International Journal of Forecasting 39 (2023) 1820-1838
ELSEVIER
Contents lists available at ScienceDirect
International Journal of Forecasting
journal homepage: www.elsevier.com/locate/ijforecast
Macroeconomic forecasting in the euro area using predictive
combinations of DSGE models^
Jan Čapek3
, Jesus Crespo Cuaresmab
'a
'c
'd
'*, Niko Hauzenbergere , a
,
Vlastimil Reichela
^Masaryk University, Czech Republic
b
Vienna University of Economics and Business, Austria
c
11ASA, Austria
" W1F0, Austria
e
University of Salzburg, Austria
A R T I C L E I N F O
Keywords:
Forecasting
Model averaging
Prediction pooling
DSGE models
Macroeconomic variables
A B S T R A C T
W e provide a comprehensive assessment of the predictive power of combinations
of dynamic stochastic general equilibrium (DSGE) models for G D P growth, inflation,
and the interest rate i n the euro area. W e employ a battery of static and dynamic
pooling weights based o n Bayesian model averaging principles, prediction pools, and
dynamic factor representations, and entertain six different DSGE specifications and five
prediction weighting schemes. O u r results indicate that exploiting mixtures of DSGE
models produces competitive forecasts compared to individual specifications for both
point and density forecasts over the last three decades. Although these combinations do
not tend to systematically achieve superior forecast performance, w e find improvements
for particular periods o f time a n d variables w h e n using prediction pooling, dynamic
model averaging, and combinations o f forecasts based o n Bayesian predictive synthesis.
© 2022 The Author(s). Published by Elsevier B.V. on behalf of International Institute of
Forecasters. This is an open access article under the CC BY license
(http://creativecommons.Org/licenses/by/4.0/).
1. Introduction
Dynamic stochastic general equilibrium (DSGE) models
have become the workhorse of modern macroeconomic
" The authors would like to thank two anonymous referees for
very helpful comments on an older version of the paper. Financial
support from the Czech Science Foundation, Grants 17-14263S and
21-10562S, is gratefully acknowledged. This work was supported by
the Ministry of Education, Youth, and Sports of the Czech Republic
through the e-INFRA CZ (ID: 90140). Jesus Crespo Cuaresma gratefully
acknowledges funding from IIASA, Austria and the National Member
Organizations, Austria that support the institute. Niko Hauzenberger
gratefully acknowledges financial support from the Jubilaumsfonds of
the Oesterreichische Nationalbank (OeNB, grant no. 18718).
* Corresponding author at: Vienna University of Economics and
Business, Austria.
E-mail address: jcrespo@wu.ac.at (J. Crespo Cuaresma).
research, due to their internal consistency and their ability
to assess the effects of policy shocks i n a rigorous
manner.
In spite of their importance i n modern economic analysis,
the existing results concerning their out-of-sample
forecasting ability are mixed. A series of studies have
assessed the predictive ability of different types of DSGE
models. Christoffel et al. (2011) examine the
out-of-sample predictive ability of the European Central
Bank's N e w Area-Wide Model ( N A W M ) , the DSGE model
used to create projections of macroeconomic variables
by the monetary authorities of the euro area. The results
in Christoffel et al. (2011) indicate that this DSGE
model, as compared to other alternative reduced-form
specifications, provides good predictions for a set of 12
different macroeconomic variables. The predictive accuracy
of DSGE models, however, does not necessarily remain
stable over time. Del Negro et al. (2016) provide
https://doi.org/10.1016/j.ijforecast.2022.09.002
0169-2070/© 2022 The Author(s). Published by Elsevier B.V. on behalf of International Institute of Forecasters. This is an open access article under
the CC BY license (http://creativecommons.Org/licenses/by/4.0/).
J. Čapek, J. Crespo Cuaresma, N. Hauzenberger et al.
evidence that forecasts produced using a Smets-Wouters
type of DSGE model (Smets & Wouters, 2003, 2007) w i t h
financial frictions perform particularly well in periods of
financial turmoil (in particular in the Great Recession),
but that the predictive accuracy of the model tends to
suffer in tranquil periods. The forecast quality of DSGE
structures that include financial frictions has also been assessed
by Kolasa and Rubaszek (2015), and improvements
in forecast ability are reported in episodes of financial
turmoil when housing market frictions are included in
the model, although no systematic gains in predictive
performance are found in more stable periods.
Another strand of the literature on macroeconomic
forecasting has shown interest in analyzing predictive
combinations based on a wide range of models, rather
than focusing on a single specification, an idea that dates
back to the work by Bates and Granger (1969). Amisano
and Geweke (2017), for instance, find improvements in
out-of-sample prediction errors for macroeconomic variables
in the US by pooling forecasts from different
macroeconomic models using Bayesian predictive distri­
butions.
In this study, we evaluate the forecast ability of
weighted combinations of six different DSGE models for
GDP growth, inflation, and the interest rate in the euro
area, making use of several prediction combination techniques.
Our analysis expands the work by Wolters (2015),
which assesses the forecast accuracy of four DSGE models
for the US, as well as the potential predictive gains
obtained by using combinations of these. W e entertain
six different DSGE specifications for the euro area and
five forecast combination methods, both static and dynamic,
and evaluate point forecasts as well as density
predictions. Our set of prediction combination techniques
contains some of the forecast pooling techniques entertained
in existing studies for DSGE models (Wolters, 2015,
for example), as well as more novel methods based on
the optimization of weights, that can potentially be timevarying
and evolve according to flexible laws of motion.
In particular, we use static weights based on principles
of Bayesian model averaging and prediction pools, and
dynamic weights that build upon dynamic (latent) factor
representations of the variables of interest.
The combination techniques employed in our analysis
result in significantly different weighting schemes
across models. While dynamic Bayesian model averaging
and combinations based on dynamic factors lead to
pooled forecasts which assign positive weights to all of
the DSGE specifications, the technique based on prediction
pools acts as a dynamic model selection tool, assigning
weights close to zero to most individual model
predictions over the out-of-sample period. The potential
gains in predictive accuracy that can be exploited are specific
to sub-periods, variables, and forecasting horizons,
with no one-size-fits-all predictive combination strategy,
ensuring systematic improvements in all situations.
The rest of this paper is organized as follows. Section 2
introduces the DSGE models used in the analysis, and
Section 3 presents the predictive density combination
methods. Section 4 shows the results of the out-of-sample
forecasting exercise, and Section 5 concludes.
International Journal of Forecasting 39 (2023) 1820-1838
2. The battery of DSGE models
2.1. Individual DSGE models
For our empirical analysis, we use a battery of DSGE
models for the euro area. Their specifications differ in size,
complexity, and the particular features highlighted. Since
the analysis is conducted on a set of three core macroeconomic
variables (GDP growth, inflation, and the interest
rate), we ensure that these three observable variables are
common across all models. The sparsest model entertained
is a basic three-equation New Keynesian model,
which serves as a benchmark in terms of simplicity. The
model presented in Cogley et al. (2010) also requires
only three basic observable variables, but introduces two
additional shocks and allows the inflation target to change
over time. The specification by Christensen and Dib (2008)
adds investment and money as additional observable variables.
This group of models is extended w i t h three more
complex specifications that share the set of observable
variables of the model by Smets and Wouters (2007): GDP
growth, inflation, the interest rate, consumption growth,
investment growth, real wage growth, and hours worked.
The specification by Justiniano et al. (2011) contains the
relative price of consumption to investment as the eighth
observable variable, whereas Del Negro et al. (2015) add
spread and inflation expectations as observable variables
to the modeling framework and allow the shocks to be of
a non-stationary nature. The group of DSGE specifications
used spans model structures which differ in the mechanisms
highlighted for the transmission of macroeconomic
shocks. Tracking the predictive ability of such models over
time can thus help us grasp changes in the relative importance
of particular theoretical channels as determinants of
macroeconomic dynamics in the euro area.
Table 1 lists the models entertained, together with
their corresponding abbreviations (which are used in the
description of the results of the analysis and in all subsequent
figures and tables), and summarizes information
about the number of observable variables, the number of
exogenous shocks, and the main features of each model.
The particular observable variables included in each one
of the DSGE models are presented in Table 2.
2.2. Data
The models in Table 1 are estimated using quarterly
data for the euro area in its 19-country composition.
The database spans information from 1970Q3 to 2019Q1
and thus contains 195 quarterly observations. The core
of the database is sourced from the Area W i d e Model
(AWM) presented in Fagan et al. (2005) and updated and
extended by Brand and Toulemonde (2015). The original
A W M database is updated using data from the European
Central Bank or Eurostat since the 1990s and is extended
by population and hours worked from the Total Economy
Database and Eurostat. Data on monetary aggregates are
obtained directly from the OECD. W e use time series
compiled by Gilchrist and Mojon (2018) for the interest
rate spread variable. Inflation expectations are sourced
from the European Central Bank's Survey of Professional
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J. Capek, J. Crespo Cuaresma, N. Hauzenberger et al. International Journal of Forecasting 39 (2023) 1820-1838
Table 1
Euro area DSGE models used in the forecasting exercise.
