HIAS-E-38

Trend Inflation and Exchange Rate Dynamics:
A New Keynesian Approach
Takashi Kano
Hitotsubashi University

December 2016 (revised April, 2023)

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
tel:+81 42 580 8668 http://hias.hit-u.ac.jp/
HIAS discussion papers can be downloaded without charge from:https://hdl.handle.net/10086/27202
https://ideas.repec.org/s/hit/hiasdp.html

Trend Inflation and Exchange Rate Dynamics:
A New Keynesian Approach
Takashi Kano

Graduate School of Economics
Hitotsubashi University
Naka 2-1, Kunitachi, Tokyo
186-8601, JAPAN
Email: tkano@econ.hit-u.ac.jp
First Draft: February 15, 2016
Current Draft: April 26, 2023
Abstract
This study examines the exchange rate implications of trend inflation within a two-country New Keynesian(NK) model. An NK Phillips curve generalized by trend inflation makes the inflation differential smoother,more persistent, and less sensitive to the real exchange rate. A Bayesian analysis with post-Bretton Woods
data for Canada and the U.S. shows that the model’s equilibrium, which relies on Taylor rules with apersistent trend inflation shock and strong policy inertia, mimics empirical regularities in exchange rates
that are difficult to reconcile within a standard NK model. Trend inflation helps explain the empirical
puzzles of the exchange rate dynamics.
Key Words: Real and Nominal Exchange Rates; Trend Inflation; New Keynesian Model;
Bayesian analysis
JEL Classification Numbers: E31, E52, F31, F41

I would like to thank anonymous referees, Martin Berka, Hafedh Bouakez, Pablo Guerron-Quintana, Ippei Fujiwara, Kazuko
Kano, Robert Kollmann, Takushi Kurozumi, Anella Munro, Jim Nason, Moto Shintani, Taka Tsuruga, Youichi Ueno, Martin
Uribe, and the conference and seminar participants at the 2nd Hitotsubashi Summer Institute (HSI 2016), 10th International
Conference on Computational and Financial Econometrics (CFE 2016), 3rd International Workshop on Financial Markets andNonlinear Dynamics, 23rd International Conference of the Society for Computational Economics (CEF 2017), 2017 Workshopof the Australasian Macroeconomics Society, 2018 Canadian Economic Association Meetings, 2018 European Meetings ofthe Econometrics Society (ESEM 2018), Institute for Monetary and Economic Studies at the Bank of Japan, Kyoto University,
Reserve Bank of New Zealand, University of Tokyo, and Tsinghua School of Economics and Management for their helpfulcomments and insightful discussions. I am grateful for the grant-in-aid for scientific research from the Japan Society for the
Promotion of Science (numbers 17H02542 and 17H00985) and financial support from the Hitotsubashi Institute for Advanced
Study and the Mitsubishi Economic Research Institute. I am solely responsible for the errors and misinterpretations of this paper.
†

1. Introduction
The empirical reality of exchange rates still confounds international finance researchers. Real exchange rates are persistent and volatile. Near random-walk nominal exchange rates have no significant dependence on current or past information on macroeconomic fundamentals. The real and nominal exchange
rates closely move together. These stylized facts jointly refute the theoretical challenges of canonical openeconomy models with price stickiness. The elucidation of the complicated life of exchange rates in theliterature on New Keynesian (NK) open-economy models is still far from satisfactory.1
In this study, I examine the crucial role of trend inflation in exchange rate dynamics.2 My investigation is based on an otherwise standard two-country NK model, as in Benigno (2004) and Engel (2014,2019), equipped with the Calvo-type time-dependent pricing behaviors of monopolistic competitive final
good firms conducting pricing-to-market strategies in terms of local currencies, Taylor rule-type monetary
policies with high interest-rate smoothing, and incomplete international financial markets with debt-elastic
risk premiums. I then extend the standard NK model by allowing for trend inflation that fluctuates stochastically around a positive long-run mean in the two countries. To the best of my knowledge, this study is thefirst attempt to embed trend inflation in a two-country NK model explicitly and extract its implications forexchange rate dynamics.3 Such a natural modification of the canonical two-country NK model regarding
the low-frequency property of inflation significantly impacts exchange rate dynamics.
Following Ireland (2007) and Cogley et al. (2010), I specify the source of the fluctuations in trend
inflation in this model by using a time-varying inflation target set by the central banks of each country
within the Taylor rule.4 As in Cogley et al. (2010), the inflation target obeys a persistent stochastic processwith a positive long-run mean. On the one hand, a positive long-run mean of trend inflation is undebatable
1

Burstein and Gopinath (2014) and Engel (2014) provide excellent surveys of the recent empirical findings on international
real and nominal relative prices.
2
Ascari and Sbordone (2014) offer an excellent summary of recent research on trend inflation, focusing on the U.S. inflation
dynamics.
3
Zhao (2022) also examines the implications of trend inflation in an open economy, focusing mainly on the welfare implication of positive long-run inflation in a calibrated two-country NK model without stochastic variations in trend inflation. Zhao
(2022), however, does not investigate the effects of a time-varying stochastic inflation target on exchange rate dynamics.
4
Haque (2022) also investigates the role of a time-varying inflation target in the Taylor rule for the postwar U.S. inflation
dynamics with the estimated closed-economy NK model with positive long-run trend inflation.

1

for post-war data in the world economy. On the other hand, there are three rationales behind the stochastic variation in the inflation target. First, as Cogley et al. (2010) discuss, a central bank only imperfectly
knows the actual economic structure, and its learning process generates fluctuations in the inflation target.
Second, private economic agents may suffer from imperfect information about inflation targets. Third, apersistent shock to the inflation target is observationally equivalent to a persistent monetary policy disturbance in the Taylor rule with high interest-rate smoothing, the importance of which the literature on
monetary policy frequently emphasizes. Coibion and Gorodnichenko (2012) provide empirical evidence
that the high persistence observed in the actual data of the short-run nominal interest rate in the U.S. stems
largely from policy inertia, but not from the exogenous persistence of the monetary policy disturbance in
the Taylor rule. Importantly, their empirical result suggests that the misspecification problem could leadto a high estimate of the serial correlation of monetary policy disturbances unless the Taylor rule correctly
specifies the time-varying inflation target. Therefore, including a persistent shock to the inflation target in
the model might resolve the Taylor rule’s misspecification problem and better identify the critical role ofmonetary policy in exchange rate dynamics.
Allowing for persistent trend inflation with a positive long-run mean within a two-country NK model
produces fundamentally richer equilibrium dynamics of real and nominal exchange rates. A positive longrun mean of trend inflation substantially alters the shape of the log-linearized New Keynesian Phillips curve(NKPC) in each country, as Cogley and Sbordone (2008) and Ascari and Sbordone (2014) demonstrate. In
the NKPC generalized by a positive long-run trend inflation (hereafter, generalized NKPC, or GNKPC), thecurrent inflation rate follows a higher-order expectational difference equation, which results in a persistentand smooth movement of each country’s inflation rate with persistent trend inflation variations. High policy
inertia in the Taylor rule yields near-non-stationary real interest-rate dynamics. With the real uncovered
interest parity (RUIP) condition, Engel and West’s (2005) proposition generates near-random-walk realexchange rate movements with high volatility. Hence, the proposed model reconciles with the persistence
and volatility puzzle of the real exchange rate (Chari et al. 2002).
In this study, I draw Bayesian posterior inferences of the proposed log-linearized asymmetric twocountry NK model augmented with trend inflation based on post-Bretton Woods (BW) Canada-U.S. data.
2

The data strictly require positive long-run trend inflation for the cross-equation restrictions of the trend
inflation (TI) model: the overall data fit of the TI model outperforms that of a canonical model restrictedby zero long-run trend inflation in terms of marginal likelihoods.
The estimated TI model generates a highly persistent and volatile real exchange rate, which I approximate as a near random-walk process. As recent studies on real exchange rate dynamics emphasize,
including those by Cheung and Lai (2000), Steinsson (2008), Iversen and Söderström (2014), Burstein and
Gopinath (2014), and Carvalho et al. (2018), the impulse response function (IRF) of the real exchange rate
to a reduced-form disturbance is hump-shaped with a peak around a year, and the corresponding half-lifeof the real exchange rate is longer than four years. Decomposing the IRF and forecast error variance into
structural shocks reveals that a shock to the Canadian trend inflation, even with a smaller standard deviation
than those of other structural shocks, dominates the amplification and propagation mechanism of the realexchange rate by generating a persistent hump-shaped impulse response. Thus, the estimated TI model
successfully reconciles with a significantly delayed overshooting of the exchange rate to a monetary easing
shock identified as a surprise change in the long-run inflation target, as Eichenbaum and Evans (1995) and
Bouakez and Normandin (2010) observe.
The hump-shaped real exchange rate response to a trend inflation shock stems from the theoreticaloutcome that the IRF of the real interest rate differential between the two countries flips its sign in several periods after the impact, as Steinsson (2008) demonstrates. The RUIP condition implies that the realexchange rate appreciates in the impact period, keeps appreciating for some periods, and then starts depreciating. This conditional covariance between the real exchange rate and the real interest rate differential
explains two important empirical facts. First, the purchasing power parity (PPP) puzzle (Rogoff, 1996) canbe reconciled with the strong hump-shaped IRF of the real exchange rate to an inflation target shock. Second, an inflation surprise generated by a positive shock to the inflation target corresponds to simultaneous
currency appreciation (Clarida and Waldman, 2008).
The estimated TI model also replicates the empirical regularities of the joint dynamics of real andnominal exchange rates established by Mussa (1986) and investigated by Monacelli (2004). In particular,
the real currency return (RCR) and nominal currency return (NCR) simulated by the estimated TI model3

are almost perfectly correlated, and the standard deviation ratio of the former to the latter is close to one
under flexible exchange rate regimes.
The remainder of this paper is organized as follows. Section 2 introduces the proposed model. Section 3 reports the results of the Bayesian posterior inferences of the proposed TI model. Section 4 concludes.

2. The Model
To extract the theoretical implications of trend inflation for exchange rate dynamics as clearly as
possible, I investigate a basic version of a two-country NK model. Because the NK model here is standard,
except for trend inflation, I supply the full derivation and detailed explanation of the model in the online
appendix. I focus mainly on deriving the model’s primary propagation mechanism, that is, the openeconomy GNKPC.
There are home (h) and foreign (f ) countries in this model. Throughout this paper, any variable withan asterisk (∗) corresponds to a foreign variable, while a variable without it denotes the home counterpart.
To avoid redundancy in the model’s description, I will describe only the home country’s problem below.
The same argument applies to the foreign country counterpart.
2.1. The household
The representative household maximizes the lifetime expected utility function subject to the budget
constraint by supplying labor to the competitive domestic labor market elastically and forming external
consumption habits. The representative household can access home and foreign state-non-contingent bondmarkets to smooth consumption surplus Ht ≡ Ct − hC̄t−1 across periods, where Ct is consumption andhC̄t−1 is the stock of external consumption habits with habit parameter h ∈ (0, 1). The household’s prob-

lem yields the Euler equation, utility-based uncovered interest parity condition, and intratemporal optimality conditions as first-order necessary conditions (FONCs).
Home aggregate consumption consists of home and foreign consumption bundles with constant substitution elasticity subject to home bias. The home and foreign consumption bundles comprise the con4

tinuum of domestically produced final goods. Dixit-Stiglitz-type aggregators provide home and foreign
consumption bundles. The static cost minimization problem then yields the household’s demand functions
for individual home- and foreign-produced final goods.
2.2. The final goods firms
Facing the home and foreign representative households’ demand functions, the home final good firms
act as identical monopolistically competitive price setters. When setting its current optimal price, each firm
follows a Calvo-type time-dependent pricing strategy, in which a firm cannot reset its optimal price withprobability µ ∈ (0, 1]. Moreover, to generate endogenous fluctuations in the real exchange rate in thismodel, I assume that all firms adopt a local currency pricing (LCP) strategy, as in Betts and Devereux(1996, 2000): each firm sets its optimal prices differently in the two countries in terms of local currencies.5
Under such a local-currency Calvo pricing strategy, the objective function of a home firm is

max EtPh,t ,P∗h,t

∞
!
i=0

i

µ Γt+i"#Ph,tPh,t+i$− mct+i

%#

+ Et

Ph,t
Ph,t+i∞!

$−ζµi Γt+ii=0Ch,t+i"#

St+i P∗h,tPh,t+i$− mct+i

%&

P∗h,t
∗
Ph,t+i'−ζ∗
Ch,t+i,∗∗where Ph,t , P∗h,t , Ph,t , Ph,t
, St , Ch,t , Ch,t
, mct , and Γt are the optimal price of the home-produced goodsin the home country, optimal price of the home-produced goods in the foreign country, home aggregate
price over the home-produced final goods, foreign aggregate price over the home-produced final goods,
nominal bilateral exchange rate, home consumption bundle comprising the home-produced final goods,
foreign consumption bundle comprising home-produced final goods, real marginal cost of the home firm,
and home real stochastic discount factor, respectively. In particular, the parameter ζ is the price elasticity
of demand of the final home-produced good. The home country’s real stochastic discount factor for futuret + i period is given by Γt+i ≡ β i (Ht /Ht+i ).

5

As in the standard two-country NK models of Benigno (2004) and Engel (2014,2019), the sticky LCP is the fundamental
mechanism for real exchange rate fluctuations in this model.

5The FONCs for the optimal prices the home firm sets for the home and foreign markets are
#
$−ζ
∞
!
ζ
1
i
Ph,t Etµ Γt+iCh,t+i =Etµ Γt+i mct+i
Ch,t+i ,
P
ζ
−
1
P
h,t+i
h,t+i
i=0
i=0
&
'1−ζ
&
'−ζ
#
$
∞
∞
∗
!
!
S
P
1
ζ
1
t+i
h,t+i
∗
∗
P∗h,t Et
µi Γt+iCh,t+i=Etµi Γt+i mct+i
Ch,t+i
,
∗
∗
P
P
ζ
−
1
P
h,t+i
h,t+i
h,t+i
i=0
i=0
∞
!

i

#

1

$1−ζ

∗
respectively. Given the optimal prices, the price aggregators Ph,t and Ph,tfollow the laws of motion:

1−ζ
1−ζ
Ph,t
= (1 − µ)P1−ζ
h,t + µPh,t−1 ,

and

∗1−ζ
∗1−ζ
Ph,t
= (1 − µ)P∗1−ζ
+ µPh,t−1.
h,tEach final good firm produces its product using only the labor input hired from the competitivedomestic labor market. The production function of the home goods is Yt (z) = At Lt (z), where At is thelabor productivity in the home country. In this case, the real marginal cost that the home-produced final
good firms face is mct = AtWPth,t , where Wt is the home competitive labor wage.
I assume that the logarithms of labor productivity, ln At and ln A∗t , are I(1). To guarantee a balanced
growth path of this two-country model, I assume that the labor productivity differential ln at ≡ ln At −ln A∗t is I(0). As Mandelman et al. (2011), Rabanal et al. (2011), and Ireland (2013) find, the I(1) laborproductivity and stationary productivity differential jointly imply that the labor productivity of the homecountry must be cointegrated with that of the foreign country. Following Kano (2021), I assume that the∗home and foreign growth rates of labor productivity, ln γA,t ≡ ln At −ln At−1 and ln γA,t
≡ ln A∗t −ln A∗t−1 ,∗are generated by the error correction processes ln γA,t = ln γA − λ2 ln at−1 + $A,t and ln γA,t
= ln γA∗ +λln at−1 + $∗A,t , where γA and γA∗ are the drift terms and λ ∈ [0, 1) is the speed of error correction. The
2error correction mechanism implies that the cross-country labor productivity differential is I(0) because
ln at = ln γA − ln γA∗ + (1 − λ) ln at−1 + $A,t − $∗A,t .6 Importantly, if the adjustment speed λ is sufficiently
close to zero, then the cross-country labor productivity differential can be realized near I(1).
2.3. Monetary policies with trend inflation

6∗ 2
As a result the two countries share the same unconditional mean of the labour productivity growth rate ln(γA γA
) .
1

6

The monetary policies of the two countries follow Taylor rules. The home country’s central bank sets
the short-term domestic nominal interest rate (1 + ih,t ) depending on the past interest rate level (1 + ih,t−1 ),current inflation rate γπ,t , detrended output level yt ≡ Yt /At , and nominal currency return St /St−1 . As
in the specifications of Ireland (2007) and Cogley et al. (2010), the central bank targets the time-varying
trend inflation γτ,t . The Taylor rule of the home country is(1 + ih,t ) = (1 + i)1−ρi (1 + ih,t−1 )ρi

(#

γπ,t
γτ,t

$a π

(yt )ay

#

St
St−1

as )1−ρi$ 1−asexp($i,t ),where i is the deterministic steady-state value of the home nominal interest rate and ρi ∈ (0, 1) captures

the degree of interest rate smoothing. The parameter as ∈ [0, 1) captures the degree of the managed
exchange rate regime: the larger as is, the more stringent the managed regime is. If as = 0, then the model
corresponds to a perfectly flexible exchange rate regime. If as → 1, then the model is subject to a strict

fixed nominal exchange rate regime with St = 1 for all t. If as ∈ (0, 1), then the model is characterized bya managed exchange rate regime, as specified by Monachelli (2004).

In this study, I allow the home trend inflation rate γτ,t to be stochastic with an exogenous AR(1)process in the logarithmic term:ln γτ,t = (1 − ρτ ) ln γτ + ρτ ln γτ,t−1 + $τ,t ,where γτ is the long-run mean of the home trend inflation rate, ρτ ∈ [0, 1) is the AR(1) root of the hometrend inflation rate, and $τ,t is the i.i.d. trend inflation shock.7 As in the closed-economy NK model by
Ascari and Sbordone (2014), the time-invariant long-run mean γτ makes the log-linearization exercise ofthis study tractable when characterizing the deterministic steady state.
I assume that the monetary policy disturbance $i,t is a serially uncorrelated i.i.d. shock. Coibion
and Gorodnichenko (2012) provide empirical evidence that the high persistence observed in the actual
short-run nominal interest rate data in the U.S. stems largely from the policy inertia caused by interest rate
7

Ireland (2007) allows for an implicit time-varying inflation target with endogenous responses to supply shocks. He showsthat in the US data the likelihood ratio test cannot reject the exogenous time-varying inflation target.

7

smoothing in the Taylor rule, but not from the exogenous persistence of the monetary policy disturbance, as
Rudebusch (2002, 2006) claims. In particular, their empirical result suggests that the time-varying inflation
target that the conventional Taylor rule misses could lead to a high estimate of the serial correlation of themonetary policy disturbance. Indeed, the persistent trend inflation γτ,t is indistinguishable from a persistentmonetary policy disturbance if we set $i,t to −(1 − ρi )aπ ln γτ,t . Following the above notion, I allow bothinterest rate smoothing and the time-varying inflation target in the Taylor rules.
2.4. Closing the model
To guarantee a stationary distribution of the two countries’ net foreign asset positions within incomplete international financial markets, I allow for a debt-elastic risk premium in the interest rates faced onlyby the home country. The risk premium is given as an externality: the home household does not consider
the effect of the debt position on the risk premium when maximizing its lifetime utility function. As in
Bergin (2006) and Justiniano and Preston (2010), I set the risk premium subject to the exogenous risk
premium shock vr,t , which follows an AR(1) process, vr,t = ρr vr,t−1 + $r,t , where ρr ∈ (0, 1).

The home and foreign bond markets’ equilibrium conditions require that the world net supply of

nominal bonds be zero on a period-by-period basis along an equilibrium path. The home and foreign finalgoods markets’ equilibrium conditions depend on time-varying price dispersions in the final-good markets.
The time-varying price dispersions represent resource costs. Given the output level, the higher the degreeof price dispersion, the lower the amount allocated to consumption.
Because the model contains non-stationary components, I stochastically detrend the FONCs to characterize the unique deterministic steady state, as I show in detail in the accompanying appendix. To derive the corresponding linear rational expectations (LRE) models, I take a log-linear approximation of thestochastically detrended FONCs around the unique deterministic steady state. For the variable xt , let x̂tdenote the percentage deviation from its deterministic steady-state value x; that is, x̂t ≡ (xt − x)/x. x̃t is
the deviation from the deterministic steady-state value; that is, x̃t ≡ xt − x.
2.5. GNKPC with trend inflation.

