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Switching Volatility in a Nonlinear
Open Economy
Jonathan Benchimol†and Sergey Ivashchenko‡
February 2021
Abstract
Uncertainty about an economy’s regime can change drastically around a
crisis. An imported crisis such as the global financial crisis in the euro area
highlights the effect of foreign shocks. Estimating an open-economy non-
linear dynamic stochastic general equilibrium model for the euro area and
the United States including Markov-switching volatility shocks, we show that
these shocks were significant during the global financial crisis compared with
periods of calm. We describe how US shocks from both the real economy and
financial markets affected the euro area economy and how bond reallocation
occurred between short- and long-term maturities during the global financial
crisis. Importantly, the estimated nonlinearities when domestic and foreign fi-
nancial markets influence the economy, should not be neglected. The nonlin-
ear behavior of market-related variables highlights the importance of higher-
order estimation for providing additional interpretations to policymakers.
Keywords: DSGE, Volatility shocks, Markov switching, Open economy, Finan-
cial crisis, Nonlinearities.
JEL Codes: C61, E32, F21, F41.
This paper does not necessarily reflect the views of the Bank of Israel. We thank the referees,
Robert Kollmann, John B. Taylor, Mark A. Wynne, Yossi Yakhin, and the participants at the Bank of
Israel Research Department seminar; the 3rd CEPR MMCN Annual Conference; the 15th Dynare
Annual Conference; the 5th Henan University and INFER Applied Macroeconomics Workshop;
and the 49th Money, Macro, and Finance Research Group conferences for their valuable comments.
†Bank of Israel, Jerusalem, Israel. Corresponding author. Email: jonathan.benchimol@boi.org.il
‡Russian Academy of Sciences (IREP), Financial Research Institute, and Saint-Petersburg State
University, Saint Petersburg, Russia.
1
1 Introduction
The widespread consensus in macroeconomics based on the linear new Keynesian
model was shaken by the global financial crisis (GFC). Linear closed-economy
dynamic stochastic general equilibrium (DSGE) models were not concerned with
the sharp variance changes, economic structural breaks, and distribution shifts
around the GFC. Consequently, regime-switching DSGE models have become the
natural framework for analyzing macroeconomic dynamics (Maih, 2015).
An economic regime change could be related to a severe domestic or foreign
financial crisis. The GFC started in the United States and affected the euro area
(EA), thus changing the global economic environment for both economies. This
switching process and analysis of such an international transition’s volatility are
not possible with the standard (linear) closed-economy DSGE models commonly
used in the literature. For example, while classical DSGE models cannot repro-
duce switching volatility effects at all, linear Markov-switching DSGE (MSDSGE)
models reproduce them only partially.
Indeed, linear DSGE models are useful for describing global macroeconomic
stylized facts, but not all economic dynamics can be replicated (Smets and Wouters,
2003, 2007), even though central banks frequently use them to assist forecasting
and monetary policy decisions as well as provide a narrative to the public (Edge
and Gürkaynak, 2010). A nonlinear model estimated at higher-order solutions is
thus essential for analyzing volatility shocks (Fernández-Villaverde et al., 2011),
term structure (Rudebusch and Swanson, 2012), risk premia (Andreasen, 2012),
and welfare dynamics (Garín et al., 2016).
In particular, higher-order approximations of DSGE models are crucial for de-
termining whether changing (switching) volatility is a driving force behind busi-
ness cycle fluctuations (Bloom, 2009). According to Markov processes, the volatil-
ity of several shocks can change over time. Furthermore, Markov-switching (MS)
models provide tractable ways to study agents’ expectation formation about changes
in the economy, such as those occurring during a crisis (Foerster et al., 2016).
A vast body of the literature on dynamic open-economy models has emerged
in the past two decades (Galí and Monacelli, 2005; Adolfson et al., 2007; Justiniano
and Preston, 2010). However, analyses of the dynamic impacts resulting from
regime-switching volatility changes in such a framework are scarce. Specifically,
no study has used MSDSGE models with switching volatility shocks (SVSs). Based
on the foregoing, we bridge this gap by considering the consequences of SVSs in
a two-country MSDSGE model.
One way of influencing the variance of stochastic processes driving the econ-
omy necessitates third-order approximations with the usual perturbation method
(Fernández-Villaverde and Rubio-Ramírez, 2013). Although our model is rela-
tively simple, this method would involve including more than 30 state variables
3
and 10 autoregressive exogenous processes in the model, slowing the third-order
approximation and model estimation. In addition, this approach suggests a slow
drifting of volatility, whereas high levels of volatility switching are more often
seen during crises. This characteristic is generally captured by MS processes in
which a second-order approximation is required to analyze volatility shocks (An-
dreasen, 2010). For this purpose, we use nonlinear approximation algorithms and
filters to estimate our MSDSGE models (Binning and Maih, 2015; Maih, 2015).
However, for the various reasons presented in Appendix A, we develop and use
a generalization of the quadratic Kalman filter applied to MSDSGE models.1
As domestic and foreign transmission channels were substantial during the
GFC as well as in previous crises (King, 2012; Benchimol and Fourçans, 2017), two
relevant transmission channels complete the model. Households can buy or sell
domestic or foreign bonds in the long or short term and their money holdings
increase their utility.
The model is estimated using the EA and US quarterly data compiled from
1995Q2 to 2015Q3 under three specifications: a baseline version without MS, a ver-
sion allowing MS in technology only, and another more developed version allow-
ing MS in three exogenous processes for each country, namely technology, home,
and foreign monetary policy processes. To the best of our knowledge, this study
is the first attempt to introduce long-term interest rates with embedded SVSs into
a nonlinear open-economy DSGE model.
This exercise provides several interesting results and policy implications. First,
we show and quantify that the average US and EA responses to shocks are differ-
ent, especially around 2009Q1, which is also the case from the switching volatil-
ity point of view. These differences essentially come from the nonlinearities in
economic dynamics, although our results are close to those obtained with linear
open-economy DSGE models (Chin et al., 2015). Second, we demonstrate the con-
sequences of SVSs on US and EA economic dynamics. SVSs produce a combina-
tion of short-term deflation and long-term inflation effects in line with Kiley (2014)
but with some asymmetries between the two economies. We demonstrate that
SVSs partially cause financial flows, showing that they significantly affect both
the trade-off between short- and long-term bonds and consumption around the
crisis. Third, we confirm that SVSs have a stronger impact on US monetary policy
than on EA monetary policy. The latter result has several policy implications, such
as monetary policy uncertainty switches.
Our results suggest that policymakers should use nonlinear models to address
open-economy and market-related variables, which are subject to more nonlinear
dynamics than standard closed-economy variables are. Comparing our models
and estimations, we also show that considering a common technology and both
1Appendix A presents the MS quadratic Kalman filter (MSQKF) we use.
4
domestic and foreign monetary policy SVSs better describes the US and EA dy-
namics.
The remainder of this paper is organized as follows. Section 2 presents the
model used for the estimation presented in Section 3. Section 4 presents the results
and Section 5 interprets them. Section 6 concludes, and the Appendix presents
additional results.
2 The model
Our generic model is a symmetric two-country model in which domestic (d) and
foreign ( f) households maximize their respective utilities subject to their budget
constraints (Section 2.1), firms maximize their respective benefits (Section 2.2),
and central banks follow their respective ad-hoc Taylor-type rules and budget con-
straints (Section 2.3). The model’s equilibrium (Section 2.4) and stochastic struc-
ture (Section 2.5) are also presented in this section.
2.1 Households
For each country i2fd,fg, we assume a representative infinitely lived household
seeking to maximize
Et"∞
∑
t=0
εu
i,t1Ui,t#, (1)
where εu
i,t1<1 is the exogenous process corresponding to households’ country-
specific intertemporal preferences,2and Ui,tis households’ country-specific in-
tertemporal utility function, such as
Ui,t=ˆ
Ci,thiˆ
Ci,t111
σi,c
11
σi,c
+εm
i,tˆ
Mi,t/Pi,t11
σi,m
11
σi,m
εl
i,t
L1+1
σi,l
i,t
1+1
σi,l
Ψi,t, (2)
where ˆ
Ci,tis the detrended country-specific Dixit and Stiglitz (1977) aggregator of
households’ purchases of a continuum of differentiated goods produced by firms,
ˆ
Mi,tindicates the detrended country-specific end-of-period households’ nominal
money balances (Mi,t/Zt), Ztis the common level of technological progress,3Pi,t
2At time t, households know their intertemporal preferences for t+1 but have uncertainty
about their preferences for the future. Hence, they know their preference multiplier for t+1.
While they know εu
i,tat time t, they do not know εu
i,t+1at time t. Because utilities for t+1 should
be multiplied by εu
i,t, current period utilities should be multiplied by εu
i,t1.
3The existence of a common stochastic trend (common level of technology progress) requires
stationary summands in the utility function. Consequently, the detrended consumption ( ˆ
Ci,t=
Ci,t/Zt) and real money ( ˆ
Mi,t/Pi,t) summands of this utility function satisfy the stationarity condi-
tion as in Adolfson et al. (2014). See, among others, Fagan et al. (2005), Schmitt-Grohé and Uribe
5
is the country-specific Dixit and Stiglitz (1977) aggregated price index and Ψi,tis
the country-specific cost function described by Eq. 3. σi,cis the country-specific
intertemporal substitution elasticity of habit-adjusted consumption (i.e., inverse
of the coefficient of relative risk aversion), σi,mis the country-specific partial inter-
est elasticity of money demand, and σi,lis the country-specific Frisch elasticity of
labor supply. εm
i,tand εl
i,tare the country-specific exogenous processes correspond-
ing to real money holding (liquidity) preferences and the worked hours (disutility
of labor) of households, respectively.
The country-specific household’s cost function, Ψi,t, is defined by
Ψi,t=1
2∑
j2fsr,lrg
ϕi,d,jBi,d,j,t
Pi,tCi,t1
µi,d,j2
+ϕi,f,jei,tBi,f,j,t
Pi,tCi,t1
µi,f,j2
, (3)
where 8k2fd,fgand 8j2fsr,lrg,ϕi,k,jand µi,k,jare scale parameters related
to the bonds’ rigidity,4and Bi,k,j,trepresents the j-term k-bonds bought by house-
holds in country iin period t, where krepresents the issuing country of the bond
and jits maturity (i.e., short-term (sr) or long-term (lr) bonds). ei,tis the country-
specific exchange rate relating to the number of domestic currency units available
for one unit of foreign currency at time t(i.e., ed,t=1/ef,t).
(2011), and Diebold et al. (2017) for similar detrending. A stochastic trend with drift is suggested
by the data—nonzero mean growth rate of macro-variables. Any DSGE model without trends is
unrelated to real-world statistics and any approximation of a solution in initial terms—without
removing trends—will not satisfy the Blanchard and Kahn (1980) conditions–explosive solution.
Although the use of several trends is better (Schmitt-Grohé and Uribe, 2011), it requires a much
more complicated model.
4When two agents with different intertemporal preferences trade the same security—especially
bonds—credit-borrowing constraints are mandatory to avoid agents taking unrealistic positions.
Thus, we add a quadratic portfolio adjustment rigidity for each type of bond position in the house-
hold’s utility function, which produces smoothed restrictions. To simplify, we do not modulate
such rigidity by restricting negative values. Although our approach is close to the portfolio adjust-
ment costs à la Schmitt-Grohé and Uribe (2003) or price rigidity à la Rotemberg (1982), we assume
preference costs in the utility function, while Schmitt-Grohé and Uribe (2003) assume real costs in
the budget constraint. As it is more likely that households feel disutility from deviations in their
financial position from the steady state, we do not assume that real goods are required to com-
pensate for these deviations. Schmitt-Grohé and Uribe (2003) provide four methods to eliminate
a unit root from an open-economy model. One comprises complete asset markets and identical
discount factors for domestic and foreign households. The other specifications consider an exoge-
nous foreign interest rate. As our model differentiates domestic and foreign households’ discount
factors and considers an endogenous foreign interest rate, these methods are not helpful. Our
motivation for portfolio costs in the utility function is also technical. It allows us to exclude both
the unit root and the cost from the resource constraint. We modify the utility portfolio adjustment
costs’ method to develop the model. Real portfolio adjustment costs should be considered as some
component of GDP, which hardly corresponds to the national account system. By contrast, utility
portfolio adjustment costs do not create such a problem. In the case of a first-order approximation
at a deterministic steady state, these types of costs are equivalent. However, such a modifica-
tion is necessary in the case of a higher-order approximation, while it does not affect the outcome
or propagation mechanism concerning the original adjustment cost of Schmitt-Grohé and Uribe
(2003).
6
The market consists of domestic and foreign one-period short- and long-term
bonds. Long-term bonds pay country-specific shares (Si) of their current nominal
value in each period.5In practice, Sidefines the bond duration (average time until
cash flows are received).
Then, 8i2fd,fg, the country-specific households’ budget constraint can be
expressed as follows:
Pi,tCi,t+Mi,t+∑
j2fsr,lrg
Bi,i,j,tQd,j,t+ei,tBi,i,j,tQi,j,t
=Wi,tLi,t+Mi,t1+Di,t
+Bi,i,sr,t1+Bi,i,lr,t1((1Si)Qi,i,lr,t+Si)
+ei,tBi,i,sr,t1+ei,tBi,i,lr,t1((1Si)Qi,i,lr,t+Si),
(4)
where index idenotes the other country (i.e., if i=d, then i=f; if i=f, then
i=d) and Qk,j,t=exp rk,j,tdenotes the price of rk,j,t, which is the country-
specific (k) nominal interest rate at maturity j.Wi,tis the country-specific wage
index and Di,trepresents the dividends paid by firms in country iat time t. The
online appendix provides the optimality conditions.
