Exchange Rate Disconnect
in General Eqilibrium∗
Oleg Itskhoki
Dmitry Mukhin
itskhoki@princeton.edu
mukhin@princeton.edu
August 30, 2016
UNDER CONSTRUCTION
download the most up-to-date version from:
http://www.princeton.edu/~itskhoki/papers/disconnect.pdf
Abstract
We propose a dynamic general equilibrium model of exchange rate disconnect, which simultaneously accounts for all major puzzles associated with the nominal and real exchange rates. This
includes the Meese-Rogoff puzzle, the PPP puzzle, the terms-of-trade puzzle, the Backus-Smith puzzle, and the UIP puzzle. The model has two main building blocks — the driving force (or the exogenous shock process) and the transmission mechanism — both crucial for the quantitative success
of the model. The transmission mechanism — which relies on strategic complementarities in price
setting, low degree of substitutability between domestic and foreign goods, and home bias in consumption — is tightly disciplined by the micro-level empirical estimates in the recent international
macroeconomics literature. The driving force is an exogenous small but persistent shock to international asset demand, which result in a wedge in the Euler equation for international bonds. We
prove formally that this is the only shock that can generate exchange rate disconnect in the autarky
limit. We then show that a model with this financial shock alone is quantitatively consistent with
the moments describing the dynamic comovement between exchange rates and the macro variables. The addition of heterogeneous firms and sticky wages and prices improves the quantitative
performance of the model, but does not affect its qualitative properties. Importantly, sticky prices
are not necessary to generate exchange rate disconnect, as the driving force does not rely on the
monetary shocks.
∗
We thank Ariel Burstein, Charles Engel, Robert Kollmann and Cédric Tille for insightful discussions, Mark Aguiar, Andy
Atkeson, Adrien Auclert, Giancarlo Corsetti, Emmanuel Farhi, Gita Gopinath, Pierre-Olivier Gourinchas, Galina Hale, Tarek
Hassan, Maury Obstfeld, Esteban Rossi-Hansberg, Jesse Schreger, Jón Steinsson, Eric Van Wincoop, Adrien Verdelhan, Mark
Watson, and seminar/conference participants at Princeton, Yale, CEPR, IMF, SED, NBER, and UCLA for helpful comments,
and Nastya Burya for excellent research assistantship.
1
Introduction
Exchange rate disconnect is among the most challenging international macro puzzle (Obstfeld and Rogoff 2001). The term disconnect narrowly refers to the lack of the correlation between exchange rate
and other macro variables, but the broader puzzle is more pervasive and nests a number of additional
empirical patterns, which stand at odds with the conventional international macro models. We define
the broader exchange rate disconnect to include:
1. Meese and Rogoff (1983) puzzle: nominal exchange rate follows a random walk and is not robustly
correlated, even contemporaneously, with the fundamentals (see also Engel and West 2005).
2. PPP puzzle (Rogoff 1996): real exchange rate tracks very closely the nominal exchange rate at
most frequencies (in particular, exhibiting similarly large persistence and volatility) and, if it
mean reverts, its half-life is as long as 3–5 years, much in excess of the typical duration of sticky
prices (see also Chari, Kehoe, and McGrattan 2002; henceforth CKM).
3. Terms of trade are positively correlated with the real exchange rate, yet exhibit a markedly lower
volatility, in contrast with the predictions of the standard models, suggesting a particular pattern
of law of one price violations and pricing to market (Atkeson and Burstein 2008). In addition, the
real exchange rate dynamics at most horizons is almost fully accounted for by the law of one price
violations for tradable goods, and the relative non-tradable prices explain almost no variation in
the real exchange rate (Engel 1999).
4. Backus and Smith (1993) international risk-sharing condition that relative consumption of the
country should be positively and strongly correlated with the real exchange rates (i.e., high consumption when low prices) is sharply violated in the data, with a mildly negative correlation
and markedly lower volatility of relative consumption (see also Kollmann 1995, Benigno and
Thoenissen 2008 and CKM).
5. Forward premium puzzle, or the violations of the uncovered interest rate parity (UIP) condition,
that the interest rate differentials should predict the nominal exchange rate devaluation is violated
in the data with an opposite sign, but a nearly zero R2 (see Fama 1984, Engel 1996).
In Section 2 we summarize and review the empirical patterns in greater detail, and use them as the
quantitative targets for our model in later sections.
The absolute majority of the modern general equilibrium international macro models feature these
puzzles, or attempt to address one puzzle at a time, resulting in a lack of a unified framework that
exhibits satisfactory exchange rate properties. This is a major challenge for the academic and policy
discussion, since exchange rates are the core prices in any international macro model, and failing to
match their basic properties jeopardizes the conclusions one can draw from a model, in particular when
the focus is on the international transmission of shocks.
A successful model of disconnect must satisfy two properties. First, it should specify the driving force
(shock process) of nominal and real exchange rates, which cannot simultaneously have a sharp effect
on the contemporaneous consumption, output, prices and interest rates. Second, it should specify the
transmission mechanism, which mutes the effect of volatile exchange rate fluctuations on local prices
1
and quantities. Violation of either of the properties would break the disconnect. While the literature
has provided a lot of empirical evidence on the transmission mechanism, taking the exchange rate
movements as exogenous, we currently lack direct empirical information on the details of the shock
process, which accounts for the bulk of the exchange rate fluctuations.
Given this state of the literature, we adopt the following strategy. On one hand, we tightly discipline
the transmission mechanism with the empirical estimates from the recent literature, which leaves us
with little uncertainty, or degrees of freedom. In contrast, we initially impose no restrictions on the
nature of the shock process, saturating the model with all possible shocks, or wedges. We then show
that there exists a single type of shocks, which can satisfy the exchange rate disconnect properties.
In particular, we impose the requirement that the shock should produce a volatile exchange rate
with a vanishing effect on the economy’s quantities, prices and interest rates, as the economy becomes
closed to trade. Indeed, in the limit of the closed economy, any exchange rate volatility (real or nominal)
should be completely inconsequential for the allocations. Not surprisingly, productivity and monetary
shocks, as well as the majority of other shocks, violate this intuitive requirement. The one shock that
satisfies this requirement is the shock to the international asset demand. We later show that this shock
can have a variety of microfoundations in the financial market, including noise trading with limits to
arbitrage, heterogeneous beliefs, and financial frictions.
We further show that the model with a single international asset demand shock is consistent both
qualitatively and quantitatively with the exchange rate disconnect. In particular, small persistent shocks
to international asset demand result in a volatile random-walk-like behavior of the nominal exchange
rate, needed to equilibrate the international asset market, simultaneously being consistent with the
intertemporal budget constraints of the countries. As the economy becomes more closed to international trade, this shock still generates volatile nominal exchange rate fluctuations, which however have
a vanishingly small effect on the rest of the economy.
Furthermore, the transmission mechanism in the model ensures that the nominal exchange rate
exhibits the empirically relevant comovement properties with the other variables, even when the economy is open to trade. The transmission mechanism relies on three main features:
1. pricing to market and law of one price violations due to strategic complementarities in price
setting (using estimates from Amiti, Itskhoki, and Konings 2016), which limit the response of
prices (terms of trade) to the exchange rate movements;
2. low elasticity of substitution between home and foreign goods (using estimates from Feenstra,
Luck, Obstfeld, and Russ 2014), which limit the extent of expenditure switching conditional on
the terms of trade movements;
3. significant home bias in consumption (consistent with the trade share in GDP for the US, Japan,
and the European Union as a whole), which limits the effect of expenditure switching on aggregate consumption, employment and output.
This provides the quantitative discipline for the transmission mechanism in our model. In particular, it
allows the model to reproduce the real exchange rate behavior that tracks closely that of the nominal
exchange rate, including the volatility and persistence, which are in line with the PPP puzzle literature.
2
Importantly, this does not rely on price or wage stickiness, which we assume away in the baseline
model. Indeed, the nominal stickiness is not needed, as the driving process does not rely on the monetary shocks. The model, instead, relies on strategic complementarities in price setting, which generate
the empirical patterns of deviations from the law of one price. The firms in the model adjust markups
in response to the exchange rate movements, absorbing part of the shock in their profit margins. In an
extension, we show that sticky wages further improve the quantitative performance of the model, but
do not change any of our qualitative conclusions.
The mechanism in the model is as follows. A small persistent increase in demand for foreign bonds
(e.g., due to its liquidity or safety properties, which we leave outside the model) results in a sharp depreciation of the home currency and a slow but persistent appreciation thereafter, as the shock gradually
dies out. Such behavior of the exchange rate ensures that both the asset demand and the intertemporal
budget constraint of the country are satisfied. The more persistent is the shock, the closer is the behavior of the nominal exchange rate to a random walk, with the unexpected exchange rate innovation
accounting for all of the exchange rate movements in the limit.1
In response to the exchange rate devaluation, there is domestic price inflation and a reduction
in the home real wage, and simultaneously an expenditure switching towards the goods produced at
home. To sustain a simultaneous drop in wages and an increase in labor demand, home consumption
falls, supported by the increase in the home interest rate. Since the exchange rate is devalued, but
expected to appreciate, the model reproduces both low consumption in periods of low prices (i.e., the
deviation from the Backus-Smith condition) and expected depreciation in periods of high interest rates
(i.e., the UIP puzzle). The transmission mechanism—with substantial home bias and low pass-through
into prices and quantities—ensures that the movements in consumption and interest rates are very
mild, much smaller than those in the nominal exchange rate, consistent with the empirical patterns.
In particular, the model reproduces the close to zero R2 in the Fama regression. In an extension, we
show that adding small productivity shocks largely leaves unchanged the quantitative properties of the
model, and additionally allows it to match the close to zero correlations between the exchange rate and
the other macro variables.
Literature review [to be completed]
• Kouri (1976, 1983), Masson (1981), Henderson and Rogoff (1982), Branson and Henderson (1985),
Blanchard, Giavazzi, and Sa (2005), Gourinchas (2008): balance-of-payment view of the exchange
rate, international demand shocks for assets and goods
• Jeanne and Rose (2002), Gourinchas and Tornell (2004), Bacchetta and van Wincoop (2006), Alvarez, Atkeson, and Kehoe (2009), Brunnermeier, Nagel, and Pedersen (2009), Farhi and Gabaix
(2016), Gabaix and Maggiori (2015), Valchev (2015), Lustig and Verdelhan (2016), Lustig, Stathopoulos, and Verdelhan (2016), Engel (2016): financial models of exchange rate
1
In the limit of the closed economy, the exchange rate devaluation simply ensures the drop in the country’s wealth relative
to the foreign, as there cannot be actual trade in assets (non-zero capital account) in the absence of the offset trade in the goods
market (zero current account). The physical allocations, however, remain unchanged despite the reduction in the nominal
wealth of the country relative to the rest of the world.
3
• Jeanne and Rose (2002), Kollmann (2005), Monacelli (2004) on Mussa (1986, 1990) puzzle
• Kollmann (2005), Devereux and Engel (2002), Farhi and Werning (2012): risk-premia (Euler equation) shocks
• Benigno and Thoenissen (2008), Corsetti, Dedola, and Leduc (2008): Backus-Smith under incomplete markets
• Chari, Kehoe, and McGrattan (2007), Eaton, Kortum, and Neiman (2015): wedge accounting
• DSGE literature: . . .
• Other Kollman, Devereux-Engel papers?
The rest of the paper is organized as follows. In Section 2 we review the stylized facts and puzzles
related to the exchange rates, and re-calculated the main empirical moments in the recent samples to
use as the targets for the model to match in later section. In Section 3 we describe the baseline model
and prove that only the international asset demand shocks can be consistent with the exchange rate
disconnect. In Section 4 then explores the qualitative and quantitative properties of the model with the
international asset demand shocks alone. In Section 5.2 we allow for multiple shocks and estimate the
model using the moments documented in Section 2; using the estimated model, we provide a variance
decomposition of the exchange rate process into the contribution of various shocks. Section 5 describes
the extensions, including a sticky wage model, a model with heterogeneous firms, and a multi-country
model. Section 6 concludes, and the appendix provides detailed derivations and proofs.
2
Stylized Facts and Puzzles
[to be completed]
3
Modeling Framework and Shocks
We start with a flexible modeling framework that can nest most standard international macro models,
which allows us in what follows to consider various special cases and extensions. There are two countries, home (Europe) and foreign (US, denoted with a ∗). Each country has a nominal unit of account,
with the wage rates given by Wt euros at home and Wt∗ dollars in foreign, and Et denotes the nominal
exchange rate, i.e. the price of one dollar in euros. Therefore, an increase in Et signifies a nominal
devaluation of the home currency (the euro). We allow for a variety of shocks hitting the economy,
with some of the shocks taking the form of wedges (following Chari, Kehoe, and McGrattan 2007),
proxying in some cases for unmodelled market imperfections. We then explore which of these shocks
and wedges can and which cannot account for the exchange rate disconnect, as we formally define it
below in Section 3.2.
4
3.1
Model setup
Households A representative home household maximizes the discounted expected utility over consumption and labor:
E0
∞
X
t χt
βe
t=0
1
eκt
1+1/ν
1−σ
,
C
−
L
1−σ t
1 + 1/ν t
(1)
where (χt , κt ) are the utility shocks, σ is the relative risk aversion parameter and ν is the Frisch elasticity of labor supply.2 Table 1 below summarizes all shocks/wedges and model parameters. The flow
budget constraint is given by:
P t Ct +
∗ E
Bt+1 Bt+1
t
+ ψt ∗ ≤ Bt + Bt∗ Et + Wt Lt + Πt + Tt ,
Rt
e Rt
(2)
where Pt is the consumer price index, (Bt , Bt∗ ) are the quantities of the home and foreign bonds paying out next period one unit of the currency of the issuing country, and (Rt , Rt∗ ) are their discounts
(i.e., 1/Rt and 1/Rt∗ are their prices); Πt are the dividends and Tt are lump-sum transfers. Lastly, ψt is
the (inverse) cost shock to the international bond trading ability of the domestic households, e.g. a
capital controls tax/wedge, as we further discuss below.3
The households are active in three markets. First, they supply labor according to the standard static
optimality condition:
1/ν
eκt Ctσ Lt
=
Wt
,
Pt
(3)
where the preference shock κt can be alternatively interpreted as the labor wedge, playing an important
role in the closed-economy business cycle literature and capturing the departures from the neoclassical
labor market dynamics due to search frictions or sticky wages (see e.g. Shimer 2009). In addition, we
denote Wt ≡ ewt and interpret wt as the shock to the nominal value of the unit of account, which
captures monetary shocks in our framework.4
Second, the households choose their bond positions according to the dynamic optimality conditions:
1 =Rt Et Θt+1
where
1=e
and
∆χt+1
Θt+1 ≡ βe
ψt
Rt∗ Et
Ct+1
Ct
Et+1
Θt+1
Et
−σ
,
(4)
Pt
Pt+1
is the stochastic discount factor. The change in the time preference shock, ∆χt+1 , affects the consumptionsavings decision and acts as the intertemporal demand shifter for both types of bonds. The ψt shock
instead acts as the demand shock for the foreign bond. We discuss in Section 4.1 the various models of
2
Our results are robust to alternative utility specifications, such as the GHH utility without income effects on labor supply.
Incompleteness of international asset markets is important for some results, as we explain in Section 4. However, the
assumption that a risk-free bond is the only internationally-traded asset is not essential, and we adopt it only because we rely
on a log-linearization for the analytical solution of the model. Some of the results, in particular Proposition 1, do not rely at
all on the assumptions about the (in)completeness of the international asset market.
4
In Section 5.1 we provide an explicit model with nominal wage and local-currency price stickiness and conventional
Taylor rules, which offers an example of one typical source of (wt , κt , µt , ηt ) shocks (with µt and ηt defined below).
3
5
Table 1: Model parameters and shocks/wedges
Parameters†
wt
Shocks/wedges
nominal wage rate
(shock to the value of the unit of account)
β = 0.99
discount factor
at
productivity shock
σ=2
relative risk aversion (inverse of IES)
gt
government spending shock
ν=1
Frisch elasticity of labor supply
χt
intertemporal preference shock
γ = 0.07
foreign share (home bias) parameter
κt
labor wedge (sticky wages)
θ = 1.5
elasticity of substitution
µt
markup shock (sticky prices)
α = 0.4
strategic complementarity elasticity
ηt
law-of-one-price shock (local currency pricing)
φ = 0.5
intermediate share
ξt
international good demand shock
ρ = 0.97
persistence of the shock
ψt
international asset/currency demand shock
†
We report the baseline parameter values used in Sections 4–5.
asset demand which result in a similar reduced form as in (4), and in what follows we refer to ψt as the
international asset/currency demand shock.5
Lastly, the households allocate their within-period expenditure between home and foreign goods:
Pt Ct = PHt CHt + PF t CF t ,
and we assume the good demand is homothetic and symmetric, and given by:
CHt = (1 − γ)e
−γξt
h
PHt
Pt
Ct
and
(1−γ)ξt
CF t = γe
h
PF t
Pt
Ct ,
(5)
where ξt is the relative demand shock for the foreign good and γ is the home bias parameter, with
the demand for the foreign good collapsing to zero as γ → 0. The function h(·) controls the curvature of the demand schedule, satisfies h0 (·) < 0 and h(1) = 1, and we denote its point elasticity by
θ ≡ − ∂ log h(x) . The demand formulation in (5) emerges from a homothetic and separable Kimball
∂ log x
x=1
(1995) demand aggregator, as we show in Appendix A.1, where we also derive an explicit expression
for the price index Pt . In our analysis, we focus on the behavior of the economy around a symmetric
steady state with ξ = 0 and P = PH = PF , and make use of the following three properties of this
demand system (see Appendix A.1):
Lemma 1 (Properties of Demand) In a symmetric steady state with ξ = 0 and P = PH = PF :
(i) The expenditure share on foreign goods (the foreign share for brief), defined as
P F t CF t
P t Ct ,
equals the
home bias parameter γ.
5
One simple story, following Dekle, Jeong, and Kiyotaki (2014), directly places foreign bonds into the utility function of
home households (in parallel with the money-in-the-utility models). In particular, if we add eψt Bt∗ to the flow utility in (1)
and drop ψt term from the budget constraint (2), we obtain an identical log-linearized reduced form of the model. More
spelled-out micro theories of asset demand model explicitly (a) the demand for liquidity or safety, (b) limits to arbitrage,
heterogeneous beliefs or financial frictions in the asset markets, or (c) risk premia due to disasters or long-run risk (reflected
in the second moments of the stochastic discount factor). We view the ψt shock/wedge as a stand-in for all such mechanisms.
6
(ii) The log-linear approximation to demand (5) around the steady state is given by:
cHt = −γξt − θ(pHt − pt ) + ct
cF t = (1 − γ)ξt − θ(pF t − pt ) + ct .
and
(6)
Therefore, cF t − cHt = ξt − θ(pF t − pHt ), and the elasticity of substitution between home and
foreign goods, defined as
∂ log(CF t /CHt )
∂ log(PHt /PF t ) ,
equals the point elasticity of the demand schedule θ.
(iii) The log-linear approximation of the consumer price index around the steady state is given by:
pt = (1 − γ)pHt + γpF t ,
(7)
where small letter pt ≡ log Pt − log P̄ denotes the log deviation of the price index from its steady
state value P , and similarly for the other variables.
We show below that the values of γ (trade openness) and θ (elasticity of substitution between home
and foreign goods) play the central role in the quantitative properties of the transmission mechanism.
Production and prices Output is produced by a given pool of identical firms according to a CobbDouglas technology in labor Lt and intermediate inputs Xt :
Yt = eat Lt1−φ Xtφ ,
(8)
where at is the productivity shock and φ is the elasticity of output with respect to intermediates, which
determines the equilibrium expenditure share on intermediate goods. Intermediates are the same bundle of home and foreign varieties as in the final consumption, and hence their price index is also given
by Pt . Therefore, the marginal cost of production is:
−at
M Ct = e
Wt
1−φ
1−φ Pt
φ
φ
,
(9)
and the firms optimally allocate expenditure between labor and intermediates according to the following input demand conditions:
Wt Lt = (1 − φ)M Ct Yt
and
Pt Xt = φM Ct Yt .
(10)
The expenditure on intermediates Xt is further split between the domestic and foreign varieties, XHt
and XF t , in parallel with the consumption expenditure by the households in (5). The profits of the
domestic firms (distributed to the domestic households) are given by:
∗
∗
Πt = (PHt − M Ct ) YHt + (PHt
Et − M Ct ) YHt
,
(11)
∗ .6
where total production is split into output delivered to the home and foreign markets, Yt = YHt + YHt
6
For analytical tractability, we focus on a constant-return-to-scale production without capital, until Section 5 where we
show the robustness of our results to an extension with capital and adjustment costs. We further assume no entry or exit
7
We postulate the following price setting:
PHt = eµt M Ct1−α Ptα ,
∗
PHt
= eµt +ηt M Ct /Et
(12)
1−α
Pt∗α ,
(13)
where α ∈ [0, 1) is the strategic complementarity elasticity, µt is the markup shock, and ηt is the law
of one price (LOP) shock. Given these prices, the firms satisfy the resulting demand in both markets.
