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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(</