Some DSGE models include a single variable for the lump-sum tax and divi-
dends in the budget constraint (Schmitt-Grohé and Uribe, 2011), whereas others
use two separate variables (Smets and Wouters, 2007). To simplify our model, we
do not include a lump-sum tax and report only the dividends instead.
Money and the money demand shock do not influence the economy in the
case of separable (additive) money in the utility function (Galí, 2015). However,
the nonexistence of a lump-sum tax in our model that controls the bond position
changes this mechanism. Our model has no such restrictive lump-sum taxation,
which leads to the influence of money (and the money demand shock) on the
economy.
2.2 Firms
The continuum of identical firms, in which each firm produces a differentiated
good using identical technology, is represented by the following production func-
tion:
YF,i,t(j)=Ai,tLi,t(j), (5)
where Ai,t=AiZtis the country-specific level of technology, assumed to be com-
mon to all firms in country iand evolving exogenously over time, and Aiis a
country-specific total factor productivity scale parameter.
5A long-term bond with a nominal value of one domestic currency unit produces Sdunits of
the domestic currency in the first period, Sd(1Sd)in the second period, Sd(1Sd)2in the third
period, and so on. Because inflation-linked bonds are relatively rare and have lower liquidity in
the United States and EA, we price bonds in nominal terms.
7
As in Galí (2015), to simplify our analysis, we do not include the capital accu-
mulation process in this model, which appears to play a minor role in the business
cycle (Backus et al., 1992) , and assume constant returns to scale for simplifica-
tion purposes.6The exogenous process Ztintroduces a stochastic trend into the
model to explain the nonzero steady-state growth of the economy (Chaudourne
et al., 2014; Diebold et al., 2017). Although alternative techniques to introduce a
unit root exist (Schmitt-Grohé and Uribe, 2011), they complicate the model. For
instance, Smets and Wouters (2007) reconstruct the deterministic component of
the trend, which reduces the model accuracy.
All firms face an identical isoelastic demand schedule and take the country-
specific aggregate price level, Pi,t, and aggregate consumption index, Ci,t, as given.
Following Rotemberg (1982), our model features monopolistic competition and
staggered price setting and assumes that a monopolistic firm faces a quadratic
cost of adjusting nominal prices measured in terms of the final good given by
1
Pi,tZt
Et2
6
6
6
4
∞
∑
s=0
Di,t+sϕi,pPi,t+s(j)
¯
Ps,i,tPi,t+s1(j)12Pi,t+sYi,t+s
s1
∏
k=0
Ri,t+k
3
7
7
7
5, (6)
where ¯
Ps,i,t=exp (viπi+(1vi)πi,t+s1)represents the country-
specificweighted average between country-specific steady-state inflation, πi,
and country-specific previous inflation, πi,t1, in period t, where viis the
country-specific weight and πi,t=ln (Pi,t/Pi,t1).
Pi,t(j)is the price of goods jfrom firms in country iin period t,Ri,t=exp (ri,t)
is the short-term nominal interest rate, and ϕi,p0 is the degree of nominal price
rigidity in country i. The country-specific adjustment cost, which accounts for the
negative effects of price changes on the customer–firm relationship in country i,
increases in magnitude with the size of the price change and with the overall scale
of the country-specific economic activity Yi,t.
In each period t, the firm’s budget constraint requires
Di,t+Wi,tLi,t=Pi,t(j)Yi,t(j), (7)
where YF,i,t(j)represents firms that manufacture goods jin country iin period t.
Firms cannot make any investment (Eq. 7) and distribute all their benefits through
dividends (Eq. 6).
6In this simple case, we also do not consider money in the production function. Several exam-
ples exploring this particular set-up are available in the literature (Benchimol, 2015; Gorton and
He, 2016). Given the complexity of our model and empirical exercise, we assume long-term exoge-
nous growth in a model without capital. Further research should analyze the benefits of capital as
a factor of production to explain long-term growth.
8
The final consumption good is a constant elasticity of substitution composite of
domestically produced and imported aggregates of intermediate goods that pro-
duces demand for firm output, such as
YF,i,t+s(j)=ωiYi,t+sPi,t+s
Pi,t+s(j)εp
i,t+s+(1ωi)Yi,tei,t+sPi,t
Pi,t+s(j)εp
i,t+s, (8)
where the exogenous process εp
i,t+srepresents the country-specific price markup
shock (elasticity of demand in country i), and the parameter ωidefines a country-
specific preference for local demand.
The aggregate country-specific price level also follows the usual constant elas-
ticity of substitution aggregation, such as
P1εp
i,t
i,t=ωiPi,t(j)1εp
i,t+(1ωi) (ei,tPi,t(j))1εp
i,t, (9)
where the local price index includes domestic and foreign prices as is usual in
open-economy models.
2.3 Central bank
Central banks follow a Taylor (1993)-type rule, such as
Ri,t=εr
i,tRρi,r
i,t1ˆ
πρi,π(1ρi,r)
i,tˆ
yρi,y(1ρi,r)
i,tˆ
eρi,e(1ρi,r)
i,t, (10)
where εr
i,tcaptures the country-specific monetary policy shocks, ˆ
πi,tis the country-
specific inflation gap expressed as the ratio between country-specific CPI and its
corresponding steady state, ˆ
yi,tis the country-specific output gap expressed as
the ratio between country-specific output (normalized by technological progress)
and its corresponding steady state, and ˆ
ei,tis the country-specific real exchange
rate gap expressed as the ratio between the real exchange rate of country iand its
corresponding steady state.
The parameter ρi,rcaptures interest rate-decision smoothing, and ρi,π,ρi,y, and
ρi,ecapture the weight placed by the monetary authority of country ion the infla-
tion gap, output gap, and real exchange rate, respectively.
A standard budget constraint applies to the debt bought by central banks, such
as Bi,g,t
Ri,t
=Bi,g,t1+Mi,tMi,t1, (11)
where Bi,g,trepresents the country-specific nominal bonds bought by the local cen-
tral bank in period t.
In our model, we assume that central banks can buy only short-term bonds, as
9
was the case in the United States and EA before the GFC.
2.4 Equilibrium
In the equilibrium, country-specific demand consists merely of consumption, such
as
Yi,t=Ci,t, (12)
and each bond should be bought, requiring that
Bi,i,sr,t+Bi,i,sr,t+Bi,g,t=0, (13)
and
Bi,i,lr,t+Bi,i,lr,t=0. (14)
The country-specific demand presented in Eq. 12, Yi,t, is different from the
country-specific supply presented in the production function (Eq. 5), YF,i,t. As
in Berka et al. (2018) which also has only one source of demand, this simplifica-
tion (Eq. 12) substantially decreases the number of variables, which is crucial for
running a nonlinear estimation.
2.5 Stochastic structure
The exogenous processes we use are defined as 8i2fd,fgand 8j2
fu,m,l,p,r,yg,
φj
i,t=ηi,jφj
i,t1+1ηi,j¯
ηi,j+ξi,j,t, (15)
where the parameter ¯
ηi,jdefines the country-specific steady state of exogenous
process j,ηi,jthe country-specific autocorrelation level, and ξi,j,tthe country (i)
shock-specific (j) white noise (zero-mean normal distribution).
The demand elasticity exogenous process is defined by φp
i,t=εp
i,t, the in-
tertemporal preference exogenous process by φu
i,t=ln εu
i,t/εu
i,t1, technological
progress by φy
t=ln (Zt/Zt1), and other exogenous processes by 8i2fd,fgand
8j2fm,l,rg,φj
i,t=ln εj
i,t.
Appendix B summarizes the variables used in the model.
3 Methodology
In this section, we present the dataset used for the estimations (Section 3.1) as well
as the estimation (Section 3.2) and computation of the nonlinear impulse response
functions (IRFs) (Section 3.3).
10
3.1 Data
We estimate our model with quarterly EA (domestic) and US (foreign) data from
1995Q2 to 2015Q3 taken from the Organisation for Economic Co-operation and
Development. In addition, we use the euro/dollar (EUR/USD) exchange rate
from the European Central Bank (ECB) and Federal Reserve Bank of St. Louis
(FRED) economic data for the exchange rate before the creation of the EA in
1999. The 11 observed variables are as follows: real gross domestic product (GDP)
growth rate (EA and US), GDP deflator (EA and US), ratio of domestic demand to
GDP (EA and US), 3-month interbank rate (EA and US), 10-year interest rate (EA
and US), and EUR/USD growth rate.
With five country-specific shocks and one joint total factor productivity shock,
the number of shocks is equal to the number of observed variables. Our model and
empirical investigation include the long-term interest rate, allowing us to capture
long-term bond demand/supply effects through their interest rates in both coun-
tries. We also capture monetary aggregate dynamics and negative interest rates.
The use of the 3-month interbank rate from the Organisation for Economic Co-
operation and Development database makes the zero lower bound problem less
critical, as it becomes negative for the European Monetary Union in several peri-
ods. Consequently, although we do not explicitly model unconventional monetary
policies, our data highlight some unconventional monetary policy effects.
3.2 Estimation
Our switching (two-regime) model is estimated in three ways with maximum like-
lihood techniques. First, we estimate a baseline version of our model without SVSs
(i.e., without switching). As the productivity shock remains the main source of un-
certainty in the business cycle (Bloom et al., 2018), another version is estimated by
considering only one SVS in Zt(hereafter, 1SVS). A third version considers both
the productivity and the monetary policy SVSs: εr
d,t,εr
f,t, and Zt(hereafter, 3SVS).
The 3SVS model aims to capture the volatility regime switches during the GFC
in both the United States and the EA, as suggested by Mavromatis (2018). Mone-
tary policy and productivity shocks are the main driving forces of business cycles.
Additional SVSs are feasible in theory; however, in practice, they require signifi-
cant additional computing resources and may not change the results or make the
model more realistic.
The model solution approximation is computed with the efficient second-order
perturbation method developed by Maih (2015). We use the MSQKF described
in Appendix A, which is an extension of the QKF for the MS case (Ivashchenko,
2014). The switching volatility and second-order approximation features consti-
tute the nonlinearities of our models. We use the first four quarters as a presample
11
of our three estimations and jackknife bootstrapping for robustness purposes.7
The estimation results of these three models in Appendix C show that the 3SVS
model, which includes switching volatility in the technology and monetary policy
shocks, is the best model to explain current and forecasted aggregate and individ-
ual (observable) dynamics.
The share of steady-state inflation indexation (vi) differs across regions as well
as in the different versions of the model. The coefficient for the United States is
close to that of Smets and Wouters (2007). The version without switching has a
larger share of steady-state inflation indexation. The other models could produce
lower estimated values of the viparameter, which are close to the 1SVS result for
the EA, and even smaller for Canada, which is close to the EA results in the version
without switching (Justiniano and Preston, 2010). The share of steady-state infla-
tion indexation for the EA is much smaller. The 3SVS version produces the closest
values of the corresponding parameters. Thus, volatility switching might influ-
ence inflation persistence, of which the share of past inflation indexation (1 vi)
is one of the key elements.
For the model with variance switching under multiple exogenous
shocks,regime 2 has higher variance of Zt. However, in this case, several
variances in the second state are smaller.
Fig. 1 presents the filtered values of regime 1 probabilities and three selected
exogenous processes (εp
d,t,εp
f,t, and Zt). This figure shows Prob (rt=1)conditional
on the data probability, where Prob (rt=1)corresponds to the probability of being
in regime 1 in period t.
Only moderate differences exist between the filtered values of the exogenous
processes. In addition, the differences in state probabilities are linked to the state
of the 1SVS model, whereas the state probabilities of the 3SVS model are more reli-
able. The latter correspond to the actual main crises that occurred during the sam-
ple period. The difference between the filtered values of the exogenous processes
is generally smaller before the GFC, whereas it is larger a few years after the be-
ginning of the GFC. Economic driving forces are generally unaffected by SVSs,
except at certain points in time, especially during crises. This is also the case when
monetary policy shocks are considered.
7Our table of observations has 11 columns (observables) and 82 rows (periods). We randomly
discard four observations from this table and perform maximum likelihood estimation. We repeat
this process more than 100 times and receive a robust variance estimation. Our methodology
(i.e., jackknife bootstrapping) is different from prefiltering, as it does not use the likelihood values
corresponding to the first four quarters for all the variables. Jackknife bootstrapping suggests
discarding four observations randomly and combining the variable and period.
12
95Q1
96Q3
97Q4
99Q3
00Q4
02Q3
04Q1
05Q3
06Q4
08Q3
09Q4
11Q3
12Q4
14Q3
16Q1
0.2
0.4
0.6
0.8
95Q1
96Q3
97Q4
99Q3
00Q4
02Q3
04Q1
05Q3
06Q4
08Q3
09Q4
11Q3
12Q4
14Q3
16Q1
0
5
10
15
20
10
-3
95Q1
96Q3
97Q4
99Q3
00Q4
02Q3
04Q1
05Q3
06Q4
08Q3
09Q4
11Q3
12Q4
14Q3
16Q1
-5
-4
-3
-2
-1
0
1
210
-3
95Q1
96Q3
97Q4
99Q3
00Q4
02Q3
04Q1
05Q3
06Q4
08Q3
09Q4
11Q3
12Q4
14Q3
16Q1
-2
0
2
4
6
10
-3
1SVS 3SVS Without switchi ng
Figure 1: Regime probability, technology (Zt), US (εp
d,t), and EA (εp
f,t) price markup
shocks.