Equations (12)–(13) are ad hoc yet general pricing equations, as the markup terms (together with a
flexible choice of α) allow them to be consistent with a broad range of price setting models, including
both monopolistic and oligopolistic competition models under both CES and non-CES demand. Furthermore, if the time path of (µt , ηt ) is not restricted, these equations are also consistent with dynamic
price setting models, and in particular the sticky price models (with either producer, local or dollar
currency pricing).7
Strategic complementarities in price setting (α > 0) reflect the tendency of the firms to set prices
closer to their local competitors, a pattern which is both pronounced in the data and emerges in a
variety of models (see Amiti, Itskhoki, and Konings 2016), and we emphasize in our analysis below its
role for the international transmission of shocks. Appendix A.1 discusses a model of Kimball (1995)
demand that is simultaneously consistent with our choices of elasticity of substitution θ and strategic
complementarity elasticity α. Lastly, we note that the violations of the law of one price:
QHt ≡
∗ E
PHt
t
= eηt Qαt ,
PHt
where
Qt ≡
Pt∗ Et
,
Pt
(14)
arise either due to the LOP shock ηt (capturing, for example, local currency pricing) or due to α > 0
(capturing pricing-to-market), where Qt is the real exchange rate, which reflects the differences in the
price levels across the two markets.
Government and transfers The government uses lump-sum taxes to finance an exogenous stochastic path of government expenditure Pt Gt , where gt ≡ log Gt is the government spending shock. For
simplicity, we assume that government expenditure is allocated between the home and foreign goods
in the same way as the final consumption in (5). The government runs a balanced budget, which in
view of Ricardian equivalence is without loss of generality. As a result, the overall transfers to the
households are given by:
B∗ E
t+1 t
Tt = e−ψt − 1
− Pt e gt .
Rt∗
(15)
of firms, as our model is a medium-run one (for the horizons of up to 5 years), where empirically extensive margins play
negligible roles (see e.g. Bernard, Jensen, Redding, and Schott 2009). Instead of introducing variable trade costs and nontradables, we simply adjust the exogenous home bias parameter γ to match the trade shares in the data. The baseline model
features symmetric representative firms, and we discuss the extension with heterogeneous firms in Section 5.
7
Note that ηt can also stand in for the trade cost shocks in the goods market, playing an important role in the recent
quantitative analyses of Eaton, Kortum, and Neiman (2015) and Reyes-Heroles (2016). A combination of ηt and ξt shocks
can stand in for the world commodity price shocks, playing an important role as terms of trade shocks for the commodityexporting countries such as Canada, Australia, South America, Brazil and Chile (see e.g. Chen and Rogoff 2003).
8
The first terms reflects the proceeds from the capital controls tax, but is also consistent with the alternative interpretations of the ψt wedge.8
Foreign The foreign households are symmetric, except that the home (euro) bonds are not available
to them, and their budget constraint is given by:
Pt∗ Ct∗ +
∗F
Bt+1
≤ Bt∗F + Wt∗ L∗t + Π∗t + Tt∗ ,
Rt∗
where Bt∗F are the holdings of the foreign (dollar) bond by foreign households.9 The foreign households
supply labor and demand home and foreign goods according to the optimality condition parallel to (3)
and (5) respectively. In particular, the goods demand by the foreign households is given by:
∗
CHt
(1−γ)ξt∗
= γe
h
∗
PHt
Pt∗
Ct∗
and
CF∗ t
= (1 − γ)e
−γξt∗
h
PF∗ t
Pt∗
Ct∗ ,
(16)
where ξt∗ is the foreign demand shock for home goods. The bond demand of foreign households is
given by:
1=
Rt∗
Et Θ∗t+1 ,
where
Θ∗t+1
∆χ∗t+1
≡ βe
∗
Ct+1
Ct∗
−σ
Pt∗
∗ .
Pt+1
(17)
Lastly, the foreign firms are also symmetric, demand foreign labor and home and foreign intermediates,
and charge prices according to the counterparts of (12)–(13) with their own markup and LOP shocks
µ∗t and ηt∗ (see Appendix A.2).
Equilibrium conditions ensure equilibrium in the asset, product and labor markets, as well as the
intertemporal budget constraints of the countries. The labor market clears when Lt is consistent simultaneosly with labor supply in (3) and labor demand in (10), and symmetrically for L∗t in foreign.
∗ , where
The goods market clearing requires Yt = YHt + YHt
PF t
YHt = CHt + XHt + GHt = (1 − γ)e−γξt h
[Ct + Xt + Gt ] ,
Pt
∗ PHt
∗
∗
∗
∗
(1−γ)ξt∗
YHt = CHt + XHt + GHt = γe
h
[Ct∗ + Xt∗ + G∗t ] ,
Pt∗
(18)
(19)
and symmetric conditions hold for YF t + YF∗t = Yt∗ . The bonds market clearing requires Bt = 0 for
the home-currency bond, as it is in zero net supply and not traded internationally, and Bt∗ + Bt∗F = 0
for the foreign-currency bond, which is in zero net supply internationally.
Lastly, we combine the household budget constraint (2) with profits (11) and taxes (15) to obtain
8
For example, if ψt is the cost of the financial intermediation, this term reflects the profit transfer from intermediaries to
the households. The model with bonds in the utility function is also exactly isomorphic to this formulation. Alternatively, if
the international cost ψt is purely waisted, it needs to be additionally subtracted from the resource constraint, but it changes
neither qualitative, nor quantitative results.
9
We consider this asymmetric formulation between home and foreign for simplicity, but provide a symmetric version of
the model with a financial sector in Section 5.
9
the country budget constraint:
∗ E
Bt+1
t
− Bt∗ Et = N Xt ,
∗
Rt
where
∗
∗
N Xt = Et PHt
YHt
− P F t YF t
(20)
is the net exports of the home country (in the home currency). Note that the relative prices at which
the home country exchanges its exports for imports is the terms of trade:
St ≡
PF t
∗ E .
PHt
t
(21)
This completes the description of the model environment and the equilibrium system (further summarized in Appendix A.2), and we now turn to the selection of shocks.
3.2
Shock selection: disconnect in the limit
This section uses the general modeling framework to prove two theoretical results, which narrow down
the set of the shocks that can be consistent with the empirical exchange rate disconnect properties. In
particular, we study the behavior of the equilibrium system around the autarky limit, which we use as
our diagnostic device. The autarky limit (as foreign share γ → 0) is interesting for two reasons:10
1. Autarky (γ = 0) offers a model of complete exchange rate disconnect. When countries are in
autarky (in both good and asset markets), the nominal exchange rate is of no consequence, and
can take any values as an outcome of arbitrary sunspot equilibria. Therefore, it can be arbitrary
volatile, yet have no relationship with any macro variables in the two economies (the MeeseRogoff puzzle). Since price levels do not respond to this volatility, the real exchange rate comoves
perfectly with the nominal exchange rate, and in particular can have arbitrary persistence (the
PPP puzzle). This is possible because in autarky the real exchange rate does not affect allocations.
2. As the economies become open and move further away from autarky (i.e., γ increases), the responses of macro variables to exchange rate increase together with openness γ, and the macro
variables become more volatile and less disconnected. Therefore, if the economy does not exhibit
exchange rate disconnect properties near autarky (for γ ≈ 0), it cannot exhibit such properties
away from autarky (for γ 0).
We now extend the autarky logic to study circumstances under which a near-closed economy features
a near-complete exchange rate disconnect. While such continuity requirement may appear natural as the
equilibrium dynamics is continuous in γ, it nonetheless offers a sharp selection criterion for exogenous
shocks. This is because a limiting economy with γ > 0 acts as a refinement on equilibria when γ = 0,
as it rules out the sunspot equilibria with volatile exchange rate dynamics.11
We start by formalizing the notion of the exchange rate disconnect in the autarky limit:
10
Note that γ = 0 implies autarky in both goods and assets, while financial autarky alone is insufficient for our results.
Indeed, for γ > 0, the equilibrium in the model of Section 3.1 is unique and excludes the possibility of sunspot dynamics
in the nominal and real exchange rates.
11
10
Definition 1 (Exchange rate disconnect in the limit) Denote with Zt ≡ (Wt , Pt , Ct , Lt , Yt , Rt )
a vector of all domestic macro variables (wage rate, price level, consumption, employment, output, interest
rate) and with
εt ≡ V 0 Ω t ,
Ωt = {wt , wt∗ , χt , χ∗t , κt , κ∗t , at , a∗t , gt , gt∗ , µt , µ∗t , ηt , ηt∗ , ξt , ξt∗ , ψt }
an arbitrary combination of shocks hitting the economy, where V is a vector of loadings. We say that
an open economy (with γ > 0) exhibits exchange rate disconnect in the autarky limit (as γ → 0), if the
impulse responses have the following properties:
dZt+j
= 0 for all j ≥ 0
γ→0 dεt
lim
A corollary of condition (22) is that limγ→0
and
d log Et+j −d log Qt+j
dεt
lim
γ→0
dEt
6= 0.
dεt
(22)
= 0 for all j ≥ 0.
A model, defined by its structure and the set of shocks, exhibits exchange rate disconnect in the autarky
limit if the shocks have a vanishingly small effect on the macro variables, yet result in a volatile equilibrium exchange rate, as captured by the two conditions in (22).12 This property captures the exchange
rate disconnect in its narrow Meese-Rogoff sense. However, as the corollary points out, this property
also implies the PPP-puzzle behavior for the real exchange rate, which in the limit comoves perfectly
with the nominal exchange rate.
Definition 1 immediately allows us to exclude a large number of candidate shocks:
Proposition 1 The model of Section 3.1 cannot exhibit exchange rate disconnect in the autarky limit (22),
if the combined shock εt = V0 Ωt has a weight of zero on the subset of shocks {ηt , ηt∗ , ξt , ξt∗ , ψt }.
In other words, this proposition states that the following shocks alone:
∗
∗
∗
∗
∗
∗
Ω∅
t ≡ {wt , wt , χt , χt , κt , κt , at , at , gt , gt , µt , µt },
in any combination and with arbitrary correlations, cannot reproduce an exchange rate disconnect
property even as the economy approaches autarky. We provide a formal proof in Appendix A.3, yet
the intuition behind this result is straightforward: Any of the shocks in Ω∅
t will have a direct effect
on real allocations, prices, and/or interest rates, and thus cannot result in a volatile exchange rate
without having a direct effect on the macro variables of the same order of magnitude.13 Therefore, as
an economy subject to these shocks approaches autarky, the disconnect property (22) is necessarily
violated. The proof of this proposition does not rely on the international risk sharing condition, and
therefore this result is robust to the (in)completeness of the international asset markets.
12
dZ
/dε
t
A weaker, yet still suitable, condition is limγ→0 dEt+j
= 0 for all j ≥ 0.
t /dεt
Intuitively, the unit of account wt shocks results in wage inflation, the markup µt shocks result in price inflation, the
labor wedge κt shocks result in changes in either employment or consumption, the productivity at shocks result in changes in
either employment or output, the government spending gt shocks result in changes in either consumption or output, and the
intertemporal preference shocks χt result in changes in the interest rate. We illustrate this logic in Figure ??. The formal proof
in Appendix A.3 establishes further that no combination of these shocks can be consistent with the disconnect property (22).
13
11
Proposition 1 can be viewed as pessimistic news for both the International RBC and the New Open
Economy Macro (NOEM) models of the exchange rate. However, it does not imply that, for example,
productivity cannot be an important source of exogenous shocks. Instead, it suggests that productivity
shocks at are unlikely to be the dominant drivers of exchange rate movements if the model is to exhibit
exchange rate disconnect.14 The same applies to monetary shocks in a model with nominal stickiness,
which we study in detail in Section 5.1.
In other words, we view Proposition 1 as a diagnostic tool suggesting that the shocks in Ω∅
t are
unlikely to be successful at reproducing the empirical exchange rate behavior even away from the
autarky limit. Therefore, we should first explore the other three types of shocks — namely, the LOP
deviation (or trade cost) shocks ηt , the international good demand shocks ξt , and/or the international
asset demand shocks ψt — as the likely key drivers of the exchange rate dynamics. The distinctive
feature of these shocks is that they affect the equilibrium system exclusively through the international
equilibrium conditions: ψt affects international risk sharing (condition (25) below), while ηt and ξt affect
the country budget constraint (20) through their impact on export prices (13) and export demand (19)
respectively.15 The impact of shocks to these equilibrium conditions on the macro variables is vanishingly small as the economy becomes closed to international trade in goods and assets, yet such shocks
can have substantial effect on the equilibrium exchange rates and terms of trade even when γ is close
to zero.
Proposition 1 does not allow us to discriminate between the remaining three types of shocks, as they
all satisfy the autarky-limit disconnect condition (22). Yet, these shocks differ in the implied comovement between exchange rates and macro variables, which we now use as a further selection criterion.
In particular, we explore the comovement between the exchange rates and respectively terms of trade,
relative consumption, and the interest rate differential, near the autarky limit (as γ → 0). Since these
shocks are already consistent with the Meese-Rogoff and the PPP puzzles by virtue of Proposition 1, the
additional moments correspond to the three remaining exchange rate puzzles. We prove the following
result (see Appendix A.3):
Proposition 2 Near the autarky limit (for γ → 0), the international asset demand shock ψt is the only
shock among (ηt , ηt∗ , ξt , ξt∗ , ψt ) that simultaneously and robustly produces:
(i) a positive correlation between the terms of trade and the real exchange rate;
(ii) a negative correlation between relative consumption growth and real exchange rate depreciation;
(iii) deviations from the UIP and a negative Fama coefficient.
The main conclusion is that both the LOP deviation (trade cost) shocks ηt and the international
good demand shocks ξt produce the conventional counterfactual comovement between the exchange
14
Productivity shocks can have two additional indirect effects, either acting as news shocks about future productivity or by
affecting the risk premium (e.g., rare-disaster or long-run-risk shocks). As the direct effect of the productivity shock becomes
vanishingly small relative to indirect effects, this shock becomes indistinguishable from an Euler-equation shock (as it does
not affect the static equilibrium conditions), and we view it as a special case of the ψt shock. Proposition 1 nonetheless applies
as long as the direct effect of the productivity shock is non-trivial. See Section 4.1 and Appendix A.5 for further discussion.
15
The ξt and ηt shocks are additionally featured in the goods market clearing (18)–(19) and in the price level (7), but in
both cases their effect on these conditions is proportional to trade openness γ, and thus vanishes in the autarky limit.
12
rates and respectively relative consumption (the Backus-Smith puzzle) and interest rate differential (the
Forward Premium puzzle). In contrast, the international asset demand shock ψt is instead consistent
with both of these empirical patterns, as we explain in detail in Section 4.
To summarize, Propositions 1 and 2 explain why almost any shock has hard times at reproducing
the empirical exchange rate properties, and hence why these properties are labeled as puzzles in the
literature. These propositions favor the international asset demand shock ψt as the only likely shock
to generate exchange rate disconnect in an equilibrium model. While these propositions are concerned
with the autarky limit, the continuity of the model in trade openness γ suggests that the near-disconnect
properties should hold for γ > 0 but small. In what follows, we explore both qualitative and quantitative
properties of the model away from the autarky limit.
4
Baseline Model of Exchange Rate Disconnect
Our baseline model features a single shocks — the financial shock ψt — and the transmission mechanism, which emphasizes home bias in expenditure (γ), strategic complementarities in price setting (α),
and elasticity of substitution between home and foreign goods (θ). The other parameters, including the
intertemporal elasticity of substitution (IES) and the elasticity of labor supply, prove to be less consequential for the results, as we discuss below. Even more surprisingly, nominal rigidities turn out to be
of little importance for generating the quantitative exchange rate disconnect properties (in response to
a ψt shock). Therefore, our baseline model does not feature any nominal rigidities, and for simplicity
we assume that the monetary authorities adopt policy rules which fully stabilize respective wage inflations at zero (specifically, we have Wt ≡ 1 and Wt∗ ≡ 1). We show the robustness of our results to
nominal frictions and conventional Taylor rules in Section 5.1.
The model of this section is analytically tractable (upon log-linearization), and most of the results
can be easily obtained with pen and paper, which allows us to fully explore the intuitions behind various
mechanisms. At the same time, we emphasize the quantitative objective of this section. That is, our
goal is to establish whether a simple one-shock model can be quantitatively consistent with a rich set
of moments describing the dynamic comovement between exchange rates and the macro variables.
In doing so, we tie our hands from the start, and calibrate the parameters of the model on which we
have direct and reliable empirical evidence. In particular, we set γ = 0.07 to be consistent with the
0.28 trade (imports plus exports) to GDP ratio of the United States, provided the intermediate input
share φ = 0.5.16 We further use the estimate of Amiti, Itskhoki, and Konings (2016) of the elasticity of
strategic complementarities α = 0.4, which is also in line with much of the markup and pass-through
literature and corresponds to the own cost shock pass-through elasticity of 1 − α = 0.6 (see survey
16
This value of the trade-to-GDP ratio is also characteristic of the other large developed economies (Japan and the Euro
Zone). Appendix A.2.1 derives the relationship between the value of the trade-to-GDP ratio and the value of γ (steady state
imports-to-expenditure ratio), which is typically four times smaller. Intuitively, imports in a symmetric steady state are half
of total trade (imports plus exports), and GDP (final consumption) is about one half of the total expenditure with the other
half allocated to intermediate inputs (φ = 0.5). This value of φ is consistent with both aggregate input-output matrices and
firm level data on intermediate expenditure share in total sales. The decomposition of gross exports for the U.S., the E.U. and
Japan in Koopman, Wang, and Wei (2014) suggests this proportion holds for trade flows as well.
13
in Gopinath and Itskhoki 2011). In contrast, the value of the elasticity of substitution between home
and foreign goods θ is more contested. We follow here the recent estimates of Feenstra, Luck, Obstfeld,
and Russ (2014) and set θ = 1.5, which is also the number used in the original calibrations of Backus,
Kehoe, and Kydland (1994) and Chari, Kehoe, and McGrattan (2002).17
For the remaining parameters, we set the relative risk aversion σ = 2, the Frish elasticity of labor
supply ν = 1 and the (quarterly) discount factor β = 0.99, as we summarize in Table 1. We consider
robustness to the alternative parameter values in Section 4.7.18 Before describing the qualitative and
quantitative results of the baseline model in Sections 4.2–4.7, we first offer a brief discussion of the
various microfounded theories of the financial shock ψt .
4.1
Models of financial shock ψt
The financial shock ψt is commonly referred in the literature as the UIP shock. Indeed, combining the
asset home and foreign asset demand of the domestic households in (4), the no arbitrage condition upon
log-linearization yields:
it − i∗t = Et ∆et+1 + ψt ,
(23)
where it = log Rt , i∗t = log Rt∗ and et = log Et . It follows that the (uncovered) interest rate parity
it − i∗t − Et ∆et+1 can deviate from zero by the magnitude of the financial shock ψt , which is the key
shock of interest in our analysis. We now review the main macro-finance theories which give rise to
this type of shocks (see also Cochrane 2016):
1. Exogenous preference for international assets, where ψt is an ad hoc shock to the utility from
holding the foreign bond, as a result of which domestic households are willing to hold the foreign
asset with a negative excess return (UIP deviation), as in Dekle, Jeong, and Kiyotaki (2014).19
2. Noise traders and limits to arbitrage in currency markets, as in Jeanne and Rose (2002), where all
international trade in assets needs to go through risk-averse intermediaries, who are willing to be
exposed to the currency risk only if they are offered a sufficient compensation (positive expected
return). A noise trader shock ψt thus needs to be accommodated by a UIP deviation. The two
useful features of this model is that it generates endogenous volatility of UIP deviations, and
hence can speak to the Mussa puzzle (see Section ??), and also can be easily nested in a general
equilibrium environment (see Section 5.3).
3. A related class of models relies on financial frictions to generate upward sloping supply in the
currency market, where ψt represents the shock to the (exogenous or endogenous) risk-bearing
17
The macro elasticity of substitution between the aggregates of home and foreign goods is indeed the relevant elasticity
for our analysis, while the estimates of the micro elasticity at more disaggregated levels are typically larger (around 3). The
quantitative performance of our model does not deteriorate significantly for elasticities of substitution as high as 2.5.
18
We normalize and fix the shocks/wedges of Section 3.1, other that ψt , as follows: wt = at = χt = κt = ηt = ξt ≡ 0,
gt ≡ g and µt ≡ µ, and symmetrically for foreign. Our results, however, are robust to any common shock processes in the
two countries (at = a∗t , etc.). In our baseline, we set g = µ = 0, and show robustness to g = log(0.3 · GDP ) and µ = 0.2.
19
This approach additionally requires incompleteness of international asset markets, and is closely related to the nonoptimizing portfolio balance models of the 1970-80s (e.g. Kouri 1976, 1983, Henderson and Rogoff 1982, Branson and Henderson 1985), revived recently by Blanchard, Giavazzi, and Sa (2005) and Gourinchas (2008).
