ISSN (Print) 0473-453X

Discussion Paper No. 1121

ISSN (Online) 2435-0982

THE AGGREGATE REAL EXCHANGE
RATE
Mario J. Crucini
Mototsugu Shintani
Takayuki Tsuruga

March 2021

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanA Behavioral Explanation for the Puzzling Persistence of
the Aggregate Real Exchange Rate∗
Mario J. Crucini†, Mototsugu Shintani‡and Takayuki Tsuruga§
This version: March 2021

Abstract
At the aggregate level, the evidence that deviations from purchasing power parity(PPP) are too persistent to be explained solely by nominal rigidities has long been apuzzle (Rogoff, 1996). Another puzzle from the micro price evidence of the law of one
price (LOP), which is the basic building block of PPP, is that LOP deviations are lesspersistent than PPP deviations. To reconcile the empirical evidence, we adapt the modelof behavioral inattention in Gabaix (2014, 2020) to a simple two-country sticky-pricemodel. We propose a simple test of behavioral inattention and find strong evidence inits favor using micro price data from US and Canadian cities. Calibrating behavioral
inattention with our estimates, we show that our model reconciles the two puzzles relating to the PPP and LOP. First, the PPP deviations are more than twice as persistent as
PPP deviations explained only by sticky prices. Second, the LOP deviations decrease
to less than two-thirds of the PPP deviations in the degree of persistence.
JEL Classification: E31; F31; D40
Keywords: Real exchange rates, Law of one price, Rational inattention, Trade cost∗

We thank David Cook, Mick Devereux, Ippei Fujiwara, Yasuo Hirose, Jinill Kim, Akihisa Shibata, Kwanho
Shin and conference and seminar participants at the 5th International Macroeconomics and Finance Conference, the North American Winter Meeting of the Econometric Society, Keio University, Otaru University ofCommerce, Renmin University of China, Tokyo Institute of Technology, The University of Tokyo, Vanderbilt
University, and Yokohama City University for their helpful comments and suggestions. Mototsugu Shintani
and Takayuki Tsuruga gratefully acknowledge the financial support of Grant-in-aid for Scientific Research(15H05728, 15H05729, 18K01684, 20H01482, 20H05631, and 20H05633), the Joint Usage/Research Center at
the Institute of Social and Economic Research (Osaka University), the Kyoto Institute of Economic Research(Kyoto University), and the International Joint Research Promotion Program in Osaka University.
†
Department of Economics, Vanderbilt University and NBER; e-mail: mario.crucini@vanderbilt.edu.
‡
Faculty of Economics, The University of Tokyo; e-mail: shintani@e.u-tokyo.ac.jp.
§
Institute of Social and Economic Research, Osaka University;
and CAMA; e-mail:tsuruga@iser.osaka-u.ac.jp.

11Introduction

It is well known that the aggregate real exchange rate, namely the deviation from purchasing
power parity (PPP), is highly persistent. In a description of this empirical anomaly known
as the PPP puzzle, Rogoff (1996) states “Consensus estimates for the rate at which PPP
deviations damp, however, suggest a half-life of three to five years, seemingly far too long
to be explained by nominal rigidities” (p. 648). A closely related empirical fact of realexchange rates is the gap in persistence between the PPP deviations and the deviations fromthe law of one price (LOP), as the basic building block for PPP. For example, Imbs et al.
(2005) and Carvalho and Nechio (2011) argue that the aggregate real exchange rate (the PPP
deviations) is likely to be much more persistent than the good-level real exchange rate (theLOP deviations).1 These previous studies emphasize the role of heterogeneity in the speedof price adjustment. As Imbs et al. (2005) argued, “It is this heterogeneity that we find to
be an important determinant of the observed real exchange rate persistence since it gives rise
to highly persistent aggregate series while relative price persistence is low on average at adisaggregated level” (p. 3).
In this paper, we aim to explain two empirical anomalies simultaneously: (1) the gap
between the observed persistence of the PPP deviations and the persistence predicted fromthe sticky-price model (e.g., Rogoff, 1996); and (2) the gap between the observed persistenceof the PPP deviations and the LOP deviations (e.g., Imbs et al., 2005). To this end, we
incorporate behavioral inattention along the lines of Gabaix (2014, 2020) into a simple twocountry sticky-price model. In this framework, firm managers bear the cost of paying attention
to the aggregate component of the marginal cost of their products. As a result, the full
attention to the state of the economy is no longer optimal when firms choose the prices ofgoods.
The key to solving the PPP puzzle is the complementarity between the LOP and PPP
deviations. After deriving a reduced-form solution for the LOP deviations, we show that the
LOP deviations are affected by the PPP deviations when firms pay only partial attention
to marginal cost. Thus, an increase in the persistence of the PPP deviations makes the
LOP deviations more persistent. At the same time, through the aggregation, the more
persistent LOP deviations lead to more persistent PPP deviations, further strengthening thelink between the PPP and LOP deviations.
1

See Crucini and Shintani (2008) for a comprehensive empirical analysis of the persistence in the LOP
deviations.

2The reduced-form solution leads to a direct testable implication. Using US and Canadianmicro price data, we implement a simple test for the null hypothesis of full attention against
an alternative hypothesis of partial attention. This test is equivalent to asking whether the
good-level real exchange rate is uncorrelated with the aggregate real exchange rate after
controlling for the common driving forces, such as the nominal exchange rate and countryspecific productivity. Using various specifications, we strongly reject the null in favor of our
proposed model of behavioral inattention. We also find the estimated degree of attention to
be around 0.17, much less than the value of 1.0 under full attention.
In the first theoretical result, our model of behavioral inattention ensures that the persistence in the aggregate real exchange rate exceeds the persistence implied solely by nominal
rigidities. While the setting differs greatly, this mechanism is consistent with Ball and Romer(1990) and Woodford (2003), who show that even small frictions in nominal price adjustment
lead to a persistent output gap when real rigidities or strategic complementarities are present.
In our model with behavioral inattention, only small nominal frictions are needed to generatea highly persistent aggregate real exchange rate. Based on our estimates of the degree ofattention, the persistence of the aggregate real exchange rate rises to 0.74 from 0.34 in thefull attention case.
In the second theoretical result of our model, we explain the gap between the highly
persistent PPP deviations and the less persistent LOP deviations. This gap arises from the
combination of the complementarity and the presence of idiosyncratic real shocks to theindividual price of goods. We show that both the aggregate and the good-level real exchange
rates are more persistent when complementarities are present. In contrast, real shocks at
the good level (but not country-specific real shocks) reduce persistence only for the LOP
deviations and not for the PPP deviations. This is because the aggregation across goods
eliminates the effect of real shocks at the good level. As a result, our estimates of the degreeof attention also imply a substantial gap in persistence between the PPP and LOP deviations.
In fact, we predict the persistence of the LOP deviations to be 57 percent lower than thepersistence of the PPP deviations.
Our results relate to an extensive literature that has contributed to a better understandingof persistent aggregate real exchange rates. For instance, Chari et al. (2002) argue that while
the sticky-price model can explain the volatility of the aggregate real exchange rate, thepersistence is less than that suggested by the data. Benigno (2004) emphasizes the role ofmonetary policy rules rather than the degree of price stickiness in accounting for the persistent
aggregate real exchange rate. Later, Engel (2019) revisits Benigno (2004) and argues for the3

importance of both monetary policy rules and price stickiness. Blanco and Cravino (2020)
introduce a concept of the real exchange rate using only newly reset prices and find that this“reset” exchange rate explains almost all fluctuations in the aggregate real exchange rate. Our
solution also closely relates to Bergin and Feenstra (2001) and Kehoe and Midrigan (2007),
who introduce strategic complementarity in pricing to two-country sticky-price models in
explaining the high persistence of real exchange rates. Our explanation of the PPP puzzle
does not rule out these explanations. Instead, we focus on the importance of behavioral
inattention in pricing of firms.
A popular explanation for the gap in persistence between aggregate (CPI-based) andgood-level real exchange rates is the heterogeneity in price adjustment at a good level, which
generates a positive bias when aggregating prices. Imbs et al. (2005) point out a positive
aggregation bias in dynamic heterogeneous panels, and Carvalho and Nechio (2011) consider
the theoretical implication of aggregation based on the sticky-price model in which the degreeof price stickiness differs across sectors. Indeed, both statistical aggregation bias and multisector sticky-price models would help toward increasing the persistence of real exchange rates.
In contrast, our solution can explain the gap even if the persistence of the LOP deviations
is restricted to be common across goods. In this sense, the mechanism in our paper further
enhances the ability of existing workhorse models in the macroeconomics literature.
The remainder of the paper is structured as follows. In Section 2, we present a simple
two-country open economy model with Calvo pricing and introduce behavioral inattention.
Section 3 introduces the reduced-form solution for the LOP deviations and discusses theimplications of behavioral inattention. In Section 4, we implement a test of behavioral inattention and quantify its importance. In Section 5, we assess how much the estimated degreeof behavioral inattention can improve model predictions. Section 6 concludes.

2

The model

The economy consists of two countries. In what follows, the US and Canada represent thehome and foreign countries, respectively. Following Kehoe and Midrigan (2007) and Crucini etal. (2010b, 2013), there is a continuum of goods and brands of each good. Goods are indexedby i and brands are indexed by z. For each good, US brands are indexed by z ∈ [0, 1/2] andCanadian brands are indexed by z ∈ (1/2, 1].
We assume that US and Canadian consumers have identical preferences over brands ofa particular good and also across goods in the aggregate consumption basket. US prefer4

ences over the brands of good i are given by the constant elasticity-of-substitution (CES)
ε
hR
i ε−1
ε−1
1
ε
index for good i ∈ [0, 1]. The US consumption of good i is cit = z=0 cit (z) dz
ε
hR
i ε−1
ε−1
1
and the aggregation across goods gives aggregate consumption ct = i=0 cit ε di
, where
ε
hR
i
ε−1
ε−1
1
∗
ε dz
ε > 1. For Canada, we have the analogous equations c∗it =
c
(z)
and
z=0 it
ε
hR
i
ε−1
ε−1
1
c∗t = i=0 c∗it ε di.

2.1

Households

The objective of the US agent is to maximize E0P∞t
t=0 δ U (ct , nt ) = E0P∞t
t=0 δ (ln ct − χnt ),subject to the budget constraint given by:
Mt + Et (∆t,t+1 Bt+1 ) = Wt nt + Bt + Mt−1 − Pt−1 ct−1 + Tt + Πt ,(1)

and a cash-in-advance (CIA) constraint, Mt ≥ Pt ct . Here, E0 (·) denotes the expectation
operator conditional on the information available in period 0, δ ∈ (0, 1), and χ > 0. In
addition, we suppress the state contingencies for notational convenience. On the right-handside of the budget constraint (1), the household supplies hours worked nt , receives nominal
wages Wt for hours worked, carries bonds Bt into period t, as well as any cash that remainedin period t − 1, Mt−1 − Pt−1 ct−1 . The household also receives nominal transfers from the US
government, Tt , and nominal profits from US firms, Πt . In (1), the aggregate price Pt is given1R 1−ε  1−ε
Pit diby Pt =
, where Pit is the price index for good i. This is a CES aggregate over
 1
R
Pit (z)1−ε dz 1−ε . The left-hand side of (1) represents the nominal value ofbrands: Pit =the total wealth of the household. The household allocates the wealth into state-contingent
nominal bond holdings Bt+1 brought into period t + 1 and money balances Mt . The CIA
constraint requires nominal money balances for expenditure which is made at the end of theperiod t. The CIA always binds with equality in equilibrium.
Canadian households solve the analogous maximization problem. We assume complete
markets for state-contingent financial claims across the US and Canada and the financial
claims are denominated in US dollars. Thus, we convert their US dollar bond holdings into
Canadian dollars at the spot nominal exchange rate, St . The Canadian households are subject
to budget constraint,Mt∗ +∗Et (∆t,t+1 Bt+1)
B∗∗∗c∗t−1 + Tt∗ + Π∗t .
= Wt∗ n∗t + t + Mt−1
− Pt−1
St
St

5

(2)

and the CIA constraint, Mt∗ ≥ Pt∗ c∗t .
The first-order conditions are standard. For the US households, we have Wt /Pt = χctand ∆t,t+1 = δ[(ct+1 /ct )−1 (Pt /Pt+1 )], where ∆t,t+1 is the nominal stochastic discount factor.
∗
For Canadian households, we have Wt∗ /Pt∗ = χc∗t and ∆t,t+1 = δ[(c∗t+1 /c∗t )−1 St Pt∗ /(St+1 Pt+1)].

The consumption Euler equations differ because Canadians buy state-contingent bonds denominated in US dollars.
Define the aggregate real exchange rate as qt = St Pt∗ /Pt . The Euler equations imply
qt+1 (c∗t+1 /ct+1 ) = qt (c∗t /ct ) = ... = q0 (c∗0 /c0 ). Normalizing q0 (c∗0 /c0 ) to unity yields2

qt =

2.2

ct
c∗t


.