3.3 Impulse response functions
To analyze the response of the variables to economic shocks, we compute for
each variable its IRF to each shock. The standard definition, such as presented in
Dynare (Adjemian et al., 2011), defines the IRF as the expected difference between
the trajectory with one shock in a single period one standard deviation higher and
the usual trajectory. More precisely, we express this as
IRFt(x,ξ)=E[xtjξ1N(σ(ξ),σ(ξ))] E[xtjξ1N(0, σ(ξ1))] , (16)
where xtis the value of the variable of interest for which the IRF is computed in
period t,ξ1is the shock of interest that deviates in period 1, σ(.)is the standard
error operator, E[.]is the expectation operator, and Nis the normal law.
We generalize this definition in the nonlinear case by making the magnitude
and sign of the shock more important. Such a generalization requires the introduc-
tion of the parameter sin Eq. 16 to determine the number of standard deviations
13
in the shock, such as
IRFt,s(x,ξ)=E[xtjξ1N(σ(ξ1)s,σ(ξ1))] E[xtjξ1N(0, σ(ξ1))]
s. (17)
In addition, we compute the IRFs conditional on the state variables’ vector Xt
to show the differences between the IRFs at different states of the world, such as
IRFt,s(xt,ξjX0)=E[xtjξ1N(σ(ξ1)s,σ(ξ1)) ;X0]E[xtjξ1N(0, σ(ξ1)) ;X0]
s,
(18)
where X0is a vector of the state variables before the shock.
The IRF for the switching shock is
IRFt(x,v0,v1)=E[xtjr0=v0;r1=v1]E[xtjr0=v0], (19)
where rtis the regime variable at time t, and v0and v1are the switching values of
the regime of interest.
To compute the expectations, we use a simulation with the same exogenous
shocks for both parts of the IRF equation. We use 50,000 draws for averaging and
100 presample draws for the unconditional IRF.8
4 Results
In this section, we present the responses of our model after an SVS (Section 4.1)
and a monetary policy shock (Section 4.2). Further, we present and analyze some
nonlinearities (Section 4.3). The other results are available upon request. Appen-
dix C presents additional performance measures showing the advantages of the
volatility switching (i.e., 3SVS) model over the other models.
4.1 Switching volatility shock
Fig. 2 presents the IRFs of the SVSs from states 1 to 2 (with higher volatility for
Zt) for the 1SVS model. We compute the unconditional IRF and plot the mean IRF
and +/- two standard deviations (std) of the IRF.
Fig. 2 shows that the regime probability effect disappears without strong per-
sistence (around 10 periods). However, the effect on the model’s variables is much
more persistent and differs by region. Following an SVS, inflation increases in the
two regions during the first periods, involving an increase in the US short-term
nominal interest rate, while the EA’s short-term nominal interest rate remains sta-
ble. The picture changes drastically in later periods when the long- and short-term
8We consider the steady state as the initial point and we draw the trajectory for 100 periods.
The shock occurs in period 101, and we repeat this 50,000 times.
14
10 20 30 40
-0.5
0
10 20 30 40
-5
0
5
10-7
10 20 30 40
-10
-5
0
510-6
10 20 30 40
-15
-10
-5
010-7
10 20 30 40
-10
-5
0
5
10-6
10 20 30 40
-8
-6
-4
-2
010-6
10 20 30 40
-15
-10
-5
0
510-5
10 20 30 40
-2
-1
010-5
10 20 30 40
0
5
10
15 10-6
10 20 30 40
-5
0
510-5
10 20 30 40
-1
0
1
10-6
10 20 30 40
-3
-2
-1
0
110-6
IRF + 1 std. shock IRF + 2 std. shock IRF - 2 std. shock
Figure 2: Unconditional IRFs to an SVS to regime 2 (1SVS).
interest rates in the United States and EA’s both decrease with the inflation rates.
Only GDP growth and the exchange rate are stabilized after several periods.
The US long-term rate decreases more smoothly a few quarters after the shock
compared with the EA long-term nominal interest rate. This difference can be
explained by the different durations of the long-term bonds in the EA (sd=0.6)
and United States (sf=0.06).
In addition, monetary policy weights, by generating different short-term in-
terest rates, could explain this phenomenon. The United States has a stronger
response to inflation and a smaller smoothing coefficient than the EA. Conse-
quently, the US short-term nominal interest rate decreases with inflation and in-
creases later, while that for the EA increases slightly. This difference in monetary
policy produces fluctuations in the exchange rate and ratio of domestic demand
to GDP.
Fig. 3 provides a more robust picture than Fig. 2.
Indeed, the 1SVS model suggests only a few differences between regimes (the
standard deviations are close), implying a small effect on the economy of switch-
ing, which explains the low values obtained in Fig. 2. However, the 3SVS model
15
10 20 30 40
-0.5
0
10 20 30 40
-0.02
0
0.02
0.04
0.06
10 20 30 40
-0.1
-0.05
0
10 20 30 40
-2
0
2
10-3
10 20 30 40
-0.03
-0.02
-0.01
10 20 30 40
-0.05
0
10 20 30 40
-0.2
0
10 20 30 40
0
0.05
10 20 30 40
-0.04
-0.02
0
10 20 30 40
0
0.1
0.2
10 20 30 40
-0.05
0
0.05
10 20 30 40
0
0.05
0.1
IRF + 1 std. shock IRF + 2 std. shock IRF - 2 std. shock
Figure 3: Unconditional IRFs to an SVS to regime 2 (3SVS).
suggests much larger differences and a substantial impact of switching shocks on
the economy.
Fig. 3 highlights that SVSs affect US inflation and nominal interest rates in both
the short and long terms, while the impact on the EA economy is less significant.
Such SVSs durably influence US long-term interest rates, whereas this is not the
case for the EA’s long-term interest rates.
Uncertainty around the EA’s short-term nominal interest rate, measured as the
gap between -2 std and +2 std around the IRF, is stronger than that around the US
short-term nominal interest rates.
In addition, the demand-to-GDP ratios of the two regions display substantial
uncertainty, showing that the SVSs in the monetary policy shocks of the two re-
gions have important economic implications.
In Fig. 3, the economy switches to regime 2, which means a substantial in-
crease in the volatility of both foreign and domestic monetary policy shocks and a
decrease in total factor productivity shock volatility. Higher uncertainty means
higher interest rates. However, the central bank controls interest rates, buys
bonds, and prints money that leads to higher inflation. As the economy is open,
16
domestic changes are substantial, and foreign households buy more domestic
bonds. Foreign households work more and sell more goods to the domestic coun-
try. Moreover, foreign investment in the domestic market makes foreign currency
cheaper. Thus, foreign households increase investments and hold more money. As
this effect is powerful, foreign inflation decreases, leading to lower foreign interest
rates.
The average effect of unconditional SVSs might differ from that of conditional
IRFs. For example, Fig. 4 compares unconditional IRFs with conditional IRFs for
2009Q1 and 2003Q4. Here, we use the filtered values of the variable vector for the
corresponding dates as the condition (the initial point for a draw).
10 20 30 40
-0.5
0
10 20 30 40
-5
0
5
10-4
10 20 30 40
-8
-6
-4
-2
0
2
410-3
10 20 30 40
-3
-2
-1
010-4
10 20 30 40
-2
-1
0
110-3
10 20 30 40
-15
-10
-5
010-4
10 20 30 40
-0.03
-0.02
-0.01
0
0.01
10 20 30 40
-4
-2
010-3
10 20 30 40
0
1
2
10-3
10 20 30 40
-0.01
0
0.01
10 20 30 40
-1
0
110-3
10 20 30 40
-2
-1
0
10-3
Unconditional 2009Q1 2003Q4
Figure 4: Conditional and unconditional IRFs to an SVS to regime 2 (1SVS).
These IRFs are different in several aspects. Indeed, the regime probability IRF
differs in the configurations shown in Fig. 4, where we compare the best expansion
(2003Q4) with the worst recession (2009Q1) periods. This could be because the
economy is in regime 2 when the shock occurs in the case of the unconditional
IRF, while the conditional IRF could be in regime 1 before the shock.
Further, the 3SVS model highlights the significant differences between the
crises as well as between the United States and EA (Fig. 5).
17
10 20 30 40
-0.5
0
10 20 30 40
-0.02
0
0.02
0.04
0.06
10 20 30 40
-0.1
-0.05
0
10 20 30 40
-1
0
1
2
10-3
10 20 30 40
-0.03
-0.02
-0.01
10 20 30 40
-0.05
0
10 20 30 40
-0.4
-0.2
0
10 20 30 40
0
0.05
10 20 30 40
-0.04
-0.02
0
10 20 30 40
0
0.1
0.2
10 20 30 40
-0.1
-0.05
0
0.05
10 20 30 40
0
0.05
0.1
0.15
Unconditional 2009Q1 2003Q4
Figure 5: Conditional and unconditional IRFs to an SVS to regime 2 (3SVS).
These differences are more reliable than in the 1SVS model. For instance, the
EA short-run nominal interest rate was not similarly affected by the switching
during good and bad times (e.g., the subprime crises) and their corresponding
SVSs. Furthermore, significant differences are observed for the ratio of domestic
demand to GDP in both regions.
An SVS has a stronger impact on the EA’s demand-to-GDP ratio than on that
of the United States, at least after the dot-com crisis. In addition, Fig. 5 shows
that the EA consumes more, while the United States consumes less. At the same
time, US inflation and interest rates decrease slightly more than the unconditional
points and the EA.
The influence of the SVS is significant. For instance, in the long run, the US
long-term interest rate change caused by an SVS is about 0.4% over ten years (Fig.
5). The EA demand to GDP changes of about 0.05% for ten years after an SVS,
which indicates a 0.5% cumulative change of trade (in terms of EA GDP). In the
short run, the US GDP growth change resulting from the SVS is about 0.15%.
Fig. 6 compares the consequences of regime switching for both models. As
expected, the IRFs are significantly different, mainly due to the switch of multiple
18
variances. The response of the 3SVS model is less monotonic and the magnitudes
of the IRFs are different for most of the economic variables.
10 20 30 40
-0.5
0
10 20 30 40
-0.02
0
0.02
0.04
10 20 30 40
-0.1
-0.05
0
10 20 30 40
-5
0
5
10
10-4
10 20 30 40
-0.03
-0.02
-0.01
0
10 20 30 40
-0.04
-0.02
0
10 20 30 40
-0.3
-0.2
-0.1
0
10 20 30 40
0
0.02
0.04
0.06
10 20 30 40
-0.03
-0.02
-0.01
0
10 20 30 40
0
0.1
0.2
10 20 30 40
-0.05
0
0.05
10 20 30 40
0
0.05
0.1
1SVS 3SVS
Figure 6: Unconditional IRFs to an SVS (to regime 2) for the 1SVS and 3SVS mod-
els.
Fig. 6 shows that the switch of multiple variances significantly affects the ex-
change rate as well as US long-run nominal interest rates, while short- and long-
term nominal interest rates in the EA are less affected. However, the EA’s demand-
to-GDP ratio is more affected than the US ratio.
The 3SVS model captures several dynamics that a 1SVS model without switch-
ing cannot, such as the decreasing short-term nominal interest rates in the United
States and oscillating inflation in the EA.
Fig. 7 shows that the 1SVS model influences the financial variables, uncondi-
tionally and conditionally, compared with the reference dates.
Although the magnitude of these IRFs is relatively low, some conclusions can
be drawn. The 1SVS responses are clearly different during the subprime crisis
and after the dot-com crisis, unconditional on time (Fig. 7). An SVS increases the
US bonds bought by EA and US households as well as the EA bonds bought by
the ECB (in the short run). Following such an SVS, the exchange rate, the EA’s
19
10 20 30 40
-10
-5
0
10-6
10 20 30 40
0
2
4
6
10-4
10 20 30 40
-1
-0.5
010-3
10 20 30 40
-2
-1
010-5
10 20 30 40
0
2
4
10-4
10 20 30 40
0
1
2
310-5
10 20 30 40
-4
-2
0
10-7
10 20 30 40
-5
0
5
10-5
10 20 30 40
-10
-5
0
10-4
10 20 30 40
0
2
4
10-5
10 20 30 40
-1
0
1
10-3
10 20 30 40
-2
-1
0
110-5
Unconditional 2009Q1 2003Q4
Figure 7: Conditional and unconditional IRFs to an SVS (to regime 2) for the fi-
nancial variables (1SVS)
short- and long-term bonds bought by EA households and the Federal Reserve,
and money held by US households all decrease.
The main problem in this scenario is that it assumes that the aftermaths of the
dot-com and subprime crises are similar, at least in terms of the IRFs and the im-
pact of an SVS on the financial variables. However, this was not the case; indeed,
the financial transmission channels during these two crises were fundamentally
different.
Fig. 8 shows a more coherent picture with significant and reliable differences
in the IRF after the dot-com crisis and during the GFC.
Indeed, the 3SVS model during the GFC increased the US bonds bought by US
households and EA bonds bought by the ECB, while this was not the case after
the dot-com crisis or unconditionally. Such shocks also decreased the exchange
rate and money held by EA households in all cases, while the money held by US
households increased.
