A Behavioral Explanation for the Puzzling Persistence of
the Aggregate Real Exchange Rate∗
Mario J. Crucini†, Mototsugu Shintani‡, and Takayuki Tsuruga§
This version: February 2023

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 beena puzzle (Rogoff, 1996). In contrast, the micro price evidence for the law of one price(LOP) has consistently shown that the LOP deviations are less persistent than the PPP
deviations. To reconcile this macroeconomic and microeconomic empirical evidence, we
adapt the model of behavioral inattention in Gabaix (2014, 2020) to a simple twocountry sticky-price model. We propose a simple test of behavioral inattention and findstrong evidence in its favor using micro price data. Calibrating behavioral inattention
using our estimates, we show that our model reconciles the two puzzles relating to PPP
and the LOP. First, PPP deviations are more than twice as persistent as those impliedsolely by nominal rigidities. Second, the persistence of the LOP deviations falls to
two-thirds that of the PPP deviations.
JEL Classification: E31; F31; D40
Keywords: Real exchange rates, Law of one price, Purchasing power parity, Rational
inattention, Trade cost∗

We thank David Cook, Mick Devereux, Ippei Fujiwara, Yasuo Hirose, Jinill Kim, Akihisa Shibata, Kwanho
Shin, and participants at numerous conferences and seminars for their helpful comments and suggestions. We
also thank Erwan Gautier for kindly providing us with the data on the frequency of price changes in the Euro
area. Shintani and Tsuruga gratefully acknowledge the financial support of Grants-in-aid for Scientific Research (JP17H02510, JP18K01684, JP20H05631, JP20H05633, and JP21H04397), the Joint Usage/ResearchCenter 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.
†
Krannert School of Management, Purdue University and NBER; e-mail: mcrucini@purdue.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, that is, 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 feature of realexchange rates is the gap in persistence between PPP deviations and deviations from thelaw 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 good-level real exchange rate (the LOP
deviations) is likely to be much less persistent than the aggregate real exchange rate (thePPP deviations).1 These previous studies emphasize the role of heterogeneity in the speed ofprice adjustment. As Imbs et al. (2005) have 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
a disaggregated level” (p.3).
In this paper, we simultaneously explain two empirical anomalies: (1) the gap between
the observed persistence of PPP deviations and the persistence predicted from the stickyprice model (e.g., Rogoff, 1996), and (2) the gap between the observed persistence of 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 two-country stickyprice model. In this framework, firm managers bear the cost of paying attention to theaggregate component of the marginal cost of their products. As a result, full attention to thestate of the economy is no longer optimal when firms choose the prices of goods.
The key to solving the PPP puzzle is then the complementarity between the PPP and
LOP deviations. After deriving a reduced-form solution for the LOP deviations, we show thatthey are affected by the PPP deviations when firms pay only partial attention to marginalcost. Thus, an increase in the persistence of the PPP deviations makes the LOP deviations
more persistent. At the same time, through aggregation, more persistent LOP deviations
lead to more persistent PPP deviations, further strengthening the link between the PPP and
LOP deviations.
The reduced-form solution leads to a direct testable implication. Using micro price datafrom the US, Canada, and European countries, we implement a simple test for the null1

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

2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 theaggregate real exchange rate after controlling for common driving forces, such as the nominal
exchange rate and country-specific productivity. Using various specifications, we strongly
reject this null in favor of our proposed model of behavioral inattention. We also find theestimated degree of attention to be around 0.15, 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 that implied solely by nominal rigidities.
Although the setting differs, this mechanism is consistent with Ball and Romer (1990) and
Woodford (2003). They show that even small frictions in nominal price adjustment lead to apersistent output gap when real rigidities or strategic complementarities are present. In ourmodel of behavioral inattention, only small nominal frictions are needed to generate a highly
persistent aggregate real exchange rate. Based on our estimates of the degree of attention,
the aggregate real exchange rate is more than twice as persistent as that implied solely by
nominal rigidities. In terms of the half-lives of the aggregate real exchange rate, the estimateddegrees of attention suggest that behavioral inattention increases the half-life of aggregate
real exchange rates by approximately 2.4 years, relative to the full attention, sticky-pricebenchmark.
In the second theoretical result, our model explains the gap between the highly persistent
PPP deviations and the less persistent LOP deviations. This gap arises from the combination of complementarity and the presence of idiosyncratic real shocks to the individual
price of goods. We show that both the PPP and LOP deviations are more persistent whencomplementarities are present. In contrast, real shocks at the goods level (but not countryspecific 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 goods level. As a result, our estimates of the degree of attention imply a substantial
gap in persistence between the PPP and LOP deviations. In fact, our model predicts thatthe persistence of the LOP deviations decreases to less than two-thirds of the persistence of
the PPP deviations when inattention is included in an otherwise standard sticky-price model.
The fact that the persistence in PPP deviations exceeds the persistence implied by nominal
rigidities relates to an extensive literature that has already contributed to a better understanding of persistent aggregate real exchange rates. For instance, Chari et al. (2002) argue
that while the sticky-price model with monetary shock can explain the volatility of the ag-

3

gregate real exchange rate, it substantively underpredicts the level of persistence. Benigno(2004) emphasizes the role of monetary policy rules rather than the degree of price stickinessin accounting for the persistent aggregate real exchange rate. Later, Engel (2019) revisits
Benigno (2004) and argues for the importance of both monetary policy rules and price stickiness. More recently, Itskhoki and Mukhin (2021) emphasize the dominant role of financial
shocks rather than monetary shocks in helping to resolve the PPP puzzle. In our model, aconventional monetary shock remains the main driver of PPP deviations. However, idiosyncratic productivity shocks are necessary to address that fact that LOP deviations are lesspersistent than the PPP deviations. Thus, Itskhoki and Mukhin (2021) and this paper sharethe view that monetary shocks alone are not sufficient for addressing the PPP puzzle when
LOP deviations are the key source of exchange rate fluctuations.
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.2 Importantly, our explanation of
the PPP puzzle does not directly challenge the underlying price adjustment mechanisms inthese studies. Instead, we offer an empirically defensible alternative, namely the importanceof behavioral inattention in firms’ pricing.
A standard explanation for the fact that the persistence in the aggregate real exchange rate
exceeds the persistence in the good-level real exchange rates is heterogeneity in the speed ofprice adjustment at the goods level, which generates a positive bias when prices are aggregatedin the construction of the consumer price index (CPI). Imbs et al. (2005) point out a positive
aggregation bias in dynamic heterogeneous panels, and Carvalho and Nechio (2011) consider
the theoretical implications of aggregation using a sticky-price model in which the degree of
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 being common across all goods. In this sense, the mechanism in our paper
further enhances the ability of existing workhorse models in the macroeconomics literature.
Furthermore, our model including behavioral inattention can also explain two related findings:
LOP deviations are more persistent than the degree of price stickiness implies (Kehoe andMirdigan 2007), and the LOP deviations are as persistent as the PPP deviations when we
2

Blanco and Cravino (2020) focus on the real exchange rate using only newly reset prices and find thatfluctuations in this “reset” exchange rate account for almost all fluctuations in the aggregate real exchange
rate. As argued in their paper, strategic complementarity somewhat raises the contribution of the resetexchange rate to the aggregate real exchange rate in fluctuations.