Reference Name Observables Shocks Features
3-equation basic NI< model NKModel 3 3 IS, PC, Taylor rule
Cogley et al. (2010) CPS 2010 3 5 Inflation target can change over time
Christensen and Dib (2008) CD 2008 5 5 Financial frictions as in Bernanke et al. (1999)
Smets and Wouters (2007) SW 2007 7 7 Deterministic growth rate driven by labor-augmenting
technological progress
Justiniano et al. (2011) JPT 2011 8 8 Two investment shocks
Del Negro et al. (2015) DNGS 2015 9 9 Smets and Wouters (2007) with financial frictions +
spread and inflation expectations as observables
Table 2
DSGE models: Observable variables.
CPS 2010
NKModel
CD 2008 SW 2007 JPT 2011 DNGS 2015
Output / / / / /
Inflation / / / / /
Interest rate / / / / /
Consumption / / /
Investment / / / /
Hours worked / / /
Wage / / /
Money supply (Ml) /
Relative investment price /
Spread /
Inflation expectations /
Forecasters. The longest-term forecast available was selected,
which spans four to five years ahead. Growth rates
are calculated as quarter-on-quarter differences of logs,
and the interest rate is calculated per quarter. Details
on the sources of the different variables are provided in
Appendix A.
The data transformations performed to the model variables
correspond to those used in Smets and Wouters
(2007). Real consumption, investment, and GDP are d i vided
by population and transformed to growth rates.
Hours worked are divided by population and logged. Inflation
is defined as the growth rate of the GDP deflator.
The nominal wage is deflated by the GDP deflator and
transformed to growth rates. The interest rates are shortterm
market interest rates. The monetary aggregate M l
is deflated by the GDP deflator, divided by population,
and transformed to growth rates. Finally, the relative price
of investment is calculated as the investment deflator
divided by the consumption deflator, and transformed to
growth rates.
2.3. Detrending macroeconomic variables
The macroeconomic variables used in the estimation
of DSGE specifications are often highly persistent and
need to be detrended using methods that are consistent
with the theoretical framework used in the model.
For some existing models, the authors specify the particular
filter employed to detrend the variables, while
in other cases, these details are not specified (see, e.g.,
Gorodnichenko & Ng, 2010). Delle Chiaie (2009) investigates
the effects of detrending observable variables with
the Hodrick-Prescott (HP) filter and a linear trend in
the model by Smets and Wouters (2003), and finds that
structural parameter estimates are rather sensitive to
the choice of a particular filtering method. Consequently,
forecasting performance may be significantly affected by
the choice of a detrending approach.
The original contributions on which we base our individual
specifications use different detrending methods
for the macroeconomic variables. While Christensen
and Dib (2008) use the HP filters, Smets and Wouters
(2007)—and models that build upon a similar structureintroduce
some of the observable variables in first differences
when estimating the parameters of the DSGE
specification. Gorodnichenko and Ng (2010) offer a perspective
of detrending approaches commonly used in a
broader literature by compiling the detrending methods
employed in 21 different models. The list of data filters
used in various DSGE models shows a predominance of
linear detrending, HP filtering, and first difference transformations.
Our analysis employs several approaches used
in the literature, while keeping the detrending method
identical across all models considered. By doing so, we
aim to separate the influence on forecasting performance
of core model features, such as financial frictions or flexible
inflation targets, from that of the trend formulation. In
our baseline detrending specification, we use the data for
GDP (and its sub-components) in first differences. Time
series which present higher persistence are filtered using
one-sided HP filters.1
Alternatively, we also assess the forecasting performance of our
models employing the filtering strategy proposed by Del Negro et al.
(2015), Justiniano et al. (2011), and Smets and Wouters (2007) and
find evidence that our baseline detrending approach leads to superior
forecasting performance in the majority of cases (see Table C.4 in
Appendix C). We also perform the analysis using different detrending
approaches, such as using the (one-sided) HP filter for all data series,
employing the regression-based data filter introduced in Hamilton
(2018), and demeaning the times series in the models. The results for
these alternative detrending specifications can be found in Appendix C.
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J. Čapek, J. Crespo Cuaresma, N. Hauzenberger et al.
2.4. Estimation and predictive densities
Each one of the models employed in the forecasting
exercise is estimated recursively using Bayesian methods,
starting with a sample size composed by 78 observations
(corresponding to the time frame from 1970Q3-1989Q4)
and adding one quarter at a time to the sample up to a
m a x i m u m of 195 observations (corresponding to the full
sample, w h i c h spans the period from 1970Q3-2019Q1).
Additionally, we perform the forecasting exercise of estimating
the models w i t h a rolling w i n d o w of 60 observations.
The models are estimated using a m i n i m u m of a half
million Metropolis-Hastings replications in two chains for
the NKModel, one million replications for CD 2008 and
CPS 2010, two million replications for JPT 2011 and S W
2007, and a m i n i m u m of four million replications for the
DNGS 2015 model. To ensure convergence of the Markov
chain, the checks in Brooks and Gelman (1998) are performed
and, if these fail, the number of replications is
increased until convergence of the posterior distributions
is achieved. W e use a Monte Carlo-based optimization
routine to ensure that the optimal acceptance ratio of the
Metropolis-Hasting algorithm is reached, and we discard
90% of the replications as burn-ins.2
Forecasts are computed using 10,000 draws from the
posterior distribution for every estimated model and each
in-sample period. In each instance, we calculate one- to
four-step-ahead out-of-sample forecasts of GDP growth,
inflation, and the interest rate, which correspond to periods
ranging from 1990Q1 to 2018Q4. The analysis of
forecasts is conducted after imposing back the trend of
the observable variables, so as to ensure comparability
across detrending approaches.
3. Predictive combinations of DSGE models
In this section, we outline the forecast combination
methods employed to average the predictions of our set
of models. Each DSGE model typically includes a different
set of observables, targets a specific feature of the
economy, and thus provides its o w n characterization of
the economy by imposing different (structural) dynamics
on the macroeconomic variables of interest. Some smallscale
DSGE models abstract from the interaction between
developments in the real economy, the labor market, and
the financial sectors, while others include features and
mechanisms related to these linkages. W e concentrate
exclusively on one- and four-step-ahead predictive densities
of GDP growth, inflation, and the interest rate, which
are common to all the specifications used. W e assess and
combine the joint predictive density of these three variables,
as well as their corresponding marginal predictive
densities.
In the following, we illustrate the methods we use to
combine predictive densities by focusing on a scalar time
series y t + 1 and the one-step-ahead horizon. W i t h only m i nor
adjustments, these techniques work analogously for
the joint predictive densities and for the multi-step-ahead
horizon. In our analysis, we therefore consider predictive
All models are estimated using Dynare (Adjemian et al., 2011).
International Journal of Forecasting 39 (2023) 1820-1838
densities for y t + 1 , w h i c h are available from K different
DSGE models. Each DSGE model, indexed by j = 1 , . . . , K,
incorporates information up to time r to generate a predictive
density p/(yt +i |Zj(t)) for period t +1. The information
set Xj(t) typically consists of the target variable y] : t =
iy\, • • • ,yt)'< as well as of the information up to time t
provided by additional variables specific to that particular
DSGE model. W e aim to combine the K predictive
densities {Pj(yt+i |2j(t))}jli using a K x 1 weights vector
oot+i = (o>i,t+i, • • •, o>K,t+\)' that is specific to the onestep-ahead
forecast horizon and potentially time-varying.
The combined predictive density f o r y t + i is then given by
K
P ( y t + 1 | i 1 ( t ) , . . . ,iK(t)) = J2w
it+m (yt+il^jCO) • ( i )
j = l
Eq. (1) directly relates to the Bayesian predictive synthesis
of McAlinn et al. (2019), McAlinn and West (2019),
where w t + i is described as a dynamic synthesis func­
tion.3
This synthesis function can incorporate different
objectives based on policy targets and historical performance
up to period r, and nests traditional approaches to
forecast combination, such as prediction pools (Geweke
& Amisano, 2011; Hall & Mitchell, 2007) and Bayesian
dynamic model averaging (Koop & Korobilis, 2012, 2013;
Raftery et al., 2010). W e start by discussing a simple static
weighting scheme implying w t + 1 = co, and then turn to
more general approaches based on using dynamic weights
for the predictive densities.
Equal static weights
An obvious starting point to combine predictions from
different DSGE models, which provides a benchmark to
evaluate different weighting schemes, is to use
Since &>j,t+i > 0 and 5Z/Li ^l/.t+i = 1> the combination
of predictive densities also constitutes a predictive density
(Geweke & Amisano, 2011; Hall & Mitchell, 2007).
This agnostic approach neglects the fact that different
models might not be equally suitable for prediction at
different time periods, and does not provide updates of
the corresponding weights as information is gained about
the differential predictive ability of model specifications.