Fig. 8 shows that the response of US short-term bonds is due to an increase
in the Federal Reserve’s bond position, while other agents decrease their bond
20
10 20 30 40
0
2
4
10-4
10 20 30 40
-2
-1
010-3
10 20 30 40
-2
0
2
410-3
10 20 30 40
-5
010-4
10 20 30 40
-2
0
2
10-3
10 20 30 40
0
5
10-4
10 20 30 40
-4
-2
010-5
10 20 30 40
-3
-2
-1
010-3
10 20 30 40
-2
-1
0
10 20 30 40
-15
-10
-5
0
10-4
10 20 30 40
0
0.02
0.04
10 20 30 40
-0.03
-0.02
-0.01
Unconditional 2009Q1 2003Q4
Figure 8: Conditional and unconditional IRFs to an SVS (to regime 2) for the fi-
nancial variables (3SVS)
position. In the EA, the picture is different: the ECB slightly increases its bond po-
sition, and both European and US households decrease their EA long-term bond
positions.
Then, because US and EA households are selling their US bonds, the construc-
tion of our model suggests that the Federal Reserve must buy them after the dot-
com crisis. Such a result is close to the reality of the past decade.
Moreover, Fig. 7 shows that following such SVSs, both regions’ households
hold more money after several periods and sell EA long-term bonds. This re-
sult is a direct consequence of the increase in the short-term EA bond position
and consumption. US households increase their overall bond position and money
holdings, such that euros return to the EA and US dollars return to the United
States.
Another interesting result lies in the differences between the 1SVS and 3SVS
models. The 1SVS model (Fig. 7) hardly discriminates between the two condi-
tional IRFs (2003Q4 and 2009Q1), while the 3SVS model (Fig. 8) differentiates
between these two dates, which are economically (and financially) substantially
21
different. Consequently, the 3SVS model could match the stylized financial facts
better than the 1SVS model (and a fortiori compared with the baseline model with-
out switching).
In terms of the IRF levels, the 3SVS model brings about higher volatility to
the responses of the economic variables, especially for the exchange rate, money
holdings, and bond quantities. Volatility shocks were essential drivers of the GFC
and, as we see hereafter, nonlinearities also affect economic dynamics.
Fig. 8 demonstrates the increasing real exchange rate difference of about 0.5%
over 10 years. Such differences between conditional and unconditional IRFs show
how nonlinearities are significant.9In the long run, an SVS leads to a change of
about 2% in the real exchange rate. The SVS effect is very persistent with a sub-
stantial consequence, in that the difference between conditional and unconditional
IRFs for the real exchange rate exceeds 0.2% over more than eight years (Fig. 8).
The importance is also related to the duration of effect. For instance, if EA exports
and imports represent about 53% of GDP,10 the cumulative effect of a 0.2% change
in exchange rates over eight years would lead to a flow of money representing
0.85% of yearly GDP (direct influence11).
4.2 Monetary policy shock
Fig. 9 shows the consequences of an EA monetary policy shock for each model.
The responses are similar except that the US long-term interest rate is lower under
the 3SVS model, while the price of long-term bonds is higher.
An EA monetary policy shock leads to higher inflation in Fig. 9. Hence, the real
interest rate increases, leading to a lower money position and a higher bonds po-
sition. The government budget means that it creates additional income for house-
holds. This means higher consumption, which increases imports. Importation
growth then leads to a cheaper national currency, and thereby inflation growth
and domestic production growth.
However, the responses of a US monetary policy shock differ depending on the
model (Fig. 10), especially for the demand-to-GDP ratio, long-term interest rates,
and GDP growth in the first quarters. US inflation responses are more pronounced
in the 3SVS model than in the model without SVSs.
In addition, EA and US growth rates are significantly different in the first quar-
ters, showing that the model without switching allows more variability to US and
EA growth in the first periods, with different signs at some points in time.
9See Section 4.3 for an analysis of nonlinearities.
10The EA national accounts show that the share of exports is 28.2% of GDP in 2018. The share
of imports is 24.7% of GDP during the same period. The exports and imports represented 53% of
GDP.
11The exchange rate influenced both export and import payments leading to a total effect would
be 0.002 80.53 =0.85%.
22
10 20 30 40
0
0.1
10 20 30 40
0
0.1
10 20 30 40
0
5
10-3
10 20 30 40
0
0.05
0.1
10 20 30 40
0
2
4
6
10-4
10 20 30 40
0
0.05
10 20 30 40
0
0.005
0.01
10 20 30 40
0
2
410-3
10 20 30 40
-3
-2
-1
010-3
10 20 30 40
-0.2
-0.1
0
10 20 30 40
0
0.1
0.2
10 20 30 40
-4
-2
0
2
410-3
1SVS 3SVS Without switching
Figure 9: Unconditional IRFs to a positive EA monetary policy shock (one stan-
dard deviation).
A foreign monetary policy shock leads to lower inflation in Fig. 10. The central
bank places significant weight on inflation. Lower inflation expectations lead to
lower inflation and interest rates, which then motivates households to increase
money and decrease bonds. This produces an additional cash flow that is spent
on consumption. Additional demand leads to higher imports. This makes the
national currency relatively cheap and domestic production rises to some extent.
Interestingly, long-term interest rates have different responses in the United
States and EA. While the US long-term nominal interest rate decreases sharply in
the 3SVS model, the decrease in the EA long-term nominal interest rate is less pro-
nounced. Without switching, the US long-term nominal interest rate decreases less
than in the 3SVS model, while the EA long-term nominal interest rate increases
more than in the 3SVS model. Thus, SVSs could provide relevant information for
monetary policy decisions.
In line with the stylized facts, a symmetric monetary policy shock does not
have similar consequences if it is in the EA or the United States.
23
10 20 30 40
-3
-2
-1
0
10 20 30 40
-0.02
0
0.02
0.04
10 20 30 40
-1
-0.5
0
10 20 30 40
-10
-5
010-3
10 20 30 40
-0.1
-0.05
0
10 20 30 40
-0.04
-0.02
0
10 20 30 40
-1
-0.5
0
10 20 30 40
-0.05
0
10 20 30 40
0
0.05
10 20 30 40
0
1
2
10 20 30 40
0
0.05
0.1
0.15
10 20 30 40
0
0.5
1
1SVS 3SVS Without switchi ng
Figure 10: Unconditional IRFs to a positive US monetary policy shock (one stan-
dard deviation).
4.3 Nonlinearities
The previous IRF figures considered only a one standard deviation positive shock.
However, in a nonlinear world, responses are also nonlinear. How should these
nonlinearities be quantified? Fig. 11 to Fig. 14 present the unconditional IRFs after
monetary policy shocks of different magnitudes to assess the importance of these
nonlinearities.
Fig. 11 presents the IRFs after an EA monetary policy shock according to the
1SVS model.
While +1 std and +3 std are similar, one crucial nonlinearity resides in -3 std,
which is also similar to that for positive shocks. This nonlinearity is easily un-
derstandable mathematically (power 2), avoiding a symmetric response, which is
standard in DSGE models’ IRFs linearized at the first order.
However, this negative EA monetary policy shock (-3 std) has a lower response
than the other positive shock, even though the direction is similar. Nonlinearities
could lower the efficiency of monetary policy shocks, which is an important result
24
10 20 30 40
0
10
20 10-4
10 20 30 40
0
2
410-4
10 20 30 40
0
1
2
10-5
10 20 30 40
0
0.5
110-3
10 20 30 40
0
5
10-6
10 20 30 40
0
5
10-4
10 20 30 40
0
1
10-4
10 20 30 40
0
2
4
10-5
10 20 30 40
-3
-2
-1
010-5
10 20 30 40
-2
-1
0
110-3
10 20 30 40
0
2
4
10-4
10 20 30 40
-2
-1
0
1
10-5
IRF +1 std. shock IRF - 3 std. shock IR F +3 std. shock
Figure 11: Unconditional IRFs to EA monetary policy shocks of different magni-
tudes (1SVS). std. stands for standard deviation.
for monetary authorities using simple linear models to assess economic situations
and take monetary policy decisions.
Further, nonlinearities are more visible in the economy, and such a picture, as
presented in Fig. 11, has a shallow impact (the scale is always between 103and
105).
Fig. 12 presents the IRFs after an EA monetary policy shock according to the
3SVS model.
Unlike Fig. 11, Fig. 12 presents the magnitudes of higher nonlinearities, and
these results are more in line with those in the literature (An and Schorfheide,
2007), especially for exchange rates (Altavilla and De Grauwe, 2010).
Furthermore, nonlinearities influence EA inflation uncertainty in an interesting
way. While the negative monetary policy shock (-3 std) has an important impact
on EA inflation in the first periods, it exceeds +1 (+3 std) after several periods,
highlighting the nonstandard perspective allowed by nonlinearities.
Moreover, Fig. 13 highlights an important result for policymakers. Responses
to a US monetary policy shock have different nonlinearities than responses to an
25
10 20 30 40
0
0.05
0.1
0.15
10 20 30 40
0
0.05
0.1
10 20 30 40
0
2
4
6
10-3
10 20 30 40
0
0.05
10 20 30 40
0
5
10-4
10 20 30 40
0
0.05
10 20 30 40
0
0.01
10 20 30 40
-1
0
1
2
3
10-3
10 20 30 40
-20
-10
0
10-4
10 20 30 40
-0.1
0
0.1
10 20 30 40
-0.1
0
0.1
10 20 30 40
-5
0
510-3
IRF + 1 std. shock IRF -3 std. shock IRF +3 s td. shock
Figure 12: Unconditional IRFs to EA monetary policy shocks of different magni-
tudes (3SVS). std. stands for standard deviation.
EA monetary policy shock (Fig. 11). EA inflation and the demand-to-GDP ratios
of both the United States and the EA behave almost nonlinearly following a US
monetary policy shock (at least in the first periods). In these cases, -3 std and
+3 std are asymmetric, whereas we find small nonlinearities (asymmetries) in the
previous case (EA monetary policy shock).
Interestingly, +1 std and +3 std US monetary policy shocks do not have the
same consequences for EA growth (Fig. 13). This finding shows that nonlinear-
ities help explain why strong monetary policy reactions do not have the same
consequences as small monetary policy reactions. The same comment applies to
the US demand-to-GDP ratio.
Thus, policymakers should analyze economic decisions, including their own,
through the spectrum of nonlinear models to optimize the magnitude of their
monetary policy reaction function.
Fig. 14 presents the IRFs after a US monetary policy shock according to the
3SVS model.
Nonlinearities are present in the case of a US monetary policy shock (Fig. 14).
26
10 20 30 40
-0.03
-0.02
-0.01
10 20 30 40
0
10
20
10-5
10 20 30 40
-2
-1
010-3
10 20 30 40
-10
-5
010-5
10 20 30 40
-10
-5
0
10-4
10 20 30 40
-4
-2
010-4
10 20 30 40
-0.01
-0.005
0
10 20 30 40
-15
-10
-5
0
10-4
10 20 30 40
0
5
10 10-4
10 20 30 40
0
0.01
0.02
10 20 30 40
-5
0
5
10
15 10-4
10 20 30 40
0
1
2
310-3
IRF + 1 std. shock IRF -3 std. shock IRF + 3 std. shock
Figure 13: Unconditional IRFs to US monetary policy shocks of different magni-
tudes (1SVS). std. stands for standard deviation.
EA inflation displays the same phenomenon as presented previously (Fig. 12),
again denoting strong differences between the linear and nonlinear models.
Fig. 14 confirms that the persistent difference in the demand-to-GDP ratio de-
pends on the shock’s magnitude and sign. This difference is larger than 0.05%
over ten years, meaning the cumulative change of trade equals 0.5% of GDP. The
short-term difference in EA GDP growth rates exceeds 0.5% due to the shift in
growth peak timing. More significant is the response of US GDP growth of about
1% after a US monetary policy shock of different magnitude.
Moreover, the role of SVSs is significant, especially concerning demand-to-
GDP ratios and exchange rates. Overall, the nonlinearities and SVSs on monetary
policy shocks affect not only the magnitudes of the considered dynamics but also
the actual dynamics as well as their orders.
Such a result is fundamental for policymakers and economists willing to model
economies around a crisis. Open-economy models are suitable for such nonlin-
earities (Altavilla and De Grauwe, 2010) and masking nonlinearities using linear
models to analyze such economies could lead to inadequate economic interpreta-
27
10 20 30 40
-3
-2
-1
0
10 20 30 40
0
0.02
0.04
0.06
0.08
10 20 30 40
-1
-0.5
0
10 20 30 40
-10
-5
010-3
10 20 30 40
-0.1
-0.05
0
10 20 30 40
-0.04
-0.02
0
10 20 30 40
-1
-0.5
0
10 20 30 40
-0.15
-0.1
-0.05
0
10 20 30 40
0
0.05
0.1
10 20 30 40
0
1
2
10 20 30 40
-0.2
0
0.2
0.4
10 20 30 40
0
0.5
1
IRF + 1 std. shock IRF -3 std. shock IRF + 3 std. shock
Figure 14: Unconditional IRFs to US monetary policy shocks of different magni-
tudes (3SVS). std. stands for standard deviation.
tions and policy decisions.