4focus only on macroeconomic shocks (Bergin et al. 2013).
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 world economy consists of two countries. For ease of exposition, we express the US
and Canada as the home and foreign countries, respectively. Following Kehoe and Midrigan(2007) and Crucini et al. (2010b, 2013), there is a continuum of goods and brands of eachgood. Goods are indexed by i and brands are indexed by z. For each good, US brands areindexed by z ∈ [0, 1/2] and Canadian brands are indexed by z ∈ (1/2, 1].
We assume that US and Canadian consumers have identical preferences over brands of aparticular good and across goods in the aggregate consumption basket. US preferences overthe brands of good i are given by the constant elasticity-of-substitution (CES) index for good
ε
hR
i ε−1
ε−1
1
i ∈ [0, 1]. The US consumption of good i is cit = z=0 cit (z) ε dz
and the aggregation
ε
hR
i
ε−1
ε−1
1
across goods gives aggregate consumption ct = i=0 cit ε di
, where ε > 1. For Canada,
ε
ε
hR
i
i ε−1
hR
ε−1
ε−1
1
1
∗
∗
∗ ε−1
∗
ε
ε
we have the analogous equations cit = z=0 cit (z) dz
and ct = i=0 cit 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 two constraints, an intertemporal 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. The left-hand sideof (1) represents the total nominal value of household wealth. The household allocates its
wealth into money balances Mt for the purchase of consumption goods and state-contingent

5

nominal bond holdings Bt+1 brought into period t + 1. Here, ∆t,t+1 denotes the nominal
stochastic discount factor. On the right-hand side of the budget constraint (1), the household
receives a nominal wage, Wt , per hour of work, nt , carries bonds Bt into period t, as well as
any cash that remained in period t − 1, Mt−1 − Pt−1 ct−1 . The household also receives nominaltransfers from the US government, Tt , and nominal profits from US firms, Πt . In (1), the1R 1−ε  1−εaggregate price Pt is given by Pt =Pit di, where Pit is the price index for good i.
R 1This, in turn, is a CES aggregate over US and Canadian brands: Pit =Pit (z)1−ε dz 1−ε .
The CIA constraint requires nominal money balances for expenditure, which is made at theend of the period 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 US dollar bond holdings into Canadian dollars at the spot nominal exchange rate, St . The Canadian households are subject tothe budget constraint,∗)
Et (∆t,t+1 Bt+1
Bt∗
∗
∗ ∗
∗
∗
Mt +
= Wt nt ++ Mt−1− Pt−1c∗t−1 + Tt∗ + Π∗t .
St
St

(2)

and 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 )]. For Canadian households, we have Wt∗ /Pt∗ = χc∗t and∗)]. The consumption Euler equations differ because∆t,t+1 = δ[(c∗t+1 /c∗t )−1 St Pt∗ /(St+1 Pt+1

Canadians buy state-contingent bonds denominated in US dollars.
The aggregate real exchange rate is defined 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 yields3

qt =

2.2

ct
c∗t


.

(3)

Firms

For each good, US firms produce the first half of the continuum, z ∈ [0, 1/2] of good i andemploy nit (z) hours of labor, and Canadian firms produce the second half of the continuum,z ∈ (1/2, 1] and employ n∗it (z). The production function of the US firms is given by yit (z) =ait nit (z), whereas that of the Canadian firms is given by yit∗ (z) = a∗it n∗it (z). Here, ait anda∗it are labor productivity specific to good i. In the US and Canada, all firms that produce
3This condition relies on our preference assumptions, which we relax in Section 4.

6

varieties of the same goods share the same productivity, but productivity across countries
differs by good.
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), forz ∈ [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), forz ∈ (1/2, 1].

2.3

(5)

Price setting

We introduce the behavioral inattention of firms into an otherwise standard two-country
model with Calvo pricing. Firms can change their prices with a constant probability, 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
the pricing problem of Canadian firms is analogous, we limit our exposition to the 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 is 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. Combined with the assumption that steady-state7

inflation is 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 +

k
X

(6)

!)
πt+l − âit+k

cit,t+k (z),

l=1

where

cit,t+k (z) =p̄i (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.4 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,
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 from 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∗ (z)/Pt∗ . A
fully attentive US firm chooses p̂∗it (z) to maximize5
vit∗ (z) = Et∞
X

λk δt,t+k qt+k

(8)

k=0

P∗
× ∗t
Pt+k
where

(

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

!)

( "
#)
 ∗ −ε
k
X
p̄
(z)
i
∗
c∗it,t+k (z) =exp −ε p̂∗it (z) −πt+l− p̂∗it+kc∗it+k .
∗p̄il=1c∗it,t+k (z),(9)∗
Here, πt∗ = ln(Pt∗ /Pt−1) and p∗it = Pit∗ /Pt∗ . In (8), the second line represents the real profits

in each period. The cost of providing a unit of the good to a Canadian consumer is higher
4
5

The derivation is provided in Appendix A.1.
The derivation is again in Appendix A.1.

8

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 with 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 its objective function. This problem is called the “sparse max”
because Gabaix (2014) originally developed a model in which the economic agents respondto 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 thereal wage and the real exchange rate, as well as microeconomic variables, such as goodspecific productivity shocks. We assume that the firm is fully attentive to its productivity
but possibly less attentive to the aggregate variables. It is worth noting at this point that theconceptually relevant departure is that the firm finds it costly to assess the precise relevanceof the aggregate state variable in its profit maximization problem.
Toward this end, let us introduce the “attention-augmented” objective function. DefinemH ∈ [0, 1] as the degree of attention to set prices of home-produced goods in the US
market, where the subscript H represents the place of production.6 The attention-augmented
objective function is given byvHi (p̂it (z), µ̂Ht , mH ) = Et∞
X

λk δt,t+k

k=0

×

Pt
{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 +Pk7
l=1 πt+l .

(10)

In the limit case of mH = 0,

managers 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,
Likewise, we define m∗H as the degree of attention to set prices of home-produced goods in the Canadianmarket. We represent the degree of attention to set prices of foreign-produced goods by m∗F when selling thegoods in the Canadian market and by mF when selling the goods in the US market.
7
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.
69there is a unit coefficient on âit+k .8
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 under mH = 1 in the third argument, but it is evaluated at the
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]
V ar (µ̂Ht ) .
ΛH = −∂ 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.9
8

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
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

10

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.10 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 into the idiosyncratic productivity and derive
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 ΛH for
the aggregate component of the marginal cost, meaning that the degree of attention to theidiosyncratic 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, suchas monetary shocks. Thus, the degree of attention to the idiosyncratic variable is again closer
to unity. Finally, our test of behavioral inattention in Section 4 can still detect inattention to
the aggregate variable even if we allow inattention of firms to their idiosyncratic productivity.
In other words, our test is fully robust to the presence of inattention of firms to their idiosyncratic productivity. In this sense, full attention to idiosyncratic productivity is a convenientassumption 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 ,

the Canadian and US markets.
10
Gabaix (2014) discusses the properties of the selected degree of attention using not only the quadratic
cost function but also other functional forms of the cost function.

11

∗

M
where εMare zero-mean i.i.d. shocks. Importantly, the stochastic processes, comt 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∗M
CIA constraints. This equation leads to ln St = ln St−1 + εnt , where εnt = εM
is thet − 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:11
ln 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
R R1
R
R
1
2
nt = i z=0
nit (z)dzdi and n∗t = 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 ,∗, for Canadian households); (ii) prices
Bt+1 , for US households and {c∗it (z)}i,z , n∗t , Mt∗ , Bt+1

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
11
Later, we consider an alternative stochastic process for productivity, but the empirical results from thetest of behavioral inattention are unaffected.

12

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 .

(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, the13degrees 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 ,(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 )∗Mand εnt = εM
∼ i.i.d.(0, σn2 ), respectively.
t − 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 = 0.12 To gain some intuition behind
12Note that ψ = 0 if τ = 0. The absence of trade cost (i.e., τ = 0) implies no home bias (i.e., ω = 1/2)

14

(31), recall that ln qit = ln St + ln Pit∗ − ln Pit . Suppose that the money supply increases
unexpectedly in the US. Although 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, whereas the adjustment of Pit is slow because of sticky prices. As a result, theincrease in Pit only partially offsets the increase in St . The extent of the offsetting effectdepends on λ. If λ → 0, a change in Pit perfectly offsets the increase in St , meaning that thenominal shock is irrelevant for the real exchange rate. If λ → 1, Pit never moves, meaningthat the good-level real exchange rate tracks the nominal exchange rate, which in turn 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 < 1.13 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 .

(32)

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 prices and the aggregate prices
leads to a link between the good-level and 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 the
because ω = 1/(1 + (1 + τ )1−ε ). In this case, ψ = 2ω − 1 = 0.
13Note 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.