14
capacity of the financial sector, for example a shock to the net worth of the financial intermediaries, which limits the size of the positions that they can absorb (see e.g. Gabaix and Maggiori
2015, Hau and Rey 2006, Brunnermeier, Nagel, and Pedersen 2009, Adrian, Etula, and Shin 2015).
4. Incomplete information, heterogeneous beliefs and expectational errors in the currency market,
as in Evans and Lyons (2002), Gourinchas and Tornell (2004) and Bacchetta and van Wincoop
(2006), where ψt represents the deviations from the full-information rational expectations. The
advantage of heterogeneous beliefs models is that they can additionally explain the large volumes
of daily currency trade, much in excess of fundamental trade flows, however these models often
come at a substantial cost in terms of tractability.
5. Time-varying risk premia models, where ψt corresponds to shocks to the second moments of the
stochastic discount factor, such as rare disasters (Farhi and Gabaix 2016), long-run risk (Colacito
and Croce 2013), or habits (Verdelhan 2010). For convenience, these models typically assume
complete markets, while an alternative approach relies on modeling segmented markets, where
the SDF shocks emerge from the participation margin (see e.g. Alvarez, Atkeson, and Kehoe 2009).
Given the chosen set of the empirical exchange rate disconnect moments, all these approaches to modeling the financial shock ψt are isomorphic, as they all result in a version of equation (23), and hence
we cannot distinguish between them. The specific models of ψt can be discriminated based on a richer
set of asset market moments, e.g. a term structure of carry trade returns, or a comovement of exchange
rate with returns across various asset classes, which we leave outside the scope of this paper (see e.g.
Engel 2016, Lustig and Verdelhan 2016, Du, Tepper, and Verdelhan 2016).
For concreteness and tractability, we assume the financial shock ψt follows an exogenous AR(1)
process with persistence ρ ∈ [0, 1]:
ψt = ρψt−1 + εt ,
(24)
and the variance of innovations given by σε2 . For our quantitative analysis, we assume that ψt shocks
are small, but persistent, with ρ close to but smaller than 1 and σψ2 = σε2 /(1 − ρ2 ) close to 0.20 In
particular, we set ρ = 0.97 as our benchmark, consistent with the empirical persistence of the interests
rate differential (see Section 4.6).
4.2
Equilibrium exchange rate dynamics
Combining the two Euler equations for the international bonds, (4) and (17), we obtain the international
risk-sharing condition, which we rewrite in the following form:
ψt Et+1
∗
Et Θt+1 e
− Θt+1 = 0.
Et
(25)
20
The model admits a solution as long as βρ < 1, and the βρ → 1 limit allows to obtain sharp characterizations, which
we show provide a good quantitative approximation for our case of interest.
15
where the home and foreign nominal SDFs are given respectively by:
Θt+1 ≡ β
Ct+1
Ct
−σ
Pt
Pt+1
and
Θ∗t+1
≡β
∗
Ct+1
Ct∗
−σ
Pt∗
∗ .
Pt+1
The second equation we use is the home country’s budget constraint (20). We show in Appendix A.4
that the only effect of shock ψt on the values of SDFs Θt+1 and Θ∗t+1 is indirectly through the exchange
rate depreciation Et+1 /Et , and therefore (25) specifies the equilibrium relationship between ψt and the
expected change in the exchange rate Et {Et+1 /Et }. Similarly, the only effect of ψt on net exports N Xt
in (20) is indirect through Et , and a depreciation causes an improvement in N Xt due to the expenditure
switching effects.
Using this information, we log-linearize (20) and (25) around a symmetric steady state to obtain a
dynamic system for et ≡ log Et as a function of the exogenous shock process ψt :21
ψt = −d1 Et ∆et+1 ,
(26)
βb∗t+1 − b∗t = γd2 et ,
(27)
where the coefficients d1 ≥ 1 (with limγ→0 d1 = 1) and d2 ≥ 0 depend on the structural model
parameters other than β and ρ, and are defined explicitly in Appendix A.4. Equations (26)–(27), together
with the shock process (24), allow us to solve for the equilibrium process for the exchange rate:
Proposition 3 (Equilibrium exchange rate process) If the only shock affecting the economy is the
international asset demand shock ψt , described by an AR(1) process (24) with persistence ρ, then the log exchange rate et follows in equilibrium an ARIMA(1,1,1) process, or equivalently ∆et follows an ARMA(1,1),
with an AR root ρ, a non-invertible MA root 1/β, and innovation equal to
β/d1
∆et = ρ∆et−1 +
1 − βρ
1
εt − εt−1 .
β
β/d1
1−βρ εt :
(28)
This constitutes the unique equilibrium exchange rate path, and non-fundamental solutions do not exist.
We provide a formal proof of this proposition in Appendix A.4, and here offer an intuitive discussion.
Note from (28) that a positive international asset demand shock εt > 0 results in an instantaneous
depreciation of the home currency to balance the international asset market, making the home country
relatively poorer.22 As this shock gradually wears out, due to the mean reversion in ψt in (24), the
home currency is expected to be gradually appreciating in the future, according to the risk-sharing
condition (26). This is akin to the famous overshooting mechanism of Dornbusch (1976). Since trade
21
The equilibrium system is very closely approximated by the log-linear approximation, which we confirm by comparing
the exact non-linear numerical solution with the solution of the log-linear system, which we find nearly indistinguishable.
Log-linearization allows us to obtain analytical characterization of the equilibrium exchange rate process, as well as derive
simple equilibrium relationships between the exchange rates and other variables.
22
In the limit of the closed economy, when the asset position of the home country cannot change, this part of the mechanism
is pushed to the limit: home becomes relatively poorer to the extent which fully offsets an increase in international asset
demand. This is akin to the safe asset mechanism in Caballero and Farhi (2013).
16
1.1
1.1
∆et
et
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.1
-0.1
0
4
8
12
16
∆et
et
1
20
0
Quarters
4
8
12
16
20
Quarters
Figure 1: Impulse response of the exchange rate et to the international asset demand shock ψt
Note: In the left panel, ρ = 0.96 and β = 0.99 (quarterly); in the right panel, ρ = 0.99 and β = 0.995. The size of the
shock ε0 is normalized in each case to produce a unit devaluation on impact.
balance in (27) is decreasing in the strength of the home currency, the intertemporal budget constraint
determines the size of the initial depreciation, given the persistence of the shock ρ and the discount
factor β (which determines the steady state interest rate). The more persistent is the shock (larger ρ)
and the smaller is discounting (larger β), the greater is the initial unexpected depriciation. In the longrun, the home currency is expected to appreciate relative to the before-shock equilibrium value, yet the
required time for this to happen is increasing in ρ and β, and can be very long.
Figure 1 plots the impulse response of the exchange rate to the international asset demand shock for
two sets of parameter values (ρ, β). The model produces a large depreciation on impact, followed immediately by small and persistent appreciations in all future periods. As β and ρ both increase towards
1, the impulse response converges to that of random walk (or, equivalently, iid ∆et ). Indeed, when
agents are patient and the ψt shock is persistent, the exchange rate process becomes indistinguishable
from a random walk, and in particular the contribution of the predictable component to the exchange
rate variation becomes vanishingly small relative to the innovation (i.e., the unexpected component),
as we show in Figure 2. Interestingly, the impulse response of the process does not depend on the structural parameters of the model other than ρ and β. Only the volatility of the exchange rate innovation
relative to εt depends on the other parameters of the model (through d1 ), and in particular achieves its
maximum in the limit of the closed economy (d1 decreases towards 1 as γ → 0). We summarize this
discussion in:
Corollary 1 Equilibrium exchange rate has the following quantitative properties:
1. The volatility of the innovation for ∆et relative to that of the innovation for ψt is increasing in ρ
and β, becoming unbounded as βρ → 1; for any given values of ρ and β, it reaches its maximum
in the limit of the closed economy (as γ → 0).
17
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
4
8
12
16
20
0
Quarters
4
8
12
16
20
Quarters
Figure 2: The contribution of the innovation (unexpected component) to the variance of ∆et+1
var (et+k −et−1 )−Et−1 {et+k −et−1 }
Note: The figure plots the variance decomposition
for various horizons k ≥ 0. The
var et+k − et−1
analytical expression for the value of this decomposition for k = 0 is provided in (29). As in Figure 1, in the left panel
ρ = 0.96 and β = 0.99, and in the right panel ρ = 0.99 and β = 0.995. The shaded area is the 95% bootstrap confidence
interval in a sample of 40 quarters.
2. The contribution of the innovation (unpredictable component) to the variance of ∆et+1 is given by:23
β 2 (1 − ρ2 )
var(∆et+1 − Et ∆et+1 )
=
−−−→ 1.
var(∆et+1 )
(1 − β)2 + 2β(1 − ρ) βρ→1
(29)
Furthermore, the autocorrelation of ∆et is given by:
ρ̂∆et ≡
cov(∆et+1 , ∆et )
1 − ρ2
=ρ−
−−−→ 0.
var(∆et )
β(1 − 2ρβ + β 2 ) βρ→1
(30)
More generally, as βρ → 1, the et process becomes arbitrary close to a random walk.
3. The long run impulse response of the exchange rate to the international asset demand shock reverses
1−β
into a depreciation, limj→∞ ∂et+j /∂εt = − d11 (1−ρ)(1−ρβ)
< 0, however the period of reversion to
zero can be arbitrary long as β and ρ increase towards 1.24
For plausible values for β and ρ, it is impossible to distinguish the exchange rate process from a
random walk when samples are of the typical size available in the data.25 Therefore, the model fits the
23
Note that the expression in (29) gives a lower bound for the contribution of the unpredictable component, as the MA root
of the et process is non-invertible, yet we assumed that εt , εt−1 , . . . is part of the information set at time t.
24
Appendix XXX discusses a slight modification of the model, in which et mean reverts in the long run, and is long-run
stationary, yet the short-to-medium run dynamics of the exchange rate process is quantitatively indistinguishable from the
one analyzed here.
25
This property of our model is similar to that of Engel and West (2005), where exchange rate can also becomes arbitrary
close to a random walk. However, there are important differences. In particular, the exchange rate dynamics in our model
is critically shaped by the budget constraint and the discount factor β, while Engel and West (2005) consider the complete
markets case, and as a result their weight on the future fundamentals does not depend on β, but instead on the elasticity of
money demand with respect to the interest rate, which needs to become arbitrary large for the limit to take effect.
18
first observation of Meese and Rogoff (1983) about the exchange rates (namely, the random walk behavior), and we show in Section 4.4–4.6 that it also reproduces the disconnect with other macroeconomic
variables. The ψt shock results in an arbitrary volatile exchange rate process, especially as the economy becomes increasingly closed to international trade. In Section 4.6, we explore the plausible size of
the ψt shocks, disciplining it with the equilibrium behavior of the interest rate. However, it is already
clear that by choosing ρ and β close to 1, we can ensure arbitrary large exchange rate volatility even
for small εt shocks. Indeed, small international asset demand shocks cause unexpected home currency
depreciations and small expected future appreciations, with the size of the unexpected depreciation
relative to the future expected movements increasing without bound in the persistence of the shock.
4.3
Real exchange rate and the PPP puzzle
We next explore the equilibrium dynamics of the real exchange, defined in (14)), which we write as:
qt ≡ et + p∗t − pt .
We prove the following result:
Proposition 4 (Real exchange rate) In the economy with ψt shocks only, the equilibrium real exchange
rate equals:
qt =
1
1+
2γ et ,
1
1−φ 1−2γ
(31)
and therefore follows the same ARIMA process as the nominal exchange rate, but a with a proportionally
smaller innovation. As the economy becomes closed to trade (γ → 0), the real and nominal exchange rates
become indistinguishable. For any values of ρ and β, the estimated AR(1) coefficient for qt converges to 1
in large samples, and the corresponding half-life estimate increases without bound.
A proof is contained in Appendix A.4, yet expression (31) is intuitive and follows from combining
the price setting equations (12)–(13) with the definition of the price index (7). Upon log-linearization,
we have:26
p t = wt +
1
γ
qt .
1 − φ 1 − 2γ
(32)
That is, domestic price level is high either when the cost of domestic labor is high or when the real
exchange rate is depreciated, provided the economy relies on foreign goods in consumption (γ > 0)
and in production (when, in addition, φ > 0). Combining with a foreign counterpart for p∗t , and using
the definition of qt , we arrive at (31).27 The implication of (31) is that the real and nominal exchange
26
Interestingly, the expression for pt does not depend on α, as pass-through into CPI does not depend on the extent of
strategic complementarities when firms are homogenous in α, due to the exactly offsetting effects of cost pass-through and
competitor price pass-through (see Amiti, Itskhoki, and Konings 2016).
27
We use the normalization wt = wt∗ = 0, so that et corresponds to the relative price of a unit of foreign labor in units of
home labor, or the wage-based real exchange rate.
19
(a) Autocorrelation
(b) Half life, quarters
1
30
25
0.95
20
0.9
15
0.85
10
0.8
5
0.75
0.9
0.92
0.94
0.96
0.98
1
0
0.9
0.92
ρ, persistence of the ψt shock
0.94
0.96
0.98
1
ρ, persistence of the ψt shock
Figure 3: Persistence of the real exchange rate qt
Note: Left panel: OLS estimated ρ̂q from projection qt = ρq qt−1 +qt . Right panel: corresponding half-life of an AR(1) process
with autorcorrelation ρ̂q , calculated according to HLq = − log 2/ log ρ̂q . Based on 10,000 simulations with 120 quarters (30
years) each, where the solid lines plot the median estimates and the areas are the 90% bootstrap sets. The dotted lines in the
right panel indicate the conventional 3–5 year half life estimates in the data (see Rogoff 1996).
rates follow the same stochastic process, with real exchange rate exhibiting a smaller volatility:
std(∆qt )
1
=
2γ < 1.
1
std(∆et )
1 + 1−φ 1−2γ
As the economies become more closed to international trade (γ → 0), the wedge between the real and
the nominal exchange rate shrinks, and the two become indistinguishable in the limit.
To summarize, the model reproduces the PPP puzzle, which we broadly interpret as the close comovement between the nominal and the real exchange rates, and the near random walk behavior for
both variables. In particular, if one were to fit an AR(1) process for the real exchange rate, as is conventionally done in the PPP puzzle literature (surveyed in Rogoff 1996), one would be challenged to
find evidence of mean reversion and would infer very long half-lives for the real exchange rate process.
Proposition 4 confirms this as an asymptotic result in large samples, and Figure 3 illustrates this property in small samples of the size conventionally used in empirically studies. Importantly, the model
reproduces a near random walk property for the nominal exchange rate et , which translates into persistent fluctuations in the real exchange rate qt , without relying on price stickiness.
In the conventional RBC and sticky price models, the productivity and monetary shocks introduce
a wedge between the nominal and the real exchange rate, at least in the long-run as prices become
flexible. As empirical duration of prices is relatively short, this results in the PPP puzzle. Our model
does not create a wedge between the two exchange rates, as both are moved together by the same force
in the financial market, and as a result a gap between the two exchange rates does not need to open up
over time. Therefore, there is no PPP puzzle from the point of view of our model.
20
4.4
Exchange rates and prices
In this section we explore the joint properties of the equilibrium real exchange rate (based on consumer
prices), terms of trade and producer prices. As emphasized in Atkeson and Burstein (2008), the conventional models imply a counterfactually volatile terms of trade and producer prices relative to consumer
prices. In the data, consumer- and producer-based real exchange rates are equally volatile, while the
terms of trade is substantially more stable (about three times less volatile).
Our model admits a block recursive structure, and the price block of the model is the first autonomous block, which we can fully solve without making use of the rest of the equilibrium system.
This block takes the dynamic process for the exchange rate (characterized in Section 4.2) as given, and
solves for the resulting behavior of terms of trade and producer prices. The price block contains the
following equations: definition of the consumer price index in (7), the marginal cost in (9), price setting
equations in the home and foreign markets (12) and (13), and their foreign counterparts. We also make
use of the definitions of the real exchange rate in (14) and terms of trade in (21), as well as the following
definition of the producer-price real exchange rate:
QPt ≡
PF∗ t Et
.
PHt
(33)
We log-linearize and solve this block of the equilibrium system in Appendix ??. In what follows, we
use corresponding small letters to denote log deviations from the steady state values, and refer by RER
and ToT to real exchange rates and terms of trade respectively.
Directly from the definitions of RERs, ToT and price indexes, we obtain the following relationship
between our three variables of interest:
qt = (1 − γ)qtP − γst ,
(34)
Indeed, domestic consumer prices are low (i.e., qt is high, corresponding to depreciated RER) when
either producer prices are low (high qtP ), or home faces favorable terms of trade (low st ). The relative
strength of the latter effect is proportional to the openness of the country, as measured by γ in our
model.28
The second equilibrium relationship between this variables expresses the terms of trade as the
home producer prices adjusted by the law of one price (LOP) deviations in international trade. Using
the price setting equations (12) and (13), we showed in (14) that LOP deviations are proportional to the
real exchange rate Qαt , in the presence of pricing to market (or strategic complementarities in price
setting with local competitors), as captured by α > 0. As a result, we have:
st = qtP − 2αqt .
(35)
That is, the terms of trade st reflect the producer prices qtP corrected for LOP deviations measured by
28
Recall that γ corresponds to the expenditure share on foreign goods in the steady state around which we approximate
the model.
21
αqt , for each of home imports and exports (hence the factor of 2 in (35)). Relatively low home consumer
prices (i.e., high qt ), mean that import prices are also lowered (and export prices are symmetrically
raised) due to strategic complementarities, resulting in favorable ToT (low st ).
Combining (34) and (35) together, we solve for the equilibrium relationship between the three variables of interest:29
Proposition 5 The consumer- and producer-based real exchange rates and the terms of trade are linked
by the following equilibrium relationship:
qtP =
1 − 2αγ
qt
1 − 2γ
and
st =
1 − 2α(1 − γ)
qt .
1 − 2γ
In the absence of pricing to market (α = 0), these relationships become st = qtP =
(36)
1
1−2γ qt .
Note that these relationships depend only on two parameters, α and γ, and in particular do not
depend on either demand elasticity θ or intermediate share φ. The conventional models, without pricing
to market (or local currency pricing), have α = 0, and as a result imply std(∆st ) = std(∆qtP ) >
std(∆qt ). Intuitively, these models feature no LOP deviations, and terms of trade simply reflect relative
producer prices, while relative consumer prices are smoothed via diversification of the consumption
bundle across the domestically and foreign produced goods (when γ > 0).
These predictions, however, are at stark contrast with the data, where we observe std(∆qt ) ≈
std(∆qtP ) std(∆st ). Proposition 5 suggests, in line with the analysis of Atkeson and Burstein
(2008), that pricing to market due to strategic complementarities α > 0 can improve the performance
of the model on both margins. When
γ
1
α<
1−γ
2(1 − γ)
(37)
we obtain the empirically relevant case, in which the RER qt is both more volatile than ToT st (left
inequality) and positively correlated with ToT (right inequality). Positive α smoothes out the response
of ToT to changes in the producer prices qtP (see (35)).
In contrast, too high an α makes ToT overshoot in the empirically wrong direction. In particular,
α = 1 is akin to the extreme form of LCP, where a RER appreciation (low qt ) is associated with a ToT
deterioration (high st ). This is because in this case qt = qtP , as foreign firms fully mimic the prices
of home firms in the domestic market (full PTM). Therefore, import prices are high when domestic
producer prices are high, and export prices are low when foreign producer prices are low, resulting in
a negative correlations between terms of trade and both RERs. Obstfeld and Rogoff (2000) criticized the
LCP models based on these grounds, as ToT and RER tend to be mildly positively correlated in the data.
We see, however, that for intermediate values of α (e.g, α ≈ 0.4–0.5, which are the values estimated
by Amiti, Itskhoki, and Konings 2016, in the data), the performance of the model is good on both
margins, and both inequalities in (37) are satisfied. Note that both inequalities in (37) are more likely to
29
In fact, these relationships are more general, and apply for all other shocks, but (ηt , ηt∗ ), which introduce an additional
wedge in the LOP equations (14), potentially corresponding to the LCP price stickiness.
22
0.5
var(st ) > var(qt )
Openness (expenditure share), Λ
0.4
0.3
corr(st , qt ) < 0
0.2
φ=0
0.1
with φ > 0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pricing-to-market, α
Figure 4: Regions for relative terms of trade volatility
Note: the yellow region satisfies joint requirement std(st ) < std(qt ) and corr(st , qt ) > 0, which are weak realistic restrictions imposed by the data. The red area imposes the tighter empirical restrictions that std(st )/std(qt ) ∈ [0.1, 0.4]. REDO
THE FIGURE USING NEW NOTATION AND ADD FIGURE FOR PPI-RER...
hold when the economy is relatively closed to international trade and γ is low. This case, with positive
α and small γ, exactly corresponds to the circumstances when std(∆qt ) ≈ std(∆qtP ), as is the case
in the data. We illustrate these insights in Figure 4, and explore more systematically the quantitative
implications of the model in Section 4.7.