An equal weighting scheme is commonly found to be a
good competitor in terms of out-of-sample forecasting
accuracy, as it tends to hedge against large forecast errors
of single specifications (see Timmermann, 2006).
Dynamic Bayesian model averaging
A natural choice of model weights can be achieved by
pooling forecasts according to particular model selection
criteria (for example, based on the predictive marginal
likelihood or past forecast performance). For a given set
i
Del Negro et al. (2016) and McAlinn and West (2019) provide
a formal treatment of the decision problem concerning the choice of
time-varying weights o ) t + i .
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J. Capek, J. Crespo Cuaresma, N. Hauzenberger et al. International Journal of Forecasting 39 (2023) 1820-1838
of priors over specifications, traditional Bayesian model
averaging (BMA) approaches give models w i t h a higher
marginal likelihood more support while downweighting
models w i t h deficient predictive characteristics. Following
Raftery et al. (2010) and Koop and Korobilis (2012,
2013), we consider posterior weights for individual specifications
based on their (discounted) historical predictive
likelihood, a procedure known as dynamic model averaging
(DMA). According to this literature, D M A consists of a
prediction equation
co
m
3,t+\\t
%t\t
2-,k=\ M
k,t\t
and an updating equation
;,t+i|t
Here, w t + 1 | t = (a>ij t + i|t <oK,t+i\tY denotes a K x
1 vector of predictive weights at period t + 1 based
on historical forecast performance up to r, wt+i|t+i =
(»i,t+i|t+i, • • •, o>K,t+\\t+\)' is a If x 1 vector of updated
weights, and pj (y^\Xj(t)j refers to the one-step-ahead
predictive density for model j evaluated at the realized
value y£li (i.e., the predictive likelihood).4
Moreover, a
forgetting factor 8 e {0,1) discounts past predictive
performance more heavily, while more recent predictive
likelihoods receive more weight. In the empirical application,
we set S = 0.95, implying that the predictive
likelihood four quarters (i.e., one year) in the past receives
around 80% of the weight of the predictive likelihood of
the most recent quarter.5
The D M A algorithm, moreover,
is easy to implement without the need for any simulation
techniques.
Prediction pools
Recent approaches to forecast combination assess the
set of model-specific forecasts as if it was a portfolio of
predictions, w h i c h must be chosen optimally w i t h respect
to a particular loss function (see, inter alia, Geweke &
Amisano, 2011, 2012; Hall & Mitchell, 2007; Pettenuzzo &
Ravazzolo, 2016). Following Geweke and Amisano (2011),
the loss function is defined as a function of historical log
predictive scores, w h i c h gives rise to optimal weights after
minimization. Similar to B M A and D M A methods, this
approach ensures that forecasts from DSGE models with
poor predictive abilities are downweighted, and those
computed from specifications that predict more successfully
receive higher weights. Information up to time r is
available in order to choose the predictive weight wt+i|t
By construction, both <w,.t+i|t and COJ
5 , t + i | t + !
, for j = 1, , K, are
non-negative, and the elements in both o ) t + i | t and ( u t + i | t + i sum to one.
•* This choice is consistent with Koop and Korobilis (2012, 2013),
who suggest defining S e [0.95, 1]. If S = 1, past predictive performance
is not discounted and the weights are defined according to the
(predictive) marginal likelihood.
optimally. The negative weighted historical log predictive
scores/likelihoods are minimized w i t h respect to the
weights vector co:
o>t + i|t = m m
J = 1
where S again denotes a discount factor that serves the
same purpose as in the D M A procedure by assigning
increasing weight to the most recent predictive performance.
W e additionally impose the restriction that
weights are non-negative and sum to one. Note that
we use standard numerical optimization algorithms for
prediction pools, w h i c h are therefore easy to implement
and computationally fast.
Bayesian predictive synthesis with a dynamic factor model
As noted by Del Negro et al. (2016), the predictive ability
of particular specifications may be affected by structural
breaks in the parameters governing the dynamics
of macroeconomic variables. Such changes in predictive
power should be addressed when combining the K predictive
densities over time, and thus the mapping from
the forecasts of each model to the combined predictive
density should be adjusted accordingly. Eq. (1) can be
directly related to a dynamic factor model, as proposed
by McAlinn and West (2019) in the context of dynamic
Bayesian predictive synthesis (BPS) methods, by defining
the synthesis function as
where we define the latent factors F t + ] = (yi,t +i, • • •,
y/c,t+i)' w i t h y J j t + i , for j = \,...K, being a draw from the
one-step-ahead predictive density pj{yt+\\Xj(t)) of each
model j for period t + 1. Further, a>t + ] refers to timevarying
loadings, and the shock in the observation equation
et+i is Gaussian w i t h zero mean and variance §. The
latent loadings (or states), that relate the draws from the
predictive distributions to the realized value y[r
^ evolve
according to a random walk:
wt+i =(*t + ?t+i> Vt+i ~ M 0 ,
where j j t + ] refers to a K x 1 vector of Gaussian state
innovations, w h i c h are centered on zero and feature a
K x K variance-covariance matrix In contrast to equal
weighting, DMA, and predictive pooling, the weights a>t + ]
are no longer necessarily non-negative and do not need
to sum up to one. a>t + ] are thus to be interpreted as
(time-varying) calibration parameters relating draws from
the predictive densities to the actual realization y^_v A
further difference from other weighting schemes is that
we consider a measurement error e t + i in the observation
equation that explicitly accounts for model incompleteness
(see, e.g., Aastveit et al., 2018; Hoogerheide
et al., 2010; McAlinn & West, 2019). Moreover, the latent
weights ( » t + ] are allowed to be correlated among models
via a full variance-covariance matrix <f, w h i c h not
only determines the amount of time variation introduced
in a>t+i, but also takes into account the dependencies
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J. Capek, J. Crespo Cuaresma, N. Hauzenberger et al. International Journal of Forecasting 39 (2023) 1820-1838
between individual predictive specifications that share
similar characteristics.
W e use weakly informative priors, w h i c h are standard
in the literature for dynamic factor models. This implies
the use of a multivariate normal prior for w 0 , an inverse
Gamma prior for §, and an inverse Wishart prior for
We repeat this procedure for R draws from the predictive
density and explicitly account for a potentially non-trivial
form of the predictive densities of the DSGE models. To
estimate the model we rely on standard Bayesian estimation
techniques used for time-varying parameter models.
In particular, we use a Gibbs sampler w h i c h iterates
through these R draws. Conditional on all other quantities,
we update the latent states a>t + ] w i t h a standard forward
filtering backward sampling (FFBS) algorithm (Carter &
Kohn, 1994; Fruhwirth-Schnatter, 1994). In a next step,
conditional on the time-varying calibration parameters,
we independently draw the observation equation variance
f and the state equation variance-covariance matrix
All steps involve standard conditional posteriors (for
details, see McAlinn & West, 2019). Moreover, by using
the filtering step in the FFBS algorithm, we directly obtain
the predictive weights wt+i|t. w h i c h are used to combine
the most recent predictive densities when the realization
is not yet available. The M C M C algorithm of the
dynamic factor model is somewhat more computationally
demanding than the approximate procedure of D M A and
the numerical optimization used for the pooling approach.
However, compared to sequential Monte Carlo techniques
such as particle filters (see, e.g., Billio et al., 2013; Del
Negro et al., 2016), the computational burden can still be
considered light.
The DECO approach
In addition to the combination methods outlined
above, we consider the dynamic predictive density combination
(DECO) approach of Billio et al. (2013). Like BPS,
DECO allows for the specification of time-varying weights
that evolve according to a flexible law of motion, and
accounts for model incompleteness:
yfl, = F ' t + 1 w t + 1 + € t + u e t + i ~ Af{0, §), (2)
with o ) t + ] relating draws from the predictive densities to
the actual realization y[r
^ and considering a Gaussiandistributed
measurement error € t + \ .
The main difference from BPS lies in the state equation
that governs the evolution of the weights a>t + ] and thus
the learning mechanism used in prediction. Instead of assuming
that the weights evolve according to a multivariate
random walk with a full variance-covariance matrix
a non-linear link function between the elements in
wt +i and K independent dynamic latent processes £ t + ] =
(fi,t+i, • • •, fr,t+i)' is introduced:
exp(&
E ^ e x p ( £ ,
forj = 1 K.
t+ij
to be non-negative and sum to one. These restrictions thus
effectively result in a non-linear state-space model, where
Eq. (2) can be interpreted as a dynamic location mixture
with a fixed variance. In what follows, £ t + ] encodes the
learning mechanism and governs the weight dynamics.
Each element in £ t + ] evolves according to independent
random walks:
?j,t + i = ft, + »7j,t + i, »7j,t+i ~ Af(0, Vo), for )=-[,..., K.
Here, fy-.t+i denotes element-specific state innovations
with zero mean and variance i/fj. In DECO, the state innovation
variances 1/0 encode the learning mechanism and
depend on a scoring rule preselected by the researcher,
a discount factor S, and a number of past observations r
considered. For example, if the scoring rule indicates that
the predictive performance of some particular model has
deteriorated for the past realized values, the mechanism
allows for the corresponding adjustment of the weights
by increasing tp-j, thus introducing time variation in cojtt+\.
