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Switching Volatility in a Nonlinear Open Economy

  • The Institute of Regional Economy Problems (Russian Academy of Sciences);Financial Research Institute

Abstract and Figures

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 nonlinear 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 financial markets influence the economy, should not be neglected. The nonlinear behavior of market-related variables highlights the importance of higher-order estimation for providing additional interpretations to policymakers.
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Switching Volatility in a Nonlinear Open
Jonathan Benchimolyand Sergey Ivashchenkoz
October 8, 2020
Uncertainty about an economy’s regime can change drastically around a
crisis. An imported crisis such as the global …nancial crisis in the euro area
highlights the e¤ect 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 signi…cant during the global …nancial crisis com-
pared with periods of calm. We describe how US shocks from both the
real economy and …nancial markets a¤ected the euro area economy and how
bond reallocation occurred between short- and long-term maturities during
the global …nancial crisis. Importantly, the estimated nonlinearities when
domestic and foreign …nancial markets in‡uence the economy, should not
be neglected. The nonlinear behavior of market-related variables highlights
the importance of higher-order estimation for providing additional interpre-
tations to policymakers.
Keywords: DSGE, Volatility Shocks, Markov Switching, Open Economy,
Financial Crisis, Nonlinearities.
JEL Classi…cation: C61, E32, F21, F41.
This paper does not necessarily re‡ect the views of the Bank of Israel. We thank the ref-
erees, 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 Macroeco-
nomics Workshop; and the 49th Money, Macro, and Finance Research Group conferences for
their valuable comments.
yBank of Israel, Jerusalem, Israel. Corresponding author. Email:
zRussian Academy of Sciences (IREP), Financial Research Institute, and Saint-Petersburg
State University, Saint Petersburg, Russia.
Please cite this paper as:
Benchimol, J., and Ivashchenko, S., 2021. Switching volatility in a non-
linear open economy. Journal of International Money and Finance, 101,
1 Introduction
The widespread consensus in macroeconomics based on the linear new Keynesian
model was shaken by the global …nancial 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
nancial crisis. The GFC started in the United States and a¤ected 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 e¤ects 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 business
cycle ‡uctuations (Bloom, 2009). According to Markov processes, the volatility 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. Specically,
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 inuencing 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
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 (Andreasen,
2010). For this purpose, we use nonlinear approximation algorithms and …lters 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 generaliza-
tion of the quadratic Kalman …lter 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 speci…cations: a baseline version without MS,
a version allowing MS in technology only, and another more developed version
allowing 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 …rst 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 derent,
especially around 2009Q1, which is also the case from the switching volatility point
of view. These di¤erences essentially come from the nonlinearities in economic dy-
namics, although our results are close to those obtained with linear open-economy
DSGE models (Chin et al., 2015). Second, we demonstrate the consequences of
SVSs on US and EA economic dynamics. SVSs produce a combination of short-
term de‡ation and long-term in‡ation e¤ects in line with Kiley (2014) but with
some asymmetries between the two economies. We demonstrate that SVSs par-
tially cause …nancial ‡ows, showing that they signi…cantly a¤ect both the trade-o¤
between short- and long-term bonds and consumption around the crisis. Third,
we conrm 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 domes-
1Appendix A presents the MS quadratic Kalman …lter (MSQKF) we use.
tic and foreign monetary policy SVSs better describes the US and EA dynamics.
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), …rms maximize their respective bene…ts (Section 2.2),
and central banks follow their respective ad-hoc Taylor-type rules and budget
constraints (Section 2.3). The model’s equilibrium (Section 2.4) and stochastic
structure (Section 2.5) are also presented in this section.
2.1 Households
For each country i2 fd; f g, we assume a representative in…nitely lived household
seeking to maximize
where "u
i;t1<1is the exogenous process corresponding to households’country-
speci…c intertemporal preferences,2and Ui;t is households’ country-speci…c in-
tertemporal utility function, such as
Ui;t =^
Ci;t hi^
i;t ^
Mi;t=Pi;t 11
L1+ 1
1 + 1
where ^
Ci;t is the detrended country-speci…c Dixit and Stiglitz (1977) aggregator of
households’purchases of a continuum of di¤erentiated goods produced by …rms,
Mi;t indicates the detrended country-speci…c 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 "u
i;t at time t, they do not know "u
i;t+1 at time t. Because utilities for t+ 1
should be multiplied by "u
i;t, current period utilities should be multiplied by "u
3The existence of a common stochastic trend (common level of technology progress) requires
stationary summands in the utility function. Consequently, the detrended consumption ( ^
Ci;t =
Ci;t=Zt) and real money ( ^
Mi;t=Pi;t ) summands of this utility function satisfy the stationarity
is the country-speci…c Dixit and Stiglitz (1977) aggregated price index and i;t is
the country-speci…c cost function described by Eq. 3. i;c is the country-speci…c
intertemporal substitution elasticity of habit-adjusted consumption (i.e., inverse of
the coe¢ cient of relative risk aversion), i;m is the country-speci…c partial interest
elasticity of money demand, and i;l is the country-speci…c Frisch elasticity of labor
supply. "m
i;t and "l
i;t are the country-speci…c exogenous processes corresponding to
real money holding (liquidity) preferences and the worked hours (disutility of labor)
of households, respectively.