5 Interpretation
Section 2 presents an original model featuring households, firms, and the cen-
tral banks of two economies, with households able to buy domestic or foreign
short-term bonds. Following Kiley (2014), we show that the short-term nominal
interest rate has a more substantial effect on the overall economy than the long-
term nominal interest rate and that both short- and long-term interest rates are key
determinants of consumption. However, our results also highlight that the EA’s
long-term interest rates comove strongly with US long-term rates rather than with
short-term rates (Chin et al., 2015). This result is confirmed with the 3SVS model
(Fig. 9) in which the US long-term nominal interest rate reacts more strongly to an
EA short-term nominal interest rate shock.
Following Chin et al. (2015), we find that US disturbances influence EA
28
economies markedly (Fig. 13). These results are confirmed by the variance de-
compositions of the variables with respect to the shocks (Appendix D) and dis-
tance correlations between the variables (online appendix).
In addition, we find that US money shocks affect the EA real variables in the
long run as well as the financial markets in the EA and United States (Appendix
D). We therefore extend the literature by highlighting new transmission channels
of money compared with other studies using linear closed-economy DSGE models
with money (Benchimol and Fourçans, 2012, 2017; Benchimol and Qureshi, 2020).
Unlike this body of the literature, we show the role of money in the economy
without assuming nonseparability between money and consumption (Benchimol,
2016), or a cash-in-advance constraint (Feenstra, 1986) or money in the production
function (Benchimol, 2015). The money holdings from households and central
banks needed to buy bonds involve such a role of money (Eq. 11). Although the
additive separable utility function (Eq. 2) excludes real money balances from the
IS curve (Jones and Stracca, 2008), money has a role through the money-in-the-
utility function and households’ budget constraints because of the direct effect (Eq.
3) highlighted by Andrés et al. (2009).
An interesting result on inflation’s variance decomposition is that the price
markup shock (demand elasticity shock) plays a critical role in the EA, while this
shock explains only a small share of the inflation dynamics in the United States, il-
lustrating how EA and US economies behave differently during crises. In the long
run, this is explained by the strength of price markup shocks explaining domestic
as well as foreign wage dynamics. This smaller effect of the price markup shock
on the inflation rate in the short run can be caused by nonlinear dynamics, which
are missing from standard closed-economy models.
Another result relates to the intertemporal preferences shock, showing that it
has minor short-term explanatory power, whereas it becomes one of the most im-
portant shocks for explaining some variable dynamics in the long run (Appen-
dix D), such as the part of inflation dynamics not explained by the price markup
shock. This fact is essential for models with domestic and foreign preference
shocks. The absence of such a shock in the literature on open-economy DSGE
models could conceal additional dynamics that could complete the economic sce-
narios developed by policymakers, such as on foreign and domestic bonds or pri-
vate consumption.
Hence, our nonlinear open-economy DSGE model with several SVSs allows
us to enrich the dynamics of interest rate markets for different maturities (Section
4.1). Fig. 2 shows that the US response to inflation is stronger than that of the
EA. How can one conciliate this with the stabilization objectives of the Federal
Reserve and ECB? The official objective of the Federal Reserve is to react to both
inflation and output growth or unemployment, while the ECB’s is to mainly react
29
to inflation. However, these objectives differ from the concrete reaction to these
variables. First, the existence of an additional component in the Federal Reserve’s
objectives does not mean a lower response to US inflation. We have shown that
the Federal Reserve responds substantially more (in absolute values) to the output
gap and exchange rate than the ECB, which could compensate for its response to
inflation. As an active central bank, this stronger response to economic changes
leads to faster stabilizing effects with more significant interest rate fluctuations
compared with the ECB. The ECB’s smaller responses smooth interest rates dur-
ing a more extended (stabilization) period. Both monetary policies correspond to
the official objectives but have significant differences in the preferences across the
components of these objectives. Other explanations relate the weaker response of
the ECB to inflation dynamics compared to those of the Federal Reserve. Tensions
within the ECB Governing Council, a change in the post-GFC inflation target and
objectives, quantitative easing, and the zero lower bound could also explain this
lower inflation coefficient compared with the Federal Reserve.
Including several SVSs could, at least during crises, more accurately explain
the changes in US and EA inflation as well as in US and EA interest rates at differ-
ent maturities. The possibility of switching in different elements of the economy,
such as technology and monetary policy (and not only technology), is essential
during crises. It is natural to capture such stylized facts by including several SVSs.
Each SVS could capture specific switching volatility that can change the regime of
the overall economy for a specific sector. Fig. 6 clearly shows that such modeling
is more appropriate for capturing changes in inflation and interest rates than a
model with only one SVS for technological progress.
In addition, such shocks are important for capturing changes in an open econ-
omy; for instance, Fig. 10 shows that after a US monetary policy shock, EA in-
flation is assumed to decrease just after the shock in the model without SVSs,
whereas this is not the case in reality. Then, models with one or several SVSs could
capture reality more accurately, especially during crisis periods when macroeco-
nomic and financial variables are not well explained. Section 4.2 discusses the
transmission channels.
Lastly, our models can shed light on nonlinear IRFs, highlighting the signifi-
cant nonlinear behaviors of market-related variables such as exchange and interest
rates. Such dynamics are absent from most policymakers’ models for such rea-
sons as technical complexity, material limitations, and time and computational
costs. However, our policy recommendation resulting from the results of this
study is that nonlinear models should be used when addressing open-economy
and market-related variables, which can be subject to highly nonlinear dynamics
compared with more standard closed-economy variables.
30
6 Conclusion
In this study, a two-country open-economy MSDSGE model was developed to un-
derstand several stylized events that occurred during the GFC, such as how the
regime-specific SVS impacts between the EA and the United States were transmit-
ted to real and financial variables.
Using a second-order approximation and Markov SVS, we showed that SVSs
are the main driving force of the shock transmissions during crises. We showed
that SVSs affect the US and EA economies and involve i) money transfers between
economies and ii) interest rate maturity trade-offs that could produce structural
changes in the economy. Hence, SVSs affect US and EA consumption in opposite
ways.
Further, price markup and money shocks behave differently to in standard lin-
ear models. Owing to direct effects (Andrés et al., 2009), the roles of both domestic
and foreign real money holdings are significant in the long run as well as the short
run, especially for bond variables and rate-related variables.
Furthermore, the difference between the average response of SVSs and re-
sponse on specific dates illustrates that SVSs are relevant during crises but less
so in calm times. Unlike EU monetary policy, which is less impacted by SVSs, US
monetary policy is significantly influenced by such shocks.
The main policy implication relates to the way monetary authorities model the
economy, especially in an open-economy world with interlinked financial mar-
kets. Our models showed that it is important for policymakers to consider nonlin-
ear models and SVSs during crisis periods (or when uncertainty about a current
regime increases). If policymakers continue to use standard linear models and
ignore SVSs, they might also overlook some nonlinear dynamics as well as the
underlying interactions between financial markets and the economy. SVSs could
thus be a promising feature included in the next generation of macroeconomic
models.
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Appendix
A Markov switching quadratic Kalman filter
This appendix presents the fast-deterministic filter used for the estimation of our
nonlinear MSDSGE model. The collapsing rule of the sigma-point Kalman filters
developed by Binning and Maih (2015) is unusual. This filter family uses vari-
ance equal to the weighted average of variance conditional on the regime. Such
a formula holds for raw moments but not for central moments.12 Our MSQKF
fixes this property by correcting the formulas for variances.13 The MSQKF is a
Gaussian-assumed filter that uses collapsing before forecasting as in Binning and
Maih (2015).
Particle filter approaches have the advantage of an unbiased likelihood esti-
mation. However, these approaches produce a stochastic estimation of likelihood,
which is a substantial disadvantage. They do not allow standard optimization al-
gorithms to be used. Moreover, fixed random draws are required for optimization
algorithms with particle filters. However, this mitigates the main advantage of
particle filters. Markov chain Monte Carlo inefficiency increases significantly: the
required number of draws should be 10 (from 5 to 400 depending on the num-
ber of particles) times higher for the same accuracy of Markov chain Monte Carlo
methods (Pitt et al., 2012). An additional disadvantage of particle filters is their
computational costs. They require a large number of particles to be comparable
with deterministic filters and are about 100 times slower than deterministic non-
linear filters (Andreasen, 2013; Ivashchenko, 2014; Kollmann, 2015). For all these
reasons, we do not use particle filters.
The purpose of a filter in DSGE models is to compute the model variable vec-
tor, Xt, density conditional on the vectors of the observed variables Y1, ..., Ytand
density and likelihood of the observed variables Y1, ..., Yt. Computing the density
means computing the parameters of the density approximation. In certain cases,
this approximation is equal to the density (e.g., the normal distribution).
Most filters loop the following steps:
1. Computation of the initial density of Xt;
12Let us consider two regimes, the probabilities of which, p(rtjrt+1), are p(1j1)=p(2j2)=
0.95 (0.6)and p(1j2)=p(2j1)=0.05 (0.4). The mean conditions on these regimes, x(rt), are
x(1)=1 and x(2)=1, and the variance condition on each regime is 1. Hence, the variance
condition on the future regime, V(rt+1), would be V(1)=V(2)=1.19 (1.96), while the formula
from Binning and Maih (2015) gives 1 in both cases. This demonstrates that their formula generates
substantial errors in the case of regime uncertainty.
13The description, properties, and comparisons of the MSQKF are detailed in Ivashchenko (2014,
2016).
36
2. Computation of the density of Ytas a function of the density of Xt(see Ap-
pendix A.1);
3. Computation of the likelihood of Yt(see Appendix A.2);
4. Computation of the conditional density of XtjYt(see Appendix A.3);
5. Computation of the density of Xt+1as a function of the density of XtjYt(see
Appendix A.4); and
6. Return step 2.
Our MSQKF assumes a Gaussian density approximation in Step 5, unlike the
sigma-point one. The sigma-point one is easier to implement for any type of state-
space model. The Gaussian one produces a better quality of filtration when the
densities are close to the Gaussian ones (Ivashchenko, 2014).
The suggested model of the data-generating process is determined by Eq. 20
to Eq. 22 and a discrete MS process for the regime variable, rt, where Xstate,tis
the vector of the state variables (a subset of the model variable vector Xt), and εt
and utare the vectors of independent shocks (model innovations and measure-
ment errors) that have a zero-mean normal distribution. δis a constant equal to
one and related to the perturbation with respect to uncertainty. The second-order
approximation of the MSDSGE model is computed with the RISE toolbox (Maih,
2015):
Yt=HXt+ut, (20)
Zt=hXstate,tδ εti, (21)
Xt+1=A0,rt+1+A1,rt+1Zt+A2,rt+1(ZtZt), (22)
where is the Kronecker product.
The difference from the usual DSGE model second-order approximation is the
existence of regime dependence. Each filtering step is described below. The non-
linear filters (including the suggested ones) use some approximations. Computa-
tions within the filters that use approximations are highlighted.
A.1 Density of Ytas a function of the density of Xt
The initial information for this step is that the density of Xtis a normal mixture.
The linear equation for the observed variables, Eq. 20, presents the density of
Ytas a normal mixture with the same probabilities of regimes and the following
expectations and variances (conditional on the regime):
Es[Yt]=Es[HXt+ut]=HEs[Xt], (23)
37
Vs[Yt]=Vs[HXt+ut]=HVs[Xt]H.0+Vs[ut], (24)
where Es[.]and Vs[.]denote the expectation and variance operators conditional
on regime s.
A.2 Likelihood of Yt
The initial information for this step is that the density of Ytis a normal mixture.
This means that the likelihood can be determined as
L[Yt]=
NS
∑
s=1
p(rt=s)L[Ytjrt=s]
=
NS
∑
s=1
p(rt=s)e1
2(YtEs[Yt])0(Vs[Yt])1(YtEs[Yt])
(2π)NY
2jVs[Yt]j1
2
, (25)
where L[.]is the likelihood, NSthe number of regimes, and NYthe number of
observed variables.
A.3 Conditional density of XtjYt
The initial information for this step is the vector of observation Ytand that the
density of Xtis a normal mixture. The linear Eq. 20 allows a computation condi-
tional on the regime and observation density in the same way as the Kalman filter
in Eq. 26 to Eq. 28:
K0
s=(Vs[Yt])1HVs[Xt], (26)
Es[XtjYt]=Es[Xt]+Ks(YtEs[Yt]) , (27)
Vs[XtjYt]=(INXKsH)Vs[Xt] (INXKsH)0, (28)
p(rt=sjYt)=p(rt=s;Yt)
p(Yt)=p(Ytjrt=s)p(rt=s)
p(Yt). (29)
Eq. 29 shows the probability of regime sconditional on the observed variables.
p(Yt)is the likelihood (computed in Appendix A.2), and p(Ytjrt=s)has a normal
density.
A.4 Density of Xt+1as a function of the density of XtjYt
The initial information for this step is the density for the vector of the model vari-
ables Xt(normal mixture).
The first step is the computation of the expectation (Es,1) and variance (Vs,1) of
38
vector Xtconditional on the future state, such as
Es,1 =E(Xtjrt+1=s)=
Ns
∑
k=1
p(rt=k)p(rt+1=sjrt=k)
p(rt+1=s)Ek(Xt), (30)
Vs,1 =Es,1 (Es,1)0(31)
+
Ns
∑
k=1
p(rt=k)p(rt+1=sjrt=k)
p(rt+1=s)Ek(Xt)Ek(Xt)0+Vk(Xt).