(3)

Firms

The US firm specializes in the production of brand z ∈ [0, 1/2] of good i and employs nit (z)hours of labor. They will produce output according to the production function yit (z) =ait nit (z), where ait is labor productivity specific to good i and to all US firms that producethat good. US firms produce brands on the first-half of the unit continuum, z ∈ [0, 1/2].
Thus, for example, all firms that produce beer in the US share the same labor productivity.
The production function of Canadian firms is yit∗ (z) = a∗it n∗it (z), for z ∈ (1/2, 1].
Goods that are shipped between the US and Canada are subject to iceberg trade costs,
τ . In addition, all goods are perishable. Thus, production of good i undertaken in the US is
exhausted between US and Canadian consumption, with Canadian consumption bearing theiceberg trade cost:cit (z) + (1 + τ )c∗it (z) = yit (z), for z ∈ [0, 1/2] .

(4)

Similarly, production of good i undertaken in Canada is exhausted between Canadian and
US consumption, with US consumption bearing the iceberg trade cost:(1 + τ )cit (z) + c∗it (z) = yit∗ (z), for z ∈ (1/2, 1].

2.3

(5)

Price setting

We introduce the inattention of firms into an otherwise standard two-country model with
Calvo pricing. Firms have the opportunity to change their prices with a constant probability,
2

The condition relies on our preference assumptions. Later, we relax these assumptions in Section 4.

6

as in Calvo (1983) and Yun (1996). Firms set prices in the buyers’ currency, referred to in theliterature as local currency pricing. We first present the pricing decision of fully attentive firms
and then relax this assumption following the approach in Gabaix (2014, 2020). Because thepricing problem of Canadian firms is analogous, we limit our exposition to pricing decisions
made by US firms.
2.3.1

Fully attentive firms

We first specify the fully attentive firm’s pricing decision. Let xt be a generic variable. We
define the log deviation of xt from the steady-state level as x̂t = ln xt − ln x̄, where x̄ isthe steady-state level of xt , so that we express xt = x̄ exp(x̂t ). Using this expression, we
write the US firm’s real profits of selling goods in the US market as [pit (z) − wt /ait ]cit (z) ={p̄i (z) exp[p̂it (z)]− w̄ exp(ŵt −âit )}cit (z), where pit (z) = Pit (z)/Pt is the relative price of brandz of good i and wt = Wt /Pt is the real wage. The demand by US consumers for a particularbrand of good i, cit (z) = (Pit (z)/Pit )−ε cit . In terms of the log deviation, this equation is
written as cit (z) = (p̄i (z)/p̄i )−ε {−ε[exp(p̂it (z)) − exp(p̂it )]}cit , where pit = Pit /Pt .
We assume that the firms cannot change their price with a probability λ. This parameter
captures the degree of price stickiness. Along with the assumption that steady-state inflationis zero, a fully attentive US firm chooses p̂it (z) to maximize the objective function:vit (z) = Et∞
X

λk δt,t+k

k=0

Pt
×
Pt+k

(
p̄i (z) exp [p̂it (z)] − w̄ exp ŵt+k +(6)!)πt+l − âit+k

cit,t+k (z),

l=1

where

cit,t+k (z) =kXp̄i (z)
p̄i

−ε

(

"

exp −ε p̂it (z) −kX#)πt+l − p̂it+k

cit+k

(7)

l=1

is the demand for brand z of good i in period t + k, conditional on the firm having last reset
the price in period t.3 Here, vit (z) is the present discount value of real profits accruing to thefirm producing brand z of good i in the US, conditional on the firm having last reset its pricein period t. In (6), the second line represents the real profits in each period. The marginal
cost is the real wage divided by the labor productivity specific to that good. However, becausePof sticky prices, real wages are adjusted with kl=1 πt+l accumulated from periods t to t + k,
3

The derivation is provided in Appendix A.1.

7

where πt = ln(Pt /Pt−1 ) denotes inflation. Real profits in each period are discounted by thestochastic discount factor δt,t+k = δ k (ct+k /ct )−1 satisfying δt,t+k Pt /Pt+k = ∆t,t+k . In (7),relative prices are also adjusted by inflation accumulated from period t to t + k. Note that
this objective function is for the US firms indexed by z ∈ [0, 1/2].
The US firm’s real profits of selling goods in the Canadian market are analogously defined.
Let p∗it (z) be the relative price in Canadian markets given by p∗it (z) = Pit∗ /Pt∗ . A fully attentive
US firm chooses p̂∗it (z) to maximize4
vit∗ (z) = Et∞
X

λk δt,t+k qt+k

(8)

k=0

P∗
× ∗t
Pt+k
where

(

w̄
p̄∗i (z) exp [p̂∗it (z)] − (1 + τ ) exp
q̄

ŵt+k − q̂t+k +

k
X

!)
∗
πt+l
− âit+kc∗it,t+k (z),

l=1

#)
( "
 ∗ −ε
k
X
p̄
(z)
i
∗
πt+l
− p̂∗it+kc∗it+k .
c∗it,t+k (z) =exp −ε p̂∗it (z) −
p̄∗i
l=1

(9)

∗
Here, πt∗ = ln(Pt∗ /Pt−1) and p∗it (z) = Pit∗ (z)/Pt∗ . In (8), the second line represents real profits

in each period. The cost of providing a unit of the good to a Canadian consumer is higher
by the amount of the iceberg trade cost τ . The real exchange rate in the second line of theequation converts the cost in terms of Canadian goods to compare it to the relative pricep∗it (z). When discounting the US firm’s real profits in each period, qt+k in the first line of (8)converts these profits in terms of the US goods.
2.3.2

Inattentive firms

We now consider the firm’s maximization problem when a firm is less than fully attentive tothe state variables that enter into its objective function. This problem is called the “sparsemax” because Gabaix (2014) originally developed the model in which the economic agents
respond to only a limited number of variables out of numerous variables.
In our model, a firm’s marginal cost is a function of aggregate variables including the real
wage and the real exchange rate, as well as microeconomic variables, such as good-specificproductivity shocks. For simplicity, we assume that the firm is fully attentive to its own
productivity but possibly less attentive to the aggregate variables.
Let us augment (6) with the degree of attention mH ∈ [0, 1] and introduce the “attention4

The derivation is again in Appendix A.1.

8

augmented” objective function:vHi (p̂it (z), µ̂Ht , mH ) = Et∞
X

λk δt,t+k

k=0Pt{p̄i (z) exp [p̂it (z)] − w̄ exp (mH µ̂Ht+k − âit+k )} cit,t+k (z),×Pt+kwhere µ̂Ht = (µ̂Ht , µ̂Ht+1 , ...)0 and µ̂Ht+k = ŵt+k +Pk5
l=1 πt+l .

(10)

In the limit case of mH = 0,

firms fully ignore changes in the aggregate components of the firm’s cost function, µ̂Ht+k . In
the opposite limit case of mH = 1, the attention-augmented objective function reduces to (6),namely, the full attention case. Because the firm is fully attentive to its own productivity,
there is a unit coefficient on âit+k .6
In the sparse max, the inattentive firm sets its optimal price to maximize (10):p̂Hi (µ̂Ht , mH ) = arg max vHi (p̂it (z), µ̂Ht , mH ) ,(11)p̂it (z)given mH .
In Gabaix (2014), agents choose the degree of attention endogenously. More attentivenessincreases expected profits, a benefit, but being more attentive is costly. We employ thequadratic cost function,
C (mH ) =

κ 2
m ,
2 H

where κ ≥ 0. Given the cost function, the firm chooses the optimal allocation of attention bysolving
max E {vHi [p̂Hi (µ̂Ht , mH ), µ̂Ht , 1] − C (mH )} ,mH ∈[0,1](12)

where E (·) represents the unconditional expectations. In (12), we evaluate vHi (·) at p̂it (z) =p̂Hi (µ̂Ht , mH ) in the first argument and at mH = 1 in the third argument. That is, the profit
function is the true function since mH = 1 in the third argument but it is evaluated at the5

For the aggregate component
costs ofPselling goods in the other markets, the P
definitions
Pk of∗ the marginal
k
k
∗
∗
∗
∗
,
are µ̂∗Ht+k = ŵt+k − q̂t+k + l=1 πt+l
, µ̂∗F t+k = ŵt+k+ l=1 πt+l
, and µ̂F t+k = ŵt+k+ q̂t+k + l=1 πt+l
respectively.
6
Pk In the attention-augmented objective function, we do not explicitly introduce mH as a coefficient on
l=1 πt+l in (7). This is because we examine the log-linearized first-order condition for the optimal prices.
When we take the log-linearization, the presence of mH in (7) does not matter for the first-order terms.
Further, nor do we explicitly introduce “cognitive discounting” as in Gabaix (2020). Gabaix (2020) assumes
that the effect of k period-ahead economic variables on the agent’s expectations is weakened relative to therational agent’s expectations, in addition to the degree of attention. In the present model setup, however, we
can show that the presence of cognitive discounting does not matter for our results.

9inattentive firm’s action because mH in p̂Hi (µ̂Ht , mH ) is not equal to one in general.
Following Gabaix (2014), we define the sparse max for vit (z) as follows. The firm’s choices
divide into two steps. In the first step, the firm chooses the degree of attention mH based on
the linear-quadratic approximation of (12):wheremH = arg min1κ(1 − mH )2 ΛH + m2H ,mH ∈[0,1] 2
2

(13)

 2

∂ vHi [p̂Hi (0, 1) , 0, 1]
ΛH = −V ar (µ̂Ht ) .
∂ p̂2Hit(14)

The solution of the first step is given by mH = ΛH /(ΛH + κ). In the second step, the firmchooses the optimal price (11), given the solution of the first step.7
In this sparse max, the choice of mH = ΛH /(ΛH +κ) = 0 is excluded as long as V ar(µ̂Ht ) >0, which implies ΛH > 0. Gabaix (2014) showed that in the case of a quadratic cost function,
the selected degree of attention is zero if and only if there is no uncertainty in the variables
to which the economic agents pay only partial attention.8 In addition, in the special case ofκ = 0, mH = ΛH /(ΛH +κ) = 1 is selected because κ = 0 means that there is no cost of paying
attention. For these reasons, in the following analysis, we focus on the case of mH ∈ (0, 1].
As we discuss later, these assumptions are convenient for our objective of accounting for the
PPP puzzle because they ensure the stationarity of the PPP and LOP deviations.
Note that we can also introduce inattention to the idiosyncratic productivity and derive
the expressions similar to (13) and (14). Nevertheless, we focus on the case that firms are fully
attentive to their productivity but are inattentive to the aggregate component for three reasons. First, in general, the level of uncertainty matters for the size of the degree of attention.
Naturally, the volatility of the idiosyncratic shock can be much higher than the aggregate
shocks. In this case, ΛH for the idiosyncratic productivity would be much larger than ΛHfor the aggregate component of the marginal cost, meaning that the degree of attention to
the idiosyncratic variable is closer to unity than that to the aggregate variable. Second,firms may have easier access to information on their variables rather than the macroeconomic
variables. In this case, κ for their productivity may be much lower than κ for the aggregate
shock, such as monetary shocks. Thus, the degree of attention to the idiosyncratic variable is
7

In Appendix A.2, we derive (13) and (14) that are relevant to US firms selling in US markets. The
appendix also describes the remaining sparse max for US firms selling abroad and Canadian firms selling in
the Canadian and US markets.
8
Gabaix (2014) discusses the properties of the selected degree of attention using not only the quadratic
cost function but also the other functional forms of the cost function.

10

again closer to unity. Finally, our test of behavioral inattention in Section 4 can still detect
the inattention to the aggregate variable even if we allow inattention to their idiosyncratic
productivity. In other words, our test is fully robust to the presence of inattention to theiridiosyncratic productivity. In this sense, the full attention to idiosyncratic productivity is aconvenient assumption for focusing on the presence of inattention to the aggregate variable.

2.4

Equilibrium

The monetary authority in each country determines the national stock of money. Following
Kehoe and Midrigan (2007), we assume that the log of the money supply follows a randomwalk:ln Mt = ln Mt−1 + εM
t ,

(15)

M∗

(16)

∗
ln Mt∗ = ln Mt−1
+ εt ,∗

M
are zero-mean i.i.d. shocks. Importantly, the stochastic processes, comwhere εMt and εtbined with (3) and the CIA constraints, imply a nominal exchange rate that follows a random
walk, which is empirically plausible. In particular, we have St = Mt /Mt∗ from (3) and the∗Mis theCIA constraints. This equation leads to ln St = ln St−1 + εnt , where εnt = εMt − εtshock to the nominal exchange rate. We simply call εnt the nominal shock.
For simplicity, we assume that the log labor productivity also follows a zero-mean i.i.d.
process:9ln ait = εit ,(17)

ln a∗it = ε∗it .

(18)

The difference in labor productivity is ln(ait /a∗it ) = εrit , where εrit = εit − ε∗it . We refer to theshock to the difference in productivity as the real shock.
The profits of US (Canadian) firms accrue exclusively to US (Canadian) households. In
R R 1/2
R R1
other words, Πt = i z=0 Πit (z)dzdi and Π∗t = i z=1/2 Π∗it (z)dzdi, where Πit (z) and Π∗it (z)are the total nominal profits of firms producing brand z. Monetary injections are assumed
to equal nominal transfers from the government to domestic residents: Tt = Mt − Mt−1∗
for the US, and Tt∗ = Mt∗ − Mt−1
for Canada. The labor market-clearing conditions are
9

Later we consider an alternative stochastic process for productivity, but the empirical results from thetest of behavioral inattention are unaffected.