The country-speci…c household’s cost function, i;t, is de…ned by
i;t =1
'i;d;j Bi;d;j;t
i;d;j 2
+'i;f;j ei;tBi;f;j;t
i;f;j 2
where 8k2 fd; fgand 8j2 fsr; lrg,'i;k ;j and i;k;j are scale parameters re-
lated to the bonds’rigidity,4and Bi;k;j;t represents the j-term k-bonds bought by
condition as in Adolfson et al. (2014). See, among others, Fagan et al. (2005), Schmitt-Grohé
and Uribe (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 di¤erent 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
household’s utility function, which produces smoothed restrictions. To simplify, we do not mod-
ulate such rigidity by restricting negative values. Although our approach is close to the portfolio
adjustment 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) as-
sume real costs in the budget constraint. As it is more likely that households feel disutility
from deviations in their …nancial position from the steady state, we do not assume that real
goods are required to compensate 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
speci…cations consider an exogenous foreign interest rate. As our model di¤erentiates domes-
tic 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 …rst-order approximation at a deterministic steady state, these types
of costs are equivalent. However, such a modi…cation is necessary in the case of a higher-order
approximation, while it does not a¤ect the outcome or propagation mechanism concerning the
original adjustment cost of Schmitt-Grohé and Uribe (2003).
households 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;t is the
country-speci…c 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).
The market consists of domestic and foreign one-period short- and long-term
bonds. Long-term bonds pay country-speci…c shares (Si) of their current nominal
value in each period.5In practice, Sidenes the bond duration (average time until
cash ‡ows are received).
Then, 8i2 fd; fg, the country-speci…c households’budget constraint can be
expressed as follows:
Pi;tCi;t +Mi;t+P
Bi;i;j;tQd;j;t +ei;t Bi;i;j;tQi;j;t
=Wi;tLi;t +Mi;t1+Di;t
+Bi;i;sr;t1+Bi;i;lr;t1((1 Si)Qi;i;lr;t+Si)
+ei;tBi;i;sr;t1+ei;tBi;i;lr;t1((1 Si)Qi;i;lr;t+Si);
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;t)denotes the price of rk;j;t, which is the country-
speci…c (k) nominal interest rate at maturity j.Wi;t is the country-speci…c wage
index and Di;t represents the dividends paid by …rms 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 in‡uence 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 in‡uence of money (and the money demand shock) on the
2.2 Firms
The continuum of identical …rms, in which each …rm produces a derentiated good
using identical technology, is represented by the following production function:
YF;i;t (j) = Ai;tLi;t (j);(5)
5A long-term bond with a nominal value of one domestic currency unit produces Sdunits
of the domestic currency in the …rst period, Sd(1 Sd)in the second period, Sd(1 Sd)2in
the third period, and so on. Because in‡ation-linked bonds are relatively rare and have lower
liquidity in the United States and EA, we price bonds in nominal terms.
where Ai;t =AiZtis the country-speci…c level of technology, assumed to be com-
mon to all …rms in country iand evolving exogenously over time, and Aiis a
country-speci…c total factor productivity scale parameter.
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 simpli…ca-
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 …rms face an identical isoelastic demand schedule and take the country-
speci…c 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 …rm faces a quadratic
cost of adjusting nominal prices measured in terms of the …nal good given by
Di;t+s'i;p Pi;t+s(j)
Ps;i;t = exp (vii+ (1 vi)i;t+s1)represents the country-speci…c weighted
average between country-speci…c steady-state in‡ation, i, and country-speci…c
previous in‡ation, i;t1, in period t, where viis the country-speci…c weight and
i;t = ln (Pi;t=Pi;t1).
Pi;t (j)is the price of goods jfrom …rms in country iin period t,Ri;t = exp (ri;t )
is the short-term nominal interest rate, and 'i;p 0is the degree of nominal price
rigidity in country i. The country-speci…c adjustment cost, which accounts for the
negative e¤ects of price changes on the customer–rm relationship in country i,
increases in magnitude with the size of the price change and with the overall scale
of the country-speci…c economic activity Yi;t.
In each period t, the …rm’s budget constraint requires
Di;t +Wi;tLi;t =Pi;t (j)Yi;t (j);(7)
6In this simple case, we also do not consider money in the production function. Several
examples 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
exogenous growth in a model without capital. Further research should analyze the bene…ts of
capital as a factor of production to explain long-term growth.
where YF;i;t (j)represents …rms that manufacture goods jin country iin period
t. Firms cannot make any investment (Eq. 7) and distribute all their bene…ts
through dividends (Eq. 6).