The next step is the approximation (collapsing rule): the density of vector Xt
is a normal mixture with regime probabilities p(rt+1=s)and Gaussian densities
with moments Es,1 and Vs,1 .
The conditional density of Xtprovides the density of Zt. This allows us to com-
pute the conditional moments of the future vector of the variables Xt+1(Xt+1,rt+1
is the future vector of the model variables conditional on future regime rt+1):
Z0,t,rt+1=Zt,rt+1Ert+1Zt,rt+1, (32)
Xt+1,rt+1=A0,rt+1+A1,rt+1Zt,rt+1+A2,rt+1Zt,rt+1Zt,rt+1
=B0,rt+1+B1,rt+1Z0,t,rt+1+B2,rt+1Z0,t,rt+1Z0,t,rt+1(33)
EXt+1,rt+1=B0,rt+1+B2,rt+1vec VZt,rt+1=B0,rt+1+B2,rt+1vec Vrt+1, (34)
vec VXt+1,rt+1 =B1,rt+1B1,rt+1vec Vrt+1(35)
+B2,rt+1B2,rt+1 vec Vrt+1vec Vrt+1+
+vec vec Vrt+1Vrt+1!,
where vec f.gis the vectorization operator.
Eq. 32 to Eq. 35 are similar to the equations developed in Ivashchenko (2014).
The difference is that these formulas become formulas for moments, conditional
on the regime.
The last action of this step is an approximation. The density of Xt+1is a normal
mixture with moments according to Eq. 34 to Eq. 35.
B Summary of the variables
Table 1 summarizes the variables used in our model, showing the equations in
which the variable is used.
39
Variable Description Equations
Bi,j,k,tBonds bought by households i
in currency jwith maturity k3, 4, 13, 14
Bi,g,tBonds bought by the central bank
or government in country i11, 13
Ci,tConsumption of households of country i2, 3, 4, 12
Di,tDividends of firms from country i6, 7
etExchange rate (number of units of the domestic
currency per unit of the foreign currency) 3, 4, 8, 9, 10
Li,tLabor in country i2, 4, 5, 7
Mi,tMoney stock in country i2, 4, 11
Pi,tAggregate price level in country i2, 3, 4, 6, 7, 8,9,10
Pi,t(j)Price of goods of firms jin country i6, 7, 8, 9
Ri,k,tInterest rate in currency iwith maturity k4, 6, 10, 11
Wi,tWage in country i4, 7
Yi,tDemand in country i6, 8, 10, 12
YF,i,t(j)Production of firms jin country i5, 7, 8
εj
i,tExogenous process of type jin country i1, 2, 8, 9, 10, 15
ZtExogenous technology process 2, 5, 6, 15
Table 1: Summary of the variables used in the model’s equations
C Estimation results
Table 2 presents the median absolute error (MAE) and log predictive score (LPS)
for each observed variable to assess the forecast quality of our models and il-
lustrate the importance of switching volatility. We compute the LPS based on
Gaussian density, which is suggested by the MSDSGE model.14
MAE LPS
No SVS 1SVS 3SVS No SVS 1SVS 3SVS
EA GDP deflator 0.408% 0.366% 0.384% 3.66 3.70 3.67
US GDP deflator 0.156% 0.147% 0.150% 4.58 4.60 4.59
EA 3m rate 0.040% 0.040% 0.041% 5.68 5.68 5.88
US 3m rate 0.086% 0.085% 0.083% 5.35 5.33 5.36
EA demand to GDP 0.611% 0.597% 0.596% 3.43 3.40 3.40
US demand to GDP 0.580% 0.577% 0.587% 3.61 3.63 3.62
EA GDP growth 0.775% 0.734% 0.741% 3.30 3.32 3.31
US GDP growth 0.424% 0.448% 0.416% 3.45 3.46 3.48
EA 10y rate 0.065% 0.067% 0.068% 5.64 5.64 5.64
US 10y rate 0.062% 0.059% 0.054% 5.26 5.29 5.29
Table 2: Forecasting performance for the one-step ahead forecasts.
14When based on Gaussian mixture density, the LPS should equal the log-likelihood divided by
the number of periods (for the multivariate measure).
40
We also compute these statistics for the 3SVS model in the cases that the volatil-
ity state is always in regime 1 and always in regime 2 to illustrate the importance
of MS.15 We show that the 3SVS model when always in regime 1 is much worse
in terms of forecasting than that always in regime 2. At the same time, the 3SVS
model produces the best density forecasts. The difference in the sum of individ-
ual LPSs is relatively small because it does not take into account the correlation
between forecasts, which increases the advantage of the 3SVS model.
Tables 3 to 5 present the estimation results for each model. Our results are gen-
erally in line with those in the DSGE literature. The persistence of monetary pol-
icy shocks is lower than that of other shocks, as explained by Smets and Wouters
(2007). The coefficient of relative risk aversion is close to unity and lower than that
found in the literature (Benchimol, 2014).
We do not compare the models using their respective log-likelihood ratios for
several reasons. First, the difference between the log-likelihood values of the two
models does not mean that we must disregard the model with the lowest log-
likelihood even if the advantage is statistically significant. For instance, the lat-
ter model could still be used to perform forecasting in changing environments
(Benchimol and Fourçans, 2017, 2019). Second, whatever the log-likelihood, the
model is designed to capture only specific characteristics of the data. It is an open
question as to whether log-likelihood is an adequate measure to evaluate how well
the model accounts for particular aspects of the data.
Nevertheless, we report the log-likelihood values and corresponding likeli-
hood ratio tests hereafter. The log-likelihood values of the 0SVS, 1SVS, and 3SVS
models are 3581.70, 3593.35, and 3611.67, respectively. This means that the p-value
of the likelihood ratio test of 1SVS vs. 0SVS is 3.51e-05, 3SVS vs. 1SVS is 1.1e-08,
and 3SVS vs. 0SVS is 1.25e-11. Consequently, a more flexible model explains sig-
nificantly more of the data. Our estimation of the covariance matrix allows us to
construct a Laplace approximation of the marginal likelihood (maximum likeli-
hood estimation is equivalent to a Bayesian one with flat priors).
The results are sensitive to the approximation methodology. We use the RISE
function “solve_accelerate.” If we try to compute the approximation without this
function—and compute the likelihood—the resulting values would be 3413.21
(0SVS), 3530.66 (1SVS), and 2515.20 (3SVS). This is probably due to the iterative
nature of the MSDSGE solution approximation that converges to a slightly differ-
ent solution. The sharp likelihood of the nonlinear approximation transforms this
small difference into a significant difference in the likelihood. Thus, even small
details of the solution algorithm can be crucial in a nonlinear world.
15See Fig. 1 for the estimated probabilities.
41
Priors Posteriors Priors Posteriors
LB UB Mean Std. LB UB Mean Std.
ηd,u-0.01 0.00 -0.006 0.000 Af-20.0 20.0 0.000 0.002
ηd,m-20.0 20.0 -6.493 0.263 ϕd,d,sr 0.00 1000 0.195 0.004
ηd,r0.00 0.01 0.000 0.000 ϕd,f,sr 0.00 1000 0.083 0.000
ηd,p1.00 20.0 3.281 0.020 ϕd,d,lr 0.00 1000 16.26 0.413
ηf,u-0.01 0.00 -0.008 0.000 ϕd,f,lr 0.00 1000 0.236 0.003
ηf,m-20.0 20.0 -7.558 0.001 ϕf,d,sr 0.00 1000 0.001 0.000
ηf,r0.00 0.01 0.001 0.000 ϕf,f,sr 0.00 1000 0.000 0.000
ηf,p1.00 20.0 11.28 0.008 ϕf,d,lr 0.00 1000 0.012 0.001
ηy0.00 0.01 0.001 0.000 ϕf,f,lr 0.00 1000 0.000 0.000
ηd,u-1.00 1.00 0.978 0.004 sd0.00 1.00 0.655 0.006
ηd,l-1.00 1.00 0.975 0.001 sf0.00 1.00 0.063 0.001
ηd,m-1.00 1.00 0.891 0.007 ϕd,P0.00 1000 591.4 21.637
ηd,r-1.00 1.00 0.180 0.021 ϕf,P0.00 1000 227.8 4.087
ηd,p-1.00 1.00 0.948 0.001 vd0.00 1.00 0.473 0.048
ηf,u-1.00 1.00 0.462 0.030 vf0.00 1.00 0.787 0.008
ηf,l-1.00 1.00 0.914 0.002 hd,c0.00 0.90 0.669 0.000
ηf,m-1.00 1.00 0.950 0.003 hf,c0.00 0.90 0.743 0.000
ηf,r-1.00 1.00 0.756 0.009 ρd,r0.00 0.99 0.974 0.000
ηf,p-1.00 1.00 0.953 0.002 ρd,p1.00 5.00 1.004 0.025
ηy-1.00 1.00 0.960 0.001 ρd,y-5.00 5.00 0.065 0.005
σξd,u0.00 10.0 0.001 0.000 ρd,e-20.0 20.0 0.005 0.000
σξd,l0.00 10.0 0.152 0.007 ρf,r0.00 0.99 0.796 0.000
σξd,m0.00 10.0 0.124 0.004 ρf,p1.00 5.00 2.906 0.001
σξd,r0.00 10.0 0.001 0.000 ρf,y-5.00 5.00 -0.097 0.003
σξd,p0.00 10.0 1.342 0.030 ρf,e-20.0 20.0 -0.029 0.000
σξf,u0.00 10.0 0.007 0.001 bd,d,sr -20.0 20.0 0.236 0.006
σξf,l0.00 10.0 0.048 0.001 bd,f,sr -20.0 20.0 1.859 0.004
σξf,m0.00 10.0 0.106 0.002 bd,d,lr -20.0 20.0 -0.006 0.000
σξf,r0.00 10.0 0.002 0.000 bd,f,lr -20.0 20.0 -0.319 0.006
σξf,p0.00 10.0 0.635 0.021 cd-20.0 20.0 -0.421 0.000
σξy0.00 10.0 0.001 0.000 cf-20.0 20.0 -1.593 0.000
1/σd,c-20.0 20.0 0.727 0.001 e-20.0 20.0 1.531 0.001
1/σd,l-20.0 20.0 9.821 0.645 pd0.00 0.01 0.000 0.000
1/σd,m-20.0 20.0 2.723 0.027 pd(j)-20.0 20.0 0.003 0.000
Ad-20.0 20.0 -0.070 0.143 pf0.00 0.01 0.003 0.000
1/σf,c-20.0 20.0 -0.688 0.000 pf(j)-20.0 20.0 0.000 0.000
1/σf,l-20.0 20.0 -1.758 0.044 rd,lr 0.00 0.03 0.027 0.000
1/σf,m-20.0 20.0 0.794 0.000
Table 3: Estimation results for the model without an SVS. LB and UB stand for lower
bound and upper bound, respectively.
42
Priors Posteriors Priors Posteriors
LB UB Mean Std. LB UB Mean Std.
pregt=2jre gt1=10.04 0.35 0.056 0.064 sd0.00 1.00 0.599 0.007
pregt=1jre gt1=20.04 0.35 0.045 0.028 sf0.00 1.00 0.062 0.001
ηd,u-0.01 0.00 -0.006 0.000 ϕd,P0.00 1000 953.4 15.45
ηd,m-20.0 20.0 -6.613 0.015 ϕf,P0.00 1000 241.1 4.540
ηd,r0.00 0.01 0.000 0.000 vd0.00 1.00 0.543 0.043
ηd,p1.00 20.0 3.255 0.039 vf0.00 1.00 0.779 0.010
ηf,u-0.01 0.00 -0.008 0.000 hd,c0.00 0.90 0.670 0.001
ηf,m-20.0 20.0 -7.556 0.000 hf,c0.00 0.90 0.741 0.000
ηf,r0.00 0.01 0.001 0.000 ρd,r0.00 0.99 0.974 0.000
ηf,p1.00 20.0 11.61 0.021 ρd,p1.00 5.00 1.002 0.012
ηy0.00 0.01 0.001 0.000 ρd,y-5.00 5.00 0.077 0.005
ηd,u-1.00 1.00 0.979 0.001 ρd,e-20.0 20.0 0.003 0.000
ηd,l-1.00 1.00 0.971 0.001 ρf,r0.00 0.99 0.796 0.000
ηd,m-1.00 1.00 0.897 0.005 ρf,p1.00 5.00 2.903 0.000
ηd,r-1.00 1.00 0.197 0.029 ρf,y-5.00 5.00 -0.087 0.003
ηd,p-1.00 1.00 0.959 0.000 ρf,e-20.0 20.0 -0.030 0.000
ηf,u-1.00 1.00 0.443 0.021 bd,d,sr -20.0 20.0 0.249 0.004
ηf,l-1.00 1.00 0.912 0.005 bd,f,sr -20.0 20.0 1.877 0.010
ηf,m-1.00 1.00 0.952 0.006 bd,d,lr -20.0 20.0 -0.007 0.000
ηf,r-1.00 1.00 0.736 0.006 bd,f,lr -20.0 20.0 -0.324 0.003
ηf,p-1.00 1.00 0.952 0.002 cd-20.0 20.0 -0.421 0.000
ηy-1.00 1.00 0.959 0.002 cf-20.0 20.0 -1.602 0.001
1/σd,c-20.0 20.0 0.733 0.001 e-20.0 20.0 1.536 0.000
1/σd,l-20.0 20.0 13.11 0.665 pd0.00 0.01 0.000 0.000
1/σd,m-20.0 20.0 2.856 0.039 pd(j)-20.0 20.0 0.003 0.000
Ad-20.0 20.0 0.571 0.157 pf0.00 0.01 0.003 0.000
1/σf,c-20.0 20.0 -0.688 0.000 pf(j)-20.0 20.0 0.000 0.000
1/σf,l-20.0 20.0 -1.774 0.048 rd,lr 0.00 0.03 0.027 0.000
1/σf,m-20.0 20.0 0.793 0.000 σξd,u0.00 10.0 0.000 0.000
Af-20.0 20.0 -1.872 0.574 σξd,l0.00 10.0 0.204 0.008
ϕd,d,sr 0.00 1000 0.192 0.018 σξd,m0.00 10.0 0.112 0.003
ϕd,f,sr 0.00 1000 0.082 0.000 σξd,r0.00 10.0 0.001 0.000
ϕd,d,lr 0.00 1000 17.51 0.438 σξd,p0.00 10.0 1.184 0.017
ϕd,f,lr 0.00 1000 0.244 0.001 σξf,u0.00 10.0 0.007 0.000
ϕf,d,sr 0.00 1000 0.001 0.000 σξf,l0.00 10.0 0.051 0.001
ϕf,f,sr 0.00 1000 0.000 0.000 σξf,m0.00 10.0 0.107 0.002
ϕf,d,lr 0.00 1000 0.008 0.001 σξf,r0.00 10.0 0.002 0.000
ϕf,f,lr 0.00 1000 0.000 0.000 σξf,p0.00 10.0 0.636 0.010
sd0.00 1.00 0.599 0.007 σξyjreg.=10.00 10.0 0.001 0.000
sf0.00 1.00 0.062 0.001 σξyjreg.=20.00 10.0 0.001 0.000
Table 4: Estimation results for the 1SVS model. LB and UB stand for lower bound and
upper bound, respectively, pregt=ajregt1=bfor the probability of switching to regime ain period tif
in regime bin period t1, and σ(ξyjreg.=a)for the standard deviation of the corresponding shock
in regime a.