11

nt =

R R1

n (z)dzdi and n∗t =i z=0 it
2

R R1
i

z=1/2

n∗it (z)dzdi.

An equilibrium of the economy is a collection of allocations and prices such that (i) households’ allocations are solutions to their maximization problem (namely, {cit (z)}i,z , nt , Mt ,∗Bt+1 , for US households and {c∗it (z)}i,z , n∗t , Mt∗ , Bt+1
, for Canadian households); (ii) prices

and allocations of firms are solutions to their sparse max for vit (z) and vit∗ (z) where z ∈ [0, 1](namely, {Pit (z), Pit∗ (z), nit (z), yit (z)}i,z∈[0,1/2] for US firms and {Pit (z), Pit∗ (z), n∗it (z), yit∗ (z)}i,z∈(1/2,1]
for Canadian firms); (iii) all markets clear; (iv) the productivity, money supply, and transfers
satisfy the specifications discussed earlier.

3

Theoretical implications for LOP deviations

In this section, we derive the reduced-form solution to the good-level real exchange rate. Taking the first-order condition with respect to p̂it (z) from (10) and log-linearizing the condition
around the steady state yield the optimal price:∞X
p̂Hi (µ̂Ht , mH ) = (1 − λδ)Et(λδ)k (mH µ̂Ht+k − âit+k )

(19)

k=0

= mH ŵt − (1 − λδ)âit .

(20)

Here, the expression reflects the forward-looking properties in the Calvo pricing and λ affects
the extent to which firms place weights on the expected marginal cost. Derivation of thesecond equality is provided in Appendix A.3. The optimal price set by US firms for the
Canadian market is given by:p̂∗Hi (µ̂∗Ht , m∗H ) = m∗H (ŵt − q̂t ) − (1 − λδ)âit .

(21)

Similarly, the prices set by Canadian firms for the Canadian market and the US market are
respectively given by:p̂∗F i (µ̂∗F t , m∗F ) = m∗F ŵt∗ − (1 − λδ)â∗it ,(22)p̂F i (µ̂F t , mF ) = mF (ŵt∗ + q̂t ) − (1 − λδ)â∗it .

(23)

and

Turning to the price index for good i, we log-linearize the CES index for good i sold in
the US market:
p̂it = λ(p̂it−1 − πt ) + (1 − λ)p̂opt
it .
12

(24)

Here, p̂opt
it denotes the weighted average of the optimal reset prices:p̂opt
it = ω p̂Hi (µ̂Ht , mH ) + (1 − ω) p̂F i (µ̂F t , mF ) ,(25)

where ω = (1 + (1 + τ )1−ε )−1 ∈ [1/2, 1] is the degree of home bias. The home bias is strictly
larger than 1/2 in the presence of the iceberg trade costs (τ > 0). The log-linearized priceindex for good i sold in the Canadian markets isp̂∗it = λ p̂∗it−1 − πt∗ + (1 − λ) p̂opt∗
it ,

(26)

p̂opt∗= ω p̂∗F i (µ̂∗F t , mH ) + (1 − ω) p̂∗Hi (µ̂∗Ht , mF ) .
it

(27)

where

In (27), we employ the assumption of a symmetry between the US and Canada. That is, thedegrees of attention in the production of domestically consumed goods are identical in the
US and Canada, such that m∗F = mH . Likewise, the degrees of attention in the productionof exported goods are also identical, such that m∗H = mF .
Recall that the PPP deviation, or the aggregate real exchange rate, is defined by qt =St Pt∗ /Pt . Similarly, the LOP deviation, or the good-level real exchange rate, is defined by
qit = St Pit∗ /Pit . Using pit and p∗it , q̂it is expressed asq̂it = q̂t + p̂∗it − p̂it .

(28)

We combine (20) - (28), and the CIA constraints to obtain the expression for the goodlevel real exchange rate. The following proposition summarizes the dynamics of the good-level
real exchange rate.
Proposition 1 Under the preferences given by U (c, n) = ln c − χn, the CIA constraints,
the stochastic processes of money supply (15) and (16), the stochastic processes of the laborproductivity (17) and (18), and the Calvo pricing with the degree of price stickiness λ ∈ (0, 1),
the stochastic process of the good-level real exchange rate is given by:ln qit = λ ln qit−1 + (1 − m)(1 − λ) ln qt + λεnt + (1 − λ)(1 − λδ)ψεrit ,

13

(29)

where m ∈ (0, 1] represents the degree of attention:m = ωmH + (1 − ω)mF .

(30)

and ψ = 2ω − 1. The two random shocks εrit and εnt are given by εrit = εit − ε∗it ∼ i.i.d.(0, σr2 )∗M∼ i.i.d.(0, σn2 ), respectively.
and εnt = εMt − εtProof. See Appendix A.4.
This stochastic process for the good-level real exchange rate generalizes the simple stochastic process considered by Kehoe and Midrigan (2007) who emphasized the importance of
nominal shocks. They showed that under the fully attentive rational expectations model, thegood-level real exchange rate follows an autoregressive process of order one (AR(1)) driven
by the nominal shock εnt :ln qit = λ ln qit−1 + λεnt .

(31)This equation is a special case of (29) with m = 1 and ψ = 0.10 To gain some intuition
behind (31), recall that ln qit = ln St + ln Pit∗ − ln Pit . Suppose that money supply increases
unexpectedly in the US. While the unexpected increase in the domestic money supply keeps
Pit∗ constant, it increases St and Pit . Notice that the nominal exchange rate is free to adjust,while the adjustment of Pit is slow because of sticky prices. As a result, the increase in Pit
only partially offsets the increase in St . The extent of the offsetting effect depends on λ. If
λ → 0, a change in Pit perfectly offsets the increase in St , meaning that the nominal shock isirrelevant for the real exchange rate. If λ → 1, Pit never moves, meaning that the good-level
real exchange rate keeps track of the nominal exchange rate, which follows a random walk.
Let us compare the stochastic processes for the good-level real exchange rates between
the cases m = 1 and 0 < m < 1.11 For comparison purposes, we maintain the assumption ofψ = 0. If firms are only partially attentive to the aggregate component of the marginal cost(i.e., 0 < m < 1), the good-level real exchange rate has the aggregate real exchange rate on
the right-hand side:ln qit = λ ln qit−1 + (1 − m)(1 − λ) ln qt + λεnt .
10

(32)

Note that ψ = 0 if and only if τ = 0 (i.e., no trade cost) from the definition of ω = 1/(1 + (1 + τ )1−ε ).
Note that m is the mean of the degrees of attention mH and mF . Because 0 < ω < 1 holds for τ ∈ [0, ∞),m = 1 holds only if all US and Canadian firms are completely attentive to the aggregate component of their
marginal costs.
11

14

The intuition behind the appearance of the aggregate real exchange rate in (32) lies in the
responses of relative prices p̂it and p̂∗it to aggregate shocks. If firms become less attentive tothe aggregate components of the marginal cost, relative prices are more invariant to aggregate
shocks. The more invariant a relative price, the more the firm anchors its nominal prices tothe aggregate price level. The link between the good-level and the aggregate price indicesleads to a link between the good-level and the aggregate real exchange rates.
It should be noted that there is a single common driving force in both (31) and (32)
because the aggregate real exchange rate that additionally enters in (32) is also driven by thenominal shock. Indeed, aggregating ln qit over i yields12
ln qt =

λλln qt−1 +εnt .
1 − (1 − m)(1 − λ)1 − (1 − m)(1 − λ)(33)

Using (33), we can see that the impact multiplier of nominal shocks on the good-level real(1−m)(1−λ)
exchange rate increases from λ in (31) to λ × (1 + 1−(1−m)(1−λ)) in (32). In other words, thebehavioral inattention changes the stochastic process of the good-level real exchange rate but
not the source of its variations.
When ψ > 0, a real shock represented by εrit appears in the stochastic process as anadditional driving force. Stressing the importance of real shocks, Crucini et al. (2010b, 2013)
extended the Kehoe and Midrigan (2007) model to incorporate idiosyncratic productivity
shocks. Under the fully attentive rational expectations, their model implies:
ln qit = λ ln qit−1 + λεnt + (1 − λ)(1 − λδ)ψεrit .

(34)

This equation is a special case of (29) under m = 1. To understand the intuition behind therole of real shocks, again recall that ln qit = ln St + ln Pit∗ − ln Pit . Positive productivity shocks
in the US firms producing good i reduce both Pit∗ and Pit because the US firms sell theirgoods in both countries. However, the home bias generated by trade cost will decrease Pit
more than Pit∗ . This effect results in an appreciation of qit . For the case of 0 < m < 1, (33)continues to hold unless aggregate real shocks are introduced. In the process of aggregating
R1
To derive the stochastic process, we integrate (29) across good i. In aggregation, i=0 ln qit di = ln qt holds
from the definition of the good-level real exchange rate. From the definition of qit , ln qit = ln qt + ln p∗it − ln pit .
R1
R1
For the US relative prices, the integral of the relative price over i is zero because i=0 ln pit di = i=0 ln Pit di −R1
ln Pt = 0. The same result holds for the Canadian relative price so that i=0 ln p∗it di = 0. These results leadR1
to i=0 ln qit di = ln qt . The resulting equation is ln qt = λ ln qt−1 + (1 − λ)(1 − m) ln qt + λεnt . Simplifying theabove equation, we obtain (33).
1215the good-level real exchange rates, all idiosyncratic real shocks are washed out in the integral
R1over i because i=0 εrit di = 0.
To summarize, behavioral inattention generates a new term that affects the good-level realexchange rate, namely, the aggregate real exchange rate. Time-dependent pricing models of
the good-level and aggregate real exchange rates without the behavioral inattention havebeen theoretically developed and empirically assessed by Kehoe and Midrigan (2007) andCrucini et al. (2010b, 2013), among many others. However, the importance of the behavioral
inattention has not been tested in the context of LOP deviations. In the next section, we use
a rich international dataset on good-level real exchange rates to test the model of behavioral
inattention.

4

A test of behavioral inattention

4.1

Methodology

In this section, we consider testing the null hypothesis of m = 1 (full hattention), against ithealternative hypothesis of m < 1 (partial attention). Define ln q̃it = ln qit / (qit−1 St /St−1 )λ =ln qit − λ ln qit−1 − λ∆ ln St and ln q̃t = ln(qt /qtλ ) = (1 − λ) ln qt . We estimate the following
panel regression model:ln q̃it = α + β ln q̃t + γ 0 Xit + uit ,(35)

where α, β, and γ are regression coefficients, Xit is a vector of the control variables, and uit
is the error term. This regression is motivated by (29) because it can be written asln q̃it = (1 − m) ln q̃t + (1 − λ)(1 − λδ)ψεrit ,(36)

where the nominal shock εnt in (29) is replaced by ∆ ln St because the (log) nominal exchange
rate follows a random walk with an increment εnt . To implement the regression, we constructln q̃it and ln q̃t using the micro evidence of λ obtained by Nakamura and Steinsson (2008).13
The error term uit represents an i.i.d. real shock εrit and is uncorrelated with the regressorln q̃t = (1 − λ) ln qt because εrit does not show up in (33). Therefore, we estimate (35) using
ordinary least squares (OLS). The control variables Xit here may include time-invariant fixed
effects or other time-varying components, such as common productivity differentials across
13

Alternatively, we can use the estimates of λ obtained by Bils and Klenow (2004) and Klenow and Kryvtsov(2008). However, our main findings are robust, even if our λ is replaced by their estimates.

16

countries, which we discuss later.
The key idea is the equivalence of testing the full attention hypothesis and checking thestatistical significance of the coefficient on ln q̃t in (35) because (36) suggests that β = 1−m =0 if firms are fully attentive. When the null hypothesis of β = 0 is rejected in favor of thealternative hypothesis of β > 0, the data are consistent with the presence of inattentive firms.
Note that because the nominal exchange rate is the common driving force of the good-level
and aggregate real exchange rates (ln qit and ln qt ), the two variables are expected to be highly
correlated to each other. In our regression, however, both good-level and real exchange rates
are modified so that two variables (ln q̃it and ln q̃t ) are correlated only when the degree ofattention is less than unity. As an important by-product of the regression (35), the degree ofattention m can be obtained as m̂ = 1 − β̂, where β̂ is an OLS estimator of β.
Some remarks are in order. First, we can generalize the stochastic process of labor productivity from a simple i.i.d. process to a more realistic process that allows for a nonstationary
stochastic trend and a stationary but serially correlated component. Let us assume that laborproductivity is given by:ln ait = ξt + ηt + εit ,(37)

ln a∗it = ξt + ηt∗ + ε∗it .