15

nominal shock. Indeed, aggregating ln qit over i yields14
ln qt =

λλln qt−1 +εn .
1 − (1 − m)(1 − λ)1 − (1 − m)(1 − λ) t

(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,

behavioral 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. A strictly positive trade cost (i.e., τ > 0) leads to home bias in theprice indexes.15 The friction allows the real shock εrit to affect the good-level real exchange
rate. Stressing the importance of real shocks, Crucini et al. (2010b, 2013) extended the
Kehoe and Midrigan (2007) model to incorporate idiosyncratic productivity shocks. Under
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
the role 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
their goods in both countries. However, the home bias generated by trade cost will decrease
Pit more than Pit∗ . This results in the 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
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
real exchange rate, namely, the aggregate real exchange rate. Time-dependent pricing models of the good-level and aggregate real exchange rates without 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 behavioral
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 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).
15Home bias is reflected in (25) and (27).
14

16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 attention), against thealternative hypothesis of m < 1 (partial attention). To derive our panel regression model,
recall (29):ln qit = λ ln qit−1 + (1 − m)(1 − λ) ln qt + λεnt + (1 − λ)(1 − λδ)ψεrit .
In this equation, the nominal shock εnt in (29) is replaced by ∆ ln St because the (log) nomn
inal exchange rate followsh a random walk withi an increment εt . Define ln q̃it = ln qit −λ ln qit−1 − λ∆ ln St = ln qit / (qit−1 St /St−1 )λ and ln q̃t = (1 − λ) ln qt = ln(qt /qtλ ). Using
these definitions, the above equation can be rewritten asln q̃it = (1 − m) ln q̃t + (1 − λ)(1 − λδ)ψεrit .

(35)

Our panel regression is given byln q̃it = α + β ln q̃t + γ 0 Xit + uit ,(36)

where α, β, and γ are regression coefficients, Xit is a vector of control variables, and uit isthe error term. To implement the regression, we rely on the micro evidence of λ to constructln q̃it and ln q̃t . The error term uit = (1 − λ)(1 − λδ)ψεrit arises from an i.i.d. real shock andis uncorrelated with the regressor ln q̃t = (1 − λ) ln qt because εrit does not appear in (33).
Therefore, we estimate (36) using ordinary least squares (OLS). The control variables Xit
here include time-invariant fixed effects or other time-varying components, such as common
productivity differentials across 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 (36) because (35) 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.
17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 byproduct of the regression (36), the degree ofattention m can be obtained as m̂ = 1 − β̂, where β̂ is the OLS estimator of β.
The above regression analysis can be extended in two directions. 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 labor productivity 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 but
share a common stochastic trend (or the two variables are cointegrated). Because only 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 (36) 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∗ , (35) is modified toln q̃it = (1 − m) ln q̃t +(1 − λ)(1 − λδ) rψηt + (1 − λ)(1 − λδ)ψεrit ,1 − λδρη

(39)

where ηtr = ηt −ηt∗ . Equation (39) now includes the new control variable ηtr and the coefficienton ln q̃t remains unchanged. We can obtain a similar equation even if we include additionallags in the process of the country-specific component.
Second, we can drop the assumption of common λ and introduce heterogeneity in the
degree of price stickiness λ in testing the null hypothesis of m = 1. In particular, we canreplace λ with λi and use the following transformations: ln q̃it = ln qit − λi ln qit−1 − λi ∆ ln St
for the dependent variable of regressions and ln q̃ti = (1 − λi ) ln qt for the explanatory variable.
18

In the subsequent section of estimation results, we test the null hypothesis of m = 1 basedon both cases of common λ and good-specific λ.
As a methodological remark, we emphasize that our regression framework of testing behavioral inattention (to the aggregate variable) remains valid even if we additionally introduce
inattention into idiosyncratic productivity. 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 the coefficient on ln q̃t and thus
our regression equation (36) remains unchanged.

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 from a variety of cities. 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 thetypical urban consumption basket tabulated by national statistical agencies.16 Recent studies
using these data include Engel and Rogers (2004), Crucini and Shintani (2008), Bergin etal. (2013), Crucini and Yilmazkuday (2014), Andrade and Zachariadis (2016), Crucini andLandry (2019), and Crucini and Telmer (2020).
Our analysis focuses on the US–Canadian city pairs and the UK–Euro area city pairs.
For the US–Canadian city pairs, the data contain the prices of 274 goods and services in
multiple cities from 1990 to 2015. There are 16 US and four Canadian cities.17 This resultsin 64 unique cross-country city pairs. However, because some US cities have many missing
values in the early 1990s, our data comprise an unbalanced panel.18 Nevertheless, the total
number of observations available for our regressions exceeds 350,000. For the UK–Euro area
city pairs, there are two UK cities and 18 Euro area cities.19 The data include 301 goods and16

See Rogers (2007) for details on the comparison between the EIU data and the CPI data from national
statistical agencies.
17
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.
18
In particular, the survey data in 1990 and 1991 do not include the price data of goods and services in
Honolulu, whereas Lexington and Minneapolis have only been included in the list of cities since 1998.
19
The UK cities are London and Manchester. The Euro area cities are Amsterdam, Barcelona, Berlin,
Brussels, Dublin, Dusseldorf, Frankfurt, Hamburg, Helsinki, Lisbon, Luxembourg, Lyon, Madrid, Milan,
Munich, Paris, Rome, and Vienna. We drop the data for Athens from the sample because inflation there in
the 1990s before adopting the Euro exceeded 10 percent on average, substantially higher than in other Euro
area countries. Likewise, we remove the data on Bratislava from the sample because the Slovak koruna to

19

services from 1990 to 2015. As in the US–Canadian city pairs, the panel is unbalanced. The
number of observations exceeds 200,000 from the 36 UK–Euro area city pairs.
We compute the log of qijt for each year (t = 1990, ..., 2015), each good (i = 1, 2, ...), andeach international city pair (j = 1, 2, ...). The prices used to construct the good-level realexchange rates are the prices in a city expressed in the local currency unit. We use the spot
nominal exchange rates from the EIU data to convert prices to common currency units. The
EIU records the nominal exchange rate vis-à-vis the US dollar at the end of the week of theprice survey. Thus, the nominal exchange rate may not necessarily be common across cities
in the same country if the timing of the price survey differs across cities. We confirm thatthe timings of the price survey in Calgary differ from those in the remaining Canadian cities
from 2003 to 2014.20 The nominal exchange rates in the cities of other countries are common
in the EIU data.
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 forthe last year of the sample (2015). The upper panel of the figure shows the distribution of
the good-level real exchange rates for the US–Canadian city pairs, and the lower panel shows
those for the UK–Euro area city pairs. For our regressions and empirical tests that follow,
we augment the micro price data with the aggregate bilateral real exchange rate computedfrom the official consumer price 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 two countries from OECD.Stat.
Based on (36), we construct ln q̃ijt and ln q̃t and calibrate λ therein. We use values suggested by previous studies. Nakamura and Steinsson (2008) report that the median frequency
of price changes in the US is 8.7 percent. Gautier et al. (2022) find that the average frequency of price changes is 8.5 percent for consumer prices in the 11 Euro area countries.21 We
transform monthly frequencies of price changes into the infrequencies of price changes at anannual rate to compute the value of λ. The transformation leads to a value of λ around 0.34
in both cases.22 Because the degree of price stickiness at the macroeconomic level is similar
UK pound exchange rate greatly appreciated before the adoption of the Euro in 2009.
20
As we discuss later, we adjust our regressions to account for this difference in timing.
21
Both Nakamura and Steinsson (2008) and Gautier et al. (2022) remove the impact of sales on thefrequencies of price changes. In addition, Nakamura and Steinsson (2008) remove the impact of productsubstitutions on the frequency of price changes.
22
We transform the monthly frequency of price changes into the annual infrequency of price changes as20

between the US and the Euro area countries, our regression (36) assumes that λ = 0.34.
When allowing for heterogeneity in the frequency of price changes, we require the monthly
frequency of price changes for each good. For the US–Canadian city pairs, we use the goodspecific monthly frequencies of price changes reported by Nakamura and Steinsson (2008).
They report the frequencies of price changes based on the Entry Level Item (ELI) of the CPI
in the US. We match goods and services in the ELI with those in the EIU data and assign themonthly frequency of price changes to goods and services in the EIU data. For the UK–Euro
area city pairs, we use the good-specific monthly frequencies of price changes calculated by
Gautier et al. (2022). They calculate frequencies of price changes based on the Classificationof Individual Consumption by Purpose (COICOP) and aggregate them using country weightsof the Euro area consumer prices. As in the case of the US–Canadian city pairs, we assign
the frequencies at the COICOP level to the EIU data.