43
Priors Posteriors Priors Posteriors
LB UB Mean Std. LB UB Mean Std.
pregt=2jre gt1=10.04 0.35 0.042 0.001 ϕd,P0.00 1000 796.4 33.12
pregt=1jre gt1=20.04 0.35 0.131 0.017 ϕf,P0.00 1000 218.5 8.054
ηd,u-0.01 0.00 -0.006 0.000 vd0.00 1.00 0.615 0.065
ηd,m-20.0 20.0 -6.807 0.054 vf0.00 1.00 0.763 0.026
ηd,r0.00 0.01 0.000 0.000 hd,c0.00 0.90 0.666 0.000
ηd,p1.00 20.0 3.253 0.027 hf,c0.00 0.90 0.742 0.001
ηf,u-0.01 0.00 -0.008 0.000 ρd,r0.00 0.99 0.972 0.000
ηf,m-20.0 20.0 -7.564 0.068 ρd,π1.00 5.00 1.003 0.016
ηf,r0.00 0.01 0.001 0.000 ρd,y-5.00 5.00 0.067 0.004
ηf,π1.00 20.0 11.44 0.100 ρd,e-20.0 20.0 0.003 0.001
ηy0.00 0.01 0.001 0.000 ρf,r0.00 0.99 0.796 0.000
ηd,u-1.00 1.00 0.979 0.001 ρf,p1.00 5.00 2.901 0.004
ηd,l-1.00 1.00 0.969 0.002 ρf,y-5.00 5.00 -0.092 0.008
ηd,m-1.00 1.00 0.883 0.010 ρf,e-20.0 20.0 -0.030 0.000
ηd,r-1.00 1.00 0.184 0.006 bd,d,sr -20.0 20.0 0.233 0.006
ηd,p-1.00 1.00 0.954 0.003 bd,f,sr -20.0 20.0 1.867 0.016
ηf,u-1.00 1.00 0.447 0.027 bd,d,l r -20.0 20.0 -0.006 0.000
ηf,l-1.00 1.00 0.907 0.006 bd,f,l r -20.0 20.0 -0.327 0.005
ηf,m-1.00 1.00 0.954 0.004 cd-20.0 20.0 -0.424 0.003
ηf,r-1.00 1.00 0.743 0.014 cf-20.0 20.0 -1.597 0.002
ηf,p-1.00 1.00 0.951 0.005 e-20.0 20.0 1.536 0.004
ηy-1.00 1.00 0.958 0.001 pd0.00 0.01 0.000 0.000
1/σd,c-20.0 20.0 0.734 0.001 pd(j)-20.0 20.0 0.003 0.000
1/σd,l-20.0 20.0 10.20 0.790 pf0.00 0.01 0.003 0.000
1/σd,m-20.0 20.0 2.774 0.038 pf(j)-20.0 20.0 0.000 0.000
Ad-20.0 20.0 -0.015 0.017 rd,lr 0.00 0.03 0.028 0.000
1/σf,c-20.0 20.0 -0.687 0.002 σξd,u0.00 10.0 0.000 0.000
1/σf,l-20.0 20.0 -1.650 0.043 σξd,l0.00 10.0 0.176 0.012
1/σf,m-20.0 20.0 0.793 0.000 σξd,m0.00 10.0 0.120 0.006
Af-20.0 20.0 8.600 4.547 σξd,p0.00 10.0 1.235 0.035
ϕd,d,sr 0.00 1000 0.182 0.004 σξf,u0.00 10.0 0.007 0.001
ϕd,f,sr 0.00 1000 0.082 0.000 σξf,l0.00 10.0 0.047 0.003
ϕd,d,lr 0.00 1000 17.33 1.478 σξf,m0.00 10.0 0.103 0.003
ϕd,f,lr 0.00 1000 0.216 0.007 σξf,p0.00 10.0 0.651 0.014
ϕf,d,sr 0.00 1000 0.002 0.000 σξyjreg.=10.00 10.0 0.002 0.000
ϕf,f,sr 0.00 1000 0.000 0.000 σξd,rjreg.=10.00 10.0 0.000 0.000
ϕf,d,lr 0.00 1000 0.010 0.000 σξf,rjreg.=10.00 10.0 0.001 0.000
ϕf,f,lr 0.00 1000 0.000 0.000 σξyjreg.=20.00 10.0 0.001 0.000
sd0.00 1.00 0.618 0.010 σξd,rjre g.=20.00 10.0 0.001 0.000
sf0.00 1.00 0.063 0.002 σξf,rjre g.=20.00 10.0 0.003 0.000
Table 5: Estimation results for the 3SVS model. LB and UB stand for lower bound and
upper bound, respectively, pregt=ajregt1=bfor the probability of switching to regime ain period tif
in regime bin period t1, and σ(ξyjreg.=a)for the standard deviation of the corresponding shock
in regime a.
44
D Variance decompositions
Tables 6 to 8 present the short- and long-run variance decompositions of the vari-
ables with respect to the shocks for each model. The variance decompositions for
the nonlinear models require additional comments. The variance decomposition
coefficients, 8i2fd,fgand 8j2fu,m,l,p,r,yg, are computed with respect to the
following function:
VDtx,ξi,j=1Ehx2
tj8s,ξi,j,sN(0, 0)iEhxtj8s,ξi,j,sN(0, 0)i2
Ehx2
tj8s,ξi,j,sN0, σξi,j,siEhxtj8s,ξi,j,sN0, σξi,j,si2,
(36)
where xtis the value of the variable of interest (for which the variance decomposi-
tion is computed) in period t,ξI,jis the shock of interest, σ(.)is the standard error
operator, E[.]is the expectation operator, and Nis the Normal law. This formula
expresses the proportion by which the variance in the variable of interest is lower
if the shock of interest is equal to zero in all periods.
The sum of VDt(x,ξ)for each shock gives 1 in the case of the linear model.
However, this does not hold for nonlinear models. The sum is close to 1 for most
of the variables, but there are exceptions.
All the models present variance decompositions in line with the literature. The
domestic price markup shock (ξd,p) plays a predominant role in domestic prices
(pd,t) in both the short run and the long run. However, the domestic preference
shock (ξd,u) plays a substantial role in domestic wages (wd,t) in the short run and
an important role in domestic consumption (cd,t) in the long run, which should be
greater if we do not consider domestic firms’ production, yd,t(j).
Foreign shocks play a role in the dynamics of several variables, especially the
foreign preference shock (ξf,u). This shock drives the dynamics of several foreign
as well as domestic variables in the long run, showing that the EA (domestic) is
still dependent on the US economy (foreign) and US households’ preferences.
Moreover, financial markets are almost entirely dependent on US economy
(foreign) shocks and technology progress shocks in all versions of the model in
the long term. Foreign shocks play an important role in domestic long-term inter-
est rates, but not for the corresponding bonds in short-term horizons. Thus, the
model reproduces the domination by the United States in financial markets.
Interestingly, foreign money demand shocks (ξf,m) play an important role in
the dynamics of several domestic variables in the long run (rather than in the
short run), such as the exchange rate, domestic long-run interest rates, and most
domestic bond quantities, including those bought by central banks. This finding
shows the importance of US money demand shocks for the EA’s economic dy-
namics, in line with the closed-economy and linear DSGE literature (Benchimol
45
and Fourçans, 2017).
In addition, foreign worked hours’ shocks have a substantial impact on domes-
tic economic dynamics, particularly for bonds’ positions and the exchange rate.
The explanatory power of the technological progress shock (ξy) is relatively
small in the short term. This shock explains 4–6% of domestic output growth
and 14–17% of foreign output growth, depending on the model version. This is
substantially smaller than the 25.4% for Europe (Lombardo and McAdam, 2012).
The long-term explanatory power is 35–43% for domestic output growth and 59–
62% for foreign output growth, depending on the model version, which is larger
than that found by Lombardo and McAdam (2012) (27.3% for a 20-quarter hori-
zon). A similar picture is related to inflation’s explanatory power. The short-term
values are 5.7–9.3% for domestic inflation and 13.7–22.5% for foreign inflation,
while the long-term values are 19.2–24.7% and 41.8–45.1%, respectively. This dif-
fers from the 26.9% and 27.7%, respectively, in the models of the EA (Lombardo
and McAdam, 2012). Small open-economy models produce similar long-term val-
ues (16âe”21%) for Canada, Spain, and Sweden (Guerrón-Quintana, 2013). How-
ever, the long-term explanatory power of the technological progress shock for in-
flation differs significantly by country: from 13% for Australia to 54% for Belgium
(Guerrón-Quintana, 2013).
The variance decomposition of foreign inflation is slightly unusual. The
markup shocks are usually crucial for inflation: 64.9% for EA inflation (Guerrón-
Quintana, 2013) and more than 80% for US inflation (Smets and Wouters, 2007).
However, our model explains 44.9–50.6% of short-term domestic inflation with a
domestic markup shock (ξd,p) and 0.4–1.9% of short-term foreign inflation with a
foreign markup shock (ξf,p).
There are few differences between the models in terms of the variance decom-
position of the variables with respect to structural shocks, except when consider-
ing domestic worked hours and domestic money demand shocks. These shocks
affect the domestic bonds’ positions of households and central banks in different
manners, showing that including SVSs in domestic and foreign monetary poli-
cies diminishes the role of domestic money demand while increasing the role of
worked hours in domestic bonds’ positions.