Sequential Monte Carlo techniques are commonly used
for such a non-linear state-space model. For the empirical
implementation of DECO, we specify the key learning
hyperparameters according to the following standard setting:
we use the Kullback-Leibler scoring rule, set the
number of past realized values to r = 9, and the discount
factor 8 = 0.95. The remaining parameters are estimated
from the data. For the particle filter, moreover, we define
50 particles and use an additional smoothing factor of
0.01.6
4. Forecasting macroeconomic variables in the euro
area using DSGE models
W e start by quantitatively assessing the predictive
ability differences across DSGE models, before moving to
the analysis of the potential improvements in forecasting
quality from combining the predictions of individual
models, and of the dynamics of predictive weights over
the out-of-sample period.
4.1. Overall forecast performance of individual DSGE models
The top panel of Table 3 presents the forecasting performance
of individual DSGE models, w h i c h are estimated
recursively over the out-of-sample period. W e present the
root mean squared forecast error (RMSE) ratios, as well as
the average log predictive Bayes factors (LPBFs), defined
as the difference in average log predictive scores (LPSs),
for one-step-ahead and four-step-ahead predictions. For
the RMSEs, Table 3 also shows the results of DieboldMariano
tests of equal predictive performance (Diebold &
Mariano, 1995), and for the LPSs, those of the A m i s a n o Giacomini
tests (Amisano & Giacomini, 2007). In both
cases, the equality of predictive ability is tested using
the SW2007 model as the benchmark specification. The
results of this predictive ability analysis based on rolling
w i n d o w estimation (instead of parameter estimates based
This logistic link function does not allow for the use
of unconstrained calibration parameters via a synthesis
function, as in BPS, since it restricts the elements in a>t + ]
6 An efficient algorithm for this approach is implemented in the
DeCo toolbox in Matlab (see Casarin et al., 2015) for one-step-ahead
forecasts.
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J. Capek, J. Crespo Cuaresma, N. Hauzenberger et al. International Journal of Forecasting 39 (2023) 1820-1838
Table 3
Forecasting performance of recursively estimated DSGE models and combinations of these models.
Target variable(s) DSGE model
CD 2008 CPS 2010 DNGS 2015 JPT 2011 NKmodel SW 2007
One step ahead
Joint 1.246*" 0.929** 1.069** 1.196*** 0.889** 0.320
(-0.194) (0.340***) (-0.063**) (-0.452) (0.159*) (0.125)
GDP growth 1.277*** 0.938 1.069** 1.206*** 0.885* 0.511
(-0.236**) (0.064*) (-0.069) (-0.224***) (0.102***) (-0.777)
Inflation 1.033 0.858*** 1.064 1.107** 0.896** 0.194
(-0.013) (0.117***) (0.036***) (-0.089) (0.019*) (0.022)
Interest rate 1.122 0.973* 1.088 1.291*** 1.016 0.085
(-0.011*) (0.138***) (-0.017) (-0.167***) (0.012) (0.819)
Four steps ahead
Joint 1.115 0.981 1.017 1.176*** 0.941** 0.379
(0.228) (0.419***) (0.308***) (-0.255***) (0.418***) (-1.219)
GDP growth 1.115 0.996 1.037* 1.148** 0.963 0.573
(-0.162) (-0.041) (-0.043) (-0.241***) (-0.026) (-0.865)
Inflation 1.099 0.863 0.938 1.375*** 0.775** 0.220
(0.268***) (0.388***) (0.268***) (-0.006) (0.331***) (-0.396)
Interest rate 1.128 0.988 0.968 1.148*** 0.940 0.234
(0.051***) (0.122***) (0.089***) (-0.038*) (0.162***) (-0.101)
Combination method
EQ DMA POOL BPS DECO
One step ahead
Joint 1.061* 0.926 0.936 0.993 0.941
(0.085) (0.328***) (0.330***) (0.164***) (0.226***)
GDP growth 1.074* 0.928 0.942 1.003 0.949
(-0.006) (0.072***) (0.085***) (-0.011) (-0.145***)
Inflation 0.958 0.877** 0.884" 0.929** 0.872***
(0.028) (0.082) (0.115***) (0.109***) (0.307***)
Interest rate 1.102 1.086 0.994 0.925 0.987
(0.017) (0.102***) (0.123***) (0.118***) (0.158***)
Four steps ahead
Joint 1.070** 0.999 0.985 0.926
(0.321***) (0.440***) (0.448***) (0.673***)
GDP growth 1.089** 1.015 1.005 0.974
(-0.020) (0.008) (-0.037) (-0.013)
Inflation 1.019 0.896 0.871 0.831**
(0.234***) (0.365***) (0.386***) (0.396***)
Interest rate 0.997 0.991 0.961 0.667**
(0.108**) (0.143***) (0.150***) (0.409**)
Notes: The table shows root mean squared errors (RMSEs), and average log predictive Bayes factors (LPBFs) in parentheses,
relative to the SW 2007 model. Bold numbers indicate the best performing DSGE model as well as the best performing
combination method that obtains the smallest RMSE ratio (largest LPBF). The SW 2007 column (highlighted in gray) shows
the actual RMSEs and log predictive scores (LPSs) of our benchmark. Asterisks indicate statistical significance relative to the
SW 2007 model at the \% (***), 5% (**), and 10% (*) significance levels in terms of Diebold and Mariano (1995) tests for RMSEs
and Amisano and Giacomini (2007) tests for LPSs.
on enlarging the in-sample period recursively) can be
found in Appendix B, and the results based on alternative
detrending methods are presented in Appendix C. The
forecast error measures are presented for the joint vector
of GDP growth, inflation, and the interest rate, as well as
for these three variables individually.
W e start by considering the overall forecasting ability
for the group of macroeconomic variables, reflected in the
characteristics of the joint predictive distribution. The results
in the top panel of Table 3 for the full out-of-sample
period indicate that the simple N K M o d e l has particularly
good predictive ability compared to other DSGE specifications
w i t h more complex model structures. In terms of
the joint accuracy of point forecasts (i.e., for the full vector
of variables) as measured by the average RMSEs, this
specification outperforms all other DSGE models for both
one-step-ahead and four-step-ahead predictions. Considering
each variable individually, the quality of point predictions
of the N K M o d e l appears particularly high for
four-step-ahead predictions.
The quality of point forecasts from the N K M o d e l partially
translates to good performance in density forecasting
(as measured by the LPBFs) in both of the prediction
horizons considered. The joint density predictions of
the N K M o d e l specification, however, appear less accurate
than those of the CPS2010 model, w h i c h includes five
structural shocks instead of the three of the N K M o d e l . The
focus of the CPS2010 specification on offering a structural
modeling framework for inflation dynamics (based on the
inclusion of changes in the inflation target in the model)
is successful at improving out-of-sample density predictions
for this variable compared to the rest of the DSGE
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J. Čapek, J. Crespo Cuaresma, N. Hauzenberger et al.
models entertained. Furthermore, the average predictive
performance of the CPS2010 model for short-run density
forecasts of the interest rate is also the best among the
set of models considered.
The particularly good forecasting ability of models that
include a small number of observable variables is broadly
robust to the use of different detrending methods and to
the use of parameter estimates based on a rolling sample
instead of on recursive estimation (see Appendices B and
C).
4.2. Overall forecast performance of predictive combinations
The comparison of results concerning predictive ability
presented in Table 3 indicates that using forecast combinations
can lead to improvements in average predictive
ability over the full out-of-sample period. The best individual
models in terms of forecasting ability at short
horizons outperform all of the combination methods for
GDP growth and jointly for all three variables. Concentrating
on point prediction performance and both analyzed
horizons, individual DSGE models predict GDP and inflation
better than any combination scheme considered. The
combinations, on the other hand, outperform individual
DSGE specifications at predicting the interest rate. Combining
predictions of DSGE models also delivers better
results for density forecasting inflation, and yields the
best results when evaluating the longer horizon of joint
predictive performance.
Since the forecasting ability results of single DSGE
specifications and their combinations for the full sample
may be driven by differences in out-of-sample predictive
quality in sub-periods of the out-of-sample interval
chosen, a more detailed analysis of the dynamics
of the weights that combination methods assign to different
DSGE models appears necessary. In the following
sub-section, we analyze the dynamics of the predictive
weights for the different averaging methods entertained,
thus moving beyond average forecast quality and turning
to the assessment of changes in predictive accuracy over
time.
4.3. The dynamics of predictive weights
We start by assessing the dynamics in the relative
predictive ability of DSGE models by studying the evolution
of predictive weights along the hold-out sample for
our three target variables: GDP growth, inflation, and the
interest rate. For each observable variable, we combine
the predictions from DSGE models using statistics based
on marginal predictive densities rather than on the joint
predictive density of all target variables. One key advantage
of this approach is that the weights used to combine
predictive densities are thus specific to each variable
and reflect changes in the relative forecasting ability of
each DSGE specification for that particular phenomenon.
