NBER WORKING PAPER SERIESACCOUNTING FOR PERSISTENCE AND VOLATILITY OF GOOD-LEVEL REAL EXCHANGE RATES:THE ROLE OF STICKY INFORMATION
Mario J. Crucini
Mototsugu Shintani
Takayuki Tsuruga
Working Paper 14381
http://www.nber.org/papers/w14381

NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
October 2008

We would like to thank Eiji Fujii, seminar and conference participants at Duke, Hokkaido, North Carolina
State and Osaka University, the Federal Reserve Bank of Dallas, 2008 Midwest Macro Meetings, 2007
9th TCER Macroeconomic Conference for helpful comments and discussions. Mario J. Crucini gratefully
acknowledges the financial support of National Science Foundation. The views expressed in the paper
are those of the authors and are not reflective of those of the Bank of Japan or the National Bureau
of Economic Research.
NBER working papers are circulated for discussion and comment purposes. They have not been peerreviewed or been subject to the review by the NBER Board of Directors that accompanies official
NBER publications.
© 2008 by Mario J. Crucini, Mototsugu Shintani, and Takayuki Tsuruga. All rights reserved. Shortsections of text, not to exceed two paragraphs, may be quoted without explicit permission provided
that full credit, including © notice, is given to the source.

Accounting for Persistence and Volatility of Good-Level Real Exchange Rates: The Roleof Sticky Information
Mario J. Crucini, Mototsugu Shintani, and Takayuki Tsuruga
NBER Working Paper No. 14381
October 2008
JEL No. D40,E31,F31
ABSTRACT
Volatile and persistent real exchange rates are observed not only in aggregate series but also in theindividual good level data. Kehoe and Midrigan (2007) recently showed that, under a standard assumption
on nominal price stickiness, empirical frequencies of micro price adjustment cannot replicate the time-series
properties of the law-of-one-price deviations. We extend their sticky price model by combining good
specific price adjustment with information stickiness in the sense of Mankiw and Reis (2002). Undera reasonable assumption on the money growth process, we show that the model fully explains both
persistence and volatility of the good-level real exchange rates. Furthermore, our framework allows
for multiple cities within a country. Using a panel of U.S.-Canadian city pairs, we estimate a dynamic
price adjustment process for each 165 individual goods. The empirical result suggests that the dispersion
of average time of information update across goods is comparable to that of average time of price adjustment.

Mario J. Crucini
Department of Economics
Vanderbilt University
Box 1819 Station B
Nashville, TN 37235-1819
and NBER
mario.j.crucini@vanderbilt.edu
Mototsugu Shintani
Vanderbilt University
mototsugu.shintani@vanderbilt.edu

Takayuki Tsuruga
Faculty of Economics, Kansai University
3-3-35 Senri-cho #105
Suita, 564-8680
Japan
ttsuruga@kansai-u.ac.jp

1

Introduction
Aggregate real exchange rates are among the most scrutinized of economic variables because

their persistence and volatility are much higher than what economists believe is consistent with aplausible degree of price rigidity. The time-dependent pricing model offers a convenient theoreticallink between price stickiness and the stochastic properties of real exchange rates. Chari, Kehoe, andMcGrattan (2002, CKM) show that to generate the observed persistence of CPI-based aggregate
real exchange rates, prices need to be exogenously fixed for at least one year. This degree of pricestickiness, however, appears implausible based on recent evidence by Bils and Klenow (2004) who
find a median duration between price changes of only 4.3 months in U.S. micro-data.
An emerging literature using international micro-data finds the half-life of deviations from theLaw of One Price (LOP) for the median good in the neighborhood of 18 months, considerably
lower than the consensus 3-5 year half-lives of aggregate real exchange rates (Crucini and Shintani(2008)). This evidence suggests that studies using prices of individual goods, rather than priceindices, is a promising approach for evaluating time-dependent pricing models and understanding
short-run international relative price dynamics. An important contribution along this line is Kehoeand Midrigan (2007) who allow different price stickiness across individual goods and show thatthe persistence in LOP deviations is equal to ‘the Calvo parameter,’ the probability of price nonadjustment at the good level. Their empirical analysis using real exchange rates of 66 individual
goods shows that the frequency of no price adjustment is higher for goods that exhibit more
persistent deviations from the LOP, as suggested by the theoretical model. However, the persistence
puzzle is still not resolved in the sense that the observed frequencies of micro price changes are
too high to replicate the persistence of real exchange rates for most goods in the cross-section.
In addition, the model does not match the time series variability of LOP deviations observed in
the micro-data. These theoretical and empirical results point to the need to break the tight link
between the frequency of price adjustment and the LOP persistence parameter characterizing thestandard Calvo-type sticky price model.
We break this tight link by extending the Kehoe-Midrigan model to allow for information stickiness. That is, in addition to the standard Calvo pricing, we assume only a fraction of firms update
their information set each month. Thus price dynamics become a convolution of price adjustment
timing and information updating. In the macroeconomic literature, Mankiw and Reis (2002) showthat a model of information stickiness, or inattentiveness, is capable of explaining the observed slow2