4.5
Exchange rate and quantities
The second block of the equilibrium system determines the equilibrium values of consumption, labor
supply and output, given exchange rates and prices. In particular, an important property of the equilibrium system is that we can express all this variables as functions of the real exchange rate, as the
exogenous shock ψt has no direct on this variables, and its effect is indirect through the fluctuations
in the real exchange rate qt . This block of the equilibrium system relies on the labor supply and labor
demand conditions ((3) and (10)), the goods market clearing conditions (??)–(19), the solution for the
price index (32), and their foreign counterparts. Appendix ?? log-linearizes this block of the equilibrium
system and Appendix A.4.2 describes the solution. In order to simplify the solution, we adopt a useful
trick from the NOEM literature following Obstfeld and Rogoff (1995), by denoting with x̃t ≡ (xt −x∗t )/2
for any pair of variables (xt , x∗t ), where again small letters denote the log deviations from the steady
state values for corresponding variables.30
First, we combine the labor demand and labor supply conditions, (3) and (10), together with the
solution for the price index (32), and subtract from them their foreign counterparts, to obtain the folIn addition to x̃t , we also define x̄t ≡ (xt + x∗t )/2. The ψt shock is a relative shock in that, while it affects x̃t , it does
not affect x̄t , and we have c̄t = ȳt = `¯t = 0 when the economy experiences ψt shocks only. Note that in this case we simply
have xt = x̄t + x̃t = x̃t and x∗t = x̄t − x̃t = −x̃t .
30
23
lowing two equilibrium relationships:
1˜
γ
1
`t = −σc̃t −
qt ,
ν
1 − φ 1 − 2γ
γ
φ
qt .
`˜t = ỹt +
1 − φ 1 − 2γ
The first of these states that labor supply is high when either consumption is low (income effect) or
the real exchange rate is appreciated (qt is low). The latter is the substitution effect, as low qt reflects
a high real wage wt − pt (see (32)). Recall from Proposition 4 that there is an incomplete pass-through
from wage-based (nominal) exchange rate et into real exchange rate qt : that is, when et is low, qt is
also low, but less than proportionally, which is equivalent to wt − pt being high. The intuition is that
et affects the relative cost for all domestically produced goods, but qt captures the relative consumer
prices, which partly (with weight γ) reflect the costs of the goods produced abroad.
The second of the equations above states that labor demand is high when either output yt is high or
when the real wage is low (as reflected in high qt ), as in this case the firms substitute away from intermediates and towards labor in their input mix. Combining the two equations together, which ensures
equilibrium in the labor market, we obtain the first equilibrium relationship between consumption,
output and the real exchange rate:
ỹt + σνc̃t = −
ν+φ γ
qt .
1 − φ 1 − 2γ
(38)
A depreciated real exchange rate (high qt ) and the associated low real wage, through the labor market
equilibrium, imply either low output (due to low labor supply) or low consumption (to ensure high
labor supply). Note that, intuitively, the effect of the real exchange rate on the labor market equilibrium
vanishes together with the openness of the economy (γ → 0).
The second equilibrium relationship between consumption, output and the real exchange rate
comes from the goods market clearing. Combining together (??)–(19), and subtracting their foreign
equivalents, we obtain after simplification:
ỹt =
1−γ
2θ(1 − α) 1−2γ
− (1 − ζ) γ
ςζ
c̃
+
qt ,
t
2γ
2γ
1 − 2γ
ζ + 1−2γ
ζ + 1−2γ
(39)
where ζ denotes the steady-state GDP-output ratio (ζ ≡ (C + G)/Y ) and ς denotes the steady-state
share of consumption in GDP (ς ≡ C/(C + G)). According to goods market clearing (39), demand for
output yt is high when either domestic consumption ct is high, or the real exchange rate is depreciated
(high qt ). There are two effects on output demand from the depreciated real exchange rate. The first
effect is the conventional expenditure switching towards the domestic goods away from the foreign
goods, and this effect is proportional to the product of exchange rate pass-through into prices (1 − α)
and the elasticity of substitution θ between the home and the foreign goods. The second, offsetting,
effect is due to the production input substitution away from the domestic intermediates towards the
domestic labor, as high qt implies low real wage wt − pt (recall the discussion above). This input mix
24
switching effect is proportional to the share of intermediates in total output given by (1 − ζ), and
under plausible parameter values is weaker than the expenditure switching effect, which however is
not necessary for our results.31
Combining together the equilibrium in the labor and goods markets, (38) and (39), we can solve for
the relative consumption c̃t and the relative output ỹt as a function of the real exchange rate qt :
c̃t = −
ỹt =
1−γ
2θ(1 − α) 1−2γ
− (1 − ζ) +
2θ(1
ν+φ
1−φ
2γ
σν
(σν + ς)ζ + 1−2γ
1−γ
ζς ν+φ
− α) 1−2γ − (1 − ζ) − σν 1−φ
2γ
(σν + ς)ζ + 1−2γ
σν
ζ+
2γ
1−2γ
γ
qt ,
1 − 2γ
γσν
qt .
1 − 2γ
(40)
(41)
While the relative consumption c̃t necessarily decreases in qt , the effect of qt on ỹt may be ambiguous, but it is positive for the plausible parameter value. We summarize the main insights from this
analysis in:
Proposition 6 The effect of the ψt shock on consumption and output operates only indirectly through the
devaluation of the real exchange rate qt , and it becomes vanishingly small, as the economy becomes closed
to international trade (γ → 0). When γ > 0, a real exchange rate devaluation (high qt ) is associated with
a decline in the relative consumption at home (low c̃t ).
Importantly, our model violates the empirically-counterfactual Backus and Smith (1993) condition,
and consistently with the data predicts a negative correlation between the real exchange rate and the
relative consumption. In other words, consumption is low when price are low, which violates the
logic of international risk-sharing, but is the pattern observed in the data (see estimates across countries in Benigno and Thoenissen 2008). This property of our model stands in stark contrast with both
the productivity-shock-driven international RBC models and the monetary-shock-driven international
New Keynesian (NOEM) models, even when those models do not rely on complete markets and hence
the Backus-Smith risk sharing conditions holds only in expectations. The logic is that both RBC and
New Keynesian models generate a real depreciation as a response to the abundance of domestically
produced goods—either because of the high home productivity or the domestic monetary stimulus
supressing domestic markups and prices.32 In both cases the real depreciation is associated with a
31
The strength of the two effects can be reversed in the extreme case of pricing to market (α → 1) or under strong
complementarity between home and foreign goods (θ 1). More realistically, α ≤ 0.5 and θ ≈ 1, in which case
1−γ
& 1 > (1 − ζ). For Proposition 6, however, we need a weaker condition,
2θ(1 − α) 1−2γ
2θ(1 − α)
1−γ
ν+φ
− (1 − ζ) +
1 − 2γ
1−φ
µ
ζ+
2γ
1 − 2γ
> 0,
which is always satisfied since ζ ≡ 1 − e− 1−α φ ≥ 1 − φ, where µ ≥ 0 is the steady state log markup.
32
We can see these effects in the general version of our model with a positive productivity at shock (or negative labor
wedge shock κt ) or a negative markup µt shock (possibly coupled with a positive LOP shock ηt to mimic LCP). The previous
literature addressed the Backus-Smith puzzle by introducing additional static ingredients (together with incomplete asset
markets assumption): e.g., non-traded goods and Balassa-Samuelson effect (Benigno and Thoenissen 2008), complementarity
between home and foreign goods (Corsetti, Dedola, and Leduc 2008), home production (Karabarbounis 2014), search frictions
in goods market (Bai and Ríos-Rull 2015).
25
0.2
Openness (expenditure share), Λ
α = 0.4
θ = 2.3
Atkeson-Burstein
region
0.15
std(ct −c∗t )
std(qt )
0.1
> 0.40
0.05
std(ct −c∗t )
std(qt )
< 0.25
0
0
1
2
3
4
5
Pass-through into quantities, θ(1 − α)
Figure 5: Exchange rate disconnect: relative consumption volatility
Note: the yellow region satisfies −
std(ct −c∗
t)
std(qt )
∈ [−0.4, −0.25] and in the red region this relative volatility is in [−0.25, 0].
domestic consumption boom, which is empirically counterfactual.
The mechanism in our ψt -model is different. A real depreciation is not caused by increased supply
of domestic goods, but instead by increased demand for foreign assets. A response to this international
asset demand is a reduction in the domestic real wage, as well as a conventional expenditure switching
towards domestic goods. This puts pressure on the domestic labor market—a simultaneous reduction
in labor supply from the substitution effect and an increase in labor demand to satisfy the increased
world good demand (in the absence of an expansion in the domestic production possibilities) can be
only balanced out by a reduction in the domestic consumption.33 Therefore, greater international asset
demand is associated with a depreciated real exchange rate, a reduction in the domestic real wages and
consumption, and an increase in the domestic production, consistent with the patterns in the data.34
Note that this logic is fully static, and the dynamic international risk-sharing condition (the combined
Euler equations (25)) plays little role in shaping the response of consumption to the real exchange rate
shocks. Indeed, in our model the expected component of the exchange rate movement is negligible
relative to the unexpected component, which in turn does not factor in the international risk sharing
under incomplete markets.
Having established the qualitative properties of the RER-consumption comovement, we now turn
to a preliminary quantitative exploration of the exchange rate disconnect. While a negative correlation
between consumption and real exchange rate is a desirable property of the model, a more salient feature of the data is a much greater volatility of the exchange rate relative to the other macroeconomic
33
Note that a nominal devaluation, driven by the demand shock for international assets, reduces the relative wealth of the
home economy, and results in a decline in the relative home consumption.
34
The same logic for consumption movement applies in response to an increased relative demand for domestic goods
(ξt∗ shock, or a negative ξt shock), however it is associated with increased relative prices of domestic goods, and hence a real
appreciation, again resulting in a counterfactual positive correlation between c̃t and qt .
26
variables, and in particular consumption.35 From (40), we have:
c∗t )
std(ct −
std(qt )
=
1−γ
2θ(1 − α) 1−2γ
− (1 − ζ) +
(σν + ς)ζ +
ν+φ
1−φ
2γ
1−2γ σν
ζ+
2γ
1−2γ
2γ
,
1 − 2γ
(42)
which tends to zero as the economy becomes closed to international trade (γ → 0). Outside this
limit, Figure 5 plots the relative consumption volatility in (42) in the (θ(1 − α), γ) space for the given
conventional values of the other parameters, which we show play a secondary role in shaping this
moment.36 In summary, the model can reproduce high relative volatility of the real exchange rate
provided low values of θ(1 − α) and γ, which are however in line with empirical estimates, as we
discuss in Section 4.7.
4.6
Exchange rate and interest rates
The final block of the equilibrium system concerns the behavior of the interest rates. Combining together the log-linearized dynamic optimality conditions (Euler equations) of the home households (??)–
(4) for the home and foreign bonds respectively, we have:
it − i∗t = Et ∆et+1 + ψt ,
(43)
which is the familiar UIP condition, modified by the international asset demand shock ψt , which acts
similarly to a risk-premium shock common in the literature. The intuition behind (23) is that, up to
the first order, the difference between the stochastic discount factors pricing home and foreign assets is
the expected exchange rate depreciation, Et ∆et+1 , while the ψt -term reflects the premium (ore, more
precisely, discount) at which the home households are willing to hold the foreign bonds in view of their
asset demand shock.
Next, we return to the international risk-sharing condition (25), which combines the dynamic optimality of home and foreign households with respect to holding foreign bonds. The log-linear version
of (25) can be written as:
2σEt ∆c̃t+1 = Et ∆qt+1 − ψt .
(44)
That is, in the absence of ψt -shocks, the expected relative consumption growth in the two countries
is proportional to the expected real devaluation, while it shifts down with the increased international
asset demand by the home households. Using the solutions above for relative consumption in (40), and
the relationship between real and nominal exchange rates in (31), we can solve (44) for the expected
35
Another way this manifests itself in the data is through a very weak negative correlation between consumption and
real exchange rate. In a single-shock economy all correlations are either plus or minus one (or zero in knife-edge cases),
and therefore our metric of disconnect is relative standard deviations std(ct − c∗t )/std(qt ). We return to the discussion of
correlations in a multi-shock model of Section 5.2.
36
The baseline values of the other parameters are: σ = 2, ν = 1, φ = 0.5, µ = 0. Appendix Figure ?? shows robustness
of our findings to varying this parameters. Larger σ and ν, and a smaller φ help with the consumption moment, but not
dramatically, and a smaller σ and/or ν still result in similar quantitative conclusions (cf. Chari, Kehoe, and McGrattan 2002).
27
change in the nominal exchange rate, already anticipated in (26). We rewrite this condition as:
Et ∆et+1 = −
1
ψt ,
d1
(45)
where recall that d1 > 1 for γ > 0 and limγ→0 d1 = 1.
Substituting this expression into the UIP condition (23), we obtain the equilibrium solution for the
interest rate differential:
it − i∗t =
d1 − 1
ψt .
d1
(46)
Therefore, the interest rate differential inherits the dynamic properties of the asset demand shock
ψt , i.e. it follows an AR(1) process with persistence ρ, and with the size of the innovation given by
(d1 − 1)/d1 · σε . That is, as the economy becomes more closed to international trade (γ → 0), the size
of the innovation to the interest rate differential becomes vanishingly small, consistent with our notion
of the necessary condition for exchange rate disconnect defined in Section 3.2. Away from the closed
economy limit, for γ > 0, we take advantage of Proposition 3 and Corollary 1 describing the exchange
rate process, to formulate the following result (see Appendix A.4.4):
Proposition 7 (Exchange rate and interest rates) The projection of the exchange rate change ∆et+1
on the interest rate differential (it − i∗t ), i.e. the Fama regression, has a negative coefficient:
E{∆et+1 |it − i∗t } = βF (it − i∗t ),
where
βF ≡ −
1
< 0.
d1 − 1
Furthermore, as the patience of the households and the persistence of the shock grow larger (βρ → 1):
1. the R2 in the Fama regression of ∆et+1 on (it − it ) becomes arbitrary small (goes to zero);
2. the volatility of ∆et+1 relative to (it − it ) becomes arbitrary larger (goes to infinity);
3. the persistence of ∆et+1 relative to (it − it ) becomes arbirtary small (goes to zero);
4. the Sharpe ratio of the carry trade, investing in the high interest rate currency, is arbitrary small
(goes to zero).
This approximates well the observed empirical patterns, as in the data positive interest rate differentials predict expected exchange rate appreciations, a pattern of the UIP deviations known as the
Forward premium puzzle (Fama 1984). Indeed, in our model, when the home currency offers a high
return (it > i∗t ), it signals the international asset demand shock ψt > 0, which causes a sharp contemporaneous unexpected devaluation (not reflect in the current interest rates set at t − 1), but a gradual
expected appreciation, as this shock mean-reverts.37 As we discussed in Section 4.2, this mechanism
is akin to the famous Dornbusch (1976) overshooting, but in the context of our model results in the
empirically-observed pattern of the puzzling UIP deviations. However, the predictive ability of the
interest rate differentials for future devaluations is very weak in the data (e.g., see Table 1 in Valchev
2015), and our model captures this with a vanishingly small R2 in the Fama regression, as the ψt shocks
37
The negative Fama-regression coefficient βF = −1/(1 − d1 ) < 0 follows immediately form combining (45) and (46).
28
(b) Fama regression R2
(a) Fama regression coefficient
(c) Carry trade Sharpe ratio
0.12
0
0.4
0.35
0.08
-5
0.3
0.25
0.2
0.04
-10
0.15
-15
0.9
0.92
0.94
0.96
ρ, persistence of the ψt shock
0.98
0
0.9
0.92
0.94
0.96
0.98
0.1
0.9
ρ, persistence of the ψt shock
0.92
0.94
0.96
0.98
ρ, persistence of the ψt shock
Figure 6: Deviations from the Uncovered Interest Rate Parity
Note: Monte Carlo study based on 10,000 simulations of the model with 120 quarters (30 years). The solid lines plot the
median estimates across simulations, the areas represent 90% bootstrap sets, and the red dotted lines are the asymptotic
values from the text. Panel (a) plots βF from the Fama regression of ∆et+1 on (it − i∗t ), while panel (b) plots the R2 from
these regressions. Panel (c) plots the within-sample Sharpe ratio calculated as the coefficient of variation for the carry return
ψt · (it − i∗t − ∆et+1 ). The parameter values for calibration are as described in footnote 36.
become more persistent. Indeed, recall from Corollary 1, that in this case the unexpected changes dominate the dynamics of the exchange rate, while the expected changes play a vanishingly small role. In
Figure 6, we illustrate the small-sample estimates of the Fama regression coefficient and the associated
R2 for the non-limiting values of the model parameters. Finally, in the data, the interest rate differentials, unlike the exchange rate changes, are very smooth and very persistent, as is again captured by
our model when ψt shocks are persistent.38
In closing this section, we briefly address the expected returns and the sharp ratio associated with
the carry trade of investing into the (temporarily) high interest rate. In parallel with Lustig and Verdelhan (2011) and Hassan and Mano (2014), we consider a zero-capital strategy of being long in the high
interest rate currency with an offsetting short position in the low interest rate currency, with the size
of the position proportional to the expected return. Formally, we denote with
xt ≡ it − i∗t − Et ∆et+1
the size of the position (long in the home bond when xt > 0 and vice versa). Note from (23), that
xt = ψt , and this carry trade strategy effectively takes the offsetting position to that of the home
households demanding the international bonds. The expected return to this strategy is given by:
n
o
n
o
r̄C ≡ E xt · (it − i∗t − ∆et+1 ) = E ψt · Et {it − i∗t − ∆et+1 } = σψ2 ,
(47)
where σψ2 ≡ Eψt2 is the unconditional variance of the ψt shock. The more volatile are the asset demand
shocks, the greater is the average return on the carry trade. We show in Appendix A.4.4 that the
We provide in the text only the solution for the interest rate differential (it − i∗t ), while Appendix A.4.4 also discusses
the solution for the individual interest rates, which can be recovered from the Euler equations of the home and foreign
households with respect to correspondingly home and foreign bonds (log-linearized versions of (??) and (17) respectively).
The weak response of consumption and prices to the exchange rate movements is also inherited by the interest rates.
38
29
variance of this strategy is also increasing in σψ2 , and the associated Sharpe Ratio is given by:
SRC ≡
r̄C
=
std(rC )
1
2+
2 /σ 2
σ∆e
ψ
1/2 ,
(48)
where
2
2
σ∆e
≡ var ∆et+1 − Et ∆et+1 = Et ∆et+1 − Et ∆et+1
is the variance of the unexpected exchange rate changes. Corollary 1 to Proposition 3 implies that
2 /σ 2 → ∞ as βρ → 1, and therefore the Sharpe ratio of the carry trade can be arbitrary small,
σ∆e
ψ
as the asset demand shock becomes fully persistent. Figure 6 illustrates the carry trade Sharpe ratios
away from this limit and shows that they vary between 0.15 and 0.3, in line with the empirical patters
documented in Hassan and Mano (2014).
4.7
Quantification
The results above show good limiting behavior of the model with the ψt shock when either γ → 0
(Propositions 1 and 2), or when βρ → 1 (Propositions 3 and 7). We now check the quantitative property
of the simple benchmark model away from these limiting cases. In particular, we set γ = 0.07 and
ρ = 0.97, and explore the robustness of the quantitative results to various parameter values. The
openness parameter γ is set to match the import to GDP ratio of 0.14, as GDP is about half of total
economy’s production (or, alternative, the cost share of the intermediate inputs is 0.5). We also set the
baseline values of θ = 1.5 following the estimates of Feenstra, Luck, Obstfeld, and Russ (2014) and
α = 0.4 following the estimates of Amiti, Itskhoki, and Konings (2016).
We report the resulting findings in Table 2, which contrasts the empirical moments with the moments obtained from the simulation of our model with ψt shocks alone. Overall, the model fits a variety
of moments remarkably well. The first and second panel shows that it captures the near-random walk
behavior of the nominal exchange rate and the long half live of the real exchange rate. For comparison,
an LCP price stickiness model of the type considered by Chari, Kehoe, and McGrattan (2002) would
require a price duration of over 2.5 years to obtain the same level of real exchange rate persistence.
In the model, the real exchange rate is about three-quarters as volatile as the nominal exchange rate,
while this ratio is closer to 1 in the data. Adding LCP price stickiness to our model further increases
RER volatility, however even in the absence of any price or wage stickiness we obtain empirical-like behavior of the real exchange rate. The third panel of the table further shows that the model captures the
relative volatilities of the terms of trade, the producer-price real exchange rate and the consumer-price
real exchange rate.
The fourth panel of Table 2 considers the volatility of the real variables, such as consumption,
output, GDP and net exports, relative to that of the real exchange rate. The model captures the large
gap in relative volatility of the real exchange rate and the real variables, which are about four times less
volatile.39 In addition, the model reproduces the negative correlation between the real exchange rate
39
In response to ψt shock, ct and c∗t are perfectly negatively correlated so that ∆c − ∆c∗ = 2∆c. Therefore, while
30
Table 2: Baseline model: quantitative properties
1.