4.3

Estimation results

Table 1 provides the estimation results of (36) for the test of behavioral inattention. The left
panel shows the results for the US–Canadian city pairs, whereas the right panel presents thosefor the UK–Euro area city pairs. The table reports the estimated coefficients on ln q̃t withthe standard errors. We include the good-specific fixed effects in the regressions by default.
This is because variations in the good-specific fixed effect are substantially larger in the LOP
deviations than in the city-pair-specific fixed effect.23 For robustness, we allow for adding
the city-pair-specific fixed effects to the regression and/or controlling for the country-specificcomponent of labor productivity ηtr , motivated by (39). In regressions for the US–Canadiancity pairs, we also 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, ..., or2014.24
follows. Let f be the monthly frequency of price changes. If the price of a good is kept unchanged for 12months under our assumption of sticky prices, the probability of not being able to change prices within ayear is (1 − f )12 . We substitute f = 0.087 in Nakamura and Steinsson (2008) and f = 0.085 in Gautier et al.
(2022) into the above formula.
23
Crucini and Telmer (2020) emphasize the importance of good-specific fixed effects using the analysis ofvariance of the EIU data.
24
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 from 2003 to 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 arising from the difference intiming of the price survey in Calgary.

21

Overall, β̂ is around 0.85, or equivalently, the estimated degree of attention m̂ = 1 − β̂
is around 0.15. The standard error of the coefficient indicates that the test strongly rejects
the null hypothesis of full attention (β = 0) against the alternative hypothesis of partial
attention (β > 0).25 Comparisons between the left and right panels reveal that the estimatedcoefficients on ln q̃t for the US–Canadian city pairs are close to those for the UK–Euro area
city pairs. Taking specification (1) as an example, the first row of Table 1 indicates thatthe estimated β is 0.84 for the US–Canadian city pairs whereas the estimated β is 0.86 for
the UK–Euro area city pairs. Our results for the test of behavioral inattention are robust
to the presence of city-pair-specific fixed effects (see specifications (2) and (4)) and to theinclusion of the log-difference in labor productivity as a control variable (see specifications(3) and (4)).26 Interpreted through the lens of our theoretical model, these results suggestthat firms are not fully attentive to the aggregate components of marginal costs in making
their pricing decisions. The bottom of the table also reports that the estimated degrees ofattention, m̂t , are similar between the US–Canadian and the UK–Euro area city pairs. For
example, in specification (1), m̂ = 0.16 in the US–Canadian city pairs and m̂ = 0.14 in the
UK–Euro area city pairs.
Table 2 points to the estimation results when we drop the assumption of the common
λ and introduce the heterogeneity of λ across goods.27 Even when we allow for the goodspecific degree of price stickiness, the null hypothesis of full attention is again significantly
rejected. Regarding the estimated degrees of attention, m̂ tends to decline when we allow for
the good-specific λ. For example, if we take specification (1) for comparison, m̂ reduces from
0.16 to 0.11 for the US–Canadian city pairs and from 0.14 to 0.13 for the UK–Euro area city
pairs.
We confirm that the null hypothesis of full attention is robustly rejected. As we discuss inAppendix A.5, the null hypothesis is rejected in the case of the more general constant-relativerisk-aversion (CRRA) form. In this case, the estimation equation becomes complicated, anda test of behavioral inattention requires an instrumental variables estimator.28 Appendix A.5
25

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. Likewise, our main findings are robust, even if we replace
λ with the values reported by previous studies on price dynamics such as Bils and Klenow (2004) and Klenow
and Kryvtsov (2008) for the US–Canadian city pairs and Álvarez et al. (2006) for the UK–Euro area city
pairs.
26
While we do not report the result to conserve the space, we also allow for a fixed effect specific to both
good i and city pair j. We find that the estimated β is not significantly different.
27
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.
28
In the case of the more general CRRA form, firms expect a dynamic path for the labor supply from thetime of price setting to the infinite future. As a result, the estimation equation includes the one-period ahead

22derives the estimation equation together with the empirical results.
We also implement an alternative test for m = 1 as a robustness check. We regress ln qit
directly on ln qt with additional 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. In Appendix A.6, we discuss that full attention is not supported by the
data.

5

Explaining 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.

5.1

Persistence of the aggregate real exchange rate

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 as 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.
good-level real exchange rate and aggregate real exchange rate. Thus, our test requires the instrumental
variable estimator because of the correlation of explanatory variables with forecast errors embedded in theerror term.

23

Proposition 2 implies that the presence of behavioral inattention assists 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 thanλ. In the extreme case of m → 0, the aggregate real exchange rate can even follow a random
walk, since 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 prices (λ = 0) in Propositions 1 and 2 because (41)suggests that λ = 0 leads to no PPP deviations, even in the short run (i.e., ln qt = 0 for allt). Our model thus requires nominal rigidities as the external source of the persistence of
the aggregate real exchange rate. We can best appreciate this feature of our model in thecontext of real rigidities in Ball and Romer (1990) or strategic complementarity 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 and a small friction in the nominal price adjustment matters for the real effect of anominal shock. In our model, a combination of behavioral inattention and a small friction in
the nominal price adjustment matters for the persistent aggregate real exchange rate.
Figure 2 shows how the persistence of aggregate real exchange rate changes as m changes.
The left panel plots ρq against m ∈ (0, 1] when λ is calibrated at 0.34. For reference, thefigure also plots the line of the lower bound of ρq : λ = 0.34. Starting from ρq = λ whenm = 1, ρq increases monotonically as m decreases. The persistence becomes closer to unity
as m approaches zero. The right panel 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 figure
indicates that the ρq to λ ratio can be quite large depending on m.
The estimated degrees of attention suggest that behavioral inattention makes the PPP
deviations more than twice as persistent as what is predicted only by the degree of pricestickiness. In the left panel of Figure 2, ρq = 0.34 if m = 1. However, the left panel of Figure
2 indicates that ρq = 0.76 if we employ m = 0.16 in specification (1) of Table 1 as a calibrated
value for the US–Canadian city pairs. We also see from the right panel of the same figurethat these calibrated values generate the ρq to λ ratio that exceeds two. In particular, the ρq
to λ ratio is 2.24 when m = 0.16. When we take m = 0.14 using specification (1) of Table 1
24

for the UK–Euro area city pairs), ρq becomes 0.79 and the ρq to λ ratio is 2.31.
Let us evaluate the half-life of the aggregate real exchange rate. The upper panel of Table
3 compares the predicted half-lives of the aggregate real exchange rate with the half-lives
observed in our data. Taking the aggregate real exchange rate used for our regressions, weestimate half-lives from the AR(1) model for ln qt .29 As shown in the third column (headed“Data”), the observed half-life for the US–Canadian city pairs is 4.92 years. The aggregate
real exchange rate for the UK–Euro area city pairs exhibits a lower half-life of 2.40 years.30
How much can the estimated degree of inattention explain the observed persistence of theaggregate real exchange rate? Regarding the US–Canadian city pairs, the predicted half-lifeis 2.62 years when we use m = 0.16 in specification (1) in Table 1. In the second column ofTable 3 (the column denoted by “95% CI”), we allow for estimation uncertainty of m̂ basedon its standard errors. In this case, the predicted half-life for the US–Canadian city pairs
ranges from 1.99 to 4.01 years. Thus, the predicted half-life under m = 0.16 falls short of theobserved half-life of 4.92 years. However, if we use m = 0.11 in specification (1) of Table 2,
the model is more successful than the previous case. In the present case, where m reduces to
0.11, the half-lives predicted by the model with behavioral inattention become longer. The
predicted half-life is 3.70 years, and its range is from 2.52 to 7.61 years, which includes theobserved half-life of 4.92 years.
For the UK–Euro area city pairs, the predicted half-life is 2.81 years ranging from 1.90 to
6.13 years when we use m = 0.14, the estimate in specification (1) of Table 1. The half-life
predicted from the point estimate exceeds the observed half-life of 2.40 years. However, therange of the predicted half-life that allows for estimation uncertainty contains the observed
half-life. Therefore, the model successfully explains the observed half-life for the UK–Euro
area city pairs. We only observe a small reduction in m̂ from 0.14 to 0.13 when we use theestimate in specification (1) of Table 2. Thus, the model continues to explain the observed
half-life for the UK–Euro area city pairs.
We emphasize that the model with behavioral inattention outperforms the model withfull attention. When m = 1, the first-order autocorrelation of the aggregate real exchange
rate is only 0.34 because ρq = λ = 0.34. This low persistence of the aggregate real exchange
rate translates into a very short half-life of just 0.64 years. Given that the half-lives predicted
29

We calculate the half-lives for the AR(1) process from the standard formula given by − ln(2)/ ln ρ, whereρ is the AR(1) coefficient.
30
Note that we have multiple aggregate real exchange rates for the UK–Euro area city pairs because theconsumer price indices differ across Euro area countries. The half-life of 2.40 years reported in Table 3 for
the UK–Euro area city pairs results from the mean of the estimated half-lives in each country pair to which
the UK–Euro area city pairs belong.