response of aggregate inflation to monetary shocks much better than sticky prices alone. When theinformation stickiness augments the Calvo-type sticky price mechanism, less frequent information
updating leads to higher price persistence, at a given frequency of price adjustment (Dupor, Kitamura, and Tsuruga (2008, DKT)). With plausible assumptions on international money growth
processes, a similar effect takes place to increase both the persistence and volatility of real exchange
rates.
In addition to the generalization of the sticky price model to allow for the information stickiness, our analysis differs from Kehoe and Midrigan (2007) in several ways. First, our empiricalanalysis is based on an international retail price survey which records local currency prices forhighly disaggregated individual goods and services spanning most of the CPI basket. Using thissurvey we expand the number of products from 66 products used in Kehoe and Midrigan (2007)
to 165.

Another advantage of this data is that the survey is conducted by a single agency, theEconomist Intelligence Unit, so we expect more comparability in the quality of the products amonginternational cities. An important limitation of our data is its annual frequency and relative short
time-span, from 1990 to 2005. As in the case of Crucini and Shintani (2008), the difficulty of
estimating persistence with short time-series is mitigated by utilizing the dynamic panel feature of
the data.
Second, our theoretical model allows for the presence of multiple cities in each country and forlong-run price deviations between the cross-border city pairs to differ by good and city pair. For
each good, we use the panel of 52 U.S.-Canadian city pairs to estimate a dynamic panel model andto compute the volatility under an error components model framework.
Third, we examine the effect of the exclusion of sales on the performance of sticky price models
in explaining real exchange rate dynamics. Recently, Nakamura and Steinsson (2008) claim thatthe evidence of the fast price adjustment reported by Bils and Klenow (2004) may be strongly
influenced by the presence of sales, or other temporary price reductions. Nakamura and Steinsson(2008) define the regular price change by excluding sales from the observed price change, and reportthat the median frequency of regular price changes increases to the range of 8 to 11 months. Since
prices are stickier based on this alternative definition of price change, it elevates the Calvo model’sability to account for important features of the data. This improvement is subject to the caveatthat our model does not explicitly model sales.
The main conclusions of Kehoe and Midrigan (2007) are robust to the change in the data.
Both persistence and volatility are much higher in the EIU data than the prediction of a standard3

Calvo-type sticky price model even if we use (i) more disaggregated retail price data, (ii) panel dataconsisting of multiple cities in the U.S. and Canada, and (iii) adjust the frequency of price changes
for temporary sales.
In contrast to the standard Calvo model, the extended model with information stickiness fully
accounts for both persistence and volatility when the average duration between information updates
is 14 to 17 months if sales are not removed and 9 to 12 months if sales are removed. The ability of
our model to replicate the observed persistence and volatility contrasts to another possible extensionof the Calvo model allowing for pricing complementarities. Kehoe and Midrigan (2007) show thatcomplementary leads to a modest improvement in explaining the persistence and little improvement
in explaining the variance.
The finding that the dispersion of price adjustment frequency and information updating frequency acoss goods are comparable is suggestive of joint endogeneity, raising a challenging identification issue. While the existing micro evidence on information stickiness is quite limited relative
to that on the frequency of price adjustment, our estimates of the information delay parameterare broadly consistent with the available survey evidence on the frequency with which firms conduct major information updates. Consequently, a deeper understanding of their separate roles will
require decision theory to explicitly account for the costs of collecting, updating and interpreting
economic data in real time. Such an approach is likely to offer insights beyond the traditional menu
cost story.1
This paper is organized as follows: Section 2 presents our model as a generalization of Kehoe andMidrigan’s model. In Section 3 we examine the model implications for the time series properties of
the good-level real exchange rates. Section 4 describes our data and how we use it to evaluate themodel. We also compare the benchmark sticky price model and our extended model. The study
ends with a discussion of future research in Section 5 .

2

The model

Trade is over a continuum of goods between two countries with multiple cities located in eachcountry. Under monopolistic competition, firms set prices in local currency to satisfy demand fora particular good in a particular city. A representative agent in each country chooses consumption
over an infinite horizon subject to a cash-in-advance (CIA) constraint. In what follows, the U.S.
1

Examples includes Sims (2003), Woodford (2008) and Gorodnichenko (2008).