ρ(∆e)
Data
Baseline
0.00
−0.02
θ = 2.5
α=0
Robustness
γ = .15 ρ = 0.9
σ=1
−0.05
(0.09)
ρ(q)
0.94
0.93
0.87
(0.04)
2.
HL(q)
12.0
9.9
4.9
(6.4)
σ(∆q)/σ(∆e)
0.98
0.75
3.
σ(∆s)/σ(∆q)
σ(∆q P )/σ(∆q)
0.34
0.95
0.30
1.10
4.
σ(∆c−∆c∗ )/σ(∆q)
σ(∆y−∆y ∗ )/σ(∆q)
σ(∆gdp−∆gdp∗ )/σ(∆q)
σ(∆nx)/σ(∆q)
0.25
—
0.25
0.12
0.31
0.12
0.19
0.25
Fama βF
.0
−8.1
0.54
0.42
0.36
0.44
1.16
1.16
0.46
1.26
0.42
0.36
0.81
0.34
0.49
0.65
0.48
0.01
0.07
(4.7)
5.
Fama
R2
0.02
0.04
0.07
(0.02)
σ(i−i∗ )/σ(∆e)
0.10
0.03
(0.01)
Carry SR
0.20
0.21
0.29
(0.04)
Notes: Baseline parameters: γ = 0.07, α = 0.4, θ = 1.5, ρ = 0.97, σ = 2, ν = 1, φ = 0.5, µ = 0, β = 0.99, and the
results are robust to changing ν, φ, µ and β (see appendix). The robustness panel of the table shows only the moments that
are sensitive to the change in the parameter values.
Notation: σ(·) stands for standard deviation, ρ(·) for autocorrelation, and HL(·) for half-life.
The dynamic moments are calculated as the median off the in-sample estimates over 10,000 simulations with 30 years (120
quarters) with the standard deviation of estimates across simulations reported in brackets. The asymptotic values of the
estimates are similar to the medians except for ρ(q) → 1, HL(q) → ∞ and βF → −4.6.
and the relative consumption growth rates. A linearized model with a single shock, however, implies
that all correlations are perfect (positive or negative), and we evaluate the ability of the model match
the empirical imperfect correlation in Section 5.2, where we augment the model with additional shocks.
The last panel of Table 2 illustrates the success of the model in capturing the joint behavior of the
nominal exchange rate and the interest rates. First, the interest rate differential in the model is highly
persistent (has persistence ρ), yet it is over 10 times less volatile that the nominal exchange rate, as in
the data. The model generates a negative Fama-regression coefficient and the Fama-regression R2 is
less than 0.05, as in the data.40 The Sharpe ratio associated with the dynamic carry trade strategy (i.e.,
investing in the temporarily high interest rate currency) is only 0.2, as in the data.
∆c − ∆c∗ is about three times less volatile than ∆q, ∆c alone is six times less volatile than ∆q.
40
The in-sample Fama regression coefficient is −8 with huge variation covering zero within two standard deviations. When
the psit is combined with other shocks, the median coefficient becomes closer to zero, as in the data (see Section 5.2).
31
Lastly, Table 2 offers robustness with respect to the main parameter values. The main insight is
that the model requires a high ρ in order to capture the dynamic properties of the real exchange rate. A
lower ρ results in a less persistent nominal and real exchange rates and in more predictable exchange
rate changes with high carry trade Sharpe ratios. The model also needs a lot of home bias (a low γ),
as when γ is doubled (corresponding to a 60% trade to GDP ratio), the model predicts a considerably
more volatile response of the real variables to the real exchange rate, in contrast with the data. Having
a low θ and a high α is important for the fit of the model, while the model is more robust with respect
to variation in other parameters, including risk aversion σ and Frisch elasticity ν. In particular, the
success of the model does not hinge on the very high values of σ.
5
5.1
Extensions
A monetary model with nominal rigidities
We now consider the robustness of our findings in Sections 3 and 4 in a fully-specified monetary model
with nominal rigidities, which is arguably a salient feature of the world. First, we demonstrate that
nominal shocks per se cannot reproduce the empirical exchange rate behavior, as suggested by Proposition 1. Second, we show that the financial shock ψt has similar quantitative properties in the monetary
model, as in the real model of Section 4, despite the differences in the transmission mechanism. Finally,
we show that the monetary model with nominal rigidities allows can explain a significant portion of
the Mussa puzzle.
We introduce nominal rigidities as in the standard New Keynesian model (see e.g. Woodford 2003),
while leaving the structure of international financial markets as in the baseline model. We focus on a
cashless-limit economy and abstract from ZLB, commitment problems and multiplicity of equilibria. In
particular, the nominal interest rate is set by a central bank according to the Taylor rule:
it = ρm it−1 + (1 − ρm ) δπ πt + εm
t .
(49)
Firms use the Calvo price-setting rule with a probability of price adjustment equal to 1−λ. We maintain
the assumption that the optimal static price depends both on own marginal cost and competitor prices,
as in (12)–(13) with the exogenous markup shocks shut down. We further assume that exporters set
prices in the local currency (LCP) and the wages are flexible, which facilitates the comparison with the
literature (e.g., Chari, Kehoe, and McGrattan 2002, Devereux and Engel 2002). Appendix A.6 provides a
full description of the model and its solution, as well as several extensions with PCP, sticky wages and
alternative Taylor rules.
To calibrate the model, we keep the same values for most parameters, as in the baseline case discussed in Section 4.7. The prices are assumed to adjust on average once a year, i.e. λ = 0.75 (Nakamura
and Steinsson 2008). For the Taylor rule, we use the estimates from Clarida, Gali, and Gertler (2000),
setting ρr = 0.8 and δπ = 2.15, and assume that the monetary (Taylor rule) shocks εm
t follow an iid
process, whenever present.
The results of our analysis are summarized in Table 3, where we contrast the moments in the data
32
Table 3: Comparison of single-shock models
Moment
Data
ψ shock
Baseline Monetary
m shock
NOEM
a shock
IRBC
1-2. PPP Puzzle and Meese-Rogoff:
ρ(q)
0.94
ρ(∆e)
0.93
0.92
0.65
0.92
(0.04)
(0.05)
(0.07)
(0.04)
−0.02
−0.04
−0.15
0.55
(0.09)
(0.09)
(0.08)
(0.15)
0.98
0.75
1.00
0.94
38.7
0.34
0.30
−0.80
−0.91
1.16
−0.31
−0.19
0.50
0.64
−8.1
−2.0
1.1
1.06
(4.7)
(1.7)
(0.3)
(0.07)
0.00
σ(∆q)
σ(∆e)
3. Terms of trade:∗
σ(∆s)
σ(∆q)
4. Backus-Smith:∗
σ(∆c − ∆c∗ )
−0.25
σ(∆q)
5. Forward premium puzzle:
Fama β
Fama
R2
i∗ )
σ(i −
σ(∆e)
Carry SR
∗
.0
0.02
0.10
0.20
0.04
0.02
0.10
0.75
(0.02)
(0.02)
(0.04)
(0.07)
0.03
0.08
0.29
0.83
(0.01)
(0.03)
(0.03)
(0.09)
0.21
0.21
0
0
(0.04)
(0.04)
Moments 3 and 4 report the ratio of standard deviations times the sign of the correlation (which is either 1 or −1 in a
single-shock model). m-shock standard for the Taylor rule shock εm in (49).
and in various versions of the model with a single source of exogenous shocks. In particular, we first
report the moments in the baseline real model and in the monetary model with LCP sticky prices,
both subject exclusively to financial shock ψt . We then report the moments in the New Open Economy
Macro (New Keynesian) model subject to the Taylor-rule shocks εm
t , and with the ψt shocks shut down.
We also provide the moments from an international RBC (IRBC) model subject to only a productivity
shock at , which follows a persistent AR(1) process.
The results of Table 3 have a number of implications. First, the monetary shock alone (NOEM) fails
to reproduce several empirical patterns. In particular, both nominal and real exchange rates are less
persistent than in the data, in line with the findings of the earlier literature (see e.g. Chari, Kehoe, and
McGrattan 2002). Also, as expected, it results in no violation of either the Backus-Smith condition or
uncovered interest parity, both counterfactual. Similar failures are characteristic of the IRBC model,
which fails to produce deviations from either UIP or Backus-Smith.41 In contrast to NOEM, the IRBC
41
Corsetti, Dedola, and Leduc (2008) show that a model with a lower elasticity of substitution and a more persistent produc-
33
model produces a persistent real exchange rate process, but this comes at a cost of a non-random walk
behavior for the nominal exchange rate, which in addition is much less volatile than the real exchange
rate or the terms of trade.
Second, the quantitative results for the financial shock in the monetary model are surprisingly close
to the baseline case of the real model, even despite the differences in the transmission mechanism.
Indeed, in the real model, the interest rates settle down at levels that clear the asset markets, while in
the monetary model, the path of the interest rates is chosen by the government according to the Taylor
rule (49). The reason is that, in both models, a ψt shock results in a sharp nominal depreciation, in order
to be simultaneously consistent with equilibrium asset demand and intertemporal budget constraint.
A nominal depreciation results in a mild home inflation as the prices of the foreign good are adjusted
upwards. In a monetary model, the central bank responds by slightly raising the nominal interest rate.42
The households respond by increasing their savings and cutting current consumption expenditures,
thus enabling the model to reproduce the Backus-Smith puzzle. Similarly, the model can still explain
the forward premium puzzle: the interest rate is higher at Home after a depreciation shock, when the
exchange rate is expected to appreciate. It is important to emphasize that this similarity in the behavior
of the real and the monetary models is not granted in general, but instead is conditional on the specific
Taylor rule, which we disciplined with the empirical estimates of Clarida, Gali, and Gertler (2000).
Alternative Taylor rules can substantially alter the quantitative properties of the monetary model, as
we discuss in Appendix A.6.
Furthermore, nominal rigidities result in a dynamic price adjustment process, which acts to increase the volatility of the real exchange rate, bringing it closer to the empirical counterpart, and hence
improving the fit of the model. The persistence of the real exchange rate stays largely unchanged,
with a median small-sample estimated half-life of around 2.5 years (as described in Figure 3). On the
other hand, as pointed out by Obstfeld and Rogoff (2000), the model with LCP implies a counterfactual
negative correlation between the exchange rate and the terms-of-trade, which however can be fixed
by replacing LCP sticky prices with sticky wages (see Appendix A.6). We conclude that a monetary
model with the financial shock can somewhat improve the fit of the real exchange rate, but to a first
approximation has strikingly similar quantitative properties to the baseline real model.
Mussa puzzle
The extension with nominal rigidities allows us to address the Mussa puzzle. Com-
paring empirical moments for several developed countries before and after the end of Bretton Woods
system, the literature has emphasized the following stylized facts:
i. The volatility of the real exchange rate has increased almost as much as the volatility of the
nominal exchange rate (Mussa 1986).
ii. There are almost no differences in the output or consumption volatilities between the two periods
(Baxter and Stockman 1989, Flood and Rose 1995).
iii. The Backus-Smith risk-sharing condition and the UIP hold much better in the data during the
tivity process can reproduce the empirical negative Backus-Smith correlation. We explain the mechanism in Appendix A.5.
42
Recall that the response of the interest rate is similar in the real model, albeit for a different reason (see Section 4.6).
34
Bretton Woods period Colacito and Croce (2013).
We check whether the model can reproduce these findings by comparing the benchmark floating exchange regime under the Taylor rule (49), with an alternative Taylor rule:
it = ρm it−1 + (1 − ρm ) δe et ,
(50)
where parameter δe = 0.45 is calibrated to match the empirical volatility of the nominal exchange rate.
We find (see Appendix A.6 for details) that the volatility of the real exchange rate falls about two
times less than the volatility of the nominal exchange rate, which is in line with the estimates in Monacelli (2004). A decrease in the correlation between the nominal and the real exchange rates from 1.00 to
0.61 is also very close to the empirical estimate of 0.66. Furthermore, the volatility of the relative consumption remains almost unchanged. The model, however, cannot reproduce the third stylized fact, as
it suggests that the Backus-Smith and the UIP puzzles remain unchanged across the nominal exchange
rate regimes. This suggests that the financial exchange rate shock ψt might have been smaller (less
volatile) during the Bretton Woods period. As was shown by Jeanne and Rose (2002), pegged exchange
rate may render arbitrageurs more active in the currency markets, thereby offsetting a larger portion
of the noise trading shock. Indeed, reducing the volatility of the ψt shock (and adjusting δe = 0.13 to
maintain the same equilibrium volatility of the nominal exchange rate) allows the model to be consistent with all empirical patterns, both under the float and under the peg.
5.2
Multiple Shocks and Decomposition of Exchange Rate Variation
A natural deficiency of any one-shock model is that it can only speak to the relative volatilities of
variables, while it implies counterfactual perfect correlations between them. We, therefore, consider
an extension with multiple shocks combined simultaneously to study whether the model is successful in
reproducing imperfect empirical correlations between exchange rates and the macro variables. Another
benefit of a calibrated multi-shock model is that it allows to obtain a decompositions of the variance
of exchange rates across various sources of exogenous variation. This way we can assess the relative
importance of the productivity, monetary and financial shocks for driving the exchange rate dynamics.
We study both a real model with a productivity shock at and a nominal model with a monetary
(Taylor rule) shock εm
t (which we denote for brevity as an m-shock). In all cases, we include the
financial shock ψt along with a demand shock for foreign goods ξt .43 For the exchange rate moments
we target, we only need to specify the relative magnitudes of these shocks across countries (that is,
ãt = (at − a∗t )/2, and similarly for other shocks). We keep the parameters of the model as in the
baseline above, but we now need to calibrate the relative variances of the shocks, with the overall level
of variance kept to match the volatility of the nominal exchange rate. To do so, we target the relative
standard deviations and the correlations of exchange rates with consumption, output and net exports.
Table 4 reports the results.
43
We have also experimented with LOP-deviation shocks ηt and nearly-equivalent trade cost shocks (denoted τt for iceberg
trade costs in goods), but find them largely redundant as long as the ξt shock is included, which has superior quantitative
properties for the moments we have chosen.
35
Table 4: Contribution of various shocks to exchange rate fluctuations
Data
Baseline
(ψ only)
0.00
0.94
0.98
0.98
−0.02
0.93
0.75
1
−0.02
0.93
0.79
0.96
−0.03
0.93
1.00
1.00
0.25
−0.28
0.31
−1
0.30
−0.22
0.18
−0.42
σ(∆nx)/σ(∆q)
ρ(∆nx, ∆q)
0.12
&0
0.25
1
0.30
−0.01
0.28
−0.02
Fama βF
Fama R2
σ(i−i∗ )/σ(∆e)
Carry SR
.0
0.02
0.10
0.20
−8.1
0.04
0.02
0.21
−0.6
0.01
0.04
0.16
−0.1
0.00
0.09
0.19
ψ-shock
100%
53%
64%
ξ-shock
—
39%
33%
a-shock
—
8%
—
m-shock
—
—
3%
ρ(∆e)
ρ(q)
σ(∆q)/σ(∆e)
ρ(∆q, ∆e)
σ(∆c−∆c∗ )/σ(∆q)
ρ(∆c−∆c∗ , ∆q)
Real Model
(ψ, ξ, a)
Monetary Model
(ψ, ξ, m)
Decomposition of var(∆qt )
Note: m-shock standard for the Taylor rule shock εm in (49). The variance decomposition for ∆e is identical in the monetary
model, while in the real model the a-shock contributes nearly zero to the dynamics of ∆e (with a 57-43 decomposition
between ψ and ξ shocks).
36
The models with three shocks are, in general, successful at matching most moments, both relative volatilities and correlations. The resulting decomposition of the contribution of the shocks to the
volatility of the exchange rate reveals in both cases that ψt plays the dominate role, while the contribution of the productivity and monetary shocks is minimal, as suggested by Proposition 1. The role of the
ξt shock in both cases is also very important, offsetting the strong correlations between exchange rates
and the macro variables arising from the ψt shock, without inducing counterfactually strong movements in consumption and other macro variables. [to be completed]
5.3
Multiple foreign assets, global imbalances and valuation effects
[to be completed]
6
Conclusion
[to be completed]
37
A
A.1
Appendix
Demand structure
Consider the general separable homothetic (Kimball 1995) demand aggregator Ct defined implicitly by:
CHt
CF t
ΩHt g
+ ΩF t g
= 1,
(A1)
ΩHt Ct
Ω F t Ct
where g 0 > 0, g 00 < 0 and g(1) = g 0 (1) = 1 (a normalization), and
ΩHt ≡ (1 − γ)e−γξt
and
ΩF t ≡ γe(1−γ)ξt
are the weights which the required properties. Then expenditure minimization results in the following
demand schedules:
CHt = (1 − γ)e
−γξt
h
PHt
Dt
Ct
and
(1−γ)ξt
CF t = γe
h
PF t
Dt
Ct ,
(A2)
where h(x) ≡ g 0−1 (x), which implies h0 < 0 and h(1) = 1, and where Dt is the Lagrange multiplier
on the aggregator (A1). We can solve for Dt by substituting demand (A2) into the aggregator (A1):
PHt
PF t
−γξt
(1−γ)ξt
(1 − γ)e
g h
+ γe
g h
= 1,
(A3)
Dt
Dt
and also define the price index in this economy from the expenditure per unit of Ct :
PHt
PF t
PHt CHt + PF t CF t
−γξt
(1−γ)ξt
PHt h
PF t h
= (1 − γ)e
+ γe
.
Pt =
Ct
Dt
Dt
Proof of Lemma 1
(A4)
It is immediate to check from (A3)–(A4), using g(1) = h(1) = 1, that when ξt = 0
and PHt = PF t , we have Pt = Dt = PHt = PF t , which corresponds to the symmetric steady state.
Using demand (A2), we immediately have that the symmetric steady state foreign share is:
P F t CF t
γCt
=
= γ,
PHt CHt + PF t CF t ξt =0, PHt =PF t
(1 − γ)Ct + γCt
where we again use h(1) = 1. We next use (A3)–(A4) to obtain the log-linear approximation for Pt
and Dt around the symmetric steady state:44
pt = dt = (1 − γ)pHt + γpF t ,
(A5)
where pt and dt are the log deviations from the steady state values. Note that the taste shock ξt 6= 0
does not affect the first order approximation to the prices index (due to the way it enters the weights
θ−1
1
In the CES case, which obtains with g(z) = θ−1
θz θ − 1 , we have Pt = Dt , while for a more general demand
Pt and Dt different by a second order term around a symmetric steady state. Since our analysis relies on the first order
approximation to the equilibrium system, we replace Dt with Pt in the demand equations (5) in the text.
44
38
ΩHt and ΩF t ). Finally, log-linearizing (A2), we have:
cHt = −γξt − θ(pHt − dt ) + ct
and
cF t = (1 − γ)ξt − θ(pF t − dt ) + ct ,
0
where θ ≡ − h (x)x
= − ∂ log∂xh(x) x=1 . Together with (A5), these expressions results in (6). Subx
x=1
tracting, we have cF t − cHt = ξt − θ(pF t − pHt ), which implies that the elasticity of substitution is
indeed θ.
Monopolistic competition and price setting
Consider now a unit continuum of symmetric domestic
firms with marginal cost M Ct and a unit continuum of symmetric foreign firms with marginal cost
Et M Ct∗ monopolistically competing in the domestic market. We generalize the consumption aggregator Ct to be defined in the following way:
Z 1
Z 1
CHt (i)
CF t (i)
ΩHt g
di +
ΩF t g
di = 1,
ΩHt Ct
Ω F t Ct
0
0
(A6)
with taste shocks (ΩHt , ΩF t ) determined as above by a common home bias parameter γ and a common
demand shifter ξt for all varieties i ∈ [0, 1]. The households choose {CHt (i), CF t (i)} to maximize Ct
given prices and total expenditure:
Z
1
Et = Pt Ct =
Z
PHt (i)CHt (i)di +
0
1
PF t (i)CF t (i)di.
(A7)
0
This expenditure minimization results in individual firm demand as in (A2). A representative home firm
takes (Ct , Pt , Dt ) as given and sets its price to maximize profits from serving the domestic market:
PHt (i)
−γξt
PHt (i) = arg max (PHt (i) − M Ct )(1 − γ)e
h
Ct ,
Dt
PHt (i)
which results in the standard markup pricing rule, with the markup Mt determined by the elasticity
of the demand curve h(·). Since all domestic firms are symmetric, we have CHt = CHt (i) and PHt =
PHt (i) for all i ∈ [0, 1]. Similar price setting rule is used by symmetric foreign firms with marginal
costs Et M Ct∗ , and we also have CF t = CF t (i) and PF t = PF t (i) for all i ∈ [0, 1]. Following the proof
of Lemma 1, the elasticity of demand in a symmetric steady state equals θ, and therefore the steady
state markup is given by M = θ/(θ − 1) for both home and foreign firms.