(38)

Here, the labor productivity consists of three components: a global component ξt , a countryspecific component ηt (or ηt∗ ), and a good-specific component εit (or ε∗it ). In this genP∞ξeralized setting, global and country-specific components follow ξt − ξt−1 =j=0 bj εt−j ,
P∞
P
η∗
ξ
η
η∗
η
∗
ηt = ∞
j=0 dj εt−j , respectively, where εt , εt , and εt are i.i.d. shocks.
j=0 dj εt−j , and ηt =This error structure implies that the productivities in both countries are nonstationary, butshare a common stochastic trend (or the two variables are cointegrated). Because only the
relative labor productivity ln ait −ln a∗it matters in the dynamics of LOP deviations, the global
component becomes irrelevant in our analysis. However, this is not the case for the countryspecific component, in which case, regression (35) requires modification. For example, if ηtand ηt∗ each follow an AR(1) process with AR coefficient ρη and firms are fully attentive toηt and ηt∗ , (36) should be modified toln q̃it = (1 − m) ln q̃t +(1 − λ)(1 − λδ) rψηt + (1 − λ)(1 − λδ)ψεrit ,1 − λδρη

17(39)

where ηtr = ηt − ηt∗ . The only difference between (36) and (39) is that the latter includes the
new control variable ηtr . We can obtain a similar equation even if we include additional lagsin the process of the country-specific component.
Second, even if we additionally introduce the inattention to the idiosyncratic productivity,
our regression framework of testing behavioral inattention (to the aggregate variable) remains
valid. The stochastic process for the good-level real exchange rate (29) needs to be slightly
modified because ln qit becomes less sensitive to εrit . A smaller coefficient on εrit in (29),
however, does not change our regression equation (35).
Finally, an alternative way of testing m = 1 is to regress ln qit directly on ln qt withadditional regressors ln qit−1 and ∆ ln St . The estimation equation is given by:ln qit = α + β ln qt + γ 0 Xit + uit ,(40)

where β in (40) corresponds to (1 − m)(1 − λ) in (29). The control variables Xit in (40) now
include ln qit−1 and ∆ ln St . Note that β = (1 − m)(1 − λ) = 0 corresponds to m = 1 provided
λ < 1. Therefore, a test of β = 0 against β > 0 in (40) is equivalent to the test of the fully
attentive hypothesis. Unlike the case of (35), however, the presence of a lagged dependent
variable in the right-hand side of (40) implies a dynamic panel structure. Therefore, dynamic
panel regression estimators, such as the generalized method of moments (GMM) estimatorof Arellano and Bond (1991), need to be employed in place of OLS (see e.g., Crucini and
Shintani, 2008 and Crucini et al., 2010a).14

4.2

Data

We use the retail price data from the Worldwide Cost of Living Survey compiled by the
Economist Intelligence Unit (EIU), which entails an extensive annual survey of international
retail prices. The survey reports all the prices of individual goods in local currency terms,
conducted by a single agency in a consistent manner over time. The coverage of goods
and services is substantial in breadth and thus overlaps with the typical urban consumption
basket tabulated by national statistical agencies.15 In our analysis, the number of goods andservices is 274. Recent studies using these data include Engel and Rogers (2004), Crucini
and Shintani (2008), Bergin et al. (2013), Crucini and Yilmazkuday (2014), Andrade and
14

In Appendix A.5, we report the empirical results based on this alternative strategy.
See Rogers (2007) for details on the comparison between EIU data and the consumer price index datafrom national statistical agencies.
15

18

Zachariadis (2016), and Crucini and Landry (2019).
The data contain prices in multiple cities over 1990–2015. Our analysis focuses on US–
Canadian city pairs. We use these city pairs from the two countries. In our data, there are
16 US and 4 Canadian cities.16 This results in 64 unique cross-country city pairs. However,
because some of the US cities have many missing values in the early 90s, our data are an unbalanced panel.17 Nevertheless, the total number of observations available for our regressions
exceeds 350,000.
We compute the log of qijt for each year (t = 1990, ..., 2015), each good (i = 1, ..., 274),
and each international city pair (j = 1, ..., 64). The prices used to construct the good-level
real exchange rates are the prices in a US city (expressed in US dollars) and the prices in a
Canadian city (expressed in Canadian dollars). We use the spot USD/CAD dollar exchange
rates taken from the EIU data to convert prices to common currency units. The EIU records
the nominal exchange rate at the end of the week of the price survey. This means that thenominal exchange rate may differ across Canadian cities if the timing of the price survey
differs between the two cities. We confirm that the timings of the price survey in Calgary andin the remaining Canadian cities were different from 2003 to 2014. Across other Canadiancities, the USD/CAD exchange rate is common in all periods.18 Figure 1 plots two kernel
density estimates of the bilateral good-level real exchange rates pooling all goods and services,
one for the first year of the sample (1990) and the other for the last year of the sample (2015).
For our regression and empirical tests that follow, we augment the micro price data with theaggregate bilateral real exchange rate computed from the official CPI indices, which the EIU
also reports.
When we allow for the general stochastic process of labor productivity (37) and (38), we
need to control for the difference in the country-specific components in the labor productivityηtr (= ηt − ηt∗ ) in (39). As a proxy for ηtr , we utilize the difference in real GDP per hour worked
between the two countries taken from OECD.Stat.
16

The US cities are Atlanta, Boston, Chicago, Cleveland, Detroit, Honolulu, Houston, Lexington, Los Angeles, Miami, Minneapolis, New York, Pittsburgh, San Francisco, Seattle, and Washington DC. The Canadiancities are Calgary, Montreal, Toronto, and Vancouver.
17
In particular, the survey only includes the data for Honolulu in 1992 and Lexington and Minneapolis
have only been included in the list of cities since 1998.
18
As we discuss later, we adjust our regressions to account for this difference in timing.

19

4.3

Estimation results

Table 1 provides the estimation results for (35). In computing ln q̃ijt and ln q̃t , we calibrate
λ at 0.34. This value of λ is obtained by transforming the median monthly frequencies
of price changes for the US economy reported in Nakamura and Steinsson (2008) into theinfrequencies of price changes at an annual rate.19
The table reports the estimates of β from regression (35) with the standard errors. To
control for unobserved heterogeneity in the LOP deviations, we include two types of fixed
effects. One is the good-specific fixed effects and the other is the city-pair-specific fixed
effects. Thus, columns (1)–(4) differ from each other in terms of alternative combinationsof these fixed effects.20 In addition, we control for the country-specific component of laborproductivity ηtr in columns (5)–(8) of the table, motivated by (39). In all regressions, we
control for the difference in timing of the price survey in Calgary by adding dummy variables
that take a value of one if a city pair includes Calgary in 2003, 2004, ..., or 2014.21
Overall, β̂ is around 0.8, and much larger than zero, which is the value under the full
attention hypothesis. Because the standard error of the coefficient is around 0.03, the null
hypothesis of full attention (β = 0) is significantly rejected against the alternative hypothesisof partial attention (β > 0).22 Our results for the test of behavioral inattention are robust to
the presence of fixed effects (see columns (2)–(4)), and to the inclusion of the log-difference inlabor productivity as a control variable (see columns (5)–(8)). Interpreted through the lens ofour theoretical model, these results suggest that firms are not fully attentive to the aggregate
components of marginal costs in making their pricing decisions. The table also reports theestimated degree of attention m̂ (= 1 − β̂). The point estimates of m range from 0.15 to 0.20.
19

We transform the monthly median frequency of price changes in Nakamura and Steinsson (2008) into theannual infrequency of price changes λ as follows. Let f be the (monthly) median frequency of price changes
calculated in Nakamura and Steinsson (2008). If the price of a good is kept unchanged for 12 months under12
our assumption of sticky prices, the probability of not being able to change prices within a year is (1 − f ) .
Nakamura and Steinsson (2008) report that the median frequency of price changes in all sectors during 1998–
2005 is 8.7 percent. Using the above formula, the annual infrequency of price changes λ is calculated as
1212
λ = (1 − f ) = (1 − 0.087) = 0.34.
20
While we do not report the result, we also allow for a fixed effect that is specific to both good i and city
pair j. We find that the estimated β is not substantially different.
21
The difference in timing of the price survey causes the aggregate real exchange rate to be city-pairand year-specific. More specifically, let qtk and Stk be the aggregate real exchange rate and the nominal
exchange rate for a city pair k that involves Calgary in a year during 2003–2014. Here, ln qtk is given bykk
ln qtk = ln Stk + ln Pt∗ − ln Pt . We can express ln qtk as ln qtk = (ln Stk − ln St ) + ln qt and ln qijt
as ln qijt=k(ln St − ln St ) + ln qijt where the variables without the superscript k are variables in the other city pairs.
Therefore, this dummy variable can control for the presence of ln Stk − ln St in our samples.
22
We report the standard errors clustered by goods, but the null hypothesis is also rejected even if thestandard errors are clustered by city pairs or years.

20

Table 2 points to the estimation results when the calibrated λ differs across goods. While
our model assumes the degree of price stickiness common across goods, the previous empiricalstudies on sticky prices report the heterogeneity in the frequency of price changes. Thus, wetransform each good-specific monthly frequency of price changes reported in Nakamura andSteinsson (2008) into the good-specific infrequency of price changes and construct ln q̃ijt =ln qijt − λi ln qijt−1 − λi ∆ ln St and ln q̃ti = (1 − λi ) ln qt for good i.23 Even when we allow
for the good-specific degree of price stickiness, the null hypothesis of full attention is again
significantly rejected and m̂ ranges between 0.11 and 0.24.

5Explaining the PPP puzzle

In the previous section, we provided strong evidence for behavioral inattention using micro
price data. We now turn to the implications of this finding for the PPP puzzle.
Let ρq be the first-order autocorrelation of aggregate real exchange rates. Because the AR
coefficient in (33) corresponds to the first-order autocorrelation, let us rewrite (33) as:ln qt = ρq ln qt−1 + ρq εnt ,(41)

where ρq = λ/[1 − (1 − m)(1 − λ)]. In the following proposition, we now discuss Rogoff’s(1996) PPP puzzle.
Proposition 2 Under the same assumptions in Proposition 1,
ρq ≥ λ,(42)

provided m ∈ (0, 1] and λ ∈ (0, 1). The equality holds if and only if m = 1
Proof. It follows from the fact that (1 − m)(1 − λ) ≤ 1, where (42) holds with the equality
if and only if m = 1.
Proposition 2 implies that the presence of behavioral inattention helps with the resolutionof Rogoff’s (1996) PPP puzzle. That is, the aggregate real exchange rate is more persistentthan the degree of price stickiness implies. Without behavioral inattention (i.e., m = 1), ρq
is equal to λ. However, if firms are inattentive (i.e., m < 1), ρq becomes strictly greater than23

See also Crucini et al. (2010a, 2010b, 2013), Hickey and Jacks (2011), and Elberg (2016) who emphasize
heterogeneity in price stickiness in research on the LOP.

21

λ. In the extreme case of m → 0, the aggregate real exchange rate can even follow a random
walk, as ρq → 1. Therefore, even when the nominal frictions are small, the model with asmall m can explain a highly persistent aggregate real exchange rate.
We rule out the case of flexible price (λ = 0) in Propositions 1 and 2 because (41) suggeststhat λ = 0 leads to no PPP deviations, even in the short run (ln qt = 0 for all t). Our model
thus requires nominal rigidities as the external source of the persistence of the aggregate realexchange rate. We can understand this feature of our model in the context of real rigidities
in Ball and Romer (1990) or as a form of strategic complementarity as in Woodford (2003).
Using a closed-economy model, Ball and Romer (1990) show that real rigidities are insufficient
to create real effects of nominal shocks. They argue that a combination of real rigidities anda small friction in the nominal price adjustment matters for the real effect of a nominal shock.
In our model, a combination of behavioral inattention and a small friction in the nominal
price adjustment could generate a substantially persistent aggregate real exchange rate.
The left panel of Figure 2 plots ρq against m ∈ (0, 1]. We report the cases of λ = 0.34
(the solid line) and 0.68 (the dashed line) to assess the impact of changes in λ. Starting fromρq = λ when m = 1, ρq increases monotonically as m increases. The persistence becomescloser to unity as m approaches zero. For example, the curve for λ = 0.34 shows that ρq = 0.34
when m = 1. However, if we use m = 0.17, the mean of the estimated degrees of attention in
Tables 1 and 2, the same line indicates that ρq = 0.74.
The right panel of the same figure illustrates the ρq to λ ratio, which is defined as:
ρq
1
=
.
λ
1 − (1 − m) (1 − λ)(43)This ratio measures the extent to which inattention amplifies the persistence of the aggregate
real exchange rate explained solely by nominal rigidities under full attention. The figuresuggests that the ρq to λ ratio can be quite large depending on m. When m = 0.17, the PPP
deviations are more than twice as persistent as what is predicted only by the degree of pricestickiness (ρq /λ = 2.20).
Let us evaluate how much the estimated degree of inattention can explain the half-life of
the aggregate real exchange rate. The upper panel of Table 3 compares the half-lives and thefirst-order autocorrelations of ln qt between the models with and without full attention.24 We
also report the half-lives and the first-order autocorrelations reported in previous empiricalstudies in the column headed “Data.” In particular, Rogoff (1996) concluded that the half-life
24

The half-lives are calculated using the formula for the AR(1) process given by − ln(2)/ ln ρq .

