HIAS-E-19

Exchange Rates and Fundamentals:
A General Equilibrium Exploration
Takashi Kano
Graduate School of Economics, Hitotsubashi University, Naka 2-1, Kunitachi, Tokyo 186-8601, Japanrevised in April, 2019

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
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Exchange Rates and Fundamentals:
A General Equilibrium Exploration
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 15, 2019

Abstract
Engel and West (2005) show that the observed near random-walk behavior of nominal exchange
rates is an equilibrium outcome of a partial equilibrium asset approach when economic fundamentals
follow exogenous first-order integrated processes and the discount factor approaches one. In thispaper, I argue that the unit market discount factor creates a theoretical trade-oﬀ within a twocountry general equilibrium model. The unit discount factor generates near random-walk nominal
exchange rates, but it counterfactually implies perfect consumption risk sharing and flat money
demand. Bayesian posterior simulation exercises based on post-Bretton Woods data from Canada
and the United States reveal diﬃculties in reconciling the equilibrium random-walk proposition
within the canonical model; in particular, the market discount factor is identified as being muchsmaller than one. A relative money demand shock is identified as the main driver of nominal
exchange rates.
Key Words: Exchange rate; Present-value model; Economic fundamental; Random walk; Two-country model; Incomplete market; Cointegrated TFPs; Perfect risk sharing.
JEL Classification Number : E31, E37, and F41†

This is a revised version of a paper previously circulated under the title of “Exchange Rates and Fundamentals:
Closing a Two-country Model.” I would like to thank the editor, Kenneth West; anonymous referees, Kosuke Aoki,
Martin Berka, Dave Cook, Mario Crucini, Charles Engel, Hiroshi Fujiki, Ippei Fujiwara, Shin-ichi Fukuda, Ichiro
Fukunaga, Shinsuke Ikeda, Susumu Imai, Kazuko Kano, Munechika Katayama, Junko Koeda, Oleksiy Kryvtsov,
Takushi Kurozumi, Enrique Martı́nez-Garcı́a, Kiminori Matsuyama, Toshi Mukoyama, Tomoyuki Nakajima, Jim
Nason, Makoto Nirei, Michel Normandin, Yoshiyasu Ono, Ricardo Reis, Makoto Saito, Moto Shintani, Etsuro Shioji,
Christoph Thoenissen, and Takayuki Tsuruga; and seminar participants at Hitotsubashi University, the Institutefor Monetary and Economic Studies at the Bank of Japan, the Institute of Social and Economic Research at Osaka
University, Kyoto University, the National Graduate Institute of Policy Studies, the University of Tokyo, Victoria
University of Wellington, the University of Auckland, the Australian National University, the Reserve Bank of New
Zealand, the 2013 Summer Workshop on Economic Theory at Hokkaido University, the 2014 Canadian Economic
Association Meetings, the 2014 Japanese Economic Association Spring Meetings, the 2014 International Conferenceof the Society for Computational Economics (CEF 2014), the Conference on Macroeconomic Issues at Sogang University, the second international workshop for financial markets and nonlinear dynamics (FMND 2015), and the
9th International Conference on Computational and Financial Econometrics (CFE 2015) for their discussions anduseful suggestions. Hiroshi Morita provides excellent research assistance. I wish to thank the Seimeikai foundation,
the Hitotsubashi Institute for Advanced Study, and a grant-in-aid for scientific research from the Japan Society for
the Promotion of Science (numbers 24330060, 17H02542, and 17H00985) for their financial support. I am solely
responsible for any errors and misinterpretations in this paper.

1. Introduction
Few equilibrium models for nominal exchange rates systematically beat a naı̈ve randomwalk counterpart in terms of out-of-sample forecast performance. Since the study of Meese andRogoﬀ (1983), this robust empirical property of nominal exchange rate fluctuations has stubbornly
resisted theoretical attempts to understand the behavior of nominal exchange rates as an equilibriumoutcome of observed fundamentals. Many open-economy dynamic stochastic general equilibrium(DSGE) models also suﬀer from this problem. They fail to generate random-walk nominal exchange
rates along an equilibrium path because their exchange rate forecasts are closely related to observed
macroeconomic fundamentals.1
Engel and West (2005, hereafter EW) shed a new light on this vexing empirical property of
nominal exchange rates. Using a partial equilibrium asset approach, in which nominal exchange
rates reflect the present discounted values of expected future economic fundamentals, they showthat if economic fundamentals are integrated of order one (hereafter I(1)) and the discount factor
approaches one, then a nominal exchange rate follows a near random-walk process in equilibrium.
This equilibrium random-walk property is attributable to the fact that only the Beveridge-Nelson
trend components of the I(1) economic fundamentals are reflected in the present-value calculation
at the limit of the unit discount factor.
The I(1) property is likely to hold in actual data of economic fundamentals, particularly
monetary fundamentals, which are identified in neoclassical open-economy monetary models as themost important driver of nominal exchange rates.2 Subsequent studies, thus, have focused on theempirical validity of the requirement that the discount factor be close to one. Based on the monetary
fundamentals, Sarno and Sojli (2009) and Balke et al. (2013, hereafter BMW) identify a discountfactor using partial equilibrium asset models similar to that of EW, examine data on diﬀerent
currencies and sample periods, and estimate the discount factor to be distributed near to one. In
1Engel (2014) provides a recent survey of studies on nominal exchange rates.
For example, the reduced-form regression exercises of Mark (1995), Rapach and Wohar (2002), Mark and Sul(2001), and Cerra and Saxena (2010) show that the monetary fundamentals contain economically significant predictive
components of nominal exchange rates. The model of this paper would oﬀer a general equilibrium foundation forthese empirical results.
2

1

particular, the Bayesian unobservable component (UC) model of BMW estimates relative money
demand shocks to be a dominant underlying driver of a long sample of the British pound/U.S. dollar
rate. This empirical finding supports the conjecture of EW that persistent unobservable economic
fundamentals and a discount factor near one play significant roles in near random-walk nominal
exchange rates.
Nason and Rogers (2008, hereafter NR) examine EW’s proposition in a neoclassical twocountry monetary dynamic stochastic general equilibrium (DSGE) model. NR rely only on a subsetof the first-order necessary conditions (FONCs) of the two-country model and estimate Bayesianposteriors using a restricted UC model with post-Bretton Woods data of Canada and the UnitedStates. Characterizing the steady-state market discount factor as the relevant discount factor for
nominal exchange rates, they confirm EW’s proposition, finding that the market discount factor isclose to one. NR identify a shock to I(1) relative consumption to be the main driver of the nominal
exchange rate.
In this paper, I use a neoclassical two-country monetary DSGE model to explore EW’s proposition that unobserved, integrated economic fundamentals, and a discount factor near one, generaterandom-walk behavior in nominal exchange rates. My focus on the model is motivated by the paststudies that find the monetary fundamentals crucial for nominal exchange rates. In particular,
BMW and NR infer the unit discount factor depending on neoclassical open-economy monetarymodels. To identify relative money demand shocks, which are estimated by BMW to be the dominant unobservable persistent economic fundamental for nominal exchange rates, I also scrutinize
an open-economy general equilibrium model with explicit money demand.3
The general equilibrium approach in this paper indicates that a unit discount factor has
3

In contrast, the recent literature of exchange rates such as Chari et al. (2002), Benigno (2004), Steinsson (2008),
Iversen and Söderström (2014), and Engel (2014, 2018) commonly adopt open-economy new Keynesian (NK) models
with the Taylor type monetary policy as the data generating process of real and nominal exchange rates. The NK
framework typically determines the equilibrium dynamics of the real exchange rate, based on three equations: theNK Phillips curve, the risk sharing condition/the real uncover interest parity (RUIP) condition, and the Taylor rule.
As shown by Benigno (2004) and Engel (2014, 2018), what matter for the endogenous persistence of real exchange
rates are the degrees of price stickiness and monetary policy inertia. Common in the recent literature of NK models,the market discount factor is calibrated to a higher value around 0.99, but not estimated, because the restrictions
imposed on exchange rate data by the NK models identify weakly the underlying subjective discount factor.

2

implications for the joint dynamics of monetary fundamentals, such as consumption, money supply,and interest rates, that do not sit easily with observed data. The unit discount factor is suﬃcient,but not necessary to generate near random-walk behavior in the nominal exchange rate. In generalequilibrium, a unit market discount factor generates near random-walk nominal exchange rates but,when the consumption Euler equations are incorporated as a FONC, implies an implausibly highcorrelation between relative consumption and the real exchange rate (Backus and Smith 1993). A
near unit discount factor also implies inelastic money demand, which requires implausibly large
money supply and demand shocks. To reconcile the joint behavior of exchange rates and monetary
fundamentals, estimation in general equilibrium environment with a full set of FONCs may informon the relative roles of integrated, near random-walk fundamentals and a discount factor near one.
Because BMW and NR do not impose the consumption Euler equation on their UC models, thefull consideration of the FONCs explored by the general equilibrium approach in this paper mightyield diﬀerent inferences about the unit discount factor and the main drivers of nominal exchange
rates.
Bayesian posterior simulation exercises with a restricted UC model, based on post-Bretton
Woods data from Canada and the United States, reconcile the equilibrium random-walk exchange
rate behavior and the observed behavior of relative consumption and interest rates with a discountfactor estimated to be much smaller than one with the posterior mean of 0.537, and identify apersistent unobserved, money demand diﬀerential as the main source of persistence in the nominal
exchange rate.4 The estimated low value of the discount factor stems from the model’s failure in
explaining the data of the monetary fundamentals with the discount factor near one. The dominantrole for money demand shocks echoes the findings of EW, BMW, and Sarno and Schmelling (2014)— economic fundamentals of near random-walk nominal exchange rates should be unobservable

4

In the literature of the open-economy NK models such as Kollmann (2002) and Itsukhoki and Mukuhin (2017),
additive exogenous disturbances to the UIP condition are frequently identified as the dominant driver of real andnominal exchange rates. BMW observe that exogenous risk premium shocks play only a minor role in nominal
exchange rate fluctuations once relative money demand shocks are correctly identified. This finding also echoes theresults of the likelihood-based inferences of the open-economy NK models by Bergin (2006) and Lubik and Schorfheide(2006).

3and nominal, such as money demand shocks.5
The posterior inference of such a low discount factor suggests significant misspecification inthis canonical open economy model. The empirical result of this paper, hence, uncovers diﬃculties
that the literature needs to overcome in explaining data variations in nominal exchange rates andthe corresponding macroeconomic fundamentals, jointly and consistently, through the lens of openeconomy general equilibrium models.6
Potential approaches to reconciling the finding of this paper should generate a weak association
between the real exchange rate and relative consumption at the unit discount factor. A promising
future extension of the canonical monetary DSGE model in this paper, thus, allows for home bias
in preference with a high degree of trade elasticity, as shown by Corsetti et al.(2008) as a resolution
for Backus and Smith’s (1993) observation.
The remainder of this paper is organized as follows. In the next section, I introduce the twocountry incomplete market model employed in this paper. Section 3 then derives and discusses theequilibrium random-walk property of nominal exchange rates and the Backus and Smith puzzle ofthe perfect correlation between relative consumption and the real exchange rate at the limit of theunit market discount factor. After reporting the main results of the Bayesian exercises in section
4, I conclude in section 5.

2. A two-country incomplete markets model
2.1. The model
In this paper, I investigate a canonical incomplete markets model with two countries, home(h) and foreign (f ). Each country is populated by a representative household whose objective is
5

It is worth noting that the standard open-economy NK models with the Taylor rule are still absent from unobserved, near I(1) nominal economic fundamentals. I will discuss this point in the conclusion.
6
As related research to this paper, Kano and Morita (2015) apply the model of this paper to post-Plaza Accord
data of the Japanese yen/U.S. dollar to understand the anecdotal “Soros chart” (i.e., the observed high correlation
between the near random-walk Japanese yen/U.S. dollar rate and the two-country diﬀerential in the monetarybase). Modeling the reserve markets and the money creation processes of the two countries, their Bayesian posterior
simulation exercise finds a better match of the model to the data: the posterior mean of the subjective discountfactor is 0.96.

4

the lifetime utility"#$%
Mi,t+j
j
β
E
ln
C
+
φ
ln
for i = h, f , where β, Ci,t , Mi,t , and
t
i,t+j
i,t+j
j=0
Pi,t+j!∞Pi,t represent the subjective discount factor, the ith country’s consumption, money stock, and priceindex, respectively. As in NR, money-in-the utility delivers a tractable money demand functionthat characterizes an equilibrium nominal interest rate. I assume that the utility value of money issubject to a persistent money demand shock φi,t .
The representative home and foreign households maximize their lifetime utility subject to the
f
f
f
h
h
h
+ St (1 + rh,t−1)Bh,t−1+home budget constraint Bh,t
+ St Bh,t+ Ph,t Ch,t + Mh,t = (1 + rh,t−1)Bh,t−1

Mh,t−1 + Ph,t Yh,t + Th,t , and its foreign counterpart

h
h
Bf,t
Bf,t−1
f
h
+
B
+
P
C
+
M
=
(1
+
r
)
+
f,t
f,t
f,t
f,t−1
f,t
St
St

f
f
l
l
(1 + rf,t−1)Bf,t−1+ Mf,t−1 + Pf,t Yf,t + Tf,t respectively, where Bi,t, ri,t, Yi,t , Ti,t , and St denote the ith

country’s holdings of the lth country’s nominal bonds at the end of time t, the ith county’s returns
on the lth country’s bonds, the ith country’s output level, the ith country’s government transfers,
and the level of the bilateral nominal exchange rate, respectively. Each country’s output Yi,t is given
as an exogenous endowment following a stochastic process Yi,t = yi,t Ai,t , where yi,t is the transitory
component and Ai,t is the permanent component. I interpret the permanent component Ai,t as the
TFP in the underlying production technology.
Each country’s government transfers the seigniorage to the household as a lump-sum. Hence,
the government’s budget constraint is Mi,t − Mi,t−1 = Ti,t , for i = h, f . The money supply Mi,t isτ
specified to be driven by permanent and transitory components, respectively Mi,tand mi,t : Mi,t ≡τmi,t Mi,tfor i = h, f .