8

The online appendix provides the entire derivation of the GNKPC for the home good price inflationin the home country, ∆ ln Ph,t , from the log-linearized FONC for the optimal price setting of the home final
good firm. Let γ̄ ≡ γτ /(γA γA∗ ) 2 denote the steady-state home gross inflation rate. With the definitions
1

−1

ϕ1
of the parameters, ϕ1 ≡ βµγ̄ ζ−1 ∈ (0, 1), ϕ2 ≡ βµγ̄ ζ ∈ (0, 1), and ϕ0 ≡ 1−β, the corresponding
β −1 ϕ1

GNKPC is∞ϕ0 (ϕ1 − ϕ2 ) ! i∆ ln Ph,t = βγ̄Et ∆ ln Ph,t+1 + ϕ0 (1 − ϕ2 )m̂ct +ϕ1 Et ∆ ln Ph,t+i
ϕ1
i=1

∞
∞
ϕ0 (ϕ1 − ϕ2 ) ! i
ϕ0 (ϕ1 − ϕ2 ) ! ih−ζ
ϕ1 Et ∆ ln Pt+i −ϕ1 Et ∆(ĉt+i − ĥt+i ) + v̂p,t
, (1)
ϕ1
ϕ1
i=1
i=1

where ∆ ln Pt is the home consumer price index (CPI) inflation rate, ∆ĉt is the change in the detrended
consumption, and ∆ĥt is the change in the detrended consumption surplus. Therefore, as Cogley and
Sbordone (2008) argue, GNKPC (1) depends on the higher-order leads of the home good price inflation as
well as the home CPI inflation. The changes in consumption and consumption surplus affect the GNKPC
through the final goods demand and stochastic discount factor.
hMoreover, v̂p,t
is a reduced-form shock additive to the GNKPC, which follows an AR(1) process,
h
h
h
v̂p,t
= ρhp v̂p,t−1+ $hp,t , where ρhp ∈ (0, 1). Steinsson (2008) interprets v̂p,t
as a mixture of different types ofshocks, including labor supply shocks, government spending shocks, shocks to world demand, and costh
push shocks. His calibration study shows that v̂p,t
is the primary driver of the hump-shaped impulse rehsponse of the real exchange rate. The GNKPC shock v̂p,t, however, does not obey any cross-equation re-strictions of the model and has no strict structural interpretation. As Lubik and Schorfheide (2006) argue,
this nonstructural shock reflects the extent of model misspecification.
Note that the coefficients ϕ0 , ϕ1 , and ϕ2 are given as nonlinear functions of the primitives β, µ, andγ̄. In particular, higher steady-state inflation γ̄ implies a smaller weight on the current real marginal costm̂ct and larger weights on the expected future home good price inflation Et ∆ ln Ph,t+i and the expected
future home CPI inflation Et ∆ ln Pt+i . This is because positive steady-state inflation affects the expected
future profits of monopolistic competitive producers through its effect on the relative prices Ph,t+i /Pt+i in
the demand functions toward an infinite future. The higher the steady-state inflation, the larger the weight
9

that producers place on expected future profits when they decide the current optimal price. Producers placea smaller weight on the current variable, such as the real marginal cost.
2.6. The open-economy GNKPCThe online appendix also provides the full derivation of the GNKPC for the home good price in∗flation in the foreign country, ∆ ln Ph,t
. To obtain an economic intuition for the role of the GNKPCs in

the equilibrium of this two-country model, I tentatively assume that the two countries are symmetric in∗
all structural parameters. Subtracting the GNKPC of ∆ ln Ph,tfrom that of ∆ ln Ph,t then provides thefollowing GNKPC for the inflation differential between the two countries, ∆ ln Ptd ≡ ∆ ln Pt − ∆ ln Pt∗ :ϕ0 ϕ2 (1 − ϕ1 )(q̂t − ât )ϕ1∞∞
ϕ0 (ϕ1 − ϕ2 )(1 − ζ) ! i
ϕ0 (ϕ1 − ϕ2 ) ! id+
ϕ1 Et ∆ ln Pt+i −ϕ1 Et ∆ĉdt+i
ϕ1
ϕ
1
i=1
i=1

d
∆ ln Ptd = βγ̄Et ∆ ln Pt+1
+ ϕ0 (1 − ϕ2 )ât +∞ϕ0 (ϕ1 − ϕ2 )(1 − ϕ1 ) ! id+
ϕ1 Et (q̂t+i − ât+i ) + v̂p,t
, (2)
ϕ1
i=1

where qˆt is the real exchange rate, ∆ĉdt+i is the change in the cross-country differential of detrended condsumption, and v̂p,t
is the cross-country differential of the GKNPC shocks.

An open-economy GNKPC (2) implies that the inflation differential becomes more persistent andless volatile in response to persistent trend inflation shocks, and less sensitive to the current real exchange
rate than does the standard NKPC with γ̄ = 1. With positive steady-state inflation γ̄ > 1, the firms place
more weight on future variables than current ones in its optimal price setting because of the implied higher
discount factor ϕ1 . The inflation differential becomes less sensitive to the current variables, including thereal exchange rate, but is more dependent on the present discounted values of the expected future variables.
Therefore, only the persistent slow-moving components of the expected future variables matter for the
inflation differential.8
8Under the condition γ̄ = 1, the GNKPC (2) is degenerated to the standard NKPC:dd∆ ln Ptd = βEt ∆ ln Pt+1
+ κµ q̂t + v̂p,t,10

3. Bayesian posterior inferences with Canadian and U.S. data.
3.1. Data and prior distributions of the structural parameters
In this section, I empirically assess the proposed asymmetric two-country NK model with trend inflation by drawing formal Bayesian posterior inferences, as in Lubik and Schorfheide (2005) and Justiniano
and Preston (2010).9 Following the latter study, I analyze post-BW Canada-U.S. quarterly data, which contain nine time-series: Canadian and U.S. inflation rates, Canadian and U.S. real consumption growth rates,
Canadian and U.S. hours worked, Canadian and U.S. short-term nominal interest rates, and the bilateralreal exchange rate of the Canadian dollar per US dollar. The quarterly data cover Q1:1982 to Q1:2007. I
use the first four quarters to initialize the Kalman filter. The appendix discusses the data sources and theconstruction.
Table 1 shows the prior distributions of the structural parameters of the model. The third, fourth, andfifth columns represent the corresponding density, mean, and standard deviation, respectively. These are
fairly common across other studies that adopt Bayesian inference methods for dynamic stochastic generalequilibrium (DSGE) models. I calibrate the subjective discount factor β to 0.990 for both countries and thedebt elasticity of risk premium ψ to 0.00001 for Canada because these parameters are not well identified.
Recognizing that the U.S. is a large foreign country, I also calibrate the U.S. home bias parameter τ ∗ to
0.999, which implies that U.S. households consume only the U.S. aggregate consumption basket.
Several parameters require further examination. I adopt normal prior densities for the drift parameters of the home and foreign labor productivities, γA and γA∗ . I set the prior means of these parameters to
the unconditional means of the Canadian and U.S. gross quarterly consumption growth rates, 1.0050 and
1.0061, along with an identical prior standard deviation of 0.0005. The prior means of the home and foreign steady-state gross inflation rates γ̄ and γ̄ ∗ stem from the unconditional means of the Canadian and U.S.
gross inflation rates of 1.0072 and 1.0077, corresponding to 2.87% and 3.12%, respectively, at the annual
where κµ ≡ (1−µ)(1−βµ)∈ (0, 1). In this restrictive case, the current inflation differential depends on the one-period-aheadµ
expected inflation differential and real exchange rate. This is the same representation of inflation differential dynamics as in
Benigno (2004). The current inflation differential becomes sensitive to current changes in the real exchange rate.
9
I thank Martin Uribe for his suggestion for the Bayesian estimation of the fully-fledged asymmetric two-country NK modelwith trend inflation.

11

rates. The model’s crucial parameters are the unconditional means of the home and foreign unobservable
trend inflation, γτ and γτ∗ , respectively. For each country, I construct the prior mean of this parameter bytaking the sum of the unconditional mean of the steady-state inflation rate and the unconditional mean of
the consumption growth rate. The resulting prior density of γτ is normal with a mean of 1.0122 and a small
standard deviation of 0.0005, whereas the prior density of γτ∗ is normally distributed with a mean of 1.0138
and a standard deviation of 0.0005.
Recall that the NK literature conventionally calibrates the price elasticity of demand by fitting the
NK model to a markup rate of approximately 10.00-12.00%. GNKPC (1) implies that the steady-statehome markup rate is the inverse of the real marginal cost:

mc =

#

ζ −1
ζ

$#

1 − βµγ̄ ζ
1 − βµγ̄ ζ−1$#

1 − µγ̄ ζ−1
1−µ

1
$ 1−ζ

.

Given the calibrated value of β and the prior means of µ, γτ , and γA , a price elasticity of demand ζ equal
to 22.00 provides a steady-state home markup rate of 8.66%. The same calculation with the foreign counterpart of ζ ∗ equal to 22.00 yields a steady-state foreign markup rate of 11.34%. Although the implied
Canadian markup rate is slightly lower than the conventional value, I use an identical gamma density witha mean of 22.00 and a standard deviation of 1.000 as the prior distributions of ζ and ζ ∗ .
I construct the prior distributions of the parameters of the home and foreign Taylor rules as follows. I
adopt identical gamma densities with a mean of 3.500 and a standard deviation of 0.500 for the inflation gap
parameters aπ and a∗π , identical gamma densities with a mean of 0.100 and a standard deviation of 0.010
for the output gap parameters ay and a∗y , and an identical beta density with a mean of 0.900 and a standard
deviation of 0.200 for the interest rate smoothing parameters ρi and ρ∗i . The prior means of aπ , a∗π , ρi ,and ρ∗i are slightly larger than the conventional values under the specification for zero-trend inflation in theliterature. Ascari and Ropele (2009) and Coibion and Gorodnichenko (2011) show in a closed-economy
NK model that high values of these Taylor rule parameters are necessary for equilibrium determinacy withpositive trend inflation. I further allow for asymmetric policy attitudes toward nominal currency returns
between the two central banks. On the one hand, I exploit the beta density with a mean of 0.250 with a12standard deviation of 0.200 as the prior density of the Bank of Canada’s policy sensitivity to the nominal
currency return as . On the other hand, I use a beta density with a mean of 0.050 and a standard deviationof 0.010 as the prior density of the U.S. counterpart a∗s , as the central bank of the large country, the Fed,puts a negligible weight on the nominal currency return when setting the short-term policy rate.
Lastly, I determine the prior distributions of the 11 exogenous impulses as follows. I assume that all
structural shocks are normally distributed. The error correction speed of the cointegrated labor productivity
process, λ, has a beta prior distribution with a mean of 0.010 and a standard distribution of 0.010 to reflect
slow cross-country productivity diffusion. I set the prior distributions of the AR(1) coefficients of the homeand foreign trend inflation processes ρτ and ρ∗τ , respectively, to identical beta distributions with a mean of
0.990 and a small standard distribution of 0.001. The resulting tight prior of the high AR(1) coefficient acts
as an identification restriction to extract the permanent components of the two countries’ inflation rates. To
construct the prior distribution of the AR(1) coefficient of the risk premium shock, ρr , I rely on Justiniano
and Preston (2010) by adopting a beta density with a mean of 0.500 and a standard deviation of 0.100. As
there is no prior information, I also apply the same beta prior density to the AR(1) coefficients of the fourff∗
GNKPC shocks, ρhp , ρh∗p , ρp , and ρp . Following the conventional exercise in the literature on Bayesianinferences of DSGE models, I adopt inverse gamma densities for the prior distributions of the standard
deviations of the 11 exogenous impulses, as summarized in Table 1. I refer to the above prior configuration
the TI model.
To identify the role of trend inflation in the empirical performance of the TI model, I prepare anotherprior configuration by restricting the model with no long-run trend inflation γ̄ = γ̄ ∗ = 1. With these prior
restrictions, the TI model degenerates to a standard two-country NK model. I also reset the prior meanof the price elasticity of demand ζ to 11.00, fitting the model to an empirical markup rate of 10.00%, asin the conventional literature. Throughout the rest of this paper, I refer to this alternative restricted priorconfiguration the non-TI model. The sixth and seventh columns of Table 1 summarize the prior densitiesof the structural parameters of the non-TI model.1010The non-TI model sets smaller prior standard deviations of the habit, home-bias, and Calvo parameters than those in the
TI model because larger prior standard deviations lead the posterior means of these structural parameters to the lower or upperboundaries of the admissible [0 1) interval, respectively.

13Given the prior distributions of the structural parameters, I draw the posterior inferences of the TI
model and the non-TI model through a Markov chain Monte Carlo (MCMC) method implemented with therandom-walk Metropolis–Hastings algorithm. During the MCMC sampling, I solve the rational expectations equilibrium of the log-linearized model, derive its state-space representation using the QZ algorithm
proposed by Sims (2001), and construct the corresponding restricted unobserved component model thatgenerates the likelihood function given the data. I keep only the MCMC samples that yield the equilibrium
determinacy. In addition, to ensure that the posterior samplers are distributed around the conventional
12.00% markup rate, I prepare the prior distributions of the home and foreign markup rates with an identical gamma density with a mean of 1.120 and a standard deviation of 0.005.11 The posterior distributions of
the model’s structural parameters are approximated with 500,000 MCMC samples, where the first 200,000
draws are burned in to guarantee posterior convergence.
3.2. Bayesian posterior inferences of the two models
The second, third, and fourth columns of Table 2 report the means, standard deviations, and 95%
Bayesian highest probability density (HPD) intervals of the posterior distributions of the TI model’s structural parameters, respectively. We should note at least five important results. First, the “Markup rate” rows
show that the posterior sampler guarantees 10.92 % and 11.96 % gross markup rates for Canada and the
U.S., respectively. Second, the posterior means of γτ , γA , γτ∗ , and γA∗ imply that the posterior Canadian and
U.S. long-run annual inflation rates γ̄ and γ̄ ∗ are 2.92% and 3.61%, respectively. These positive long-run
inflation rates support the empirical relevance of the GNKPCs in the TI model.
Third, the TI model reveals a smaller degree of Canadian home bias. The posterior mean of theCanadian home bias parameter τ is 0.7221, with a posterior standard deviation of 0.0121. Together withthe calibration of perfect U.S. home bias (τ ∗ = 0.999), this result reflects the small Canadian economy,
relative to the large U.S. economy.
Fourth, the TI model does not depend much on the persistence of the risk premium and GNKPC
11

The markup rate mc is a non-linear function of the other structural parameters θ; mc = mc(θ). Let p(mc) denote the priordistribution of the markup rate. Then with the prior distributions of the markup rate and the structural parameters, the posterior
distribution of the structural parameters p(θ|Y) is given by p(θ|Y) ∝ p(mc(θ))p(θ)p(Y|θ),where Y is the data and p(Y|θ)
is the likelihood function.

14

shocks. The highest posterior mean among the AR(1) roots of these exogenous shocks is, at best, 0.9251
for ρh∗p . Past studies depend on these reduced-form shocks to explain the highly persistent real exchange
rate data.12 In contrast, to explain the real exchange rate data, the TI model in this study relies on the highpersistence of neither the risk premium shock nor the GNKPC shocks.
Finally, as the fifth result, the posterior means of the standard deviations of the trend inflation shocks,στ and στ∗ , remain at 0.0009 and 0.0006, which are smaller than those of the other shocks. With the highposterior means of the AR(1) coefficients of the trend inflation shocks, this result is consistent with the usual
conjecture that changes in the inflation target are small and long-lasting. Figure 1(a) plots the posteriormean of the Kalman-smoothed inference of the Canadian trend inflation γτ,t as the solid blue line, withthe corresponding 68% HPD interval as the dotted blue lines and the actual data of the Canadian inflation
rate as the solid black line, while Figure 1(b) shows the U.S. counterparts. The Canadian smoothed trend
inflation is less volatile with a small posterior uncertainty and can capture the actual inflation rate’s longswing; indeed, it is highly correlated with the Hodrick-Prescott filtered trend component of the actual
inflation rate with the correlation coefficient of 0.8236. In particular, the sharp change in the direction of
the smoothed Canadian trend inflation around 1991 precisely detects the Bank of Canada’s introductionof the inflation targeting policy. The U.S. smoothed trend inflation depicted in Figure 1(b) tracks the ups
and downs of the slow-moving component of the actual U.S. inflation over the sample period. Thus, the TI
model can plausibly identify the permanent components of the two countries’ actual inflation rates.
The fifth, sixth, and seventh columns of Table 2 describe the posterior means, posterior standard
deviations, and 95% HPD intervals of the structural parameters of the non-TI model. There are three crucial
differences in the posterior distributions between the TI and non-TI models. First, the non-TI model infers
a much higher degree of Canadian home bias than that of the TI model: the posterior mean of τ , estimatedby the former model, is 0.9781, whereas that estimated by the TI model is 0.7221. This result casts doubt
on the non-TI model’s identification of the small openness of the Canadian economy.
12

For instance, using quarterly Canada-U.S. data, Justiniano and Preston (2010) estimate the posterior mean of the AR(1)coefficient of the risk premium to be 0.980 and report that the risk premium shock accounts for roughly 90% of the forecast
error variance of the real exchange rate. Lubik and Schorfheide (2005) also show the heavy dependence of their two-country
open-economy DSGE model on reduced-form PPP shocks when accounting for real exchange rate variations in actual Euro and
U.S. dollar data.

15

Second, the non-TI model estimates much smaller posterior mean of the Canadian habit parameter
than do the TI model: the posterior mean of h for the former model is 0.1631, while that for the lattermodel is 0.6598. This inference about negligible Canadian consumption habit is inconsistent with the posterior inference by Justiniano and Preston (2010). Third, the non-TI model estimates a significantly larger
posterior mean of the standard deviation of the NKPC shock than do the TI model: the posterior mean ofσph∗ for the former model is 0.0374, while that for the latter model is 0.0007. Therefore, the posterior inferences above imply that the non-TI model fails to identify endogenous propagation mechanisms in Canada.
Instead, the non-TI model relies heavily on a reduced-form NKPC shock to explain the real exchange rate
dynamics, as Steinsson (2008) shows in his calibration exercise.
The TI model beats the non-TI model in terms of the overall data fit. The last row of Table 2 showsthat the TI and Non-TI models’ log marginal likelihoods are 2905.906 and 2801.289, respectively, estimatedusing Geweke’s (1999) harmonic mean estimator. This significant log difference in marginal likelihood is“strongly” favorable for the TI model, according to Jeffreys’ criterion (Kass and Raftery 1995). The positive
long-run trend inflation estimated by the TI model significantly improves the overall empirical fit of theotherwise standard two-country NK model to the Canada-U.S. data.
Tables 3 (a) and (b) summarize the posterior means of the forecast error variance decompositions(FEVDs) of the real exchange rates estimated by the TI and non-TI models, respectively. On the one hand,
Table 3(a) conveys the most outstanding finding from this study’s FEVD exercise: the Canadian trend
inflation shock explains the dominant portions of the stochastic variation in the real exchange rate for all
horizons. In the impact period, 23.5% of the FEVD of the real exchange rate is due to the Canadian trend
inflation shock. The same shock generates 25.2%, 28.2%, and 36.5% of the FEVDs of the real exchange
rate for the 1-, 3-, and 10-year horizons, respectively. The TI model also admits other structural shocks,
especially the Canadian productivity and monetary policy shocks and the U.S. trend inflation shock, whichplay relatively small but non-negligible roles in real exchange rate variations. The other reduced-form
shocks, that is, the risk premium shock and the GNKPC shocks, jointly generate between 26.9% and 44.6%
of stochastic variations in the real exchange rate over all the forecast horizons.
On the other hand, as Table 3(b) shows, the non-TI model estimates that the NKPC shocks pre16

dominantly explain the stochastic variations in the real exchange rate. The NKPC shocks jointly generate
between 40.0% and 66.5% of the total variations in the real exchange rate across all the forecast horizons.
In particular, the NKPC shock $hp,t alone explains about 50% of the FEVD of the real exchange rate forlonger horizons.
The results of the FEVD exercise strongly support this study’s hypothesis that stochastic variation intrend inflation is fundamental for exchange rate dynamics and suggest that the Bank of Canada’s inflation
targeting policy, which crucially influenced Canada’s long-run trend inflation, might have played a significant role in the Canada-U.S. real exchange rate after 1991. Figure 2 displays the demeaned Canada-U.S.
RCR data (on the right axis) as a solid black line and the TI model’s historical decompositions of the RCR
into the six structural shocks (on the left axis) as the yellow bars for the Canadian trend inflation shocks,
the purple bars for the U.S. trend inflation shocks, the blue bars for the Canadian productivity shocks, thered bars for the U.S. productivity shocks, the green bars for the Canadian monetary policy shocks, and thelight blue bars for the U.S. monetary policy shocks, respectively. The vertical dashed black line indicatesthe period Q1:1991 for the Bank of Canada’s introduction of the inflation-targeting policy.13 Observe that
the Canadian trend inflation shocks played quite significant roles in the historical movements of the RCR
over the whole sample period; the absolute values of the RCR historical decompositions to the Canadiantrend inflation shocks record around 20 % in many periods. Before Q1:1991, the historical decompositions
for the Canadian trend inflation shocks went up and down toward positive and negative values. However, inmost periods after Q1:1991, the historical decompositions for the Canadian trend inflation shocks remained
positive until 2002. These results suggest that because the Bank of Canada’s inflation targeting policy successfully anchored the Canadian long-run inflation toward the inflation target, the resulting sharp and steadydeclines of the Canadian inflation rate significantly contributed to the Canadian real depreciation realized
by 2002.
Figure 3 plots the TI and non-TI models’ IRFs of the percentage deviation of the real exchange ratefrom the long-run steady state to the eleven structural shocks: panel (a) for the Canadian productivity
shock, panel (b) for the U.S. productivity shock, panel (c) for the Canadian trend inflation shock, panel
13

Each historical decomposition for a structural shock is represented as a percentage ratio to the sum of the absolute valuesof the historical decompositions for all the eleven shocks.