4and Canada represent the home and foreign country, respectively, and the unit of time is one month.
The lowest level of aggregation is the brand, z of a particular good. U.S. brands of each good
are indexed z ∈ [0, 1/2] while those in Canada are indexed z ∈ (1/2, 1]. Integrating over brands,
we have the CES indices for consumption of good j in a U.S. city l and a Canadian city l∗ , givenbyZ 1
ct (j, l) =ct (j, l, z)

θ−1
θ

θ
 θ−1

(1)

dz

0

and
c∗t (j, l∗ ) =

Z 1

θ−1
c∗t (j, l∗ , z) θ dz

θ
 θ−1

,

(2)

0

where ct (j, l, z) is consumption of a brand z of good j in U.S. city denoted l and c∗t (j, l∗ , z) is theanalog consumption of that brand for a Canadian city, l∗ .
CES aggregation across cities l ∈ [0, 1] and l∗ ∈ [0, 1], gives national consumption of good jwithin the U.S.
Z
ct (j) =ct (j, l)

θ−1
θ

 θ

θ−1

dl

,

(3)

and Canada,c∗t (j) =Zθ−1c∗t (j, l∗ ) θ dl∗

 θ

θ−1

,

(4)

respectively.
CES aggregation across goods in each country gives aggregate consumption in the U.S., ct ,Zct =ct (j)

θ−1
θ

 θ

θ−1

dj

and Canada, c∗t ,
c∗t =

2.1

Z

θ−1
c∗t (j) θ dj

(5)

 θ

θ−1

.

(6)

Households

As in Kehoe and Midrigan (2007) complete markets for state-contingent money claims exist. Agentsdecide how many one-period nominal bonds to hold in each state of the world in period t + 1. U.S.
∗
households hold Bt+1 while Canadians hold Bt+1
(both denominated in the U.S. dollars).2 The

price of a bond issued at date t, maturing at date t + 1 is denoted by Qt,t+1 . Also, Qt,t+h is the2

As Kehoe and Midrigan (2007) argue, it does not matter if foreign (Canadian) consumers hold complete andstate-contingent one-period nominal bonds denominated in the foreign currency (Canadian dollars). It would be
simply a redundant assumption under state-contingent bond markets.

5nominal stochastic discount factor by which all firms, regardless of their country of origin, discountprofits earned in period t + h back to the present period t.
Households in the each country maximize the discounted sum of U (ct , nt ) = ln ct − χnt (χ > 0)
subject to an intertemporal budget constraint and a CIA constraint. The maximization problem
for U.S. households is:
E0∞X

β t U (ct , nt ),

(7)

t=0

s.t.

Mt + Et (Qt,t+1 Bt+1 ) = Rt−1 Wt−1 nt−1 + Bt + (Mt−1 − Pt−1 ct−1 ) + Tt + Πt ,(8)

Mt ≥ Pt ct ,(9)where β is the discount factor of the household satisfying 0 < β < 1 and Et (·) denotes the expectation operator conditional on the information available in period t.
The left hand side of the intertemporal budget constraint (8) represents the nominal value of
total wealth of the household brought into the beginning of period t + 1. It consists of cash holding
Mt and bond holdings Bt+1 . As shown in the right hand of (8), the household receives nominal
labor income Wt−1 nt−1 in period t − 1 which earns gross nominal interest Rt−1 per unit of labor
income until period t in terms of U.S. currency.3 Households carry nominal bonds in amount Bt
and cash holding remaining after consumption expenditures (Mt−1 − Pt−1 ct−1 ) into period t; Pt is
the aggregate price index defined below. Finally, Tt and Πt are nominal lump sum transfers fromthe U.S. government and nominal profits of firms operating in the U.S., respectively.4
R
1
The equation (9) is the CIA constraint. The aggregate price Pt is given by Pt = [ Pt (j)1−θ dj] 1−θ ,where Pt (j) is the aggregate price index for good j; it is a CES aggregate over city-specific prices
R
1
for that good: Pt (j) = [ Pt (j, l)1−θ dl] 1−θ . The price index for good j in a particular city l used inthis aggregation is given byZ
Pt (j, l) =Pt (j, l, z)

1−θ

 1

1−θ

dz

.

