2021 A BVAR TOOLKIT TO ASSESS MACROFINANCIAL RISKS IN BRAZIL AND MEXICO - DOCUMENTOS OCASIONALES N.º 2114
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A BVAR toolkit to assess macrofinancial risks in Brazil 2021 and Mexico Documentos Ocasionales N.º 2114 Erik Andres-Escayola, Juan Carlos Berganza, Rodolfo Campos and Luis Molina
A BVAR toolkit to assess macrofinancial risks in Brazil and Mexico
A BVAR toolkit to assess macrofinancial risks in Brazil and Mexico (*) Erik Andres-Escayola, Juan Carlos Berganza, Rodolfo Campos and Luis Molina BANCO DE ESPAÑA (*) We thank Pedro del Río, Javier Pérez, and Jacopo Timini for their valuable comments and suggestions. The views expressed in this paper are those of the authors and do not represent the view of the Banco de España or the Eurosystem. Corresponding authors: erik.andres@bde.es; jcberganza@bde.es; rodolfo.campos@bde.es; lmolina@bde.es. Documentos Ocasionales. N.º 2114 June 2021
The Occasional Paper Series seeks to disseminate work conducted at the Banco de España, in the performance of its functions, that may be of general interest. The opinions and analyses in the Occasional Paper Series are the responsibility of the authors and, therefore, do not necessarily coincide with those of the Banco de España or the Eurosystem. The Banco de España disseminates its main reports and most of its publications via the Internet on its website at: http://www.bde.es. Reproduction for educational and non-commercial purposes is permitted provided that the source is acknowledged. © BANCO DE ESPAÑA, Madrid, 2021 ISSN: 1696-2230 (on-line edition)
Abstract This paper describes the set of Bayesian vector autoregression (BVAR) models that are being used at Banco de España to project GDP growth rates and to simulate macrofinancial risk scenarios for Brazil and Mexico. The toolkit consists of large benchmark models to produce baseline projections and various smaller satellite models to conduct risk scenarios. We showcase the use of this modelling framework with tailored empirical applications. Given the material importance of Brazil and Mexico to the Spanish economy and banking system, this toolkit contributes to the monitoring of Spain’s international risk exposure. Keywords: macroeconomic projections, risk scenarios, Bayesian vector autoregressions. JEL classification: C32, C53, F44, F47.
Resumen Este documento describe el conjunto de modelos vectoriales autorregresivos bayesianos (BVAR) que se utilizan en el Banco de España para elaborar proyecciones de crecimiento del PIB y simular escenarios de riesgo macrofinanciero para Brasil y México. Esta herramienta está compuesta por un modelo base para elaborar proyecciones de referencia y varios modelos satélite para llevar a cabo escenarios de riesgo. Ilustramos el uso de este marco de modelización con aplicaciones empíricas específicas. Dada la importancia material de Brasil y México para la economía española y su sistema bancario, este conjunto de modelos contribuye al seguimiento de la exposición internacional de España. Palabras clave: proyecciones macroeconómicas, escenarios de riesgo, vectores autorregresivos bayesianos. Códigos JEL: C32, C53, F44, F47.
Contents
Abstract 5
Resumen 6
1 Introduction 8
2 Methodology 10
2.1 Framework 10
2.2 General models 11
2.3 Conditional forecasts 13
2.4 Satellite models 14
2.5 Auxiliary models 16
2.6 Data 18
3 Empirical Results 20
3.1 Validations 20
3.2 Applications 22
4 Concluding Remarks 26
References 27
Appendix A: Forecasting Exercice 30
Appendix B: Historical Decompositions and Potential Output 35
Appendix C: Technical Assumptions 39
Appendix D: Data Details 401 Introduction
The analysis of the Latin American economies has a long tradition at the Banco de España, as
1 Introduction
evidenced by the continued publication of a half-yearly report on the region since September
2003. Over time, the analysis has become more sophisticated which has led to the use
The analysis of the Latin American economies has a long tradition at the Banco de España, as
of macroeconometric
the Eurosystem4 and tools tailored obtained
the forecasts for each country.
from theirAmonguse were thealso
newnottools in use, vector
compatible with
evidenced by the continued publication of a half-yearly report on the region since September
autoregression
assumptions (VAR) models
underlying have proved
the forecasts for the to Spanish
be a suitable
economy andthatflexible
are kit to study
regularly the co-
published
2003. Over time, the analysis has become more sophisticated which has led to the use
movement of macroeconomic variables in these economies.
by Banco de España. The current methodology solves this problem. Moreover, from a
1
Starting with the half-yearly
of macroeconometric tools tailored for each country. Among the new tools in use, vector
report of 2020 (Banco
methodological point ofdeview,
España (2020)),BVAR
the current VAR models
modelshave been and
for Brazil usedMexico
to guide and inform
follow closely
autoregression (VAR) models have proved to be a suitable and flexible kit to study the co-
forecasts
the for the two
methodology largest
of the suiteeconomies
of BVAR modelsof the region:
for the Brazil
Spanish and Mexico.which
economy, In thisis paper,
describedwe
movement of macroeconomic variables in these economies. Starting with the half-yearly 1
in detail the
describe suite of Bayesian
by Leiva-Leon (2017).VAR (BVAR) models that are currently in use to inform those
report of 2020 (Banco de España (2020)), VAR models have been used to guide and inform
projections. 2
These
The toolkit for Brazil two countries
and Mexico are the only countries—other
contains a relatively large than set ofSpain—for
variables (10 which Banco
variables)
forecasts for the two largest economies of the region: Brazil and Mexico. In this paper, we
de España
and includespublishes regularassumptions
all technical forecasts. The advantage
in the of theseB/MPE
Eurosystem’s models overthat global structural
are relevant for
describe the suite of Bayesian VAR (BVAR) models that are currently in use to inform those
modelsand
Brazil such as NiGEM
Mexico. A largeorbenchmark
Global Economic
model that Model 3
includesis all
that they are
variables better
is used adapted toa
to determine
projections. These two countries are the only countries—other than Spain—for which Banco
2
the characteristics
baseline projection for of the
GDP economies of the region
growth, whereas smaller and allow versions
reduced for an introduction
of it are usedoftorelevant
predict
de España publishes regular forecasts. The advantage of these models over global structural
idiosyncratic
the impact ofvariables.
changes ofFurther,
technical thisassumptions
flexibility and empirical
or the nature on
steady-state of BVARs
the growth makes them
forecast.
models such as NiGEM or Global Economic Model is that they are better adapted to
3
better suited
Following for designing
Leiva-Leon (2017),adequate
we define anda reasonable
number of risk scenarios.
satellite models for Brazil and Mexico
the characteristics of the economies of the region and allow for an introduction of relevant
thatThere have
capture thebeen prior in-house
mechanisms of theefforts of using
particular risk macroeconometric
scenario to be simulated. tools to Thestudy Latin
satellite
idiosyncratic variables. Further, this flexibility and empirical nature of BVARs makes them
American
models foreconomies. For example,
scenarios concerning Estrada et
deviations in al.
the(2020) describe
technical BVAR models
assumptions that were
are structurally
better suited for designing adequate and reasonable risk scenarios.
estimated via
identified for sign
Brazil, Mexico, Turkey,
restrictions, Chile, by
as described andUhlig
Peru.(2005).