17(d) for the U.S. trend inflation shock, panel (e) for the Canadian monetary policy shock, panel (f) for the
U.S. monetary policy shock, panel (g) the Canadian GNKPC shock to the Canadian price, panel (h) for
the U.S. GNKPC shock to the Canadian price, panel (i) for the Canadian GNKPC shock to the U.S. price,panel (j) for the U.S. GNKPC shock to the U.S. price, and panel (k) for the risk premium shock. In all
panels, the solid blue lines are the posterior mean of the IRFs estimated by the TI model, accompanied by
the 95% HPD interval as dotted blue lines. Similarly, the solid red lines are the posterior means of the IRFs
estimated by the non-TI model, accompanied by the 95% HPD interval as dotted red lines. To compare themagnitudes of the propagation mechanisms between the two models, I set the IRFs to one-unit shocks.
The most apparent finding of the TI model is persistent, nearly permanent IRFs. The Canadianproductivity, trend inflation, and monetary policy shocks, shown in panels (a), (c) and (e), the U.S. trend
inflation shock, shown in panel (d), and several GNKPC shocks, shown in panels (g), (h), and (i), generatestrong hump-shaped IRFs. The two countries’ trend inflation shocks in particular dominate the IRFs of thereal exchange rate to the other structural shocks in terms of magnitude. The non-TI model, on the other
hand, fails to generate persistent hump-shaped IRFs; the IRFs to most structural shocks quickly converge
toward zero. Moreover, the magnitudes of the non-TI model’s IRFs are much smaller than those of the TI
model, except for the U.S. monetary policy shock in panel (f).
These IRF exercise results clearly show that the TI model possesses powerful amplification andpropagation mechanisms, especially for the two countries’ trend inflation shocks for the real exchange rate,
which are absent in the standard two-country NK model.
3.3. Posterior predictive analysis of exchange rate moments
This subsection evaluates the TI and non-TI models’ empirical performance in terms of empiricalmoments of the real and nominal exchange rates through posterior predictive checks. The first and secondrows of Table 4 summarize the empirical moments I target in this study. The first empirical moment is
the sum of the autoregressive coefficients of the AR(5) process of the real exchange rate, denoted by α.
Applying the augmented Dickey-Fuller (ADF) regression with five lags for the U.S. trade-weighted realexchange rate, Steinsson (2008) estimates α to be 0.954 with a 90% confidence interval between 0.879 and18

1.000.
Steinsson (2008) and Burstein and Gopinath (2014) report a half-life measure of the real exchange
rate. The estimated ADF equation provides the IRFs of the real exchange rate to a unit reduced-form disturbance. I calculate the second empirical moment, denoted by HL, as the maximum period before the IRF
reaches 0.500; that is, HL = T such that IRF(T-1) > 0.500 and IRF(T) ≤ 0.500. Burstein and Gopinath(2014) report HLs for eight advanced countries: Canada, France, Germany, Italy, Japan, Switzerland, theU.K., and the U.S. The cross-country average HL is 4.425 years, with a minimum of 1.600 years for Switzerland and a maximum of 6.000 years for the U.S.
The third empirical moment is the correlation coefficient between the RCR and NCR. As Mussa(1986) observes, the real and nominal exchange rates move closely. Burstein and Gopinath (2014) calculate
the correlation coefficients between the RCR and NCR for these eight advanced countries. The correlation
coefficient, denoted as Corr, is 0.932 on average, with a minimum of 0.820 for Italy and a maximum of0.990 for Japan. The tight comovement between the RCR and NCR implies that the exchange rate dynamics
remain disconnected from the inflation differential, at least in the short run. Burstein and Gopinath (2014)
Std(∆ ln qt )also provide the standard deviation ratios of the RCR to the NCR, Std(∆for these eight economies. The
ln St )

cross-country average is 0.956, with a minimum of 0.870 for France and a maximum of 1.040 for the U.K. A
standard deviation ratio close to one implies that the exchange rate is much more volatile than the inflationdifferential. These moments confirm that the RCR and NCR move almost one-to-one and are much morevolatile than the inflation differential. The third row of Table 4 reports the corresponding sample moments
for the Canada-U.S. data I use in this study.
The fourth row of Table 4 presents the posterior means of the targeted moments inferred by the
TI model, accompanied by the corresponding 95% HPD intervals in the fifth row. Given a particularposterior draw of the structural parameters, I simulate the state variables of 100 periods from the statespace representation of the TI model and calculate the model counterparts of the targeted moments. I
generate ten repetitions of the targeted moments and keep the ensemble average to mitigate possible Monte
Carlo errors. Repeating this process over 300,000 MCMC samplers constructs the posterior distributionsof the targeted moments. The TI model infers a large AR root of the real exchange rate; the posterior mean19

of α is 0.939. The HPD interval between 0.912 and 0.963 covers the empirical counterparts of 0.954 forthe cross-country average and 0.961 for the Canada-U.S. data. The TI model also estimates the posteriormean of the HL measure to be 4.615 years, closely mimicking the empirical counterpart of 4.425 years forthe cross-country average and 5.250 for the Canada-U.S. data. The corresponding HPD interval between
3.365 and 6.000 covers the two empirical HL measures. Hence, the TI model successfully generates thehighly persistent real exchange rate observed in the actual data. The TI model is also consistent with
Mussa’s (1986) observations of one-to-one comovement between the real and nominal exchange rates. The
TI model estimates a high positive correlation coefficient between the RCR and NCR, and the posteriormean of the correlation coefficient of 0.980 is close to its empirical and Canada-U.S. counterparts. The
TI model also replicates the relative volatility of the RCR to the NCR, as in the actual data. The TI model
estimates the posterior mean of the standard deviation ratio of the RCR to NCR to be 1.029, with a 95%
HPD interval between 1.022 and 1.046. This posterior inference reasonably matches the Canadian and
U.S. data counterpart of 1.037.
The sixth and seventh rows of Table 4 report the targeted moments inferred by the non-TI model.
First, the sum of the AR(5) coefficients, α, decreases sharply to 0.747 in the non-TI model from 0.939 in
the TI model. Furthermore, the posterior mean of the HL measure of 1.245 years indicates low persistence in the real exchange rate, as this model implies. The non-TI model also fails to reconcile Mussa’s(1986) observations. The posterior mean of the correlation coefficient between the RCR and NCR is 0.883,
which is far below the corresponding Canada-U.S. point estimates of 0.957. The reason behind the non-TI
model’s apparent failure in Mussa’s (1986) observations is the implied excess volatility of the two countries’ inflation differential; the standard NKPC cannot generate smoothed inflation dynamics observed in
actual data.
Figure 4 plots, as the solid blue line, the posterior mean of the IRF of the real exchange rate to a
reduced-form disturbance from the ADF regression inferred by the TI model. The two dotted blue lines
represent the corresponding 95% HPD interval. The inferred IRF is hump-shaped with a peak around ayear and monotonically, but slowly, declining toward zero over time. The solid red line, on the other hand,displays the non-TI model’s posterior mean of the IRF of the real exchange rate from the ADF regression
20

with the corresponding 95% HPD interval as the dotted red lines. Note that the IRF monotonically andquickly declines toward zero with no hump. The HL measure’s significant difference between the twomodels reported in Table 4 reflects this difference in the IRF of the real exchange rate.
The same figure plots the point estimate of the IRF of the real exchange rate for the Canada-U.S. data
as a solid black line. The grey-shaded area represents the corresponding 95% bootstrapping confidence
interval. The hump-shaped IRF estimated by the TI model is included in the shaded area. However, its
absolute size is much smaller than that of the Canadian and U.S. counterparts. This result is an apparent
failure of the TI model; the propagation mechanism embedded in the TI model is not sufficiently strong tomimic the large hump-shaped real exchange rate response observed in the Canada-U.S. data.
3.4. Why can the TI model replicate the exchange rate dynamics?
To intuitively capture the economic mechanism underlying the successful outcomes of the TI modelin terms of real exchange rate dynamics, it is useful to solve the RUIP forward:q̂t = −∞!
j=0∗
Et {(1 + r̂t+j ) − (1 + r̂t+j)} + lim Et q̂t+j ,
j→∞

(3)

where (1 + r̂t ) and (1 + r̂t∗ ) are the home and foreign real interest rates, respectively. The last limiting
term on the right-hand side (RHS) converges toward zero. Hence, the real exchange rate response depends
negatively on the expected present values of the future real interest rate differentials with the unit discountfactor.14
To understand the high persistence of the real exchange rate implied by the TI model, suppose thatthe real interest rate differential (1 + r̂t )d ≡ (1 + r̂t ) − (1 + r̂t∗ ) follows an AR(1) process exogenous to the14

Eq.(3) is a restricted case of the fundamental asset pricing equation in Engel and West (2005).
zt = (1 − b)∞!

b Et (a$1 x1,t+i ) + b
j

j=0

∞
!

bj Et (a$2 x2,t+j ),j=0where zt is the asset price, b is the discount factor, x1,t is a vector of the observable economic fundamentals, x2,t is a vector of
the unobservable economic fundamentals, and a1 and a2 are the corresponding coefficient vectors. I obtain Eq.(3) by imposing
a1 = 0 and b → 1 in the fundamental asset pricing equation. As Engel and West (2005) prove, if an element of the economic
fundamentals, which is the real interest rate differential in this case, is I(1), then the real exchange rate converges to a random
walk with an infinite variance of the RCR.

21

real exchange rate: (1 + r̂t )d = ρ(1 + r̂t−1 )d + ut , where ρ ∈ (0, 1) and ut is an i.i.d. shock. The RUIP
condition (3) implies that if the real interest rate differential (1 + r̂t )d approaches a unit root process, thenthe real exchange rate approximately follows a random walk:
*

+
1
1
d
lim ∆qt = − lim
ut + ρ(1 + r̂t−1 ) = − lim
ut ,ρ→1ρ→1 1 − ρ
ρ→1 1 − ρ1where the second equality stems from the fact that limρ→1 1−ρ
ut dominates (1+ r̂t−1 )d in terms of volatility

at the limit ρ → 1. Therefore, the TI model implies that a persistent real interest rate differential is sufficientto yield the near random-walk behavior of the real exchange rate accompanied by high volatility. Largeand permanent IRFs of the real exchange rates to the two-countries’ trend inflation shocks, displayed in
Figure 3, indeed confirm this near random-walk property that the TI model reflects the real exchange rate
dynamics.
Why can the TI model generate the highly persistent real interest rate differential, independent ofreal exchange rate movements? Figures 5(a) and (b) display the posterior means of the IRFs of selectedvariables to a unit standard deviation shock to the Canadian trend inflation in the TI model. In particular,
Figure 5(a) depicts the posterior means of the IRFs of the real interest rate differential (solid red line),
expected inflation rate differential (circle-bar green line), inflation gap differential (cross-bar blue line),and nominal interest rate differential (solid black line). Figure 5(b) displays the IRF of the real exchange
rate (solid blue line).
Observe the persistent response of the nominal interest rate differential. In response to a positive
Canadian trend inflation shock, the nominal interest rate differential rises in the impact period and continues
to increase for 28 consecutive quarters thereafter. The nominal interest rate differential then begins to
decline toward the long-run value at a slow rate, and even after ten years, the corresponding IRF stays at
93.7% of the peak value.
This highly persistent response of the nominal interest rate differential stems from the slow-moving∗∗
dynamics of the two countries’ inflation gaps, γ̂π,t − γ̂τ,t and γ̂π,t− γ̂τ,t, which are an inherent outcome of

the GNKPCs. Moreover, the inflation gaps are insensitive to any developments in the current real exchange

22rate because of the primary property of GNKPCs, as discussed in Section 2. The Taylor rules then propagate the inflation gaps into the nominal interest rate differential; the high degrees of policy inertia in the
Taylor rules then cumulatively and persistently feed these positive responses of the inflation gaps into thenominal interest rate differential. In summary, the GNKPCs and high policy inertia jointly imply the highpersistence of the nominal interest rate differential, exogenous to real exchange rate developments. The
expected inflation rate differential response is also highly persistent. Thus, the persistence of the nominalinterest rate differential results in the real interest rate differential. Indeed, the IRF of the latter is almost amirror image of that of the former.
The IRF of the real interest rate differential in Figure 5(a) further reveals an outstanding TI model
property. In their complete market sticky-price models, Steinsson (2008) and Iversen and Söderström(2014) state that to generate a hump-shaped response of the real exchange rate, the response of the realinterest rate differential needs to flip its sign within several quarters of the impact period. Figure 5(a)confirms that this is the case for the TI model. The IRF of the nominal interest rate differential has apersistent hump shape. While the IRF of the expected inflation rate differential is greater than that of thenominal interest rate differential up to 5.5 years, the latter gradually overcomes the former over time. In
particular, as Steinsson (2008) and Iversen and Söderström (2014) claim, the IRF of the real interest rate
differential is negative for up to 5.5 years but turns positive subsequently for long periods. Therefore, as
in Figure 5(b), the real exchange rate appreciates in the impact period because the expected present valueof the future real interest rate differential is positive. The real exchange rate keeps appreciating when the
response of the real interest rate differential remains negative; when the response of the real interest rate
differential becomes zero, the real exchange rate stops appreciating, and when the response of the realinterest rate differential turns positive, the real exchange rate starts depreciating. Consequently, the model
generates a hump-shaped IRF of the real exchange rate to a positive shock to the Canadian trend inflation.
The TI model implies that, other things being equal, monetary easing with a positive trend inflation
shock in Canada results in a sudden rise in the inflation rate differential and real exchange rate appreciationin the impact period. This conditional covariance between the inflation rate differential and the real exchange rate is consistent with the well-known empirical finding of Clarida and Waldman (2008): bad news
23

about inflation is good news for the real exchange rate. In the Canada-U.S. data, the correlation coefficient
between the RCR and inflation differential is -0.417.
Figures 5(c) and (d) show the non-TI model counterparts of the selected IRFs to the Canadian trend
inflation shock. Figure 5(d) shows the non-hump-shaped IRF of the real exchange rate in the non-TI model.
Figure 5(c) reveals that this failure of the non-TI model regarding the IRF of the real exchange rate stems
from the monotonically increasing IRF of the real interest rate differential to zero without sign flipping.
The lack of sign flipping of the IRF of the real interest rate differential results from a sharp increase in
the expected inflation rate differential and a quick convergence of the nominal interest rate to the expected
inflation rate differential due to the weak persistence of the inflation gap differential. The non-TI model
fails to reconcile Clarida and Waldman’s (2008) findings.

4. Conclusion
This paper argues that allowing trend inflation in an otherwise standard two-country NK model
essentially changes the exchange rate dynamics. Positive long-run trend inflation implies a more persistent,
but less volatile, inflation differential with the GNKPCs of the two countries. Simultaneously, the GNKPCs
weaken the link between the inflation differential and the real exchange rate. The RUIP condition, jointlywith the Taylor rule with high policy inertia, generates a nearly permanent real exchange rate with highvolatility. Combined with the less volatile inflation differential, the RCR and NCR move almost one-toone in the equilibrium of the model.
The results of this study highlight the crucial role of monetary policy frameworks in exchange rate
dynamics. In particular, the model identifies small stochastic variation in trend inflation around the long-run
inflation target as the primary driver of volatile and persistent exchange rate fluctuations: under positive
trend inflation, revisions in market participants’ perceptions of the long-term policy goal of the central
bank could be the primary fundamental for the real and nominal exchange rates. This result stands in stark
contrast to that of the influential calibration study by Steinsson (2008), which advocates real shocks to
the Phillips curve to resolve the PPP puzzle. Future research is thus needed to delve into the structuralinterpretations of time-varying trend inflation and inflation targets.
24

The model I present in this paper still lacks many of the theoretical features used to approach exchange rate anomalies emphasized in the recent literature on international relative prices. For example,
even when equipped with strong home bias, the TI model estimates a positive correspondence between thereal exchange rate and the consumption differential in incomplete international financial markets. Thus, it
is essential to address Backus and Smith’s (1993) anomaly. Moreover, the risk-neutral UIP condition nearlyholds along the equilibrium of the TI model. This property implies that the TI model cannot describe an
auto-covariance structure between the expected excess currency return and the nominal interest rate differential frequently and robustly observed in actual data of advanced currencies. To reconcile the empirical
puzzles of Fama (1984) and Engel (2016), future work should augment the proposed TI model to generate
an endogenous currency risk premium consistent with the empirical reality of exchange rates.

Appendix. Data description and construction.
Most data for Canada and the U.S. are distributed by the Federal Reserve Economic Data (FRED), operated by
the Federal Reserve Bank of St. Louis (http://http://research.stlouisfed.org/fred2/). I construct the U.S. consumption
data as the sum of real personal consumption expenditure on non-durables and services. However, FRED distributes
only the nominal values of the two categories of personal consumption expenditure as Personal Consumption Expenditure on NonDurables and Personal Consumption Expenditure on Services. To construct real total personal
consumption expenditure, I first calculate the share of the two nominal consumption categories in nominal total
personal consumption expenditure (Personal Consumption Expenditure) and then multiply real total personal consumption expenditure (Real Personal Consumption Expenditure at Chained 2005 Dollars) by the calculated share.
The U.S. aggregate price data are provided by the Consumer Price Index for All Urban Consumers: All Items in
U.S. City Average. The U.S. nominal interest rate is provided by the Effective Federal Funds Rate. The U.S. labor
supply is provided by Weekly Hours Worked for Manufacturing. All variables except for the nominal interest rate areseasonally adjusted at annual rates and converted into the corresponding per-capita terms by the Total Population.
The Canadian dollar/U.S. dollar nominal exchange rate is distributed as Canada/U.S. Foreign Exchange Rate.
FRED also distributes the Canadian CPI (Consumer Price Index: Total, All Items for Canada) and policy rate
(Immediate Rates: Less than 24 Hours: Central Bank Rates for Canada). The consumption data for Canada are
distributed by Statistics Canada (http://www5.statcan.gc.ca/cansim/). I construct the real consumption data as thesum of Personal Expenditure on NonDurables at Chained 2002 Dollars, Personal Expenditure on Semi-Durables at
Chained 2002 Dollars, and Personal Expenditure on Services at Chained 2002 Dollars. The Canadian labor supply isprovided by Business Sector Average Hours Worked. All variables except for the nominal interest rate are seasonallyadjusted at annual rates and converted into the corresponding per-capita terms by the Estimate of Total Population.

25

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28

Table 1: Prior Distributions: Structural Parameters
Parameters

Density

TI modelMean
Std

Non-TI model
Mean
Std

NormalNormal
Beta
Beta
Beta
Gamma
Gamma
Gamma
Gamma
Beta
Beta
Beta
Beta
Beta
Beta
Beta
InvGamma
InvGamma
InvGamma
InvGamma
InvGamma
InvGamma

1.0122
1.00500.80000.50000.70002.000022.0003.50000.10000.25000.90000.01000.99000.50000.50000.50000.00500.00070.00100.00100.00100.00100.00050.00050.10000.20000.05000.50001.00000.50000.01000.20000.20000.01000.00100.10000.10000.10000.00500.00005
0.00050.00050.00050.0005—1.00500.80000.50000.70002.000011.0003.50000.10000.25000.90000.01000.99000.50000.50000.50000.00500.00070.00100.00100.00100.0010—0.00050.01000.05000.03000.50001.00000.50000.01000.20000.20000.01000.00100.10000.10000.10000.00500.00005
0.00050.00050.00050.0005

NormalNormal
Beta
Beta
Gamma
Gamma
Gamma
Gamma
Beta
Beta
Beta
Beta
Beta
InvGamma
InvGamma
InvGamma
InvGamma
InvGamma

1.01381.00610.80000.50002.000022.0003.50000.10000.05000.90000.99000.50000.50000.00500.00070.00100.00100.00100.00050.00050.10000.20000.50001.00000.50000.01000.01000.20000.00100.10000.10000.00500.00005
0.00050.00050.0005—1.00610.80000.50002.000011.0003.50000.10000.05000.90000.99000.50000.50000.00500.00070.00100.00100.0010—0.00050.01000.20000.50001.00000.50000.01000.01000.20000.00100.10000.10000.00500.00005
0.00050.00050.0005

Canadian Parameters
γτ
γA
µ
h
τ
η
ζ
aπ
ay
as
ρi
λ
ρτ
ρr
ρhp
ρfp
σA
στ
σi
σr
σph
σpf

Mean Trend Inflation Rate
Mean Productivity Growth
Calvo Probability of No Price Resetting
Consumption Habit
Home Bias
Labor Supply Elasticity
Price Elasticity of Final Goods Demand
Taylor Rule Parameter for the Inflation Gap
Taylor Rule Parameter on the Output Gap
Taylor Rule Parameter for NCR
Interest Rate Smoothing Parameter
Error Correction Speed of Productivity Shock
Trend Inflation AR(1) Coefficient (Coef)
Risk Premium Shock AR(1) Coef
Home Market GNKPC Shock AR(1) Coef
Foreign Market GNKPC Shock AR(1) Coef
Productivity Shock Standard Deviation (Std)
Trend Inflation Shock Std
Monetary policy shock Std
Risk Premium Shock Std
Home Market GNKPC Shock Std
Foreign Market GNKPC Shock Std
U.S. Parameters

γτ∗
∗
γA
∗

µ
h∗
η∗
ζ∗
a∗π
a∗y
a∗s
ρ∗i
ρ∗τ
ρh∗
p
ρfp ∗
∗
σA
στ∗
σi∗
σph∗
σpf ∗

Mean Trend Inflation Rate
Mean Productivity Growth
Calvo Probability of No Price Resetting
Consumption Habit
Labor Supply Elasticity
Price Elasticity of Final Goods Demand
Taylor Rule Parameter for the Inflation Gap
Taylor Rule Parameter for the Output Gap
Taylor Rule Parameter on NCR
Interest Rate Smoothing Parameter
Trend Inflation AR(1) Coef
Home Market GNKPC Shock AR(1) Coef
Foreign Market GNKPC Shock AR(1) Coef
Productivity Shock Std
Trend Inflation Shock Std
Monetary Policy Shock Std
Home Market GNKPC Shock Std
Foreign Market GNKPC Shock Std

Note: The subjective discount factor β is calibrated to 0.99 for both countries. The sensitivity of the debt-elastic risk premium ψ is calibrated to
0.00001. The U.S. home bias τ ∗ is calibrated to 0.999 by assuming that the U.S. is a large, nearly closed country.