To close the model, I allow for a debt-elastic risk premium in the interest rates faced only bylllτ¯ − 1}] with d¯ ≤ 0 and ψ > 0 for l = h, f ,the home country: rh,t
= rw,t[1 + ψ{exp(−Bh,t
/Ml,t
+ d)
l
where rw,tis the equilibrium risk-free interest rate of the lth country’s bond. The debt-elastic

risk premium prevents the home household from running up infinite debt, and addresses the nonstationarity associated with an incomplete international financial market.7 The risk premium is
7

As characterized by Schmitt-Grohé and Uribe (2003) in a small open-economy model and Boileu and Normandin(2008) in a two-country international business cycle model, a debt-elastic risk premium has served as a popular
instrument to induce the stationarity of the net foreign asset distribution. A non-exhaustive list of studies that
adopt a debt-elastic risk premium as a device to avoid the non-stationarity problem in open-economy DSGE models
includes Nason and Rogers (2006), Adolfson et al. (2007), Kano (2009), Justiniano and Preston (2010), Garcı́a-Cicco

5

given as an externality: the household does not internalize the eﬀect of the debt position on therisk premium when maximizing the lifetime utility function. The foreign country’s interest rate arellrf,t= rw,tfor l = h, f .8

Following EW and BMW, I assume that purchasing power parity (PPP) holds only up toa persistent PPP deviation shock ln qt : St Pf,t = Ph,t qt , where the PPP deviation qt follows anexogenous stochastic process. The market-clearing conditions of the two countries’ bond marketsffhh
are Bh,t+ Bf,t= 0 and Bh,t
+ Bf,t= 0. Along the equilibrium path, the world net supply of nominal

bonds is zero on a period-by-period basis.
I assume that the logarithms of the total factor productivity (TFP) and the permanentτ
component of the money supply, respectively ln Ai,t and ln Mi,t, are I(1) for i = h, f , and theττcross-country diﬀerential in the permanent component of money supply, ln Mh,t
− ln Mf,t, is alsoτ
I(1). I specify each country’s monetary growth rate ∆ ln Mi,tto be an independent AR(1) process:ττ∆ ln Mi,t= (1 − ρM ) ln γM + ρM ∆ ln Mi,t−1+ ϵiM,t for i = h, f , where ln γM and ρM are the mean

and AR root, respectively, of the money supply growth rate common to the two countries.
To guarantee the balanced growth path of the general equilibrium model, I assume thatthe cross-country TFP diﬀerential, ln at ≡ ln Ah,t − ln Af,t , is integrated of order zero (I(0)).9
The stationary TFP diﬀerential implies that the TFP of the home country must be cointegrated
with that of the foreign country. ln Ah,t and ln Af,t are cointegrated with the cointegrated vector[1, −1] and have the error correction models: ∆ ln Ah,t = ln γA − λ2 (ln Ah,t−1 − ln Af,t−1 ) + ϵhA,t and∆ ln Af,t = ln γA + λ2 (ln Ah,t−1 −ln Af,t−1 )+ϵfA,t , where γA > 1 is the common drift term and λ ∈ [0, 1)
is the adjustment speed of the error correction mechanism.
The stochastic process of the logarithm of the transitory output component for each country,
ln yi,t , is specified as the following AR(1) process: ln yi,t = (1−ρy ) ln yi +ρy ln yi,t−1 +ϵiy,t , for i = h, f .
Similarly, the stochastic process of the logarithm of the transitory money supply component for each
et al. (2010), and Bodenstein (2011).
8
A one-sided debt-elastic risk premium is suﬃcient to close the model.
9
I allow the stochastic trends of the two countries, which are interpreted as their TFPs, to be cointegrated, asemphasized in recent papers by Mandelman et al. (2011), Rabanal et al. (2011), and Ireland (2013) in the context
of international business cycles. In this case, because the TFP diﬀerential is stationary in population, a balanced
growth path is guaranteed to exist in equilibrium.

6

country, ln mi,t , is specified as the following AR(1) process: ln mi,t = (1 − ρm ) ln mi + ρm ln mi,t−1 +ϵim,t , for i = h, f . The three other structural shocks, the home and foreign money demand shocks
φh,t and φf,t , respectively, and the PPP shock qt , follow persistent stationary processes. Specifically,
they are characterized by AR(1) processes as follows: ln φi,t = (1 − ρφ ) ln φ + ρφ ln φi,t−1 + ϵiφ,t , fori = h, f and ln qt = ρq ln qt−1 +ϵq,t . I assume that all structural shocks are distributed independently.
2.2. The log-linear approximation of the stochastically de-trended systemτlτDefine stochastically de-trended variables as ci,t ≡ Ci,t /Ai,t , pi,t ≡ Pi,t Ai,t /Mi,t, bli,t ≡ Bi,t
/Ml,t
,
i
i
τ
τ
τ
τ
γA,t
≡ Ai,t /Ai,t−1 , γM,t≡ Mi,t/Mi,t−1, and st ≡ St Mf,t/Mh,t. I construct the stochastically de-trended system of the FONCs, as reported in the accompanying online appendix. The resulting tenffhhequations determine the ten endogenous variables ch,t , cf,t , ph,t , st , bhh,t , bfh,t , rh,t
, rh,t, rw,t, and rw,t
,
f
f
h
h
given nine exogenous variables γM,t, γM,t, γA,t, γA,t, at , mh,t , mf,t , yh,t , and yf,t .

Let x̂ denote a percentage deviation of any variable xt from its deterministic steady statevalue x∗ , x̂ ≡ ln xt − ln x∗ . Additionally, let x̃ denote a deviation of x from its deterministic
steady state, x̃ = x − x∗ . The online appendix reports the results of the ten equations of the
log-linearized FONCs. Under the assumption of the symmetric two-countries with d¯ = 0 (i.e.,the deterministic steady state value of bond holdings is assumed to be zero), the resulting linear
rational expectations (LRE) system is further simplified as a three-equation representation with
two-country relative variables. To see this, let ct , yt , mt , and φt denote the de-trended consumption
ratio, the de-trended output ratio, the transitory money supply ratio, the money demand shock
ratio between the two countries, ct ≡ ch,t /cf,t , yt ≡ yh,t /yf,t , mt ≡ mh,t /mf,t , and φt ≡ φh,t /φf,t ,respectively. Furthermore, let Mtτ denote the ratio of the permanent money supplies of the homeττ
and foreign countries Mh,t/Mf,t; let Mt denote the ratio of money supplies of the home to the foreigncountries Mh,t /Mf,t ≡ mt Mtτ ; let γM,t denote the ratio of the permanent money supply growth ratefh
γM,t ≡ γM,t/γM,t; let Ct denote the ratio of the consumptions of the home and foreign countries

Ch,t /Cf,t . The log-linearized FONCs then reduce to the following three expectational diﬀerence
equations with respect to the three endogenous variables ŝt , ĉt , and b̃t , given the six exogenous

7

variables γ̂M,t , m̂t , ât , ŷt , φ̂t , and q̂t :ŝt = κEt ŝt+1 − (1 − κ)ĉt + (1 − κ)(m̂t − φ̂t + q̂t − ât ) + κEt γ̂M,t+1 − ψκ(1 − κ)b̃t ,ŝt + ĉt − q̂t + ât = κEt (ŝt+1 + ĉt+1 − q̂t+1 + ât+1 ) + (1 − κ)(m̂t − φ̂t ) + κEt γ̂M,t+1 ,b̃t = β −1 b̃t−1 + p∗h y ∗ (ŷt − ĉt ),(1)

where y ∗ = y/4 and y = yh = yf under the assumption of the symmetric two countries.
The first equation of LRE system (1) represents the stochastically de-trended uncovered
interest rate rarity (UIP) condition; the second equation is the cross-country diﬀerence in the Euler
equation; and the third equation is the law of motion of the net foreign asset position, after solving
for the interest rate diﬀerential through the money demand functions of the two countries. As
shown by the first equation above, in this model the steady-state nominal market discount factor,κ ≡ 1/(1 + r∗ ) = β/γM , is the discount factor that governs the stochastic process of the de-trended
nominal exchange rate ŝt .

3. A general equilibrium analysis of random-walk exchange rates

3.1. Equilibrium random-walk property of nominal exchange rates
I will now explore how the equilibrium random-walk property of the exchange rate holds inthis two-country model. Rewriting the first equation of LRE system (1) by restoring stochastic
trends and solving the resulting expectational diﬀerence equation of the nominal exchange rateln St by forward iterations under a suitable transversality condition provides the PVM:
ln St = (1 − κ)

∞
&
j=0

#
$
κj Et ln Mt+j − ln Ct+j − ψκb̃t+j − ln φt+j + ln qt+j .

(2)

If the sum of economic fundamentals in the right-hand side of PVM (2) is I(1), so is the exchange
rate. The online appendix shows that, after rearranging PVM (2) in several steps, the currency
return is8∆ ln St =1−κ(ln St−1 − ln Mt−1 + ln Ct−1 + ln φt−1 − ln qt−1 ) + ψ(1 − κ)b̃t−1 + us,t ,κ

(3)where us,t is an i.i.d., rational expectations error. NR’s argument that PVM (2) implies an errorcorrection representation of the currency return ∆ ln St , in which ∆ ln St depends on the lagged
error correction term ln St−1 − ln Mt−1 + ln Ct−1 , holds in this model.
Rewriting the second equation of LRE system (1) by restoring stochastic trends and solving
the resulting expectational diﬀerence equation by forward iterations under a suitable transversality
condition provides the following cross-equation restriction (CER):
ln St − ln Mt + ln Ct − ln qt =κρMκ(1 − ρm )1−κγM,t −
ln mt −
ln φt .
1 − κρM
1 − κρm
1 − κρφ(4)

Imposing CER (4) on error-correction process (3) provides the equilibrium currency return∆ ln St = ψ(1 − κ)b̃t−1 +(1 − κ)ρM
γM,t−11 − κρM(1 − κ)(1 − ρφ )(1 − κ)(1 − ρm )+ln φt−1 −ln mt−1 + us,t . (5)1 − κρφ1 − κρm

Equation (5) shows that any dependence of the currency return on past information emerges throughthe persistence of the net foreign asset position, the money supply growth diﬀerential, the transitory
money demand shock diﬀerential, and the transitory money supply diﬀerential.
The important implication of equilibrium currency return (5) is that the logarithm of thenominal exchange rate follows a Martingale diﬀerence sequence at the limit of κ → 1 becauselimκ→1 Et ∆ ln St+1 = 0. Therefore, in this model, the exchange rate behaves like a random walkwhen the market discount factor approaches one along the equilibrium path of the two-country
model. Equation (5) exhibits no dependence of the equilibrium currency return on past informationin this case. Hence, the equilibrium random-walk property of the exchange rate, as found in EW
and NR, is also present in this model.
In the limiting case of a unit market discount factor, the equilibrium currency return isdominated by the i.i.d. rational expectations error us,t = (Et − Et−1 )∆ ln St = (Et − Et−1 )ŝt + ϵM,twhere ϵM,t ≡ ϵhM,t − ϵfM,t denotes the relative permanent money supply shock. The online appendix
9

shows that, in the special case of two symmetric countries, assuming d¯ = 0 and yh = yf , theequilibrium de-trended exchange rate is a linear function of b̃t−1 , ln at , ln mt , ln φt , ln yt , ln qt , andγ̂M,t :ŝt =βη − 1
1 − βη1−κ
1−κb̃ −ln at +
ln mt −ln φt
∗ ∗ t−1βph y
1 − βη(1 − λ)1 − κρm
1 − κρφ1 − βη
1 − βη
κρM−
ln yt +
ln qt +
γM,t , (6)1 − βηρy1 − βηρq
1 − κρM

where the constant η, which is less than one, approaches one at the limit of κ → 1.10 The rational

1
expectations error, us,t , is then given as a linear function of the structural shocks: us,t = 1−κρϵM,t −
M
1−βη
1−βη
1−βη
1−κ
1−κ
ϵ + 1−κρ
ϵm,t − 1−κρϵφ,t − 1−βηρ
ϵy,t + 1−βηρ
ϵq,t , where ϵA,t ≡ ϵhA,t − ϵfA,t , ϵm,t ≡ ϵhm,t − ϵfm,t ,1−βη(1−λ) A,tmyqφϵφ,t ≡ (ϵhφ,t − ϵfφ,t ), and ϵy,t ≡ ϵhy,t − ϵfy,t denote shocks to the relative TFP, the relative transitory
money supply, the relative money demand shock, and the relative transitory income.
The steady-state gross growth rate of money is empirically close to one, γM ≈ 1.11 Therefore,
the limit of κ ≡ β/γM → 1, by construction, implies that the subjective discount factor, β, mustalso be close to one. In this limiting case, the shock to the monetary supply growth diﬀerential,
ϵM,t , dominates the rational expectations error, us,t , and, as a result, the random walk behaviorof the exchange rate: limκ→1 ∆ ln St = limκ,β,η→1 us,t = 1−ρ1 M ϵM,t . Therefore, no transitory shockmatters for the random-walk behavior of the nominal exchange rate.
Equation (5), also, reveals that a market discount factor close to one is suﬃcient, but notnecessary, for the random-walk exchange rate.12 Regardless of the value of κ, if the permanentmoney supply diﬀerential, the transitory money supply diﬀerential, or the money demand shock
diﬀerential follows a near random-walk process, so does the nominal exchange rate. In that case,1−βη
the rational expectations error us,t implies limρM →0, ρφ , ρm →1 ∆ ln St = ϵM,t +ϵm −ϵφ − 1−βη(1−λ)ϵA,t −1−βη
1−βη
ϵ + 1−βηρ
ϵq,t . A lower market discount factor allows the model to generate the near random1−βηρy y,tqwalk nominal exchange rate by a richer and more flexible impulse structure, with multiple structural10

As defined in the online appendix, the constant η is one of the two roots of the expectational diﬀerence equationof the de-trended net foreign asset position b̃t . A simple calculation shows that the equilibrium currency return (5)can be derived directly from the CER once the approximated relation ŝt ≈ ln St − ln Mtτ is recognized.
11
The sample mean of the M1 money supply’s gross growth rate is 1.016 for Canada and 1.014 for the UnitedStates.
12
I appreciate the editor, Kenneth West, for pointing out this implication of the model.