However, Forthese prior models
the scenario did not
that changes
There have been prior in-house efforts of using macroeconometric tools to study Latin
closely
the tie the forecasts
steady-state of GDPtogrowth,the technical
we useassumptions
an auxiliarythat modelare toused for producing
estimate potentialforecasts
output
American economies. For example, Estrada et al. (2020) describe BVAR models that were
within
using athe Eurosystem.
production Therefore,
function. In this their results
paper, we were difficult
simulate to square
scenarios with
related tothe forecasts
fluctuations
estimated for Brazil, Mexico, Turkey, Chile, and Peru. However, these prior models did not
for foreign
in Brazil demand,
and Mexico in thein (Broad)
changes Macroeconomic
the commodity Projections
prices, shifts Exercise
in domestic (B/MPE)
economic of
policy
closely
1 tie the forecasts to the technical assumptions that are used for producing forecasts
the See for example
Eurosystem the the
and half-yearly
forecastsreport Banco de
obtained España
from their(2016) or thealso
useoutput
were half-yearly
notcouldreport Banco
compatible de
witness with
4
uncertainty
España (2017).
and financial tensions, and long-term effects on as we with
within
2 the Eurosystem. Therefore, their results were difficult to square with the forecasts
We consider
assumptions these two the
underlying Latinforecasts
Americanfor economies as they
the Spanish are of major
economy that importance
are regularly for the Spanish
published
the COVID-19 pandemic.
economy and they represent the largest exposure of the Spanish banking system among emerging economies,
for Brazil and Mexico in the (Broad) Macroeconomic Projections Exercise (B/MPE) of
see Box
by Banco 2 in de
Banco de España (2020).
The rest
3 ofEspaña.
the paper The current methodology
is structured as follows: solves this 2problem.
in section we explainMoreover,
1 These are two popular models used in central banks from the National Institute of Economic and Social
the methodolog-from a
See for example the half-yearly report Banco de España (2016) or the half-yearly report Banco de
Research and Oxford
methodological Economics.
point of view,
España
ical2
(2017).
framework, the details of the
thecurrent BVAR
theoretical models
BVAR for Brazil
models used and Mexico
to carry outfollow closely
the baseline
We consider these two Latin American economies as they are of major importance for the Spanish
the methodology
economy and they
projections of risk
the the suite
and represent the of BVAR
largest
scenarios, models
exposure
and the the for
ofdata thefor
Spanish
used Spanish
banking economy,
system which
among
the empirical is
Indescribed
emerging
exercise. economies,
section 3
see Box 2 in Banco de España (2020).
in
wedetail
3
Theseby
present areLeiva-Leon
the (2017).
twoestimation
popular models usedand
results in central banks
discuss from the National Institute of Economic and Social
Research and Oxford Economics. 1our findings, and in section 4 we conclude.
The
4 toolkitonfor
For details theBrazil and Mexico
Eurosystem’s contains
projection a relatively
exercises large
please refer setwebsite
to the of variables
here. (10 variables)
and includes all technical assumptions in the Eurosystem’s B/MPE that are relevant for
8
1 includes all variables is used to determine a
Brazil and Mexico. A large benchmark model that
BANCO DE ESPAÑA DOCUMENTO OCASIONAL N.º 2114
baseline projection for GDP growth, whereas smaller reduced versions of it are used to predictthe Eurosystem4 and the forecasts obtained from their use were also not compatible with
assumptions underlying the forecasts for the Spanish economy that are regularly published
by Banco de España. The current methodology solves this problem. Moreover, from a
methodological point of view, the current BVAR models for Brazil and Mexico follow closely
the methodology of the suite of BVAR models for the Spanish economy, which is described
in detail by Leiva-Leon (2017).
The toolkit for Brazil and Mexico contains a relatively large set of variables (10 variables)
and includes all technical assumptions in the Eurosystem’s B/MPE that are relevant for
Brazil and Mexico. A large benchmark model that includes all variables is used to determine a
baseline projection for GDP growth, whereas smaller reduced versions of it are used to predict
the impact of changes of technical assumptions or the steady-state on the growth forecast.
Following Leiva-Leon (2017), we define a number of satellite models for Brazil and Mexico
that capture the mechanisms of the particular risk scenario to be simulated. The satellite
models for scenarios concerning deviations in the technical assumptions are structurally
identified via sign restrictions, as described by Uhlig (2005). For the scenario that changes
the steady-state of GDP growth, we use an auxiliary model to estimate potential output
using a production function. In this paper, we simulate scenarios related to fluctuations
in foreign demand, changes in the commodity prices, shifts in domestic economic policy
uncertainty and financial tensions, and long-term effects on output as we could witness with
the COVID-19 pandemic.
The rest of the paper is structured as follows: in section 2 we explain the methodolog-
ical framework, the details of the theoretical BVAR models used to carry out the baseline
projections and the risk scenarios, and the data used for the empirical exercise. In section 3
we present the estimation results and discuss our findings, and in section 4 we conclude.
4
For details on the Eurosystem’s projection exercises please refer to the website here.
2
BANCO DE ESPAÑA 9 DOCUMENTO OCASIONAL N.º 21142 Methodology
In this section, we define the framework and models of the toolkit used for producing baseline
2 Methodology
projections and risk scenarios. We also refer to the data included in the models for the
empirical results.
In this section, we define the framework and models of the toolkit used for producing baseline
projections and risk scenarios. We also refer to the data included in the models for the
2.1 Framework
empirical results.
Our modelling strategy to conduct growth projections and simulate risk scenarios entails
two
2.1 steps. The first step consists of defining a benchmark model capturing most of the
Framework
fundamental macrofinancial relations of the economies of Brazil and Mexico. We then run
Our modelling strategy to conduct growth projections and simulate risk scenarios entails
conditional forecasts where the conditions are anchored to the technical assumptions of the
two steps. The first step consists of defining a benchmark model capturing most of the
Eurosystem’s B/MPE.5 This benchmark model is used to carry out the baseline projection of
fundamental macrofinancial relations of the economies of Brazil and Mexico. We then run
GDP growth. Then, four satellite models are defined which are in essence reduced versions
conditional forecasts where the conditions are anchored to the technical assumptions of the
of the benchmark model. Fewer variables enable for better identification and to focus on
Eurosystem’s B/MPE.5 This benchmark model is used to carry out the baseline projection of
the mechanisms for the particular scenario considered. For each of the satellite models,
GDP growth. Then, four satellite models are defined which are in essence reduced versions
we first simulate a central forecast that includes the same technical assumptions or steady-
of the benchmark model. Fewer variables enable for better identification and to focus on
state as the benchmark model. The risk scenario then includes deviations of the technical
the mechanisms for the particular scenario considered. For each of the satellite models,
assumptions for some specific variables or a different steady-state. Finally, we compute the
we first simulate a central forecast that includes the same technical assumptions or steady-
difference between the central and the scenario forecasts in the satellite model to obtain the
state as the benchmark model. The risk scenario then includes deviations of the technical
net effect that we add to the baseline projection. By isolating this effect, and integrating
assumptions for some specific variables or a different steady-state. Finally, we compute the
it into the baseline projection, we measure the impact of each scenario separately and shed
difference between the central and the scenario forecasts in the satellite model to obtain the
light on the sensitivity of our projections.