29

Table 2: Posterior Distributions: Structural Parameters
Parameter

Mean

TI modelStd
95% HPD

Mean

Non-TI model
Std
95% HPD—
1.00490.82970.1631
0.97812.88969.24883.26750.10640.5344
0.94140.00000.99080.50700.69920.68140.00760.00130.00330.00090.00730.00511.1214—0.00020.01070.02050.00330.00910.24370.06700.01060.02200.00630.00000.00070.00830.00710.00470.00060.00010.00030.00040.00090.00080.0037—1.00490.81710.68291.65388.92523.23050.0906
0.05470.9627
0.99200.85550.47940.00500.00090.00200.03740.00301.1260——0.0002 [1.0045 1.0053]
0.0092 [0.8000 0.8356]
0.0173 [0.6506 0.7140]
0.0074 [1.6424 1.6660]
0.2804 [8.3021 9.4182]
0.0468 [3.1491 3.3272]
0.0093 [0.0724 0.1088]
0.0109 [0.0339 0.0760]
0.0041 [0.9546 0.9707]
0.0005 [0.9909 0.9929]
0.0044 [0.8473 0.8640]
0.0048 [0.4711 0.4879]
0.0005 [0.0040 0.0060]
0.0000 [0.0008 0.0010]
0.0002 [0.0016 0.0024]
0.0054 [0.0281 0.0492]
0.0003 [0.0025 0.0036]
0.0042 [1.1191 1.1364]
2801.289

Canadian parameters
γτ
γA
µ
h
τ
η
ζ
aπ
ay
as
ρi
λ
ρτ
ρr
ρhp
ρfp
σA
στ
σi
σr
σph
σpf
Markup rate

1.01201.00350.81120.65980.72212.878222.5393.45880.08210.42050.97370.00120.99140.51630.57420.71590.02170.00090.00310.00100.00320.01281.10920.00020.00030.00440.01770.01210.00100.01050.00800.00840.02800.00260.00020.00080.00280.02580.01160.00170.00010.00030.00050.00030.00110.0033[1.0116 1.0125][1.0028 1.0041][0.8023 0.8186][0.6318 0.6882][0.6985 0.7466][2.8765 2.8797][22.521 22.557][3.4421 3.4732][0.0665 0.0994][0.3747 0.4681][0.9682 0.9783][0.0007 0.0017][0.9897 0.9930][0.5114 0.5213][0.5307 0.6179][0.6946 0.7377][0.0184 0.0250][0.0008 0.0010][0.0025 0.0036][0.0003 0.0020][0.0026 0.0038][0.0107 0.0152][1.1029 1.1159]—[1.0045 1.0053][0.8084 0.8503][0.1172 0.2016][0.9716 0.9842][2.8704 2.9010][8.7312 9.6414][3.1576 3.3637][0.0858 0.1259][0.4862 0.5719][0.9282 0.9525][0.0000 0.0000][0.9896 0.9923][0.4961 0.5229][0.6853 0.7082][0.6737 0.6888][0.0065 0.0088][0.0012 0.0014][0.0026 0.0040][0.0004 0.0016][0.0056 0.0090][0.0035 0.0066][1.1155 1.1291]

U.S. Parameters
γτ∗
∗
γA
∗

µ
h∗
η∗
ζ∗
a∗π
a∗y
a∗s
ρ∗i
ρ∗τ
ρh∗
p
ρfp ∗
∗
σA
στ∗
σi∗
σph∗
σpf ∗
Markup rate
Log Marginal Likelihood

1.01371.00600.77740.5593
1.618222.392
3.51750.0841
0.05300.91700.98720.92510.47620.00170.00060.00130.00070.0023
1.11960.0002 [1.0132 1.0142]
0.0004 [1.0053 1.0067]
0.0046 [0.7679 0.7857]
0.0210 [0.5285 0.5957]
0.0008 [1.6167 1.6197]
0.0165 [22.361 22.420]
0.0230 [3.4776 3.5571]
0.0097 [0.0654 0.1029]
0.0105 [0.0332 0.0736]
0.0072 [0.9030 0.9288]
0.0010 [0.9852 0.9891]
0.0084 [0.9086 0.9403]
0.0163 [0.4384 0.4982]
0.0004 [0.0009 0.0025]
0.0000 [0.0006 0.0007]
0.0001 [0.0011 0.0015]
0.0001 [0.0005 0.0010]
0.0002 [0.0019 0.0026]
0.0035 [1.1122 1.1259]
2905.906

Note 1: “HPD” represents the Bayesian highest probability density intervals.
Note 2: The marginal log likelihoods are estimated based on Geweke’s (1999) harmonic mean estimator.

30

Table 3: Posterior Mean FEVDs (%): Real Exchange Rate
Horizon

+A

+∗A

+∗τ

+τ

+i

+∗i

+hp

+fp

+h∗
p

+fp ∗

+r

2.18
0.85
0.28
0.09

10.4
12.3
6.06
1.99

16.3
17.6
13.0
4.92

7.55
10.0
12.6
14.2

10.43
7.48
2.92
0.95

0.00
0.00
0.00
0.00

23.8
15.3
13.6
13.2

33.1
50.4
49.5
48.3

1.99
5.05
7.26
7.24

0.45
1.44
4.14
5.05

4.67
6.04
5.64
5.50

0.00
0.00
0.00
0.00

(a) TI model
Impact
1 yr
3 yr
10 yr

5.38
13.0
21.3
22.9

0.02
0.02
0.06
0.08

23.5
25.2
28.2
36.5

4.67
7.65
11.3
14.5

19.3
5.75
4.17
3.45

(b) Non-TI model
Impact
1 yr
3 yr
10 yr

0.13
0.21
0.43
1.22

1.10
0.52
0.54
0.86

3.19
1.97
1.83
1.86

5.38
3.68
3.31
3.27

26.1
15.3
13.7
13.3

Note: Part (a) reports the posterior means of the FEVDs of the real exchange rate in the structural shocks at the impact period, 1-, 3-,
and 10-year forecast horizons inferred by the TI model; and Part (b) reports those for the non-TI model.

Table 4: Posterior Predictive Analysis for Exchange Rate Moments
Std(∆ ln qt )Std(∆ ln St )α
0.954
[0.879 1.000]

HL
4.425
{1.600 6.000}

Corr
0.932
{0.820 0.990}

0.956{0.870 1.040}

0.961

5.250

0.957

1.037

TI model95% HPD

0.939[0.912 0.963]

4.615[3.365 6.000]

0.980[0.973 0.986]

1.029[1.022 1.046]

Non-TI model
95% HPD

0.747[0.684 0.808]

1.245[1.050 1.450]

0.883[0.858 0.907]

1.043[0.989 1.101]

Empirical
90% CI or {min, max}
Canada-U.S. data

Note 1. α is the sum of the AR coefficients in the AR(5) process of ln qt estimated using the ADF regression.
Note 2. Half-life (HL) is the maximum period at which the IRF of ln qt is greater than 0.5, measured in years.
Note 3. Corr represents the correlation coefficient between ∆ ln qt and ∆ ln St .
Std(∆ ln qt )Note 4. Std(∆
represents the ratio of the standard deviations of ∆ ln qt to ∆ ln St .
ln St )
Note 5. In the “Empirical” row, α comes from Steinsson’s (2008) estimate using the U.S. trade-weighted real exchange rate. The otherstatistics are from Burstein and Gopinath (2014). Both report the average point estimates for eight advanced countries.

31

(a) Canada

0.035

0.03

0.03

0.025

0.025

0.02

0.02

0.015

0.015

0.01

0.01

0.005

0.005

0

0

-0.005-0.005

-0.01

-0.01

-0.015
1982:Q2

1987:Q1

1992:Q1(b) U.S.

0.035
Actual Inflation Data
Posterior Mean
95% HPD
95% HPD

1997:Q1

2002:Q1

2007:Q1-0.015
1982:Q2

1987:Q1

1992:Q1

1997:Q1

2002:Q1

2007:Q1

Figure 1: Posterior Inferences of Kalman-smoothed Trend Inflation

0.08
Real Currency Return: Data
Trend Inflation: CAN
Trend Inflation: US
Productivity: CAN
Productivity: US
Monetary policy: CAN
Monetary policy: US

60

40

0.06

20

0.02

0

0

-0.02

-20

-0.04
-40

-0.06
-60

1982:Q2

1987:Q1

1991:Q1

1997:Q1

2002:Q1-0.08
2007:Q1

Figure 2: Historical Decomposition of Real Currency Return into Structural Shocks

32

Real Currency Return (demeaned)

Historical decomposition (%)

0.04

% Deviation From Steady State

0

10

20

30

40

10

20

30

40

30

40

periods (quarter)

30

40

0

20

30

40

30

-0.04
0

0

20

30

20

30

periods (quarter)

10

TI modelNon-TI model(k) Risk Premium Shock10

Figure 3: IRFs of the Real Exchange Rate to Structural Shocks

periods (quarter)

10

-2

20

-2

10

0

-1.5

0

-0.02

2

-1

-0.03

-0.01

4

-0.5

0

6

(j) US GNKPC Shock to US Price

-6

20

-2

0

2

0
10

20

40

40

40

0

10

20

30

40

-10

0

10

20

30

40

(d) US Trend Inflation Shock

0

10

20

30

40

(g) CN GNKPC Shock to CN Price (h) US GNKPC Shock to CN Price

-4

0

10

0

-30
-40

10

-20

30

40

20

0

(c) CN Trend Inflation Shock

-10

0

10

2

4

6

(f) US Monetary Policy Shock

80(b) US Productivity Shock0(i) CN GNKPC Shock to US Price

-6

-4

-2

0

2

4

(e) CN Monetary Policy Shock

40

-1

30

0
20

-0.5

0.5

10

0

1

0

0.5

(a) CN Productivity Shock

1.5% Deviation From Steady State% Deviation From Steady State

33

Data
TI modelNon-TI model

Figure 4: Reduced-Form IRFs of the Real Exchange Rate(a) TI Model% Deviation From Steady State

2

Real Exchange Rate

1.5

-5

1

-10

0.5

-15

0

-20

-0.5

Nominal Interest Rate
Expected Inflation Rate
Real Interest Rate
Inflation Gap

-1
-1.5

0

5

10

15

20

25

30

35

(c) Non-TI Model

2% Deviation From Steady State(b) TI Model

0

-25
-30
-35

0

5

10

15

20

25

30

35

30

35

(d) Non-TI Model

3.5
3

1.5

2.5

1

2
0.5
1.5
0

1

-0.5
-1

0.5
0

5

10

15

20

25

30

35

0

0

5

10

15

20

25

Figure 5: IRFs of Two-country Differentials in the Nominal Interest Rate, Expected Inflation Rate,
Inflation Gap, and Real Interest Rate to a Canadian Trend Inflation Shock34

Online Appendix
Trend Inflation and Exchange Rate Dynamics:
A New Keynesian Approach
Takashi Kano

Graduate School of Economics
Hitotsubashi University
Naka 2-1, Kunitachi, Tokyo
186-8608, JAPANTel/fax: +81-42-580-8283
Email : tkano@econ.hit-u.ac.jp
Current Draft: April 26, 2023
Not intended for publication

Appendix A: The two-country new Keynesian model
A.1. Household sectors
There are home and foreign countries in this model. Throughout this paper, any variable with an asterisk corresponds to a foreign variable, while a variable without an asterisk denotes the home counterpart.
The two countries are endowed with representative households whose objectives are the following lifetime
utility functions:∞!
j=0

β j Et"(Nt+j )1+ηln(Ct+j − hC̄t+j−1 ) −
1+η

#

and

∞
!

β j Et

j=0

$

∗
(Nt+j
)1+η
∗
∗
ln(Ct+j− h∗ C̄t+j−1)−
1 + η∗∗%,

for the home and foreign countries, respectively, where Ct and Nt represent the home country’s consumption basket and hours worked, and Ct∗ and Nt∗ represent their foreign country’s counterparts. The twocountries share the same subjective discount factor β ∈ (0, 1) and have different Frisch elasticities of labor
supply η and η ∗ > 1.1
The representative households in the two countries consume both home (h) and foreign (f ) final
goods. Home consumption basket Ct consists of home and foreign product aggregators Ch,t and Cf,t ,∗∗while foreign consumption basket Ct∗ consists of home and foreign product aggregators Ch,tand Cf,t
:
& 1
' ζ
ζ−1
ζ−1 ζ−11ζζζζCt = τ (Ch,t )+ (1 − τ ) (Cf,t )

and

Ct∗ =

&

∗

(1 − τ )

1
ζ∗

∗
(Ch,t
)

ζ ∗ −1
ζ∗

+τ

∗ ζ1∗

∗
(Cf,t
)

ζ ∗ −1ζ∗∗' ζ ∗ζ−1

where ζ and ζ ∗ > 0 are the elasticities of substitution between the home and foreign product aggregators,and τ and τ ∗ are the degrees of home bias. The static cost minimization problems of the home and foreign∗∗
households derive the demand functions for Ch,t , Cf,t , Ch,t
, and Cf,t:
Ch,t = τ

(

Ph,t
Pt

∗
Ch,t
= (1 − τ ∗ )

(

)−ζ

∗
Ph,t
Pt∗

Cf,t = (1 − τ )

Ct ,

)−ζ ∗

and

Ct∗ ,

(

Pf,t
Pt

∗
Cf,t
= τ∗

(

)−ζ

∗
Pf,t
Pt∗

Ct ,

)−ζ ∗

Ct∗ ,

where Pt and Pt∗ are the aggregate consumer price indices (CPIs) of the home and foreign countries:*+ 1
Pt = τ (Ph,t )1−ζ + (1 − τ )(Pf,t )1−ζ 1−ζ

*+ 1∗ 1−ζ ∗∗ 1−ζ ∗ 1−ζ ∗Pt∗ = (1 − τ ∗ )(Ph,t)+ τ ∗ (Pf,t
)
.

and

Here, Ph,t is the aggregate price over the home goods in the home country, Pf,t the aggregate price over1

Because of the stochastic trends due to the permanent labor productivity shocks in the two countries, the model assumes
the log-utility functions over consumption to guarantee the existence of a balanced growth path in the two-country equilibrium.

1

,

∗
the foreign goods in the home country, Ph,t
the aggregate price over the home goods in the foreign country,∗
and Pf,t
the aggregate price over the foreign goods in the foreign country.
∗∗
The home and foreign aggregators of the home and foreign final goods, Ch,t , Cf,t , Ch,t
, and Cf,t,consist of a continuum of the home and foreign final goods, each of which is produced by a monopolisticallycompetitive firm residing either in the home or in the foreign countries. The home and foreign final good
aggregators in the home country are of the following Dixit–Stiglitz type: for j = {h, f },Cj,t =(, 1

Cj,t (z)

ζ−1
ζ

dz

0

ζ
) ζ−1

∗
and Cj,t
=

(, 1
0

∗
Cj,t
(z)

ζ ∗ −1ζ∗dz∗) ζ ∗ζ−1,∗
where Cj,t (z) and Cj,t(z) are home and foreign demand for the particular home and foreign final goodindexed by z ∈ [0, 1], and ζ and ζ ∗ > 0 represent the home and foreign price elasticities of demand foreach final good.
The static cost minimization problems of the representative households in the two countries derive
the two countries’ demand functions for the home and foreign final goods. Given the home and foreign∗∗
prices of the home and foreign goods indexed by z, Ph,t (z), Pf,t (z), Ph,t(z), and Pf,t(z), they are

Cj,t (z) =(Pj,t (z)
Pj,t

)−ζ

Cj,t

and

∗
Cj,t
(z) =

(

∗
Pj,t
(z)
∗
Pj,t

)−ζ ∗

∗
Cj,t
,

∗
∗
for j = {h, f }. Price indices Ph,t , Pf,t , Ph,t
, and Pf,t
are then given as

Pj,t =(, 1
0

Pj,t (z)

1−ζ

dz

1
) 1−ζ

and

∗
Pj,t
=

(, 1
0

∗
∗
Pj,t
(z)1−ζ dz) 1−ζ1 ∗,for j = {h, f }.
The home representative household maximizes the lifetime utility subject to the following budget
constraint:
Bh,t + St Bf,t + Pt Ct ≤ (1 + ih,t−1 )Bh,t−1 + St (1 + if,t−1 )Bf,t−1 + Wt Nt + Λt ,where Bh,t , Bf,t , ih,t , if,t , Wt , Λt , and St denote the home country’s holdings of the home country’s nominal
bonds, the home country’s holdings of the foreign country’s nominal bonds, the home country’s nominalinterest rate for the home country’s bonds, the home country’s nominal interest rate for the foreign country’s bonds, the home country’s nominal wage, the static profits from the home country’s monopolisticallycompetitive firms, and the bilateral nominal exchange rate of the home currency per the foreign country,
respectively. Similarly, the representative household in the foreign country maximizes the lifetime utility

2subject to the following budget constraint:∗∗
Bh,tBh,t−1∗∗+ Bf,t+ Pt∗ Ct∗ ≤ (1 + i∗h,t−1 )+ (1 + i∗f,t−1 )Bf,t−1+ Wt∗ Nt∗ + Λt∗ ,
St
St
∗
∗
where Bh,t
, Bf,t, i∗h,t , i∗f,t , Wt∗ , and Λ∗t denote the foreign country’s holdings of the home country’s nominal bonds, the foreign country’s holdings of the foreign country’s nominal bonds, the foreign country’s
nominal interest rate for the home country’s bonds, the foreign country’s nominal interest rate for the foreign country’s bonds, the foreign country’s nominal wage, and the static profits from the foreign country’s
monopolistically competitive final good firms.
∗
Define habit-adjusted consumption Ht ≡ Ct − hC̄t−1 and Ht∗ ≡ Ct∗ − h∗ C̄t−1. The first-ordernecessary conditions (FONCs) for the home and foreign households’ lifetime utility maximization problems
consist of the home and foreign Euler equations:11= β(1 + ih,t )Et
Pt H tPt+1 Ht+1

and1
Pt∗ Ht∗= β(1 + i∗f,t )Et

1
∗
∗
Pt+1
Ht+1

;

the home and foreign utility-based uncovered interest parity conditions (UIPs):1
Et
Pt+1 Ht+1"#
St+1(1 + ih,t ) − (1 + if,t )
=0
St

and

1
Et ∗
∗
Pt+1 Ht+1

"
#
St
∗
∗
(1 + if,t ) − (1 + ih,t )
= 0;
St+1

and the home and foreign intratemporal optimality conditions for hours worked
Ntη =

Wt
Pt H t∗and Nt∗η =Wt∗.
Pt∗ Ht∗Suitable transversality conditions for international bond holdings should be satisfied.
A.2. Final good sectors
Facing the corresponding demand functions in the home and foreign countries, a final good firmwith index z ∈ [0, 1] acts as a monopolistically competitive price setter. When setting its current optimal
price, each firm follows a Calvo-type time-dependent pricing strategy: in each period, a firm cannot reset
its optimal price with a probability µ ∈ (0, 1]. Moreover, to generate endogenous fluctuations of the realexchange rate in this model, I assume that each firm adopts the local currency pricing strategy as in Betts
and Devereux (1996, 2000): each firm sets its optimal price differently between the two countries in termsof the local currencies.
Under such a local currency pricing–Calvo pricing strategy, the objective function for the home firm’s

3profit maximization problem is
max EtPh,t ,P∗h,t∞!

iµ Γt+i

i=0

"(

Ph,t
Ph,t+i)− mct+i

#(

+ Et

Ph,t
Ph,t+i∞!

)−ζµi Γt+ii=0Ch,t+i"(

St+i P∗h,tPh,t+i)− mct+i

#-

P∗h,t
∗
Ph,t+i.−ζ ∗∗Ch,t+i,where Ph,t , P∗h,t , mct , and Γt are the optimal price of the home good in the home country, the optimal priceof the home good in the foreign country, the real marginal cost of the home firm, and the home country’s
real stochastic discount factor Γt+i ≡ β i (Ht /Ht+i ), respectively. Similarly, the objective function for theforeign firm’s profit maximization problem is

max EtPf,t ,P∗f,t∞!

µ∗i Γ∗t+i

i=0

$-

Pf,t
∗
St+i Pf,t+i

.