We calibrate the weights for each forecast combination
scheme with at least eight quarters (1990Q1 to 1991Q4)
for the first period of our hold-out sample (1992Q1) and
employ S = 0.95.
Figs. 1 and 2 show the weights obtained for each
model and target variable in the hold-out sample period
International Journal of Forecasting 39 (2023) 1820-1838
for one-step-ahead (Fig. 1) and four-step-ahead forecasts
(Fig. 2). The weighting schemes across forecast horizons
are relatively similar, indicating a certain degree of stability
of the predictive power of DSGE models across
forecast horizons. In spite of the fact that the loss functions
in the D M A and prediction pool methods are both
based on log predictive scores, we observe substantial
differences in the magnitude of the weights obtained for
these two approaches. The weights in the prediction pool
approach typically suggest a dynamic model selection
scheme where single models tend to receive a weight
close to one in a given period of time, while D M A usually
assigns positive weights to forecasts from all different
DSGE models. For the combination approach based on
Bayesian predictive synthesis, weights (corresponding to
factor loadings) are positive and relatively similar across
models for the majority of periods. However, during the
financial crisis, individual negative factor loadings can be
observed, implying a reversal of the sign of the prediction
of the respective DSGE model in the combined forecast for
these quarters.
Focusing on one-step-ahead weights, the first row of
panels in Fig. 1 shows the results for the different combination
techniques for GDP growth. For DMA, we observe
that CPS2010 and NKModel tend to dominate in terms of
predictive ability prior to the financial crisis. In the subsequent
years, and in particular after the debt crisis in the
euro area, the relevance of CPS2010 within the group of
combined predictions decreases in favor of SW2007. For
prediction pooling, the distribution of weights shows the
importance of predictions from CPS2010 and NKModel for
the combined forecast in particular periods, with SW2007
gaining importance only in the aftermath of the debt
crisis. Both the D M A and DECO combination schemes give
high weights to predictions from CPS2010 and NKModel,
and the weights from DECO reflect the importance of forecasts
from DNGS2015 until the mid-2000s. The distribution
of weights implied by Bayesian predictive synthesis
is much more uniform and stable over time.
The second row of panels in Fig. 1 depicts the dynamics
of weighting schemes for inflation as a target variable for
one-step-ahead forecasts. Using DMA, the highest weights
are assigned to CPS2010 and DNGS2015, with the latter
gaining importance during the financial crisis. Both of
these models are designed with a focus on tracking inflation
dynamics: CPS2010 features a time-varying inflation
target, and DNGS2015 includes inflation expectations, operationalized
by making use of data from the Survey of
Professional Forecasters. W i t h prediction pools, a qualitatively
similar scheme appears, with weights close to
unity alternating between these two DSGE models, and
predictions from DNGS2015 being particularly important
during the financial crisis years. Bayesian predictive synthesis
and DECO assign practically identical stable weights
across models for the full period.
For interest rate predictions, the resulting weighting
schemes are presented in the third row of panels in
Fig. 1. In general, for the interest rate we observe a more
persistent pattern in the weighting scheme, similar to
that found for inflation. The D M A method leads to large
and stable weights for CPS2010 throughout the hold-out
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J. Čapek, J. Crespo Cuaresma, N. Hauzenberger et al.
B P S D E C O
International Journal of Forecasting 39 (2023) 1820-1838
D M A P O O L
C D 2 0 0 8
C P S 2 0 1 0
D N G S 2 0 1 5
J P T 2011
N K m o d e l
S W 2 0 0 7
C D 2 0 0 8
C P S 2 0 1 0
D N G S 2 0 1 5
J P T 2011
N K m o d e l
S W 2 0 0 7
C D 2 0 0 8
C P S 2 0 1 0
D N G S 2 0 1 5
J P T 2011
N K m o d e l
S W 2 0 0 7
2018Q4 199201 2018Q4 199201
Fig. 1. Evolution of model weights over the hold-out sample for one-step-ahead predictions. Notes: The figure shows four different weighting schemes
for the three target variables: GDP growth, inflation, and the interest rate. For BPS and DECO we use the posterior mean as a point estimate.
sample, w i t h the exception of the period corresponding
to the financial crisis, when DNGS2015 and NKModel
receive relatively larger weights. The results from prediction
pools are qualitatively similar, with forecasts from
CPS2010 receiving weights close to unity throughout the
period, except for in the mid-1990s and during the financial
crisis, where predictions from SW2007 and DNGS2015
play a small role. As in the case of inflation, for interest
rates, Bayesian predictive synthesis and DECO assign
stable and similar weights to the individual model
predictions throughout the hold-out sample.
For four-step-ahead forecasts of GDP growth, Fig. 2
shows a partly similar evolution of the weights for D M A
combinations, but w i t h weights that are more spread
across DSGE specifications, especially before the financial
crisis. In contrast to one-step-ahead predictions, for the
longer horizon, the forecasts of GDP growth from SW2007
gain importance during the euro area debt crisis period,
and weights in the last part of our hold-out sample are
more uniformly spread across DSGE specifications. For
output, the combination chosen by prediction pooling
leads to a more erratic weighting scheme prior to the
financial crisis as compared to one-step-ahead predictions.
Output growth forecasts from CD2008 gain relevance
right before the financial crisis, as do those from
NKModel and SW2007 in the aftermath of the debt crisis
i n the euro area. The weights from the combination
method based on Bayesian predictive synthesis for fourstep-ahead
forecasts roughly resemble those found for
one-step-ahead predictions.
The evolution of weighting schemes along the holdout
sample for inflation predictions at the four-step-ahead
horizon is relatively similar to that for the one-step-ahead
predictions. The pooling combination scheme selects the
CPS2010 model for almost the whole time period under
study, as in the case of the shorter prediction horizon.
More notable differences across prediction horizons
can be found for D M A combinations. For the longer prediction
horizon, the JPT2011 and SW2007 models are
assigned almost zero weight, while DNGS2015 receives
higher weight in the aftermath of the debt crisis in the
euro area. The particular characteristics of the DNGS2015
model, which includes financial frictions and aims to explain
the dynamics of output and inflation after financial
shocks, make it conceptually adequate for predictions
in the environment of debt distress. The Bayesian predictive
synthesis combination method results in roughly
uniformly distributed weights across models.
Finally, the results for interest rate predictions at the
four-step-ahead horizon, presented in the last row of
Fig. 2, differ strongly from those obtained for one-stepahead
forecasts. The predictions of the CPS2010 model,
which obtained the highest weights using D M A and prediction
pools for the shorter-term horizon, now receive
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J. Capek, J. Crespo Cuaresma, N. Hauzenberger et al. International Journal of Forecasting 39 (2023) 1820-1838
B P S D M A P O O L
C D 2008
C P S 2 0 1 0
D N G S 2 0 1 5
J P T 2011
NKmodel
S W 2 0 0 7
C D 2008
C P S 2 0 1 0
D N G S 2 0 1 5
J P T 2011
NKmodel
S W 2 0 0 7
C D 2008
C P S 2 0 1 0
D N G S 2 0 1 5
J P T 2011
NKmodel
S W 2 0 0 7
II
I
I.
I
I II
2018Q41992Q1 2000Q1 2018Q41992Q1
- 0 . 2 5 0.00 0.25 0.50 0.75 1.00
Fig. 2. Evolution of model weights over the hold-out sample for four-step-ahead predictions. Notes: The figure shows three different weighting
schemes for the three target variables: GDP growth, inflation, and the interest rate. For BPS we use the posterior mean as a point estimate. Note
that DECO is only used for the one-step-ahead horizon.
low weights over the hold-out sample and are replaced by
the NKModel for the majority of the hold-out period, with
the weights for CD2008 and DNGS2015 being prominent
during the outbreak of the financial crisis.
The results of the analysis of the evolution of weight
estimates for combinations of DSGE model predictions
illustrate the stark differences in weights across forecast
pooling methods and over time. The fact that the combination
method based on prediction pools acts as a dynamic
model-selection device contrasts w i t h the
weighting schemes resulting from the other approaches
entertained in the exercise, which tend to lead to composite
predictions w i t h positive weights for all specifications.
The relative predictive performance of these combination
approaches along the hold-out sample, as well as that of
individual model forecasts, is explored in more detail in
the next section.7
4.4. Predictive ability of individual specifications and forecast
combinations: Variation over time
In this section, we examine the variation over time of
the predictive performance of the individual DSGE models
and the forecast combinations. W e concentrate on the
7
The evolution of predictive weights across methods and over time
for rolling samples can be found in Appendix B.