The …nal consumption good is a constant elasticity of substitution compos-
ite of domestically produced and imported aggregates of intermediate goods that
produces demand for …rm output, such as
YF;i;t+s(j) = !iYi;t+sPi;t+s
+ (1 !i)Yi;t ei;t+sPi;t
where the exogenous process "p
i;t+srepresents the country-speci…c price markup
shock (elasticity of demand in country i), and the parameter !ide…nes a country-
speci…c preference for local demand.
The aggregate country-speci…c price level also follows the usual constant elas-
ticity of substitution aggregation, such as
i;t =!iPi;t (j)1"p
i;t + (1 !i) (ei;tPi;t (j))1"p
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 ="r
i;t1^i;(1i;r )
i;t ^yi;y(1i;r )
i;t ^ei;e(1i;r )
i;t ;(10)
where "r
i;t captures the country-speci…c monetary policy shocks, ^i;t is the country-
speci…c in‡ation gap expressed as the ratio between country-speci…c CPI and its
corresponding steady state, ^yi;t is the country-speci…c output gap expressed as the
ratio between country-speci…c output (normalized by technological progress) and
its corresponding steady state, and ^ei;t is the country-speci…c real exchange rate
gap expressed as the ratio between the real exchange rate of country iand its
corresponding steady state.
The parameter i;r captures interest rate-decision smoothing, and i;,i;y ,
and i;e capture the weight placed by the monetary authority of country ion the
in‡ation 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
=Bi;g;t1+Mi;t Mi;t1;(11)
where Bi;g;t represents the country-speci…c nominal bonds bought by the local
central bank in period t.
In our model, we assume that central banks can buy only short-term bonds, as
was the case in the United States and EA before the GFC.
2.4 Equilibrium
In the equilibrium, country-speci…c demand consists merely of consumption, such
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)
Bi;i;lr;t +Bi;i;lr;t = 0:(14)
The country-speci…c demand presented in Eq. 12, Yi;t , is di¤erent from the
country-speci…c 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 simpli…cation
(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 dened as 8i2 fd; fgand 8j2 fu; m; l; p; r; yg,
i;t =i;j j
i;t1+1i;j i;j +i;j;t ;(15)
where the parameter i;j de…nes the country-speci…c steady state of exogenous
process j,i;j the country-speci…c autocorrelation level, and i;j;t the country (i)
shock-speci…c (j) white noise (zero-mean normal distribution).
The demand elasticity exogenous process is de…ned by p
i;t ="p
i;t, the in-
tertemporal preference exogenous process by u
i;t = ln "u
i;t1, technological
progress by y
t= ln (Zt=Zt1), and other exogenous processes by 8i2 fd; f gand
8j2 fm; l; rg,j
i;t = ln "j
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).
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 de‡ator (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 …ve country-speci…c 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 cap-
ture long-term bond demand/supply e¤ects through their interest rates in both
countries. 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 e¤ects.
3.2 Estimation
Our switching (two-regime) model is estimated in three ways with maximum likeli-
hood 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: "r
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).
Monetary policy and productivity shocks are the main driving forces of business
cycles. Additional SVSs are feasible in theory; however, in practice, they require
signi…cant additional computing resources and may not change the results or make
the model more realistic.
The model solution approximation is computed with the e¢ cient 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 constitute
the nonlinearities of our models. We use the …rst four quarters as a presample
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 individual
(observable) dynamics.
The share of steady-state in‡ation indexation (vi) ders across regions as well
as in the di¤erent versions of the model. The coe¢ cient for the United States is
close to that of Smets and Wouters (2007). The version without switching has a
larger share of steady-state ination 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 in‡ation indexation for the EA is much smaller. The 3SVS version produces
the closest values of the corresponding parameters. Thus, volatility switching
might in‡uence in‡ation persistence, of which the share of past in‡ation indexation
(1vi) 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 …ltered values of regime 1 probabilities and three selected
exogenous processes ("p
f;t, and Zt). This …gure shows P rob (rt= 1) conditional
on the data probability, where P rob (rt= 1) corresponds to the probability of being
in regime 1 in period t.
Only moderate di¤erences exist between the …ltered values of the exogenous
processes. In addition, the derences in state probabilities are linked to the state
of the 1SVS model, whereas the state probabilities of the 3SVS model are more
reliable. The latter correspond to the actual main crises that occurred during
the sample period. The di¤erence between the …ltered values of the exogenous
processes is generally smaller before the GFC, whereas it is larger a few years after
the beginning of the GFC. Economic driving forces are generally una¤ected 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 di¤erent from pre…ltering, as it does not use the likelihood values
corresponding to the …rst four quarters for all the variables. Jackknife bootstrapping suggests
discarding four observations randomly and combining the variable and period.
1SVS 3SVS Without switching
Figure 1: Regime probability, technology (Zt), US ("p
d;t), and EA ("p
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 denition, such as presented in
Dynare (Adjemian et al., 2011), de…nes the IRF as the expected di¤erence 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; ) = E[xtj1N(();  ())] E[xtj1N(0;  (1))] ;(16)
where xtis the value of the variable of interest for which the IRF is computed in
period t,1is the shock of interest that deviates in period 1,(:)is the standard
error operator, E[:]is the expectation operator, and Nis the normal law.