22

estimated by previous studies is 3–5 years. This range of half-life is roughly equivalent to0.79–0.87 in terms of the first-order autocorrelation.
We reconfirm that the model with full attention performs poorly. When m = 1, the firstorder autocorrelation of the aggregate real exchange rate is only 0.34 because ρq = λ = 0.34.
This low first-order autocorrelation translates into a very short half-life of 0.64 years. These
predictions are inconsistent with the estimated persistence of the aggregate real exchange
rates as the half-life of 0.64 years is far below the half-life of 3–5 years in the aggregate realexchange rate.
By contrast, the model with a relatively low degree of attention explains the persistenceof the aggregate real exchange rate quite well. For example, when we use the lowest point
estimate (specification (2) in Table 2), m = 0.11, the model predicts that the half-life of theaggregate real exchange rate is 3.7 years, which falls in the range of 3–5 years.
We next turn to the good-level real exchange rate. We let ρqi be the first-order autocorrelation of the good-level real exchange rate implied by (29). The following proposition
describes the relationship between the persistence of the good-level real exchange rates andthat of the aggregate real exchange rate, as predicted by the behavioral model.
Proposition 3 Under the same assumptions in Proposition 1,
ρq ≥ ρqi ,(44)

provided m ∈ (0, 1], λ ∈ (0, 1), τ ∈ [0, ∞), ε ∈ (1, ∞), and σr /σn ∈ [0, ∞). The equality
holds if m = 1, τ = 0, or σr /σn = 0.
Proof. See Appendix A.6.
Proposition 3 explains the stylized fact that good-level real exchange rates are much lesspersistent than the aggregate real exchange rate. (Imbs et al. 2005; Crucini and Shintani
2008). Importantly, we obtain this aggregation result without relying on the “aggregationbias” pointed out by Imbs et al. (2005). They emphasized that the heterogeneity in thepersistence of the good-level real exchange rates induces a positive bias in the persistence of
the aggregate real exchange rate. Using multisector sticky-price models with heterogeneity
in the degree of price stickiness, Carvalho and Nechio (2011) successfully explain the positivebias. By contrast, our model intentionally assumes homogeneity in the persistence acrossgoods. Nevertheless, our model can qualitatively explain the gap in persistence between theaggregate and the good-level real exchange rates.
23

Once again, the value of m plays a crucial role in generating the gap between ρq and ρqi .
This point can be further investigated from the ρq to ρqi ratio defined by:
ρq
1
=
,
A
ρqi
1 − (1 − m) (1 − λ) 1+A
where

1 − ρ2q
A = (1 − λ) (1 − λδ) ψ 2
ρq (1 − λ2 )
2

2

2



(45)

σr
σn

2
.

(46)

The derivation is in Appendix A.6. Similar to the ρq to λ ratio in (43), the ρq to ρqi ratio
indicates that ρq = ρqi if m = 1. Therefore, combined with the result from (43), full attention
leads to the complete failure to explain the PPP puzzle: ρq = ρqi = λ. If firms are inattentive(i.e., m < 1), ρqi is strictly less than ρq .
What is necessary for explaining the gap between ρq and ρqi is real friction. More specifically, trade cost (τ ) needs to be strictly positive and the elasticity of substitution acrossbrands (ε) needs to be larger than one for the ρq to ρqi ratio (45) to be strictly greater than
one. If τ = 0 or ε → 1, there is no home bias (ω = 1/ 1 + (1 + τ )1−ε = 1/2) so thatψ = 2ω − 1 = 0. According to (46), either τ = 0 or ε → 1 makes A zero and thus (45)becomes one. Likewise, σr /σn , namely the standard deviation ratio of real shocks (εrit ) tonominal shocks (εnt ), in (46) needs to be strictly positive. If the nominal shock fully dominates the real shock such that σr /σn → 0, A is again zero, such that the model fails togenerate the gap between ρq and ρqi .
To assess the effect of m on the gap between ρq and ρqi , we calibrate the parameters in(45) and (46). For the parameters of real frictions, we set τ to 74 percent from Anderson andvan Wincoop (2004) and ε to 4 from Broda and Weinstein (2006).25,26 Using these values, we
obtain the degree of home bias ω of 0.84, which is roughly consistent with the parameter forhome bias used in the literature.27 The resulting calibrated value of ψ becomes 0.68. Crucini
et al. (2013) found that σr /σn = 5 is a sensible estimate of the standard deviation ratio,
based on the sectoral real exchange rate data in Europe. The households’ discount factor δ
25

Using US data, Anderson and van Wincoop (2004) argue that the transportation costs are 21 percent andthat the border-related trade barriers are 44 percent. Using these values, they calculate total international
trade costs as 0.74(= 1.21 × 1.44 − 1).
26
Broda and Weinstein (2006) report that the medians of the elasticities of substitution during 1990–2001
are 3.1 at the seven-digit level of the Standard International Trade Classification (SITC) and 2.7 at the
five-digit level of the SITC.
27
For example, Chari et al. (2002) calibrate the degree of home bias as 0.76, while Steinsson (2008) uses
0.94.

24

is set at 0.98 and the degree of price stickiness λ is set to 0.34.
The left panel of Figure 3 plots ρqi against m ∈ (0, 1] in the dashed line. It also includes thecurve for ρq with λ = 0.34 taken from the solid line in Figure 2. As suggested by Proposition
3, the curve for ρqi is always located below the curve for ρq . Recall that the lower bound of
ρq is λ(= 0.34) at m = 1. This property is preserved for ρqi because ρq = ρqi = λ hold at
m = 1.
The right panel of Figure 3 indicates that the ρq to ρqi ratio is hump shaped againstm ∈ (0, 1]. The ρq to ρqi ratio is one, when m → 0 or m = 1. We reconfirm this from the left
panel of the same figure. When m is either zero or one, we have ρq = ρqi so that ρq /ρqi = 1
holds. However, when 0 < m < 1, the ρq to ρqi ratio exceeds unity. Indeed, when m is set to
0.17, the mean of m̂, the ρq to ρqi ratio amounts to 1.57, indicating that the LOP deviations
are 57 percent less persistent than the PPP deviations.
Using the above-calibrated parameters and the estimated degree of attention, we assess
how much the model can explain the gap in persistence between the aggregate and the goodlevel real exchange rates. The lower panel of Table 3 presents the predicted half-lives and thefirst-order autocorrelations of the good-level real exchange rate. In the rightmost column, we
report the half-lives and the first-order autocorrelations of the good-level real exchange rates
taken from Crucini and Shintani (2008). The observed half-lives range between 1.0 and 1.6
years and the corresponding first-order autocorrelations range between 0.51 and 0.65.
We can see from the table that the model with behavioral inattention (m < 1) performs
much better than the model with full attention in explaining the half-life of the good-level
real exchange rates. In contrast to 0.64 years under m = 1, the half-life is 0.93 years underm = 0.17. If we set m at 0.11, the half-life is 1.22 years. Thus, the model well explains the
gap in persistence between the aggregate and the good-level real exchange rates. In the data,
the half-life of the LOP deviations is at most 1.6 years, but that in the PPP deviations is
at least 3 years so that there seems to be a gap of around 2 years between them. Based on
the estimated values of m, the half-lives predicted from our model are between 0.82 and 1.22
years for the former and between 1.83 and 3.70 years for the latter, suggesting the gap of 1.01
to 2.48 years between them.
Before closing this section, two remarks are in order. First, it is straightforward to combine
Propositions 2 and 3 to obtain the ρqi to λ ratio that measures the amplification from λ to25ρqi . In particular, using (43) and (45), we haveA
1 − (1 − m) (1 − λ) 1+A
ρqi
=
≥ 1.
λ
1 − (1 − m) (1 − λ)(47)

Given that the equality is excluded as long as A > 0 and m < 1, the persistence of thegood-level real exchange rate exceeds λ. We can reconfirm this inequality from the left panelof Figure 3. The dashed line for ρqi is always located above the dotted line for λ. The result
is consistent with Kehoe and Midrigan’s (2007) finding that the observed persistence of thegood-level real exchange rate often exceeds the degree of price stickiness.
Second, our model assumes quasi-linear preferences U (c, n) = ln c − χn, but the estimatesof m are robust even when we replace these with the more general constant-relative-riskaversion (CRRA) form. In this case, firms expect a dynamic path for the labor supply fromthe time of price setting to the infinite future, and the good-level real exchange rate does
not have a simple reduced-form solution. However, as shown in Appendix A.7, we can still
conduct a test for behavioral inattention and obtain the degree of attention by employing
the instrumental variables estimator under CRRA preferences. Our test rejects the null
hypothesis of full attention and the estimated values of degree of attention are very close to
the results shown in Tables 1 and 2.

6

Conclusion

In this paper, we offer a possible explanation for two empirical anomalies. First, the observed
PPP deviations are much more persistent than the theoretical predictions given by the standard model of nominal rigidities in prices. Second, the micro price evidence suggests that thedeviations from the LOP are often less persistent than the PPP deviations. To reconcile the
PPP and LOP evidence, we adapt the model of behavioral inattention in Gabaix (2014) to asimple two-country, sticky-price model. We show that pricing by inattentive firms generates
the complementarity between the LOP and PPP deviations, which is the key to accounting
for the puzzling behavior of real exchange rates.
Using international price data for US and Canadian cities, we implement a test of behavioral inattention and quantify its importance. We find strong evidence consistent withbehavioral inattention. Our model produces an aggregate real exchange rate that is morethan twice as persistent as the real exchange rate explained only by sticky prices under theestimated degree of attention. With additional calibrated parameters, our model also predicts
26that the persistence of the LOP deviations is less than two-thirds of the persistence of the
PPP deviations.
Based upon our examination of the behavioral inattention hypothesis, it seems plausiblethat it plays a comparable role to other real rigidities in the existing real exchange rate
literature while also amplifying some prominent existing mechanisms such as sticky prices.
The avenues for further exploration appear to be quite promising.

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29

A

Appendix

A.1

The derivation of the objective function for the pricing decision

To derive (6) and (7), we begin with the standard expression.28 The objective function of US
firms that sell their brand in US markets is given by:
∞
X


Wt+k
cit,t+k (z),vit (z) = Et
λ δt,t+k (1/Pt+k ) Pit (z) −
a
it+k
k=0


k

(48)

subject to the demand function by US consumers for brand z of good i conditional on thefirm having last reset its price in period t:

Pit (z)cit,t+k (z) =
Pit+k

−ε
cit+k ,

(49)

where z ∈ [0, 1/2].
Using the definitions of pit (z), wt , and pit , we rewrite (48) as:∞X
wt+k Pt+kPt
pit (z) −cit,t+k (z).
vit (z) = Et
λ δt,t+k
Pt+k
ait+k Pt
k=0
k

(50)

Next, for a generic variable xt , we express xt as xt = x̄ exp(x̂t ), where x̂t = ln xt − ln x̄
and x̄ is the steady-state value of xt . In addition, by assumption, Pt+k /Pt and ait are both
unity in the steady state. Rewriting (50) yields (6):vit (z) = Et

∞
X

∞
X

"

λk δt,t+kk=0where Pt+k /Pt =Pt
πt+l − âit+kp̄i (z) exp [p̂it (z)] − w̄ exp ŵt+k +
Pt+k
l=0

Qk

l=1 Pt+l /Pt+l−1 = exphPkl=1 ln (Pt+l /Pt+l−1 )

i

= exp

!#
cit,t+k (z),

hP
k

i
π
. For thet+ll=1

demand function, we can rewrite (49) as cit,t+k (z) = [Pit (z)/Pit+k ]−ε cit+k = [(Pit (z)/Pt )/(Pit+k /Pt+k )×(Pt /Pt+k )]−ε cit+k = [(pit (z)/pit+k )(Pt /Pt+k )]−ε cit+k . Using the log deviation, we can derive(7):
cit,t+k (z) =28p̄i (z)
p̄i

−ε

(

"

exp −ε p̂it (z) −

k
X
l=1

For example, see Galı́ (2015).

30#)
πt+l − p̂it+k

cit+k .

(51)

We next work on the derivation of (8) and (9). When US firms sell their brands in
Canadian markets, they set the price in the local currency. Under this assumption, theobjective function of these firms isvit∗ (z) = Et

∞
X



Wt+k ∗∗
λ δt,t+k (1/Pt+k ) St+k Pit (z) − (1 + τ )cit+k (z),
a
it+k
k=0
k

(52)

subject to the demand function by Canadian consumers:c∗it+k (z) =Pit∗ (z)
∗
Pit+k

−ε

c∗it+k ,(53)

where z ∈ [0, 1/2].
Using the definitions of p∗it (z) = Pit∗ (z)/Pt∗ and p∗it = Pit∗ /Pt∗ , we rewrite (48) as follows:
∞
X

 ∗

∗
St+k Pt+k
Pit (z) Pt∗
Pt+k Wt+k /Pt+k− (1 + τ )cit,t+k (z)vit (z) = Et
λ δt,t+k∗∗Pt+kPt∗ Pt+k
St+k Pt+k
ait+k
k=0


∞
X
Pt∗
wt+k
k
∗
λ δt,t+k qt+k pit (z) ∗ − (1 + τ )= Et
cit,t+k (z)Pt+kqt+k ait+k
k=0

∞
∗ 
X
wt+k Pt+kPt∗∗kpit (z) − (1 + τ )cit,t+k (z).
= Et
λ δt,t+k qt+k ∗∗Pq
t+k ait+k Pt
t+k
k=0
k

Again, using xt = x̄ exp (x̂t ) and assuming the zero-inflation steady state, we obtain (8):vit (z) = Et∞
X

λk δt,t+k qt+k

k=0

Pt∗
× ∗
Pt+k

(
p̄∗i (z) exp [p̂∗it (z)] − (1 + τ )

k
X

w̄
∗
exp ŵt+k − q̂t+k +πt+l− âit+kq̄
l=1!)
cit,t+k (z).