We next take a log-linear approximation to the optimal price PHt around the symmetric steady state:
pHt = −Γ(pHt − pt ) + mct ,
where we use the approximation dt = pt and Γ denotes the elasticity of the markup Mt with respect
to the relative price of the firm, evaluated at the symmetric steady state. Note that this equation is the
counterpart to (12) in the text with α ≡
Γ
1+Γ
and µt = 0.
Lastly, we provide further details about the primitive determinants of θ and α (see Amiti, Itskhoki,
and Konings 2016, for a more indepth exposition). Define the demand elasticity function θ̃(x) ≡
− ∂ ∂loglogh(x)
x , so that θ ≡ θ̃(1). Then the markup function is given by M̃(x) ≡
39
θ̃(x)
,
θ̃(x)−1
and the elas-
M̃(x)
ticity of the markup is given by Γ̃(x) ≡ − ∂ log∂x
, with M = M̃(1) and Γ = Γ̃(1). Manipulating
these definitions, we can represent
Γ̃(x) =
˜(x)
,
θ̃(x) − 1
where
ε̃(x) ≡
∂ log θ̃(x)
∂ log x
is the elasticity of elasticity (or super-elasticity) of demand. Therefore, Γ =
θ−1 ,
where = ˜(1), and
we further have:
Γ
=
.
1+Γ
+θ−1
To the extent and θ are controlled by independent parameters, we can decouple the elasticity of
α=
substitution θ from the strategic complementarity elasticity α. Indeed, θ is a characteristic of the slope
(the first derivative) of demand h0 , while is a characteristic of the curvature (the second derivative) of
demand h00 . Formally, we have:
h0 (x)x θ=−
h(x) x=1
and
h00 (x)x ∂ log θ̃(x)
h0 (x)x h00 (x)x = 1+θ+ 0
= 1−
+ 0
=
.
∂ log x
h(x)
h (x) x=1
h (x) x=1
We assume that the demand schedule h(·) is log-concave, that is ≥ 0, and therefore α ∈ [0, 1), since
θ > 1 is the second order requirement for price setting optimality. An appropriate choice of produces
any required value of α for any given value of θ. A suitable parametric example can be found in Klenow
and Willis (2006) and Gopinath and Itskhoki (2010), where h(x) = [1 − log(x)]θ/ for some elasticity
parameter θ > 1 and super-elasticity parameter > 0.
40
A.2
Equilibrium system
We summarize here the equilibrium system of the general model from Section 3.1 by breaking it into
blocks:
1. Labor supply (3) and its exact foreign counterpart.
2. Labor demand in (10), used together with the definition of the marginal cost (9), and its exact
foreign counterpart.45
3. Demand for home and foreign goods:
∗
Yt = YHt + YHt
and
Yt∗ = YF t + YF∗t ,
(A8)
where the sources of demand for home good are given in (18) and (19), and the counterpart
sources of demand for foreign good are given by:
PF t
(1−γ)ξt
h
YF t = γe
[Ct + Xt + egt ] ,
Pt
∗ h
i
PF t
∗
−γξt∗
∗
∗
gt∗
,
YF t = (1 − γ)e
h
C
+
X
+
e
t
t
Pt∗
(A9)
(A10)
where Xt and Xt∗ satisfy the intermediate good demand in (10) and its foreign counterpart.
4. Supply of goods: given price setting (12)–(13) and their foreign counterparts given by:
∗
∗
PF t = eµt +ηt (M Ct∗ Et )1−α Ptα ,
µ∗t
PF∗ t = e M Ct∗1−α Pt∗α ,
(A11)
(A12)
output produced is determined by the demand equation (A8).
∗ , P , P ∗ ), equation (7) defines the price level P as a log-linear approxGiven prices (PHt , PHt
t
Ft
Ft
imation, and a similar equation defines Pt∗ .46
5. Asset demand by home and foreign households (4) and (17), which can be rewritten as an international risk sharing condition and a no-arbitrage condition:
Et+1
ψt
∗
Et e Θt+1
− Θt+1 = 0,
Et
ψt ∗ Et+1
Et Θt+1 e Rt
− Rt
= 0,
Et
(A13)
(A14)
with the stochastic discount factors Θt+1 and Θ∗t+1 defined in the text.
6. Home-country flow budget constraint (20), with its foreign counterpart redundant by Walras Law.
45
Note that the input demand equations (10) together with the marginal cost (9) imply the production function equation (8).
Log-linear expression for pt in (7) can be replaced with two non-linear expressions (A3)–(A4) defining (Pt , Dt ), and
Pt should be replaced with Dt in demand equations (18)–(19) and (A9)–(A10). The rest of the equilibrium system stays
unchanged. However, these adjustments do not have first order consequences, as Pt and Dt are the same up to second order
terms, and therefore the log-linearized system in Appendix A.2.2 is unchanged.
46
41
A.2.1
Symmetric steady state
In a symmetric steady state, B ∗ = B ∗F = 0, and the shocks (defined in Table 1) take the following
values:
ψ = ξ = ξ ∗ = η = η ∗ = χ = χ∗ = 0,
and we normalize W = W ∗ = 1 (corresponding to w = w∗ = 0). We let the remaining shocks take
arbitrary (zero or non-zero) symmetric values:
a = a∗ ,
g = g∗,
κ = κ∗
µ = µ∗ .
and
We start with the equations for prices. In a symmetric steady state, exchange rates and terms of
trade are equal to 1:
E = Q = S = 1,
(A15)
and therefore we can evaluate the prices using the equilibrium conditions described above:
µ
"
e 1−α −a
= PF =
(1 − φ)1−φ φφ
P = P ∗ = PH = PF∗ = PH∗
with the marginal costs given by M C =
M C∗
e−a P φ
(1−φ)1−φ φφ
=
=
φµ
#
1
1−φ
−a
e 1−α
(1−φ)1−φ φφ
,
(A16)
1
1−φ
.
Next we use these expressions together with production function, labor demand and labor supply
to obtain two relationships for (C, Y, L):
φ
µ
L = e 1−φ 1−α
C σ L1/ν =
e−κ
P
a
− 1−φ
a
= e 1−φ
φ
φ
− 1−φ
Y,
µ
− (1−α)(1−φ)
−κ
(A17)
φ
(1 − φ)φ 1−φ .
(A18)
−µ
Substituting prices (and using h(1) = 1) and intermediate good demand X = φ MPC Y = e 1−α φY into
the goods market clearing, we obtain an additional relationship between C and Y :
h
i
µ
C + eg = 1 − e− 1−α φ Y.
(A19)
We further have Y = Y ∗ , and YH = YF∗ = (1 − γ)Y and YH∗ = YF = γY .
The asset demand conditions imply that R = R∗ = 1/β.
Lastly, we define the following useful ratios:
µ
P (C + G)
GDP
=
= 1 − e− 1−α φ,
Output
PH Y
P F YF
P F YF
Import
γ≡
=
=
= γ,
Expenditure
PH YH + PF YF
PH Y
EPH∗ YH∗ + PF YF
Import+Export
2γ
=
=
.
GDP
P (C + G)
ζ
ζ≡
42
(A20)
(A21)
(A22)
A.2.2
Log-linearized system
We log-linearize the equilibrium system (summarized above in Appendix A.2) around the symmetric
steady state (described in Appendix A.2.1). We split the equilibrium system into three blocks — prices,
quantities and dynamic equations — and solve them sequentially, as the equilibrium system is blockrecursive.
Exchange rates and prices
The price block contains the definitions of the price index (7) and its foreign
counterpart:
p∗t = γp∗Ht + (1 − γ)p∗F t ,
(A23)
as well as the price setting equations (12)–(13) and (A11)–(A12), in which we substitute the marginal
cost (9) and its foreign counterpart and log-linearize to obtain:
pHt = µt − (1 − α)at + (1 − α)(1 − φ)(wt − pt ) + pt ,
p∗Ht
p∗F t
= µt + ηt − (1 − α)at + (1 − α)(1 − φ)(wt − pt ) + (1 − α)pt +
=
µ∗t
− (1 −
α)a∗t
+ (1 − α)(1 −
φ)(wt∗
−
p∗t )
+
(A24)
αp∗t ,
p∗t ,
pF t = µ∗t + ηt∗ − (1 − α)a∗t + (1 − α)(1 − φ)(wt∗ − p∗t ) + (1 − α)p∗t + αpt .
(A25)
(A26)
(A27)
In addition, we use the logs of the definitions of the real exchange rate and terms of trade (14) and (21):
qt = p∗t + et − pt ,
(A28)
st = pF t − p∗Ht − et .
(A29)
First, it is useful to define the log LOP deviations (as in equation (14) and in its foreign counterpart):
qHt ≡ p∗Ht + et − pHt = ηt + αqt ,
qF t ≡
p∗F t
+ et − pF t =
−ηt∗
+ αqt ,
(A30)
(A31)
where the expression on the right-hand side are obtained by using (A24)–(A27) together with (A28).
Then, we combine (A28)–(A29) together with these expressions, to obtain:
st = qtP − 2η̃t − 2αqt ,
(A32)
qt = (1 − γ)qtP − γst ,
(A33)
where qtP = p∗F t +et −pHt is the producer-price-based real exchange rate and we use the tilde notation
x̃t ≡ (xt − x∗t )/2 for any pair of variables (xt , x∗t ). Lastly, we solve for qtP and st as function of qt :
1 − 2αγ
2γ
qt −
η̃t ,
1 − 2γ
1 − 2γ
1 − 2α(1 − γ)
2(1 − γ)
st =
qt −
η̃t .
1 − 2γ
1 − 2γ
qtP =
(A34)
(A35)
Next, we use these solutions together with the expressions for price indexes (7) and (A23), to solve
43
for:47
γ
(1 − α)γ
γ
(pF t − pt ) = γ(pHt − pF t ) = −
qt −
η∗,
1−γ
1 − 2γ
1 − 2γ t
(1 − α)γ
γ
γ
(p∗Ht − p∗t ) = γ(p∗F t − p∗Ht ) =
qt −
ηt .
p∗F t − p∗t = −
1−γ
1 − 2γ
1 − 2γ
pHt − pt = −
Combining these expression with (A24) and (A26), we can solve for the price levels:
"
#
γ
µt + 1−2γ
ηt∗
1
γ
pt = wt +
− at +
qt ,
1−φ
1−α
1 − 2γ
" ∗
#
γ
µ
+
η
t
1
γ
t
1−2γ
p∗t = wt∗ +
− a∗t −
qt ,
1−φ
1−α
1 − 2γ
which together allow to solve for the relationship between qt and nominal exchange rate et :
2γ
2γ
η̃t
2µ̃t
(1 − φ) +
+
.
qt = (1 − φ)et − (1 − φ)2w̃t + 2ãt −
1 − 2γ
1 − α 1 − 2γ 1 − α
Real exchange rate and quantities
(A36)
(A37)
(A38)
(A39)
(A40)
The supply side is the combination of labor supply (3) and labor
demand (10) (together with marginal cost (9)), which we log-linearize as:
κt + σct + ν1 `t = wt − pt ,
`t = −at − φ(wt − pt ) + yt .
Combining the two to solve out `t , and using (A38) to solve out (wt − pt ), we obtain:48
"
#
γ
∗
1+ν
ν + φ µt + 1−2γ ηt
γ
νσct + yt =
at −
+
qt − νκt .
1−φ
1−φ
1−α
1 − 2γ
(A41)
(A42)
(A43)
Symmetrically, the same expression for foreign is:
" ∗
#
γ
µ
+
η
t
ν
+
φ
γ
1
+
ν
t
1−2γ
νσc∗t + yt∗ =
a∗ −
−
qt − νκ∗t .
1−φ t
1−φ
1−α
1 − 2γ
Adding and subtracting the two we obtain:
γ
1+ν
ν + φ µ̄t + 1−2γ η̄t
āt −
− ν κ̄t ,
1−φ
1−φ
1−α
"
#
γ
1+ν
ν + φ µ̃t − 1−2γ η̃t
γ
νσc̃t + ỹt =
ãt −
+
qt − ν κ̃t ,
1−φ
1−φ
1−α
1 − 2γ
νσc̄t + ȳt =
47
Note from (7) that pHt − pt = γ(pHt − pF t ), and we use the following steps to solve for:
pHt − pF t = −(pF t − p∗Ht − et ) − (p∗Ht + et − pHt ) = −(st + qHt ) = −(st + αqt + ηt ),
in which we then substitute (A35) to solve out st . Similarly, we solve for p∗F t − p∗t .
48
A useful interim step is: νσct + yt = (ν + φ)(wt − pt ) + at − νκt .
44
(A44)
(A45)
where x̄t ≡ (xt + x∗t )/2 and x̃t ≡ (xt − x∗t )/2 for any pair of variables (xt , x∗t ).
The demand side is the goods market clearing (A8) together with (18)–(19), which we log-linearize as:
∗
yt = (1 − γ)yHt + γyHt
,
yHt = −γξt − θ(pHt − pt ) + ζ[ςct + (1 − ς)gt ] + (1 − ζ) (1 − φ)(wt − pt ) − at + yt ,
∗
yHt
= (1 − γ)ξt∗ − θ(p∗Ht − p∗t ) + ζ[ςc∗t + (1 − ς)gt∗ ] + (1 − ζ) (1 − φ)(wt∗ − p∗t ) − a∗t + yt∗ ,
where ς ≡ C/(C + G), ζ ≡ P (C + G)/(PH Y ), and we used expression (10) and (9) to substitute for
Xt (and correspondingly for Xt∗ ). Combining together, we derive:
2(1 − γ)
∗
− (1 − ζ) qt
(A46)
yt − (1 − ζ)[yt − 2γ ỹt ] − ζς[ct − 2γc̃t ] = γ θ(1 − α)
1 − 2γ
1−ζ γ
1−ζ
1−ζ
η̃t − 2γ(1 − γ)ξ˜t ,
[µt − 2γ µ̃t ] −
ηt∗ − 2γ θ − γ
+ ζ(1 − ς)[gt − 2γg̃t ] −
1−α
1 − α 1 − 2γ
1−α
where we have slowed out (wt − pt ) and (wt∗ − p∗t ) using (A38)–(A39) and solved out (pHt − pt ) and
(p∗Ht − p∗t ) using (A36)–(A37). Adding and subtracting the foreign counterpart, we obtain:49
1−ζ
γ
[1 − (1 − 2γ)(1 − ζ)]ȳt = (1 − 2γ)ζς c̄t + ζ(1 − ς)ḡt −
µ̄t +
η̄t ,
(A47)
1−α
1 − 2γ
1−ζ
[1 − (1 − 2γ)(1 − ζ)]ỹt = (1 − 2γ)ζς c̃t + (1 − 2γ) ζ(1 − ς)g̃t −
µ̃t
(A48)
1−α
1−ζ
2(1 − γ)
˜
− 2γ θ − γ
η̃t − 2γ(1 − γ)ξt + γ θ(1 − α)
− (1 − ζ) qt .
1−α
1 − 2γ
An immediate implication of (A44) and (A47) is that (ȳt , c̄t ) depends only on (āt , ḡt , κ̄t , µ̄t , ν̄t ) and
does not depend on the real exchange rate qt . In particular, if āt = ḡt = κ̄t = µ̄t = ν̄t = 0, then
ȳt = c̄t = 0. This is the case we focus on throughout the paper, since as we see below the variation in
(āt , ḡt , κ̄t , µ̄t , ν̄t ) does not affect qt . Combining (A47) and (A47) we can solve for ỹt and c̃t . For example,
the expression for c̃t is:
#
γ
1+ν
ν + φ µ̃t − 1−2γ η̃t
(1−2γ)ζ(νσ+ς) + 2γνσ c̃t = [(1−2γ)ζ + 2γ]
ãt −
− ν κ̃t − (1−2γ)ζ(1−ς)g̃t (A49)
1−φ
1−φ
1−α
2(1 − γ)
1−ζ
1−ζ
ν+φ 1
1+ν
+ (1 − 2γ)
µ̃t + 2γ θ − γ
η̃t + 2γ(1 − γ)ξ˜t − γ θ(1 − α)
+
−
(1 − ζ) qt .
1−α
1−α
1 − 2γ
1 − φ 1 − 2γ
1−φ
h
i
"
Lastly, we provide the linearized expression for net exports:
∗
nxt = γ yHt
− yF t − s t ,
where nxt =
49
1
PH Y
N Xt is linear deviation of net exports from steady state N X = 0 relative to the
The foreign counterpart is obtained from combining together and rearranging:
yt∗ = (1 − γ)yF∗ t + γyF t ,
yF∗ t = −γξt∗ − θ(p∗F t − p∗t ) + ζ[ςc∗t + (1 − ς)gt∗ ] + (1 − ζ) (1 − φ)(wt∗ − p∗t ) − a∗t + yt∗ ,
yF t = (1 − γ)ξt − θ(pF t − pt ) + ζ[ςct + (1 − ς)gt ] + (1 − ζ) (1 − φ)(wt − pt ) − at + yt .
45
∗ and y , we obtain:
total value of output. Substituting in the expressions for st , yHt
Ft
2(1 − γ)
1−α
2γ(1 − ζ)
qt
(A50)
nxt = γ θ(1 − α)
+α−
+
1 − 2γ
1 − 2γ
1 − 2γ
#
"
γ
h
i
µ̃t − 1−2γ
η̃t
+ 2γ (1 − θ)η̃t + (1 − ζ)
− (1 − γ)ξ˜t − 2γ ζ[ς c̃t + (1 − ς)g̃t ] + (1 − ζ)ỹt .
1−α
The log-linear approximation to the flow budget constraint (20) is given then by:
βb∗t+1 − b∗t = nxt ,
E
PH Y
of B ∗
(A51)
where b∗t =
Bt∗ is the linear deviation of the net foreign asset (NFA) position from its steady
state value
= 0 relative to the total value of output (both in foreign currency, using steady
state exchange rate of E = 1). Note that the dynamics of Et and Rt∗ has only second order effects on
the returns on NFA (and hence drops out from the linearized system), as we approximate around a
symmetric steady state with zero NFA position. Equations (A51) is part of the dynamic block.
Exchange rate and interest rates
It only remains now to log-linearize the asset demand conditions (4)
and (17), which pins down the equilibrium interest rates, as well as provide an international risk sharing
condition:
it = Et {σ∆ct+1 + ∆pt+1 − ∆χt+1 } ,
i∗t = Et {σ∆ct+1 + ∆pt+1 − ∆et+1 − ∆χt+1 } − ψt ,
i∗t = Et σ∆c∗t+1 + ∆p∗t+1 − ∆χ∗t+1 ,
where it ≡ log Rt − log R and i∗t ≡ log Rt∗ − log R∗ . We combine the first two to obtain a no-arbitrage
(UIP) condition, the last two to obtain a risk-sharing (Backus-Smith) condition, and the first with the
third to solve for the interest rate differential:
it − i∗t = Et ∆et+1 + ψt ,
Et σ(∆ct+1 − ∆c∗t+1 ) − ∆qt+1 = ψt + Et {2∆χ̃t+1 } ,
ĩt ≡ 12 (it −
i∗t )
= Et {σ∆c̃t+1 + ∆p̃t+1 − ∆χ̃t+1 } .
(A52)
(A53)
(A54)
Substituting out ∆ct+1 − ∆c∗t+1 = 2∆c̃t+1 in (A53) using (A49), we obtain an equation characterizing the expected real depreciation Et ∆qt+1 as a function of exogenous shocks. Together with (A51),
in which we substitute (A50), it forms a system of two dynamic equations that describe the equilibrium
dynamics of the real exchange rate given the exogenous dynamic processes for the shocks.
46
A.3
Autarky Limit and Proofs for Section 3.2
Proof of Propositions 1
The strategy of the proof is to evaluate the log deviations of the macro
variables zt ≡ (wt , pt , ct , `t , yt , it ) from the deterministic steady state (described in Appendix A.2.1)
in response to a shock εt = V0 Ωt 6= 0.50 In particular, we explore under which circumstances
limγ→0 zt = 0. It is sufficient to consider the log-linearized equilibrium conditions described in Appendix A.2.2, as providing a counterexample is sufficient for the prove (hence, the focus on the small
log deviations is without loss of generality). Furthermore, the proof does not rely on the international
risk sharing conditions, and hence does not depend on the assumptions about the (in)completeness of
the international asset markets.
To prove the propositions, consider any shock εt with the restriction that
ηt = ηt∗ = ξt = ξt∗ = ψt ≡ 0.
(A55)
We now go through the list of requirements imposed by the first part of the condition (22):
1. No wage response limγ→0 wt = 0 implies wt = 0, i.e. the unit of account shocks cannot lead to
the exchange rate disconnect in the limit.
2. No price level response implies, using (A38) and (A55):
1
µt
lim pt = wt +
− at = 0,
γ→0
1−φ 1−α
which in light of wt = 0 requires µt = (1 − α)at , i.e. the markup shocks must offset the productivity shocks to avoid variation in the price level.
When the same requirements are imposed for foreign, it ensures limγ→0 {qt − et } = 0, as immediartely follows from the the definition of the real exchange rate qt = p∗t +et −pt (see also (A40)).