We generalize this de…nition in the nonlinear case by making the magnitude and
sign of the shock more important. Such a generalization requires the introduction
of the parameter sin Eq. 16 to determine the number of standard deviations in
the shock, such as
IRFt;s (x; ) = E[xtj1N((1)s;  (1))] E[xtj1N(0;  (1))]
In addition, we compute the IRFs conditional on the state variables’vector Xt
to show the di¤erences between the IRFs at di¤erent states of the world, such as
IRFt;s (xt; jX0) = E[xtj1N((1)s;  (1)) ; X0]E[xtj1N(0;  (1)) ; X0]
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 e¤ect disappears without strong per-
sistence (around 10 periods). However, the e¤ect on the model’s variables is much
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.
Figure 2: Unconditional IRFs to an SVS to regime 2 (1SVS).
more persistent and di¤ers by region. Following an SVS, in‡ation increases in the
two regions during the …rst periods, involving an increase in the US short-term
nominal interest rate, while the EAs short-term nominal interest rate remains sta-
ble. The picture changes drastically in later periods when the long- and short-term
interest rates in the United States and EA’s both decrease with the in‡ation 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 di¤erence can be
explained by the di¤erent 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 di¤erent short-term in-
terest rates, could explain this phenomenon. The United States has a stronger
response to ination and a smaller smoothing coe¢ cient than the EA. Conse-
quently, the US short-term nominal interest rate decreases with in‡ation and in-
creases later, while that for the EA increases slightly. This di¤erence in monetary
policy produces ‡uctuations in the exchange rate and ratio of domestic demand to
Fig. 3 provides a more robust picture than Fig. 2.
Figure 3: Unconditional IRFs to an SVS to regime 2 (3SVS).
Indeed, the 1SVS model suggests only a few di¤erences between regimes (the
standard deviations are close), implying a small e¤ect on the economy of switching,
which explains the low values obtained in Fig. 2. However, the 3SVS model
suggests much larger di¤erences and a substantial impact of switching shocks on
the economy.
Fig. 3 highlights that SVSs a¤ect US in‡ation and nominal interest rates
in both the short and long terms, while the impact on the EA economy is less
signi…cant. Such SVSs durably in‡uence US long-term interest rates, whereas this
is not the case for the EAs 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
regions 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 in‡ation. As the economy is open, domestic
changes are substantial, and foreign households buy more domestic bonds. For-
eign households work more and sell more goods to the domestic country. Moreover,
foreign investment in the domestic market makes foreign currency cheaper. Thus,
foreign households increase investments and hold more money. As this e¤ect is
powerful, foreign in‡ation decreases, leading to lower foreign interest rates.
The average e¤ect of unconditional SVSs might di¤er from that of conditional
IRFs. For example, Fig. 4 compares unconditional IRFs with conditional IRFs
for 2009Q1 and 2003Q4. Here, we use the …ltered values of the variable vector for
the corresponding dates as the condition (the initial point for a draw).
These IRFs are di¤erent in several aspects. Indeed, the regime probability IRF
di¤ers in the con…gurations 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 signicant derences between the crises
as well as between the United States and EA (Fig. 5).
These di¤erences are more reliable than in the 1SVS model. For instance, the
EA short-run nominal interest rate was not similarly a¤ected by the switching
during good and bad times (e.g., the subprime crises) and their corresponding
SVSs. Furthermore, signi…cant di¤erences 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 in‡ation and interest rates decrease slightly more than the unconditional
points and the EA.
The in‡uence of the SVS is signi…cant. 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,
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Unconditional 2009Q1 2003Q4
Figure 4: Conditional and unconditional IRFs to an SVS to regime 2 (1SVS).
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 signi…cantly di¤erent, mainly due to the switch of multiple
variances. The response of the 3SVS model is less monotonic and the magnitudes
of the IRFs are di¤erent for most of the economic variables.
Fig. 6 shows that the switch of multiple variances signi…cantly a¤ects the
exchange rate as well as US long-run nominal interest rates, while short- and
long-term nominal interest rates in the EA are less a¤ected. However, the EA’s
demand-to-GDP ratio is more a¤ected 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 in‡ation in the EA.
Fig. 7 shows that the 1SVS model in‡uences the …nancial variables, uncondi-
Figure 5: Conditional and unconditional IRFs to an SVS to regime 2 (3SVS).