25

from the point estimates are roughly three years, we conclude that behavioral inattention
increases the half-life of aggregate real exchange rates by 2.4 years.

5.2

Persistence of the good-level real exchange rate

We next turn to the good-level real exchange rate. We let ρqi be the first-order autocorrelationof the good-level real exchange rate implied by (29). The following proposition describes therelationship between the persistence of the good-level real exchange rates and that of theaggregate real exchange rate, as predicted by the model.
Proposition 3 Under the same assumptions as 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.7.
Proposition 3 explains the stylized fact that good-level real exchange rates are much lesspersistent than the aggregate real exchange rate. Importantly, we obtain this aggregation
result without relying on the “aggregation bias” pointed out by Imbs et al. (2005). They emphasized that heterogeneity in the persistence of the good-level real exchange rates induces apositive bias in the persistence of the aggregate real exchange rate. Using multisector stickyprice models with heterogeneity in the degree of price stickiness, Carvalho and Nechio (2011)
successfully explain the positive bias. By contrast, our model intentionally assumes homogeneity in the persistence across goods. Nevertheless, our model can qualitatively explain the
gap in persistence between the aggregate and the good-level real exchange rates.
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.7. Similar to the ρq to λ ratio in (43), the ρq to ρqi ratio26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.
While the behavioral inattention (m < 1) is necessary for ρqi < ρq , it is not sufficientfor explaining the gap between ρq and ρqi . What is additionally needed is real friction.
More specifically, trade cost (τ ) needs to be strictly positive and the elasticity of substitution
across brands (ε) needs to be larger than one for the ρq to ρqi ratio (45) to be strictly greaterthan one. If τ = 0 or ε → 1, there is no home bias (ω = 1/ 1 + (1 + τ )1−ε = 1/2) sothat ψ = 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 )to nominal 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).31,32 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.33 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 δis set to 0.98, and the degree of price stickiness λ is again set to 0.34.
Figure 3 illustrates the extent to which the good-level real exchange rate becomes lesspersistent than the aggregate real exchange rate against the degree of attention. The left
panel plots ρqi against m ∈ (0, 1] in the dashed line. It also includes the curve for ρq takenfrom 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 represents
the ρq to ρqi ratio along with the lower bound of unity. The panel indicates that the ρq to ρqi
ratio is hump shaped against m ∈ (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
31

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).
32
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.
33
For example, Chari et al. (2002) calibrate the degree of home bias as 0.76, whereas Steinsson (2008) uses
0.94.

27

ρq = ρqi so that ρq /ρqi = 1 holds.34 However, when 0 < m < 1, the ρq to ρqi ratio exceeds
unity.
The estimated degrees of attention suggest that inattention reduces the persistence of thegood-level real exchange rate to less than two-thirds of that of the aggregate real exchange
rate. Suppose that m = 0.16, which is the estimated degree of attention in the US–Canadiancity pairs. The left panel of Figure 3 shows that ρqi is around 0.49 whereas ρq is 0.76. The
right panel indicates that the ρq to ρqi ratio is 1.55. Equivalently, ρqi is less than two-thirds ofρq (i.e., 0.49/0.76 < 2/3). We confirm that the estimated degree of attention in the UK–Euro
area city pairs generates similar results. When m = 0.14, ρqi = 0.51, ρq = 0.79. Thus, ourmodel predicts that ρqi is also less than two-thirds of ρq (0.51/0.79 < 2/3) in the UK–Euro
area city pairs.
The lower panel of Table 3 presents the predicted half-lives of the good-level real exchange
rate, together with the half-lives observed in the data. The rightmost column reports themedian half-lives of the good-level real exchange rates estimated from our dataset.35 In thedata over 1990–2015, we find that the half-life of the median goods is 1.61 years for the US–
Canadian city pairs and 1.18 years for the UK–Euro area city pairs, both of which are much
shorter than the half-lives of the aggregate real exchange rate shown in the same column ofthe upper panel. The estimated half-lives are also consistent with previous studies using EIU
data. For example, Crucini and Shintani (2008) find the half-life of median goods to range
from 1.03 to 1.61 years based on the EIU data from 1990–2005. Bergin et al. (2013) also
use the EIU data and construct the good-level real exchange rates of 20 cities in industrial
countries (including 16 European cities) relative to New York City between 1990 and 2007.
When they estimate the AR(1) model for the good-level real exchange rates, the averagehalf-life is 1.15 years.
How much can the estimated degree of inattention explain the observed persistence ofthe good-level real exchange rates? For the US–Canadian city pairs, m = 0.16 (the estimate
from specification (1) of Table 1) is again insufficient to explain the observed half-life of thegood-level real exchange rate for the US–Canadian city pairs. The predicted half-life is 0.98
34

Analytically, the results can be understood as follows. When m = 1, ρq /ρqi = 1 immediately follows from(45). When m → 0, ρq → 1 holds from the definition of ρq = λ/[1 − (1 − m)(1 − λ)]. As ρq → 1, A → 0,
which leads to ρq /ρqi → 1.
35
We estimate the panel AR(1) model of ln qijt for each good i, using the generalized method of moments
estimator of Arellano and Bond (1991). The first-order autocorrelation estimated from the panel AR(1) modelis transformed into the half-life. Typically, the good-by-good panel consists of more than 1,400 observationsin the US–Canadian city pairs and more than 700 observations for the UK–Euro area city pairs. Our medianhalf-lives reported in Table 3 are calculated from half-lives in which the number of observations exceeds 500
for the US–Canadian city pairs or 250 for the UK–Euro area city pairs.

28years and ranges between 0.85 and 1.29 years. The range of predicted half-life that allows for
estimation uncertainty thus falls short of the observed half-life of 1.61 years. However, themodel under m = 0.11 successfully explains the data for the US–Canadian city pairs. Recall
that in the previous section, we observed that the reduction in m increased the predictedhalf-life of the aggregate real exchange rate. This result also applies to the good-level realexchange rate. In particular, when m decreases from 0.16 to 0.11, the half-life for the US–
Canadian city pairs is now 1.22 years rather than 0.98 years, and the range of the predictedhalf-life includes the observed half-life of 1.61 years.
For the UK–Euro area city pairs, the predicted half-lives are 1.02 under m = 0.14 and1.07 years under m = 0.13. In both cases, the range of the predicted half-life contains theobserved half-life, so the model explains the persistence of the good-level real exchange rates
fairly well.
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 to
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)

As long as A > 0 and m < 1, the persistence of the good-level real exchange rate exceeds
λ. The result is also consistent with Kehoe and Midrigan’s (2007) finding that even thepersistence of the good-level real exchange rate is more persistent than what is predicted
only by the degree of price stickiness. Together with the result from (45), we can summarize
the relationship as ρq > ρqi > λ.
Second, our model of behavioral inattention can reproduce the findings by Bergin et al.
(2013), who analyze the persistence of good-level real exchange rate conditional on shocks.
Using a vector error correction model for each good, they find that the good-level real exchange rate is as persistent as the aggregate real exchange rate, conditional on macroeconomic
shock. We can analyze q̂it conditional on macroeconomic shock by setting σr = 0. As wediscuss earlier, σr = 0 implies A = 0. Therefore, (45) and (47) implies that ρq = ρqi > λ,which is consistent with the empirical finding by Bergin et al. (2013).