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J. Capek, J. Crespo Cuaresma, N. Hauzenberger et al. International Journal of Forecasting 39 (2023) 1820-1838
a.) 1-step-ahead
Joint Output growth Inflation Interest rate
1992Q1 2000Q1 2010Q1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2 0 1 0 Q 1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2 0 1 0 Q 1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2 0 1 0 Q 1 2 0 1 8 Q 4
b.) If.-step-ahead
Joint Output growth Inflation Interest rate
1992Q1 2 0 0 0 Q 1 2 0 1 0 Q 1 2 0 1 8 Q 4 1992Q1 2000Q1 2 0 1 0 Q 1 2 0 1 8 Q 4 1992Q1 2000Q1 2 0 1 0 Q 1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2010Q1 2 0 1 8 Q 4
Fig. 3. Evolution of average log predictive Bayes factors (LPBFs) relative to the SW 2007 model. Combination methods. Notes: The gray shaded areas
indicate OECD recessions for the euro area. Note that DECO is only used for the one-step-ahead horizon.
analysis of the evolution of log predictive Bayes factors, as
a measure for the marginal likelihood, over the hold-out
sample.
Fig. 3 presents the predictive performance of forecasts
based on the different weighting schemes across variables
and forecast horizons by means of log predictive
Bayes factors relative to the SW2007 model. In panel a) of
Fig. 3, the results for one-step-ahead forecasts are shown.
The overall evolution of the predictive ability of forecast
combination methods at this prediction horizon presents
similar dynamics across most of the approaches, w i t h i m provements
in predictive ability over the hold-out sample
and a relatively stable forecasting performance at the end
of the out-of-sample period. A notable exception is the
DECO scheme, especially for output growth and inflation.
Practically all forecast combination methods tend to perform
poorly at the very beginning of our hold-out sample
compared to the SW2007 benchmark, a feature that is
likely related to the imprecise estimation of weights.8
Considering the joint set of macroeconomic variables
of interest as a whole, the predictive ability of prediction
pooling and D M A tends to be similar and to dominate
all other combination methods after the mid-1990s, a
result which is mostly driven by their ability to provide
precise predictions of GDP growth. Combinations of forecasts
based on the DECO method, on the other hand,
We also perform the exercise based on rolling samples instead
of a recursive reestimation scheme, and the results are presented
in Appendix B. The relative forecasting ability of individual models
does not change qualitatively, while the performance of combination
schemes with respect to the SW2007 benchmark tends to worsen, thus
lending support to this conclusion.
dominate the other combination alternatives when predicting
inflation and interest rates after the mid-1990s. In
contrast to the results obtained for the shorter-term horizon,
the Bayesian predictive synthesis method of forecast
averaging systematically outperforms the other predictive
combinations for the joint group of observable macroeconomic
variables after the mid-1990s at the longer horizon.
The predictive quality shown by this method is fueled
by its performance at predicting interest rates in the
longer term, while in the other two variables, the forecast
error appears comparable to that of other combination
methods.
In Fig. 4 we present the log predictive Bayes factors
of individual specifications over the hold-out period with
respect to the benchmark model, SW2007. A comparison
across DSGE models reveals a systematically good relative
predictive performance of the CPS2010 model (in
particular after the mid-1990s) that extends to all three
variables and to both forecasting horizons. In addition, a
worsening in forecast ability of some specifications with
respect to the SW2007 benchmark during the financial
crisis and in its aftermath can be observed for many
of the individual DSGE specifications. This is particularly
the case for CD2008 at both horizons, but the loss of
predictive quality also takes place in other specifications
and is asymmetric across macroeconomic variables, with
GDP growth forecasts being the most affected. The loss
of predictive power triggered by the financial crisis is
in many cases persistent, and relative predictive scores
(as measured by the log predictive Bayes factor) do not
always reach the level they had prior to the crisis. A n
interesting exception to this stylized fact is the inflation
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J. Capek, J. Crespo Cuaresma, N. Hauzenberger et al. International Journal of Forecasting 39 (2023) 1820-1838
a.) 1-step-ahead
Joint Output growth Inflation Interest rate
- 1 . 2 - L i 1 1 1-1
! - i 1 1 1-1
L-i 1 1 H '-i 1 1 H
1992Q1 2000Q1 2 0 1 0 Q 1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2010Q1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2010Q1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2010Q1 2 0 1 8 Q 4
b.) If.-step-ahead
Joint Output growth Inflation Interest rate
3 I 1 1 I 1
L_, , , , I J , , , I—I , , ,
1992Q1 2000Q1 2 0 1 0 Q 1 2 0 1 8 Q 4 1992Q1 2000Q1 2 0 1 0 Q 1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2010Q1 2 0 1 8 Q 4 1992Q1 2000Q1 2010Q1 2 0 1 8 Q 4
Fig. 4. Evolution of average log predictive Bayes factors (LPBFs) relative to the SW 2007 model. DSGE models. Notes: The gray shaded areas indicate
OECD recessions for the euro area.
predictions from the DNGS2015 model, whose specification
incorporates a more sophisticated assessment of
inflation expectations than the rest of the DSGE models
used, and whose predictive ability for this variable
improves in the crisis period.
A comparison of the predictive ability of forecast combinations
and individual DSGE models over the hold-out
period reveals that in some periods and for particular
variables, weighted averages of forecasts achieve higher
and less volatile log predictive Bayesian factors. However,
the results show that it is not possible to find a onesize-fits-all
method to combine predictions from DSGE
models that would provide systematically superior predictions
for all variables under scrutiny and over the full
period studied. The difficulty in finding such a forecast
averaging method for our sample is related to the particular
characteristics of the economic area being studied.
The existence of cross-country heterogeneity in shock
transmission mechanisms and macroeconomic outcomes
across euro area economies, in particular since the onset
of the sovereign bond crisis, is widely documented in the
literature (see Burriel & Galesi, 2018; Holton & d'Acri,
2018, just to name two recent examples). The difference
in shock propagation between countries in the euro area
aggregate poses particular challenges in terms of how
they can be accommodated in DSGE specifications such
as those entertained in our analysis.
5. Conclusions
The results of our analysis show that combining forecasts
from DSGE models does not systematically lead to
improvements in predictive ability for macroeconomic
variables for the euro area over the full period under
scrutiny, which spans the last three decades. For some
variables and periods, predictive weighting schemes are
able to reach superior forecasting performance over individual
DSGE specifications. In particular, the gains in
the predictive ability of forecast combinations of DSGE
models are larger in the last part of our sample.
The weighting schemes implied by the combination
methods employed are fundamentally different across
techniques. Weighting based on prediction pools tends
to lead to forecasts based on dynamic model selection,
assigning zero weights to many individual model predictions
over the out-of-sample period. D M A and weighting
based on dynamic factors, on the other hand, results
in combined forecasts with positive weights for practically
all of the DSGE specifications. The forecasting performance
of individual DSGE models and combinations
thereof systematically worsens during the financial crisis
with respect to the benchmark, although the loss
of predictive power and the volatility of forecast errors
appear larger in individual specifications as compared to
predictive combinations.
The results of our analysis may be significantly affected
by the focus on the euro area economy, which is characterized
by differences in the propagation of macroeconomic
shocks across the countries that compose it. The
suite of DSGE models employed in our forecasting exercise
does not contain any specification that explicitly
addresses the differential structural characteristics of the
euro area. In this context, the results of our analysis
should be considered very conservative estimates of the
potential of predictive combination methods combined
with forecasts from DSGE models. Refining the theoretical
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J. Capek, J. Crespo Cuaresma, N. Hauzenberger et al.
structure of the models employed for predictive combinations
to address the particularities of the euro area is
likely to be a fruitful avenue of further research building
upon the analysis presented here.
Declaration of competing interest
The authors declare that they have no k n o w n c o m peting
financial interests or personal relationships that
could have appeared to influence the w o r k reported in
this paper.
International Journal of Forecasting 39 (2023) Í820-Í838
Appendix A. Data
See Table A . l .
Appendix B. Forecasting performance based on rolling
window estimation
See Table B . l and Figs. B.1-B.4.
Appendix C. Forecasting performance based on alternative
detrending schemes
See Tables C.1-C.4.
Table A.1
Source of data.
Source Database, mnemonic
Output AWM, Eurostat AWM:YER, Eurostat:namq_10_gdp (Q.CLV10_MEUR.SCA.B1GQ.EA19)
Inflation AWM, Eurostat AWM:YED, Eurostat:namq_10_gdp (Q.PD10_EUR.SCA.B1GQ.EA19)
Interest rate AWM, Eurostat AWM:STN, Eurostat:irt_st_q (Q.IRT_M3.EA)
Consumption AWM, Eurostat AWM:PCR, Eurostat:namq_10_gdp (O_CLV10_MEUR.SCA.P31_S14_S15.EA19)
Investment AWM, Eurostat AWM:ITR, Eurostat:namq_10_gdp (O_CLV10_MEUR.SCA.P51G.EA19)
Hours worked Conference Board, CB:Total Economy Database ("Total Hours Worked"), Eurostat:namq_10_al0_e
Eurostat (Q.THS_HW.TOTAL.SCA.EMP_DC.EA19)
Wage AWM, Eurostat AWM:WIN, Eurostat:namq_10_al0 (0_.CP_MEUR.SCA.TOTAL.Dl.EA19)
Money supply ( M l ) OECD MANMM101*
Relative investment price AWM, Eurostat AWM:PCD, ITD, Eurostat:namq_10_gdp (Q.PD10_EUR.SCA.P31_S14_S15.EA19,
aPD10_EUR.SCA.P51G.EA19)
Spread Gilchrist and spr_nfc_bund_ea
Mojon (2018)
Inflation expectations ECB SPF - Survey of Professional Forecasters (SPF.Q.U2.HICP.POINT.LT.Q.AVG)
Population Eurostat demo_pjanbroad (ANR.Y15-64.T), lfsq_pganws (Q.THS.T.TOTAL.Y15-64.POP.EA19)
Notes: 'Although the time series of the monetary aggregate M l is described as seasonally adjusted in the OECD database, some parts of the series
still exhibit a clear seasonal pattern, which we removed making use of the TRAMO-SEATS method in JDemetra+.