net effect that we add to the baseline projection. By isolating this effect, and integrating
The ten model variables are split up into four global and six local ones. For the group
it into the baseline projection, we measure the impact of each scenario separately and shed
of global variables, we construct a measure of foreign demand (EXD) for each country as
light on the sensitivity of our projections.
a weighted average of real imports from the main trade partners. The weights are based
The ten model variables are split up into four global and six local ones. For the group
on bilateral exports. Further, we include oil prices to proxy commodity prices (OIL), a
of global variables, we construct a measure of foreign demand (EXD) for each country as
5
measure for global
For more financial
details on how we turbulences (VIX)
adopt the B/MPE and the
technical US short-term
assumption interest exercises
to our projection rate to proxy
at the
a weighted
Banco average
de España, of real
see Box imports
3 in the from
report on the the
Latinmain tradeeconomy,
American partners. The
Banco weights(2020).
de España are based
foreign monetary policy (FFU). We assume that global variables affect local variables, but
on bilateral exports. Further, we include oil prices to proxy commodity prices (OIL), a
not 5 the other way around (i.e., we assume block 3 exogeneity). In this way, we can better
For more details on how we adopt the B/MPE technical assumption to our projection exercises at the
Banco de the
capture España, see Box 3 inspillover
international the reporteffects
on the Latin
that American economy,
potentially affectBanco de España
the growth (2020).
projections of
Brazil and Mexico. The local variables are real GDP growth (GDP), CPI core inflation
10
BANCO DE ESPAÑA 3
DOCUMENTO OCASIONAL N.º 2114
(INF), bilateral exchange rate vis-à-vis the USD (FXR), the EMBI+ as a measure of the
sovereign spread that proxies external financing costs (EMB), a measure of economic policymeasure
measure for
for global
global financial
financial turbulences
turbulences (VIX)
(VIX) and
and the
the US
US short-term
short-term interest
interest rate
rate to
to proxy
proxy
foreign
foreign monetary
monetary policy
policy (FFU).
(FFU). We
We assume
assume that
that global
global variables
variables affect
affect local
local variables,
variables, but
but
not
not the
the other
other way
way around
around (i.e.,
(i.e., we
we assume
assume block
block exogeneity).
exogeneity). In
In this
this way,
way, we
we can
can better
better
capture
capture the
the international
international spillover
spillover effects
effects that
that potentially
potentially affect
affect the
the growth
growth projections
projections of
of
Brazil
Brazil and
and Mexico.
Mexico. The
The local
local variables
variables are
are real
real GDP
GDP growth
growth (GDP),
(GDP), CPI
CPI core
core inflation
inflation
(INF),
(INF), bilateral
bilateral exchange
exchange rate
rate vis-à-vis
vis-à-vis the
the USD
USD (FXR),
(FXR), the
the EMBI+
EMBI+ as
as aa measure
measure of
of the
the
sovereign
sovereign spread
spread that
that proxies
proxies external
external financing
financing costs
costs (EMB),
(EMB), aa measure
measure of
of economic
economic policy
policy
uncertainty
uncertainty (EPU),
(EPU), and
and the
the policy
policy interest
interest rate
rate (INT)
(INT) for
for the
the two
two Latin
Latin American
American countries.
countries.
In
In Figure
Figure 11 we
we illustrate
illustrate our
our approach
approach to
to producing
producing the
the baseline
baseline projections
projections and
and the
the risk
risk
scenarios.
scenarios.
Figure
Figure 1:
1: Summary
Summary of of the
the modelling
modelling framework
framework to
to produce
produce the
the GDP
GDP growth
growth baseline
baseline pro-
pro-
jections and the risk scenarios of Brazil and Mexico
jections and the risk scenarios of Brazil and Mexico
Notes: The
Notes: The benchmark
benchmark model
model is
is used
used for
for the
the baseline
baseline projection
projection which
which is
is conditioned
conditioned on
on technical
technical assump-
assump-
tions.
tions. The different satellite models are used to produce a central and a scenario forecast that makes
The different satellite models are used to produce a central and a scenario forecast that makes up
up the
the
net
net effect of the risk scenario at hand. This net effect is added to the baseline projection to measure the
effect of the risk scenario at hand. This net effect is added to the baseline projection to measure the
impact
impact of
of the
the risk
risk scenario.
scenario.
44
2.2 General Models
For the projection exercises we estimate large VARs using Bayesian inference. These models
have become an essential empirical tool for central banks to conduct macroeconomic analysis.
As explained in detail in Bańbura et al. (2010) and Giannone et al. (2015) larger systems
and Bayesian estimation are extremely appealing for forecasting applications. Based on
BANCO DE ESPAÑA 11 DOCUMENTO OCASIONAL N.º 2114
marginal likelihoods and in-sample criteria, we specify models with four lags and Minnesota-
type priors. These priors are the most common ones used in the literature and they assumehave become an essential empirical tool for central banks to conduct macroeconomic analysis.
As explained in detail in Bańbura et al. (2010) and Giannone et al. (2015) larger systems
and Bayesian estimation are extremely appealing for forecasting applications. Based on
marginal likelihoods and in-sample criteria, we specify models with four lags and Minnesota-
type priors. These priors are the most common ones used in the literature and they assume
2.2 the
that General Models
VAR coefficients behave according to a normal distribution.6 Then, it is left to
the researcher to specify the values for the mean and covariance of the distribution, known
For the projection exercises we estimate large VARs using Bayesian inference. These models
as the hyperparameters. For the Minnesota-type prior the hyperparameter values follow a
have become an essential empirical tool for central banks to conduct macroeconomic analysis.
certain rationale.7
As explained in detail in Bańbura et al. (2010) and Giannone et al. (2015) larger systems
Finally, Bayes’ formula is applied to combine information of the prior distribution and
and Bayesian estimation are extremely appealing for forecasting applications. Based on
the likelihood function, resulting in a posterior distribution. From this latter distribution
marginal likelihoods and in-sample criteria, we specify models with four lags and Minnesota-
one obtains draws to compute the functions and estimates of interest (i.e., impulse response
type priors. These priors are the most common ones used in the literature and they assume
functions (IRFs) and forecasts).8 In practical terms, the estimation is implemented using
that the VAR coefficients behave according to a normal distribution.6 Then, it is left to
Markov chain Monte Carlo (MCMC) algorithms (i.e., Gibbs sampling).9
the researcher to specify the values for the mean and covariance of the distribution, known
The multivariate time-series model for periods t = 1, . . . , T consists of a n × 1 vector of
as the hyperparameters. For the Minnesota-type prior the hyperparameter values follow a
variables {yt }Tt=1 , n × n matrices of coefficients Ai , i = 1, . . . , p capturing the dynamics of the
certain
pth rationale.