10shocks. Near random-walk nominal exchange rate behavior could be driven by all the exogenous
economic fundamentals in this model.
3.2. Backus and Smith’s puzzle at the limit
This model has an unrealistic implication for the equilibrium dynamics of consumption diﬀerential, ln Ct , when the discount factor approaches one. At the limit of the unit discount factor, theperfect correlation between relative consumption and the real exchange rate (i.e., the one-to-one
co-movement in consumption in terms of a common currency between the two countries) emerges,
even under incomplete international financial markets. To observe this property, taking the firstdiﬀerence of CER (4), substituting equilibrium currency return (5) into the result, and exploiting
rational expectations error us,t yields the following equilibrium consumption diﬀerential dynamics:∆ ln Ct = ∆ ln qt − ψ(1 − κ)b̃t−1 +1 − βη
1 − βη
1 − βη
ϵA,t +
ϵy,t −
ϵq,t .
1 − βη(1 − λ)1 − βηρy1 − βηρq(7)

Except through the net foreign asset position, no monetary shock directly matters for the change
in the equilibrium consumption diﬀerential. As in the standard international business cycle model,
only real shocks to the endowments and the PPP deviation aﬀect the equilibrium consumption
allocation between the two countries.
Taking the limit of equation (7) above when κ → 1 and γM ≈ 1 (which imply that η → 1 andβ ≈ 1) results in limκ,β,η→1 ∆ ln Ct = ∆ ln qt . The consumption diﬀerential becomes unrelated toany shocks to the endowments of the two countries, but is perfectly correlated with the exogenous
real exchange rate. The intuition behind this result is as follows. In this incomplete markets model,
consumption in each country is determined by splitting the global aggregate endowment between
the two countries in each period. The portion of the global aggregate endowment allocated to one
country is simply given as the present discounted values of the expected future relative endowmentsof this country to the other. Because the endowment diﬀerential is stationary, due to the balanced
growth restriction, the unit discount factor and γM ≈ 1 make the portion converge to a constant,which is one-half in the case of two symmetric countries. Consumption in both countries, hence,
responds to any endowment shocks in the same fashion. As a result, when the discount factor is11close to one, relative consumption depends neither on permanent nor transitory endowment shocks.
The only shock that aﬀects relative consumption is the real exchange rate qt .
3.3. Inelastic money demand
The two-country model of this paper also characterizes the analytical closed-form solutionsof the nominal interest rates along the equilibrium path. The online appendix shows that thefhequilibrium interest rate diﬀerential (1 + rt ) ≡ (1 + r̂h,t) − (1 + r̂f,t
) is
'
(
ρM
1 − ρm1 − ρφ(1 + rt ) = (1 − κ)γM,t −
ln mt +
ln φt .
1 − κρM
1 − κρm
1 − κρφ(8)

If the AR root of the money supply growth diﬀerential, ρM , is close to zero, then the main determinants of the nominal interest rates are the money demand shock diﬀerential and the transitory
money supply diﬀerential, ln φt and ln mt .
At the limit of the unit market discount factor, the interest rate diﬀerential becomes insensitive
to domestic monetary shocks because money demand functions are perfectly flat. In fact, under
the presumption of ρM = 0, taking the limit of equation (8) gives (1 + rt ) ≈ limκ→1 (1 − κ)(ln φt −ln mt ). The interest rate diﬀerential, hence, becomes dependent only on the transitory money
supply diﬀerential ln mt and the money demand shock diﬀerential ln φt with very small sensitivities.
Inelastic money demand means that the two monetary shocks would have to be extremely volatile
to explain the actual data variations in the nominal interest rates.

4. A Bayesian unobserved component approachThis section empirically explores the question of how the unrealistic restrictions implied by
the unit market discount factor aﬀect posterior inferences about the two-country model. I utilizea Bayesian UC approach to estimate the two-country model, using data of the nominal exchange
rate, the consumption diﬀerential, the money supply diﬀerential, the output diﬀerential, and theinterest rate diﬀerential. The online appendix describes the Bayesian UC approach used in thispaper, in detail.
124.1. Data and prior construction
The two countries that I empirically examine in this paper are Canada and the United States,as the model’s home and foreign countries, respectively. I examine post-Bretton Woods quarterly
data for these two countries because these data are likely to satisfy the model assumption of aflexible exchange rate regime under nearly perfect international capital movements between the twocountries. The data span the period from Q1:1973 to Q4:2007.13
Table 1 reports the prior distributions of the structural parameters of the two-country model.
Since an important goal of this paper’s empirical investigation is to draw a posterior inference on themarket discount factor κ ≡ β/γM , I use a uniform prior distribution of κ and let the data dictate theposterior position of κ, given the identification of the restricted UC model. The prior distributionof the mean gross monetary growth rate, γM , is intended to tightly cover its sample counterpartsin both countries through the Gamma distribution, with a mean of 1.015 and standard deviation of
0.005. In contrast, the prior distribution of the subjective discount factor, β, is uniformly distributed
between zero and one. As a result, the prior distribution of the market discount factor κ is wellapproximated as the uniform distribution spread over the support of the unit interval.
To guarantee the stationarity of the de-trended net foreign asset position, b̃t , the debt elasticityof the home risk premium, ψ, should be positive. I therefore set the prior distribution of ψ to theGamma distribution, with a mean of 0.010 and standard deviation of 0.001. Closing the model
also requires the technological diﬀusion speed, λ, to be positive but less than one. This necessary
condition for the equilibrium-balanced growth path elicits the prior distribution of λ as the Beta
distribution, with a mean of 0.010 and standard deviation of 0.001. The slow technological diﬀusion,
which the prior mean of λ implies, is intended to capture the slow-moving time-series properties
observed in the actual consumption and output diﬀerentials between Canada and the United States.
The prior distribution of the mean monetary demand shock φ follows the Gamma distribution, witha mean of 1.000 and small standard deviation of 0.010. The small standard deviation assumes, a

13

The appendix below provides a detailed description of the source and construction of the data examined in thispaper.

13

priori, that the monetary demand shock has no eﬀect on the deterministic steady state.
I admit a small persistence of the permanent money growth rate by setting the prior distribution of the AR(1) coeﬃcient ρM to the Beta distribution, with a mean of 0.100 and standard
deviation of 0.010. The PPP deviation shock (i.e., the real exchange rate shock) is presumed to bevery persistent, as observed by many past empirical studies on the real exchange rate. The AR(1)coeﬃcient of the real exchange rate, ρq , is then accompanied by the Beta prior distribution, witha mean of 0.850 and standard deviation of 0.100. This prior distribution mimics fairly well theposterior distribution of the same structural parameter reported in Figure 3 of BMW, who used along annual sample of data from the United Kingdom and the United States.14 On the other hand,
there is no robust empirical consensus on the extent of the persistence of the money demand shock.
Hence, I allow the prior distribution of the AR(1) coeﬃcient of the money demand shock, ρφ , tobe distributed around 0.850 following the Beta distribution, with a mean of 0.850 and a large standard distribution of 0.100. The prior distribution’s 95% coverage is [0.607 0.983], which also covers
the corresponding posterior distribution displayed within Figure 3 of BMW. To better identify thepermanent components of the money supplies and TFPs of both countries, while avoiding overprametarizing the model, I assume that the corresponding transitory components are white noiseby setting the prior mass points of the AR(1) coeﬃcients ρm and ρy to zero. Following NR, I also
allow for the deterministic time trend in the exchange rate, γS , with the normal prior distribution
with the zero mean and the large standard deviation of 1.500. Finally, the prior standard deviationsof all the structural shocks are assumed to share the identical inverse-Gamma distribution, witha mean of 0.010 and standard deviation of 0.010. Below, I refer to this prior specification as the
Benchmark model.
4.2. Main Results
The second, third, and fourth columns of Table 2 describe the posterior distributions of thestructural parameters of the Benchmark model. The most striking posterior inference conveyed by
14

In fact, the 95% interval of [0.607 0.983] includes the most inferences on real exchange rate persistence establishedin major past studies (see, e.g., Rogoﬀ 1996 and Lothian and Taylor 2000).

14

these columns is that the market discount factor κ is identified as being far below one. As displayedin the first row, the data pins down the location of κ very tightly around the posterior mean of 0.537,
with the standard deviation of 0.041. This posterior distribution of the market discount factor is
too low to guarantee the second necessary condition of the equilibrium random-walk exchange rateestablished by EW and NR (i.e., that the market discount factor is suﬃciently close to one). The
other significant posterior inference of the Benchmark model is found on the stochastic process ofthe money demand shock diﬀerential, ρφ and σφ . The data show a more persistent and volatilemoney demand shock diﬀerential compared to the prior specification of the Benchmark model.
Notice that the posterior mean of ρφ is 0.997 and almost 10% larger than its prior mean value; theposterior mean of σφ is 0.027 and 17% larger than its prior mean value. The very persistent money
demand diﬀerential shock provides evidence that such a structural shock could play a significant
role in actual exchange rate movements.
Does this lower market discount factor impair the model’s fit to the actual exchange rate
movements? The answer is clearly no. The estimated Benchmark model is indeed successful inexplaining the historical trajectory of the exchange rate. Figure 1(a) plots the actual Canadiandollar against the U.S. dollar as the solid black line. The figure also displays the posterior mean of
the in-sample prediction of the exchange rate by the Benchmark model as the dot-dashed block line
accompanied by the corresponding 95% Bayesian highest probability density (HPD) interval as thegray area. Observe almost no visual discrepancy between the actual outcome and the prediction of
the exchange rate. The HPD interval is very narrow and includes the actual exchange rate in almost
all the sample periods. The Benchmark model, hence, tracks the actual exchange rate closely.
A natural but crucial question is as follows: Why does the Benchmark model fit the seemingly
random-walk exchange rate fairly well even with such a low market discount factor? To answer thisquestion, it is worth remembering the theoretical implication of the equilibrium currency return
equation (5): if the three exogenous stochastic impulses — the permanent money supply diﬀerential, the transitory money supply diﬀerential, and the money demand shock diﬀerential — follow
near random-walk processes, so does the exchange rate. Note that the transitory money supply

15

diﬀerential is assumed to be white noise with ρm = 0 for identification purposes. As shown in Table
2, the data require in the Benchmark model that the permanent money supply diﬀerential and themoney demand shock diﬀerential follow near random walks; the posterior mean of ρM is 0.088 andthat of ρφ is 0.997. Therefore, even with such a low market discount factor, the Benchmark model
replicates the near random-walk property of the exchange rate data.
To verify the above conjecture graphically, Figure 1(b) displays the in-sample prediction of theexchange rate simulated by the Benchmark model but fixing ρφ and ρM to their prior values of 0.850
and 0.100. The figure plots the posterior mean of the in-sample prediction of the exchange rate as
the dot-dashed black line and the corresponding 95% HPD interval as the gray area accompanied by
the actual exchange rate data, as the solid black line. Observe that the mean in-sample prediction
is far above the actual exchange rate. The 95% HPD interval is very wide and does not containthe actual exchange rate over the whole sample period. The near random-walk processes of themoney demand shock diﬀerential and permanent money supply diﬀerential, therefore, result in the
successful in-sample prediction of the exchange rate of the Benchmark model even with such a low
discount factor.
Table 3 reports the forecast error variance decompositions (FEVDs), estimated in the Benchmark model, of the three endogenous variables, the exchange rate, the consumption diﬀerential, andthe interest rate diﬀerential, into the six structural shocks to the permanent money supply diﬀerential shock ϵM,t , the TFP diﬀerential shock ϵA,t , the transitory money supply diﬀerential shock ϵm,t ,the transitory output diﬀerential shock ϵy,t , the PPP deviation shock ϵq,t , and the money demand
diﬀerential shock ϵφ,t , over the forecast horizons of the impact, one year, three years, five years, andten years. Window (a) corresponds to the exchange rate; window (b) the consumption diﬀerential;
and window (c) the interest rate diﬀerential.
Window (a) in Table 3 clearly shows that the most important structural shock for the nearrandom-walk exchange rate between Canada and the United States is the persistent money demand
diﬀerential shock in conjunction with the permanent money supply diﬀerential shock; about 65%of the FEVD of the exchange rate is attributed to the former shock, and about 30% to the latter16

shock over all the forecast horizons. This inference about the main drivers of nominal exchange rates
echoes the findings of the past studies by EW, BMW, and Sarno and Schmelling (2014): economic
fundamentals of near random-walk exchange rates should be unobservable and nominal, such as amoney demand diﬀerential shock.
Windows (b) and (c) of Table 3 display the FEVDs of the consumption diﬀerential andthe interest rate diﬀerential in the Benchmark model. Observe the dominant roles of the TFP
diﬀerential shock in the consumption diﬀerential and the transitory money supply diﬀerential shockin the interest rate diﬀerential. The former result fully reflects the data aspect that the consumption
diﬀerential shares the same stochastic trend as the output diﬀerential. The latter result is consistent
with the theoretical implication of the Benchmark model for the equilibrium interest rate diﬀerential,
given a highly persistent money demand diﬀerential shock.
4.3. Understanding the lower discount factor: The High Discount Factor model
Why does the Benchmark model draw the sharp inference of such a lower discount factor?
To dig deeper into this question, in the online appendix, I derive two posterior predictive densitiesof the one-period-ahead forecast errors implied by the following two models. The first posterior
predictive density simply exploits the posterior distributions of the structural parameters of the
Benchmark model. The second one, on the other hand, uses the same posterior distributions of
the structural parameters of the Benchmark model except that I draw the subjective discount factor
β from a new prior distribution that is the Beta distribution with the mean of 0.999 and standard
deviation of 0.001. I refer to this new specification as the High Discount Factor (HDF) model.15
Comparing the two posterior predictive densities of the forecast errors uncovers the data aspects inwhich the high discount factor of 0.999 deteriorates the fit of the Benchmark model significantly.
The HDF model yields much larger forecast errors for the consumption diﬀerential and themoney supply diﬀerential than does the Benchmark model. The large forecast error of the HDF
model in the consumption diﬀerential results from the model’s implying tight positive correlation
15

Note that this prior construction of the subjective discount factor is almost the same exercise as calibrating themarket discount factor to the unconditional mean of the nominal interest rates observed in actual data.

17

between the consumption diﬀerential and the PPP deviation shock. It is the high dependence ofthe consumption diﬀerence on the PPP deviation shock that leads to the significant failure in the
HDF model to explain the actual data of the consumption diﬀerential. The data, as a result, forces
the subjective discount factor to be much lower to increase the empirical fit as in the Benchmark
model.
Understanding the diﬃculty of the HDF model in explaining the money supply diﬀerential
requires several steps. When the discount factor κ approaches one, the interest rate diﬀerential
becomes dependent only on the transitory money supply diﬀerential and the money demand shock
diﬀerential with very small sensitivities. Because of the flat money demand function, the forecast of
the money demand shock diﬀerential should be very volatile to explain the actual data of the interest
rate diﬀerential. Transitory exchange rate equation (6), then, predicts that under the HDF model
the forecast of the transitory exchange rate should be dominated by the volatile forecast of themoney demand shock diﬀerential. To keep the fit of the model to the actual ln St good, the volatile
forecast of ln st should be oﬀset by a volatile forecast of ln Mtτ . The forecast of the money supply
diﬀerential ln Mt , therefore, should be counterfactually dominated only by the volatile forecast of
the permanent money supply diﬀerential ln Mtτ .16
In sum, the high discount factor, as in the HDF model, imposes counterfactual restrictions
on the consumption diﬀerential and the money supply diﬀerential. In the Benchmark model, thetwo aspects of the data require β to be much lower than the prior value of the HDF model. These
restrictions, indeed, are so strictly binding that they jointly pin down the low value of β with a highdegree of precision, as shown by the corresponding small posterior standard deviation reported inTable 2.
Finally, I estimate the HDF model. The fifth, sixth, and seventh columns of Table 2 correspond to the posterior distributions of the structural parameters under the HDF model. Observethat the resulting posterior distributions of both the market and subjective discount factors aremuch closer to one, with posterior means of 0.950 and 0.998, respectively. Crucial changes in the16

The online appendix explains why the HDF model fails in relative money supply in more detail.