%(

)−ζ
Pf,t
Cf,t+iPf,t+i$.
%.−ζ ∗
∞
∗
∗
!
P
P
f,t
f,t
∗
+ Et
µ∗i Γ∗t+i− mc∗t+i
Cf,t+i
,
∗
∗
P
P
f,t+i
f,t+i
i=0

− mc∗t+i

where Pf,t , P∗f,t , mc∗t , and Γ∗t are the optimal price of the foreign good in the home country, the optimal
price of the foreign good in the foreign country, the real marginal cost of the foreign firm, and the foreign∗country’s real stochastic discount factor Γ∗t+i = β i (Ht∗ /Ht+i), respectively.
The FONCs for the optimal prices that the home firm sets for the home and foreign countries are
(
)−ζ
∞
!
ζ
1
i
Ph,t Etµ Γt+iCh,t+i =Etµ Γt+i mct+i
Ch,t+i ,Ph,t+iζ − 1 i=0
Ph,t+ii=0.1−ζ ∗
.−ζ ∗
(
)∞
∞
∗
∗
!
!
S
P
1
ζ
1
t+i
h,t+i
∗
∗
P∗h,t Et
µi Γt+iCh,t+i= ∗Etµi Γt+i mct+i
Ch,t+i
,
∗
∗
P
P
ζ
−
1
P
h,t+i
h,t+i
h,t+i
i=0
i=0
∞
!

i

(

1

)1−ζ

respectively. Similarly, the FONCs for the optimal prices that the foreign firm sets for the home and foreigncountries are
.(
)1−ζ
(
)−ζ
∞
∞
!
!
P
1
ζ
1
f,t+i
∗i ∗
∗i ∗
∗
Pf,t Etµ Γt+iCf,t+i =Etµ Γt+i mct+i
Cf,t+i ,∗St+i Pf,t+i
Pf,t+iζ − 1 i=0
Pf,t+ii=0.1−ζ ∗
.−ζ ∗
∞
∞
∗
!
!
1
ζ
1
∗
∗
∗
P∗f,t Etµ∗i Γ∗t+i
Cf,t+i= ∗Etµ∗i Γ∗t+i mct+i
Cf,t+i
.
∗
∗
P
ζ
−
1
P
f,t+i
f,t+i
i=0
i=0

4

∗
∗
Given the optimal prices, price indices Ph,t , Pf,t , Ph,t
, and Pf,tfollow the laws of motion:
1−ζ
1−ζ
Pj,t
= (1 − µ)P1−ζ
j,t + µPj,t−1 ,

∗

∗

∗

∗1−ζ
∗1−ζ
and Pj,t= (1 − µ∗ )P∗1−ζ+ µ∗ Pj,t−1,
j,twhere j = {h, f }.
Each firm produces its final good by using labor input hired from a domestic competitive labormarket. The production functions of the home and foreign goods are Yt (z) = At Lt (z) and Yt∗ (z) =A∗t L∗t (z), where At and A∗t are the labor productivities in the home and foreign countries. In this case, thereal marginal costs that the home and foreign firms face are
mct =

Wt
At Ph,tand mc∗t =Wt∗.
∗
A∗t Pf,t

The home and foreign static profits are, 1∗∗{Ph,t (z)Ch,t (z) + St Ph,t(z)Ch,t(z) − Wt Nt (z)}dz0()−ζ( ∗ )−ζ ∗
Ph,t
Ph,t
∗
∗
= Ph,t τ
Ct + St Ph,t (1 − τ )Ct∗ − Wt Nt ,
∗
Pt
Pt
, 1
∗
∗
St Λ∗t ≡{Pf,t (z)Cf,t (z) + St Pf,t(z)Cf,t(z) − St Wt∗ Nt∗ (z)}dz0()−ζ( ∗ )−ζ ∗
Pf,t
Pf,t
∗ ∗
= Pf,t (1 − τ )Ct + St Pf,t τCt∗ − St Wt∗ Nt∗ ,
∗
Pt
Pt

Λt ≡

respectively.
I assume that the logarithms of labor productivities, ln At and ln A∗t , are of I(1). To guarantee abalanced growth path of the two-country model, the labor productivity differential ln at ≡ ln At − ln A∗tshould be of I(0). As investigated by Mandelman et al. (2011), Rabanal et al. (2011), and Ireland (2013),
the I(1) labor productivities and stationary productivity differential jointly imply that the labor productivityof the home country must be cointegrated with that of the foreign country. Following Kano (2020), I∗
assume that the home and foreign growth rates of labor productivity, γA,t ≡ ln At − ln At−1 and γA,t≡ln A∗t − ln A∗t−1 , are generated by error correction processes:
ln γA,t = ln γA −λln at−1 + 'A,t ,
2

and

∗
ln γA,t= ln γA∗ +λln at−1 + '∗A,t ,2

where γA , γA∗ > 1 are the drift terms and λ ∈ [0, 1) are the speed of error correction. The error correction
mechanism implies that the cross-country labor productivity differential is of I(0) because
ln at = ln γA /γA∗ + (1 − λ) ln at−1 + 'A,t − '∗A,t .
5

Importantly, if the adjustment speed λ is sufficiently close to zero, the cross-country labor productivity
differential follows a near I(1) process.
A.3. Monetary policies with trend inflation
The monetary policies in the two countries are characterized by Taylor rules. The central banks ofthe two countries set their short-term domestic nominal interest rate as (1 + ih,t ) and (1 + i∗f,t ) dependingon the past interest rate levels (1 + ih,t−1 ) and (1 + i∗f,t−1 ), current inflation rates γπ,t ≡ Pt /Pt−1 and∗∗γπ,t≡ Pt∗ /Pt−1, and detrended output levels yt ≡ Yt /At and yt∗ ≡ Yt∗ /A∗t . In this paper, I allow for∗∗∗stochastic trend inflation γτ,t ≡ Pτ,t /Pτ,t−1 and γτ,t≡ Pτ,t/Pτ,t−1in the two countries, where Pτ,t and∗Pτ,t are the exogenous permanent components of the aggregate price levels of the two countries generated
by trend inflation. As specified by Ireland (2007) and Cogley et al. (2010), the central banks of the twocountries target the time-varying trend inflation levels in the Taylor rules:(1 + ih,t ) = (1 + i)1−ρi (1 + ih,t−1 )ρi


/(

γπ,t
γτ,t

)a π

(yt )ay(qt γπ,t∗qt−1 γπ,tas 01−ρi) 1−asexp('i,t ), ∗( ∗ )a∗π()− a∗s ∗ 1−ρi
1−aγsqt γπ,t∗∗∗π,t(1 + i∗f,t ) = (1 + i∗ )1−ρi (1 + i∗f,t−1 )ρi  ∗(yt∗ )ay
exp('∗i,t ),∗γτ,tqt−1 γπ,twhere i and i∗ are the deterministic steady-state values of the home and foreign nominal interest rates andρi and ρ∗i ∈ (0, 1) capture the home and foreign degrees of interest rate smoothing. In this paper, the trend∗
inflation rates γτ,t and γτ,tfollow exogenous AR(1) processes in the logarithmic term:ln γτ,t = (1 − ρτ ) ln γτ + ρτ ln γτ,t−1 + 'τ,t

and

∗
∗
ln γτ,t= (1 − ρ∗τ ) ln γτ∗ + ρ∗τ ln γτ,t−1+ '∗τ,t ,where γτ and γτ∗ are the long-run means of the home and foreign trend inflation rates, ρτ and ρ∗τ ∈ [0, 1) are
the AR(1) roots of the home and foreign trend inflation rates, and 'τ,t and '∗τ,t are the i.i.d. trend inflation
shocks.2 As in a closed-economy NK model by Ascari and Sbordone (2014), the time-invariant unconditional means γτ and γτ∗ make the log-linearization exercise of this study tractable when characterizing thedeterministic steady state.3
I assume that the monetary policy disturbances in the home and foreign countries, 'i,t and '∗i,t , are i.i.d.
shocks. Coibion and Gorodnichenko (2012) provide a series of empirical evidence that the large persistence
observed in the actual data on the short-run nominal interest rate in the United States stems largely from2

Cogley et al. (2010) discuss as a primary reason for the stochastic variations in the central banks’ long-run inflation targets that the central banks only imperfectly know the true economic structure and their learning process generates endogenous
updating of the inflation targets.
3
Ascari and Sbordone (2014) do not allow for stochastic variations in trend inflation.

6the policy inertia caused by interest rate smoothing following the Taylor rule, but not from the exogenous
persistence of the monetary policy disturbance, as claimed by Rudebusch (2002, 2006). In particular, theirempirical result suggests that the time-varying inflation target that the conventional Taylor rule misses
could lead to a high estimate of the serial correlation of the monetary policy disturbance. Indeed, thepersistent trend inflation shock γτ,t is indistinguishable from a persistent monetary policy disturbance if 'i,tis set to −(1 − ρi )aπ ln γτ,t . Following the above notion, I allow for both interest rate smoothing and thetime-varying inflation target in the Taylor rules in this study.
A.4. Market clearing and productivity shocks
To guarantee a stationary distribution of the net foreign asset positions of the two countries within
incomplete international financial markets, I allow for a debt-elastic risk premium in the interest rates faced
only by the home country: for j = {h, f },¯ − 1}] exp(vr,t ),ij,t = i∗j,t [1 + ψ{exp(−bj,t + d)d¯ ≤ 0,

ψ > 0,

where bj,t is the transitory component of the home country’s holdings of country j’s bonds, which is precisely defined below. The home country needs to pay a risk premium over the interest rate level the foreign
household faces when the transitory component of the home country’s net borrowing positions bj,t < 0 is
¯ The risk premium is given as an externality: the household does not take into
beyond its threshold level d.
account the effect of the debt position on the risk premium when maximizing its lifetime utility function.
As in Bergin (2006) and Justiniano and Preston (2010), the risk premium is also subject to exogenous risk
premium shock vr,t , which follows an AR(1) process:ln vr,t = ρr ln vr,t−1 + 'r,t .
On the contrary, I do not attach a risk premium to the foreign country’s interest rates.4
The market-clearing conditions of the two countries’ bond markets are∗
Bh,t + Bh,t=0∗
and Bf,t + Bf,t
= 0.

In other words, along an equilibrium path, the world net supply of nominal bonds is zero on a period-by-

4

The elasticity of the risk premium toward the debt position, ψ, is set to a small number 0.00001.

7

period basis. The market-clearing conditions of the home and foreign final goods are
At Nt =A∗t Nt∗ =, 1∗∗{Ch,t (z) + Ch,t(z)}dz = Ωh,t Ch,t + Ω∗h,t Ch,t
,

,0 1
0

∗
∗
{Cf,t (z) + Cf,t(z)}dz = Ωf,t Cf,t + Ω∗f,t Cf,t,where the variables Ωh,t , Ω∗h,t , Ωf,t , and Ω∗f,t , which are defined byΩj,t ≡, 1(
0

Pj,t (z)
Pj,t

)−ζ

dz,

and

Ω∗j,t ≡

, 1(
0

∗
Pj,t
(z)
∗
Pj,t

)−ζ ∗

dz,

for j = h, f

capture the degrees of price dispersion in the four final good markets. As discussed by Ascari and Sbordone(2014), the price dispersion variables are greater than one under price stickiness, but are one if all the
prices are identical within each final good market in each country under flexible price adjustments. The
market-clearing conditions then imply that variables Ωj,t and Ω∗j,t represent the resource costs of pricedispersion: given the output level, the higher the price dispersion variable, the lower the amount allocatedto consumption. It is shown that the price dispersion variables follow the transitions
(

)−ζ
(
)ζ
Pj,t
Pj,t
Ωj,t = (1 − µ)+µΩj,t−1 ,Pj,tPj,t−1( ∗ )−ζ ∗( ∗ )ζ ∗
Pj,t
Pj,t
∗
∗
∗
Ωj,t = (1 − µ )+µΩ∗j,t−1 ,∗∗Pj,tPj,t−1for j = h, f .5

Appendix B: log-linear approximation of FONCs∗
Because the model contains non-stationary components, At , A∗t , Pτ,t , and Pτ,t
, I stochastically detrend the FONCs by these stochastic trend components to characterize the unique deterministic steadystate. In doing so, I define the stochastically detrended versions of home consumption by ct ≡ Ct /At ;foreign consumption c∗t ≡ Ct∗ /A∗t ; home consumption habit ht ≡ Ht /At ; foreign consumption habith∗t ≡ Ht∗ /A∗t ; the home price of home goods ph,t ≡ Ph,t At /Pτ,t ; the home price of foreign goods∗∗
pf,t ≡ Pf,t At /Pτ,t ; the foreign price of home goods p∗h,t ≡ Ph,tA∗t /Pτ,t; the foreign price of foreign goods∗∗p∗f,t ≡ Pf,tA∗t /Pτ,t; the optimal home price of home goods ph,t ≡ Ph,t At /Pτ,t ; the optimal home price∗of foreign goods pf,t ≡ Pf,t At /Pτ,t ; the optimal foreign price of home goods p∗h,t ≡ P∗h,t A∗t /Pτ,t; the∗
optimal foreign price of foreign goods p∗f,t ≡ P∗f,t A∗t /Pτ,t; the home CPI pt ≡ Pt At /Pτ,t ; the foreign
5

The full derivation of the transition equations of the price dispersion variables is found in Ascari and Sbordone (2014).

8

∗
CPI p∗t ≡ Pt∗ A∗t /Pτ,t; the home holdings of the home bond bh,t = Bh,t /Pτ,t ; the home holdings of the∗∗foreign bond bf,t = Bf,t /Pτ,t; the foreign holdings of the home bond b∗h,t = Bh,t/Pτ,t ; the foreign hold∗∗∗ings of the foreign bond bf,t ≡ Bf,t /Pτ,t ; the home nominal wage wt ≡ Wt /Pτ,t ; the foreign nominal∗∗
wage wt∗ ≡ Wt∗ /Pτ,t; and the nominal exchange rate st ≡ St Pτ,t/Pτ,t . The real exchange rate is given by∗∗
qt ≡ St Pt /Pt = st at pt /pt .

B.1. Stochastically detrended FONCs
The stochastically detrended versions of the FONCs for the home country consist of the budget
constraint(1 + ih,t−1 )bh,t−1 (1 + if,t−1 )st bf,t−1
pt ct +bh,t +st bf,t =++ph,t τ
∗
γτ,t
γτ,t

(

ph,t
pt

)−ζ

∗
ct +st ph,t(1−τ ∗ )( ∗ )−ζ ∗
ph,t
c∗t ;
p∗t
(B.1)

the home intratemporal optimality conditionNtη =the home Euler equation1= β(1 + ih,t )Et
pt h tthe home UIP conditionst (1 + ih,t )Et(1
pt+1 ht+1 γτ,t+1)

wt;
pt h t((B.2)

1γτ,t+1 pt+1 ht+1

= (1 + if,t )Et

(

)

(B.3)

;

st+1
∗
pt+1 ht+1 γτ,t+1

)

;

(B.4)

the debt elastic risk premium that the home country faces¯ − 1}] exp(vr,t ),ih,t = i∗h,t [1 + ψ{exp(−bh,t + d)

¯ − 1}] exp(vr,t );and if,t = i∗f,t [1 + ψ{exp(−bf,t + d)
(B.5)

the market clearing condition for the home country
Nt = Ωh,t ch,t + Ω∗h,t c∗h,t a−1
t ;

9

(B.6)

the transition equations of price dispersions Ωh,t and Ω∗h,tΩh,t = (1 − µ)

(

ph,t
ph,t

)−ζ

+µ

(

and

ph,t
ph,t−1

)ζ (

γτ,t
γA,t

)ζ

Ω∗h,t = (1 − µ∗ )-Ωh,t−1 ,

p∗h,t
p∗h,t

.−ζ ∗

+ µ∗

-

p∗h,t
p∗h,t−1.ζ ∗ -

∗
γτ,t
∗
γA,t

.ζ ∗

Ω∗h,t−1 ; (B.7)

the optimal price setting rules of the home firm for the home and foreign markets
(
)
(
)
∞
∞
!
ph,t !
ch,t+iζ
ch,t+ii ζ−1i ζEt(βµ) φh,t+i=Et(βµ) φh,t+i mct+i
, and
ph,t
h
ζ
−
1
h
t+i
t+i
i=0
i=0
( ∗ )
( ∗ )
∞
∞
∗
∗
!
!
ph,t
s p
ch,t+ich,t+iζ∗
i t+i h,t+i ∗ζ ∗ −1i ∗ζ ∗ mct+iEt(βµ)φh,t+i= ∗Et(βµ) φh,t+i,∗
ph,tph,t+iht+i
ζ − 1 i=0
at+i
ht+i
i=0

(B.8)
(B.9)

where cumulative inflations φt , φh,t , and φ∗h,t are defined as
φt+i ≡ Πis=1

(

Pt+s
Pt

)

(

)
(
) (
)
(
)
γτ,t+sPh,t+sph,t+iγτ,t+sii=, φh,t+i ≡ Πs=1=
Πs=1,
γA,t+sPh,tph,tγA,t+s
. .
.
∗
∗
∗
P
p
γ
h,t+s
h,t+i
τ,t+s
and φ∗h,t+i ≡ Πis=1
=
Πis=1
;
(B.10)
∗
∗
Ph,t
p∗h,t
γA,t+s
pt+i
pt

)

Πis=1

(

the home and foreign demand functions for the home goodch,t = τ

(

ph,t
pt

)−ζ

ct ,

and

c∗h,t = (1 − τ ∗ )( ∗ )−ζ ∗
ph,t
c∗t ;
∗
pt

(B.11)

the home CPI1−ζ
p1−ζ
= τ p1−ζth,t + (1 − τ )pf,t ;(B.12)

the laws of motion of the home final good prices at the home and foreign countries
1−ζ
1−ζ
p1−ζ
h,t = (1−µ)ph,t +µph,t−1

(

γA,t
γτ,t

)1−ζ

,

and

∗
∗
∗
p∗1−ζ
= (1−µ)p∗1−ζ+µp∗1−ζ
h,t
h,t
h,t−1

the real marginal cost of the home firm
mct =

10

wt
;
ph,t

( ∗ )1−ζ ∗γA,t; (B.13)∗γτ,t(B.14)

the Taylor rule the home central bank follows(1 + ih,t ) = (1 + i)1−ρi (1 + ih,t−1 )ρi

/(

pt −1
γ
pt−1 A,t

)a π

a
Nt y

(

qt
qt−1

as (
) 1−a
s

γπ,t
∗
γπ,t

as 01−ρi) 1−asexp('i,t ); (B.15)

and home consumption habit(B.16)−1
ht = ct − hγA,tct−1 .

Similarly, the stochastically detrended versions of the FONCs for the foreign country consist of thebudget constraint(1 + i∗h,t−1 )bh,t−1 (1 + i∗f,t−1 )st bf,t−1
qt pt c∗t
−bh,t −st bf,t = −−+pf,t (1−τ )
∗
at
γτ,t
γτ,t

(

pf,t
pt

)−ζ

ct +st p∗f,t τ ∗( ∗ )−ζ ∗
pf,t
c∗t ;
∗
pt
(B.17)

the foreign intratemporal optimality condition∗

Nt∗η =
the foreign Euler equationwt∗ st at;
qt pt h∗t

(B.18)

at s tat+1 st+1= β(1 + i∗f,t )Et ∗;∗qt p t h tγτ,t+1 qt+1 pt+1 h∗t+1(B.19)

the foreign UIP conditionst (1 + i∗h,t )Et(

at+1
qt+1 pt+1 h∗t+1 γτ,t+1)= (1 + i∗f,t )Et()
at+1 st+1. ,∗qt+1 pt+1 h∗t+1 γτ,t+1(B.20)

the market clearing condition for the foreign country(B.21)

Nt∗ = Ωf,t at cf,t + Ω∗f,t c∗f,t ;the transition equations of price dispersions Ωf,t and Ω∗f,t∗Ωf,t = (1 − µ )

(

pf,t
pf,t

)−ζ

+µ

and

∗

(

pf,t
pf,t−1

)ζ (

γτ,t
γA,t
-

Ω∗f,t = (1 − µ∗ ))ζΩf,t−1 ,

p∗f,t
p∗f,t

11

.−ζ ∗

+ µ∗

-

p∗f,t
p∗f,t−1.ζ ∗ -

∗
γτ,t
∗
γA,t

.ζ ∗

Ω∗f,t−1 ; (B.22)

the optimal price setting rules of the foreign firm
(
)
(
)
∞
∞
!
pf,t !
cf,t+iζ
cf,t+iζ−1ζ∗ i pf,t+i
∗ i
∗
Et
(βµ )
φ
=
Et
(βµ ) mct+i at+i φf,t+i,
pf,tst+i p∗f,t+i f,t+i h∗t+i
ζ − 1 i=0
h∗t+i
i=0

and

( ∗ )
( ∗ )
∞
∞
!
p∗f,t !
cf,t+icf,t+iζ∗∗ζ ∗∗ i ∗ζ ∗ −1
∗ i∗Et(βµ ) φf,t+i= ∗Et(βµ ) mct+i φf,t+i
,
∗
∗
∗
pf,t
h
ζ
−
1
h
t+i
t+i
i=0
i=0

(B.24)(B.23)

where cumulative inflations φ∗t , φf,t , and φ∗f,t are defined asφ∗t+i ≡ Πis=1

(

∗
Pt+s
Pt∗

)

.
( ∗ )
(
) (
)
(
)
∗
γ
pt+i
Pf,t+spf,t+iγτ,t+s
τ,t+s
i
i
i
=
Πs=1
, φf,t+i ≡ Πs=1
=
Πs=1
,
∗
p∗t
γA,t+sPf,tpf,tγA,t+s
. .
.
∗
∗
∗
P
p
γ
f,t+s
f,t+i
τ,t+s
and φ∗f,t+i ≡ Πis=1
=
Πis=1
;
(B.25)
∗
∗
∗
Pf,t
pf,t
γA,t+sthe demand functions for the foreign goods
cf,t = (1 − τ )

(

pf,t
pt

)−ζ

( ∗ )−ζ ∗pf,tc∗f,t = τ ∗
c∗t ;
p∗t

and

ct ,

(B.26)

the foreign CPI∗∗∗(B.27)p∗1−ζ= (1 − τ ∗ )p∗1−ζ+ τ ∗ p∗1−ζ
;
t
h,t
f,t
the laws of motion of the foreign final good prices at the home and foreign countries
1−ζ
∗
∗ 1−ζ
p1−ζ
f,t = (1 − µ )pf,t + µ pf,t−1

(

γA,t
γτ,t

)1−ζ

and

,

∗
∗
∗
p∗1−ζ
= (1 − µ∗ )p∗1−ζ+ µ∗ p∗1−ζ
f,t
f,t
f,t−1

( ∗ )1−ζ ∗
γA,t
;
∗
γτ,t
(B.28)

the real marginal cost the foreign firm faces
mc∗t =

wt∗
;
p∗f,t

(B.29)

the Taylor rule the foreign central bank follows∗∗

(1 + i∗f,t ) = (1 + i∗ )1−ρi (1 + i∗f,t−1 )ρi (

p∗t ∗−1γp∗t−1 A,t

)a∗π

∗a∗

Nt y

(

qt
qt−1

)− a∗s ∗ (
1−as

γπ,t
∗
γπ,t

)− a∗s ∗1−as1−ρ∗i

exp('∗i,t );(B.30)

and foreign consumption habit∗−1 ∗h∗t = c∗t − h∗ γA,tct−1 .