46
Short-run variance decompositions Long-run variance decompositions
ξd,uξd,lξd,mξd,rξd,pξf,uξf,lξf,mξf,rξf,pξyξd,uξd,lξd,mξd,rξd,pξf,uξf,lξf,mξf,rξf,pξy
bd,d,sr,t0,0 55,2 42,5 0,1 0,0 0,0 0,0 0,0 0,0 0,0 0,2 19,2 2,5 10,9 2,4 15,8 26,4 11,6 10,7 4,1 7,9 17,5
bd,f,sr,t0,2 0,1 0,0 0,2 0,7 44,6 17,1 32,8 3,6 2,3 17,3 3,5 0,0 0,2 1,0 0,8 30,4 17,5 40,8 2,5 12,6 48,9
bd,g,t0,0 55,3 42,4 0,1 0,0 0,2 0,0 0,1 0,1 0,0 0,0 17,6 2,2 10,0 2,1 14,8 27,4 12,5 11,5 4,2 9,4 18,9
bd,d,lr,t1,1 0,0 0,0 5,4 1,4 57,1 17,9 18,2 0,3 1,4 1,6 3,4 0,1 0,2 1,8 4,7 27,0 16,0 39,2 3,1 13,1 48,2
bd,f,lr,t0,1 0,1 0,0 0,2 0,0 70,9 20,0 0,6 4,9 2,1 21,6 4,2 0,1 0,2 1,1 3,2 27,3 15,7 42,5 2,4 15,9 47,1
bf,g,t0,5 0,0 0,0 0,5 0,0 65,5 21,0 1,9 7,8 2,0 29,5 3,7 0,0 0,0 1,1 0,2 24,3 13,1 34,6 2,9 4,9 72,3
cd,t7,8 18,7 0,9 69,4 5,6 0,2 0,1 0,1 0,0 0,9 1,5 26,5 0,4 0,1 2,3 39,5 13,2 7,2 15,0 1,6 13,2 16,1
cf,t0,2 0,0 0,0 0,1 0,5 46,0 0,3 18,0 21,3 2,0 8,7 1,6 0,1 0,0 0,0 1,0 61,4 10,4 22,2 2,7 24,5 20,0
et6,0 1,7 0,0 2,0 3,0 41,5 17,9 24,5 4,0 1,3 16,6 2,3 0,1 0,1 1,4 1,2 40,3 19,4 31,6 5,8 12,3 41,4
md,t3,2 54,4 37,5 1,5 4,2 0,4 0,0 0,1 0,1 0,3 0,0 10,3 5,1 13,9 9,7 12,4 27,2 11,5 9,4 3,6 5,9 18,9
mf,t0,3 0,0 0,0 0,5 0,0 54,5 24,9 0,4 5,7 1,7 19,2 2,6 0,1 0,1 1,5 0,8 31,8 18,3 34,7 4,3 8,5 54,8
pd,t34,8 4,2 0,7 25,8 50,6 1,0 0,2 0,3 0,1 5,0 5,8 12,7 1,4 2,4 10,5 53,5 5,7 1,9 4,4 0,4 10,1 19,2
pf,t0,6 0,0 0,0 0,2 0,0 38,4 1,9 38,3 3,3 1,9 13,7 2,2 0,1 0,1 1,4 1,0 37,6 18,9 33,0 5,2 10,4 43,8
rd,t0,6 0,1 98,6 0,6 0,5 0,0 0,0 0,0 0,0 0,0 0,3 13,7 0,3 16,5 2,5 11,4 29,5 13,0 10,3 4,4 6,8 20,5
rf,t0,6 0,0 0,0 0,5 0,0 64,3 1,6 0,0 8,8 2,1 22,3 2,4 0,1 0,0 1,4 1,0 36,8 17,9 28,1 5,4 9,5 48,4
rd,lr,t0,7 28,3 11,1 3,5 0,7 35,3 7,2 9,5 0,2 0,7 0,4 4,8 0,2 1,4 1,5 2,6 37,6 16,5 32,5 4,6 13,6 39,9
rf,lr,t2,6 0,0 0,0 0,6 1,8 16,5 3,9 12,4 0,3 1,1 63,8 2,2 0,1 0,1 1,3 1,2 38,0 19,3 29,9 5,6 10,8 45,0
wd,t64,4 2,8 0,1 14,2 2,3 1,7 0,8 1,4 0,0 15,9 1,0 5,7 0,3 0,1 4,3 72,2 3,0 2,2 7,2 0,0 17,4 7,0
wf,t0,0 0,0 0,0 0,1 0,1 62,3 0,7 10,2 15,4 3,4 3,8 1,2 0,2 0,0 0,0 0,2 40,1 3,3 23,0 7,3 30,8 19,8
yd,t(j)2,8 6,6 0,1 28,3 4,9 6,0 2,4 5,6 0,0 56,1 4,2 30,5 0,4 0,1 2,6 51,8 2,8 2,4 7,5 0,0 16,6 6,4
yf,t(j)0,5 0,2 0,0 0,0 3,8 29,3 1,6 14,1 16,1 34,7 5,6 1,8 0,3 0,0 0,3 1,3 63,4 5,9 16,0 2,2 30,9 15,3
Table 6: Short- and long-run variance decompositions for the model without an SVS
47
Short-run variance decompositions Long-run variance decompositions
ξd,uξd,lξd,mξd,rξd,pξf,uξf,lξf,mξf,rξf,pξyξd,uξd,lξd,mξd,rξd,pξf,uξf,lξf,mξf,rξf,pξy
bd,d,sr,t0,0 50,2 49,4 0,3 0,0 0,0 0,0 0,0 0,0 0,0 0,3 17,6 2,2 13,8 1,7 13,1 24,7 11,8 10,7 2,4 11,1 19,2
bd,f,sr,t0,1 0,1 0,1 0,5 0,4 43,5 16,9 30,5 4,2 1,9 16,9 0,7 0,1 0,0 1,3 0,7 28,5 16,5 36,5 3,3 13,6 55,7
bd,g,t0,0 50,1 49,3 0,3 0,0 0,2 0,1 0,1 0,0 0,0 0,4 14,7 1,9 12,2 1,6 11,7 25,9 12,4 11,9 2,8 13,0 22,5
bd,d,lr,t0,5 0,0 0,0 4,4 0,7 59,9 20,1 14,4 0,7 0,7 2,7 3,7 0,3 0,0 2,9 4,3 23,6 17,4 37,6 3,3 12,0 49,8
bd,f,lr,t0,3 0,1 0,0 0,4 0,0 68,6 16,1 0,6 5,5 0,6 24,6 1,4 0,2 0,0 2,1 2,0 25,1 16,1 39,1 3,3 15,8 51,8
bf,g,t0,6 0,0 0,0 1,2 0,0 63,1 16,8 1,8 8,9 0,6 32,0 1,2 0,0 0,0 1,5 0,9 25,7 12,2 29,9 1,7 6,0 77,0
cd,t7,0 17,4 0,6 74,2 3,2 0,3 0,1 0,0 0,0 0,5 1,3 24,3 0,9 0,1 3,8 28,3 16,3 8,5 17,0 0,7 18,1 23,0
cf,t0,2 0,0 0,0 0,2 0,4 49,8 0,5 18,1 22,2 1,6 8,8 0,7 0,1 0,1 2,0 1,4 58,8 7,4 19,4 2,4 25,5 25,3
et3,9 1,4 0,0 2,8 1,6 46,3 19,6 22,2 4,9 0,6 19,3 1,8 0,0 0,0 2,8 1,4 39,9 22,9 31,1 5,0 11,2 40,0
md,t2,8 50,1 43,5 0,9 3,4 0,4 0,1 0,1 0,0 0,2 0,2 10,8 4,3 18,6 11,4 10,3 22,5 11,3 8,1 2,3 7,8 16,4
mf,t0,5 0,0 0,0 0,8 0,1 51,8 22,0 0,6 6,6 0,5 23,8 1,5 0,0 0,0 2,3 1,1 32,6 18,7 31,7 3,4 8,0 54,1
pd,t36,0 3,7 0,0 22,3 44,9 1,8 0,0 1,0 0,0 5,3 9,3 16,5 2,2 1,5 10,6 42,0 6,8 4,1 7,6 0,6 14,5 23,8
pf,t0,5 0,0 0,0 0,8 0,1 39,2 0,8 36,2 4,6 0,4 18,6 1,8 0,0 0,0 2,5 1,1 38,8 20,9 32,7 4,7 10,0 41,8
rd,t0,5 0,0 98,8 0,3 0,5 0,1 0,0 0,0 0,0 0,0 0,2 13,5 0,0 21,8 1,4 10,1 25,9 12,3 8,6 2,7 8,6 19,0
rf,t0,5 0,0 0,0 0,9 0,1 64,6 0,4 0,2 8,8 0,6 26,0 1,3 0,0 0,0 2,3 1,0 38,7 18,7 26,5 3,9 8,3 46,6
rd,lr,t0,3 21,7 11,9 3,3 0,6 44,9 9,3 6,7 0,1 0,4 2,4 4,2 0,3 0,4 2,9 1,9 34,5 19,7 32,9 4,5 13,5 41,5
rf,lr,t1,4 0,0 0,0 1,3 0,8 14,3 3,6 10,8 0,6 0,0 69,0 1,6 0,0 0,0 2,5 1,2 38,8 21,9 29,4 4,7 9,8 42,9
wd,t66,7 1,9 0,1 11,0 0,6 1,7 0,4 2,7 0,0 16,6 1,8 11,5 1,2 0,1 6,1 57,3 5,3 3,4 9,3 0,1 25,7 9,3
wf,t0,2 0,1 0,0 0,2 0,2 64,5 0,2 10,8 16,6 3,0 5,4 1,1 0,2 0,0 1,1 0,3 42,3 4,4 18,4 6,9 32,6 20,6
yd,t(j)3,4 4,7 0,0 29,1 2,3 7,1 3,2 8,9 0,2 60,6 5,9 33,0 0,8 0,2 4,5 40,5 4,2 2,3 8,5 0,1 22,5 8,6
yf,t(j)0,3 0,3 0,0 0,6 1,8 33,3 2,0 15,1 15,3 34,0 6,3 2,2 0,0 0,0 1,1 0,2 64,6 7,3 12,9 1,4 30,6 14,4
Table 7: Short- and long-run variance decompositions for the 1SVS model
48
Short-run variance decompositions Long-run variance decompositions
ξd,uξd,lξd,mξd,rξd,pξf,uξf,lξf,mξf,rξf,pξyξd,uξd,lξd,mξd,rξd,pξf,uξf,lξf,mξf,rξf,pξy
bd,d,sr,t0,0 76,4 23,2 0,4 0,0 0,1 0,0 0,0 0,0 0,0 0,0 18,4 2,5 13,2 2,9 14,9 25,2 13,0 9,7 3,3 7,8 20,5
bd,f,sr,t0,8 0,0 0,0 0,3 0,8 51,0 18,1 19,3 4,1 2,3 19,7 3,4 0,1 0,2 2,6 0,7 31,3 22,2 39,7 2,7 13,2 51,1
bd,g,t0,0 76,5 23,0 0,4 0,0 0,0 0,0 0,1 0,0 0,0 0,1 16,2 2,2 11,9 2,8 13,7 26,6 14,2 10,6 3,7 9,6 22,4
bd,d,lr,t1,4 0,0 0,0 4,6 1,8 58,8 19,4 10,9 0,6 0,9 1,9 5,4 0,0 0,0 3,6 3,7 24,0 21,9 39,9 1,9 10,7 51,2
bd,f,lr,t0,6 0,0 0,0 0,0 0,0 68,7 16,3 0,4 5,2 2,1 25,7 5,0 0,0 0,1 3,6 1,6 25,8 22,0 41,4 2,1 15,5 50,2
bf,g,t1,2 0,0 0,0 0,2 0,0 63,6 17,7 1,4 8,6 2,0 34,1 0,3 0,1 0,2 1,1 0,6 31,6 13,2 33,5 2,1 2,9 73,4
cd,t8,4 16,0 0,2 73,5 7,3 0,6 0,1 0,0 0,0 0,2 0,9 26,0 0,6 0,5 3,9 35,2 14,6 10,3 14,4 1,1 13,7 18,6
cf,t0,1 0,0 0,0 0,3 0,6 49,2 0,8 9,9 26,6 2,8 10,9 1,3 0,1 0,2 1,5 0,6 56,6 10,3 20,2 2,5 25,0 25,2
et6,0 1,5 0,0 2,7 2,9 44,7 22,6 16,7 4,3 1,0 21,0 1,9 0,0 0,0 2,7 0,6 37,9 23,4 33,5 4,1 8,7 43,5
md,t3,1 72,8 20,7 1,6 3,9 0,0 0,0 0,1 0,0 0,6 0,4 13,5 4,6 15,9 10,8 12,0 21,4 9,9 8,4 2,7 4,9 19,9
mf,t0,8 0,0 0,0 0,3 0,0 54,0 22,1 0,7 5,9 1,6 24,5 1,1 0,0 0,1 2,0 0,4 33,0 20,7 33,9 3,3 5,0 56,2
pd,t40,9 3,1 0,6 22,7 46,3 0,2 0,0 0,2 0,1 5,4 8,7 15,3 1,7 2,0 9,6 48,9 5,0 2,4 4,9 0,5 12,1 24,8
pf,t0,9 0,1 0,0 0,7 0,1 45,1 0,7 27,6 4,8 1,8 22,5 1,6 0,0 0,0 2,1 0,4 36,8 21,3 33,1 3,8 7,7 45,1
rd,t0,9 0,1 96,0 1,0 2,4 0,0 0,0 0,0 0,0 0,0 0,1 15,1 0,0 19,0 2,3 11,7 25,0 12,0 8,6 3,5 5,5 23,1
rf,t0,6 0,0 0,0 0,4 0,0 64,4 0,3 0,4 8,7 1,7 26,5 0,9 0,0 0,1 1,9 0,3 36,1 19,6 27,7 4,6 5,6 49,7
rd,lr,t0,6 31,4 4,6 2,9 1,2 44,6 8,5 6,2 0,5 1,2 1,4 5,2 0,1 1,1 3,3 1,9 32,9 21,7 36,6 3,0 11,2 43,3
rf,lr,t2,7 0,1 0,0 0,9 1,2 15,1 2,3 7,6 0,8 0,7 72,4 1,6 0,0 0,0 2,2 0,5 36,5 22,3 30,1 4,3 7,2 46,1
wd,t73,6 1,6 0,1 9,9 1,9 1,2 0,9 1,1 0,1 17,3 1,5 7,4 0,9 0,3 6,0 63,3 2,7 2,0 6,9 0,7 22,1 9,3
wf,t0,2 0,0 0,0 0,2 0,3 65,6 0,2 7,7 16,9 2,7 6,6 0,3 0,0 0,0 0,6 0,1 41,0 2,4 21,5 10,0 33,0 19,2
yd,t(j)2,9 4,4 0,0 26,4 4,1 6,1 3,9 2,4 0,0 62,7 5,1 27,6 0,4 0,3 4,4 46,2 3,3 1,4 6,0 0,9 19,8 10,0
yf,t(j)0,8 0,2 0,0 0,4 3,5 32,8 2,3 8,7 17,9 37,0 7,4 1,1 0,0 0,1 0,9 0,9 59,1 6,7 14,0 3,0 30,2 19,4
Table 8: Short- and long-run variance decompositions for the 3SVS model
49