lagged
7
system, a n × 1 vector of constants C and a n × 1 vector of error terms εt with
6
In contrast
pth lagged to theafrequentist
system, n × 1 vector approach, Bayesian C
of constants inference
and a treats
n × 1 the data of
vector as error
deterministic
terms εand the
t with
Finally,
parameter Bayes’
space as formula
stochastic. is
This applied
type of to combine
inference has information
become of
increasingly the prior
popular
zero-mean and positive-definite n × n covariance matrix Σ. This data generating process is as adistribution
shrinkage and
method
to resolve overparametrization
zero-mean issues, noften
and positive-definite × nencountered
covarianceinmatrix
empirical
Σ.macroeconomics. Given that
This data generating the time
process is
the likelihood
series in Brazil function,
and Mexico resulting
are in
relatively a posterior
short
assumed to evolve according to the following VAR(p): compareddistribution.
to advanced From this
economies, latter
the use distribution
of a shrinkage
method is particularly
assumed appropriateto
to evolve according in the
our application.
following VAR(p):
one7obtains
Littermandraws
(1986)to compute
describes the the functions
specific and
structure estimates of interest
of hyperparameters (i.e., impulse
in the Minnesota prior.response
They are
applied to the whole block of coefficients and are therefore a global shrinkage method to reduce overfitting.
functions
For (IRFs)
the Minnesota and
yt =prior, forecasts).
+ the
8
In practical
residual covariance matrix terms,
is estimatedthe from
estimation
the εmodel
with is via
implemented
OLS, whereasusing other
(1)
C A 1 yt−1 + A2 yt−2 + · · · + Ap yt−p + εt , t ∼ N (0, Σ).
approaches treat yt is= asC unknown,
+ A1 yt−1and +A assume
y
2 t−2
an
+ ·inverse
· · + A Wishart
y
p t−p + distribution
εt , with as
εt
the
∼ prior.
N (0, We
Σ). choose the hy-
(1)
Markov chainbyMonte
perparameters runningCarlo
a grid (MCMC)
search algorithmalgorithms (i.e., Gibbs
that automatically sampling).
selects suitable9hyperparameters based
on the
Themarginal likelihood. model
open-economy Then, we partly adjust
contains the hyperparameter
variables for the global values
blockbased(G)onand this for
information.
the local
8
The multivariate
Usually, the mediantime-series
open-economy ofmodel model
contains
the posterior is for periods
variables
reported and fortused
=the1,asglobal
. . the consists
block
. , T point (G)of and
estimate. a nNote× vector
for1 that
the local
underof
Bayesian inference
block (L) consisting the model uncertainty is much better captured since one computes
of variables for Brazil (BR) and Mexico (MX). Hence, yt = {ytG , ytL }, with an entire
G distribution
L
variables
block
for the (L) t }t=1 , n ×
consisting
parameters.
{y T
matricesfor
ofnvariables of Brazil
coefficients
(BR) A and
i , i Mexico , p capturing
= 1, . . .(MX). Hence,the yt =dynamics
{yt , yt },ofwith
the
9
For the computational implementation of the models we used the
L =6 (BR, M X). Each superscript indicates a group of variables (10 variables in total, 4 in developer version of the BEAR toolbox.
= In
Refer
L tocontrast
Dieppe
(BR, M X). to al.
et the
Each frequentist
(2016)superscriptapproach,
for further Bayesian
indicates
details. inference
a group treats the (10
of variables datavariables
as deterministic
in total, and4the
in
parameter space as stochastic. This type of inference has become increasingly popular as a shrinkage method
the G group and 6 in the L group). In particular we have ytG = (EXDt , OILt , V IXt , F F Ut )⊤
G
to resolve
the G groupoverparametrization
and 6 in the L issues,group). often
In encountered
particular we in empirical
have yt macroeconomics.
= (EXDt , OILGiven that the time
t , V IXt , F F Ut )
⊤
series in
L Brazil and Mexico are relatively short compared to ⊤advanced economies, the use of a shrinkage
and ytL = (GDPt , IN Ft , F XRt , EM Bt , EP Ut , IN Tt )⊤ in the case of the benchmark model
method
and y is=particularly
(GDPt , IN appropriate
Ft , F XRin our application.
t , EM Bt , EP Ut5, IN Tt ) in the case of the benchmark model
7 t
Litterman (1986) describes the specific structure of hyperparameters in the Minnesota prior. They are
and subsets from both blocks in the case of the satellite models. The block exogeneity
applied
and to the whole
subsets from blockbothofblocks
coefficients
in theand case
are therefore
of the asatellite
global shrinkage
models.method The toblock reduceexogeneity
overfitting.
For the Minnesota prior, the residual covariance matrix is estimated from the model via OLS, whereas other
assumption entails imposing zero-restrictions on the coefficient matrices so to cancel out
approaches treat
assumption is as unknown,
entails imposingandzero-restrictions
assume an inverse on Wishart distribution as
the coefficient the prior.soWe
matrices tochoose
cancel theout
hy-
perparameters 10by running aGgrid search algorithm that automatically selects suitable hyperparameters based
past effects10 of ytL on ytG :
L
on theeffects
past marginal of likelihood.
yt on ytThen, : we partly adjust the hyperparameter values based on this information.
8
Usually, the median of the posterior is reported and used as the point estimate. Note that under
Bayesian inference themodel uncertainty
is much better captured since one computes
G an entire distribution
p
for the parameters. ytG
G
c
∑ ∑ p
a 0 t−i y G
εG
1 11,i G t
9
For the computational yt
= c1
implementation
+ of the a11,i we0used
models
theyt−i +
developer
εt . of the BEAR toolbox.
version (2)
Refer to Dieppe et al. (2016) L = +
ytL for further
c2 details. i=1 a21,i a22,i L + L .