Equation (9) can be derived from (53) in the same way as the derivation of (7) from (49).
We can similarly derive the objective function of Canadian firms indexed by z ∈ (1/2, 1].

31

When Canadian firms sell their brands in Canadian markets, their objective function is
∞
X


∗ 
Wt+k
∗
Pit (z) − ∗c∗it,t+k (z)
a
it+k
k=0



∞
∗
∗
X
wt+k
Pt+k
Pt∗
∗
∗
= Et
λk δt,t+kp(z)−c∗it,t+k (z)
it
∗
∗
∗
P
a
P
t
t+k
it+k
k=0
"
!#
k
∞
∗
X
X
P
t
∗
∗
p̄∗i (z) exp [p̂∗it (z)] − w̄∗ exp ŵt+k+
πt+l − â∗it+kc∗it,t+k (z),= Et
λk δt,t+k
∗
P
t+k
l=1
k=0
vit∗ (z) = Et∗
λk δt,t+k∗
1/Pt+kfor z ∈ (1/2, 1]. Similarly, when Canadian firms sell their brands in US markets, the objectivefunction isvit (z) = Et∞
X∗
λk δt,t+k∗
1/Pt+k

k=0
∞
X


∗ 
 Pit (z)Wt+k
− (1 + τ ) ∗cit,t+k (z)
St+k
ait+k


∗ 
∗
∗
St+k Pt+k
Wt+k
/Pt+k
Pt+k
Pit (z) Pt= Et− (1 + τ )cit,t+k (z)∗St+k Pt+k
Pt Pt+ka∗it+k
Pt+k
k=0


∞
∗
X
wt+k
qt+k
Pt
−1
k ∗
λ δt,t+k qt+k pit (z)= Et− (1 + τ ) ∗cit,t+k (z)
P
a
t+k
it+k
k=0


∞
∗
X
wt+k
qt+k Pt+k
−1 Pt
k ∗
= Et
λ δt,t+k qt+kpit (z) − (1 + τ )cit,t+k (z)
∗
P
a
P
t+k
t
it+k
k=0
= Et

∞
X

∗
λk δt,t+k−1∗
λk δt,t+k
qt+k

k=0

Pt
×
Pt+k

(
p̄i (z) exp [p̂it (z)] − (1 + τ )w̄∗ q̄ exp ŵt+k + q̂t+k +kX!)
πt+l − âit+k

cit,t+k (z),l=1

for z ∈ (1/2, 1].

A.2

The sparse max

Following Gabaix (2014), we assume that firms choose the degree of attention. Equations(13) and (14) correspond to the case of US firms that sell their goods in the US market. The
US firms’ objective function for choosing mH is based on the second-order Taylor expansion
of EvHi [pHi (µ̂Ht , mH ), µ̂Ht , 1] − EvHi [pHi (µ̂Ht , 1), µ̂Ht , 1] around µ̂Ht = 0, which is the loss of
profits of choosing the price distorted by partial attention. In this appendix, we derive (13)and (14).
32

To obtain (13) and (14), we first take the approximation of EvHi [pHi (µ̂Ht , mH ), µ̂Ht , 1].
Here, the profit of the firm is evaluated at mH = 1 (which appears in the last augment in
vHi (·), but the price is distorted by mH 6= 1. We then evaluate the approximated equation at
mH = 1. The second-order approximation of EvHi [pHi (µ̂Ht , mH ), µ̂Ht , 1] around µ̂Ht = 0 is
EvHi [p̂Hi (µ̂Ht , mH ), µ̂Ht , 1]
(∞
)
∞
2 0
X ∂ 2 v 0  ∂ p̂Hit (0, mH ) 2 X
∂
v
1
Hi
Hi
0
Eµ̂2Ht+k
+
' vHi
+
2
2 k=0 ∂ p̂it (z)2
∂ µ̂Ht+k
∂
µ̂
Ht
k=0


∞
0
X
∂ 2 vHi∂ p̂Hit (0, mH )+Eµ̂2Ht+k ,
∂
p̂
∂
µ̂
it (z)∂ µ̂Ht+k
Ht+k
k=0

(54)

0
= vHi [p̂Hi (µ̂Ht , mH ), µ̂Ht , 1]µ̂Ht =0 = vHi (p̂Hi (0, mH ), 0, 1). For the second derivawhere vHi00tives, ∂ 2 vHi/∂ p̂it (z)2 = ∂ 2 vHi (p̂Hi (0, mH ), 0, 1)/∂ p̂it (z)2 and ∂ 2 vHi/∂ µ̂2t+k = ∂ 2 vHi (p̂Hi (0, mH ), 0, 1) /∂ µ̂2t+k .

We use the first-order condition for pricing of inattentive firms to simplify (54). The
first-order condition is ∂vHi [p̂it (z), µ̂Ht , mH ]/∂ p̂it (z) = 0. Taking the partial derivative of thefirst-order conditions with respect to µ̂Ht+k for k = 0, 1, 2, ... and evaluating them at µ̂Ht = 0:
∂ 2 vHi (p̂Hit (0, mH ), 0, mH )∂ 2 vHi (p̂Hit (0, mH ), 0, mH ) ∂ p̂Hit (0, mH )=−, for k = 0, 1, 2, ....
∂ p̂it (z)∂ µ̂Ht+k∂ p̂it (z)2
∂ µ̂Ht+k(55)
Let us focus on ∂ p̂Hit (0, mH )/∂ µ̂Ht+k in the right-hand side of (55). The optimal price isgiven by p̂Hi (µ̂Ht , mH ) = mH ŵt − (1 − λδ)âit = mH µ̂t − (1 − λδ)âit (see (20)). Thus,∂ p̂Hi (µ̂Ht , mH ) n mH for k = 0=
0 for k 6= 0
∂ µ̂t+k(56)

When we evaluate the profits in (55) at mH = 1 but not the prices, (55) can be substituted
into (54). Then, together with (56), we now simplify (54) to:
EvHi [p̂Hi (µ̂Ht , mH ), µ̂Ht , 1]


∞
0
0
0
1 ∂ 2 vHi
∂ 2 vHi
1 X ∂ 2 vHi
0
2
2
m
−
2
m
E(µ̂
)
+
' vHi +
E(µ̂2Ht+k ).
HHt
2 ∂ p̂it (z)2 H∂ p̂it (z)22 k=0 ∂ µ̂2Ht+k(57)

We further need the second-order approximation of EvHi [pHi (µ̂Ht , mH ), µ̂Ht , 1] where the33price is not distorted by mH . Evaluating (57) at mH = 1 yields
∞00∂ 2 vHi
1 X ∂ 2 vHi
2
E(µ̂
)
+
E(µ̂2Ht+k ).
Ht
2 ∂ p̂it (z)22 k=0 ∂ µ̂2Ht+k

1
0
EvHi [p̂Hi (µ̂Ht , 1), µ̂Ht , 1] ' vHi
−



(58)

Combining (57) and (58), EvHi [pHi (µ̂Ht , mH ), µ̂Ht , 1] − EvHi [pHi (µ̂Ht , 1), µ̂Ht , 1] aroundµ̂Ht = 0 is
EvHi [p̂Hi (µ̂Ht , mH ), µ̂Ht , 1] − EvHi [p̂Hi (µ̂Ht , 1), µ̂Ht , 1]0
 ∂ 2 vHi
1'm2H − 2mH + 1
E(µ̂2Ht )2∂ p̂it (z)20
∂ 2 vHi1(1 − mH )2E(µ̂2Ht )=2∂ p̂it (z)2
1
= − (1 − mH )2 ΛH .
2

(59)While firms can reduce the loss of paying partial attention (59) by paying more attention,
they also have to pay costs of increasing attention, which we specify as a quadratic costfunction. Formally, the choice of attention for US firms that sell their goods in US markets
is characterized by:κ
1
min [(1 − mH )2 ΛH ] + m2H , where ΛH = −mH ∈[0,1] 22∂ 2 vHi[0, 0, 1] V ar(µ̂Ht ).
∂ p̂2it (z)The remaining sparse max can analogously be defined. The sparse max for US firms
selling their goods in Canadian markets is
1
κ ∗ 2
∗ 2 ∗
min
(1
−
m
)
Λ
+
(m ) , where Λ∗H = −HHm∗H ∈[0,1] 2
2 H

 2 ∗

∂ vHi [0, 0, 1]
V ar(µ̂∗Ht ).
∂ p̂∗2Hit

Next, the sparse max for Canadian firms selling their goods in Canadian markets is1κmin (1 − m∗F )2 Λ∗F + (m∗F )2 , where Λ∗F = −∗mF ∈[0,1] 2
2

 2 ∗

∂ vF i [0, 0, 1]
V ar(µ̂∗F t ).
∗2
∂ p̂F itBy symmetry, we can easily show that Λ∗F = ΛH , which reconfirms m∗F = mH . the sparse
max for Canadian firms selling their goods in US markets is1κmin (1 − mF )2 ΛF + m2F , where ΛF = −mF ∈[0,1] 2
2
34

 2

∂ vF i [0, 0, 1]
V ar(µ̂F t ).
∂ p̂2F it

Again, by symmetry, we have ΛF = Λ∗H and mF = m∗H .

A.3

The optimal prices under behavioral inattention

Using the definition of µ̂Ht+k , we rewrite the log-linearized first-order condition (19) aspHi (µ̂Ht , mH ) = (1 − λδ)Et

∞
∞
k
X
X
X
(λδ)k (mH ŵt+k − âit+k ) + mH (1 − λδ)Et(λδ)kπt+l , (60)k=0where we used µHt+k = ŵt+k +

k=0

l=1

Pk

l=1 πt+l .

We separately arrange the terms in the right-hand side of (60). First, note that∞X(λδ)k (mH ŵt+k − âit+k )(1 − λδ)Et
k=0

= Et

∞
X

∞
X

k=0

k=0

(λδ)k (mH ŵt+k − âit+k ) − Et(λδ)k+1 (mH ŵt+k − âit+k )= mH ŵt − âit
+Et [(λδ)1 (mH ŵt+1 − âit+1 ) − (λδ)1 (mH ŵt − âit )]+Et [(λδ)2 (mH ŵt+2 − âit+2 ) − (λδ)2 (mH ŵt+1 − âit+1 )]+...
∞X(λδ)k (mH ∆ŵt+k − ∆ait+k ).
= mH ŵt − âit + Et
k=1

Next, the remaining terms are
∞
k
X
X
k
mH (1 − λδ)Et
(λδ)
πt+l
k=0

l=1


(λδ)πt+1
2
2
= mH (1 − λδ)Et  +(λδ)3 πt+1 + (λδ)3 πt+2+(λδ) πt+1 + (λδ) πt+2 + (λδ)3 πt+3
+...
"∞
#
)
(
"∞
#
"∞
#
X
X
X
= mH (1 − λδ)Et (λδ)(λδ)k πt+1 + (λδ)2(λδ)k πt+2 + (λδ)3(λδ)k πt+3 + ...


k=0

k=0

k=0

∞
X
= mH Et(λδ)k πt+k ,k=1

where the last line uses

P∞

k
−1
k=0 (λδ) = (1 − λδ) .

35

Finally, combining the above expressions,(60) becomesp̂Hi (µ̂Ht , mH ) = (mH ŵt − âit ) + Et∞X(λδ)k {mH (∆ŵt+k + πt+k ) − ∆âit+k }.

(61)

k=1

Now, under the assumption of U (c, n) = ln c − χnt , the first-order conditions of US
households (Wt /Pt = χct ) imply ŵt = ĉt . In addition, their CIA constraint (Mt = Pt ct ) leads
to πt = ln Mt /Mt−1 − ∆ĉt . Thus, using (15), we have ∆ŵt + πt = ln Mt /Mt−1 = εM
t . As a
result, (61) becomesp̂Hi (µ̂Ht , mH ) = (mH ŵt − âit ) − Et∞X(λδ)k ∆ait+k .

k=1

If the stochastic process âit is given by (17), EtP∞kk=1 (λδ) ∆ait+k = −λδâit . Therefore,p̂Hi (µ̂Ht , mH ) = mH ŵt − (1 − λδ)âit ,which is (20) in the main text.
For the price of goods exported by US firms, we havep̂∗Hi (µ̂∗Ht , m∗H )=[m∗H (ŵt − q̂t ) − âit ] + Et∞X∗(λδ)k [mH (∆ŵt+k − ∆q̂t+k + πt+k
) − ∆âit+k ]

k=1
∞
X
∗
∗
∗
∗
(λδ)k [mH (∆ŵt+k+ πt+k) − ∆âit+k ],= (mH ŵt − âit ) + Et

(62)

k=1

where we used the log-linearized equation of (3): q̂t = ĉt − ĉ∗t = ŵt − ŵt∗ . This equation has
the same structure as (61). Using the CIA constraint, (16), and (17), the above equation canbe simplified top̂∗Hi (µ̂∗Ht , m∗H ) = m∗H (ŵt − q̂t ) − (1 − λδ)âit ,which is equivalent to (21).
The remaining optimal prices, namely p̂∗F i (µ̂∗F t , m∗F ) and p̂F i (µ̂F t , mF ) are analogously
derived.