29

6Conclusion

In this paper, we explain two empirical anomalies. First, 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 the deviations from the LOP
are often less persistent than the PPP deviations. To reconcile the PPP and LOP evidence,
we adapted the model of behavioral inattention in Gabaix (2014) to a simple two-country,
sticky-price model. We showed 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, we implemented a test of behavioral inattention and quantified its importance. We found strong evidence consistent with behavioral inattention. The
complementarity in our model with behavioral inattention produces an aggregate real exchange rate that is more than twice as persistent as the real exchange rate explained only by
sticky prices. Our model also predicts that the persistence of the LOP deviations is less than
two-thirds of the persistence of the PPP deviations. We showed that the model quantitatively
replicates the observed half-lives of both the aggregate and the good-level real exchange rates.
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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33

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.36 The objective function of US
firms that sell their brand in US markets is given by:
∞
X


Wt+k
vit (z) = Et
cit,t+k (z),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



Pt
wt+k Pt+kvit (z) = Et
λ δt,t+kpit (z) −cit,t+k (z).
Pat+kit+k Pt
k=0
k

(50)

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 inthe steady state. Rewriting (50) yields (6):
!#
"
∞
X
P
t
πt+l − âit+k
cit,t+k (z),vit (z) = Et
λk δt,t+kp̄i (z) exp [p̂it (z)] − w̄ exp ŵt+k +
Pt+k
l=0
k=0
∞
X

where Pt+k /Pt =Qk

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

i

= exp

hP
k

i
π
l=1 t+l . For thedemand 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) =p̄i (z)
p̄i

−ε

(

"

exp −ε p̂it (z) −kX#)πt+l − p̂it+k

cit+k .

(51)

l=1

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, the36

For example, see Galı́ (2015).

34

objective 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 (52) as follows:vit∗ (z)

∞
X

 ∗

∗
St+k Pt+k
Pit (z) Pt∗
Pt+k Wt+k /Pt+k
cit,t+k (z)λ δt,t+k− (1 + τ )
= Et
∗
∗
Pt+k
Pt∗ Pt+k
St+k Pt+k
ait+k
k=0


∞
X
wt+k
Pt∗
k
∗
= Et
λ δt,t+k qt+k pit (z) ∗ − (1 + τ )cit,t+k (z)Pqt+k ait+k
t+k
k=0

∞
∗ 
X
Pt∗
wt+k Pt+kk∗
λ δt,t+k qt+k ∗= Etp (z) − (1 + τ )cit,t+k (z).
Pt+k itqt+k ait+k Pt∗
k=0kAgain, 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

P∗
× ∗t
Pt+k

(

k
X
w̄
∗
∗
∗
πt+l
− âit+kp̄i (z) exp [p̂it (z)] − (1 + τ ) exp ŵt+k − q̂t+k +
q̄
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].
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
∗
∗
= Et
λk δt,t+kp̄∗i (z) exp [p̂∗it (z)] − w̄∗ exp ŵt+k+
πt+l − â∗it+kc∗it,t+k (z),
∗
P
t+k
k=0
l=1
vit∗ (z) = Et∗
λk δt,t+k∗
1/Pt+k35

for 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


∗ 
∗
∗
Wt+k
/Pt+k
St+k Pt+k
Pit (z) Pt
Pt+k− (1 + τ )cit,t+k (z)= Et∗
St+k Pt+k
Pt Pt+ka∗it+k
Pt+k
k=0


∞
∗
X
wt+k
qt+k
Pt
−1
k ∗
= Et
− (1 + τ ) ∗λ δt,t+k qt+k pit (z)cit,t+k (z)
Pt+k
ait+k
k=0


∞
∗
X
wt+k
qt+k Pt+k−1 Ptk ∗pit (z) − (1 + τ )cit,t+k (z)= Et
λ δt,t+k qt+k
∗
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).
To obtain (13) and (14), we first take the approximation of EvHi [pHi (µ̂Ht , mH ), µ̂Ht , 1]around µ̂Ht = 0. 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 next evaluate
the price in the approximated equation at mH = 1. The second-order approximation of36

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
1
∂
v
Hi
Hi
0
' vHi
+
+
Eµ̂2Ht+k2
2 k=0 ∂ p̂it (z)2
∂ µ̂Ht+k
∂
µ̂
Ht+k
k=0


∞
0
X
∂ p̂Hit (0, mH )∂ 2 vHiEµ̂2Ht+k ,
+
∂
p̂
∂
µ̂
it (z)∂ µ̂Ht+k
Ht+k
k=0

(54)

0
where vHi
= vHi [p̂Hi (µ̂Ht , mH ), µ̂Ht , 1]µ̂Ht =0 = vHi (p̂Hi (0, mH ), 0, 1). For the second deriva00tives, ∂ 2 vHi/∂ p̂it (z)2 = ∂ 2 vHi (p̂Hi (0, mH ), 0, 1)/∂ p̂it (z)2 and ∂ 2 vHi/∂ µ̂2Ht+k= ∂ 2 vHi (p̂Hi (0, mH ), 0, 1) /∂ µ̂2Ht+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 ) ∂ p̂Hit (0, mH )∂ 2 vHi (p̂Hit (0, mH ), 0, mH )=−, fork = 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 µ̂Ht − (1 − λδ)âit (see (20)). Thus,∂ p̂Hi (µ̂Ht , mH ) n mH fork = 0=
0fork 6= 0
∂ µ̂Ht+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
202mH − 2mH E(µ̂Ht ) +' vHi +
E(µ̂2Ht+k ).
22
2
2 ∂ p̂it (z)∂ p̂it (z)2 k=0 ∂ µ̂Ht+k(57)

We further need the second-order approximation of EvHi [pHi (µ̂Ht , mH ), µ̂Ht , 1] where theprice 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] around37µ̂Ht = 0 is
EvHi [p̂Hi (µ̂Ht , mH ), µ̂Ht , 1] − EvHi [p̂Hi (µ̂Ht , 1), µ̂Ht , 1]0
 ∂ 2 vHi
1m2H − 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)

where we used (14) for the last equality.
Although 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 C(mH ) = (κ/2)m2H . Formally, the choice of attention for US firms that sell theirgoods in US markets is characterized by:1κ
min [(1 − mH )2 ΛH ] + m2H , whereΛH = −mH ∈[0,1] 22

∂ 2 vHi∂ p̂2it (z) [0, 0, 1]V ar(µ̂Ht ).

The remaining sparse max can analogously be defined. The sparse max for US firms
selling their goods in Canadian markets is
κ ∗ 2
1
∗ 2 ∗
(1
−
m
)
Λ
+
(m ) , whereΛ∗H = −min
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 isκ1
min (1 − m∗F )2 Λ∗F + (m∗F )2 , whereΛ∗F = −∗mF ∈[0,1] 2
2

 2 ∗

∂ vF i [0, 0, 1]
V ar(µ̂∗F t ).
∂ p̂∗2F it

By 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 isκ1
min (1 − mF )2 ΛF + m2F , whereΛF = −mF ∈[0,1] 22

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

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

38

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
(1 − λδ)Et(λδ)k (mH ŵt+k − âit+k )
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= mH ŵt − âit + Et(λδ)k (mH ∆ŵt+k − ∆ait+k ).
k=1

Next, the remaining terms are
∞
k
X
X
k
(λδ)
πt+l
mH (1 − λδ)Et
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
(λδ)k πt+3 + ...
= mH (1 − λδ)Et (λδ)(λδ)k πt+1 + (λδ)2(λδ)k πt+2 + (λδ)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) .

39

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 − χn, 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= (m∗H ŵt∗ − âit ) + Et

∞
X

∗
∗
(λδ)k [mH (∆ŵt+k+ πt+k) − ∆âit+k ],

(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.

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 .

40

(63)

Thus, q̂it can be rewritten as:q̂it = (p̂∗it − ĉ∗t ) − (p̂it − ĉt ).

(64)

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)





opt
∗
−
ĉ
p̂opt∗
−
ĉ
−
p̂
,
t
t
it
it

(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 definedby (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 isit− ĉ∗t = −(1 − m)ĉ∗t − (1 − λδ)[ωâ∗it + (1 − ω)âit ],p̂opt∗
it41(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

The model with CRRA preferences

So far, we have assumed that the preferences of households are given by U (c, n) = ln c − χn.
In this appendix, we assume more general CRRA preferences, U (c, n) = c1−σ /(1 − σ) −χn1+ϕ /(1 + ϕ), where σ 6= 1 and ϕ 6= 0. 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 nominal exchange rate growth that is predictable using the inflation of the twocountries.37 Because this is inconsistent with the exchange-rate disconnect puzzle, we replace
this assumption with the new assumption on the money growth rate:1σ−1πt + ε M,σσ tσ − 1 ∗ 1 M∗=
πt + ε t .
σσ∆ ln Mt =(73)∆ ln Mt∗(74)

Under (73) and (74), the nominal exchange rate continues to follow a random walk.38
In particular, the nominal exchange rate growth is given by ∆st = σεnt + (1 − σ)(πt − πt∗ ), meaning that
πt − πt∗ can help forecast ∆st .
38
To see this, note that the nominal exchange rate growth is given by: ∆st = σ (∆ ln Mt − ∆ ln Mt∗ ) +(1 − σ) (πt − πt∗ ). Substituting (73) and (74) into the above equation yields ∆st = (σ − 1) (πt − πt∗ ) +M∗(1 − σ) (πt − πt∗ ) + εM
= εnt .
t − εt37

42

Using the CIA constraints, we can rewrite (73) and (74) as:σ∆ĉt+k + πt+k = εM
t+k ,
∗

∗
σ∆ĉ∗t+k + πt+k
= εM
t+k ,

(75)
(76)

for k > 0. Later, we utilize (75) and (76) for deriving the estimation equation.
A.5.1

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 ).