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 di¤erent 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
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
impact of an SVS on the …nancial variables. However, this was not the case; indeed,
the …nancial transmission channels during these two crises were fundamentally
Fig. 8 shows a more coherent picture with signi…cant and reliable di¤erences
Figure 6: Unconditional IRFs to an SVS (to regime 2) for the 1SVS and 3SVS
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
position. In the EA, the picture is di¤erent: the ECB slightly increases its bond
position, and both European and US households decrease their EA long-term bond
Then, because US and EA households are selling their US bonds, the con-
struction of our model suggests that the Federal Reserve must buy them after the
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10 20 30 40
Unconditional 2009Q1 2003Q4
Figure 7: Conditional and unconditional IRFs to an SVS (to regime 2) for the
nancial variables (1SVS)
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 result is
a direct consequence of the increase in the short-term EA bond position and con-
sumption. 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 di¤erences 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) di¤erentiates
between these two dates, which are economically (and …nancially) substantially
di¤erent. Consequently, the 3SVS model could match the stylized …nancial facts
better than the 1SVS model (and a fortiori compared with the baseline model
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10 20 30 40
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10 20 30 40
Unconditional 2009Q1 2003Q4
Figure 8: Conditional and unconditional IRFs to an SVS (to regime 2) for the
nancial variables (3SVS)
without 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 a¤ect economic dynamics.
Fig. 8 demonstrates the increasing real exchange rate di¤erence of about 0.5%
over 10 years. Such derences between conditional and unconditional IRFs show
how nonlinearities are signi…cant.9In the long run, an SVS leads to a change
of about 2% in the real exchange rate. The SVS e¤ect is very persistent with a
substantial consequence, in that the derence between conditional and uncondi-
tional IRFs for the real exchange rate exceeds 0.2% over more than eight years
9See Section 4.3 for an analysis of nonlinearities.
(Fig. 8). The importance is also related to the duration of e¤ect. For instance, if
EA exports and imports represent about 53% of GDP,10 the cumulative e¤ect of
a 0.2% change in exchange rates over eight years would lead to a ‡ow of money
representing 0.85% of yearly GDP (direct in‡uence11).
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 ination in Fig. 9. Hence,
the real interest rate increases, leading to a lower money position and a higher
bonds position. The government budget means that it creates additional income
for households. This means higher consumption, which increases imports. Im-
portation growth then leads to a cheaper national currency, and thereby ination
growth and domestic production growth.
However, the responses of a US monetary policy shock di¤er depending on the
model (Fig. 10), especially for the demand-to-GDP ratio, long-term interest rates,
and GDP growth in the …rst quarters. US in‡ation responses are more pronounced
in the 3SVS model than in the model without SVSs.
In addition, EA and US growth rates are signi…cantly di¤erent in the …rst
quarters, showing that the model without switching allows more variability to US
and EA growth in the …rst periods, with derent signs at some points in time.
A foreign monetary policy shock leads to lower in‡ation in Fig. 10. The central
bank places signi…cant weight on in‡ation. Lower in‡ation expectations lead to
lower in‡ation and interest rates, which then motivates households to increase
money and decrease bonds. This produces an additional cash ‡ow 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 di¤erent 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
pronounced. 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.
10 The 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.
11 The exchange rate in‡uenced both export and import payments leading to a total e¤ect
would be 0:002 80:53 = 0:85%.
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10 20 30 40
1SVS 3SVS Without switching
Figure 9: Unconditional IRFs to a positive EA monetary policy shock (one stan-
dard deviation).
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.
4.3 Nonlinearities
The previous IRF …gures considered only a one standard deviation positive shock.
However, in a nonlinear world, responses are also nonlinear. How should these
nonlinearities be quanti…ed? Fig. 11 to Fig. 14 present the unconditional IRFs
after monetary policy shocks of di¤erent magnitudes to assess the importance of
these nonlinearities.
Fig. 11 presents the IRFs after an EA monetary policy shock according to the
1SVS model.
Figure 10: Unconditional IRFs to a positive US monetary policy shock (one stan-
dard deviation).
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
understandable mathematically (power 2), avoiding a symmetric response, which
is standard in DSGE models’IRFs linearized at the …rst 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 e¢ ciency of monetary policy shocks, which is an important result
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
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10 20 30 40
IRF +1 std. shock IRF -3 std. shock IRF +3 std. shock
Figure 11: Unconditional IRFs to EA monetary policy shocks of di¤erent magni-
tudes (1SVS). std. stands for standard deviation.
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 in‡uence EA in‡ation uncertainty in an interesting
way. While the negative monetary policy shock (-3 std) has an important impact
on EA in‡ation in the …rst 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 di¤erent nonlinearities than responses to an
EA monetary policy shock (Fig. 11). EA in‡ation and the demand-to-GDP ratios
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10 20 30 40
IRF +1 std. shock IRF -3 std. shock IRF +3 std. shock
Figure 12: Unconditional IRFs to EA monetary policy shocks of di¤erent magni-
tudes (3SVS). std. stands for standard deviation.
of both the United States and the EA behave almost nonlinearly following a US
monetary policy shock (at least in the …rst periods). In these cases, -3 std and
+3 std are asymmetric, whereas we …nd 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 …nding shows that nonlinearities
help explain why strong monetary policy reactions do not have the same conse-
quences 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.
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10 10-4
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15 10-4
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IRF +1 std. shock IRF -3 std. shock IRF +3 std. shock
Figure 13: Unconditional IRFs to US monetary policy shocks of di¤erent magni-
tudes (1SVS). std. stands for standard deviation.