3. From the labor supply and labor demand conditions (A41)–(A42), no consumption, employment
and output response require:
n
o
lim σct + ν1 `t + pt = wt − κt = 0,
γ→0
n
o
lim yt − `t + φpt = at + φwt = 0,
γ→0
which then implies at = κt = wt ≡ 0 and by consequence µt ≡ 0 from the result above. That
is, there cannot be productivity, markup or labor wedge shocks, if the price level, consumption,
output and employment are not to respond in the autarky limit.
4. Rearranging the goods market clearing in the home market (A46), we have:
1−ζ
lim ζyt − ζςct = ζ(1 − ς)gt −
µt = 0,
γ→0
1−α
50
We do not impose any restrictions on the process for shocks in Ωt , with the exception of the mild requirement that any
innovation in Ωt has some contemporaneous effect on the value of shocks in Ωt , i.e. we rule out pure news shocks. We discuss
examples with specific time series processes for the shocks in the end of this appendix.
47
which in light of the above results requires gt ≡ 0.
5. Lastly, the home bond demand requires:
lim σ∆Et ct+1 + ∆Et pt+1 − it = Et ∆χt+1 = 0,
γ→0
therefore there cannot be predictable changes in χt and unpredictable changes in χt do not affect
allocations in a one-period bond economies, hence without loss of generality we impose χt ≡ 0.
To summarize, the first condition in (22) (combined with the absence of ηt , ξt and ψt shocks) implies:
wt = χt = κt = at = µt = gt ≡ 0,
i.e. no other shock can be consistent with limγ→0 zt+j = 0 for all j ≥ 0, however in the absence of
shocks limγ→0 et+j = 0, violating the second condition in (22). A symmetric argument for foreign
rules out the foreign counterparts of these shocks. This completes the proof.
Proof of Proposition 2
For the proof, we consider the equilibrium system in the autarky limit by only
keeping the lowest order terms in γ for each shock or variable.51 Throughout the proof we impose
wt = χt = κt = at = µt = gt ≡ 0, as well as for their foreign counterparts.
First, we consider our three moments of interest when ψt is the only shock, that is we set ηt =
ξt ≡ 0. For this purpose, it is sufficient to consider the static equilibrium conditions only, as the effect
of the ψt shock on the macro variables is exclusively indirect through qt . Specifically:
1. Consider the near-autarky comovement between the terms of trade and the real exchange rate
from (A35):
lim
γ→0
cov(∆st+1 , ∆qt+1 )
= (1 − 2α) > 0
var(∆qt+1 )
1
iff α < ,
2
since we have η̃t = 0. α < 1/2 is a necessary parameter requirement for this result, which is
borne out in the data, as we discuss in Section 4.
2. Consider the near-autarky comovement between the relative consumption and the real exchange
rate from (A49), which in the absence of all shocks but ψt simplifies to:
h
i
2(1 − γ) ν + φ 1
1+ν
(1 − 2γ)ζ(νσ + ς) + 2γνσ c̃t = −γ θ(1 − α)
+
−
(1 − ζ) qt .
1 − 2γ
1 − φ 1 − 2γ 1 − φ
Hence, we have:
2
ν+φ
1 cov ∆ct+1 − ∆c∗t+1 , ∆qt+1
lim
=−
2θ(1 − α) + ζ
− (1 − ζ) < 0,
γ→0 γ
var(∆qt+1 )
ζ(νσ + ς)
1−φ
51
For example, consider equation (A40), which we now rewrite as:
1
µ̃t
1
η̃t
qt − e t = 2
ãt −
− w̃t + 2γ
.
1−φ
1−α
1−φ1−α
Note that the gap between qt and et is zero-order in γ for shocks (ãt , µ̃t , w̃t ) and first-order in γ for shock η̃t .
48
which is negative for all parameter values since
ζ
ν+φ
ζν + ζ − (1 − φ)
− (1 − ζ) =
>0
1−φ
1−φ
as from (A20) ζ = 1 − e−µ/(1−α) φ > 1 − φ.
3. Consider the near-autarky comovement between the nominal exchange rate and the nominal
interest rate differential (the Fama coefficient) by using (A54), which we write in the limit as:
2γσ
ζ(1 − ς/σ)
∗
it − it = Et {2σ∆c̃t+1 + 2∆p̃t+1 } = −
2θ(1 − α) − 1 +
Et ∆qt+1 .
ζ(νσ + ς)
1−φ
where we used expression (A49) for c̃t and expressions (A38)–(A39) for pt and p∗t . Furthermore,
(A40) and (A53) imply Et ∆et+1 = Et ∆qt+1 = −ψt in the limit and with ψt shocks only. Therefore, the Fama regression coefficient in the limit is:52
lim γ
γ→0
cov (Et ∆et+1 , it − i∗t )
1
ζ(νσ + ς)
=−
∗
var (it − it )
2σ
2θ(1 − α) − 1 +
ζ(1−ς/σ)
1−φ
< 0,
which is always negative under a mild additional requirement that θ > 1 and σ > 1 (since ς ≤ 1
and α < 1/2), with a necessary condition being substantially weaker.53
This proves the first claim of the proposition that ψt robustly and simultaneously produces all three
empirical regularities in the autarky limit.54
Second, recall that the uncovered interest rate parity (A52) implies that the Fama regression coefficient:
βF ≡
cov(∆et+1 , it − i∗t )
=1
var(it − i∗t )
whenever
ψt ≡ 0.
Therefore, (ηt , ηt∗ , ξt , ξt∗ ) shocks that follow any joint process cannot resolve the forward premium
puzzle. This is sufficient for the second claim of the proposition that the remaining shocks cannot
deliver the empirical comovement for all three moments. Nonetheless, we explore the remaining two
moments as well.
Third, in the remainder of the proof, we focus on the ξt and ηt shocks (setting all other shocks
including ψt to zero), and impose specific time series process for these two types of shocks, which can
be viewed as providing counterexamples sufficient for our argument in Proposition 2. Specifically, we
focus on AR(1) processes for relative shocks:
ξ˜t = ρξ ξ˜t−1 + σξ εξt ,
η̃t = ρη η̃t−1 + ση εηt ,
We make use of the fact that cov (∆et+1 , it − i∗t ) = cov (Et ∆et+1 , it − i∗t ) since it − i∗t is known at t.
The Fama coefficient for the real interest rates is always negative without any further parameter restrictions, as it is
proportional to the expression for the Backus-Smith correlation, since the real interest rate rt ≡ it − Et ∆pt+1 = σEt ∆ct+1
in the absence of χt shocks.
54
It is also easy to verify that the dispersion of the (real and nominal) exchange rate is separated from zero in response
to a ψt shock since from (A53) Et ∆qt+1 = −ψt and qt needs to adjust in response to ψt to ensure intertemporal budget
constraint with net exports following (A50). We show this formally in Appendix A.4 for ψt following an AR(1) process.
52
53
49
with ρξ , ρη ∈ [0, 1] and where εξt , εηt ∼ iid(0, 1). We focus on the zero-order component of the
exchange rate dynamics in γ, as this component is non-trivial for both ξ˜t and η̃t shocks. Therefore, we drop the first and higher order components in γ, so that we have et = qt from (A40) and
Et ∆qt+1 = Et ∆et+1 = 0 from (A53) together with (A49). Hence, the dynamics of the exchange rates
is a random walk with jumps that satisfy the intertemporal budget constraints. The flow budget constraint (A51) (with net exports (A50), in which we substitute the solutions for c̃t and ỹt from (A45) and
(A48)) up to first order terms in γ is given by:
βbt+1 − bt = 2γ ϑqt − ξ˜t − (θ − 1)η̃t ,
T ∗
where ϑ ≡ θ(1 − α) − 1−2α
2 . Solving this equation forward and imposing limT β bT +1 = 0, we obtain
the solution for the equilibrium exchange rate:55
∆qt+1 =
1 1−β
θ−1 1−β
ση εηt+1 .
σξ εξt+1 +
ϑ 1 − βρξ
ϑ 1 − βρη
We can now calculate the moments using static equilibrium conditions (A35) for st and (A49) for c̃t :56

(1 − 2α) > 0,
for ξt shock

cov(∆st+1 , ∆qt+1 ) 
cov(∆ηt+1 ,∆qt+1 )
(1−2α)−2
lim
=
,
var(∆qt+1 )

γ→0
"
#
var(∆qt+1 )

2(1−α)
for
η
shock,
t
1−2α θ−1 1−βρη
=(1−2α) 1−

1 cov (∆c̃t+1 , ∆qt+1 ) 

=
lim
γ→0 γ
var(∆qt+1 )

θ−1
1−β
2θ(1−α)−(1−2α)
ζ(νσ+ς)
2θ(1−α)+ζ ν+φ
1−φ
1
1−β
θ−1
ζ(νσ+ς)
θ− 1−2α 1−βρη
2(1−α)
<0,
β(1−ρξ )
1−β
1−
−
(1−2α)+ζ ν+φ
−(1−ζ)
1−φ
2θ(1−α)−(1−2α)
θ−1
1−2α
θ− 2(1−α)
≷ 0,
ν+φ
1−β 2θ(1−α)+ζ 1−φ −(1−ζ)
1−βρη
2θ(1−α)+ζ ν+φ
1−φ
for ξt ,
> 0, for ηt ,
where for the first moment we maintain the assumption that α < 1/2 and to sign the second moment
we use the fact that ζ > 1 − φ. To see that the Backus-Smith correlation under ξt shocks can take both
signs, it is sufficient to consider the case with ρξ = 1 (when the correlation is negative) and the case
with ρξ = 0 and β ≈ 1 (when the correlation is positive). If β ≥ ρξ , under our parameterization it is
sufficient to have the quarterly discount factor β > 0.75 for the sign to be positive (with the calibrated
value of β = 0.99). This shows that the η̃t shock robustly generates counterfactual comovement with
all three macro variables, while the ξ˜t shock does not robustly deliver empirically relevant comovement
between exchange rates on one hand and interest rates and relative consumption on the other hand. 55
We describe a rigorous solution method in Appendix A.4, while here we offer a heuristic argument: the net present value
(using β as a discount factor) of any innovation to the right hand side of the flow budget constraint needs to be zero for
intertemporal budget balance. Denote εt ≡ ∆qt the (random
of the exchange rate. The net present value
P walk)j innovation
j
ξ
η
j
of the innovation to the flow budget constraint is therefore ∞
β
ϑε
−
ρ
σ
t
ξ ξ εt − (θ − 1)ρη ση εt = 0, and solving for
j=0
εt from this equation we obtain the expression in the proof.
56
In our calculations, we use the interim results that in response to ηt shocks:
2
θ−1
θ−1
1−β
1−β 2
var(∆qt+1 ) =
ση2
and
cov(∆ηt+1 , ∆qt+1 ) =
ση ,
1−2α
1
− βρη
θ(1 − α) − 2 1 − βρη
θ(1 − α) − 1−2α
2
and similarly for the ξt shock.
50
A.4
The Baseline Model with ψt Shock
wt = wt∗ = 0 and µt = µ∗t = ηt = ηt∗ = ξt = ξt∗ = gt = gt∗ = at = a∗t = κt = κ∗t = χt = χ∗t = 0
A.4.1
Price block
1 − 2αγ
qt ,
1 − 2γ
1 − 2α(1 − γ)
=
qt ,
1 − 2γ
(1 − α)γ
γ
(pF t − pt ) = −
qt ,
=−
1−γ
1 − 2γ
γ
(1 − α)γ
=−
(p∗Ht − p∗t ) =
qt ,
1−γ
1 − 2γ
γ
1
qt ,
=
1 − φ 1 − 2γ
1
γ
=−
qt ,
1 − φ 1 − 2γ
qtP = p∗F t + et − pHt =
st
pHt − pt
p∗F t − p∗t
pt
p∗t
and in addition we have qtW = et and:
qt =
A.4.2
2γ
1−2γ
1−φ
et .
+ (1 − φ)
Quantity block
We have c̄t = `¯t = ȳt = 0, where for any pair (xt , x∗t ) we denote x̄t ≡ (xt + x∗t )/2). We further have:
1
1
γ
σc̃t + `˜t = −
qt ,
ν
1 − φ 1 − 2γ
φ
γ
`˜t = ỹt +
qt ,
1 − φ 1 − 2γ
ỹt =
θ(1 − α) 2(1−γ)
(1 − 2γ)ζς
1−2γ − (1 − ζ)
c̃t +
γqt ,
2γ + (1 − 2γ)ζ
2γ + (1 − 2γ)ζ
where we used:
γ
θ(1 − α)γ
qt + ζςct + (1 − ζ) yt −
qt ,
1 − 2γ
1 − 2γ
θ(1 − α)γ
γ
=−
qt + ζςc∗t + (1 − ζ) yt∗ +
qt ,
1 − 2γ
1 − 2γ
γθ(1 − α)γ
=−
qt ,
1 − 2γ
γθ(1 − α)γ
=
qt .
1 − 2γ
yHt =
yF∗ t
yF t − yHt
∗
yHt
− yF∗ t
51
We solve for c̃t as a function of qt :
γ
φ
qt ,
1 − φ 1 − 2γ
φ+ν γ
ỹt = −σνc̃t −
qt ,
1 − φ 1 − 2γ
`˜t = ỹt +
c̃t = −
θ(1 − α)2(1 − γ) +
φ+ν
1−φ
1+ν
− (1 − 2γ)(1 − ζ) 1−φ
2γσν + (1 − 2γ)(σν + ς)ζ
γ
qt .
1 − 2γ
Lastly, we express net exports as a function of qt :
∗
nxt = γ yHt
− yF t − st
h
i
1−α
2γ(1 − ζ)
2(1 − γ)
qt − 2γ ζς c̃t + (1 − ζ)ỹt
= γ θ(1 − α)
+α−
+
1 − 2γ
1 − 2γ
1 − 2γ
"
1+ν
= θ(1 − α)2(1 − γ) + α(1 − 2γ) − (1 − α) + 2γ(1 − ζ)
1−φ
#
2γ (ζς − (1 − ζ)σν)
γ
φ+ν
1+ν
+
θ(1 − α)2(1 − γ) +
− (1 − 2γ)(1 − ζ)
qt
2γσν + (1 − 2γ)(σν + ς)ζ
1−φ
1−φ
1 − 2γ
A.4.3
Dynamic block
ψt = ρψt−1 + εt ,
ψt = Et {2σ∆c̃t+1 − ∆qt+1 } ,
βb∗t+1
− b∗t = nxt .
Using the solution for c̃t and nxt , we can rewrite:
ψt = −aEt ∆qt+1 ,
X∞
b∗0 = −γd
β t qt ,
t=0
where
φ+ν
1+ν
2σγ θ(1 − α)2(1 − γ) + 1−φ − (1 − 2γ)(1 − ζ) 1−φ
> 1,
a≡1+
1 − 2γ
2γσν + (1 − 2γ)(σν + ς)ζ
"
1+ν
d ≡ θ(1 − α)2(1 − γ) + α(1 − 2γ) − (1 − α) + 2γ(1 − ζ)
1−φ
#
2γ (ζς − (1 − ζ)σν)
φ+ν
1+ν
1
+
θ(1 − α)2(1 − γ) +
− (1 − 2γ)(1 − ζ)
.
2γσν + (1 − 2γ)(σν + ς)ζ
1−φ
1−φ
1 − 2γ
(We assume the initial condition b∗0 = 0.) Recall that
qt =
2γ
1−2γ
1−φ
et ,
+ (1 − φ)
52
2γ
1
therefore et and qt follow the same process, with the volatility of et greater by a factor of 1 + 1−φ
1−2γ .
Note that
φ+ν
1+ν
2γ
1
2σγ θ(1 − α)2(1 − γ) + 1−φ − (1 − 2γ)(1 − ζ) 1−φ
>
,
1 − 2γ
2γσν + (1 − 2γ)(σν + ς)ζ
1 − φ 1 − 2γ
as long as [θ(1 − α)2(1 − γ) − (1 − 2γ)ς/σ](1 − φ) + [1 − (1 − 2γ)ς/σ] MM−1 φ + 2γφ/M > 0, (for which
2γ 1
σ ≥ 1 and θ(1 − α) > 1/2 are weak sufficient conditions) and therefore d1 ≡ a/ 1 + 1−φ
1−2γ > 1
still (so that the same properties are satisfied for the NER as for RER).
We can then solve for the equilibrium dynamic process for et and qt :
Lemma A1 qt follows an ARIMA(1,1,1), or equivalently ∆qt follows an ARMA(1,1).
Proof:
1
Et qt+1 = qt − ψt .
a
Since (1 − ρL)ψt = εt , the solution for qt must take the form:
qt =
1 1
ψt + mt ,
1−ρa
where mt is a martingale. In the fundamental solution, the innovation in the martingale process must
be proportional to the innovation to ψt , ∆mt = ϑεt . We now solve for the equilibrium value of ϑ,
which is consistent with the intertemporal restriction imposed by the budget constraint:
X∞
ϑ
1 1 1
1 1 j
j
0=
ρ εt =
+
,
β ϑεt +
j=0
1−ρa
1 − β 1 − ρ a 1 − ρβ
which implies:
ϑ=−
1−β1 1
.
1 − ρ a 1 − ρβ
Therefore, we have:
a(1 − ρ)(1 − ρL)∆qt = ∆εt + aϑ(1 − ρ)(1 − ρL)εt
1−β
1−β
β(1 − ρ)
1
= 1−
εt − 1 − ρ
εt−1 =
εt − εt−1 .
1 − ρβ
1 − ρβ
1 − ρβ
β
Therefore, the process for ∆qt is ARMA(1,1) with the AR root ρ, the MA root 1/β, and the innovation
β/a
1−βρ εt :
β/a
1
∆qt = ρ∆qt−1 +
εt − εt−1 .
1 − ρβ
β
Non-fundamental solution, where martingale mt is driven by sunspot shocks, are not possible due to
the budget constraint, and therefore, the process described here is the unique equilibrium solution for
the exchange rate. 53
Long-run effect
β/a
,
1 − ρβ
ρβ/a
1/a
∆q1 =
−
= −1/a,
1 − ρβ 1 − ρβ
q0 =
∆qt = ρt−1 ∆q1 .
Therefore,
q∞ = lim qT = q0 +
T →∞
∞
X
β/a
1/a
1
1−β
∆q1
=
−
=−
< 0.
1−ρ
1 − ρβ 1 − ρ
a (1 − ρ)(1 − ρβ)
∆qt = q0 +
t=1
And time to reversion to 0 is given by:
0 = qT0
T0
X
β/a
1 1 − ρT0
1 β(1 − ρ) − (1 − βρ)(1 − ρT0 )
1 −(1 − β) + (1 − βρ)ρT0
qt =
= q0 +
−
=
=
,
1 − ρβ a 1 − ρ
a
(1 − ρ)(1 − βρ)
a
(1 − ρ)(1 − βρ)
t=1
and therefore
T0 =
1
1−β
log
−−−→ ∞.
log ρ
1 − βρ βρ→1
Variance decomposition
var(∆qt+1 − Et ∆qt+1 )
=
var(∆qt+1 )
β/a
1 − ρβ
2
since
2
σ∆q
=
2
ρ2 σ∆q
+
β/a
1 − ρβ
σε2
β 2 (1 − ρ2 )
β 2 (1 − ρ2 )
=
=
2
2
1 − 2ρβ + β
(1 − β)2 + 2β(1 − ρ)
σ∆q
2
ρβ/a 1/a 2
1 + β2 2
σε − 2
σ ,
2
β
1 − ρβ 1 − ρβ ε
which implies
2
σ∆q
Autocorrelation
1 − 2ρβ + β 2
=
1 − ρ2
1/a
1 − ρβ
2
σε2 .
In changes:
ρ̂∆q
2 − 1 β/a σ 2
ρσ∆q
cov(∆qt , ∆qt−1 )
1 − ρ2
β 1−βρ ε
=
=
=
ρ
−
2
var(∆qt )
β[1 − 2ρβ + β 2 ]
σ∆q
And in levels, conditional on q−1 = 0:
ρ̂q (T ) =
E−1
PT
E−1
t=1
P
T
∆qt ∆qt−1
t=0 (∆qt )
54
2
=1−
A.4.4
Interest rates (UIP)
it = Et {σ∆ct+1 + ∆pt+1 } ,
i∗t = Et σ∆c∗t+1 + ∆p∗t+1 ,
it − i∗t = Et 2σ∆c̃t+1 + ∆pt+1 − ∆p∗t+1 = Et ∆et+1 + ψt .
Recall that we have Et = −ψt /d1 with d1 ≥ 1, with a strict inequality when γ > 0. Therefore, we
have:
it − i∗t = γd3 ψt ,
γde ≡
d1 − 1
,
d1
d3 > 0.
Alternatively, the interest rate differential (it − i∗t ) can be expressed as:
it −
i∗t
σθ(1 − α)2(1 − γ)(1 − φ) + σφ − (1 − 2γ) σ(1 − ζ) + ςζ
=−
2γσν + (1 − 2γ)(σν + ς)ζ
2γ
1−2γ
2γ
1−2γ
+ (1 − φ)
Et ∆et+1
Therefore, the process for it − i∗t is proportional to ψt and, therefore, follows an AR(1) process with
persistence ρ and innovation γd3 εt .
