$\mathbf{A}_{1},\dots,\mathbf{A}_{p}$ are the $n\times n$ VAR coefficient
matrices, $\mathbf{A}_{0}$ is a full-rank contemporaneous impact matrix,
$\mathbf{0}_{n}$ is a $n\times 1$ zero vector and $\mathbf{I}_{n}$ is the
$n\times n$ identity matrix.

Given the history of observations
$\\{\mathbf{y}_{1-p},\dots,\mathbf{y}_{T}\\}$ at any time $T$, unconditional
forecasting of the $n$ variables for the next $h$ periods, namely the
$nh\times 1$ vector
$\mathbf{y}_{T+1:T+h}=(\mathbf{y}^{\prime}_{T+1},\dots,\mathbf{y}^{\prime}_{T+h})^{\prime}$,
is based on the implied form

$\mathbf{H}\mathbf{y}_{T+1:T+h}=\textrm{c}+{\bm{\varepsilon}}_{T+1:T+h},\quad{\bm{\varepsilon}}_{T+1:T+h}\sim
N(\mathbf{0}_{nh},\mathbf{I}_{nh})$ (11)

where

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

with $\mathbf{0}_{n\times n}$ the $n\times n$ zero matrix. Since
$\mathbf{A}_{0}$ is full rank and $|\mathbf{H}|=|\mathbf{A}_{0}|^{h}\neq 0$,
then $\mathbf{H}$ is non-singular and

$\mathbf{y}_{T+1:T+h}\sim N(\mathbf{m},\mathbf{M})\quad\textrm{with \
moments}\quad\mathbf{m}=\mathbf{H}^{-1}\textrm{c}\ \ \textrm{and}\ \
\mathbf{M}=(\mathbf{H}^{\prime}\mathbf{H})^{-1}.$ (12)

Now consider a set of $r<nh$ linear restrictions on the path of future
observables, namely

$\mathbf{R}\mathbf{y}_{T+1:T+h}\sim N(\mathbf{r},{\bm{\Omega}})$ (13)

where $\mathbf{R}$ is a chosen $r\times nh$ constant matrix with full row
rank, $\mathbf{r}$ is an $r\times 1$ vector mean and ${\bm{\Omega}}$ is the
corresponding $r\times r$ variance matrix related to the restrictions. This
can be regarded as a set of uncertain constraints $\mathbf{r}$ when
${\bm{\Omega}}$ is non-zero, or exact constraints
$\mathbf{R}\mathbf{y}_{T+1:T+h}=\mathbf{r}$ in the case
${\bm{\Omega}}=\mathbf{0}$.

Conditioning the forecast distribution of eqn. (12) on this constraint
information yields the updated distribution below; details follow from West
and Harrison (sect. 16.3.2 1997) and Chan et al. (2023). The updated
distribution for the path is

$\mathbf{y}_{T+1:T+h}|\mathbf{R},\mathbf{r},{\bm{\Omega}}\sim
N(\mathbf{m}^{*},\mathbf{M}^{*}),$

with

$\mathbf{m}^{*}=\mathbf{m}+\mathbf{A}(\mathbf{r}-\mathbf{R}\mathbf{m})\quad\textrm{and}\quad\mathbf{M}^{*}=\mathbf{M}+\mathbf{A}({\bm{\Omega}}-\mathbf{R}\mathbf{M}\mathbf{R}^{\prime})\mathbf{A}^{\prime}\quad\textrm{where}\quad\mathbf{A}=\mathbf{M}\mathbf{R}^{\prime}(\mathbf{R}\mathbf{M}\mathbf{R}^{\prime})^{-1}.$