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).
EA in‡ation displays the same phenomenon as presented previously (Fig. 12),
again denoting strong di¤erences between the linear and nonlinear models.
Fig. 14 con…rms that the persistent di¤erence in the demand-to-GDP ratio
depends on the shock’s magnitude and sign. This di¤erence is larger than 0.05%
over ten years, meaning the cumulative change of trade equals 0.5% of GDP. The
short-term di¤erence in EA GDP growth rates exceeds 0.5% due to the shift in
growth peak timing. More signicant is the response of US GDP growth of about
1% after a US monetary policy shock of di¤erent magnitude.
Moreover, the role of SVSs is signi…cant, especially concerning demand-to-
Figure 14: Unconditional IRFs to US monetary policy shocks of di¤erent magni-
tudes (3SVS). std. stands for standard deviation.
GDP ratios and exchange rates. Overall, the nonlinearities and SVSs on monetary
policy shocks a¤ect 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-
tions and policy decisions.
5 Interpretation
Section 2 presents an original model featuring households, …rms, and the central
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 e¤ect 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 EAs
long-term interest rates comove strongly with US long-term rates rather than with
short-term rates (Chin et al., 2015). This result is con…rmed 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 …nd that US disturbances inuence EA economies
markedly (Fig. 13). These results are con…rmed by the variance decompositions
of the variables with respect to the shocks (Appendix D) and distance correlations
between the variables (online appendix).
In addition, we …nd that US money shocks a¤ect the EA real variables in the
long run as well as the …nancial 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 e¤ect
(Eq. 3) highlighted by Andrés et al. (2009).
An interesting result on in‡ation’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 in‡ation dynamics in the United States,
illustrating how EA and US economies behave di¤erently 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 e¤ect of the price markup shock
on the in‡ation 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
important shocks for explaining some variable dynamics in the long run (Appendix
D), such as the part of in‡ation 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 scenarios developed
by policymakers, such as on foreign and domestic bonds or private consumption.
Hence, our nonlinear open-economy DSGE model with several SVSs allows us
to enrich the dynamics of interest rate markets for di¤erent maturities (Section
4.1). Fig. 2 shows that the US response to in‡ation is stronger than that of the
EA. How can one conciliate this with the stabilization objectives of the Federal
Reserve and ECB? The o¢ cial objective of the Federal Reserve is to react to both
in‡ation and output growth or unemployment, while the ECB’s is to mainly react
to in‡ation. However, these objectives di¤er 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 in‡ation. 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
in‡ation. As an active central bank, this stronger response to economic changes
leads to faster stabilizing e¤ects with more signi…cant interest rate ‡uctuations
compared with the ECB. The ECB’s smaller responses smooth interest rates during
a more extended (stabilization) period. Both monetary policies correspond to the
cial objectives but have signi…cant di¤erences in the preferences across the
components of these objectives. Other explanations relate the weaker response of
the ECB to in‡ation dynamics compared to those of the Federal Reserve. Tensions
within the ECB Governing Council, a change in the post-GFC in‡ation target and
objectives, quantitative easing, and the zero lower bound could also explain this
lower in‡ation coe¢ cient compared with the Federal Reserve.
Including several SVSs could, at least during crises, more accurately explain the
changes in US and EA in‡ation as well as in US and EA interest rates at di¤erent
maturities. The possibility of switching in di¤erent 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 speci…c switching volatility that can change the regime of the
overall economy for a speci…c sector. Fig. 6 clearly shows that such modeling is
more appropriate for capturing changes in ination 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‡ation 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 …nancial variables are not well explained. Section 4.2 discusses the
transmission channels.
Lastly, our models can shed light on nonlinear IRFs, highlighting the signi…-
cant nonlinear behaviors of market-related variables such as exchange and interest
rates. Such dynamics are absent from most policymakers’models for such reasons
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.
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-speci…c SVS impacts between the EA and the United States were trans-
mitted to real and …nancial 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 a¤ect the US and EA economies and involve i) money transfers between
economies and ii) interest rate maturity trade-o¤s that could produce structural
changes in the economy. Hence, SVSs a¤ect US and EA consumption in opposite
Further, price markup and money shocks behave di¤erently to in standard
linear models. Owing to direct e¤ects (Andrés et al., 2009), the roles of both
domestic and foreign real money holdings are signi…cant in the long run as well as
the short run, especially for bond variables and rate-related variables.
Furthermore, the di¤erence between the average response of SVSs and response
on speci…c 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 signi…cantly in‡uenced by such shocks.
The main policy implication relates to the way monetary authorities model the
economy, especially in an open-economy world with interlinked …nancial markets.
Our models showed that it is important for policymakers to consider nonlinear
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 …nancial markets and the economy. SVSs could thus be a
promising feature included in the next generation of macroeconomic models.