36

A.4

Proof of Proposition 1

We begin by (28), q̂it = q̂t + p̂∗it − p̂it . From (3), the log deviation of the real exchange rate isq̂t = ĉt − ĉ∗t .

(63)

q̂it = (p̂∗it − ĉ∗t ) − (p̂it − ĉt ).

(64)

Thus, q̂it can be rewritten as:

In what follows, we focus on p̂it − ĉt and p̂∗it − ĉ∗t to derive (29). Equation (24) impliesp̂it − ĉt = λ(p̂it−1 − πt ) + (1 − λ)p̂opt
it − ĉt= λ(p̂it−1 − ĉt−1 ) − λ(∆ĉt + πt ) + (1 − λ)(p̂opt
it − ĉt ).

(65)

Note that ∆ĉt + πt in (65) is equal to εM
t because of the CIA constraint of US households
and the money supply process (15). Substituting this result yields
optp̂it − ĉt = λ(p̂it−1 − ĉt−1 ) − λεM
t + (1 − λ)(p̂it − ĉt ).

(66)

Similarly, p̂∗it − ĉ∗t is given by:∗p̂∗it − ĉ∗t = λ(p̂∗it−1 − ĉ∗t−1 ) − λεM+ (1 − λ)(p̂opt∗− ĉ∗t ).
t
it

(67)

Substituting (66) and (67) into (64) yields an expression for q̂it :q̂it = λq̂it−1 + λεnt + (1 − λ)p̂opt∗− ĉ∗t − p̂opt,
itit − ĉt

(68)

∗

M
where εnt = εMt − εt .

We next focus on the expression inside the bracket on the right-hand side of (68). Using(20), (23), (25), and (63), we rewrite p̂opt
it as
∗
p̂opt
it = mĉt − (1 − λδ)[ωâit + (1 − ω)âit ],(69)

where the relative price index is determined by the aggregate demand ĉt and the weighted
average of labor productivity. In the equation, we also use the degree of attention m defined37

by (30). We then subtract ĉt from both sides of (69) to get∗p̂opt
it − ĉt = − (1 − m) ĉt − (1 − λδ) [ωâit + (1 − ω) âit ] .

(70)

Similarly, p̂opt∗− ĉ∗t isitp̂opt∗− ĉ∗t = −(1 − m)ĉ∗t − (1 − λδ)[ωâ∗it + (1 − ω)âit ],
it

(71)

where m = ωm∗F + (1 − ω)m∗H = ωmH + (1 − ω)mF . Combining (70) and (71), we have∗∗(p̂itopt∗ − ĉ∗t ) − (p̂opt
it − ĉt ) = (1 − m)(ĉt − ĉt ) + (1 − λδ)(2ω − 1)(âit − âit )= (1 − m)q̂t + (1 − λδ)ψεrit ,where q̂t = ĉt − ĉ∗t from (63), εrit = εit − ε∗it = ait − a∗it from (17) and (18), and ψ = 2ω − 1.
Substituting the above equation into (68) yields
q̂it = λq̂it−1 + (1 − λ) (1 − m) q̂t + λεnt + (1 − λ) (1 − λδ)ψεrit .

(72)

Here, q̂it = ln qit and q̂t = ln qt because ln q̄i = ln q̄ = 0 from the symmetry between thetwo countries. In particular, the symmetry ensures that ln q̄ = ln c̄ − ln c̄∗ = 0 and thatln q̄i = ln q̄ + p̄∗i − p̄i = 0. Therefore, (72) is equivalent to (29) in Proposition 1.

A.5

Estimation results for (40)

Table A.1 reports the estimation results for (40) by GMM. We report the results of four
specifications. In specifications (2) and (4), we impose the restriction that the coefficientson ln qit−1 and ∆ ln St as control variables are the same as each other. This is because (29)indicates that ln qit−1 and εnt = ∆ ln St have the same coefficient. Specifications (3) and (4)differ from specifications (1) and (2) in that the regressions include ηtr as a control variable.
The table indicates that, in all regressions, the null hypothesis that β = 0 in (40) is
statistically rejected. In addition, the estimates of β are all positive, consistent with thetheory. Therefore, even if we directly regress ln qijt on ln qt , the data are consistent with thepartial attention of 0 < m < 1.

38

A.6

Persistence of the good-level real exchange rate

A.6.1

Proof of Proposition 3

As a preparation, we rewrite (41) in terms of the log deviation:q̂t = ρq q̂t−1 + ρq εnt .

(73)

The variance of q̂t is given by σq2 = [ρ2q /(1 − ρ2q )]σn2 , so
σn2 =1 − ρ2q 2
σq .
ρ2q

(74)

In Appendix A.4, we have shown that the log deviation of the LOP deviations isq̂it = λq̂it−1 + θq̂t + λεnt + ψ̃εrit ,(75)

where θ = (1 − λ)(1 − m) and ψ̃ = (1 − λ) (1 − λδ)ψ.
Let the covariances denote R0 = Eq̂t q̂it and R1 = Eq̂t q̂it−1 . They are written as
R0 = λR1 + θσq2 + λρq σn2 ,(76)

R1 = ρq R0 ,(77)

which can be derived from (73) and (75).
We further simplify (76) and (77). Substitute (74) and (77) into (76) to getλ(1 − ρ2q ) 2
σq .
R0 = λρq R0 + θ +
ρq


(78)

Note that, using the definition of ρq , the expression inside the brackets can be simplified as29
λ(1 − ρ2q )θ+= 1 − λρq .
ρq

(79)

29
To see this, θ + λ(1 − ρ2q )/ρq = θ + λ(1 − ρ2q )/(λ/(1 − θ)) = θ + (1 − ρ2q )(1 − θ) = 1 − ρ2q (1 − θ). Applying
the definition of ρq to this equation again, we obtain (79).

39

Using (79), (78) and (77) become
R0 = σq2 ,(80)

R1 = ρq σq2 ,(81)

respectively.
2= Eq̂it2 andWe next work on the variance and the autocovariance that are denoted as σqi

γ1 = Eq̂it q̂it−1 , respectively. Using (75), (80), and (81), we have2
= λγ1 + θσq2 + λρq σn2 + ψ̃ 2 σr2 ,
σqi

(82)

2
γ1 = λσqi+ θρq σq2 .

(83)

We obtain (82) and (83) from tedious algebra. Regarding (82), we use (75) and (80)2σqi
= Eq̂it2 = λE(q̂it q̂it−1 ) + θE(q̂it q̂t ) + λE(q̂it εnt ) + ψ̃E(q̂it εrit )= λγ1 + θσq2 + λE(q̂it εnt ) + ψ̃E(q̂it εrit ).
Simplifying the last two terms on the right-hand side of the above equation yields (82):2σqi= λγ1 + θσq2 + λE[(λq̂it−1 + θq̂t + λεnt + ψ̃εrit )εnt ] + ψ̃E[(λq̂it−1 + θq̂t + λεnt + ψ̃εrit )εrit ]= λγ1 + θσq2 + λθE(q̂t εnt ) + λ2 σn2 + ψ̃ 2 σr2= λγ1 + θσq2 + λ(θρq + λ)σn2 + ψ̃ 2 σr2= λγ1 + θσq2 + λρq σn2 + ψ̃ 2 σr2 .
The third equality results from (73), and the fourth equality is from the definition of ρq .
Regarding (83), use (75) and (81) to get2γ1 = Eq̂it q̂it−1 = λE(q̂it−1 q̂it−1 ) + θE(q̂t q̂it−1 ) + λE(εnt q̂it−1 ) + ψ̃E(εrit q̂it−1 ) = λσqi
+ ρq θσq2 .

40

2We further simplify σqi
and γ1 . Using (74) and (83), (82) becomes2
σqi= λγ1 + θσq2 + λρq σn2 + ψ̃ 2 σr2λ(1 − ρ2q ) 2
σq + ψ̃ 2 σr2= λγ1 + θ +ρqλ(1 − ρ2q ) 2
2 2
2
σq + ψ̃ 2 σr2 .
= λ σqi + λρq θσq + θ +ρqRecall that, from (79), the expression inside the brackets is 1 − λρq . This implies,22
σqi= λ2 σqi+ [1 − λρq (1 − θ)]σq2 + ψ̃ 2 σr2
2= (1 − λ2 )σq2 + ψ̃ 2 σr2 ,(1 − λ2 )σqi
2and γ1 are
where we use 1 − λρq (1 − θ) = 1 − λ2 given ρq = λ/(1 − θ). Therefore, σqi

ψ̃ 2
2
σqi
= σq2 +
σ2.
2 r
1
−
λ
#
"
2
ψ̃
σ 2 + θρq σq2
γ1 = λ σq2 +1 − λ2 r= (λ + θρq )σq2 += ρq σq2 +(84)

λψ̃ 2 2σ
1 − λ2 rλψ̃ 2 2σ .
1 − λ2 r(85)

Now, because the first-order autocorrelation of the good-level real exchange rate is given2,by ρqi = γ1 /σqi2
ρqi = γ1 /σqi= ωρ ρq + (1 − ωρ ) λ,(86)

where ωρ is defined as
ωρ =

σq2
ψ̃ 2
2
σq2 + 1−λ2 σrbecauseA==1∈ [0, 1]
1+Aψ̃ 2 σr2≥ 0.
1 − λ2 σq2(87)

Equation (86) means that ρqi is the weighted average of ρq and λ. When we combine Proposition 2, namely ρq ≥ λ, with (86), it immediately follows that ρq ≥ ρqi ≥ λ.
41

A.6.2Derivation of (45) and (46)

Using ρq = λ/(1 − θ), eliminate λ from (86):ρqi = ωρ ρq + (1 − ωρ ) (1 − θ) ρq = ρq [1 − θ (1 − ωρ )] .

(88)

Recall (74) and the definition of ψ̃. Then, (87) becomes (46):
1 − ρ2q
A = (1 − λ) (1 − λδ) ψ 2
ρq (1 − λ2 )
2

2

2



σr
σn

2
.

(89)

From ωρ = 1/ (1 + A), 1 − ωρ = A/(1 + A). In addition, recall that θ = (1 − λ)(1 − m).
Therefore, (88) implies
1
ρq
=
.
A
ρqi
1 − (1 − λ) (1 − m) 1+A

A.7

(90)

The model with CRRA preferences

Let us assume more general CRRA preferences: U (c, n) = c1−σ /(1 − σ) − χn1+ϕ /(1 + ϕ).
We modify the first-order conditions for households to allow for the degree of relative risk
aversion. Under σ 6= 1, the first-order conditions imply St = (Mt /Mt∗ )σ (Pt /Pt∗ )1−σ .
If we maintain the assumption that the money supply follows a random walk, the equationfor St leads to the nominal exchange rate growth that is predictable through the inflation of
the two countries.30 Because this is inconsistent with the exchange-rate disconnect puzzle,
we replace this assumption by the new assumption on the money growth rate:σ−11πt + ε M,σσ tσ − 1 ∗ 1 M∗=
πt + ε t .
σσ∆ ln Mt =(91)∆ ln Mt∗(92)

Under (91) and (92), the nominal exchange rate continues to follow a random walk.31
In particular, the nominal exchange rate growth is given by ∆st = σεnt + (1 − σ)(πt − πt∗ ), meaning that
πt − πt∗ can help forecast ∆st .
31
To see this, note that the nominal exchange rate growth is given by: ∆st = σ (∆ ln Mt − ∆ ln Mt∗ ) +(1 − σ) (πt − πt∗ ). Substituting (91) and (92) into the above equation yields ∆st = (σ − 1) (πt − πt∗ ) +M∗(1 − σ) (πt − πt∗ ) + εM
= εnt .
t − εt30

42

Using the CIA constraints, we can rewrite (91) and (92) as:σ∆ĉt+k + πt+k = εM
t+k ,
∗

∗
σ∆ĉ∗t+k + πt+k
= εM
t+k ,

(93)
(94)

for k > 0. Later, we utilize (93) and (94) for deriving the estimation equation.
A.7.1

The derivation of the estimation equation

To derive the estimation equation, we follow the same procedure as the derivation of (29).
When σ 6= 1, the international risk-sharing condition (3) is replaced by qt = (ct /c∗t )σ . Combining its log-linearized expression with (28), q̂it can be written asq̂it = (p̂∗it − σĉ∗t ) − (p̂it − σĉt ).

(95)

We focus on p̂it − σĉt and p̂∗t − σĉ∗t and obtain the expression for q̂it using (95). Note that(24) remains valid even under the CRRA preferences. Therefore, we subtract σĉt from bothsides of (24) and arrange terms to getoptp̂it − σĉt = λ (p̂it−1 − σĉt−1 ) − λεM
t + (1 − λ) p̂it − σĉt ,(96)where we replace σ∆ĉt + πt by εMt using (93). Analogously, (26) remains valid under the
CRRA preferences. Using (26), we have∗p̂∗it − σĉ∗t = λ(p̂∗it−1 − σĉ∗t−1 ) − λεM+ (1 − λ)(p̂opt∗− ĉ∗t ).
t
it

(97)

Therefore, the good-level real exchange rate is− σĉ∗t ) − (p̂opt
q̂it = λq̂it−1 + λεnt + (1 − λ)[(p̂opt∗
itit − σĉt )].