12(B.31)

B.2. Deterministic steady state of the stochastically detrended system
Let ωh , ωf , ωh∗ , and ωf∗ denote the steady-state home and foreign relative prices, ph /p, pf /p, p∗h /p∗ ,and p∗f /p∗ . Let Φ and Φ∗ denote the home and foreign countries’ terms of trade ph /pf and p∗f /p∗h . Also
let Ψ and Ψ∗ denote the deviations of the home and foreign goods from the law of one price, sap∗h /ph andpf /(sap∗f ). In this paper, the deterministic steady state of the stochastically detrended system (B.1) - (B.31)
is characterized under a restriction that the law of one price holds for the imported goods of the home andforeign countries at the deterministic steady state; sap∗h = pf . Under this condition, the law of one pricedeviation of the home good is equal to the inverse of the home terms of trade; Ψ = Φ−1 . Similarly, the lawof one price deviation of the foreign good is equal to the foreign terms of trade; Ψ∗ = Φ∗−1 .
1
1
1
1
Define γ̄ ≡ γτ /(γA γA∗ ) 2 , γ̄ ∗ ≡ γτ∗ /(γA γA∗ ) 2 , h̄ ≡ h/(γA γA∗ ) 2 , h̄∗ ≡ h∗ /(γA γA∗ ) 2 , and a ≡1(γA /γA∗ ) λ . The model then determines the steady state values of the 19 detrended variables, q, c, c∗ ,N , N ∗ , ωh , ωh∗ , ωf , ωf∗ , mc, mc∗ , Ωh , Ω∗h , Ωf , Ω∗f , Ψ, Ψ∗ , Φ, and Φ∗ , by the following 19 equations:
Φ=

1
,
Ψ

1
,
Ψ∗
w∗
q=Ψ h
wh

Φ∗ =

ωhζ−1 = τ + (1 − τ )Φζ−1 ,∗∗ωh∗ζ −1 = 1 − τ ∗ + τ ∗ (ΨΨ∗ Φ)ζ −1 ,ωfζ−1 = 1 − τ + τ Φ1−ζ ,∗∗ωf∗ζ −1 = τ ∗ + (1 − τ ∗ )(ΨΨ∗ Φ)1−ζ ,(1 − h̄)cN η = ωh mc,
N = [Ωh τ ωh−ζ + Ω∗h (1 − τ )ωf−ζ ]c∗(1 − h̄∗ )c∗ N ∗η = ωf∗ mc∗ ,∗∗N ∗ = [Ωf (1 − τ ∗ )ωh∗−ζ + Ω∗f τ ∗ ωf∗−ζ ]c∗ ,
(

)(
)(
) 1
ζ −1
1 − βµγ̄ ζ
1 − µγ̄ ζ−1 1−ζmc =,ζ
1 − βµγ̄ ζ−1
1−µ
( ∗
)(
)(
) 1
∗ ∗ζ ∗∗ ∗ζ ∗ −1 1−ζ ∗
ζ
−
1
1
−
βµ
γ̄
1
−
µ
γ̄
mc∗ =
,
ζ∗
1 − βµ∗ γ̄ ∗ζ ∗ −1
1 − µ∗
13

Ωh =

(

1−µ
1 − µγ̄ ζ(

1−µ
1 − µγ̄ ∗ζ ∗Ωf =1 − µ∗
1 − µ∗ γ̄ ζ

Ω∗h =

(

(

)(

)(

1 − µγ̄ ζ−11−µζ
)− 1−ζ

∗1 − µγ̄ ∗ζ −11−µ)(

1 − µ∗ γ̄ ζ−1
1 − µ∗,

ζ∗)− 1−ζ
∗ζ)− 1−ζ,,)() ζ∗ ∗∗∗ ∗ζ ∗ −1 − 1−ζ
1
−
µ
1
−
µ
γ̄
Ω∗f =
,
1 − µ∗ γ̄ ∗ζ ∗1 − µ∗()()() 1 ( ∗ )()()− 1−ζ1 ∗∗∗ζ −1
1 − βµγ̄ ζ
1 − µγ̄ ζ−1 1−ζζ
1 − βµγ̄ ∗ζ −1
1 − µγ̄ ∗ζ −1Ψ=,
ζ
1 − βµγ̄ ζ−11−µζ∗ − 1
1 − βµγ̄ ∗ζ ∗
1−µ
1
( ∗
)(
) 1 (
)(
)(
)− 1−ζ∗ )(∗ζ −11 − βµ∗ γ̄ ∗ζ
1 − µ∗ γ̄ ∗ζ −1 1−ζ ∗ζ
1 − βµ∗ γ̄ ζ−1
1 − µ∗ γ̄ ζ−1
∗
Ψ =
.
ζ∗
1 − βµ∗ γ̄ ∗ζ ∗ −1
1 − µ∗ζ −1
1 − βµ∗ γ̄ ζ
1 − µ∗
B.3. Log-linear approximation of the stochastically detrended system
To derive the corresponding linear rational expectations (LRE) models, I take a log-linear approximation of the stochastically detrended FONCs around the unique deterministic steady state. For a variable
xt , let x̂t denote the percentage deviation from its deterministic steady-state value x, i.e., x̂t ≡ (xt − x)/x,
and x̃t the deviation from the deterministic steady-state value, i.e., x̃t ≡ xt − x.
Given the above stochastically detrended FONCs (B.1)-(B.31), the corresponding log-linearized
FONCs around the deterministic steady state are derived as follows. The log-linear approximation of thehome budget constraint is
&
'
&
'
−1 ¯
−1
−1 ¯
∗
¯
b̃h,t + b̃f,t + sdŝt = β d (1 + îh,t ) − γ̂τ,t + β b̃h,t−1 + β d (1 + îf,t ) − γ̂τ,t + ŝt + β −1 sb̃f,t−1 +*+∗∗ζ ∗ ∗1−ζ ∗ ∗τ pζ p1−ζphc ŝt + (1 − ζ ∗ )p̂∗h,t + ζ ∗ p̂∗t + ĉ∗t − pc(p̂t + ĉt );h c [(1 − ζ)p̂h,t + ζ p̂t + ĉt ] + (1 − τ )sp(B.32)
that of the home intratemporal optimality condition isη N̂t = ŵt − p̂t − ĥt ;(B.33)

p̂t + ĥt + (1 + îh,t ) = Et (p̂t+1 + ĥt+1 + γ̂τ,t+1 );(B.34)

that of the home Euler equation is14

that of the home UIP condition is(B.35)∗
Et ŝt+1 − ŝt = (1 + îh,t ) − (1 + îf,t ) − Et (γ̂τ,t+1 − γ̂τ,t+1);

those of the home risk premiums areand (1 + îf,t ) = (1 + î∗f,t ) − ψ(1 − κ)b̃f,t + (1 − κ)vr,t ,(B.36)
where κ is defined by κ ≡ β/γτ ; that of the optimal price setting rule of the home firm for the home market
is(1 + îh,t ) = (1 + î∗h,t ) − ψ(1 − κ)b̃h,t + (1 − κ)vr,t ,p̂h,t − p̂h,t + (1 − µϕ1 )(1 − µϕ2 )

∞
!
i=0

∞
!
i=0

i

(µϕ1 ) Et (ĉh,t+i − ĥt+i ) + (1 − µϕ1 )(ζ − 1)i(µϕ2 ) Et (ĉh,t+i − ĥt+i ) + (1 − µϕ2 )

∞
!
i=0

∞
!

(µϕ1 )i Et φ̂h,t+i =i=1i(µϕ2 ) Et m̂ct+i + (1 − µϕ2 )ζ

∞!

(µϕ2 )i Et φ̂h,t+i ,i=1(B.37)and that for the foreign market isp̂∗h,t − p̂∗h,t + (1 − µϕ∗1 )

∞
!
i=0

(µϕ∗1 )i Et (ĉ∗h,t+i − ĥt+i ) + (1 − µϕ∗1 )(1 − µϕ∗1 )(ζ ∗ − 1)∞!

(µϕ∗1 )i Et φ̂∗h,t+i = (1 − µϕ∗2 )

∞
!
i=0

∞
!

(µϕ∗1 )i Et (ŝt+i + p̂∗h,t+i − p̂h,t+i )+(µϕ2 )∗i Et (ĉ∗h,t+i − ĥt+i )

i=1
i=0
∞
∞
!
!
∗
∗ i
∗ ∗
+ (1 − µϕ2 )(µϕ2 ) Et (m̂ct+i − ât+i ) + (1 − µϕ2 )ζ
(µϕ∗2 )i Et φ̂∗h,t+i ,i=0
i=1(B.38)

where the parameters ϕ1 , ϕ2 , ϕ∗1 , and ϕ∗2 are defined by ϕ1 ≡ βγ̄ ζ−1 , ϕ2 ≡ βγ̄ ζ , ϕ∗1 ≡ βγ̄ ∗ζ −1 , and∗ϕ∗2 ≡ βγ̄ ∗ζ , respectively; those of the home cumulative inflation rates are∗

φ̂t+i = p̂t+i − p̂t +

i
!
s=1

φ̂h,t+i = p̂h,t+i − p̂h,t +φ̂∗h,t+i = p̂∗h,t+i − p̂∗h,t +(γ̂τ,t+s − γ̂A,t+s ) =

i
!
s=1
i
!
s=1

i
!
s=1

(γ̂τ,t+s − γ̂A,t+s ) =∗∗(γ̂τ,t+s− γ̂A,t+s)=(π̂t+s + γ̂τ,t+s − γ̂A,t+s ) =

i
!
s=1
i
!
s=1

i
!

χ̂t+s ,s=1

(π̂h,t+s + γ̂τ,t+s − γ̂A,t+s ) =∗∗∗(π̂h,t+s+ γ̂τ,t+s− γ̂A,t+s
)=

i
!
s=1
i
!

χ̂h,t+s ,χ̂∗h,t+s ,(B.39)

s=1∗
where π̂t ≡ p̂t − p̂t−1 , π̂h,t ≡ p̂h,t − p̂h,t−1 , π̂h,t≡ p̂∗h,t − p̂∗h,t−1 , χ̂t ≡ π̂t + γ̂τ,t − γ̂A,t , χ̂h,t ≡ π̂h,t + γ̂τ,t − γ̂A,t ,

15

∗
∗
∗
and χ̂∗h,t ≡ π̂h,t+ γ̂τ,t− γ̂A,t; that of the market clearing condition for the home country is(B.40)

N N̂t = Ωh ch (Ω̂h,t + ĉh,t ) + Ω∗h c∗h a−1 (Ω̂∗h,t + ĉ∗h,t − ât );those of the transition equations of price dispersions Ω̂h,t and Ω̂∗h,t areΩ̂h,t = ζµγ̄ ζ−1 Θ (π̂h,t + γ̂τ,t − γ̂A,t ) + µγ̄ ζ Ω̂h,t−1 , and
5 ∗
6
∗
∗
∗
∗
Ω̂∗h,t = ζ ∗ µγ̄ ∗ζ −1 Θ∗ π̂h,t+ γ̂τ,t− γ̂A,t+ µγ̄ ∗ζ Ω̂∗h,t−1 ,(B.41)
∗ζ ∗

1−µγ̄
1−µγ̄
∗
∗
where the parameters Θ and Θ∗ are defined by Θ ≡ γ̄ − 1−µγ̄
ζ−1 and Θ ≡ γ̄ − 1−µγ̄ ∗ζ ∗ −1 ; those of thedemand functions for the home goods areζĉh,t = ĉt − ζ (p̂h,t − p̂t ) ,that of the home CPI is5
6
and ĉ∗h,t = ĉ∗t − ζ ∗ p̂∗h,t − p̂∗t ;(B.42)(B.43)

p̂t = τ ωh1−ζ p̂h,t + (1 − τ ωh1−ζ )p̂f,t ;those of the laws of motion of the home good prices in the home and foreign countries arep̂h,t = (1 − µ)p̂∗h,t = (1 − µ)

(
(

ph
ph
p∗h
p∗h

)1−ζ

p̂h,t + µγ̄ ζ−1 [p̂h,t−1 + γ̂A,t − γ̂τ,t ] ,)1−ζ ∗

and

*
+
∗
∗
∗
p̂∗h,t + µγ̄ ∗ζ −1 p̂∗h,t−1 + γ̂A,t− γ̂τ,t;(B.44)

that of the real marginal cost of the home firm ism̂ct = ŵt − p̂h,t ;(B.45)

that of the Taylor rule of the home central bank is
7

8
as
∗
(1 + îh,t ) = ρi (1 + îh,t−1 ) + (1 − ρi ) aπ (χ̂t − γ̂τ,t ) + ay ŷt +(q̂t − q̂t−1 + χ̂t − χ̂t ) + 'i,t ; (B.46)
1 − asand that of home consumption habit is(1 − h̄)ĥt = ĉt − h̄ĉt−1 + h̄γ̂A,t ,(B.47)

where h̄ ≡ hγA−1 .
The foreign country’s counterparts are given as follows. The log-linear approximation of the foreign16

country’s budget constraint is&'&'¯ t = −β −1 d¯ (1 + î∗ ) − γ̂τ,t −β −1 b̃h,t−1 −β −1 d¯ (1 + î∗ ) − γ̂ ∗ + ŝt −β −1 sb̃f,t−1 +− b̃h,t − b̃f,t −sdŝ
h,t
f,t
τ,t
*
+
∗
∗ ∗ζ ∗ ∗1−ζ ∗(1−τ )pζ p1−ζpf
c ŝt + (1 − ζ ∗ )p̂∗f,t + ζ ∗ p̂∗t + ĉ∗t −sp∗ c∗ (ŝt +p̂∗t +ĉ∗t );f c [(1 − ζ)p̂f,t + ζ p̂t + ĉt ]+τ sp(B.48)
that of the foreign intratemporal optimality condition is(B.49)

η ∗ N̂t∗ = ŵt∗ − p̂∗t − ĥ∗t ;that of the foreign Euler equation is∗p̂t + ĥ∗t + q̂t − ât − ŝt + (1 + î∗f,t ) = Et (p̂t+1 + ĥ∗t+1 + q̂t+1 − ât+1 − ŝt+1 + γ̂τ,t+1);(B.50)

that of the foreign UIP condition is(B.51)∗
Et ŝt+1 − ŝt = (1 + î∗h,t ) − (1 + î∗f,t ) − Et (γ̂τ,t+1 − γ̂τ,t+1);

those of the optimal price setting rules of the foreign firm are∗p̂f,t − p̂f,t + (1 − µ ϕ1 )

∞
!
i=0

(µ ϕ1 ) Et (ĉf,t+i − ĥ∗t+i ) + (1 − µ∗ ϕ1 )(1 − µ∗ ϕ1 )(ζ − 1)

∗

∞
!

∗

i

(µ∗ ϕ1 )i Et φ̂f,t+i = (1 − µ∗ ϕ2 )

i=1
∞
!

(1 − µ ϕ2 )

i=0

∞
!

∞
!
i=0

i=0

(µ∗ ϕ1 )i Et (p̂f,t+i − ŝt+i − p̂∗f,t+i )+(µ∗ ϕ2 )i Et (ĉf,t+i − ĥ∗t+i )+(µ ϕ2 ) Et (m̂c∗t+i + ât+i ) + (1 − µ∗ ϕ2 )ζ
∗

i

∞
!

(µ∗ ϕ2 )i Et φ̂f,t+i , (B.52)

i=1

and
p̂∗f,t − p̂∗f,t + (1 − µ∗ ϕ∗1 )∗(1−µ ϕ∗2 )

∞
!
i=0

∗

∞
!
i=0

(µ∗ ϕ∗1 )i Et (ĉ∗f,t+i − ĥ∗t+i ) + (1 − µ∗ ϕ∗1 )(ζ ∗ − 1)(µ ϕ∗2 )i Et (ĉ∗f,t+i −ĥ∗t+i )+(1−µ∗ ϕ∗2 )

∞
!

∞
!

(µ∗ ϕ∗1 )i Et φ̂∗f,t+i =i=1(µ ϕ∗2 )i Et m̂c∗t+i +(1−µ∗ ϕ∗2 )ζ ∗

i=0

17

∗

∞
!

(µ∗ ϕ∗2 )i Et φ̂∗f,t+i ;i=1(B.53)

those of the foreign cumulative inflation rates areφ̂∗t+i = p̂∗t+i − p̂∗t +

i
!
s=1

φ̂f,t+i = p̂f,t+i − p̂f,t +φ̂∗f,t+i = p̂∗f,t+i − p̂∗f,t +∗∗(γ̂τ,t+s− γ̂A,t+s
)=
i
!
s=1
i
!
s=1

i
!
s=1

(γ̂τ,t+s − γ̂A,t+s ) =∗∗(γ̂τ,t+s− γ̂A,t+s)=∗∗∗(π̂t+s+ γ̂τ,t+s− γ̂A,t+s
)=
i
!
s=1
i
!
s=1

i
!

χ̂∗t+s ,s=1(π̂f,t+s + γ̂τ,t+s − γ̂A,t+s ) =∗∗∗(π̂f,t+s+ γ̂τ,t+s− γ̂A,t+s
)=

i
!
s=1
i
!

χ̂f,t+s ,χ̂∗f,t+s ,(B.54)

s=1∗∗∗
where π̂t∗ ≡ p̂∗t − p̂∗t−1 , π̂f,t ≡ p̂f,t − p̂f,t−1 , π̂f,t≡ p̂∗f,t − p̂∗f,t−1 , χ̂∗t ≡ π̂t∗ + γ̂τ,t− γ̂A,t, χ̂f,t ≡ π̂f,t + γ̂τ,t − γ̂A,t ,∗∗∗and χ̂∗f,t ≡ π̂f,t+ γ̂τ,t− γ̂A,t; that of the market clearing condition for the foreign country is(B.55)N ∗ N̂t∗ = Ωf cf a(Ω̂f,t + ĉf,t + ât ) + Ω∗f c∗f (Ω̂∗f,t + ĉ∗f,t );those of the transition equations of price dispersions Ω̂f,t and Ω̂∗f,t areΩ̂f,t = ζµ∗ γ̄ ζ−1 Ξ (π̂f,t + γ̂τ,t − γ̂A,t ) + µ∗ γ̄ ζ Ω̂f,t−1 , and
5 ∗
6
∗
∗
∗
∗
Ω̂∗f,t = ζ ∗ µ∗ γ̄ ∗ζ −1 Ξ∗ π̂f,t+ γ̂τ,t− γ̂A,t+ µ∗ γ̄ ∗ζ Ω̂∗f,t−1 ,(B.56)∗ ∗ζ ∗∗ ζ

1−µ γ̄
1−µ γ̄
∗∗where the parameters Ξ and Ξ∗ are defined by Ξ ≡ γ̄ − 1−µ∗ γ̄ ζ−1 and Ξ ≡ γ̄ − 1−µ∗ γ̄ ∗ζ ∗ −1 ; those of thedemand functions for the foreign goods areĉf,t = ĉt − ξ (p̂f,t − p̂t ) ,andthat of the foreign CPI is

5
6
ĉ∗f,t = ĉ∗t − ζ ∗ p̂∗f,t − p̂∗t ;∗(B.57)∗(B.58)p̂∗t = (1 − τ ∗ ωf∗1−ζ )p̂∗h,t + τ ∗ ωf∗1−ζ p̂∗f,t ;those of the laws of motion of the home good prices in the home and foreign countries are∗p̂f,t = (1 − µ )p̂∗f,t = (1 − µ∗ )

(

-

pf
pf
p∗f
p∗f

)1−ζ

p̂f,t + µ∗ γ̄ ζ−1 [p̂f,t−1 + γ̂A,t − γ̂τ,t ] ,.1−ζ ∗

and

*
+
∗
∗
∗
p̂∗f,t + µ∗ γ̄ ∗ζ −1 p̂∗f,t−1 + γ̂A,t− γ̂τ,t;(B.56)

that of the marginal cost of the foreign firm ism̂c∗t = ŵt∗ − p̂∗f,t ;18(B.59)

that of the Taylor rule of the foreign central bank is(1+ î∗f,t ) = ρ∗i (1+ î∗f,t−1 )+(1−ρ∗i )7a∗π

and that of foreign consumption habit is5 ∗6∗χ̂t − γ̂τ,t+ a∗y ŷt∗ −8a∗s∗(q̂t − q̂t−1 + χ̂t − χ̂t ) +'∗i,t ; (B.58)
1 − a∗s∗(1 − h̄∗ )ĥ∗t = ĉ∗t − h̄∗ ĉ∗t−1 + h̄∗ γ̂A,t.

(B.59)

where h̄∗ ≡ h∗ γA∗−1 ; finally, that of the real exchange rate isq̂t = ŝt + ât + p̂∗t − p̂t .

(B.60)

Comparing the home and foreign UIPs (B.35) and (B.51) reveals that the two countries must face
the same interest rate differential: (1 + îh,t ) − (1 + îf,t ) = (1 + î∗h,t ) − (1 + î∗f,t ). The home risk premium
equations (B.36) then imply that the home holdings of the home issued bond must be equal to the home
holdings of the foreign issued bond: b̂t = b̂h,t = b̂f,t in equilibrium.

Appendix C: GNKPCs with trend inflation
Now I derive the generalized new Keynesian Phillips curves (GNKPCs). In doing so, I assume thatthe parameters ϕ1 , ϕ2 , ϕ∗1 , and ϕ∗2 take values between zero and one. At the deterministic steady state, thelaw of motion of the home good price at the home market (B.13) implies that
(

ph
ph

)1−ζ

=

1 − µγ̄ ζ−1
.
1−µ

(C.1)

Substituting the above equation (C.1) into the log-linear approximation of the law of motion of the homegood price at the home market, eq.(B.44), givesχ̂h,t = ϕ0 (p̂h,t − p̂h,t ),(C.2)

where the parameter ϕ0 is defined by ϕ0 ≡ (1 − µγ̄ ζ−1 )/µγ̄ ζ−1 . Cumulative inflation rate processes (B.39)
rewrite the optimal price setting of the home firm for the home market (B.37) as
p̂h,t − p̂h,t = (1 − µϕ2 )Dh,t − (1 − µϕ1 )Zh,t ,
19

(C.3)

where random variables Dh,t and Zh,t follow the stochastic difference equations
ζµϕ2
Et χ̂h,t+1 + µϕ2 Et Dh,t+1 ,1 − µϕ2(ζ − 1)µϕ1
Zh,t = ĉh,t − ĥt +Et χ̂h,t+1 + µϕ1 Et Zh,t+1 .
1 − µϕ1

Dh,t = ĉh,t − ĥt + m̂ct +After several calculations, it is straightforward to show that eqs. (C.1) – (C.3) provide/0∞!
ϕ0 (ϕ1 − ϕ2 )χ̂h,t =
ĉh,t − ĥt − (1 − µϕ1 )(µϕ1 )i Et (ĉh,t+i − ĥt+i ) + ϕ0 (1 − µϕ2 )m̂ct
ϕ1
i=0

∞
!
ϕ0 (ϕ1 − ϕ2 )+ µϕ2 (1 + ϕ0 )Et χ̂h,t+1 −(ζ − 1)(µϕ1 )i Et χ̂h,t+i . (C.4)ϕ1i=1Further, the demand function for the home good (B.42) impliesĉh,t − ĥt − (1 − µϕ1 )

∞
!
i=0

(µϕ1 )i Et (ĉh,t+i − ĥt+i )= ĉh,t − ĉt + ĉt − ĥt − (1 − µϕ1 )
=−

∞
!

i=0

i

(µϕ1 ) Et ∆(ĉh,t+i − ĉt+i ) −

i=1
∞
!