yt−i εLt (2)
L
yt c2 i=1 a21,i a22,i yt−i εt
One might 12
BANCO DE ESPAÑA be interested in having an economic interpretation of the shocks, and for
DOCUMENTO OCASIONAL N.º 2114
One might be interested in having an economic5 interpretation of the shocks, and for
this, one needs to define the contemporaneous relationships between variables. Let Dj , j =
this, one needs to define the contemporaneous relationships between variables. Let Dj , j =
0, . . . , p capture such structural relationships, K be the constants and ηt be the structuralThe open-economy model contains variables for the global block (G) and for the local
block (L) consisting of variables for Brazil (BR) and Mexico (MX). Hence, yt = {ytG , ytL }, with
L = (BR, M X). Each superscript indicates a group of variables (10 variables in total, 4 in
the G group and 6 in the L group). In particular we have ytG = (EXDt , OILt , V IXt , F F Ut )⊤
pth lagged system, a n × 1 vector of constants C and a n × 1 vector of error terms εt with
and ytL = (GDPt , IN Ft , F XRt , EM Bt , EP Ut , IN Tt )⊤ in the case of the benchmark model
zero-mean and positive-definite n × n covariance matrix Σ. This data generating process is
and subsets from both blocks in the case of the satellite models. The block exogeneity
assumed to evolve according to the following VAR(p):
established by
assumption entails DDi , C =zero-restrictions
Ai =imposing DK, εt = Dηt on andthe DΓD⊤ , with
Σ =coefficient matrices D0−1to. To
D = so recover
cancel out
the structural
past effects10 ofmodel ytG : its reduced-form or disentangle ηt from εt , one needs to adopt an
ytL onfrom
yt = C + A1 yt−1 + A2 yt−2 + · · · + Ap yt−p + εt , with εt ∼ N (0, Σ). (1)
identification strategy to
restrict the contemporaneous
matrix D.
p
The open-economy
G
yt model ccontains
1 ∑variables
a11,i for 0 G
theglobal
yt−i blockG
εt (G)
and for the local
= + + . (2)
2.3 Conditional y L
Forecasts
block (L) consisting oftvariables 2for Brazil (BR)
c i=1 a21,iand22,i
a y L
Mexico t−i ε L
(MX). Hence,
t yt = {yt , yt }, with
G L
L = (BR,
Note MtheX).
thatmight Each projection
baseline superscriptand indicates a group of variables (10out
variables inconditional
total, 4 in
One be interested in havingthe anrisk scenarios
economic are carried
interpretation through
of the shocks, and for
the G
forecasts group and 6 in the L group). In particular we have y G
= (EXD , OIL , V IX , F F Ut ) ⊤
one in termsto of sequences of observables, asrelationships
describedt bybetween
Waggoner and ZhaLet (1999).
Dj , j In
t t t
this, needs define the contemporaneous variables. =
and L
yt = (GDP
particular, t , IN Ft , F XR
the conditional t , EMwhen
forecasts Bt , EP Ut , IN
using Tt ) in the describe
this
⊤
case of the benchmark model
0, . . . , p capture such structural relationships, K betechnique
the constants andthe ηt most
be the likely future
structural
and
path subsets from both blocks
of the unconditioned variablein given
the case of the imposed
conditions satellite for
models.
the restThe variables
block exogeneity
innovations with zero-mean and structural covariances Γ. Therefore, theofstructural (i.e.,
VAR(p)the
assumption
likely future entails imposing zero-restrictions on the coefficient matrices so to cancel out
is given by: path of GDP growth given the assumed path of the rest of the variables). As
past effectsin10Antolín-Díaz
explained of ytL on ytG : et al. (2021), the structural identification of D does not play a role
in how theDconditional arey constructed
forecasts +
using
this withη ∼
ηt , approach. N (0, Γ). structural
0 yt = K + D 1 yt−1 + D 2 t−2 + · · · + Dp yt−p t However, (3)
p
y G
t 1 c ∑ a
11,i 0 y G
t−i t ε G
identification still
helpsto= gain
insights
+ on why weobserve
this
+ forecasting
. (2)
path for the
ytL c2 i=1 a21,i a22,i L
yt−i εLt
As a result,variable
unconditioned the linkinbetween the reduced-form
the context of a “what if”VAR and the
scenario. structural
Therefore, forcounterpart
11
the benchmark is
10
This restriction
established bythe implies
DDithat
, C local variables are not Granger
and causing
DΓD⊤global
, withvariables.
model wemight
run i central forecasts based on thet technical assumptions and obtain0 .the
Tobaseline
recover
−1
11One
A
be =interested =
in DK,
havingεt =
an Dηeconomic Σ=interpretation Dthe
of = Dshocks, and for
The key difference between equation 1 and 3 is that the reduced-form VAR is a mere statistical model,
for which estimation is possible. Conversely,
the structural
projection
this, model
without
one needs fromthe
adopting
to define itsany specificthe
reduced-form structural
structural
contemporaneous
augmentation
or disentangle frommakes
identification.
relationships
theneeds
εt , one
Contrary,
ηt between modelto
economically
Letadopt
for the
variables. Dj , j an
satellite=
identification
models
. . . , p used strategy
for
capture to restrict
simulating the contemporaneous
risk scenarios,
such structural it makes
relationships, sensematrix
to identify
D. structural shocks to gain
0, 6 K be the constants and ηt be the structural
additional insights
innovations regardingand
with zero-mean the structural
impact of covariances
the scenariosΓ.on the baseline
Therefore, the projections.
structural VAR(p)
2.3 Conditional Forecasts
is given by:
2.4 Satellite
Note that Models
the baseline projection and the risk scenarios are carried out through conditional
forecasts 0 yt = of
inDterms K sequences
+ D1 yt−1 + 2 yt−2 + · · · +
ofDobservables, asD p yt−p + ηtby
described with ηt ∼ and
, Waggoner Γ). (1999). (3)
N (0,Zha In
In the spirit of Leiva-Leon (2017) we partially set-identify the satellite models using sign
particular, the
restrictions. 12 conditional forecasts when using this technique describe the most likely future
For conducting the sign restriction identification, one needs to focus on εt =
As a result, the link between the reduced-form VAR and the structural counterpart11 is
path
Dηt toofrestrict
the unconditioned variable given
the contemporaneous conditions
impact matrix imposed forcase
D. For the the rest of variables (i.e.,
of sign-identified the
models
10
This restriction implies that local variables are not Granger causing global variables.
likely
we 11
future
The
have key
D= path
P Q, of
difference GDPP growth
between
where equation
is given
1 and the
the Cholesky assumed pathofofΣthe
3 isdecomposition
that the reduced-form rest
VAR
and of
Qis is athe
a mere variables).
n ×statistical As
model,
n orthogonal
for which estimation is possible. Conversely, the structural augmentation makes the model economically
interpretableinbut
explained Antolín-Díaz
not feasible for et estimation.
al. (2021),These
the structural identification
types of models of D does
are the workhorse not playmacroe-
for empirical a role
conomics since their introduction by Sims (1980).
in 12how
There theis an
conditional forecasts
ongoing discussion arehow
about constructed
6 using
appropriate this approach.
this identification However,
strategy is, see Frystructural
and Pagan
(2011), Baumeister and Hamilton (2015), Uhlig (2017), Inoue and Kilian (2020). Yet for the purpose of our
identification
analysis this 13
BANCO DE ESPAÑA
still helpsscheme
identification to gain
DOCUMENTO OCASIONAL N.º 2114
insights
is flexible and on why we observe this forecasting path for the
well-established.