18

posterior distributions of the structural parameters from the Benchmark model are found in (i) significant increases in the posterior means of the standard deviations of the three monetary shocks,
σM , σm , and σφ , and (ii) a large increase in the posterior mean of the AR(1) coeﬃcient of the PPP
deviation shock, ρq .
The HDF model, indeed, displays a great decline in overall fit. The last row of Table 2reports the logarithms of the estimated marginal likelihood for each model.17 The HDF modelyields a smaller log marginal likelihood (1871.309) than that of the Benchmark model (2148.572).
The diﬀerence in the log marginal likelihoods of the two models indicates that forcing the discountfactor to be close to one makes the HDF model’s overall fit to the data significantly worse than thatof the Benchmark model.18
Figure 1(c) depicts the in-sample predictive density of the exchange rate implied by the HDF
model. The figure confirms that the HDF model tracks the actual near random-walk exchange rate
to almost the same degree as the Benchmark model. Table 4 exhibits the HDF model’s results for
the FEVDs of the three endogenous variables, as in Table 3. The FEVDs convey three propertiesof the HDF model: (i) the permanent money supply diﬀerential shock and the persistent money
demand diﬀerential shock jointly and dominantly explain actual exchange rate movements; (ii) the
PPP deviation shock, not the TFP shock as in the Benchmark model, is the dominant driver of theconsumption diﬀerential; and (iii) not only the transitory money supply diﬀerential shock but also
the persistent money demand diﬀerential shock explains the interest rate diﬀerential. The second
property represents the clear drawback of the HDF model.

5. Conclusion
In this paper, I explore the random-walk property of nominal exchange rates with a neoclassical two-country monetary model. A general equilibrium environment allows the joint estimation
17

This paper estimates marginal likelihoods by using Geweke’s (1999) modified harmonic mean estimator. A
marginal likelihood is the probability of data conditional on an underlying model. In general, the higher the marginal
likelihood is, the better the underlying model’s overall fit to the data.
18
The ratio of the marginal likelihood of the Benchmark model to that of the HDF model is 3.152.

19

of equilibrium dynamics of nominal exchange rates and endogenous economic fundamentals — relative consumption and interest rates. The equilibrium random-walk behavior of nominal exchange
rates can be driven by either the steady-state market discount factor near one or by the exogenous
structural shocks following near random-walk processes. A market discount factor close to one,however, implies unrealistic restrictions that do not easily sit on the observed data. A unit discountfactor implies a perfect correlation between relative consumption and real exchange rate, which is
not seen in the data (Backus and Smith, 1993), and counterfactually large volatility in the nearrandom-walk money demand shock diﬀerential due to flat money demand.
Bayesian posterior simulation exercises of the implied restricted UC model, based on postBretton Woods data from Canada and the United States, rejects a market discount factor near
one and identifies the main driver of near random-walk nominal exchange rate behavior as unobservable near random-walk behavior in relative money demand. This paper’s posterior inference ofthe low discount factor around 0.537 suggests significant misspecifications in this canonical open
economy model. The empirical result of this paper, hence, uncovers diﬃculties that the literatureneeds to overcome in explaining data variations in nominal exchange rates and the corresponding
macroeconomic fundamentals, jointly and consistently, through the lens of open-economy generalequilibrium.
There are many potential extensions of the simple canonical model in this paper such as pricestickiness, monetary policy, non-tradable goods, home bias, distributional margins, and endogenous
risk premium.19 The real exchange rate is not determined endogenously in this model, as in thetwo-country models of Benigno (2004), Benigno and Thoenissen (2008), and Corsetti et al. (2008).
For example, Corsetti et al. (2008) argue that the Backus-Smith observation could be replicated in
an international real business cycle model with a high degree of trade elasticity when each country’s
productivity shock is nearly permanent. An interesting question is whether the same result would
hold at the limit of the unit discount factor, when cointegration is incorporated into the home andforeign productivity processes of their model. Together with sticky nominal prices, the Taylor type
19

Among them, the online appendix extends the argument of this paper by introducing non-tradable goods into
the Benchmark model.

20monetary policy rule is investigated by Benigno (2004), Steinsson (2008), and Engel (2014, 2018)as a primary source of persistent and volatile moments in real exchange rates. Incorporating sucha framework into the model employed here would change its CERs significantly. Very persistent
trend inflation, as investigated by Cogley and Sbordone (2008), Coibion and Gorodnichenko (2011),
and Ascari and Sbordone (2014), might be a plausible candidate for an unobserved I(1) nominal
economic fundamental, and interpret relative money demand shocks in a better way. Finally, thelog-linear approximation of the FONCs automatically eliminates any potential relationship between
the interest rate diﬀerential and endogenous time-varying risk and liquidity premium, which isemphasized by Alvarez et al. (2007, 2009) and Engel (2016). I leave these challenging questions asavenues for future research.

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Sarno, L., Sojli, E., 2009, The feeble link between exchange rates and fundamentals: Can we blame
the discount factor?, Journal of Money, Credit, and Banking 41, 437 − 442.
Schmitt-Grohé, S., Uribe, M., 2003, Closing small open economy models, Journal of International
Economics 61, 163 − 185.
23

Steinsson, J., 2008, The dynamic behavior of the real exchange rate in sticky price models, American
Economic Review 98, 519 − 533.
Appendix. Data description and construction

All data for the United States are distributed by Federal Reserve Economic Data (FRED), operated
by the Federal Reserve Bank of St. Louis (http://http://research.stlouisfed.org/fred2/). The consumption
data are constructed as the sum of the real personal consumption expenditures on non-durables and services. FRED, however, distributes only the nominal values of the two categories of personal consumption
expenditures as Personal Consumption Expenditure on Non-Durables (PCND) and Personal Consumption
Expenditure on Services (PCESV). To construct the real total personal consumption expenditure Cus,t , I
first calculate the share of the two nominal consumption categories in the nominal total personal consumption expenditure Personal Consumption Expenditure and then multiply the real total personal consumption
expenditures, Real Personal Consumption Expenditures at Chained 2005 Dollars (PCECC96), by the calculated share. Following NR, I adopt the M1 money stock, M1SL, as the aggregate money supply Mus,t .
The nominal interest rate rus,t is provided by three-month Treasury Bill (TB3MS). All the variables except
for the nominal interest rate are seasonally adjusted at annual rates and converted to the corresponding
per capita terms by Total Population (POP).
All Canadian data are distributed by Statistics Canada (CANSIM) (http://www5.statcan.gc.ca/cansim/).
The real consumption data Ccan,t are constructed as the sum of Personal Expenditure on Non-Durables
at Chained 2002 Dollars, Personal Expenditure on Semi-Durables at Chained 2002 Dollars, and Personal
Expenditure on Services at Chained 2002 Dollars. I use the M1 money stock as the money supply Mcan,t .
The nominal interest rate rcan,t is provided by three-Month Treasury Bills. All the variables except for
the nominal interest rate are seasonally adjusted at annual rates and converted to the corresponding per
capita terms by Estimate of Total Population.
The output measures for Canada and the United States, Ycan,t and Yus,t , are constructed in a modelconsistent way. In this two-country endowment economy model, a country’s output is given by the sum of
consumption and the trade balance. To measure the bilateral trade balance between Canada and the UnitedStates, T Bt , I use the Canadian goods trade balance for the United States included in CANSIM’s balanceof international payments data (CANSIM Table 376-0005). The Canadian output Ycan,t is constructed byCcan,t + T Bt and the United States output Yus,t is constructed by Cus,t − T Bt /St , where St is the bilateralexchange rate between Canada and the United States.

24

Table 1:

β
γM
γS
ψ
λ
φ
ρM
ρq
ρφ

Prior Distributions of Structural Parameters

Parameters
Household Subjective Discount Factor
Deterministic (Gross) Money Growth
Deterministic EX Trend
Debt Elasticity of Risk PremiumTechnology Diﬀusion Speed
Mean Money Demand Shock
Permanent Money Growth AR(1) Coef.
Real EX AR(1) Coef.
Money Demand AR(1) Coef.

Distribution Mean
Uniform(0,1)
—
Gamma
1.015
Normal
0.000
Gamma
0.010
Beta
0.010
Gamma
1.000
Beta
0.100
Beta
0.850
Beta
0.850

S.D. 95 % Coverage—[0.025 0.975]
0.005 [1.005 1.024]
1.500 [-2.939 2.939]
0.001 [0.008 0.012]
0.001 [0.008 0.012]
0.010 [0.981 1.019]
0.010 [0.081 0.120]
0.100 [0.607 0.983]
0.100 [0.607 0.983]

Note 1. The AR(1) coeﬃcients of the transitory money and output shocks, ρm and ρy respectively, have mass pointsof zero for identification.
Note 2. The standard deviations of all the structural shocks, σM , σA , σm , σy , σq , and σφ , have the identical inverse
Gamma prior distribution, with a mean of 0.01 and standard deviation of 0.01 for the benchmark information set.
Note 3. The prior distribution of β is given by the Beta distribution, with a mean of 0.999 and standard deviationof 0.001 for the High Discount Factor model.

25

Table 2:

Posterior Distributions of Structural Parameters

Parameters
κ
β
γM
γS
ψ
λ
φ
ρM
ρq
ρφ
σM
σA
σm
σy
σq
σφ
Marginal Likelihood

BenchmarkMean S.D. 95 % Interval
0.537 0.041 [0.457 0.618]
0.547 0.042 [0.464 0.630]
1.019 0.005 [1.009 1.027]-0.000 0.002 [-0.005 0.002]
0.010 0.001 [0.008 0.012]
0.009 0.001 [0.008 0.011]
0.999 0.010 [0.980 1.019]
0.088 0.005 [0.083 0.097]
0.867 0.044 [0.778 0.948]
0.997 0.001 [0.995 0.999]
0.017 0.001 [0.016 0.019]
0.006 0.000 [0.005 0.007]
0.007 0.001 [0.005 0.009]
0.003 0.000 [0.002 0.003]
0.006 0.001 [0.004 0.009]
0.027 0.002 [0.024 0.031]
2148.572

HDF
Mean S.D. 95 % Interval
0.950 0.001 [0.947 0.952]
0.998 0.000 [0.998 0.999]
1.051 0.001 [1.050 1.054]
0.002 0.000 [0.002 0.003]
0.011 0.001 [0.009 0.012]
0.007 0.001 [0.006 0.009]
0.999 0.009 [0.980 1.018]
0.105 0.003 [0.099 0.108]
0.982 0.004 [0.974 0.989]
0.928 0.002 [0.924 0.931]
0.026 0.001 [0.024 0.028]
0.006 0.000 [0.005 0.007]
0.105 0.004 [0.098 0.111]
0.003 0.000 [0.002 0.003]
0.006 0.000 [0.005 0.006]
0.066 0.003 [0.061 0.070]
1871.309

Note 1: The “Benchmark” represents the Benchmark specification of the two-country model and the “HDF” represents the High Discount Factor specification.
Note 2: The marginal likelihoods are estimated based on Geweke’s (1999) harmonic mean estimator.

26

Table 3:

Forecast Error Variance Decompositions (%): Benchmark Modelhorizon

ϵM

ϵA

ϵm

ϵy

ϵq

ϵφ

(a) Exchange Rate: ln St
Impact

27
[22

[2

33]

[2

34]

[2

34]

[2

30
[24 34]

[2

1 yr

28
[23

3 yr
5 yr

[0

4]

[0

4]

[0

4]

[0

4]

[0

0
1]

[0

0]

[0

0]

[0

0]

[0

0]

[0

0

3

29
[24

1
4]

3

29
[24

10 yr

3
31]

0]

[1

0]

[1

0]

[1

0]

[1

[63

3]

[62

3]

[62

3]

[61

3]

[60

71]
66

2

0

72]
67

2

0

0

68
3]

2

0

0

3

[1

0

0

3

2
0]

71]
66

2

71]
66
70]

(b) Consumption Diﬀerential : ln Ct
Impact

0
[0

1 yr

93
0]

[91

0]

[89

0]

[73

0]

[63

0]

[53

0
[0

3 yr
5 yr
10 yr

[0

90]

[0

85]

[0

80]

[0

5
0]

[3

0]

[2

0]

[2

0]

[2

0]

[2

0

[0

7]

[0

6]

[6

6]

[11

5]

[15

[0

4]

[0

23]

[0

33]

[0

43]

[0

0]
0

22

4

0]
0

14

4

0

0
3]

2

4

0

68

2
8]

5

0

74

0
[0

96]
81

0
[0

[0

93

0
[0

0
96]

0]
0

29

0]
0
0]

(c) Interest Rate Diﬀerential : (1 + rt )Impact

6
[3

1 yr

0
10]

[0

10]

[0

10]

[0

10]

[0

10]

[0

6
[3

3 yr
5 yr
10 yr

[90

0]

[90

0]

[89

0]

[89

0
97]

[0

97]

[0

97]

[0

96]

[0

96]

[0

94

[0

0]

[0

0]

[0

0]

[0

0]

[0

[0

0]

[0

0]

[0

0]

[0

0]

[0

0]
0

0

0

0]
0

0

0

94

0
0]

0

0

94

0

0
0]

0

94

0

6
[3

0]
0

6
[3

[90

0

6
[3

94
0]

1]
0

0

1]
0
2]

Note: Part (a) reports the posterior means and the 95% Bayesian highest probability density (HPD) intervals (insquare brackets) of the forecast error variance decompositions (FEVDs) of the nominal exchange rate into the six
structural shocks over the impact, the one-year, three-year, five-year, and ten-year forecast horizons; Part (b) reports
those of the consumption diﬀerential; and Part (c) reports those of the interest rate diﬀerential.