(77)

We focus on p̂it − σĉt and p̂∗t − σĉ∗t and obtain the expression for q̂it using (77). 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 ,(78)

where we replace σ∆ĉt + πt by εMt using (75). 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

(79)

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

(80)

Equations (78)–(80) generalize (66)–(68), respectively.
We next focus on the expression inside the brackets on the right-hand side of (80). For thecase of σ 6= 1 and ϕ 6= 0, 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 ). Even in the case of σ 6= 1 and ϕ 6= 0, (61) continues to hold.
However, ŵt is no longer equal to ĉt and is now given by ŵt = σĉt + ϕn̂t . Accordingly, we43rewrite (61) 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 ,

(81)

where L is the lag operator. In the second equality, we used (75) and replaced σ∆ĉt+k + πt+kby εMt+k , which greatly simplifies the equation.
Equation (81) 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 − λδ=n̂t ,1 − λδL−1
1 − λδ
∗
∗
∗
∗
∗
∗
∗
p̂F i (µ̂F t , mF ) = mF σĉt − (1 − λδ)âit + mH ϕn̂∗t ,−1
1 − λδL
1 − λδ
∗p̂F i (µ̂F t , mF ) = mF σĉt − (1 − λδ)âit + mF ϕn̂∗t ,−1
1 − λδLp̂∗Hi (µ̂∗Ht , m∗H )m∗H σĉ∗t − (1 − λδ)âit + m∗H ϕ

(82)
(83)
(84)

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

(85)

(86)

respectively. In (86), we assumed that m∗F = mH and m∗H = mF .
Plugging (85) and (86) into (80) yields
q̂it = λq̂it−1 + (1 − λ)(1 − m)q̂t + λεnt + (1 − λ)(1 − λδ)ψεrit− ϕψm(1 − λ)(1 − λδ)(n̂t − n̂∗t ),1 − λδL−1(87)

where ψm = ωmH − (1 − ω) mF . Equation (87) differs from (72) in that the former includes
the forward-looking terms for labor supply. If ϕ = 0, these forward-looking terms disappear,44

and the equation coincides with (72). Under our assumptions, σ does not appear in (87).
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(88)

which generalizes (29).
To derive the estimation equation for our empirical analysis, we use the definition of q̃itand q̃t and further rewrite (88) 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 )(89)−(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 ,(90)

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 as 2010.
Table A.1 reports the estimation results of (90). The left panel presents the results of
the US–Canadian city pairs, while the right panel points to the results of the UK–Euro area
city pairs. In both panels, we assume a common λ in specifications (1) and (2) and thegood-specific λ in specifications (3) and (4). Specifications (2) and (4) include the city-pair-45

specific fixed effects as additional explanatory variables. We instrument ln q̃˜t by ln q̃t−1 in all
specifications. In all cases, the null hypothesis of full attention, namely β = 0, is significantly
rejected. The estimated values of m are much smaller than one, suggesting robustness tochanges in the assumption of preferences.

A.6

Estimation results for (40)

Table A.2 reports the estimation results for (40). Unlike the case of (36), the presence of alagged dependent variable on the right-hand side of (40) implies a dynamic panel structure.
Therefore, dynamic panel regression estimators, such as the generalized method of moments
estimator of Arellano and Bond (1991), need to be employed in place of OLS (see, e.g.,
Crucini and Shintani, 2008 and Crucini et al., 2010a). The left panel of the table presents
the estimation results of the US–Canadian city pairs, whereas the right panel shows thoseof the UK–Euro area city pairs. In specifications (2) and (4), we impose the restriction thatthe coefficients on ln qit−1 and ∆ ln St as control variables are the same as each other. This isbecause (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 the theory.
Therefore, even if we directly regress ln qijt on ln qt , the estimation results are consistent withthe behavioral inattention of 0 < m < 1.

A.7

Persistence of the good-level real exchange rate

A.7.1

Proof of Proposition 3

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

(91)

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

46

(92)

In Appendix A.4, we have shown that the log deviation of the LOP deviations isq̂it = λq̂it−1 + θq̂t + λεnt + ψ̃εrit ,(93)

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 ,(94)

R1 = ρq R0 ,(95)which can be derived from (91) and (93).
We further simplify (94) and (95). Substitute (92) and (95) into (94) to getλ(1 − ρ2q ) 2
σq .
R0 = λρq R0 + θ +
ρq


(96)

Note that, using the definition of ρq , the expression inside the brackets can be simplified as39θ+λ(1 − ρ2q )= 1 − λρq .
ρq

(97)

R0 = σq2 ,(98)

R1 = ρq σq2 ,(99)

Using (97), (96) and (95) become

respectively.
2
We next work on the variance and the autocovariance that are denoted as σqi= Eq̂it2 andγ1 = Eq̂it q̂it−1 , respectively. Using (93), (98), and (99), we have2
σqi
= λγ1 + θσq2 + λρq σn2 + ψ̃ 2 σr2 ,

(100)

2
+ θρq σq2 .
γ1 = λσqi

(101)

39

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 (97).

47

We obtain (100) and (101) from tedious algebra. Regarding (100), we use (93) and (98)2
σqi
= 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 (100):2= λγ1 + θσq2 + λE[(λq̂it−1 + θq̂t + λεnt + ψ̃εrit )εnt ] + ψ̃E[(λq̂it−1 + θq̂t + λεnt + ψ̃εrit )εrit ]σqi= 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 (91), and the fourth equality is from the definition of ρq .
Regarding (101), use (93) and (99) to get2γ1 = Eq̂it q̂it−1 = λE(q̂it−1 q̂it−1 ) + θE(q̂t q̂it−1 ) + λE(εnt q̂it−1 ) + ψ̃E(εrit q̂it−1 ) = λσqi
+ θρq σq2 .

2
We further simplify σqi
and γ1 . Using (92) and (101), (100) becomes2
σqi= λγ1 + θσq2 + λρq σn2 + ψ̃ 2 σr2λ(1 − ρ2q ) 2= λγ1 + θ +σq + ψ̃ 2 σr2ρqλ(1 − ρ2q ) 2
2 2
2= λ σqi + λθρq σq + θ +σq + ψ̃ 2 σr2 .
ρqRecall that, from (97), 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

48

2
where we use 1 − λρq (1 − θ) = 1 − λ2 given ρq = λ/(1 − θ). Therefore, σqi
and γ1 are

2
σqi

γ1

ψ̃ 2
=
σr2 .
2
" 1−λ
#
2
ψ̃
= λ σq2 +σ 2 + θρq σq2
1 − λ2 rσq2 +=(λ + θρq )σq2 += ρq σq2 +(102)

λψ̃ 2 2σ
1 − λ2 rλψ̃ 2 2σ .
1 − λ2 r(103)

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,(104)

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(105)

Equation (104) means that ρqi is the weighted average of ρq and λ. When we combine
Proposition 2, namely ρq ≥ λ, with (104), it immediately follows that ρq ≥ ρqi ≥ λ.
A.7.2

Derivation of (45) and (46)

Using ρq = λ/(1 − θ), eliminate λ from (104):ρqi = ωρ ρq + (1 − ωρ ) (1 − θ) ρq = ρq [1 − θ (1 − ωρ )] .

(106)

Recall (92) and the definition of ψ̃. Then, (105) becomes (46):
1 − ρ2q
A = (1 − λ) (1 − λδ) ψ 2
ρq (1 − λ2 )
2

2

49

2



σr
σn

2
.