Carry trade
The expected return and volatility of return on the carry trade strategy with exposure
xt = it − i∗t − Et ∆et+1 = ψt
and return
rtC = xt it − i∗t − ∆et+1
is given by
r̄C ≡ ErtC = E ψt · Et {it − i∗t − ∆et+1 }
= Eψt2 = σψ2 ,
2
var(rtC ) = E ψt · ψt − (∆et+1 − Et ∆et+1 )
− (r̄C )2
2
2
= Eψt4 + Eψt2 · Et ∆et+1 − Et ∆et+1 − Eψt2 .
Assuming normality of the ψt innovations, we have Eψt4 = 3 Eψt2
2
= 3σψ4 . Therefore, we can write
β/d1
1 − βρ
the Sharpe ratio of the carry trade as:
σψ2
r̄C
SR =
=
=
2 )1/2
(2σψ4 + σψ2 σ∆e
std(rtC )
σ2
2 + ∆e
σψ2
!−1/2
=
2+
2
!−1/2
(1 − ρ2 )
where we used σψ2 = σε2 /(1 − ρ2 ) and
2
β/d1
β/d1
2
σ∆e
= var ∆et+1 − Et ∆et+1 = var 1−βρ
εt = 1−βρ
σε2 .
55
−−−→ 0,
βρ→1
A.5
Integrated Productivity Shocks
Exchange rates and prices
st = (1 − 2α)qt ,
pHt − pt = γ(pHt − pF t ) = −γ(1 − α)qt ,
p∗F t − p∗t = γ(p∗F t − p∗Ht ) = γ(1 − α)qt ,
1
1
p t − wt = −
at + γ
qt ,
1−φ
1−φ
1
1
a∗t − γ
qt ,
p∗t − wt∗ = −
1−φ
1−φ
1
qt − et = 2
ãt .
1−φ
Exchange rate and quantities
1+ν
ν+φ
ãt − γ
qt ,
1−φ
1−φ
1
ỹt − ς c̃t = γ (2θ(1 − α) − (1 − ζ)) qt .
ζ
ỹt + νσc̃t =
implies
1+ν
1
ν+φ
ãt − γ
2θ(1 − α) + ζ
− (1 − ζ) qt ,
1−φ
ζ
1−φ
h
i
1 − 2α
nxt = −2γ ζς c̃t + (1 − ζ)ỹt + 2γ θ(1 − α) −
qt ,
2
ς
1+ν
1 − 2α
= −2γ
ãt + 2γ θ(1 − α) −
qt .
νσ + ς 1 − φ
2
(νσ + ς)c̃t =
Exchange rate and interest rates
Et ∆qt+1 = Et 2σ∆c̃t+1 ,
it − i∗t = Et ∆et+1 .
First equation implies in the limit:
Et ∆qt+1 = σκEt ∆ãt+1 ,
β b̂t+1 − b̂t = ϑqt − ςκãt ,
∆ãt+1 = ρ∆ãt + εt+1 ,
where b̄t ≡ b∗t /γ and:
ϑ ≡ (2θ(1 − α) − (1 − 2α))
and
56
κ≡
2 1+ν
.
νσ + ς 1 − φ
Solving with xt ≡ (qt b̂t )0 :
1
Et xt+1 =
0
!
xt + κ
ϑ/β 1/β
σρ
0
0
−ς/β
!
Ƌt
!
ãt
,
and the left eigenvector associated with eigenvalue 1/β is v = ϑ (1 − β) . Then zt ≡ vxt satisfies:
ϑqt + (1 − β)b̂t = zt = βEt zt+1 + κ [ς(1 − β)ãt − ϑσβρ∆ãt ]
=κ
∞
X
β j [ς(1 − β)Et ãt+j − ϑσβρEt ∆ãt+j ]
j=0
= κ ςãt −
βρ(ϑσ − ς)
Ƌt ,
1 − βρ
where we used the fact that:
Et ãt+j = ãt +
j
X
Et ∆ãt+k = ãt + ρ
j−1
X
ρk ∆ãt = ãt + ρ
k=0
k=1
1 − ρj
Ƌt .
1−ρ
Therefore, we have:57
βρ
Ƌt ,
1 − βρ
ρ(ϑσ − ς)
ϑσβρ − ς κ
∆ãt+1 −
Ƌt .
=−
1 − βρ ϑ
ϑσβρ − ς
β∆b̂t+1 = −κ(ϑσ − ς)
∆qt+1
The volatility of exchange rate in this case:
"
var(∆qt+1 ) = (σκ)2 ρ2 + (1 − ρ2 )
βρ − ς/(σϑ)
1 − βρ
2 #
σa2
.
1 − ρ2
Imagine the limit ρ → 1 and σa2 → 0 such that σa2 /(1 − ρ2 ) stays finite. Then var(∆qt+1 ) stays
separated from zero, while the conditional variance vart (∆ct+1 ) goes to zero (under the joint limit
γ → 0, ρ → 1), and the correlations between RER and other variables are exactly as in the case of ψt
shocks (only indirect effect of shocks on variables through qt ).
57
For comparison, if at = ρat−1 + εt , then:
β∆b̂t+1 = κ(ϑσ − ς)
β(1 − ρ)
ãt ,
1 − βρ
∆qt+1 = β∆b̂t+2 − ∆b̂t+1 + ςκ∆ãt+1
ϑσ(1 − ρ) + ςρ(1 − β)
κ ϑσβ(1 − ρ) + ς(1 − β)
=
ãt+1 −
ãt
ϑ
1 − βρ
ϑσβ(1 − ρ) + ς(1 − β)
and the two models converge to the same limit as ρ → 0 in the first case and ρ → 1 in the second case.
57
A.6
Monetary model with nominal rigidities
We outline the details of the monetary model, adopting a general enough setup to nest several extensions as special cases. In particular, we allow for both nominal wage and price rigidities. As before, we
focus on Home and symmetric relationships hold in Foreign.
Households
Consider a standard New Keynesian two country model in a cashless limit. The problem
of Home household is
1−σ
Z
∞
X
Ct
−1
κ
1+1/ν
max E β t
−
Lit
di
1−σ
1 + 1/ν
t=0
∗ E
Bt+1 Bt+1
t
+ ψt ∗ ≤ Bt + Bt∗ Et + Wt Lt + Πt + Tt
Rt
e Rt
Wit −
Lit =
Lt ,
Wt
R −1 /(−1)
, and the associate wage index is
where the last relation is demand for labor, Lt =
Lit di
R
1/(1−)
Wt =
Wit1− di
. Taking the first order conditions for asset holdings, log-linearizing them
s.t. Pt Ct +
and expressing in terms of x̃t =
1
2
(xt − x∗t ), we obtain the New Keynesian IS curve and the UIP
condition:
Et [σ4c̃t+1 + 4p̃t+1 ] = ĩt
(A56)
Et 4ẽt+1 = 2ĩt − ψt
(A57)
We also assume that the household sets wages a la Calvo and supplies as much labor as demanded
at a given wage rate. The probability of changing wage in the next period is 1 − λw . Then the first
order condition for wage setting is
E0
∞
X
1/ν
(βλw )
t=0
t
L
− 1 Ct−σ
κ it −
Wt
Pt
!
Lit = 0.
Log-linearizing, we obtain:
ŵit =
1 − βλw
1 + /ν
1
σct + `t + pt + wt
ν
ν
+ βλw Et ŵit+1 .
Note that the wage inflation can be expressed as πtw ≡ ∆wt = (1 − λw ) (ŵit − wt−1 ). Also, notice
that the price index can be written as p̃t = (1 − γ) p̃Ht − γ p̃∗Ht . Aggregate wages using these equalities
and express the wage process in terms of cross-country differences to obtain the NKPC for wages:
1˜
∗
− βEt w̃t+1 = w̃t−1 + kw σc̃t + `t + (1 − γ) p̃Ht − γ p̃Ht − [1 + β + kw ] w̃t ,
(A58)
ν
where kw =
(1−βλw )(1−λw )
λw (1+/ν) .
58
Firms
Assume that firms set prices a la Calvo with probability of changing price next period equal
1 − λp . There are two Phillips curves, one for domestic sales p̃Ht and one for export p̃∗Ht . The first order
conditions for prices in log-linearized form are
p̂Hit = (1 − βλp ) Et
∞
X
(βλp )j−t [(1 − α) (−aj + (1 − φ) wj + φpj ) + αpj ]
j=t
p̂∗Hit
= (1 − βλp ) Et
∞
X
(βλp )j−t (1 − α) (−aj + (1 − φ) wj + φpj − ej ) + αp∗j .
j=t
Note these equations do not depend on whether firms use LCP or PCP. The law of motion for home
prices and the resulting NKPC are then
πHt = (1 − λp ) (p̂Hit − pHt−1 ) =
1 − λp
(p̂Hit − pHt )
λp
−βEt p̃Ht+1 = p̃Ht−1 + kp (1 − α) [−ãt + (1 − φ) w̃t ] −
−kp γ [1 − (1 − α) (1 − γ)] p̃∗Ht − [1 + β + kp (γ + (1 − α) (1 − γ) (1 − φ))] p̃Ht
where kp =
(1−βλp )(1−λp )
.
λp
,
(A59)
On the other hand, the law of motion for export prices depends on currency
of invoicing. Assuming LCP one obtains
1 − λp ∗
∗
πHt
= (1 − λp ) p̂∗Hit − p∗Ht−1 =
(p̂Hit − p∗Ht )
λp
−βEt p̃∗Ht+1 = p̃∗Ht−1 + kp (1 − α) [−ãt + (1 − φ) w̃t − et ] −
−kp (1 − γ) [α − (1 − α) φ] p̃Ht − [1 + β + kp (1 − αγ + γφ (1 − α))] p̃∗Ht ,
(A60)
In case of PCP the law of motion of price index and NKPC are
1 − λp ∗
∗
πHt
= (1 − λp ) p̂∗Hit − p∗Ht−1 + λp 4et =
(p̂Hit − p∗Ht ) + 4et
λp
−βEt p̃∗Ht+1 + et+1 = p̃∗Ht−1 + et−1 + kp (1 − α) [−ãt + (1 − φ) w̃t ] − [1 + β + kp (1 − α)] et
−kp (1 − γ) [α − (1 − α) φ] p̃Ht − [1 + β + kp (1 − αγ + γφ (1 − α))] p̃∗Ht
Government policy and shocks
We assume that Central Bank conducts active monetary policy, while
the government chooses the fiscal policy (taxes) passively to balance the budget. The monetary policy
is represented by a standard Taylor rule:
˜ + m ,
ĩt = ρm ĩt−1 + (1 − ρm ) δπ [(1 − γ) 4p̃Ht − γ4p̃∗Ht ] + (1 − ρm ) δy gdp
t
t
(A61)
˜ = c̃ + nx. We allow persistence of the interest rate to be different from the autocorrelation
where gdp
of other shocks:
ψt = ρψt−1 + ψ
t
(A62)
ξ˜t = ρξ˜t−1 + ξt
(A63)
59
ãt = ρãt−1 + at
Market clearing
(A64)
The last dynamic equation is the country’s budget constraint:
∗
βbt+1 = bt + 2 (p̃∗Ht + ỹHt
) + et ,
(A65)
where bt is the net foreign asset position of the Home country. The static part of the model is represented by goods market equilibrium and labor demand condition:
`˜t = ỹt − ãt + φ ((1 − γ) p̃Ht − γ p̃∗Ht − w̃t )
(A66)
ỹHt = −γ ξ˜t − θγ (p̃Ht + p̃∗Ht ) + (1 − φ) c̃t + φ ((1 − φ) (w̃t − (1 − γ) p̃Ht + γ p̃∗Ht ) − ãt + ỹt ) (A67)
∗
ỹHt
= −ỹHt − ξ˜t − θ (p̃Ht + p̃∗Ht )
(A68)
∗
ỹt = (1 − γ) ỹHt + γ ỹHt
(A69)
The system (1)-(14) fully describes the equilibrium of the model.
Quantitative results
Table A1 shows the results for alternative monetary models. Columns 1 and 2
reproduce the results from Table 3 for a monetary model with a single ψ-shock and m-shock respectively. Column 3 reproduces results from Table 4 for the monetary model with three shocks (ψ, ξ, m).
In the remaining columns we keep the mix of the shocks the same (ψ, ξ, m) and consider alternative
specifications:
• Columns 4-5: a model with Peg (Taylor rule (50)), with ψ-shocks shut down in column 5, as
discussed in the main text.
• Column 6: More general Taylor rule from Clarida, Gali and Gertler (2000):
rt = ρr rt−1 + (1 − ρr ) (δπ Et πt+1 + δy gdpt ) + m
t ,
where ρr = 0.8, δπ = 2.15, δy = 0.23 and the policy depends on expected inflation.
• Column 7: A model with sticky nominal wages instead of sticky prices.
• Column 6: A monetary model with PCP and flexible wages.
60
Table A1: Alternative specifications of the monetary model
std(4q)
std(4e)
std(4s)
std(4e)
∗
std(4c−4c )
std(4q)
std(4nx)
std(4q)
∗
std(4gdp−4gdp )
std(4q)
cor (4e, 4q)
cor (4e, 4s)
cor (4c − 4c∗ , 4q)
cor (4nx, 4q)
cor (4gdp − 4gdp∗ , 4q)
Fama coefficient
R2 in Fama regression
std(r−r ∗ )
std(4e)
Sharpe Ratio
∗
acor (r − r )
acor (4e)
acor (q)
(1)
1.00
(2)
0.94
(3)
1.00
(4)
2.12
(5)
1.55
(6)
0.99
(7)
0.77
(8)
0.86
(0.00)
(0.00)
(0.00)
(0.22)
(0.16)
(0.01)
(0.01)
(0.01)
0.80
0.91
0.79
1.15
1.12
0.78
0.23
0.85
(0.01)
(0.00)
(0.01)
(0.09)
(0.06)
(0.02)
(0.00)
(0.01)
0.19
0.50
0.18
1.76
0.98
0.26
0.52
0.36
(0.00)
(0.00)
(0.02)
(0.17)
(0.10)
(0.02)
(0.04)
(0.03)
0.20
0.00
0.28
1.00
1.14
0.25
0.38
0.36
(0.00)
(0.00)
(0.03)
(0.12)
(0.10)
(0.02)
(0.03)
(0.03)
0.21
0.51
0.45
2.40
3.20
0.27
0.69
0.61
(0.00)
(0.00)
(0.05)
(0.25)
(0.31)
(0.03)
(0.06)
(0.06)
1.00
0.99
1.00
0.63
0.61
0.99
1.00
1.00
(0.00)
(0.00)
(0.00)
(0.06)
(0.07)
(0.01)
(0.00)
(0.00)
−0.93
−0.99
−0.92
−0.53
−0.82
−0.91
1.00
0.97
(0.02)
(0.00)
(0.02)
(0.07)
(0.04)
(0.02)
(0.00)
(0.01)
−0.99
1.00
−0.42
−0.62
−0.03
−0.39
−0.68
−0.58
(0.00)
(0.00)
(0.08)
(0.05)
(0.07)
(0.08)
(0.05)
(0.06)
1.00
0.98
−0.02
0.13
−0.41
0.14
0.30
0.14
(0.00)
(0.00)
(0.09)
(0.09)
(0.05)
(0.09)
(0.09)
(0.09)
1.00
1.00
−0.20
−0.35
−0.30
−0.12
−0.19
−0.19
(0.00)
(0.00)
(0.09)
(0.06)
(0.06)
(0.09)
(0.09)
(0.09)
−2.0
1.1
−0.1
0.0
1.1
0.1
0.1
0.0
(1.7)
(0.3)
(0.9)
(0.1)
(0.2)
(1.3)
(0.5)
(0.6)
0.02
0.10
0.00
0.00
0.16
0.01
0.00
0.00
(0.02)
(0.04)
(0.01)
(0.01)
(0.04)
(0.02)
(0.01)
(0.01)
0.08
0.29
0.09
0.88
0.38
0.08
0.18
0.15
(0.03)
(0.03)
(0.03)
(0.28)
(0.04)
(0.03)
(0.03)
(0.03)
0.23
-
0.19
0.71
(0.06)
(0.15)
-
(0.04)
0.21
0.23
0.21
(0.06)
(0.06)
(0.06)
0.98
0.65
0.88
0.93
0.66
0.96
0.80
0.81
(0.01)
(0.07)
(0.06)
(0.04)
(0.07)
(0.03)
(0.07)
(0.07)
−0.04
−0.15
−0.03
−0.19
−0.15
−0.01
0.03
0.00
(0.09)
(0.08)
(0.09)
(0.08)
(0.08)
(0.09)
(0.09)
(0.09)
0.92
0.65
0.93
0.98
0.98
0.92
0.95
0.94
(0.05)
(0.07)
(0.04)
(0.01)
(0.01)
(0.04)
(0.03)
(0.03)
std(4Π/GDP )
std(4q)
0.29
0.46
0.37
5.54
3.84
0.34
0.09
0.32
(0.00)
(0.00)
(0.02)
(0.45)
(0.37)
(0.01)
(0.00)
(0.03)
RER decomposition:
ψ-shocks
m-shocks
ξ-shocks
100%
—
—
—
100%
—
64%
3%
33%
59%
8%
33%
—
38%
62%
63%
1%
36%
71%
12%
17%
62%
10%
28%
61
A.7
A Model of the financial sector
Consider a financial sector in which three types of agents trade assets:
1. Home and foreign households trade their local-currency bonds, holding respectively NFA posi∗
tions Bt+1 and Bt+1
from t to t + 1.
2. N Noise traders that take a zero-capital position long Nt∗ > 0 in foreign currency and short
Nt∗
Rt∗ Et Rt
in home currency, and vice versa when Nt∗ < 0. We assume that the size of their
position:
Nt∗ = N eψ̃t − 1 ,
where ψ̃t is the noise-trader demand shock for foreign currency, which follows an exogenous
stochastic process.
3. M arbitrageurs that take a zero-capital position Dt∗ in foreign currency and short
home currency, and vice versa when
Dt∗
Dt∗
Rt∗ Et Rt
in
< 0. The arbitrageurs maximize the CARA utility of
profits:
1
max
− Et exp
Dt∗
γ
∗
Et
Dt
Rt∗ − Rt
.
Et+1 Rt∗
Note that arbitrageurs maximize nominal one-period profits in foreign currency.58 Each arbitrage
has risk aversion γ, so the effective risk-aversion in the market is given by γ/M .
The profits and losses of arbitrageurs are transferred to the foreign households, and hence the budget
constraints of the home and foreign households take the following form:
Bt+1
− Bt = N Xt ,
Rt
∗
∗
∗
Bt+1
+ Nt−1
Et−1 Dt−1
∗
∗
∗
−
B
=
N
X
+
R
−
R
.
t−1
t
t
t−1
∗
Rt∗
Et
Rt−1
The market clearing conditions in the bond market are given by:
∗
∗
∗
Bt+1
+ Dt+1
+ Nt+1
= 0,
Bt+1 +
∗
∗
Dt+1
+ Nt+1
Et Rt = 0.
Rt∗
Combining these two market clearing conditions, we can check that the home budget constraint implies
the foreign budget constraint (after noting that N Xt = −N Xt∗ Et ), and therefore we can drop one of
them from the equilibrium system (by Walras law).
We use the following approximation:
Rt∗ − Rt
Et
≈ i∗t − it + ∆et+1 ,
Et+1
and guess that the innovation of the return (equal to the innovation of ∆et+1 ) is normal. Then the
58
This assumption can be motivated by OLG of arbitrageurs hired by households with an agency problem.
62
optimal portfolio of arbitrageurs is given by:
Dt∗
i∗t − it + Et ∆et+1
=
.
2
Rt∗
γ/M · σ∆e
Combining this with the other equations, we end up with two dynamic equilibrium conditions:
Bt+1
− B t = N Xt ,
Rt
Bt+1
i∗ − it + Et ∆et+1
N ψ̃t
= t
+
e
−
1
.
2
Et Rt
Rt∗
γ/M · σ∆e
We consider a steady state with ψ̃ = 0, E = 1, B = B ∗ = N X = 0 and R = R∗ = 1/β, and
linearization around this steady state results in:
βbt+1 − bt = nxt ,
GDP · bt+1 = −
βM
∗
2 (it − it − Et ∆et+1 ) + N ψ̃t ,
γσ∆e
where GDP is steady state GDP used as normalization for net exports and NFA. We rewrite:
it − i∗t − Et ∆et+1 = δψt − χbt+1 .
where:
δ≡
N
,
M
ψt ≡
2
γσ∆e
ψ̃t
β
and
χ≡
2 GDP
γσ∆e
.
βM
Our baseline model corresponds to the limit N = M → ∞, while the case with finite N and M
generalizes our model, and the exchange rate process in this case in equilibrium follows a stationary
ARMA(2,1) instead of ARIMA(1,1,1).
63
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