27

Table 4:

Forecast Error Variance Decompositions (%): HDF Modelhorizon

ϵM

ϵA

ϵm

ϵy

ϵq

ϵφ

(a) Exchange Rate: ln St
Impact

52
[50

[0

60]

[0

72]

[0

87]

[0

87
[76 79]

[0

1 yr

58
[56

3 yr
5 yr

[2

0]

[0

0]

[0

1]

[0

1]

[0

0
2]

[0

1]

[0

0]

[0

0]

[0

0]

[0

0

0

77
[86

2
0]

0

70
[68

10 yr

0
54]

0]

[0

0]

[0

0]

[0

0]

[0

[44

0]

[39

0]

[28

0]

[21

0]

[12

43]
30

0

0

48]
41

0

0

0

46
0]

0

0

0

0

[0

0

0

0

0
0]

32]
22

0

24]
13
14]

(b) Consumption Diﬀerential : ln Ct
Impact

0
[0

1 yr

15
0]

[10

0]

[10

0]

[12

0]

[14

0]

[19

0
[0

3 yr
5 yr
10 yr

[0

24]

[0

26]

[0

34]

[0

0
0]

[0

0]

[0

0]

[0

0]

[0

0]

[2

0

[80

0]

[79

0]

[76

0]

[74

5]

[66

[0

90]

[0

88]

[0

86]

[0

81]

[0

0]
0

80

0

0]
0

82

0

0

0
90]

84

0

0

27

85
0]

0

0

20

0
[0

21]
18

0
[0

[0

16

0
[0

0
20]

0]
0

73

0]
0
0]

(c) Interest Rate Diﬀerential : (1 + rt )Impact

0
[0

1 yr

0
0]

[0

0]

[0

0]

[0

0]

[0

0]

[0

0
[0

3 yr
5 yr
10 yr

[66

0]

[52

0]

[49

0]

[48

0
87]

[0

69]

[0

55]

[0

52]

[0

50]

[0

68

[0

0]

[0

0]

[0

0]

[0

0]

[0

[12

0]

[31

0]

[45

0]

[48

0]

[49

34]
46

0

0

14]
32

0

0

49

13
0]

0

0

50

0

0
0]

0

53

0

0
[0

0]
0

0
[0

[86

0

0
[0

87
0]

48]
49

0

51]
51
52]

Note: Part (a) reports the posterior means and the 95% Bayesian highest probability density (HPD) intervals (insquare brackets) of the forecast error variance decompositions (FEVDs) of the nominal exchange rate into the six
structural shocks over the impact, the one-year, three-year, five-year, and ten-year forecast horizons; Part (b) reports
those of the consumption diﬀerential; and Part (c) reports those of the interest rate diﬀerential.

28

1990

1995

2000

2005

1980

1985

1990

1995

2000

(c) HDF Model: Posterior Predictive Densities

2005

1980

1985

1990

and

1995

(b) Lower AR Roots of ln

2000

ln M

2005

Figure 1: Exchange rates and in-sample predictions. Note: Window (a) corresponds to the Benchmark model, window (b) the Benchmark model withlower AR roots of the money demand shock diﬀerential and the permanent money supply growth rate, and window (c) the posterior high discountfactor (HDF) model. In each window, the solid black line represents the actual Canadian dollar against the US dollar; the dashed black line the meansof in-sample predictions; and the gray area the 95% Bayesian highest probability density (HPD) intervals of the in-sample predictions.

-0.1
1975

0

0.1

0.2

0.3

0.4

0.5

-0.5
1975

1985

-0.1
1975

0.5

1

1.5

2

2.5

3

0

1980

Actual Exchange Rate
Forecast
95% HPD(a) Benchmark Model

0

0.1

0.2

0.3

0.4

0.5

Online Appendix to
Exchange Rates and Fundamentals:
A General Equilibrium Exploration
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 15, 2019
Not intended for publication

Appendix A. First order necessary conditions and stochastically de-trended system

The first-order necessary conditions (FONCs) for the household’s problem in the home country
are given by the budget constraint
f
f
f
h
h
h
Bh,t
+ St Bh,t+ Ph,t Ch,t + Mh,t = (1 + rh,t−1)Bh,t−1+ St (1 + rh,t−1)Bh,t−1+ Mh,t−1 + Ph,t Yh,t + Th,t ;the Euler equation1h= β(1 + rh,t
)Et
Ph,t Ch,t!

1
Ph,t+1 Ch,t+1";

the utility-based uncovered parity condition (UIP)h(1 + rh,t)Et!

1
Ph,t+1 Ch,t+1"

f(1 + rh,t
)
=
Et
St

!

St+1
Ph,t+1 Ch,t+1";

and the money demand function
Mh,t= φh,tPh,t#h
1 + rh,thrh,t$Ch,t .

The foreign country’s counterparts of the FONCs are the budget constraint
f
f
f
h
h
h
Bh,t
+ St Bh,t+ Ph,t Ch,t + Mh,t = (1 + rh,t−1)Bh,t−1+ St (1 + rh,t−1)Bh,t−1+ Mh,t−1 + Ph,t Yh,t + Th,t ;the Euler equation1f= β(1 + rf,t)Et
Pf,t Cf,t!

1
Pf,t+1 Cf,t+1";

the utility-based uncovered parity condition (UIP)h(1 + rf,t)Et!

1
St+1 Pf,t+1 Cf,t+1"

f(1 + rf,t
)
=
Et
St

1

!

1
Pf,t+1 Cf,t+1";

and the money demand function
Mf,t= φf,tPf,t#f
1 + rf,tf
rf,t$

Cf,t ;The stochastically de-trended versions of the FONCs of the home country consist of the budget
constraintfh)st bfh,t−1(1 + rh,t−1)bhh,t−1 (1 + rh,t−1fhph,t ch,t + bh,t + st bh,t =++ ph,t yh,t ;
h
f
γM,t
γM,t

the Euler equation1h= β(1 + rh,t
)Et
ph,t ch,t#1hγM,t+1ph,t+1 ch,t+1$;

the UIP conditionhst (1 + rh,t
)Et

#

1
h
ph,t+1 ch,t+1 γM,t+1$f= (1 + rh,t
)Et

the money demand function
mh,t= φh,t ch,t
ph,t

#

h
1 + rh,t
h
rh,t

$

#

st+1
f
ph,t+1 ch,t+1 γM,t+1

$

;

;

the risk premiumshh¯ − 1}],
rh,t= rw,t[1 + ψ{exp(−bhh,t + d)andff
¯ − 1}].
rh,t= rw,t[1 + ψ{exp(−bfh,t + d)

Similarly, the stochastically de-trended versions of the FONCs of the foreign country consist of thebudget costraintfh(1 + rw,t−1)st bfh,t−1 (1 + rw,t−1)bhh,t−1 qt ph,t yf,t
qt ph,t cf,tfh
− st bh,t − bh,t = −
−
+
;
f
h
at
at
γM,t
γM,t

2

the Euler equation
at s tat+1 st+1f= β(1 + rw,t
)Et f;
qt ph,t cf,tγM,t+1 qt+1 ph,t+1 cf,t+1
the UIP conditionhst (1 + rw,t
)Et

#

at+1
h
qt+1 ph,t+1 cf,t+1 γM,t+1$f= (1 + rw,t
)Et#

at+1 st+1
f
qt+1 ph,t+1 cf,t+1 γM,t+1$;

and the money demand function
at st mf,t
= φf,t cf,t
qt ph,t#f
1 + rw,t
f
rw,t

$

.

Finally The stochastically de-trended PPP condition is st = ph,t qt /(at pf,t ).
If the TFP diﬀerential at is I(1) as assumed in NR, the above system of stochastic diﬀerence
equations becomes non-stationary through the home and foreign budget constraints and there isno deterministic steady state to converge. Notice that the cross-country permanent money supplyττ
diﬀerential ln Mh,t/Mf,tdoes not appear in the stochastically de-trended system of the FONCs. Inττcontrast to the TFP diﬀerential at , the I(1) property of ln Mh,t/Mf,tin Assumption 2 does notmatter for the closing of the model. This might be an obvious result of the model’s property thatthe super-neutrality of money holds in the money-in-utility model: Money growth does not matterfor the deterministic steady state.
Notice that at the deterministic steady state, the TFP diﬀerential a∗ is one. Because of thestationarity of the above system of equations, the deterministic steady state is characterized by

3

f∗
f∗
h∗
h∗
f∗
constants c∗h , c∗f , p∗h , s∗ , bh∗
h , bh , rh , rh , rw , and rw that satisfy

f∗
¯
bh∗
h = bh = d,r∗ ≡ rhh∗ = rff ∗ = rwh∗ = rwf ∗ = γM /β − 1,
yf (φγM )−1 r∗ + (yh + yf )(1 − β −1 )d¯s∗ =,yh (φγM )−1 r∗ − (yh + yf )(1 − β −1 )d¯p∗h yh = (1 − β −1 )(1 + s∗ )d¯ + (φγM )−1 r∗ ,p∗h c∗h = (φγM )−1 r∗ ,c∗f = s∗ c∗h .
The log-linear approximation of the stochastically de-trended home budget constraint is¯ − β −1 )s∗ ŝt + s∗ b̃fp∗h (c∗h − yh )p̂h,t + p∗h c∗h ĉh,t − p∗h yh ŷh,t + b̃hh,t + d(1h,t∗ −1 f¯ + r̂h ) − γ̂ h ] + s∗ β −1 d[(1¯ + r̂f ) − γ̂ f ] + β −1 b̃h= β −1 d[(1b̃h,t−1 ; (A.1)
h,t−1
M,t
h,t−1 + s β
M,t
h,t−1

that of the home Euler equation ishhp̂h,t + ĉh,t + (1 + r̂h,t) = Et (p̂h,t+1 + ĉh,t+1 + γ̂M,t+1
);

(A.2)

that of the home UIP condition isffhh
Et ŝt+1 − ŝt = (1 + r̂h,t) − (1 + r̂h,t) − Et (γ̂M,t+1− γ̂M,t+1
);

(A.3)

and that of the home money demand function isp̂h,t + ĉh,t − m̂h,t =1h(1 + r̂h,t) − φ̂h,t .
r∗

(A.4)

The foreign country’s counterparts are the log-linear approximation of the stochastically de-trended4foreign budget constraint¯ − β −1 )s∗ ŝt − s∗ b̃fp∗h (c∗f − yf )(p̂h,t + q̂t − ât ) + p∗h c∗f ĉf,t − p∗h yf ŷf,t − b̃hh,t − d(1
h,t
f
−1 h
¯ + r̂h ) − γ̂ h ] − s∗ β −1 d[(1¯ + r̂f= −β −1 d[(1b̃h,t−1 − s∗ β −1 b̃fh,t−1 ; (A.5)
w,t−1 ) − γ̂M,t ] − β
w,t−1
M,t

that of the foreign Euler equationff) = Et (ŝt+1 − p̂h,t+1 − ĉf,t+1 − q̂t+1 + ât − γ̂M,t+1);ŝt − p̂h,t − ĉf,t − q̂t + ât − (1 + r̂w,t(A.6)

that of the foreign UIP conditionffhh
Et ŝt+1 − ŝt = (1 + r̂w,t) − (1 + r̂w,t) − Et (γ̂M,t+1− γ̂M,t+1
);

(A.6)

and that of the home money demand function
1fŝt + m̂f,t − p̂h,t − ĉf,t − q̂t + ât = − ∗ (1 + r̂w,t) + φ̂f,t .
r

(A.7)

The log-linear approximations of the home country’s interest rates arehh(1 + r̂h,t) = (1 + r̂w,t) − ψ(1 − κ)b̃hh,t ,ffand (1 + r̂h,t) = (1 + r̂w,t) − ψ(1 − κ)b̃fh,t .

(A.8)

Notice that the home interest rates (A.8) redefine the home UIP condition (A.3) asffhh
Et ŝt+1 − ŝt = (1 + r̂w,t) − (1 + r̂w,t) − ψ(1 − κ)(b̃hh,t − b̃fh,t ) − Et (γ̂M,t+1− γ̂M,t+1
).

(A.9)

Comparing the above home UIP condition with the foreign UIP condition (A.6) implies that thehome and foreign bonds are perfectly substitutable along the equilibrium path. Hence, the equilibrium condition b̃t ≡ b̃hh,t = b̃fh,t holds.

5Appendix B. Solving the equilibrium with two symmetric countries
To understand the equilibrium transitory dynamics of the exchange rate in this model, it isinformative to scrutinize a simpler version of the model that includes two symmetric countries. For
this purpose, I set the parameter d¯ to zero and assume that the transitory output components ofthe two countries, yh and yf , are equal to y. Notice that the deterministic steady state in this caseis characterized by s∗ = 1, c∗h = c∗f = y, and p∗h = (φγM )−1 r∗ , where r∗ = γM /β − 1.
The home and foreign money demand functions, (A.4) and (A.7), and the home interest rates(A.8) yield the following interest rate diﬀerential:fh(1 + r̂w,t) − (1 + r̂w,t) = r∗ (ŝt + ĉt − m̂t + φ̂t − q̂t + ât ) + ψ(1 − κ)b̃t .

Substituting the interest rate diﬀerential into the foreign UIP condition (A.6) leads to the expectational diﬀerence equation of the de-trended exchange rate ŝt :fhŝt = κEt ŝt+1 − (1 − κ)ĉt + (1 − κ)(m̂t − φ̂t + q̂t − ât ) + κEt (γ̂M,t+1− γ̂M,t+1) − ψκ(1 − κ)b̃t .

I combine the log-linearized Euler equations of the home and foreign countries, (A.2) and(A.6), with those of the home country’s interest rates (A.8) to yield the first-order expectational
diﬀerence equation of ŝt + ĉt − q̂t + ât :ŝt + ĉt − q̂t + ât = κEt (ŝt+1 + ĉt+1 − q̂t+1 + ât+1 ) + κEt γ̂M,t+1 + (1 − κ)(m̂t − φ̂t ).
Since κ takes a value between zero and one, the above expectational diﬀerence equation has a forward
solution of ŝt + ĉt − q̂t + ât = κρM (1 − κρM )−1 γ̂M,t + (1 − κ)(1 − κρm )−1 m̂t − (1 − κ)(1 − κρφ )−1 φ̂tunder a suitable transversality condition. By exploiting this forward solution and the stochastic
processes of both countries’ TFPs (1), I rewrite the foreign UIP condition (A.6) as

Et ŝt+1 − ŝt = ψ(1 − κ)b̃t −κρM (1 − ρM )(1 − κ)(1 − ρm )(1 − κ)(1 − ρφ )γ̂M,t −m̂t +φ̂t ,1 − κρM
1 − κρm
1 − κρφ6(B.1)

Furthermore, taking a diﬀerence between the log-linearized budget constraints of the home andforeign countries, (A.1) and (A.5), I find the law of motion of the international bond holdingsp∗ y ∗ κρMp∗ y ∗ (1 − κ)p∗ y ∗ (1 − κ)b̃t = β −1 b̃t−1 +p∗h y ∗ ŝt −p∗h y ∗ (q̂t −ât )− hγ̂M,t − hm̂t + hφ̂t +p∗h y ∗ ŷt , (B.2)
1 − κρM
1 − κρm
1 − κρφ
where y ∗ = y/4 and ŷt ≡ ŷh,t − ŷf,t .
Combining equation (B.1) with equation (B.2) then yields the following second-order expectational diﬀerence equation with respect to international bond holdings:
Et b̃t+1 − [1 + β −1 + p∗h y ∗ ψ(1 − κ)]b̃t + β −1 b̃t−1 = −λp∗h y ∗ ât + p∗h y ∗ (1 − ρq )q̂t − p∗h y ∗ (1 − ρy )ŷt (B.3)
It is straightforward to show that equation (B.3) has two roots, one of which is greater than one
and the other of which is less than one.1 Without losing generality, let η denote the root thatis less than one. Solving equation (B.3) by forward iterations then shows that the equilibrium
international bond holdings level is determined by the following cross-equation restriction (CER):b̃t = η b̃t−1 + βηλp∗h y ∗= η b̃t−1 +

∞
%

j=0
∗ ∗
βηλph y(βη) Et ât+j + βηp∗h y ∗ (1 − ρy )1 − βη(1 − λ)

j

ât +

βηp∗h y ∗ (1 − ρy )1 − βηρy

ŷt −

∞
%

(βη) Et ŷt+j − βηp∗h y ∗ (1 − ρq )jj=0
βηp∗h y ∗ (1 − ρq )1 − βηρq

q̂t .