-1.5-1—, ,
1992Q1 2000Q1
a.) 1-step-ahead
Output growth
DSGE model
C D 2008
C P S 2010
D N G S 2 0 1 5
— J P T 2011
NKmodel
S W 2 0 0 7
2010Q1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2 0 1 0 Q 1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2010Q1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2010Q1 2 0 1 8 Q 4
b.) 4-step-ahead
Output growth Interest rate
0.8 i ,
1992Q1 2000Q1
• I _
• r
DSGE model
— C D 2008
C P S 2010
D N G S 2 0 1 5
— J P T 2011
NKmodel
— S W 2 0 0 7
2010Q1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2 0 1 0 Q 1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2010Q1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2010Q1 2 0 1 8 Q 4
Fig. B.l. Evolution of average log predictive Bayes factors (LPBFs) relative to the SW 2007 model. Notes: The gray shaded areas indicate OECD
recessions for the euro area. DSGE models are estimated based on a rolling window.
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J. Capek, J. Crespo Cuaresma, N. Hauzenberger et al. International Journal of Forecasting 39 (2023) 1820-1838
a.) 1-step-ahead
Output growth Interest rate
M e t h o d
— EO
— D M A
— P O O L
— B P S
D E C O
1992Q1 2000Q1 2 0 1 0 Q 1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2 0 1 0 Q 1 2 0 1 8 Q 4 1992Q1 2 0 0 0 0 1 2 0 1 0 0 1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2 0 1 0 Q 1 2 0 1 8 Q 4
b.) ^-step-ahead
Output growth Interest rate
1992Q1 2000Q1 2 0 1 0 Q 1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2010Q1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2010Q1 2 0 1 8 Q 4 1992Q1 2 0 0 0 Q 1 2 0 1 0 Q 1 2 0 1 8 Q 4
Fig. B.2. Evolution of average log predictive Bayes factors (LPBFs) relative to the SW 2007 model. Notes: The gray shaded areas indicate OECD
recessions for the euro area. Note that DECO is only used for the one-step-ahead horizon. DSGE models are estimated based on a rolling window.
B P S D E C O P O O L
C D 2 0 0 8
C P S 2 0 1 0
D N G S 2 0 1 5
J P T 2 0 1 1
N K m o d e l
S W 2 0 0 7
C D 2 0 0 8
C P S 2 0 1 0
D N G S 2 0 1 5
J P T 2 0 1 1
N K m o d e l
S W 2 0 0 7
C D 2 0 0 8
C P S 2 0 1 0
D N G S 2 0 1 5
J P T 2 0 1 1
N K m o d e l
S W 2 0 0 7
in
2018Q4 1992QI 200001 2018Q4 199201 J3CGOI 201001 201BQ4 199201 200901
Fig. B.3. Evolution of model weights over the hold-out sample for one-step-ahead predictions. Notes: The figure shows four different weighting
schemes for the three target variables: output growth, inflation, and the interest rate. For BPS and DECO we use the posterior mean as a point
estimate. DSGE models are estimated based on a rolling window.
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J. Capek, J. Crespo Cuaresma, N. Hauzenberger et al. International Journal of Forecasting 39 (2023) 1820-1838
BPS D M A P O O L
C D 2 0 0 8
C P S 2 0 1 0
D N G S 2 0 1 5
J P T 2011
N K m o d e l
S W 2 0 0 7
C D 2 0 0 8
C P S 2 0 1 0
D N G S 2 0 1 5
J P T 2 0 1 1
N K m o d e l
S W 2 0 0 7
C D 2 0 0 8
C P S 2 0 1 0
D N G S 2 0 1 5
J P T 2 0 1 1
N K m o d e l
S W 2 0 0 7
I
II
II
I
2018Q4 1992Q1 2018Q41992Q1 2000Q1
- 0 . 2 5 0.00 0.25 0.50 0.75 1.00
Fig. B.4. Evolution of model weights over the hold-out sample for four-step-ahead predictions. Notes: The figure shows three different weighting
schemes for the three target variables: output growth, inflation, and the interest rate. For BPS we use the posterior mean as a point estimate. Note
that DECO is only used for the one-step-ahead horizon. DSGE models are estimated based on a rolling window.
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J. Capek, J. Crespo Cuaresma, N. Hauzenberger et al. International Journal of Forecasting 39 (2023) 1820-1838
Table B.l
Forecasting performance of DSGE models based on rolling window estimation and combinations of these models.
Target variable(s) DSGE model
CD 2008 CPS 2010 DNGS 2015 JPT 2011 NKmodel SW 2007
One step ahead
Joint 1.181 0.947** 1.089* 1.164*** 0.888** 0.319
(-0.295) (0.196***) (-0.187) (-0.543**) (0.146***) (0.501)
GDP growth 1.215 0.957 1.097* 1.165** 0.884** 0.511
(-0.321) (-0.016) (-0.051) (-0.161**) (0.046) (-0.794)
Inflation 0.899 0.850*** 1.011 1.113 0.877*** 0.196
(0.010*") (0.135) (-0.047**) (-0.088) (0.066***) (0.226)
Interest rate 1.264*** 1.085 1.198*** 1.431*** 1.136 0.075
(-0.107) (0.066) (-0.145***) (-0.318***) (0.018) (1.030)
Four steps ahead
Joint 1.017 0.996 1.115 1.079 0.961** 0.382
(-0.328) (-0.036) (-0.155*) (-0.347*) (0.034) (-0.787)
GDP growth 1.022 1.012 1.122 1.082 0.981 0.583
(-0.384) (-0.424) (-0.135) (-0.227**) (-0.330) (-0.877)
Inflation 0.949 0.903 1.026 1.052 0.804** 0.210
(0.127*) (0.230***) (0.074***) (-0.037**) (0.218*) (-0.011)
Interest rate 1.039 0.970 1.135 1.083 0.956 0.233
(-0.170*) (0.111***) (-0.188) (-0.095*) (0.093***) (0.015)
Combination method
EQ DMA POOL BPS DECO
One step ahead
Joint 1.005 1.018 0.932** 1.036 0.951**
(0.150) (0.223***) (0.261***) (-0.010***) (-0.265***)
GDP growth 0.992 1.035 0.935* 1.054 0.962
(0.060**) (0.078*) (0.065) (0.031) (-0.296***)
Inflation 1.071 0.858*** 0.893*** 0.903*** 0.857***
(0.072***) (0.110) (0.090) (0.009***) (0.078***)
Interest rate 1.134** 1.186 1.053 1.033 1.065
(-0.015**) (0.057) (0.067) (-0.015**) (-0.013)
Four steps ahead
Joint 1.066 0.990 1.007 0.922***
(0.241) (0.271) (0.369) (0.437*)
GDP growth 1.082 1.006 1.025 0.967
(-0.013) (0.025) (0.037) (-0.033)
Inflation 0.973 0.882* 0.871 0.842**
(0.138) (0.229***) (0.227***) (0.137)
Interest rate 1.037 0.973 0.995 0.657**
(0.070***) (0.115***) (0.079***) (0.402)
Notes: The table shows root mean squared errors (RMSEs), and average log predictive Bayes factors (LPBFs) in parentheses,
relative to the SW 2007 model. Bold numbers indicate the best performing DSGE model as well as the best combination
method that obtains the smallest RMSE ratio (largest LPBF). The SW 2007 column shows the actual RMSEs and LPSs of our
benchmark. Asterisks indicate statistical significance relative to SW 2007 at the 1% (***), 5% (**), and 10% (*) significance levels
in terms of Diebold and Mariano (1995) tests for RMSEs and Amisano and Giacomini (2007) tests for LPSs.
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J. Capek, J. Crespo Cuaresma, N. Hauzenberger et al. International Journal of Forecasting 39 (2023) 1820-1838
Table C.l
Forecasting performance of recursively estimated DSGE models with HP filter detrending.