= −ζ

∞
!

i=1

(µϕ1 )i Et (π̂t+i − π̂h,t+i ) −(µϕ1 )i Et (ĉh,t+i − ĉt+i + ĉt+i − ĥt+i )

∞
!
i=1

∞
!
i=1

(µϕ1 )i Et ∆(ĉt+i − ĥt+i )(µϕ1 )i Et ∆(ĉt+i − ĥt+i ).

Substituting this relation into eq.(C.4) yields the GNKPC for the home good price at the home market
χ̂h,t = µϕ2 (1 + ϕ0 )Et χ̂h,t+1 + ϕ0 (1 − µϕ2 )m̂ct −∞ϕ0 (ϕ1 − ϕ2 ) !
ζ(µϕ1 )i Et (χ̂t+i − χ̂h,t+i )
ϕ1
i=1

∞
∞
!
ϕ0 (ϕ1 − ϕ2 )ϕ0 (ϕ1 − ϕ2 ) !
ih−(ζ − 1)(µϕ1 ) Et χ̂h,t+i −(µϕ1 )i Et ∆(ĉt+i − ĥt+i ) + v̂p,t. (C.5)
ϕ1
ϕ
1
i=1
i=1

Following Steinsson (2008), I add a reduced-form cost push shock v̂cp,h,t to the RHS of the GNKPC (C.5).
Steinsson (2008) discusses that this additive shock to the NKPC plays a crucial role in replicating the
hump-shaped impulse response of the real exchange rate. The cost push shock follows an AR(1) process
h
h
v̂p,t
= ρhp v̂p,t−1+ 'hp,t .

20

The same steps of calculation above is applicable to derive the GNKPCs for the home good priceinflation at the foreign market, χ∗h,t , the foreign good price inflation at the home market, χf,t , and the foreign
good price inflation at the foreign market, χ∗f,t as follows.

∗ϑ∗ ϕ∗ (1 − µϕ∗1 )χ̂∗h,t = µϕ∗2 (1+ϑ∗0 )Et χ̂∗h,t+1 +ϑ∗0 (1−µϕ∗2 )(m̂ct −ât )− 0 2 ∗[q̂t −ât −τ ∗ ωf∗1−ζ Ψ̂∗t −(1−τ ωh1−ζ )Ψ̂t ]
ϕ1
∞
∞
∗
∗
∗
∗
ϑ0 (ϕ1 − ϕ2 ) ∗ !
ϑ0 (ϕ1 − ϕ2 ) !
∗ i
∗
∗
+
ζ
(µϕ1 ) Et (χ̂h,t+i − χ̂t+i ) −(µϕ1 )i Et ∆(ĉ∗t+i − ĥt+i )
∗
ϕ1
ϕ1
i=1
i=1

∞

∗
ϑ∗ (ϕ∗ − ϕ∗2 )(1 − µϕ∗1 ) !
− 0 1(µϕ∗1 )i Et [q̂t+i − ât+i − τ ∗ ωf∗1−ζ Ψ̂∗t+i − (1 − τ ωh1−ζ )Ψ̂t+i ]
∗
ϕ1
i=0

∞
!
ϑ∗0 (ϕ∗1 − ϕ∗2 ) ∗h∗−(ζ − 1)(µϕ∗1 )i Et χ̂∗h,t+i + v̂p,t, (C.6)ϕ∗1i=1∗
ϑ0 ϕ2 (1 − µ∗ ϕ1 )[q̂t −ât +(1−τ ∗ ωf∗1−ζ )Ψ̂∗t +τ ωh1−ζ Ψ̂t ]ϕ1∞∞
ϑ0 (ϕ1 − ϕ2 ) ! ∗ i
ϑ0 (ϕ1 − ϕ2 ) ! ∗ i+ζ(µ ϕ1 ) Et (χ̂f,t+i − χ̂t+i ) −(µ ϕ1 ) Et ∆(ĉt+i − ĥ∗t+i )
ϕ1
ϕ
1
i=1
i=1

χ̂f,t = µ∗ ϕ2 (1+ϑ0 )Et χ̂f,t+1 +ϑ0 (1−µ∗ ϕ2 )(m̂c∗t +ât )++∞∗ϑ0 (ϕ1 − ϕ2 )(1 − µ∗ ϕ1 ) ! ∗ i(µ ϕ1 ) Et [q̂t+i − ât+i + (1 − τ ∗ ωf∗1−ζ )Ψ̂∗t+i + τ ωh1−ζ Ψ̂t+i ]
ϕ1
i=0

∞
!
ϑ0 (ϕ1 − ϕ2 )f−(ζ − 1)(µ∗ ϕ1 )i Et χ̂f,t+i + v̂p,t, (C.7)ϕ1i=1and
χ̂∗f,t = µ∗ ϕ∗2 (1 + ϕ∗0 )Et χ̂∗f,t+1 + ϕ∗0 (1 − µ∗ ϕ∗2 )m̂c∗t −∞ϕ∗0 (ϕ∗1 − ϕ∗2 ) ∗ ! ∗ ∗ iζ(µ ϕ1 ) Et (χ̂∗t+i − χ̂∗f,t+i )
ϕ∗1
i=1

∞
∞
!
ϕ∗0 (ϕ∗1 − ϕ∗2 ) ∗ϕ∗0 (ϕ∗1 − ϕ∗2 ) ! ∗ ∗ i
f∗
∗ ∗ i
∗
−
(ζ − 1)(µ ϕ1 ) Et χ̂f,t+i −(µ ϕ1 ) Et ∆(ĉ∗t+i − ĥ∗t+i ) + v̂p,t, (C.8)
∗
ϕ∗1
ϕ
1
i=1
i=1

where the parameters ϑ∗0 , ϑ0 , and ϕ∗0 are defined by ϑ∗0 ≡ (1−µγ̄ ∗ζ −1 )/µγ̄ ∗ζ −1 , ϑ0 ≡ (1−µ∗ γ̄ ζ−1 )/µ∗ γ̄ ζ−1 ,∗and ϕ∗0 ≡ (1 − µ∗ γ̄ ∗ζ −1 )/µ∗ γ̄ ∗ζ∗−1 . Similar to the GNKPC for the home good price at the home market,∗∗
eq.(C.5), the GNKPCs (C.6) - (C.8) are subject to reduced-form cost push shocks v̂cp,h,t, v̂cp,f,t , and v̂cp,f,t
∗

21

∗

which follows independent AR(1) processes
h∗
h∗
h∗
v̂p,t
= ρh∗
p v̂p,t−1 + 'p,t ,
f
f
v̂p,t
= ρfp v̂p,t−1+ 'fp,t ,
f∗
f∗
v̂p,t
= ρfp ∗ v̂p,t−1+ 'fp,t∗ .

Appendix D: The linear rational expectations model
The log-linear approximation of the stochastically detrended FONCs, eqs.(B.32)-(B.60), which isaugmented by the second-order expectational difference equations of the GNKPCs (C.5)-(C.8), provide thefollowing tractable linear rational expectations (LRE) model. The LRE model contains the UIP condition∗
Et ŝt+1 − ŝt = (1 + îh,t ) − (1 + î∗f,t ) + ψ(1 − κ)b̃t + (1 − κ)vr,t − Et (γ̂τ,t+1 − γ̂τ,t+1
);

(D.1)

the transition equation of bond holdings∗b̃t = β −1 b̃t−1 + p̄c(1 − τ ωh1−ζ )[τ ωh1−ζ (1 − ζ)Ψ̂t − τ ∗ ωf∗1−ζ (1 − ζ ∗ )Ψ̂∗t + q̂t − ât − ĉt + ĉ∗t ],(D.2)

where p̄ ≡ p/2; the home Euler equationĥt + (1 + îh,t ) = Et (ĥt+1 + χ̂t+1 + γ̂A,t+1 );

(D.3)

∗
ĥ∗t + (1 + î∗f,t ) = Et (ĥ∗t+1 + χ̂∗t+1 + γ̂A,t+1
);

(D.4)

the foreign Euler equation

the home intratemporal optimality condition
η N̂t = m̂ct + (1 − τ ωh1−ζ )Ψ̂t − ĥt ;(D.5)

the foreign intratemporal optimality condition∗η ∗ N̂t∗ = m̂c∗t + (1 − τ ∗ ωf∗1−ζ )Ψ̂∗t − ĥ∗t ;(D.6)

the home market clearing condition
N N̂t = Ωh ch (Ω̂h,t + ĉh,t ) + Ω∗h c∗h a−1 (Ω̂∗h,t + ĉ∗h,t − ât );
22

(D.7)

the foreign market clearing condition
N ∗ N̂t∗ = Ωf cf a(Ω̂f,t + ĉf,t + ât ) + Ω∗f c∗f (Ω̂∗f,t + ĉ∗f,t );(D.8)

the transition equation of price dispersion Ω̂h,tΩ̂h,t = µγ̄ ζ Ω̂h,t−1 + ζµγ̄ ζ−1 Θχh,t ;

(D.9)

∗
the transition equation of price dispersion Ω̂h,t∗∗Ω̂∗h,t = µγ̄ ∗ζ Ω̂∗h,t−1 + ζ ∗ µγ̄ ∗ζ −1 Θ∗ χ∗h,t ;(D.10)

the transition equation of price dispersion Ω̂f,tΩ̂f,t = µ∗ γ̄ ζ Ω̂f,t−1 + ζµ∗ γ̄ ζ−1 Ξχf,t ;(D.11)

the transition equation of price dispersion Ω̂∗f,tΩ̂∗f,t = µ∗ γ̄ ∗ζ Ω̂∗f,t−1 + ζ ∗ µ∗ γ̄ ∗ζ −1 Ξ∗ χ∗f,t ;(D.12)

χ̂t = τ ωh1−ζ χ̂h,t + (1 − τ ωh1−ζ )χ̂f,t ;(D.13)∗∗

the home general inflation rate

the foreign general inflation rate∗∗χ̂∗t = (1 − τ ∗ ωf∗1−ζ )χ̂∗h,t + τ ∗ ωf∗1−ζ χ̂∗f,t ;(D.14)

Ψ̂t = Ψ̂t−1 + χ̂h,t − χ̂f,t ;(D.15)

Ψ̂∗t = Ψ̂∗t−1 + χ̂∗f,t − χ̂∗h,t ;(D.16)

the home terms of trade
the foreign terms of trade
the GNKPC for χ̂h,tβµϕ2 Et χ̂h,t+2 − [µϕ1 + βγ̄ + ϕ0 µ(ϕ1 − ϕ2 )]Et χ̂h,t+1 + χ̂h,t= ϕ0 (1 − µϕ2 )m̂ct − ϕ0 (1 − µϕ2 )µϕ1 Et m̂ct+1 − ϕ0 µ(ϕ1 − ϕ2 )ζEt χ̂t+1− ϕ0 µ(ϕ1 − ϕ2 )Et ∆(ĉt+1 − ĥt+1 ) + (1 − µϕ1 ρcp,h )v̂cp,h,t ; (D.17)
23

the GNKPC for χ̂∗h,tβµϕ∗2 Et χ̂∗h,t+2 − [µϕ∗1 + βγ̄ ∗ + ϑ∗0 µ(ϕ∗1 − ϕ∗2 )]Et χ̂∗h,t+1 + χ̂∗h,t∗= ϑ∗0 (1−µϕ∗2 )(m̂ct −ât )−ϑ∗0 (1−µϕ∗2 )µϕ∗1 Et (m̂ct+1 −ât+1 )−ϑ∗0 (1−µϕ∗1 )[q̂t −ât −τ ∗ ωf∗1−ζ Ψ̂∗t −(1−τ ωh1−ζ )Ψ̂t ]∗+ ϑ∗0 µϕ∗2 (1 − µϕ∗1 )Et [q̂t+1 − ât+1 − τ ∗ ωf∗1−ζ Ψ̂∗t+1 − (1 − τ ωh1−ζ )Ψ̂t+1 ] − ϑ∗0 µ(ϕ∗1 − ϕ∗2 )ζ ∗ Et χ̂∗t+1∗− ϑ∗0 µ(ϕ∗1 − ϕ∗2 )Et ∆(ĉ∗t+1 − ĥt+1 ) + (1 − µϕ∗1 ρ∗cp,h )v̂cp,h,t; (D.18)

the GNKPC for χ̂f,tβµ∗ ϕ2 Et χ̂f,t+2 − [µ∗ ϕ1 + βγ̄ + ϑ0 µ∗ (ϕ1 − ϕ2 )]Et χ̂f,t+1 + χ̂f,t∗= ϑ0 (1−µ∗ ϕ2 )(m̂c∗t +ât )−ϑ0 (1−µ∗ ϕ2 )µ∗ ϕ1 Et (m̂c∗t+1 +ât+1 )+ϑ0 (1−µ∗ ϕ1 )[q̂t −ât +(1−τ ∗ ωf∗1−ζ )Ψ̂∗t +τ ωh1−ζ Ψ̂t ]∗− ϑ0 µ∗ ϕ2 (1 − µ∗ ϕ1 )Et [q̂t+1 − ât+1 + (1 − τ ∗ ωf∗1−ζ )Ψ̂∗t+1 + τ ωh1−ζ Ψ̂t+1 ] − ϑ0 µ∗ (ϕ1 − ϕ2 )ζEt χ̂t+1− ϑ0 µ∗ (ϕ1 − ϕ2 )Et ∆(ĉt+1 − ĥ∗t+1 ) + (1 − µ∗ ϕ1 ρcp,f )v̂cp,f,t ; (D.19)

the GNKPC for χ̂∗f,tβµ∗ ϕ∗2 Et χ̂∗f,t+2 − [µ∗ ϕ∗1 + βγ̄ ∗ + ϕ∗0 µ∗ (ϕ∗1 − ϕ∗2 )]Et χ̂∗f,t+1 + χ̂∗f,t= ϕ∗0 (1 − µ∗ ϕ∗2 )m̂c∗t − ϕ∗0 (1 − µ∗ ϕ∗2 )µ∗ ϕ∗1 Et m̂c∗t+1 − ϕ∗0 µ∗ (ϕ∗1 − ϕ∗2 )ζ ∗ Et χ̂∗t+1∗− ϕ∗0 µ∗ (ϕ∗1 − ϕ∗2 )Et ∆(ĉ∗t+1 − ĥ∗t+1 ) + (1 − µ∗ ϕ∗1 ρ∗cp,f )v̂cp,f,t; (D.20)

the demand function of ĉh,t
ĉh,t = ĉt − ζ(1 − τ ωh1−ζ )Ψ̂t ;the demand function of ĉ∗h,t∗(D.21)ĉ∗h,t = ĉ∗t + ζ ∗ τ ∗ ωf∗1−ζ Ψ̂∗t ;(D.22)

ĉf,t = ĉt + ζτ ωh1−ζ Ψ̂t ;(D.23)ĉ∗f,t = ĉ∗t − ζ ∗ (1 − τ ∗ ωf∗1−ζ )Ψ̂∗t ;∗(D.24)(1 − h̄)ĥt = ĉt − h̄ĉt−1 + h̄γ̂A,t ;(D.25)∗(1 − h̄∗ )ĥ∗t = ĉ∗t − h̄∗ ĉ∗t−1 + h̄∗ γ̂A,t;(D.26)

the demand function of ĉf,t
the demand function of ĉ∗f,t
the home habit formation
the foreign habit formation

24

the home Taylor rule
7

8
as
∗
(1+ îh,t ) = ρi (1+ îh,t−1 )+(1−ρi ) aπ (χ̂t − γ̂τ,t ) + ay N̂t +(q̂t − q̂t−1 + χ̂t − χ̂t ) +vi,t ; (D.27)
1 − asthe foreign Taylor rule
(1+ î∗f,t ) = ρ∗i (1+ î∗f,t−1 )+(1−ρ∗i )

7∗a∗π (χ̂∗t − γ̂τ,t) + a∗y N̂t∗ −

8
a∗s
∗
∗
(q̂t − q̂t−1 + χ̂t − χ̂t ) +vi,t; (D.28)
1 − a∗sthe real exchange rate
(D.29)∗∆q̂t = ∆ŝt − (χ̂t − γ̂τ,t ) + (χ̂∗t − γ̂τ,t);Appendix E. The restricted UC model and posterior simulation strategy
Let Xt denote an 66 × 1 unobserved state column vector defined as
Xt ≡ [ĉt , ĉ∗t , ĉh,t , ĉ∗h,t , ĉf,t , ĉ∗f,t , ĥt , ĥ∗t , N̂t , N̂t∗ , m̂ct , m̂c∗t , ∆p̂t , ∆p̂∗t , χ̂t , χ∗t , χ̂h,t , χ̂∗h,t , χ̂f,t , χ̂∗f,t , Ψ̂t , Ψ̂∗t ,

f
f∗
∗
∗
∗
∗
h
h∗
ω̂h,t , ω̂h,t, ω̂f,t , ω̂f,t, (1 + îh,t ), (1 + î∗f,t ), b̂t , ∆ŝt , q̂t , ât , γ̂A,t , γ̂A,t, γ̂τ,t , γ̂τ,t, 'i,t , '∗i,t , vp,t
, vp,t, vp,t, vp,t, vr,t ,Et ĉt+1 , Et ĉ∗t+1 , Et ĥt+1 , Et ĥ∗t+1 , Et m̂ct+1 , Et m̂c∗t+1 , Et χ̂t+1 , Et χ̂∗t+1 , Et χ̂h,t+1 , Et χ̂h,t+2 , Et χ̂∗h,t+1 , Et χ̂∗h,t+2 ,Et χ̂f,t+1 , Et χ̂f,t+2 , Et χ̂∗f,t+1 , Et χ̂∗f,t+2 , Et Ψ̂t+1 , Et Ψ̂∗t+1 , Et ∆ŝt+1 , Et q̂t+1 , q̂t−1 , ĉt−1 , ĉ∗t−1 ]$ .
Furthermore, let 't and ηt denote 11 × 1 and 20 × 1 random vectors consisting of structural shocks andrational expectations errors:f∗$'t ≡ ['A,t , '∗A,t , 'τ,t , '∗τ,t , 'i,t , '∗i,t , 'hp,t , 'fp,t , 'h∗p,t , 'p,t , 'r,t ]andηt ≡ [ĉt − Et−1 ĉt , ĉ∗t − Et−1 ĉ∗t , ĥt − Et−1 ĥt , ĥ∗t − Et−1 ĥ∗t , m̂ct − Et−1 m̂ct , m̂c∗t − Et−1 m̂c∗t ,χt −Et−1 χt , χ∗t −Et−1 χ∗t , χh,t −Et−1 χh,t , Et χh,t+1 −Et−1 χh,t+1 , χ∗h,t −Et−1 χ∗h,t , Et χ∗h,t+1 −Et−1 χ∗h,t+1 ,χf,t −Et−1 χf,t , Et χf,t+1 −Et−1 χf,t+1 , χ∗f,t −Et−1 χ∗f,t , Et χ∗f,t+1 −Et−1 χ∗f,t+1 , Ψ̂t , −Et−1 Ψ̂t , Ψ̂∗t −Et−1 Ψ̂∗t ,∆ŝt − Et−1 ∆ŝt , q̂t − Et−1 q̂t ]$ ,respectively. In particular, for empirical investigation purposes, I presume that the structural shock vector 't is normally distributed, with a mean of zero and a diagonal variance-covariance matrix Σ: 't ∼i.i.d.N (0, Σ) with diag(Σ) = [σA , σA∗ , στ , στ∗ , σi , σi∗ , σph , σpf , σph∗ , σpf ∗ , σr ]$ .
25

Accompanied by the stochastic processes of the exogenous forcing variables, the LRE model thenimplies that
G0 (θ)Xt = G1 (θ)Xt−1 + Q(θ)ηt + R(θ)'t ,where G0 , G1 , Q, and R are the corresponding coefficient matrices and θ is the vector of the structuralparameters. Applying Sims’s (2001) QZ algorithm to the linear rational expectations model above yields
a unique solution as the following stationary transition equation of the unobservable state vector:
Xt = F(θ)Xt−1 + Φ(θ)'t ,(E.1)

where F and Φ are confirmable coefficient matrices.
Let Yt denote the information set that consists of the nine time series:
Yt ≡ [∆ ln Pt , ∆ ln Pt∗ , ∆ ln Ct , ∆ ln Ct∗ , (1 + ih,t ), (1 + i∗f,t ), ln qt , ln Nt , ln Nt∗ ]$ .
It is then straightforward to show that the information set Yt is linearly related to the unobservable statevector Xt as
Yt = d(θ) + H(θ)Xt ,(E.2)

where d and H are confirmable coefficient vector and matrix. The transition equation of the unobserved
state (E.1) and the observation equation (E.2) jointly consist of a non-stationary state-space representationof the two-country model.
Given the data set YT ≡ {Yt }Tt=0 , applying the Kalman filter to the UC model provides model
likelihood L(YT |θ), where θ is the structural parameter vector of the two-country model. Multiplying the
likelihood by a prior probability of the structural parameters, p(θ), is proportional to the corresponding
posterior distribution p(θ|YT ) ∝ p(θ)L(YT |θ) through Bayes’ law. The posterior distribution p(θ|YT )is simulated by the random-walk Metropolis-Hastings algorithm, as implemented by Schorfheide (2000),
Bouakez and Kano (2006), Nason and Rogers (2008), and Kano (2009; 2020).
References
Ascari, G., Sbordone, A.M., 2014, The macroeconomics of trend inflation, Journal of Economic Literature
52, 679 − 739.
Bergin, P. R., 2006, How well can the New Open Economy Macroeconomics explain the exchange rate andcurrent account?, Journal of International Money and Finance 25, 675 − 701.
26