∞
%

(βη)j Et q̂t+j ,

j=0

(B.4)

Substituting equation (B.4) back into equation (B.2) provides the CER for the exchange rate (7):ŝt =βη − 1
1 − βη
1−κ
1−κ
b̃ −
ât +
m̂t −
φ̂t
∗ ∗ t−1βph y
1 − βη(1 − λ)1 − κρm
1 − κρφ1 − βη
1 − βη
κρM
−
ŷt +
q̂t +
γ̂M,t .
1 − βηρy1 − βηρq
1 − κρM

Therefore, in this symmetric case, the competitive equilibrium along the balanced growth path ischaracterized by a lower dimensional dynamic system of (ŝt , b̃t , ât , γ̂M,t , m̂t , φ̂t , ŷt , q̂t ).

1

To characterize the roots of the second-order expectational diﬀerence equation, see, for example, Sargent (1987).

7

Adding the log-linearized home and foreign budget constraints together implies the resource
constraint ĉh,t + ĉf,t = ŷh,t + ŷf,t . Since the equilibrium dynamics of the consumption diﬀerential
follow ĉh,t − ĉf,t = −ŝt + q̂t − ât + κρM (1 − κρM )−1 γ̂M,t + (1 − κ)(1 − κρm )−1 m̂t − (1 − κ)(1 − κρφ )−1 φ̂t ,the home country’s consumption obeys 2ĉh,t = (ŷh,t + ŷf,t ) − ŝt + q̂t − ât + κρM (1 − κρM )−1 γ̂M,t +(1 − κ)(1 − κρm )−1 m̂t − (1 − κ)(1 − κρφ )−1 φ̂t , while the foreign country’s is 2ĉf,t = (ŷh,t + ŷf,t ) + ŝt −q̂t + ât − κρM (1 − κρM )−1 γ̂M,t − (1 − κ)(1 − κρm )−1 m̂t + (1 − κ)(1 − κρφ )−1 φ̂t . The home country’s
price p̂h,t then is determined as follows. The Euler equation and the money demand function of theforeign country, (A.6) and (A.7), imply the expectational diﬀerence equation of ŝt − p̂h,t − ĉf,tfŝt − p̂h,t − ĉf,t − q̂t + ât = κEt (ŝt+1 − p̂h,t+1 − ĉf,t+1 − q̂t+1 + ât+1 − γ̂M,t+1) − (1 − κ)(m̂f,t − φ̂f,t ).

Solving the above equation by forward iterations and imposing a suitable transversality conditionfyields the CER ŝt − p̂h,t − ĉf,t − q̂t + ât = −κρM (1 − κρM )−1 γ̂M,t− (1 − κ)(1 − κρm )−1 m̂f,t + (1 −κ)(1 − κρφ )−1 φ̂f,t . This CER characterizes the equilibrium home price
2p̂h,t = ŝt − (ŷh,t + ŷf,t ) − q̂t + ât +

κρM
1−κ
1−κ
f
h
(γ̂M,t+ γ̂M,t)+(m̂h,t + m̂f,t ) −(φ̂h,t + φ̂f,t ).
1 − κρM
1 − κρm
1 − κρφ

The money demand functions of both countries, eqs.(A.4) and (A.7), imply that the interest rates
in the two countries are"ρM
1 − ρm1 − ρφhγ̂ −m̂h,t +
φ̂h,t
1 − κρM M,t 1 − κρm
1 − κρφ!
"ρM
1 − ρm1 − ρφff(1 + r̂w,t ) = (1 − κ)γ̂ −m̂f,t +
φ̂f,t .
1 − κρM M,t 1 − κρm
1 − κρφh(1 + r̂h,t) = (1 − κ)!

Finally, as the last endogenous variable, the risk-free nominal interest rate of the home bonds thenhhfluctuates in response to the risk premium, following (1 + r̂w,t) = (1 + r̂h,t) + ψ(1 − κ)b̃t .

Suppose that ψ = 0: There is no debt elastic risk premium in the home country’s interest
rate. It is easy to show that in this case, the second-order expectational diﬀerence equation (B.3)

8has a unit root, i.e., η = 1, and the resulting forward solution turns out to beb̃t = b̃t−1 +βλp∗h y ∗βp∗ y ∗ (1 − ρy )βp∗ y ∗ (1 − ρq )ât + hŷt − hq̂t .
1 − β(1 − λ)1 − βρy1 − βρq

Hence, the stochastic process of the de-trended international bond holding b̃t contains a permanent
unit root component and never converges to the steady state. This lack of stationarity of theequilibrium balance growth path motivates this paper to allow for a positive elasticity of the risk
premium with respect to the debt level.
Importantly, a permanent stochastic process of the de-trended international bond holding also
emerges even when κ = 1. Because the log-linearized home country’s interest rates (A.8) imply that
under κ = 1, the debt elastic risk premia in play no role in determining the interest rates faced by
the home country. As a result, the de-trended international bond holding b̃t contains a permanent
unit root component, as in the case where ψ = 0. Hence, the closing of the two-country DSGE
model in this paper requires the market discount factor to be strictly less than one.
The consumption logarithms of the home and foreign countries in terms of the home currency
can be solved as

2 ln Ch,t = ln Yh,t + ln qt Yf,t +1 − βη
1 − βη
1 − βη
1 − βη
ln at +
ln yt −
ln qt +b̃t−1 ,
1 − βη(1 − λ)1 − βηρy1 − βηρq
βp∗h y ∗2 ln qt Cf,t = ln Yh,t + ln qt Yf,t −1 − βη
1 − βη
1 − βη
1 − βη
ln at −
ln yt +
ln qt −b̃t−1 .
1 − βη(1 − λ)1 − βηρy1 − βηρq
βp∗h y ∗Each country’s consumption depends on the log-linearized global aggregate endowment ln Yh,t +1−βη
ln qt Yf,t , the log-linearized country-specific portion of the aggregate endowment [1−βη(1−λ)]ln at +1−βη
1−βη
ln yt − (1−βηρ
ln qt ,
(1−βηρy )q)

1−βηand the wealth eﬀect of the net foreign asset position βp
If∗ y ∗ b̃t−1 .
hthe discount factor approaches one, both the log-linearized country-specific portion and the wealtheﬀect of the net foreign asset position disappear and the log consumption levels become
11
ln Ch,t = (ln Yh,t + ln Yf,t ) + ln qt ,2
2

1
1
ln Cf,t = (ln Yh,t + ln Yf,t ) − ln qt .
22
9

Relative consumption then turns out to be correlated perfectly with the RER because

ln Ch,t − ln Cf,t = ln qt .

Appendix C. Derivation of the error correction representation (4)Let nt denote the fundamental of PVM (3): nt ≡ ln Mt − ln Ct − ψκb̃t − ln φt + ln qt . Consider
the currency return ∆ ln St adjusted by the fundamental (1 − κ)nt−1 : ∆ ln St + (1 − κ)nt−1 . PVM(3) then implies:∆ ln St + (1 − κ)nt−1 = (1 − κ)
− (1 − κ)

∞
%
j=0

∞
%
j=0

∞
%

κj (Et − Et−1 )nt+i + (1 − κ)∞%κj Et−1 nt+i

j=0

κj Et−1 nt+i−1 + (1 − κ)nt−1 ,∞(1 − κ)2 % i(1 − κ)2= (1 − κ)κ (Et − Et−1 )nt+i +κ Et−1 nt+i−1 −nt−1 ,
κκ
j=0
i=0= (1 − κ)

∞
%
j=0

j

κj (Et − Et−1 )nt+i +1−κ
(1 − κ)2
ln St−1 −nt−1 .
κκThis result means that the currency return has the following error correction representation, given
by equation (4):∆ ln St =1−κ(ln St−1 − ln Mt−1 + ln Ct−1 + ψκb̃t−1 + ln φt−1 − ln qt−1 )κ+ (1 − κ)∞%
j=0

κj (Et − Et−1 )nt+i .

Appendix D. The restricted UC model and posterior simulation strategy

Let Xt denote an unobserved state vector defined as
10Xt = [ŝt ĉt Et ŝt+1 Et ĉt+1 b̃t γ̂M,t ât m̂t ŷt q̂t φ̂t ]′ .
Furthermore, let ϵt and ωt denote random vectors consisting of structural shocks and rational
expectations errors: ϵt ≡ [ϵM,t ϵA,t ϵm,t ϵy,t ϵq,t ϵφ,t ]′ and ωt ≡ [ŝt − Et−1 ŝt ĉt − Et−1 ĉt ]′ , respectively.
In particular, for empirical investigation purposes, I presume that the structural shock vector ϵt
is normally distributed, with a mean of zero and a diagonal variance-covariance matrix Σ: ϵt ∼22i.i.d.N (0, Σ) with diag(Σ) = [σMσA2 σmσy2 σq2 σφ2 ]′ .

Accompanied by the stochastic processes of the exogenous forcing variables, LRE model (2)then implies that

Γ0 Xt = Γ1 Xt−1 + Φ0 ωt + Φ1 ϵt ,where Γ0 , Γ1 , Φ0 , and Φ1 are the corresponding coeﬃcient matrices. 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 = FXt−1 + Φϵt ,(D.1)

where F and Φ are confirmable coeﬃcient matrices.
To construct this paper’s UC model, I further expand the unobservable state vector Xt by
the permanent money supply diﬀerential ln Mtτ to obtain the augmented state vector Zt : Zt ≡[X′t ln Mtτ ]′ . The stochastic process of ln Mtτ and the state transition (D.1) then imply the following
non-stationary transition of the expanded state vector Zt :Zt = GZt−1 + Ψϵt ,ϵt ∼ i.i.d.N (0, Σ),

where G and Ψ are confirmable coeﬃcient matrices.

11

(D.2)

In this paper, I explore time-series data on the log of the consumption diﬀerential ln Ct , thelog of the output diﬀerential ln Yt , the log of the money supply diﬀerential ln Mt , the interest ratefh
diﬀerential rt ≡ rh,t
− rf,t, and the log of the bilateral exchange rate ln St . Let Yt denote theinformation set that consists of these five time series: Yt ≡ [ln Ct ln Yt ln Mt rt ln St ]′ . It is thenstraightforward to show that the information set Yt is linearly related to the unobservable statevector Zt as

Yt = HZt ,(D.3)

where H is a confirmable coeﬃcient matrix. The transition equation, the unobserved state (D.2),
and the observation equation (D.3) jointly consist of a non-stationary state-space representation of
the two-country model, which is the restricted UC model estimated in this paper.2
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 the 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), and Kano (2009).
Appendix E. The high discount factor (HDF) model
In Figure A.1, the five windows plot the 95% HPD intervals of the forecast errors for thefive time-series data sets, which are implied by the Benchmark model as the dark gray area andthe HDF model as the light gray area. Windows (a) and (b) of Figure A.1, especially, reveal thatthe HDF model yields much larger forecast errors for the consumption diﬀerential and the money
supply diﬀerential than does the Benchmark model.
2

The state-space form of the model, (D.2) and (D.3), decomposes the I(1) diﬀerence-stationary information set
Yt into permanent and transitory components exploiting the theoretical restrictions provided by the two-country
model. Recursion of the Kalman filter for a non-stationary state-space model is explained in detail by Hamilton(1994).

12

The large forecast error of the HDF model in the consumption diﬀerential results from themodel’s implying tight positive correlation between the consumption diﬀerential and the PPP deviation shock. Figure A.1 plots as the light gray area the 95% HPD interval of the forecast error
in the consumption diﬀerential that the HDF model draws. The same figure also displays as thedark gray area the 95% HPD interval of the forecast error in the consumption diﬀerential generated
only by the PPP deviation shock. Observe that these two HPD intervals overlap each other to quitea large degree. This simply means that the PPP deviation shock is the main contributor to thelarge variations in the forecast error in the consumption diﬀerential that the HDF model implies.
Therefore, it is the high dependence of the consumption diﬀerence on the PPP deviation shock that
leads to the significant failure in the HDF model to explain the actual data of the consumption
diﬀerential. The data, as a result, forces the subjective discount factor to be much lower to increase
the empirical fit as in the Benchmark model.
Understanding the diﬃculty of the HDF model in explaining the money supply diﬀerential
requires several steps. First, recall that the AR(1) coeﬃcient of the transitory money supply
diﬀerential, ρm , is assumed to be zero for identification. Additionally, as reported in Table 2, theposterior mean of the AR(1) coeﬃcient of the money growth rate diﬀerential, ρM , is estimated tobe small around 0.088, while that of the money demand shock diﬀerential, ρφ , is estimated to beclose to one. Equation (8), then, shows that when the discount factor κ approaches one, the interest
rate diﬀerential becomes dependent only on the transitory money supply diﬀerential and the money
demand shock diﬀerential with very small sensitivities.
Second, applying the Kalman filter to interest rate diﬀerential equation (8) yields the oneperiod-ahead forecast of the interest rate diﬀerential. Importantly, the forecast of the interest rate
diﬀerential should be dominated by the persistent money demand shock diﬀerential ln φt accompanied by the white noise forecast error ln mt . Because of the flat money demand function, theforecast of the money demand shock diﬀerential should be very volatile to explain the actual dataof the interest rate diﬀerential.
Third, transitory exchange rate equation (6), then, predicts that under the HDF model the13forecast of the transitory exchange rate should be dominated by the volatile forecast of the money
demand shock diﬀerential. Indeed, the negative correlation between the forecasts of ln st and ln φt
is almost perfect. Figure A.3(a) plots the 95% HPD intervals of the forecasts of ln st as the darkgray area and of ln φt as the light gray area, as predicted by the HDF model. The figure confirmsthat the forecast of ln φt is very volatile and is correlated negatively with that of ln st . The HDF
model, therefore, predicts the volatile forecast of the transitory exchange rate.
Fourth, the forecast of the permanent money supply diﬀerential ln Mtτ should be volatile andcorrelated negatively with that of the transitory exchange rate ln st . This is due to the stochastic
de-trending of the exchange rate ln St ≡ ln st + ln Mtτ . To keep the fit of the model to the actual
ln St good, the volatile forecast of ln st should be oﬀset by a volatile forecast of ln Mtτ . Figure A.3(b)plots the 95% HPD intervals of the forecasts of ln st and ln Mtτ as the dark and light gray areas,
respectively. The figure also displays the actual ln St as the black solid line. Observe in the figurethat the forecast of ln Mtτ is very volatile and associated negatively with that of ln st . Moreover, thevolatile forecasts of ln st and ln Mtτ oﬀset each other to fit to the actual exchange rate data ln St .
As a result, the forecast of the permanent money supply diﬀerential turns out to be as volatile asthat of the money demand shock diﬀerential.
Finally, remember the decomposition of the money supply diﬀerential into the permanent andtransitory components ln Mt ≡ ln Mtτ + ln mt . With this decomposition the Kalman filter interpretsthat the forecast of the money supply diﬀerential ln Mt should be dominated only by the volatile
forecast of the permanent money supply diﬀerential ln Mtτ .3 This is the direct reason why the HDF
model fails to explain the money supply diﬀerential.4
In sum, the high discount factor, as in the HDF model, imposes counterfactual restrictions
on the consumption diﬀerential and the money supply diﬀerential. In the Benchmark model, the3The corresponding forecast error matches the transitory money supply diﬀerential ln mt . Indeed, the Kalmansmoother of the transitory money supply shock equals the forecast errors of both the money supply diﬀerential andthe interest rate diﬀerential.
4
This result does not depend crucially on the identification assumption of the white noise transitory money
supply diﬀerential ln mt . Even when I allow for a high serial correlation of this shock, the flat money demand
function requires large volatilities of the monetary demand shock diﬀerential and results in a worse fit of the HDF
toward the money supply diﬀerential than the Benchmark model.