unconditioned variable in the context of a “what if” scenario. Therefore, for the benchmark
7
model we run the central forecasts based on the technical assumptions and obtain the baselineforecasts in terms of sequences of observables, as described by Waggoner and Zha (1999). In
particular, the conditional forecasts when using this technique describe the most likely future
path of the unconditioned variable given conditions imposed for the rest of variables (i.e., the
likely future path of GDP growth given the assumed path of the rest of the variables). As
explained in Antolín-Díaz et al. (2021), the structural identification of D does not play a role
established by Ai = DDi , C = DK, εt = Dηt and Σ = DΓD⊤ , with D = D0−1 . To recover
in how the conditional forecasts are constructed using this approach. However, structural
established
the by model
structural Ai = DDfromi , its
C= reduced-form
DK, εt = Dη and Σ = DΓD
ort disentangle , with
ηt from
⊤
εt , one D0−1 .toToadopt
D =needs recover
an
identification still helps to gain insights on why we observe this forecasting path for the
the structural strategy
identification model from its reduced-form
to restrict or disentangle
the contemporaneous ηt from
matrix D. εt , one needs to adopt an
unconditioned variable in the context of a “what if” scenario. Therefore, for the benchmark
identification strategy to restrict the contemporaneous matrix D.
model we run the central forecasts based on the technical assumptions and obtain the baseline
2.3 Conditional Forecasts
projection without adopting
2.3 Conditional any specific structural identification. Contrary, for the satellite
Forecasts
Note that the baseline projection and the risk scenarios are carried out through conditional
models used for simulating risk scenarios, it makes sense to identify structural shocks to gain
Note thatinthe
forecasts baseline
terms projection
of sequences and the risk as
of observables, scenarios areby
described carried out through
Waggoner and Zhaconditional
(1999). In
additional insights regarding the impact of the scenarios on the baseline projections.
forecasts
particular,inthe
terms of sequences
conditional of observables,
forecasts when usingasthis
described by describe
technique Waggonertheand Zhalikely
most (1999). In
future
particular,
path of the the conditional variable
unconditioned forecastsgiven
whenconditions
using this imposed
techniquefordescribe
the restthe most likely
of variables future
(i.e., the
2.4 Satellite Models
path
likelyoffuture
the unconditioned
path of GDPvariablegrowth given conditions
the assumed imposed
path of forthe
therest
restofof the
variables (i.e., the
variables). As
In the spirit of Leiva-Leon (2017) we partially set-identify the satellite models using sign
likely future
explained path of GDPetgrowth
in Antolín-Díaz giventhe
al. (2021), thestructural
assumed identification
path of the rest of Dofdoes
the not
variables).
play a roleAs
restrictions. For conducting the sign restriction identification, one needs to focus on εt =
12
explained
in how theinconditional
Antolín-Díaz et al. (2021),
forecasts the structural
are constructed using identification
this approach. of D However,
does not play a role
structural
Dηt to restrict the contemporaneous impact matrix D. For the case of sign-identified models
in how the conditional
identification still helpsforecasts are constructed
to gain insights on why using this approach.
we observe However,
this forecasting pathstructural
for the
we have D = P Q, where P is the Cholesky decomposition of Σ and Q is a n × n orthogonal
identification
unconditionedstill helpsintothe
variable gain insights
context of aon whyif”
“what wescenario.
observe Therefore,
this forecasting
for thepath for the
benchmark
interpretable but not feasible
⊤ for estimation. These types of models are the workhorse for empirical macroe-
matrix such that QQ = In , with In being the n-dimensional identity matrix. Then, to
unconditioned
model wesince
conomics runtheir variable
the central inforecasts
introductionthebycontext
Simsbasedofon
a “what
(1980). if” scenario.
the technical Therefore,
assumptions for thethe
and obtain benchmark
baseline
12
There is an ongoing discussion about how appropriate this identification
achieve identification rewrite the vector of structural impulse responses as Ξ = f (A, Σ, strategy is, see Fry and Pagan
Q),
model
(2011), we run
projection
Baumeister theand
without central
adoptingforecasts
Hamilton any
(2015), based
Uhligon
specific the technical
structural
(2017), andassumptions
Inoueidentification. and
Contrary,
Kilian (2020). obtain
Yet for the baseline
for purpose
the satellite
of our
analysisfthis
where identification
(·) is a non-linear scheme is flexible
function, A= and
(Awell-established.
1 , A2 , . . . , An ) contains the coefficient matrices, Σ
projection
models used without adoptingrisk
for simulating anyscenarios,
specific structural
it makes sense identification.
to identifyContrary,
structuralfor the satellite
shocks to gain
is the covariance of reduced-form errors, and Q is the matrix from which to obtain candidate
models
additionalused for simulating
insights regarding risk
thescenarios,
impact of it the
makes sense toonidentify
7 scenarios structural
the baseline shocks to gain
projections.
draws that satisfy the imposed sign restrictions. Refer to Kilian and Lütkepohl (2017) for
additional insights regarding the impact of the scenarios on the baseline projections.
an extensive discussion in the identification and estimation of sign-restricted models and to
2.4 Satellite Models
Arias
2.4 etSatellite
al. (2018) for the practical implementation we use for this class of models.
Models
In the spirit of Leiva-Leon (2017) we partially set-identify the satellite models using sign
Moving to the details of the specification of the satellite models, each one of them contains
In the spirit12ofFor
restrictions. Leiva-Leon
conducting (2017) we partially
the sign restrictionset-identify the satellite
identification, one needsmodels using
to focus on εsign
t =
a set of core variables shared across all satellite models and then additional variables included
Dηt to restrict For
restrictions. 12
conducting the sign
the contemporaneous restriction
impact matrixidentification,
D. For the case oneofneeds to focus on
sign-identified models
εt =
explicitly for the risk scenario. The core variables are GDP growth, inflation, and interest
we to restrict
Dηthave D = Pthe contemporaneous
Q, where impactdecomposition
P is the Cholesky matrix D. For of theΣcase
and of
Q sign-identified models
is a n × n orthogonal
rates. As is common in the literature, these three variables are used to identify demand, cost-
we have D =
interpretable butPnot where P
Q, feasible forisestimation.
the Cholesky
Thesedecomposition of Σ
types of models are theand Q is afor
workhorse n× n orthogonal
empirical macroe-
push, and
conomics monetary
since policy shocks.
their introduction by SimsThen,
(1980).we augment this core setup with scenario-specific
interpretable
12
There is but not feasible
an ongoing for estimation.
discussion about howThese types ofthis
appropriate models are the workhorse
identification forsee
strategy is, empirical
Fry andmacroe-
Pagan
variables
conomics
(2011), to identify
since theirand
Baumeister additional
introduction
Hamilton by structural
SimsUhlig
(2015), (1980).shocks.
(2017), Weand
Inoue specify
Kilian the following
(2020). Yet for satellite models:
the purpose of our
12
There
analysis thisisidentification
an ongoing discussion
scheme isabout how
flexible appropriate
and this identification strategy is, see Fry and Pagan
well-established.