The special case of exact constraints
$\mathbf{R}\mathbf{y}_{T+1:T+h}=\mathbf{r}$ has constraint uncertainty matrix
${\bm{\Omega}}=\mathbf{0}$ so that $\mathbf{M}^{*}$ reduces to
$\mathbf{M}^{*}=\mathbf{M}-\mathbf{A}\mathbf{R}\mathbf{M}\mathbf{R}^{\prime}\mathbf{A}^{\prime}.$
This is relevant in applications where it is justifiable to assume exact
constraints. In our policy setting, the targeted constraint vector
$\mathbf{r}$ is just that, a target policy path, for example, so that there
will always be some level of uncertainty. The starting point is uncertainty
represented by the variance matrix
$V(\mathbf{R}\mathbf{y}_{T+1:T+h})=\mathbf{R}\mathbf{M}\mathbf{R}^{\prime}.$
As argued in Antolin-Diaz et al. (2021), adopting
${\bm{\Omega}}=\mathbf{R}\mathbf{M}\mathbf{R}^{\prime}$ represents a position
that admits relevant and conservative levels of such uncertainty. This is used
in the application of our paper as we constrain on candidate values of the
policy path in considering conditional forecasts, hedged with uncertainty.

#### Constraints on Structural Shocks

The above development implicitly involves constraints imposed on all of the
structural shocks of the model. In conditional forecasting exercises, it is
often necessary to focus on only specific structural shocks set at restricted
values (exact or uncertain), to obtain finer control. For example, a forecast
conditional on an increasing policy rate path might result in an increase in
inflation (i.e., the “price puzzle”) as the forecast is driven by reduced form
shocks which are a mix of the structural shocks (i.e., demand, supply,
monetary). For further discussion, see Antolin-Diaz et al. (2021). Due to
these issues, we impose structural restrictions so that changes in the policy
rate are driven by monetary policy shocks (i.e., decisions from the Central
Bank) rather than as reactions to other shocks. This can be done by
considering the restrictions

$\mathbf{W}{\bm{\varepsilon}}_{T+1:T+h}\sim N(\mathbf{w},{\bm{\Psi}})$ (14)

where $\mathbf{W}$ is a $w\times nh$ full rank-selection matrix, $\mathbf{w}$
is a $w\times 1$ vector of constants and ${\bm{\Psi}}$ is a $w\times w$
covariance matrix. Substituting eqn. (11) into eqn. (14) for
${\bm{\varepsilon}}_{T+1:T+h}$ results in

$\mathbf{W}\mathbf{H}\mathbf{y}_{T+1:T+h}\sim
N(\mathbf{W}\textrm{c},\mathbf{I}_{w})$ (15)

Then combining eqn. (15) with eqn. (13) yields

$\underbrace{\begin{bmatrix}\mathbf{R}_{0}\\\
\mathbf{W}\mathbf{H}\end{bmatrix}}_{\mathbf{R}}\mathbf{y}_{T+1:T+h}\sim
N\Biggl{(}\underbrace{\begin{bmatrix}\mathbf{r}_{0}\\\
\mathbf{W}\textrm{c}\end{bmatrix}}_{\mathbf{r}},\underbrace{\begin{bmatrix}{\bm{\Omega}}_{0}&\mathbf{0}_{r_{0}\times
w}\\\ \mathbf{0}_{w\times
r_{0}}&\mathbf{I}_{w}\end{bmatrix}}_{{\bm{\Omega}}}\Biggl{)}$ (16)

where $\mathbf{R}_{0}$ is a selection matrix for observable restrictions,
$\mathbf{r}_{0}$ is a vector of restrictions on observables, and $\Omega_{0}$
is a matrix of covariance restrictions on observables. Thus, by setting
$\mathbf{R},\mathbf{r}$ and ${\bm{\Omega}}$ as indicated in eqn. (15), the
structural restrictions can be regarded as a specific case of eqn. (13). In
our applications, we set $\mathbf{r}_{0}=\mathbf{x}$ and
$\mathbf{W}=\mathbf{0}$ for all ${\bm{\varepsilon}}_{T+1:T+h}$ except those
associated with the monetary policy shock– those are kept unrestricted. This
amounts to conditioning on a proposed decision vector $\mathbf{x}$ and
assuming it is driven only by monetary policy shocks. Other conditioning
assumptions are possible. For example, in addition to assuming the driving
shock is monetary policy one could assume a vector of positive/negative demand
shocks representing different future scenarios.