14

two aspects of the data require β to be much lower than the prior value of the HDF model. These
restrictions, indeed, are so strictly binding that they jointly pin down the low value of β with a highdegree of precision, as shown by the corresponding small posterior standard deviation reported inTable 2.
Appendix F. A two-country model with non-tradable goods
F.1. The model
In this augmented model, each country is endowed with a representative household whose
objective is the lifetime money-in-utility
∞
%

&

j

β Et ln Ci,t+j + φi,t+j ln

j=0!

Mi,t+jPi,t+j"',0 < β < 1,

for i = h, f ,where Ci,t , Mi,t , and Pi,t represent the ith country’s consumption, money stock, and price index,
respectively. The money-in-utility function is subject to a persistent money demand shock φi,t .
To extend the basic model of this paper with non-tradable goods, I assume that in each country,
the representative household consumes both tradable and non-tradable goods, CT,i,t and CN,i,t fori = h, f , and raises utility from the Cobb-Douglas type consumption aggregator
Ci,t = G(CT,i,t )α (CN,i,t )1−α ,0 < α < 1,

for i = h, f ,where G is a constant.
Notice that the expenditure minimization problem across the tradable and non-tradable goods
derives the following demand functions for the two goods

CT,i,t = G

−1

!

α
1−α

"1−α !

PT,i,t
PN,i,t"α−1Ci,t ,and CN,i,t = G

−1

!

α
1−α

"−α !

PT,i,t
PN,i,t"αCi,t , (F.1)

where PT,i,t and PN,i,t are the prices of the tradable and non-tradable goods, respectively. When con( α )1−α ( α )αstant G is set to 1−α+ 1−α , the unit price of the consumption basket, Pi,t , is (PT,i,t )α (PN,i,t )1−α .
15Moreover, due to the property of the constant elastic function, the expenditure share of the tradable
good is proportional to the total consumption expenditure PT,i,t CT,i,t = ωPi,t Ci,t where ω ∈ (0, 1).
Below I follow the conventional assumption that the law of one price holds in the tradable
goods across the two countries: St PT,f,t = PT,h,t . It is the straightforward to show that the realexchange rate qt is given as*+1−αPN,f,t!
" 1−α !
" 1−α1−αα
St PT,f,t PN,f,t
PT,f,tSt Pf,tCN,h,t α
Cf,t αqt ≡=,+1−α =
1−α = *
αPN,h,tPh,tCN,f,tCh,tPT,f,t
PN,f,tPT,h,t(F.2)

where the last equality is obtained by using the demand functions of the non-tradable goods,
equation (F.1).
The representative households in the home and foreign countries maximize their lifetime utility
functions subject to the home budget constraint
f
f
f
h
h
h
Bh,t
+ St Bh,t+ Ph,t Ch,t + Mh,t = (1 + rh,t−1)Bh,t−1+ St (1 + rh,t−1)Bh,t−1+ Mh,t−1 + Ph,t Yh,t + Th,t ,and its foreign counterparthhBf,tBf,t−1ff
fh+ Bf,t+ Pf,t Cf,t + Mf,t = (1 + rf,t−1)+ (1 + rf,t−1)Bf,t−1+ Mf,t−1 + Pf,t Yf,t + Tf,t ,
St
St

l
l
respectively, where Bi,t, ri,t, Yi,t , Ti,t , and St denote the ith country’s holdings of the lth country’s

nominal bonds at the end of time t, the ith county’s returns on the lth country’s bonds, the ith
country’s output level, the ith country’s government transfers, and the level of the bilateral nominal
exchange rate, respectively.
Each country’s output Yi,t is given as an exogenous endowment that consists of the tradable
and non-tradable endowments with a Cobb-Douglas type aggregator: Yi,t = (YT,i,t )α (YN,i,t )1−α .
Notice that Pi,t Yi,t = PT,i,t YT,i,t + PN,i,t YN,i,t holds. The stochastic processes of the tradable andnon-tradable endowments are YT,i,t = yT,i,t AT,i,t and YN,i,t = yN,i,t AN,i,t , where yT,i,t and yN,i,t are
the transitory components and AT,i,t and AN,i,t are the permanent components of the tradable and16non-tradable goods. Below, I interpret the permanent components as the TFPs in the underlying
production technologies of the two goods.
The first-order necessary conditions (FONCs) of the home country’s household are given by
the budget constraint, the Euler equation1h= β(1 + rh,t
)Et
Ph,t Ch,t!

1
Ph,t+1 Ch,t+1",

the utility-based uncovered parity condition (UIP)h(1 + rh,t)Et!

1
Ph,t+1 Ch,t+1"

f(1 + rh,t
)
=
Et
St

!

St+1
Ph,t+1 Ch,t+1",

and the money demand function
Mh,t= φh,tPh,t#h
1 + rh,thrh,t$Ch,t .

The foreign country’s FONC counterparts are the budget constraint, the Euler equation1f= β(1 + rf,t)Et
Pf,t Cf,t!

1
Pf,t+1 Cf,t+1",

the utility-based uncovered parity condition (UIP)h(1 + rf,t)Et!

1
St+1 Pf,t+1 Cf,t+1"

f(1 + rf,t
)
=
Et
St

!

1
Pf,t+1 Cf,t+1",

and the money demand function
Mf,t= φf,tPf,t#f
1 + rf,tf
rf,t$

Cf,t .

Each country’s government transfers the seigniorage to the household as a lump-sum. Hence,17

the government’s budget constraint is

Mi,t − Mi,t−1 = Ti,t ,for i = h, f .

τThe money supply Mi,t is specified to consist of permanent and transitory components, Mi,tand
τ
mi,t : Mi,t ≡ mi,t Mi,tfor i = h, f .

To close the model within an incomplete international financial market, I allow for a debtelastic risk premium in the interest rates faced only by the home country:lllτ¯ − 1}],
rh,t= rw,t[1 + ψ{exp(−Bh,t
/Ml,t
+ d)

d¯ ≤ 0,

ψ > 0,

for l = h, flwhere rw,tis the equilibrium world interest rate of the lth country’s bond. The risk premium isgiven as an externality: The household does not take into account the eﬀect of the debt positionon the risk premium when maximizing the lifetime utility function. On the other hand, I do notll
attach a risk premium to the foreign country’s interest rates: rf,t
= rw,tfor l = h, f .

The market-clearing conditions of the two countries’ bond markets are
f
f
h
h
Bh,t
+ Bf,t= 0 and Bh,t
+ Bf,t= 0,i.e., along an equilibrium path, the world net supply of nominal bonds is zero on a period-by-periodbasis. Also the non-tradable good market is cleared in each country:

CN,i,t = YN,i,t ,fori = h, f.

τThe logarithm of the permanent component of the money supply, ln Mi,t
, is I(1) for i = h, f ,ττand the cross-country diﬀerential in the permanent component of money supply, ln Mh,t
− ln Mf,t,τ
is also I(1). I specify each country’s monetary growth rate ∆ ln Mi,tto be an independent AR(1)18process:ττ∆ ln Mi,t= (1 − ρM ) ln γM + ρM ∆ ln Mi,t−1+ ϵiM,t ,for i = h, f .

where ln γM and ρM are the mean and AR root, respectively, of the money supply growth ratecommon to the two countries.
I assume that on the one hand each country’s TFPs are I(1). On the other hand, the crosscountry TFP diﬀerentials, ln aT,t ≡ ln AT,h,t − ln AT,f,t and ln aN,t ≡ ln AN,h,t − ln AN,f,t , are assumed
to be I(0) to guarantee the balanced growth path. These two requirements jointly imply that the
TFPs of the home country must be cointegrated with that of the foreign country: for j = T, N
λj(ln Aj,h,t−1 − ln Aj,f,t−1 ) + ϵhj,A,t ,2
λj∆ ln Aj,f,t = ln γj + (ln Aj,h,t−1 − ln Aj,f,t−1 ) + ϵfj,A,t ,2∆ ln Aj,h,t = ln γj −where γj,A > 1 is the common drift term and λj ∈ [0, 1) is the adjustment speed of the error
correction mechanism (ECM). The ECMs imply that the cross-country TFP diﬀerentials are I(0)becauseln aj,t = (1 − λj ) ln aj,t−1 + ϵhj,A,t − ϵfj,A,t .
Importantly, if the adjustment speed λj is suﬃciently close to zero, the cross-country TFP diﬀerential can be realized near I(1), as maintained by NR.
The stochastic process of the logarithm of the transitory output component for each country,
ln yj,i,t , is specified as the following AR(1) process:ln yj,i,t = (1 − ρj ) ln yi + ρj ln yj,i,t−1 + ϵij,y,t ,for i = h, f . Similarly, the stochastic process of the logarithm of the transitory money supply
component for each country, ln mi,t , is specified as the following AR(1) process:ln mi,t = (1 − ρm ) ln mi + ρm ln mi,t−1 + ϵim,t ,19

for i = h, f . The three other structural shocks, the home and foreign money demand shocks φh,t and
φf,t follow persistent stationary processes. Specifically, they are characterized by AR(1) processes
in terms of the following logarithm:
ln φi,t = (1 − ρφ ) ln φ + ρφ ln φi,t−1 + ϵiφ,t ,Throughout this paper, I assume that all structural shocks are distributed independently.
E.2. The log-linear approximation of the stochastically de-trended system
τDefine stochastically de-trended variables as ci,t ≡ Ci,t /Ai,t cj,t,t ≡ Cj,i,t /Aj,i,t , pi,t ≡ Pi,t Ai,t /Mi,t
,
τ
l
τ
i
τ
τ
τ
τ
pj,i,t ≡ Pj,i,t Aj,i,t /Mi,t, bli,t ≡ Bi,t/Ml,t, γM,t≡ Mi,t/Mi,t−1, and st ≡ St Mf,t/Mh,t. Taking the stochas-tic de-trending of the FONCs, I construct the stochastically de-trended system of the FONCs as
follows.
The stochastically de-trended versions of the FONCs of the home country consist of the budget
constraintfh)st bfh,t−1(1 + rh,t−1)bhh,t−1 (1 + rh,t−1fhωph,t ch,t + bh,t + st bh,t =++ pT,h,t yT,h,t ;
h
f
γM,t
γM,t

the Euler equation1h= β(1 + rh,t
)Et
ph,t ch,t#1hγM,t+1ph,t+1 ch,t+1$;

the UIP conditionhst (1 + rh,t
)Et

#

1
h
ph,t+1 ch,t+1 γM,t+1$f= (1 + rh,t
)Et

the money demand function
mh,t= φh,t ch,t
ph,t

#

h
1 + rh,t
h
rh,t

$

#

st+1
f
ph,t+1 ch,t+1 γM,t+1;

the risk premiumshh¯ − 1}],
rh,t= rw,t[1 + ψ{exp(−bhh,t + d)

20

$

;

and
f
f
¯ − 1}].
rh,t= rw,t[1 + ψ{exp(−bfh,t + d)

Similarly, the stochastically de-trended versions of the FONCs of the foreign country consist of thebudget constraintfh(1 + rw,t−1)st bfh,t−1 (1 + rw,t−1)bhh,t−1fhωst pf,t cf,t − st bh,t − bh,t = −−+ st pT,f,t yT,f,t ;
f
h
γM,t
γM,t

the Euler equation11f= β(1 + rw,t
)Et f;
pf,t cf,tγM,t+1 pf,t+1 cf,t+1
the UIP conditionhst (1 + rw,t
)Et

#

1
h
st+1 pf,t+1 cf,t+1 γM,t+1$f= (1 + rw,t
)Et

#

1
f
pf,t+1 cf,t+1 γM,t+1$;

and the money demand function
mf,t= φf,t cf,t
pf,t

#

f
1 + rw,t
f
rw,t

$

.

The real exchange rate is determined by

qt =

!

cf,t
ch,t

" 1−α
!
α

yN,h,tyN,f,t

" 1−α
α

α−1

1−α

α
at α aN,t.