(98)

Equations (96)–(98) generalize (66)–(68), respectively.
We next focus on the expression inside the brackets on the right-hand side of (98). For
the case of σ 6= 1, we recalculate the log optimal prices: p̂Hi (µ̂Ht , mH ), p̂∗Hi (µ̂∗Ht , m∗H ),p̂∗F i (µ̂∗F t , m∗F ), and p̂F i (µ̂F t , mF ). While (61) continues to hold under the CRRA preferences,ŵt is no longer equal to ĉt but is now given by ŵt = σĉt + ϕn̂t . Accordingly, we rewrite (61)43

asp̂Hi (µ̂Ht , mH ) = mH (σĉt + ϕn̂t ) − âit∞X+Et(λδ)k [mH (σ∆ĉt+k + πt+k + ϕ∆n̂t+k ) − ∆âit+k ] .
k=1

= mH σĉt − (1 − λδ)âit + mH ϕ

1 − λδ
1 − λδL−1


n̂t .

(99)

where L is the lag operator. In the second equality, we used (93) and replaced σ∆ĉt+k + πt+kby εMt+k , which greatly simplifies the equation.
Equation (99) differs from (20) in the presence of the forward-looking terms for the labor
supply. Equations for p̂∗Hi (µ̂∗Ht , m∗H ), p̂∗F i (µ̂∗F t , m∗F ), and p̂F i (µ̂F t , mF ) are
1 − λδn̂t ,=1 − λδL−1
1 − λδ
∗
∗
∗
∗
∗
∗
∗
p̂F i (µ̂F t , mF ) = mF σĉt − (1 − λδ)âit + mH ϕn̂∗t ,−1
1 − λδL
1 − λδ
∗n̂∗t ,p̂F i (µ̂F t , mF ) = mF σĉt − (1 − λδ)âit + mF ϕ
−1
1 − λδLp̂∗Hi (µ̂∗Ht , m∗H )m∗H σĉ∗t − (1 − λδ)âit + m∗H ϕ



(100)
(101)
(102)

respectively.
opt∗− σĉ∗t are given by:
Using (25), (27), p̂opt
it − σĉt , and p̂it∗p̂opt
it − σĉt = −(1 − m)σĉt − (1 − λδ)[ωâit + (1 − ω)âit ]1 − λδ
+ ϕ[ωmH n̂t + (1 − ω)mF n̂∗t ] ,1 − λδL−1p̂opt∗− σĉ∗t = −(1 − m)σĉ∗t − (1 − λδ) [ωâ∗it + (1 − ω)âit ]it1 − λδ
+ ϕ[ωmH n̂∗t + (1 − ω)mF n̂t ] ,−1
1 − λδL

(103)

(104)

respectively. In (104), we assumed that m∗F = mH and m∗H = mF .
Plugging (103) and (104) into (98) yields
q̂it = λq̂it−1 + (1 − λ)(1 − m)q̂t + λεnt + (1 − λ)(1 − λδ)ψεrit− ϕψm(1 − λ)(1 − λδ)(n̂t − n̂∗t ),−1
1 − λδL(105)

where ψm = ωmH − (1 − ω) mF . Equation (105) differs from (72) in that there are theforward-looking terms for labor supply. If ϕ = 0, these forward-looking terms disappear, and44

the equation coincides with (72). Under our assumptions, σ does not appear in (105). More
importantly, σ does not affect the coefficient on the aggregate real exchange rate.
As in the proof of Proposition 2, the symmetry between the two countries implies thatq̄i = q̄ = 1, and thus q̂it = ln qit and q̂t = ln qt . Likewise, the symmetry implies the samesteady-state labor supply between the two countries: n̄ = n̄∗ , leading to n̂t − n̂∗t = ln nt −ln n∗t .
Substitution of these equations into the above equation leads toln qit = λ ln qit−1 + (1 − λ)(1 − m) ln qt + λεnt + (1 − λ)(1 − λδ)ψεrit(1 − λ)(1 − λδ)−ϕψm(ln nt − ln n∗t ) ,−1
1 − λδL(106)

which generalizes (29).
To derive the estimation equation for our empirical analysis, we use the definition of q̃itand q̃t and further rewrite (106) as
ln q̃it = (1 − m) ln q̃t + (1 − λ)(1 − λδ)ψεrit − ϕψm(1 − λ)(1 − λδ)(ln nt − ln n∗t ),−1
1 − λδLor equivalently,
ln q̃it − λδEt ln q̃it+1 = (1 − m)(ln q̃t − λδEt ln q̃t+1 )(107)−(1 − λ)(1 − λδ)ϕψm (ln nt − ln n∗t ) + (1 − λ)(1 − λδ)ψεrit ,where Et εrit+1 = 0.
Let ln q̃˜it = ln q̃it − λδ q̃it+1 and ln q̃˜t = q̃t − λδ q̃t+1 . Our estimation equation isln q̃˜it = α + β ln q̃˜t + γ 0 Xit + uit ,(108)

where Xit includes the log-difference in labor supply ln nt − ln n∗t and γ includes −(1 − λ)(1 −λδ)ϕψm as an element. Note that OLS is no longer a valid estimation because uit now
includes forecast error ln q̃it+1 − Et ln q̃it+1 and ln q̃t+1 − Et ln q̃t+1 . We thus use the instrument
for estimation. For the data source of ln nt − ln n∗t , we take the indices of total hours worked
from OECD.Stat with the base year 2010.
Table A.2 reports the estimation results based on (108). We use a common λ in specifications (1) and (2) and the good-specific λ in specifications (3) and (4). In addition, we
instrument ln q̃˜t by ln q̃t−1 in specifications (1) and (3) and ln q̃˜t−1 in specifications (2) and45

(4). In all cases, the null hypothesis of full attention, namely β = 0, is significantly rejected.
The estimated values of m are also very close to each other, suggesting robustness to changes
in the assumption of preferences.

46

0

.2

.4

density.6.8

1

Figure 1: Empirical distributions of the good-level real exchange rates

−2

−1
0
1
the log of good−level real exchange rate
1990

47

2015

2

Figure 2: Persistence of the aggregate real exchange rate and the ρq to λ ratio
1

3

0.9

2.8

= 0.68

2.4

0.7

2.2

0.6
/

0.5

2

q

q

= 0.34/ with
q
/ =1
q

2.6

0.8

/ with

q

1.8
0.4
1.6
0.3
1.4
0.2
q

0.1

q

with

= 0.34with= 0.68

1.2
1

0
0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

m

0.6

0.8

1

m

Figure 3: Persistence of the aggregate and the good-level real exchange rates and the ρq toρqi ratio
1

1.7

0.9

1.6

0.8

q

/ qi

q

/ qi = 1

1.5

0.7
1.4
/ qi

1.3

q

0.5

q

, qi,

0.6

0.4

1.2

0.3

1.1

0.2

1

q

0.1

qi

0.9

=0.34

0
0

0.2

0.4

0.6

0.8

1

0

m

0.2

0.4

0.6
m

48

0.8

1

Table 1: Regressions with common λDependent variable: ln q̃ijt with λ = 0.34
(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

0.843

0.844

0.806

0.802

0.848

0.812

0.844

0.806

(0.030)(0.030)(0.028)(0.028)(0.031)(0.029)(0.031)(0.029)Observations

389,500

389,500

389,500

389,500

389,500

389,500

389,500

389,500

Adj. R2

0.049

0.225

0.087

0.262

0.049

0.225

0.087

0.262

good FE

N

Y

N

Y

N

Y

N

Y

city-pair FE

N

N

Y

Y

N

N

Y

Y

Control for ηtr

N

N

N

N

Y

Y

Y

Y

0.156

0.194

ln q̃t49

The implied degree of attention from the regression
m̂

0.157

0.156

0.194

0.198

0.152

0.188

NOTES: Regressions estimated using panel data for 274 items and 64 US and Canadian city-pairs over 26 years. Standard errors are clusteredby goods in parentheses below the coefficients. The dependent variable is ln qijt − λ(ln qijt−1 − ∆ ln St ), where λ is calibrated at 0.34. The table
reports the regression coefficients on (1 − λ) ln qt . Each specification includes dummy variables that control for the difference in timing of the pricesurvey in Calgary over 2003–2014. In columns (5)–(8), we control for the difference in the log of real GDP per hour worked. Each column reports
the estimate of m along with the p-value for the null of m = 1. The “Adj. R2 ” denotes the adjusted R-squared.

Table 2: Regressions with good-specific λDependent variable: ln q̃ijt with good-specific λ
(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

0.793

0.894

0.766

0.862

0.765

0.883

0.763

0.88

(0.045)(0.029)(0.045)(0.028)(0.053)(0.032)(0.053)(0.033)Observations

389,500

389,500

389,500

389,500

389,500

389,500

389,500

389,500

Adj. R2

0.062

0.256

0.100

0.294

0.062

0.256

0.100

0.294

good FE

N

Y

N

Y

N

Y

N

Y

city-pair FE

N

N

Y

Y

N

N

Y

Y

Control for ηtr

N

N

N

N

Y

Y

Y

Y

0.237

0.120

ln q̃t50

The implied degree of attention from the regression
m

0.207

0.106

0.234

0.138

0.235

0.117

NOTES: The dependent variable is ln q̃ijt = ln qijt − λi [ln qijt−1 − ln(St /St−1 )], where λi is good specific and calibrated based on Nakamura andSteinsson (2008). The table reports the regression coefficients on ln q̃ti = ln(qt /qtλi ). See the notes to Table 1 for the remaining details.

Table 3: Persistence implied by the estimated degree of attention
Model with

Model with behavioral inattention

full attention

m = 0.237

m = 0.174m = 0.106

Data

The aggregate real exchange rate
HL

0.643

1.831

2.382

3.704

3-5

ρq

0.340

0.685

0.748

0.829

0.79-0.87

The good-level real exchange rate
HL

0.643

0.819

0.933

1.223

1.0-1.6

ρqi

0.340

0.429

0.476

0.567

0.51-0.65

NOTES: The table reports the half-lives (denoted “HL”) and the first-order autocorrelations (denoted ρqand ρqi ) of the real exchange rates predicted by the model. The unit of half-lives is a year. To calculate
the predicted values in the table, we use the calibrated values of λ = 0.34, τ = 0.74, ε = 4, σr /σ∆s = 5,and δ = 0.98. The computed persistence measures are based on m̂ = 0.237 (the largest estimate in Tables 1
and 2), m̂ = 0.174 (the mean of the estimates in Tables 1 and 2), and m̂ = 0.106 (the smallest estimate inTables 1 and 2), together with the model with full attention (m = 1) for reference. The rightmost column
reports estimates from previous studies for comparison. For the aggregate real exchange rate, we take therange of the half-lives from Rogoff (1996) and compute the implied first-order autocorrelations from the halflives. For the good-level real exchange rate, we take the median estimate of the half-lives and the first-order
autocorrelations of all goods provided in Table 4 of Crucini and Shintani (2008).

51

Table A.1: Estimation results from the GMM
Dependent variable: ln qijt
(1)

(2)

(3)

(4)

0.302

0.300

0.264

0.261

(0.021)(0.020)(0.025)(0.022)

Observations

371,347

371,347

371,347

371,347

P-value for Hansen’s J-test

0.932

0.925

0.922

0.926

Control for ηtr

N

N

Y

Y

ln qt

NOTES: The estimation equation is given by (40). Following Arellano and Bond (1991), we regress ∆ ln qijt
on the lagged dependent variable ∆ ln qijt−1 , ∆ ln qt , ∆2 ln St along with the control variables used in thebenchmark regressions. In specifications (2) and (4), we impose the parameter restrictions that the coefficientson ln qijt−1 and ∆ ln St are the same. In specifications (3) and (4), we control for the difference in the log ofreal GDP per hour worked. As suggested by Arellano and Bond (1991), the levels of the lagged dependent
variables are used for instruments, depending on the period of observation. The test for the over-identifying
restrictions is also reported in which the degrees of freedom for the chi-squared distribution under the nullare 299 in specifications (1) and (3) and 300 in specifications (2) and (4). See the notes to Table 1 for theremaining details.

Table A.2: Estimation results under CRRA preferences
Dependent variable: ln q̃˜ijt
(1)

(2)

(3)

(4)

λ

0.34

0.34

good-specificgood-specificln q̃˜t

0.804

0.800

0.870

0.861

(0.034)(0.037)(0.041)(0.045)Observations

371,347

371,347

371,347

371,347

good FE

Y

Y

Y

Y

city-pair FE

Y

Y

Y

Y

Instruments

ln q̃t−1ln q̃˜t−1

ln q̃t−1ln q̃˜t−1

The implied degree of attention from the regression
m

0.196

0.200

0.130

0.139

NOTES: The dependent variable is ln q̃˜ijt = ln q̃ijt − λδ ln q̃ijt+1 , where λ is calibrated at the values specified
in the first row of the table. The table reports the regression coefficients on ln q̃˜t = ln q̃t − λδ q̃t+1 and thedifference in the log of hours worked. In specifications (1) and (3), the instrument is ln q̃t−1 . In specifications(2) and (4), the instrument is ln q̃˜t−1 . See the notes to Table 1 for the remaining details.

52