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A Markov switching quadratic Kalman …lter
This appendix presents the fast-deterministic …lter used for the estimation of our
nonlinear MSDSGE model. The collapsing rule of the sigma-point Kalman …lters
developed by Binning and Maih (2015) is unusual. This …lter 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
xes this property by correcting the formulas for variances.13 The MSQKF is a
Gaussian-assumed …lter that uses collapsing before forecasting as in Binning and
Maih (2015).
Particle …lter 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, …xed random draws are required for optimization
algorithms with particle …lters. However, this mitigates the main advantage of
particle …lters. Markov chain Monte Carlo ine¢ ciency increases signi…cantly: the
required number of draws should be 10 (from 5 to 400 depending on the number
of particles) times higher for the same accuracy of Markov chain Monte Carlo
methods (Pitt et al., 2012). An additional disadvantage of particle …lters is their
computational costs. They require a large number of particles to be comparable
12 Let 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.
13 The description, properties, and comparisons of the MSQKF are detailed in Ivashchenko
(2014, 2016).
with deterministic …lters and are about 100 times slower than deterministic non-
linear …lters (Andreasen, 2013; Ivashchenko, 2014; Kollmann, 2015). For all these
reasons, we do not use particle …lters.
The purpose of a …lter in DSGE models is to compute the model variable
vector, Xt, density conditional on the vectors of the observed variables Y1; :::; Yt
and 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 …lters loop the following steps:
1. Computation of the initial density of Xt;
2. Computation of the density of Ytas a function of the density of Xt(see
Appendix 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+1 as 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 …ltration 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;t is
the vector of the state variables (a subset of the model variable vector Xt), and "t
and utare the vectors of independent shocks (model innovations and measurement
errors) that have a zero-mean normal distribution. is 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,
Zt=Xstate;t  "t;(21)
Xt+1 =A0;rt+1 +A1;rt+1 Zt+A2;rt+1 (ZtZt);(22)
where is the Kronecker product.
The di¤erence from the usual DSGE model second-order approximation is the
existence of regime dependence. Each …ltering step is described below. The nonlin-
ear …lters (including the suggested ones) use some approximations. Computations
within the …lters 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)
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] =
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
conditional on the regime and observation density in the same way as the Kalman
lter in Eq. 26 to Eq. 28:
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)
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
A.4 Density of Xt+1 as a function of the density of XtjYt
The initial information for this step is the density for the vector of the model
variables Xt(normal mixture).
The …rst step is the computation of the expectation (Es;1) and variance (Vs;1)
of vector Xtconditional on the future state, such as
Es;1=E(Xtjrt+1 =s) =
p(rt=k)p(rt+1 =sjrt=k)
p(rt+1 =s)Ek(Xt);(30)
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;1and 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+1 Ert+1 Zt;rt+1 ;(32)
Xt+1;rt+1 =A0;rt+1 +A1;rt+1 Zt;rt+1 +A2;rt+1 Zt;rt+1 Zt;rt+1
=B0;rt+1 +B1;rt+1 Z0;t;rt+1 +B2;rt+1 Z0;t;rt+1 Z0;t;rt+1 (33)
EXt+1;rt+1 =B0;rt+1 +B2;rt+1 vec VZt;rt+1 =B0;rt+1 +B2;rt+1 vec Vrt+1 ;
vec VXt+1;rt+1  =B1;rt+1 B1;rt+1 vec Vrt+1 (35)
+B2;rt+1 B2;rt+1 vec Vrt+1 vec Vrt+1 +
+vec vec Vrt+1 Vrt+1 ;
where vec f:gis the vectorization operator.
Eq. 32 to Eq. 35 are similar to the equations developed in Ivashchenko (2014).
The di¤erence 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+1 is 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.
Variable Description Equations
Bonds bought by households i
in currency jwith maturity k3, 4, 13, 14
Bonds bought by the central bank
or government in country i11, 13
Ci;t Consumption of households of country i2, 3, 4, 12
Di;t Dividends of …rms from country i6, 7
Exchange rate (number of units of the domestic
currency per unit of the foreign currency) 3, 4, 8, 9, 10
Li;t Labor in country i2, 4, 5, 7
Mi;t Money stock in country i2, 4, 11
Pi;t Aggregate price level in country i2, 3, 4, 6, 7, 8,9,10
Pi;t (j)Price of goods of …rms jin country i6, 7, 8, 9
Ri;k;t Interest rate in currency iwith maturity k4, 6, 10, 11
Wi;t Wage in country i4, 7
Yi;t Demand in country i6, 8, 10, 12
YF;i;t (j)Production of …rms jin country i5, 7, 8
i;t Exogenous 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 illustrate
the importance of switching volatility. We compute the LPS based on Gaussian
density, which is suggested by the MSDSGE model.14
EA GDP deator 0.408% 0.366% 0.384% 3.66 3.70 3.67
US GDP deator 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.