Betts, C., Devereux, M., 1996, The exchange rate in a model with pricing-to-market, European Economic
Review 40, 1007 − 1021.
Betts, C., Devereux, M., 2000, Exchange rate dynamics in a model of pricing-to-market, Journal of International Economics 50, 215 − 244.
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27

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28

Table A.1: Posterior Mean FEVDs (%): Home Inflation Rate
Horizon

"A

"∗A

"τ

"∗τ

"i

"∗i

"hp

"fp

"h∗
p

"fp ∗

"r

0.00
0.01
0.01
0.01

70.5
46.9
39.2
34.9

0.77
0.99
0.83
0.88

0.09
0.25
0.38
0.83

0.05
0.08
0.08
0.08

0.00
0.00
0.00
0.00

1.61
1.88
1.69
1.42

69.7
57.7
54.0
45.7

0.20
0.23
0.23
0.20

2.91
3.34
3.00
2.54

0.00
0.01
0.01
0.01

0.00
0.00
0.00
0.00

(a) TI model
Impact
1yr
3yr
10yr

8.01
13.2
13.3
11.9

0.00
0.00
0.00
0.00

3.40
7.77
13.7
21.7

0.29
0.46
0.41
0.98

16.8
30.2
31.9
28.5

(b) Non-TI model
Impact
1yr
3yr
10yr

0.02
0.01
0.01
0.01

0.13
0.15
0.13
0.11

6.81
13.3
19.0
29.9

2.47
3.75
4.15
4.99

16.1
19.5
17.6
14.9

Table A.2: Posterior Mean FEVDs (%): Foreign Inflation Rate
Horizon

"A

"∗A

"τ

"∗τ

"i

"∗i

"hp

"fp

"h∗
p

"fp ∗

"r

0.91
1.75
1.54
1.13

0.05
0.13
0.12
0.09

0.66
1.78
1.57
1.26

1.26
2.92
3.61
4.71

90.2
73.3
59.1
43.2

0.00
0.00
0.00
0.00

36.6
39.0
34.6
27.7

0.34
0.29
0.30
0.24

0.45
0.45
0.47
0.39

1.15
1.17
1.06
1.02

28.3
15.5
13.8
11.0

0.00
0.00
0.00
0.00

(a) TI model
Impact
1yr
3yr
10yr

0.00
0.01
0.06
0.16

0.15
0.26
0.22
0.17

0.00
0.01
0.01
0.13

6.67
19.7
33.7
48.9

0.03
0.03
0.06
0.10

(b) Non-TI model
Impact
1yr
3yr
10yr

0.00
0.00
0.00
0.00

5.10
4.49
3.90
3.12

0.04
0.04
0.04
0.04

27.8
38.9
45.6
56.3

0.06
0.05
0.05
0.04

29

Table A.3: Posterior Mean FEVDs (%): Home Consumption Growth
Horizon

"A

"∗A

"τ

"∗τ

"i

"∗i

"hp

"fp

"h∗
p

"fp ∗

"r

0.02
0.04
0.04
0.04

21.6
21.3
19.3
19.9

0.65
0.42
0.46
0.54

5.71
6.40
6.54
6.62

0.88
0.69
0.56
0.55

0.00
0.00
0.00
0.00

2.61
2.81
2.70
2.70

44.8
45.5
47.5
47.5

0.18
0.23
0.25
0.26

6.45
6.15
5.89
5.90

0.01
0.01
0.01
0.01

0.00
0.00
0.00
0.00

(a) TI model
Impact
1yr
3yr
10yr

16.1
24.2
24.5
24.0

0.00
0.01
0.01
0.01

36.7
34.1
30.1
28.8

3.61
2.28
1.86
1.84

14.6
10.4
16.5
17.5

(b) Non-TI model
Impact
1yr
3yr
10yr

12.5
11.3
10.8
10.8

0.16
0.18
0.17
0.17

3.58
3.59
3.46
3.46

2.36
2.48
2.39
2.38

27.2
27.7
26.7
26.7

Table A.4: Posterior Mean FEVDs (%): Foreign Consumption Growth
Horizon

"A

"∗A

"τ

"∗τ

"i

"∗i

"hp

"fp

"h∗
p

"fp ∗

"r

3.40
2.74
3.04
2.93

1.69
1.62
1.59
1.67

36.7
36.0
35.1
36.4

0.31
0.25
0.27
0.33

33.1
31.3
32.2
31.4

0.00
0.00
0.00
0.00

64.3
57.9
57.1
56.9

0.51
0.88
0.99
1.04

1.38
2.36
2.49
2.68

3.57
3.77
3.56
3.70

7.11
9.24
10.0
10.0

0.00
0.00
0.00
0.00

(a) TI model
Impact
1yr
3yr
10yr

0.75
0.78
0.76
0.78

0.01
0.16
0.23
0.22

0.02
0.01
0.03
0.07

22.7
25.8
25.5
24.7

1.18
1.24
1.20
1.31

(b) Non-TI model
Impact
1yr
3yr
10yr

0.00
0.01
0.01
0.01

0.65
5.58
5.95
5.92

0.00
0.01
0.01
0.01

22.3
20.1
19.7
19.6

0.02
0.07
0.07
0.08

30

Table A.5: Posterior Mean FEVDs (%): Home Interest Rate
Horizon

"A

"∗A

"τ

"∗τ

"i

"∗i

"hp

"fp

"h∗
p

"fp ∗

"r

0.20
0.13
0.10
0.08

3.30
14.5
23.0
15.3

2.73
2.34
1.93
1.23

6.90
8.62
10.2
6.74

0.31
0.38
0.44
0.35

0.00
0.00
0.00
0.00

9.59
3.62
2.23
1.23

29.5
53.9
40.2
22.3

0.19
0.08
0.13
0.08

6.33
2.31
1.43
0.79

0.19
0.10
0.06
0.03

0.00
0.00
0.00
0.00

(a) TI model
Impact
1yr
3yr
10yr

24.1
19.6
15.6
12.6

0.03
0.02
0.02
0.01

0.33
1.01
4.99
30.2

1.61
1.47
1.30
1.19

60.4
51.8
42.2
32.2

(b) Non-TI model
Impact
1yr
3yr
10yr

0.00
0.00
0.00
0.00

0.65
0.23
0.14
0.08

9.91
21.6
41.6
63.2

0.67
1.04
3.56
6.35

42.9
16.9
10.5
5.83

Table A.6: Posterior Mean FEVDs (%): Foreign Interest Rate
Horizon

"A

"∗A

"τ

"∗τ

"i

"∗i

"hp

"fp

"h∗
p

"fp ∗

"r

47.7
22.6
12.6
7.83

0.13
0.66
1.72
1.61

0.06
6.76
27.3
27.3

0.18
1.07
3.96
5.00

43.7
64.1
48.3
30.7

0.00
0.00
0.00
0.00

82.1
48.7
22.4
9.19

0.16
0.77
0.58
0.24

0.12
1.13
1.09
0.46

0.13
1.42
2.18
1.79

6.77
9.66
4.66
1.91

0.00
0.00
0.00
0.00

(a) TI model
Impact
1yr
3yr
10yr

0.29
0.23
0.16
0.78

6.91
3.20
1.78
1.10

0.04
0.07
0.19
0.24

0.02
0.47
3.39
23.6

0.80
0.66
0.44
1.68

(b) Non-TI model
Impact
1yr
3yr
10yr

0.00
0.00
0.00
0.00

6.27
3.23
1.47
0.60

0.00
0.03
0.06
0.05

4.37
34.9
67.4
85.7

0.01
0.03
0.03
0.01

31

Table A.7: Posterior Mean FEVDs (%): Home Hours Worked
Horizon

"A

"∗A

"τ

"∗τ

"i

"∗i

"hp

"fp

"h∗
p

"fp ∗

"r

0.02
0.05
0.07
0.08

17.0
11.9
4.96
3.71

3.60
5.99
4.58
3.04

0.14
1.26
5.63
17.3

0.81
0.81
0.66
0.59

0.00
0.00
0.00
0.00

2.88
1.28
1.12
1.11

58.4
81.6
83.8
82.9

0.22
0.32
0.33
0.33

5.46
2.28
2.06
3.05

0.00
0.00
0.00
0.00

0.00
0.00
0.00
0.00

(a) TI model
Impact
1yr
3yr
10yr

57.4
34.5
28.5
22.9

0.00
0.01
0.01
0.01

1.01
0.99
0.58
4.12

1.28
2.63
2.84
2.17

18.6
41.7
52.1
45.9

(b) Non-TI model
Impact
1yr
3yr
10yr

0.47
0.10
0.09
0.09

0.21
0.10
0.09
0.09

3.45
1.58
1.40
1.41

2.49
1.12
0.99
0.98

26.3
11.5
10.0
9.97

Table A.8: Posterior Mean FEVDs (%): Foreign Hours Worked
Horizon

"A

"∗A

"τ

"∗τ

"i

"∗i

"hp

"fp

"h∗
p

"fp ∗

"r

9.14
7.06
6.06
4.71

3.57
3.50
3.38
2.51

54.4
75.9
77.6
56.3

0.18
0.28
3.64
23.2

18.0
4.39
2.48
2.16

0.00
0.00
0.00
0.00

32.8
39.6
36.8
25.7

7.36
7.32
6.40
4.45

9.39
12.7
12.3
8.51

0.02
0.50
7.33
35.1

2.93
8.51
9.07
6.34

0.00
0.00
0.00
0.00

(a) TI model
Impact
1yr
3yr
10yr

0.42
0.21
0.16
0.36

4.22
1.56
1.15
0.88

0.09
0.24
0.54
0.58

9.35
6.60
4.53
7.84

0.55
0.22
0.40
1.33

(b) Non-TI model
Impact
1yr
3yr
10yr

0.03
0.02
0.02
0.03

33.8
15.0
12.7
8.89

0.21
0.19
0.18
0.13

12.2
15.0
14.3
10.1

1.15
0.90
0.79
0.55

32

% Deviation From Steady State

10

20

30

40

0

10

20

30

40

20

30

40

-0.05

0

0.05

0.1

0

20

30

periods (infuarter)

1040(i) CN GNKPC Shock to US Price

0

10

20

30

40

0

0.02

10

20

30

40

10

20

30

0

0

20

30

20

30

periods (quarter)

10

TI modelNon-TI model(k)-4Risk Premium Shock10

10

Figure 1: IRFs of the Home Inflation Rate to Structural Shocks

periods (quarter)

-3

-2

-1

0

0.06

2

0

0.5

1

1.5

2

1

0.04

0

40

(c) CN Trend Inflation Shock

40

40

-0.5

0

0.5

-1

0

1

2

0

10

20

30

40

(d) US Trend Inflation Shock

0

10

20

30

40

(g) CN GNKPC Shock to CN Price (h) US GNKPC Shock to CN Price

0

0.5

0.08

0

(j) US GNKPC Shock to US Price

0.1

-0.8

-1.5

10

-0.6

-0.4

-1

-0.5

0

0

0
-0.2

(f) US Monetary Policy Shock

0.5

0.2

-0.08

-0.06

(e) CN Monetary Policy Shock

-0.1

-0.08

-0.04

-0.06
1

1.5

-0.02

2.5

-0.04

0

(b) US Productivity Shock
2

0.02
0

0

(a) CN Productivity Shock-0.02% Deviation From Steady State% Deviation From Steady State33

% Deviation From Steady State% Deviation From Steady State

34

20

30

40

40

40

-0.1

-0.05

0

0

20

30

periods (quarter)

1040(i) CN GNKPC Shock to US Price

0

10

20

30

40

-0.5

0

0.5

1

1.5

2

0

20

30

40

10

20

30

0.02

0.04

0.06

-5

0

5

10

15

0

0

20

30

20

30

periods (quarter)

10

TI modelNon-TI model(k)-5Risk Premium Shock10

10

Figure 2: IRFs of the Foreign Inflation Rate to Structural Shocks

periods (quarter)

10(j) US GNKPC Shock to US Price

-3

-2

-1

0

40

(c) CN Trend Inflation Shock

40

40

0

0.2

0.4

0.6

0.8

1

0

1

2

3

4

5

0

10

20

30

40

(d) US Trend Inflation Shock

0

10

20

30

40

(g) CN GNKPC Shock to CN Price (h) US GNKPC Shock to CN Price

0.08

-0.05

-0.02

30

30

-0.06

20

20

0

0.05

0.1

0.15

0

10

10

(f) US Monetary Policy Shock00(b) US Productivity Shock

-0.04

-0.02

0

0.02

0

(e) CN Monetary Policy Shock

-0.4

10

-3

-0.2

-0.1

0

-0.3

0

(a)
CN Productivity Shock
10-3

-2

-1

0

1

2

0.04

% Deviation From Steady State

% Deviation From Steady State

1

20

30

40

0

10

20

30

40

20

30

40

-0.2

-0.1

0

0.1

0

20

30

periods (quarter)

1040(i) CN GNKPC Shock to US Price

0

10

20

30

40

0

10

20

30

40

-2

0

2

4

0

20

30

periods (quarter)

10

40

-2

-1

0

1

2

3

-2

-1

0

1

0

0

20

30

20

30

periods (quarter)

10

TI modelNon-TI model(k)-3Risk Premium Shock
10

10

40

40

-2

0

2

4

6

0

10

20

30

40

(d) US Trend Inflation Shock

0

10

20

30

40

(g) CN GNKPC Shock to CN Price (h) US GNKPC Shock to CN Price-10(c) CN Trend Inflation Shock

Figure 3: IRFs of the Home Consumption Growth Rate to Structural Shocks

-0.6

-0.4

-0.2

0

(j) US GNKPC Shock to US Price

0.2

-2

-4

10

-1

-2

0

0

0

1

(f) US Monetary Policy Shock2(e) CN Monetary Policy Shock

-0.2

-5

0

5

-6

10

(b) US Productivity Shock

0

-0.1

0

0.1

-4

0

(a) CN Productivity Shock

0.2

0.4

0.6

0.8

% Deviation From Steady State% Deviation From Steady State35

% Deviation From Steady State

0

0.02

0.04

0.06

0

10

20

30

40

(a) CN Productivity Shock

30

40

10

20

30

40

periods (quarter)

40

20

30

periods (quarter)

10

40

0

10

20

30

40

(c) CN Trend Inflation Shock

2

-10

-5

0

5

0

10

20

30

40

(d) US Trend Inflation Shock

-3

-2

-1

0

1

-0.6

-0.4

-0.2

0

0

0

20

30

20

30

periods (quarter)

10

TI modelNon-TI model(k)-4Risk Premium Shock
10

10

40

40

-0.5

0

0.5

1

1.5

0

10

20

30

40

(g) CN GNKPC Shock to CN Price (h) US GNKPC Shock to CN Price

0.2

-0.2

0

0.2

0.4

Figure 4: IRFs of the Foreign Consumption Growth Rate to Structural Shocks

0

-4

30

-0.2

20

-3

0

10

-2

0.2

0

-1

40

0.4

30

0

20

0.6

0.8

1

10

(j) US GNKPC Shock to US Price0(i) CN GNKPC Shock to US Price

-3

20

-0.2

10

-2

-1

0.2

0

0

0

0

(b) US Productivity Shock(f) US Monetary Policy Shock

1

-0.1

0

0.1

0.2

0.4

0.6

(e) CN Monetary Policy Shock-0.02% Deviation From Steady State% Deviation From Steady State36

% Deviation From Steady State

0

0.02

0.04

0.06

0.08

0

10

20

30

40

(a) CN Productivity Shock

0

10

20

30

40

10

20

30

40

20

30

40

0

10

20

30

40

0

10

20

30

periods (quarter)

40

0

10

20

30

40

(c) CN Trend Inflation Shock

-2

-1.5

-1

-0.5

0

-0.1

0

0.1

0.2

0.3

0

0

20

30

20

30

periods (quarter)

10

TI modelNon-TI model(k)-3Risk Premium Shock
10

10

40

40

-0.5

0

0.5

1

1.5

2

-0.5

0

0.5

1

0

10

20

30

40

(d) US Trend Inflation Shock

0

10

20

30

40

(g) CN GNKPC Shock to CN Price (h) US GNKPC Shock to CN Price

0.4

0

0.2

0.4

0.6

0.8

1

Figure 5: IRFs of the Home Nominal Interest Rate to Structural Shocks

0

0.04

0.06

0.08

-0.06

periods (quarter)

10(j) US GNKPC Shock to US Price

0.1

0

0.1

0.2

0.3

0.4

0.02

0

0

(b) US Productivity Shock(f) US Monetary Policy Shock

0

0.01

0.02

0.03

0.04

-0.04

-0.02

0

0.02

(i) CN GNKPC Shock to US Price

-0.2

0

0.2

0.4

0.6

0.8

(e) CN Monetary Policy Shock-0.02% Deviation From Steady State% Deviation From Steady State37

% Deviation From Steady State% Deviation From Steady State

38

0

10

20

30

40

(a)
CN Productivity Shock
10-3

40

10

20

30

40

-0.1

-0.08

-0.06

-0.04

-0.02

0

0

20

30

periods (quarter)

1040(i) CN GNKPC Shock to US Price

0

20

30

periods (quarter)

10

40

10

20

30

40

0

1

2

3

0

0

20

30

20

30

periods (quarter)

10

TI modelNon-TI model(k)-4Risk Premium Shock
10

10

40

40

Figure 6: IRFs of the Foreign Nominal Interest Rate to Structural Shocks

0

0.2

0.4

0.6

(j) US GNKPC Shock to US Price

0.8

0
0

0.06

0.6
0.04

0.08

0.8

0.4

0

(c) CN Trend Inflation Shock

0.8

0

0.5

1

1.5

0

10

20

30

40

(d) US Trend Inflation Shock

-0.2

0

0.2

0.4

0.6

0

10

20

30

40

(g) CN GNKPC Shock to CN Price (h) US GNKPC Shock to CN Price

0.1

0
40

30

-0.1
30

20

0.02

20

10

-0.15

-0.1

-0.05

0

0.05

0.1

0.2

10

0

(b) US Productivity Shock(f) US Monetary Policy Shock

1

0

0.1

0.2

0.3

-0.05

0

0.05

0

(e) CN Monetary Policy Shock

-6

-4

-2

0

2

4

0.1

% Deviation From Steady State

% Deviation From Steady State% Deviation From Steady State

10

20

30

40

0

0.2

0.4

0.6

0.8

1

0

20

30

periods (quarter)

1040(i) CN GNKPC Shock to US Price

10

20

30

40

0

10

20

30

40

0

0.5

1

1.5

0

20

30

40

0

10

20

30

40

(c) CN Trend Inflation Shock

-15

-10

-5

0

5

0

10

20

30

40

(d) US Trend Inflation Shock

-1

0

1

2

-4

-2

0

2

4

0

0

20

30

20

30

periods (quarter)

10

TI modelNon-TI model(k)-3Risk Premium Shock
10

10

40

40

-20

-15

-10

-5

0

0

10

20

30

40

(g) CN GNKPC Shock to CN Price (h) US GNKPC Shock to CN Price
6

-5

0

5

10

Figure 7: IRFs of the Home Hours Worked to Structural Shocks

periods (quarter)

10(j) US GNKPC Shock to US Price

-2

-10

-1

-0.5

-1.5

0

0

(b) US Productivity Shock(f) US Monetary Policy Shock

0

-8

-6

-4

-2

0

(e) CN Monetary Policy Shock

40

-0.25

30

-1

20

-0.2

-0.8

10

-0.15

-0.6

0

-0.1

0

-0.4

(a) CN Productivity Shock
-0.05

0

-0.2

% Deviation From Steady State39

% Deviation From Steady State

0

0.02

0.04

20

30

40

0

10

20

30

40

-1

-0.5

0

0.5

(e) CN Monetary Policy Shock

10

20

30

40

1

2

0

10

20

30

40

4

6

8

10

-0.5

periods (quarter)

10

20

30

0

0

20

30

20

30

periods (quarter)

10

TI modelNon-TI model(k)-4Risk Premium Shock10

10

Figure 8: IRFs of the Foreign Hours Worked to Structural Shocks

periods (quarter)

0

40

40

-1

30

30

-2

20

20

2

10

10

0

0

0

(j) US GNKPC Shock to US Price
3

-4

-1.5

-1

-0.5

0

0.5

(i) CN GNKPC Shock to US Price

-0.4

0

-0.2
0

0.5

-2
-3

1

-0.3

0

40

(c) CN Trend Inflation Shock

-10

-5

0

5

10

0

10

20

30

40

(d) US Trend Inflation Shock

40

40

-15

-10

-5

0

0

10

20

30

40

(g) CN GNKPC Shock to CN Price (h) US GNKPC Shock to CN Price

1.5

-1

(f) US Monetary Policy Shock

0

-0.1

0

-1.5

-2

10

(b) US Productivity Shock

-0.06

-1

-0.5

0

-1.5

0

(a) CN Productivity Shock

-0.04

-0.02

% Deviation From Steady State% Deviation From Steady State

40