(107)

From ωρ = 1/ (1 + A), 1 − ωρ = A/(1 + A). In addition, recall that θ = (1 − λ)(1 − m).
Therefore, (106) implies
ρq
1
=
.
A
ρqi
1 − (1 − λ) (1 − m) 1+A

50

(108)

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 (US−Canadian city pair)
2015

0

.2

density
.4

.6

.8

1990

2

−2

−1
0
1
the log of good−level real exchange rate (UK−Euro area city pair)
1990

51

2015

2

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

1

3

0.9

2.8

0.8

2.6

/ with
/ =1

q

2.4

0.7

2.2

0.6
/

0.5

2

q

q

= 0.34

q

1.8
0.4
1.6
0.3
1.4
0.2

1.2
with
q

0.1

= 0.341

lower bound ( = 0.34)

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

/ qi

q

0.8

/ qi = 1

q

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

lower bound ( = 0.34)

0

0

0.2

0.4

0.6

0.8

1

0

m

0.2

0.4

0.6
m

52

0.8

1

53
N

0.156

m̂

Y

N

0.225
Y

Y

0.262

389,500(0.029)

0.806

(4)

N

N

0.265

214,115(0.042)

0.856

(1)

N

Y

0.323

214,115(0.043)

0.851

(2)

0.188

0.194

0.144

0.149

0.147

Y

N

0.265

213,064(0.042)

0.853

(3)

The implied degree of attention from the regression
0.198

N

Y

0.262

389,500(0.029)

0.812

(3)

0.868

(4)

0.132

Y

Y

0.324

213,064(0.042)

UK–Euro area city pairs

NOTES: The left panel reports the estimation results based on 274 items and 64 US–Canadian city pairs. The right panel presents the estimation
results based on 301 items and 36 UK–Euro area city pairs. The data used for the regressions cover the period from 1990 to 2015. The table
reports the regression coefficients on (1 − λ) ln qt . The numbers in parentheses below the coefficients are the standard errors clustered by goods. Eachspecification includes good-specific fixed effects. Specifications (2) and (4) include city-pair fixed effects. In specifications (3) and (4), we control forthe difference in the log of real GDP per hour worked. Regressions for the US–Canadian city pairs in the left panel also include dummy variables that
control for the difference in timing of the price survey in Calgary from 2003 to 2014. Each column reports the estimate of m in the bottom row. The“Adj. R2 ” denotes the adjusted R-squared.

N

Control for ηtr

0.225

389,500(0.028)(0.030)389,500

0.802

(2)

0.844

city-pair FE

Adj. R

2

Observations

ln q̃t(1)

US–Canadian city pairs

Dependent variable: ln q̃ijt with λ = 0.34

Table 1: Estimation results of (36) under common λ

54
N
0.106

m̂

Y

N

0.256
Y

Y

0.294

389,500(0.033)

0.880

(4)

N

N

0.246

171,606(0.047)

0.866

(1)

N

Y

0.295

171,606(0.049)

0.834

(2)

0.117

0.120

0.134

0.166

0.136

Y

N

0.247

170,750(0.048)

0.864

(3)

The implied degree of attention from the regression
0.138

N

Y

0.294

389,500(0.032)

0.883

(3)

0.840

(4)

0.160

Y

Y

0.295

170,750(0.049)UK–Euro area city pairs

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 the previous studies.
The calibrated λi in the US–Canadian city pairs is from Nakamura and Steinsson (2008), and that in the UK–Euro area city pairs is from Gautier etal. (2022). The table reports the regression coefficients on ln q̃ti = ln(qt /qtλi ). See the notes to Table 1 for the remaining details.

N

Control for ηtr

0.256

389,500(0.028)(0.029)389,500

0.862

(2)

0.894

(1)

city-pair FE

Adj. R

2

Observations

ln q̃ti

US–Canadian city pairs

Dependent variable: ln q̃ijt with good-specific λTable 2: Estimation results of (36) under good-specific λTable 3: Half-lives implied by the estimated degree of attention
Half-lives of the aggregate real exchange rate
Predicted half-life

95% CIm̂ = 0.156

2.620[1.989, 4.010]m̂ = 0.106

3.704[2.524, 7.605]m̂ = 0.144

2.812[1.903, 6.129]m̂ = 0.134

2.998[1.905, 8.868]

Data

US–Canadian city pairs
4.922

UK–Euro area city pairs
2.398

Half-lives of the good-level real exchange rate
Predicted half-life

95% CIm̂ = 0.156

0.984[0.851, 1.292]m̂ = 0.106

1.223[0.963, 2.110]m̂ = 0.144

1.026[0.834, 1.773]m̂ = 0.134

1.066[0.834, 2.399]

Data

US–Canadian city pairs
1.606

UK–Euro area city pairs
1.182

NOTES: The table reports the predicted half-lives of the aggregate and good-level real exchange rates. The
unit of half-lives is a year, and the half-life under full attention is 0.64 years. The upper panel presents thehalf-lives of the aggregate real exchange rate, and the lower panel shows those of the good-level real exchange
rate. To calculate the predicted half-lives in the table, we use the calibrated values of τ = 0.74, ε = 4,
σr /σ∆s = 5, and δ = 0.98. In all calculations, λ is kept constant at λ = 0.34.
In each panel, we report the half-lives for US–Canadian city pairs and UK–Euro area city pairs. The
first column of the table reports the half-lives predicted by the model with partial attention, and the second
column is their 95% confidence intervals denoted by “95% CI”. We compute the half-lives from m̂ and the
95% confidence intervals of m̂ based on specification (1) of Tables 1 and 2. For comparisons, the rightmost
column presents the half-lives estimated from the EIU data. See the main text for the estimation of thehalf-lives from the EIU data.

55

56

(0.034)(0.034)

0.218

m

0.200

Y

0.206
Y

0.269

371,347(0.041)0.870

good-specific

(4)

N

0.217

201,578(0.050)

0.784

0.34

(1)

Y

0.261

201,578(0.052)

0.768

0.34

(2)

0.147

0.130

0.216

0.232

(3)

0.209

N

0.201

161,931(0.063)

0.791

good-specific

The implied degree of attention from the regression

N

0.242

371,347(0.041)

0.853

good-specific

(3)

(4)

0.265

Y

0.237

161,931(0.065)0.735

good-specific

UK–Euro area city pairs

NOTES: The dependent variable is ln q̃˜ijt = ln q̃ijt −λδ ln q̃ijt+1 , where λ is calibrated at either common or good-specific λ. In each panel, specifications(1) and (2) report the results using common λ whereas specifications (3) and (4) report those using good-specific λ. The table reports the regression
coefficients on ln q̃˜t = ln q̃t − λδ q̃t+1 . We use the instrument ln q̃t−1 to estimate parameters. See the notes in Table 1 for the remaining details.

N

0.179

371,347

0.804

0.782
371,347

0.34

(2)

0.34

city-pair FE

Adj. R

2

Observations

λln q̃˜t(1)

US–Canadian city pairs

Dependent variable: ln q̃˜ijt

Table A.1: Estimation results of (90)57
N

Control for ηtr
N

0.925
Y

0.922

371,347(0.025)

0.264

(3)

Y

0.926

371,347(0.022)

0.261

(4)

N

0.489

202,621(0.029)

0.077

(1)

N

0.485

202,621(0.026)

0.107

(2)

Y

0.488

201,578(0.029)

0.074

(3)

0.103

(4)

Y

0.496

201,578(0.027)

UK–Euro area city pairs

NOTES: The estimation equation is (40). Following Arellano and Bond (1991), we regress ∆ ln qijt on ∆ ln qt , ∆ ln qijt−1 , and ∆2 ln St along with theother control variables used in the benchmark regressions. In specifications (2) and (4), we impose the parameter restrictions that the coefficients on
ln qijt−1 and ∆ ln St are the same. In specifications (3) and (4), we control for the difference in the log of real 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. “P-valuefor Hansen’s J-test” denotes the p-value for the test of the over-identifying restrictions. The degrees of freedom for the chi-squared distribution under
the null are 299 in specifications (1) and (3) and 300 in specifications (2) and (4). See the notes to Table 1 for the remaining details.

0.932

P-value for Hansen’s J-test

371,347(0.020)(0.021)
371,347

0.300

(2)

0.302

Observations

ln qt

(1)

US–Canadian city pairs

Dependent variable: ln qijt

Table A.2: Estimation results of (40)