(i) the Baumeister
(2011), external model with y(2015),
and Hamilton Uhlig
external,t (2017),tInoue
= (EXD , GDPand Kilian
t , IN (2020).
Ft , IN where
Tt )⊤Yet for thewepurpose
identify an
of our
analysis this identification scheme is flexible and well-established.
external shock,
14
BANCO DE ESPAÑA
7 ycommodity,t = (OILt , GDPt , IN Ft , IN Tt )⊤
(ii) the commodity model with
DOCUMENTO OCASIONAL N.º 2114
7 the uncertainty model with yuncertainty,t =
where we identify a commodity shock, and (iii)
(GDPt , IN Ft , EM Bt , EP Ut , IN Tt )⊤ where we identify financial risk and economic policy riskexplicitly for the risk scenario. The core variables are GDP growth, inflation, and interest
rates. As is common in the literature, these three variables are used to identify demand, cost-
push, and monetary policy shocks. Then, we augment this core setup with scenario-specific
variables to identify additional structural shocks. We specify the following satellite models:
(i) the external model with yexternal,t = (EXDt , GDPt , IN Ft , IN Tt )⊤ where we identify an
external shock, (ii) the commodity model with ycommodity,t = (OILt , GDPt , IN Ft , IN Tt )⊤
where we identify a commodity shock, and (iii) the uncertainty model with yuncertainty,t =
(GDPt , IN Ft , EM Bt , EP Ut , IN Tt )⊤ where we identify financial risk and economic policy risk
shocks. In the fourth scenario concerning the long-term effects of output, we use (iv) the
potential output model which has the same ten variables as the benchmark model ypotential,t =
yt . Given that in this particular scenario we are concerned with deviations from the steady-
state and not in deviations from the technical assumptions, we implement an alternative
modelling strategy detailed later.
Following the literature described in Leiva-Leon (2017) we establish the structural iden-
tification schemes for the satellite models (i) - (iii):
External:
∗ ∗
∗ ηexternal,t
εexd,t
+
εgdp,t + + − − ηdemand,t
= . (4)
ε η
inf,t ∗ + + − cost−push,t
εint,t ∗ 8
+ + + ηmonetary,t
Commodity:
εoil,t + ∗ ∗
∗ ηcommodity,t
εgdp,t + + − − ηdemand,t
= . (5)
+ + − ηcost−push,t
εinf,t +
εint,t ∗ + + + ηmonetary,t
Uncertainty:
ε + − − − − η
gdp,t demand,t
εinf,t + + ∗ ∗ − ηcost−push,t
ε
emb,t = ∗ ∗ + ∗ ∗ ηf in−risk,t
. (6)
εepu,t ∗ ∗ ∗ + ∗ ηpol−risk,t
εint,t + + ∗ ∗ + ηmonetary,t
Each entry in the D matrix describes how the responses to the structural shocks will
15
be restricted in relation to the observables. That is, the candidate draws from Q have to
BANCO DE ESPAÑA DOCUMENTO OCASIONAL N.º 2114
satisfy the positive and negative sign restrictions in order to be accepted and they are thenepu,t pol−risk,t
εint,t + + ∗ ∗ + ηmonetary,t
Each entry in the D matrix describes how the responses to the structural shocks will
be restricted in relation to the observables. That is, the candidate draws from Q have to
satisfy the positive and negative sign restrictions in order to be accepted and they are then
2.5 Auxiliary Models
used to obtain Ξ from the posterior distribution. The symbol “*” denotes that the entry
Lastly,
in for theisscenario
the matrix on the long-term
left unrestricted. effects
With these on output we
sign-identified use the
satellite potential
models, output
we run the
satelliteforecasts
central model which is based
following the on an alternative
technical BVARand
assumptions model. Specifically,
the scenario since based
forecasts our goal
on
is to impose
deviations a different
from steady-state
the technical than for
assumptions thesome
one from the baseline
variables. projection
The difference we decide
between these
to depart
two from
forecasts the standard
results in the netBVAR
effect described above
that is added to and use the mean-adjusted
the baseline projection and BVAR of
measures
Villani
the (2009).
impact In this
of the risk alternative
scenario. representation of the model, the unconditional mean of the
vector of variables is E(yt ) = µ, where µ is an n × 1 vector of constants with the steady-state
2.5 Auxiliary
values.13 Models
Hence, we can now include information on the steady-state through the prior mean
of µ. for the scenario on the long-term effects on output we use the potential output
Lastly,
9
satellite model which is based on an alternativeand
Next, we obtain potential output for Brazil Mexico
BVAR usingSpecifically,
model. an auxiliarysince
model.
ourUsing
goal
historical
is data,
to impose we fit a calibrated
a different Cobb-Douglas
steady-state production
than the one from thefunction
baselineand apply theweHodrick-
projection decide
Prescott
to depart(HP)
fromfilter
the with smoothing
standard BVARparameter
describedλabove to the
and
= 1600 use production factors toBVAR
the mean-adjusted compute
of
the long-run
Villani trend
(2009). of output.
In this As explained
alternative by Tóth
representation (2021),
of the this
model, theisunconditional
a common approach
mean ofused
the
by international
vector of variablesorganizations
is E(yt ) = µ,to obtain
where µ isa an
measure of potential
n × 1 vector output.
of constants Assuming
with that the
the steady-state
time-series
values.13 considered
Hence, we cancan
nowbeinclude
decomposed into a on
information cycle
theand a trend component,
steady-state through thewe specify
prior meana
production
of µ. function with constant returns to scale:
Next, we obtain potential output for Brazil and Mexico using an auxiliary model. Using
Y ∗ = A∗ K ∗α L∗1−α , (7)
historical data, we fit a calibrated Cobb-Douglas production function and apply the Hodrick-
Prescott (HP) filter with smoothing parameter λ = 1600 to the production factors to compute
where the asterisk denotes the trend component of a variable, and Yt refers to output, At to
the long-run trend of output. As explained by Tóth (2021), this is a common approach used
Total Factor Productivity (TFP), Kt to capital, and Lt to labor. The parameter α is the
by international organizations to obtain a measure of potential output. Assuming that the
share of capital in output.
time-series considered can be decomposed into a cycle and a trend component, we specify a
We first construct the capital series using the perpetual inventory method, that is, iter-
production function with constant returns to scale:
atively applying the law of motion of capital:
Y ∗ = A∗ K ∗α L∗1−α , (7)
Kt = (1 − δ)Kt−1 + It , t ≥ 1, (8)
13
This relation is straight forward because the only exogenous component of our model is the vector of
where theC,asterisk
constants denotes
for the full the oftrend
derivation component
the model refer to of a variable,
Villani (2009). and Yt refers to output, At to
Total Factor Productivity (TFP), Kt to capital, and Lt to labor. The parameter α is the
16 10
share of capital in output.