The home price of the tradable good is

pT,h,t = ph,t,ch,t(1 − ω)yN,h,t- α−1αThe stochastically detrended PPP deviation is qt = st pf,t at /ph,t . Finally, the stochastically detrended LOP of the tradable goods is aT,t st pT,f,t = pT,h,t .
The resulting fourteen equations determine the fourteen endogenous variables ch,t , cf,t , ph,t ,
21

f
f
f
h
h
h
pf,t , pT,h,t , pT,f,t , st , qt , bhh,t , bfh,t , rh,t
, rh,t, rw,t, and rw,t
, given eight exogenous variables γM,t, γM,t,at ≡ (aT,t )α (aN,t )1−α , mh,t , mf,t , yh,t , and yf,t .
Let x̂ denote a percentage deviation of any variable xt from its deterministic steady statevalue x∗ , x̂ ≡ ln xt − ln x∗ . Also, let x̃ denote a deviation of x from its deterministic steady state,x̃ = x − x∗ .5 The log-linear approximation of the stochastically de-trended home budget constraintis¯ − β −1 )s∗ ŝt + s∗ b̃fωp∗h yh (p̂h,t + ĉh,t ) − p∗T,h yT,h (p̂T,h,t + ŷh,t ) + b̃hh,t + d(1h,t∗ −1 f¯ + r̂h ) − γ̂ h ] + s∗ β −1 d[(1¯ + r̂f ) − γ̂ f ] + β −1 b̃h= β −1 d[(1b̃h,t−1 ; (F.3)
h,t−1
M,t
h,t−1 + s β
M,t
h,t−1

that of the home Euler equation ishhp̂h,t + ĉh,t + (1 + r̂h,t) = Et (p̂h,t+1 + ĉh,t+1 + γ̂M,t+1
);

(F.4)

that of the home UIP condition isffhh
Et ŝt+1 − ŝt = (1 + r̂h,t) − (1 + r̂h,t) − Et (γ̂M,t+1− γ̂M,t+1
);

(F.5)

and that of the home money demand function isp̂h,t + ĉh,t − m̂h,t =1h(1 + r̂h,t) − φ̂h,t .
∗
r

(F.6)

The foreign country’s counterparts are the log-linear approximation of the stochastically de-trended

5

In particular, for an interest rate rt , (1 + r̂t ) = (rt − r∗ )/(1 + r∗ ).

22

foreign budget constraint¯ − β −1 )s∗ ŝt − s∗ b̃f
ωp∗f c∗f (p̂f,t + ĉf,t ) − p∗T,f yT,f (p̂T,f,t + ŷf,t ) − b̃hh,t − d(1
h,t
f
−1 h
¯ + r̂h ) − γ̂ h ] − s∗ β −1 d[(1¯ + r̂f= −β −1 d[(1b̃h,t−1 − s∗ β −1 b̃fh,t−1 ; (F.7)
w,t−1 ) − γ̂M,t ] − β
w,t−1
M,t

that of the foreign Euler equationff) = Et (ŝt+1 + p̂f,t+1 + ĉf,t+1 + γ̂M,t+1);
p̂f,t + ĉf,t + (1 + r̂w,t(F.8)

that of the foreign UIP conditionffhh
Et ŝt+1 − ŝt = (1 + r̂w,t) − (1 + r̂w,t) − Et (γ̂M,t+1− γ̂M,t+1
);

(F.9)

and that of the home money demand functionp̂f,t + ĉf,t − m̂f,t =1
f(1 + r̂w,t) − φ̂f,t .
r∗(F.10)The log-linear approximations of the home country’s interest rates arehh(1 + r̂h,t) = (1 + r̂w,t) − ψ(1 − κ)b̃hh,t ,ffand (1 + r̂h,t) = (1 + r̂w,t) − ψ(1 − κ)b̃fh,t .

(F.11)The log-linear approximation of the real exchange rate is

q̂t =

1−α
1−α
(ĉf,t − ĉh,t ) +(ŷN,h,t − ŷN,f,t ) − (1 − α)(âT,t − âN,t ),αα(F.12)The log-linear approximation of the home price of the tradable good isp̂T,h,t = p̂h,t −1−α(ĉh,t − ŷN,h,t ).
α23(F.13)

The log-linear approximation of the PPP deviation isq̂t = ŝt + p̂f,t − p̂h,t + αâT,t + (1 − α)âN,t .

(F.14)

Finally, the log-linear approximation of the LOP isâT,t + ŝt + p̂T,f,t = p̂T,h,t .

(F.15)

Notice that the home interest rates (F.11) redefine the home UIP condition (F.5) asffhh
Et ŝt+1 − ŝt = (1 + r̂w,t) − (1 + r̂w,t) − ψ(1 − κ)(b̃hh,t − b̃fh,t ) − Et (γ̂M,t+1− γ̂M,t+1).

Comparing the above home UIP condition with the foreign UIP condition (F.9) implies that thehome and foreign bonds are perfectly substitutable along the equilibrium path. Hence, the equilibrium condition b̃t ≡ b̃hh,t = b̃fh,t holds.
F.3. Equilibrium random-walk property of nominal exchange rates
Let ct , yT,t , yN,t , mt , and φt denote the de-trended consumption ratio, the de-trended tradable
output ratio, the de-trended non-tradable output ratio, the transitory money supply ratio, themoney demand shock ratio between the two countries, ct ≡ ch,t /cf,t , yT,t ≡ yT,h,t /yT,f,t , yN,t ≡yN,h,t /yN,f,t , mt ≡ mh,t /mf,t , and φt ≡ φh,t /φf,t , respectively. Furthermore, let Mtτ denote the ratioττ
of the permanent money supplies of the home and foreign countries Mh,t/Mf,t; let Mt foreign money

supplies of the home to the foreign countries Mh,t /Mf,t ≡ mt Mtτ ; let γM,t denote the ratio of thefhpermanent money supply growth rate γM,t ≡ γM,t/γM,t; let Ct denote the ratio of the consumptionsof the home and foreign countries Ch,t /Cf,t . Below, the steady state value of the nominal market
discount factor is denoted by κ ≡ 1/(1+r∗ ) = β/γM . Under the symmetric case with d¯ = 0, FONCs(F.3)-(F.15) are degenerated to the following four expectational diﬀerence equations with respect
to the four endogenous variables ŝt , q̂t , ĉt , and b̃t , given the six exogenous variables γ̂M,t , m̂t , âT,t ,24

âN,t , ŷT,t , ŷN,t , and φ̂t :ŝt = κEt ŝt+1 − (1 − κ)(ĉt − q̂t + ât ) + (1 − κ)(m̂t − φ̂t ) + κEt γ̂M,t+1 − ψκ(1 − κ)b̃t ,ŝt + ĉt − q̂t + ât = κEt (ŝt+1 + ĉt+1 − q̂t+1 + ât+1 ) + (1 − κ)(m̂t − φ̂t ) + κEt γ̂M,t+1 ,b̃t = β −1 b̃t−1 + p∗T yT∗ (âT,t + ŷT,t − ĉt + q̂t − ât ),
q̂t =

α−1
1−α
ĉt +
ŷN,t − (1 − α)(âT,t − âN,t ),αα(F.16)

where yT∗ = yT /4 and yT = yT,h = yT,f . In particular, the first equation of the linear rational
expectations (LRE) system (F.16) represents the stochastically de-treded UIP after solving theinterest rate diﬀerential through the money demand functions of the two countries; the second
equation the cross-country diﬀerence in the Euler equation; the third equation the law of motion of
net foreign asset position; and the fourth equation the real exchange rate.
The first and second equations of LRE system (F.16) result inĉt + ât − q̂t = Et (ĉt+1 + ât+1 − q̂t+1 ) + ψ(1 − κ)b̃t ,The third equation of LRE system (F.16) and the above equation implies the second order diﬀerence
equation of the net foreign asset position b̃t
Et b̃t+1 − [1 + β −1 + p∗T yT∗ ψ(1 − κ)]b̃t + β −1 b̃t−1 = −p∗T yT∗ λT âT,t + p∗T yT∗ (ρT − 1)ŷT,t .

(F.17)

It is straightforward to show that equation (F.17) has two roots, one of which is greater than one andthe other of which is less than one. Without losing generality, let η denote the root that is less thanone. Solving equation (F.17) by forward iterations then shows that the equilibrium international25

bond holdings level is determined by the following cross-equation restriction (CER):b̃t = η b̃t−1 + βηλT p∗T yT∗= η b̃t−1 +

∞
%

j=0
∗ ∗
βηλT pT yT(βη)j Et âT,t+j + βηp∗T yT∗ (1 − ρT )1 − βη(1 − λT )âT,t +

βηp∗T yT∗ (1 − ρT )1 − βηρT∞%(βη)j Et ŷT,t+j ,j=0ŷT,t .

(F.18)

Substituting the fourth equation of LRE system (F.16) into the third equation yields b̃t =()β −1 b̃t−1 + p∗T yT∗ ŷT,t + 1−αŷN,t − α1 ĉt . CER (F.18) then implies that the equilibrium consumption
αratio isĉt =α(1 − ηβ)αηβλTα(1 − ηβ)b̃t−1 −
âT,t +ŷT,t + (1 − α)ŷN,t .
∗ ∗pT yT β1 − ηβ(1 − λT )1 − ηβρT(F.19)

Using the fourth equation of LRE system (F.16), the equilibrium RER is determined asq̂t = −

(1 − α)(1 − ηβ)(1 − α)(1 − ηβ)(1 − α)(1 − ηβ)b̃t−1 −
âT,t −ŷT,t + (1 − α) ln YN,t . (F.20)∗ ∗pT yT β1 − ηβ(1 − λT )1 − ηβρT

Now a forward iteration of the second equation of LRE (F.16) toward an infinite future leadstoŝt + ĉt + ât − q̂t =
=

∞
%
i=1

i

κ Et γ̂M,t+i − (1 − κ)

∞
%
i=0

κi Et (φ̂t+i − m̂t+i ),

κρM
1−κ
1−κ
γ̂M,t −φ̂t +m̂t
1 − κρM
1 − κρφ1 − κρm

Substituting equilibrium consumption ratio (F.19) and equilibrium RER (F.20) into the above
equation provides the equilibrium de-trended nominal exchange rateŝt =

ηβ − 1κρM1−κ
1−κ
1 − ηβ1 − ηβb̃t−1 +
γ̂M,t −φ̂t +m̂t −âT,t −ŷT,t
∗ ∗pT yT β1 − κρM
1 − κρφ1 − κρm
1 − ηβ(1 − λT )1 − ηβρT(F.21)

Finally, substituting the equilibrium de-trended nominal exchange rate (F.21) into the relation26

ln St = ln Mtτ + ŝt and taking the first diﬀerence of the result yields the equilibrium currency returnρM (1 − κ)
(1 − κ)(1 − ρφ )(1 − κ)(1 − ρm )γ̂M,t−1 +φ̂t−1 −m̂t−1
1 − κρM
1 − κρφ1 − κρm11−κ
1−κ
1 − ηβ1 − ηβ+ϵM,t −
ϵφ,t +
ϵm,t −
ϵy,t −
ϵT,t . (F.22)
1 − κρM
1 − κρφ1 − κρm
1 − ηβρT
1 − ηβ(1 − λT )∆ ln St = ψ(1 − κ)b̃t−1 +where ϵM,t ≡ ϵhM,t − ϵfM,t , ϵφ,t ≡ ϵhφ,t − ϵfφ,t , ϵm,t ≡ ϵhm,t − ϵfm,t , ϵT,y,t ≡ ϵhT,y,t − ϵfT,y,t , and ϵT,A,t ≡ϵhT,A,t − ϵfT,A,t . Therefore, at the limit of the unit discount factor (κ, β, η → 1), the currency returnfollows a white noise process.
lim ∆ ln St =κ,β,η→1

1
ϵM,t .
1 − κρM

F.4. Backus and Smith’s puzzle at the limit
Recall that unwinding the stochastic trend into the equilibrium consumption ratio leads toln Ct = ln at + ĉt . Equilibrium de-trended consumption ratio (F.19) then implies

ln Ct =α(1 − ηβ)α(1 − ηβ)α(1 − ηβ)b̃t−1 +
âT,t +ŷT,t + (1 − α) ln YN,t .
∗ ∗pT yT β1 − ηβ(1 − λT )1 − ηβρT

At the limit of the unit discount factor the equilibrium consumption diﬀerential turns out to be

lim ln Ct = (1 − α) ln YN,tκ,η,β→1

Similarly at the limit of the unit discount factor the RER becomes

lim ln qt = (1 − α) ln YN,tκ,η,β→1

Therefore, the consumption diﬀerential is perfectly correlated with the RER at the limit.

References

27

Bouakez, H., Kano, T., 2006, Learning-by-doing or habit formation?, Review of Economics Dynamics 9, 508 − 524.
Kano, T., 2009, Habit formation and the present-value model of the current account: yet anothersuspect, Journal of International Economics 78, 72 − 85.
Sargent, T.J., 1987, Macroeconomic Theory, second ed. Academic Press, San Diego, California.
Schorfheide, F., 2000, Loss function-based evaluation of DSGE models, Journal of Applied Econometrics 15, 645 − 670.
Sims, C. A., 2001, Solving linear rational expectations model, Computational economics 20, 1 − 20.

28

1990

2000

1980

1990

HDF Model
2000

Benchmark Model(d) Output Differential

1980(a) Consumption Differential-0.015-0.01-0.005

0

0.005

0.01

0.015

0.02

-1.5
1990

2000

1980

1990

2000

(e) Interest Rate Differential

-0.1

-0.08

-0.06

-0.04

-0.5
-1

-0.02

0

1980

0.02

0.5
0

0.04

0.06

1

(b) Money Supply Differential

1980

1990

2000

(c) Exchange Rate

Figure A.1 : Forecast errors for observations. Note: In each window corresponding to each time series observation, the dark and light gray areas
display the 95% Bayesian highest probability density (HPD) intervals of the one-period-ahead forecast error calculated through the Kalman forward
recursion based on the state space representation of the Benchmark and high discount factor (HDF) models, respectively.

-0.02

-0.01

0

0.01

0.02

-0.4

-0.2

0

0.2

1980

HDF Model

Only PPP Deviation Shock

1985

1990

1995

2000

2005

Figure A.2 : Forecast error of consumption diﬀerential Note: The dark gray area depicts the 95% Bayesian highest probability density (HPD) intervals
attached to the forecast error of the consumption diﬀerential, which is implied by the high discount factor (HDF) model. The light gray area indicatesthe 95% HPD intervals of the forecast error of the consumption diﬀerential, which is generated only by the purchasing power parity (PPP) deviation
shock.

-1
1975

-0.8

-0.6

-0.4

-0.2

0

0.2

2005

1985

1990

1995

2000

1980

1985

1990

1995

2000

2005

Figure A.3 : Forecast errors for observations. Note: In each window corresponding to a particular observation, the dashed blue and dotted red lines
display the 95% Bayesian highest probability density (HPD) intervals of the one-period-ahead forecast error calculated through the Kalman forward
recursion based on the state space representation of the Benchmark and high discount factor (HDF) models, respectively.

1980

-5
1975

-4

-4

-5
1975

-3

-3

Permanent Money Supply Differential-2-2

Money Demand Shock Differential-1-1

Transitory Exchange Rate

00

Transitory Exchange Rate

1

1

3

(b) Transitory Exchange Rate and Permanent Money Supply Differential

2(a) Transitory Exchange Rate and Money Demand Shock Differential

23