We also compute these statistics for the 3SVS model in the cases that the
volatility 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 di¤erence in the
sum of individual 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
generally in line with those in the DSGE literature. The persistence of monetary
policy shocks is lower than that of other shocks, as explained by Smets and Wouters
(2007). The coe¢ cient 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 derence 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 signi…cant. For instance, the latter
model could still be used to perform forecasting in changing environments (Benchi-
mol and Fourçans, 2017, 2019). Second, whatever the log-likelihood, the model is
designed to capture only speci…c characteristics of the data. It is an open question
as to whether log-likelihood is an adequate measure to evaluate how well the model
14 When based on Gaussian mixture density, the LPS should equal the log-likelihood divided
by the number of periods (for the multivariate measure).
15 See Fig. 1 for the estimated probabilities.
accounts for particular aspects of the data.
Nevertheless, we report the log-likelihood values and corresponding likelihood
ratio tests hereafter. The log-likelihood values of the 0SVS, 1SVS, and 3SVS mod-
els 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 ‡exible model explains
signi…cantly more of the data. Our estimation of the covariance matrix allows
us to construct a Laplace approximation of the marginal likelihood (maximum
likelihood estimation is equivalent to a Bayesian one with ‡at 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 di¤er-
ent solution. The sharp likelihood of the nonlinear approximation transforms this
small di¤erence into a signi…cant di¤erence in the likelihood. Thus, even small
details of the solution algorithm can be crucial in a nonlinear world.
Priors Posteriors Priors Posteriors
LB UB Mean Std. LB UB Mean Std.
d;u -0.01 0.00 -0.006 0.000 Af-20.0 20.0 0.000 0.002
d;m -20.0 20.0 -6.493 0.263 'd;d;sr 0.00 1000 0.195 0.004
d;r 0.00 0.01 0.000 0.000 'd;f;sr 0.00 1000 0.083 0.000
d;p 1.00 20.0 3.281 0.020 'd;d;lr 0.00 1000 16.26 0.413
f;u -0.01 0.00 -0.008 0.000 'd;f;lr 0.00 1000 0.236 0.003
f;m -20.0 20.0 -7.558 0.001 'f;d;sr 0.00 1000 0.001 0.000
f;r 0.00 0.01 0.001 0.000 'f;f;sr 0.00 1000 0.000 0.000
f;p 1.00 20.0 11.28 0.008 'f;d;lr 0.00 1000 0.012 0.001
y0.00 0.01 0.001 0.000 'f;f;lr 0.00 1000 0.000 0.000
d;u -1.00 1.00 0.978 0.004 sd0.00 1.00 0.655 0.006
d;l -1.00 1.00 0.975 0.001 sf0.00 1.00 0.063 0.001
d;m -1.00 1.00 0.891 0.007 'd;P 0.00 1000 591.4 21.637
d;r -1.00 1.00 0.180 0.021 'f;P 0.00 1000 227.8 4.087
d;p -1.00 1.00 0.948 0.001 vd0.00 1.00 0.473 0.048
f;u -1.00 1.00 0.462 0.030 vf0.00 1.00 0.787 0.008
f;l -1.00 1.00 0.914 0.002 hd;c 0.00 0.90 0.669 0.000
f;m -1.00 1.00 0.950 0.003 hf;c 0.00 0.90 0.743 0.000
f;r -1.00 1.00 0.756 0.009 d;r 0.00 0.99 0.974 0.000
f;p -1.00 1.00 0.953 0.002 d;p 1.00 5.00 1.004 0.025
y-1.00 1.00 0.960 0.001 d;y -5.00 5.00 0.065 0.005
d;u0.00 10.0 0.001 0.000 d;e -20.0 20.0 0.005 0.000
d;l0.00 10.0 0.152 0.007 f;r 0.00 0.99 0.796 0.000
d;m0.00 10.0 0.124 0.004 f;p 1.00 5.00 2.906 0.001
d;r0.00 10.0 0.001 0.000 f;y -5.00 5.00 -0.097 0.003
d;p0.00 10.0 1.342 0.030 f;e -20.0 20.0 -0.029 0.000
f;u0.00 10.0 0.007 0.001 bd;d;sr -20.0 20.0 0.236 0.006
f;l 0.00 10.0 0.048 0.001 bd;f;sr -20.0 20.0 1.859 0.004
f;m0.00 10.0 0.106 0.002 bd;d;lr -20.0 20.0 -0.006 0.000
f;r 0.00 10.0 0.002 0.000 bd;f;lr -20.0 20.0 -0.319 0.006
f;p0.00 10.0 0.635 0.021 cd-20.0 20.0 -0.421 0.000
y0.00 10.0 0.001 0.000 cf-20.0 20.0 -1.593 0.000
1=d;c -20.0 20.0 0.727 0.001 e-20.0 20.0 1.531 0.001
1=d;l -20.0 20.0 9.821 0.645 pd0.00 0.01 0.000 0.000
1=d;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=f;c -20.0 20.0 -0.688 0.000 pf(j)-20.0 20.0 0.000 0.000
1=f;l -20.0 20.0 -1.758 0.044 rd;lr 0.00 0.03 0.027 0.000
1=f;m -20.0 20.0 0.794 0.000