BANCO DE ESPAÑA DOCUMENTO OCASIONAL N.º 2114
We first construct the capital series using the perpetual inventory method, that is, iter-Y ∗ = A∗ K ∗α L∗1−α , (7)
where the asterisk denotes the trend component of a variable, and Yt refers to output, At to
Total Factor Productivity (TFP), Kt to capital, and Lt to labor. The parameter α is the
share of capital in output.
We first construct the capital series using the perpetual inventory method, that is, iter-
atively applying the law of motion of capital:
Kt = (1 − δ)Kt−1 + It , t ≥ 1, (8)
13
This relation is straight forward because the only exogenous component of our model is the vector of
constants C, for the full derivation of the model refer to Villani (2009).
where It denotes investment and δ is the depreciation rate of capital. We initialize K0 in (8)
with the pre-sample historical average. We compute
10 TFP by the Solow residual:
Yt
At = . (9)
Ktα L1−α
t
The series for the labor input is observable, but we need to pin down values for α and δ to
determine the evolution of capital and TFP. Following the common practice in neoclassical
growth models we calibrate these parameters by computing the averages of the right-hand
side terms in the following equations:
w t Lt
α=1− , (10)
Yt
It 1
( − 1)
(11)
Yt β
δ= ,
α YItt
1
β= , (12)
1 + rt
where wt is the real wage rate and rt is the real rental rate of capital. For the forecasting
periods, we iterate forward the series of each factor input using long-term growth rates
proxied by historical averages. Subsequently, we apply the HP filter to the factor input series
and substitute their trend components into the Cobb-Douglas production function (7). In
this way, we obtain potential output Y ∗ for both countries over the historical and forecasting
periods.
Using this auxiliary model, we calculate steady-states as the average of potential output
growth rates over all periods. To compute an alternative steady-state from a simulated
scenario, we assume that factor inputs evolve at different long-term growth rates than the
BANCO DE ESPAÑA 17 DOCUMENTO OCASIONAL N.º 2114
ones assumed initially. This results in different trend components of the factor inputs and
therefore yields an alternative potential output series. The differences between these averagesUsing this auxiliary model, we calculate steady-states as the average of potential output
growth rates over all periods. To compute an alternative steady-state from a simulated
scenario, we assume that factor inputs evolve at different long-term growth rates than the
ones assumed initially. This results in different trend components of the factor inputs and
therefore yields an alternative potential output series. The differences between these averages
2.6 Data
determine the deviation of the steady-state from the baseline projection. We introduce it
into the mean-adjusted BVAR through alternative priors for µ.
We use quarterly data from 2000Q1 to 2020Q4, but the estimation sample runs until 2019Q4
and the year 2020 is included as conditional paths
11 for the forecasting exercises. We do so to
2.6 Data
avoid problems in the estimation process and the stability of the parameters in the model
We
due use quarterly
to outliers in data
mostfrom 2000Q1
variables to 2020Q4,
during but the period.
the COVID-19 estimation
14 sample
Given runs
that theuntil
focus2019Q4
of the
and theis year
paper 2020 is included
to illustrate as conditional
the functionalities paths
of the for the
toolkit andforecasting exercises.
the simulation Wescenarios
of risk do so to
avoid problems
rather in the estimation
than accurately forecastingprocess and thecurrent
GDP during stability of thethis
times, parameters
approachinisthe
themodel
most
due to outliers
appealing in most
for our variables during
applications. Yet in the
our COVID-19 reports ofGiven
half-yearly period.
14
Bancothat
dethe focus (2020)
España of the
paper is to de
and Banco illustrate
Españathe functionalities
(2021), of the
we explicitly dealtoolkit and current
with this the simulation of risk there
issue because scenarios
our
rather
growth than accurately
projections forecasting
reflect our most GDP during
updated current
stance on thetimes,
futurethis approach
evolution is the
of the most
Brazilian
appealing for economies.
and Mexican our applications. Yet on
For details in how
our half-yearly
we do this, reports
refer to of
theBanco de España (2020)
reports.
andData
Banco de España
included have(2021), we explicitly
the following dealWe
features. with thisseasonally
took current issue because
adjusted real there
GDP,our
as
growth projections
published reflect ourstatistics
by the respective most updated
offices.stance onseasonally
To get the future adjusted
evolutioncore
of the
CPIBrazilian
we use
and MexicanX-13-ARIMA-SEATS
the method economies. For detailsfrom
on how we do this,
JDemetra+. 15 refer to the reports.
For other high-frequency variables,
Data
daily includeddata
or monthly haveare
theaveraged
followingtofeatures. We took
get quarterly series.seasonally adjusted
The growth realexpressed
rates are GDP, as
published by the respective
as quarter-on-quarter statistics
percentages. offices. we
Moreover, To test
get seasonally adjusted
for stationarity usingcore
theCPI we use
augmented
the method X-13-ARIMA-SEATS
Dickey-Fuller from JDemetra+.15 For other
(ADF) and Kwiatkowski-Phillips-Schmidt-Shin high-frequency
(KPSS) variables,
tests and in the latter,
daily
there or monthlyempirical
is enough data areevidence
averagedtotoreject
get quarterly series.ofThe
the hypothesis unitgrowth rates
roots at a 5%aresignificance
expressed
as quarter-on-quarter percentages. Moreover, we test for stationarity using the augmented
level.
Dickey-Fuller
Our data is(ADF) and Kwiatkowski-Phillips-Schmidt-Shin
downloaded from international sources as well(KPSS) tests sources.
as national and in the latter,
Starting
therethe
with is enough empirical evidence
global variables, to reject
for the foreign the hypothesis
demand, of unit
we use real rootsofatgoods
imports a 5%and
significance
services
level.
from national sources. The bilateral exports data to construct the weights in the foreign
Our data
demand is downloaded
measure come fromfrom international
the IMF’s sources
Direction as well
of Trade as national
Statistics sources.
(DOTS). We Starting
consider
with the global
the main variables,for
trade partners forboth
the countries:
foreign demand, we we
in Brazil usetake
real the
imports of goods
US, Euro Area,and
andservices
China
from
14 national sources. The bilateral exports data to construct the weights in the foreign
For a more thorough treatment on how to perform estimation and forecasts with VARs in the context
of the COVID-19 pandemic see Lenza and Primiceri (2020) and Primiceri and Tambalotti (2020).
demand
15
This measure
software iscome
a toolfrom the IMF’s
specifically Direction
designed of Trade
for the seasonal Statisticsof (DOTS).
adjustment time-series We
and consider
has been
officially recommended by the European Commission. For more details visit the website here.
the main trade partners for both countries: in Brazil we take the US, Euro Area, and China
14
For a more thorough treatment on how to perform estimation and forecasts with VARs in the context
18 DOCUMENTO OCASIONAL N.º 2114
BANCO DE ESPAÑA
of the COVID-19 pandemic see Lenza and Primiceri (2020) and Primiceri and Tambalotti (2020).
15 12
This software is a tool specifically designed for the seasonal adjustment of time-series and has been
officially recommended by the European Commission. For more details visit the website here.You can also read