Target variable(s) DSGE modelTarget variable(s)
CD 2008 CPS 2010 DNGS 2015 JPT 2011 NKmodel SW 2007
One step ahead
Joint 1.078" 0.923** 1.064*** 1.103 0.881*** 0.278
(-0.092") (0.220***) (-0.221) (-0.313**) (0.085) (0.492)
GDP growth 1.082" 0.920* 1.050*** 1.107 0.853*** 0.446
(-0.066) (0.091***) (-0.030) (-0.092*) (0.154***) (-0.630)
Inflation 1.027 0.928* 1.094 1.020 1.001 0.166
(-0.042*) (0.004*) (-0.040**) (-0.058***) (-0.058***) (0.178)
Interest rate 1.202*** 0.989 1.353*** 1.325*** 1.190** 0.074
(0.021***) (0.151***) (-0.115*) (-0.186***) (-0.033) (0.873)
Four steps ahead
Joint 1.010 1.018 1.166*** 1.148** 0.989 0.300
(0.113**) (0.069) (-0.182**) (-0.201*) (0.128) (-0.455)
GDP growth 0.976 1.021 1.135*** 1.141** 0.981 0.457
(0.022) (-0.017) (-0.098**) (-0.166***) (0.015) (-0.672)
Inflation 1.134 1.025 1.255*** 1.261** 0.953 0.166
(0.070) (0.101) (0.015) (-0.054) (0.087) (-0.035)
Interest rate 1.105** 0.997 1.270** 1.094 1.064 0.187
(0.070***) (0.076) (-0.115*) (0.003***) (0.082***) (0.111)
Notes: The table shows root mean squared errors (RMSEs), and average log predictive Bayes factors (LPBFs) in
parentheses, relative to the SW 2007 model. Bold numbers indicate the best performing DSGE model that obtains
the smallest RMSE ratio (largest LPBF). The SW 2007 column shows the actual RMSEs and log predictive scores (LPSs)
of our benchmark. Asterisks indicate statistical significance relative to SW 2007 at the 1% (***), 5% (**), and 10% (*)
significance levels in terms of Diebold and Mariano (1995) tests for RMSEs and Amisano and Giacomini (2007) tests
for LPSs.
Table C.2
Forecasting performance of recursively estimated DSGE models with Hamilton filter detrending.
Target variable(s) DSGE model
CD 2008 CPS 2010 DNGS 2015 JPT 2011 NKmodel SW 2007
One step ahead
Joint 1.235*** 0.909*** 0.979 1.104*** 0.901* 0.340
(-0.428**) (0.310) (-0.023) (-0.452***) (0.020) (-0.494)
GDP growth 1.277*** 0.920** 0.988 1.092* 0.878** 0.519
(-0.214***) (0.110***) (0.014) (-0.116***) (0.141***) (-0.821)
Inflation 1.009 0.827*** 0.900* 1.057 0.889 0.250
(-0.103) (0.088**) (0.059) (-0.094) (-0.006**) (-0.124)
Interest rate 1.332*** 1.033 1.116*** 1.469*** 1.296*** 0.120
(-0.074) (0.136***) (-0.054***) (-0.232***) (-0.104***) (0.376)
Four steps ahead
Joint 1.123 0.984 0.990 1.120*** 0.979 0.418
(0.018) (0.256**) (0.188***) (-0.183***) (0.192***) (-1.780)
GDP growth 1.129 1.009* 1.035 1.139 0.993* 0.552
(-0.131) (-0.016) (-0.022) (-0.223***) (-0.030) (-0.864)
Inflation 0.979 0.825 0.786 1.056 0.779 0.315
(0.160***) (0.287***) (0.168***) (0.010) (0.203***) (-0.577)
Interest rate 1.212 1.038 1.022 1.123 1.084 0.347
(0.020) (0.094*) (0.062) (0.015) (0.098) (-0.560)
Notes: The table shows root mean squared errors (RMSEs), and average log predictive Bayes factors (LPBFs) in
parentheses, relative to the SW 2007 model. Bold numbers indicate the best performing DSGE model that obtains
the smallest RMSE ratio (largest LPBF). The SW 2007 column shows the actual RMSEs and log predictive scores (LPSs)
of our benchmark. Asterisks indicate statistical significance relative to SW 2007 at the 1% (***), 5% (**), and 10% (*)
significance levels in terms of Diebold and Mariano (1995) tests for RMSEs and Amisano and Giacomini (2007) tests
for LPSs.
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J. Capek, J. Crespo Cuaresma, N. Hauzenberger et al. International Journal of Forecasting 39 (2023) 1820-1838
Table C.3
Forecasting performance of recursively estimated DSGE models with demeaned observables.
Target variable(s) DSGE modelTarget variable(s)
CD 2008 CPS 2010 DNGS 2015 JPT 2011 NKmodel SW 2007
One step ahead
Joint 1.206"* 0.801*** 0.889*** 1.031 0.905 0.373
(-0.734***) (0.512***) (0.242) (-0.356***) (-0.112) (-0.298)
GDP growth 1.025 0.794*** 0.875*** 1.015 0.848** 0.602
(-0.057) (0.203***) (0.117***) (-0.020) (0.076*) (-0.919)
Inflation 2.136*** 0.824*** 0.965 1.066 1.216*** 0.216
(-0.598***) (0.119) (0.136**) (-0.071**) (-0.206***) (-0.124)
Interest rate 1.313*** 0.950 1.020 1.465*** 1.217*** 0.091
(-0.104*) (0.168***) (0.012) (-0.234***) (-0.098***) (0.695)
Four steps ahead
Joint 1.059 0.808*** 0.879*** 1.019 1.139** 0.474
(-0.090) (0.751***) (0.723***) (-0.018) (-0.132) (-1.998)
GDP growth 0.827** 0.809*** 0.888*** 0.958 0.988 0.703
(0.082) (0.134) (0.089) (-0.084) (-0.115**) (-1.036)
Inflation 1.978*** 0.755*** 0.834* 1.169 1.700*** 0.280
(-0.223**) (0.429***) (0.405***) (0.078**) (-0.183***) (-0.625)
Interest rate 1.046 0.837** 0.870 1.171*** 1.279*** 0.319
(0.061) (0.163***) (0.174**) (-0.015) (-0.153**) (-0.388)
Notes: The table shows root mean squared errors (RMSEs), and average log predictive Bayes factors (LPBFs) in
parentheses, relative to the SW 2007 model. Bold numbers indicate the best performing DSGE model that obtains
the smallest RMSE ratio (largest LPBF). The SW 2007 column shows the actual RMSE and log predictive scores of
our benchmark. Asterisks indicate statistical significance relative to SW 2007 at the 1% (***), 5% (**), and 10% (*)
significance levels in terms of Diebold and Mariano (1995) tests for RMSEs and Amisano and Giacomini (2007) tests
for log predictive scores (LPSs).
Table C.4
Forecasting performance of the three recursively estimated DSGE models with the baseline data filtering used for
Table 3 relative to the originally proposed model and data filtering.
Target variable(s) DNGS 2015 JPT 2011 SW 2007Target variable(s)
Baseline Original Baseline Original Baseline Original
One step ahead
Joint 0.943* 0.362 1.164*** 0.329 0.944 0.338
(0.470***) (-0.407) (0.114) (-0.441) (0.381) (-0.256)
GDP growth 0.951 0.575 1.195*** 0.516 0.950 0.538
(0.044) (-0.890) (-0.179***) (-0.822) (0.051) (-0.828)
Inflation 0.992 0.208 1.122** 0.192 0.899** 0.216
(0.099**) (-0.041) (0.025) (-0.092) (0.153) (-0.132)
Interest rate 0.660*** 0.139 0.760*** 0.143 0.999 0.085
(0.356***) (0.447) (0.242***) (0.410) (0.101***) (0.719)
Four steps ahead
Joint 0.772*** 0.500 1.027 0.434 0.875*** 0.434
(0.986***) (-1.898) (0.356) (-1.830) (0.742***) (-1.961)
GDP growth 0.918 0.648 1.091" 0.603 0.910* 0.630
(0.117) (-1.025) (-0.105) (-1.001) (0.124) (-0.989)
Inflation 0.639* 0.323 1.292 0.234 0.796* 0.277
(0.257**) (-0.385) (-0.094**) (-0.308) (0.238***) (-0.634)
Interest rate 0.476*** 0.475 0.700** 0.384 0.776** 0.301
(0.609**) (-0.621) (0.391**) (-0.530) (0.289***) (-0.390)
Notes: The table shows root mean squared errors (RMSEs), and average log predictive Bayes factors (LPBFs) in
parentheses of the baseline data filtering relative to the originally proposed data filtering in Del Negro et al. (2015),
Justiniano et al. (2011), and Smets and Wouters (2007), respectively. The columns "Original" show the actual RMSEs
and log predictive scores of these benchmarks. Asterisks indicate statistical significance of the "Baseline" relative to
the "Original" at the 1% (***), 5% (**), and 10% (*) significance levels in terms of Diebold and Mariano (1995) tests for
RMSEs and Amisano and Giacomini (2007) tests for LPSs.
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J. Čapek, J. Crespo Cuaresma, N. Hauzenberger et al.
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