Households in Canada solve the analogous optimization problem except we must convert their
U.S. dollar bond holdings into Canadian dollars at the spot nominal exchange rate, St . Thus the
Canadian-dollar intertemporal budget constraint is
Mt∗ +
3

∗ )
Et (Qt,t+1 Bt+1
St−1 Rt−1 ∗ ∗B∗∗∗=
Wt−1 nt−1 + t + (Mt−1− Pt−1c∗t−1 ) + Tt∗ + Π∗t .
St
St
St

We assume that the government pays interest rate Rt (= 1/Et Qt,t+1 ) on labor income in period t. This assumption

allows households’ intratemporal optimality condition to be undistorted.
4
We assume that government’s lump sum transfers and firms’ profits in a country go to households in that country.

6

The first order conditions of households in both countries are as follows:
Wt
Pt
Wt∗
Pt∗

= χct

(10)

= χc∗t(11)"Et Qt,t+1 = βEt"Et Qt,t+1 = βEt

ct+1
ct

−1

c∗t+1
c∗t

−1

Pt
Pt+1

#

St Pt∗∗
St+1 Pt+1

(12)
#
(13)

Mt = Pt ct(14)

Mt∗ = Pt∗ c∗t .

(15)

The equations (10) and (11) represent intratemporal substitution between labor and consumption
while (12) and (13) represent intertemporal consumption choices across adjacent months. The intertemporal conditions, (12) and (13), are slightly different because Canadians buy state-contingent
one-period nominal bonds denominated in the U.S. dollars. The CIA constraints always bind as
shown in equations (14) and (15).
The nominal wage rate in a country is proportional to the stock of money held by households
in that country. Combining the intratemporal conditions (10) and (11) with the CIA constraints
we have:
Wt = χMt ,(16)

Wt∗ = χMt∗ .

(17)

The aggregate real exchange rate is determined by combining the home and foreign intertemporal conditions:qt =St Pt∗ct= κ ∗,
Pt
ct(18)

where κ = q0 c∗0 /c0 .5
The nominal exchange rate is determined by combining (18) with the CIA constraints (14) and(15):
St = κ
5

See Appendix A.

7

Mt
.
Mt∗

(19)

2.2

Firms

The output of brand z of good j in the U.S. is equal to the number of hours allocated to that
activity:yt (j, z) = nt (j, z) .

(20)

Goods are perishable, so the consumption of each good across all cities equals output of that goodin the current period:ZZ
ct (j, l, z)dl +(1 + τ (j, l∗ ))c∗t (j, l∗ , z)dl∗ = yt (j, z).

(21)

We allow for long-run deviations from the LOP across borders through τ (j, l∗ ), an iceberg transportation cost in exporting good j from the U.S. to a Canadian city indexed by l∗ . A firm mustship (1 + τ (j, l∗ )) units of good j to city l∗ for one unit of that good to arrive at the destination.
An analogous market clearing condition holds for each of the Canadian goods:
ZZ(1 + τ (j, l))ct (j, l, z)dl +2.3c∗t (j, l∗ , z)dl∗ = yt∗ (j, z).

(22)

Price adjustment and information updatingThis section begins by reviewing Calvo pricing used by Kehoe and Midrigan (2007) and thenpresents our extension to allow for information updating as in Mankiw and Reis (2002). The
equilibrium is briefly described in each setting.
2.3.1

Calvo pricing

We model the nominal price rigidities as in Calvo (1983) and Yun (1996): each month a fraction offirms 1 − λj are randomly drawn and allowed to reset their prices. As suggested by the subscript,
the frequency of price changes varies according to the type of good j and is assumed to be the same
in both countries, good-by-good.
All U.S. firms that sell their good j in city l choose the same optimal price when they adjust
prices in period t. The price PH,t (j, l) solves the following maximization problem:
max EtPH,t (j,l)∞
X

λhj Qt,t+h [PH,t (j, l) − Wt+h ]

h=0


×

PH,t (j, l)
Pt+h (j, l)−θ Pt+h (j, l)
Pt+h (j)8−θ Pt+h (j)
Pt+h

(23)

−θ
ct+h ,

for all cities l ∈ [0, 1]. Here, we used the three demand functions as constraints:
Pt (j) −θct (j) =ctPtPt (j, l) −θ
ct (j, l) =ct (j)Pt (j)

Pt (j, l, z) −θct (j, l, z) =ct (j, l).
Pt (j, l)
The optimality condition for PH,t (j, l) is∞XPH,t (j, l) −θhEt
λj Qt,t+h
ct+h
Pt+h
h=0



∞
X
PH,t (j, l) −θ
Wt+h
θ
h
Et
λj Qt,t+h
ct+h .
=
θ−1
PH,t (j, l)
Pt+h

(24)

h=0

Similarly, all Canadian firms that export and sell their good j in city l choose the same optimal
price PF,t (j, l) when they adjust prices. The price PF,t (j, l) for these firms solves the maximization
problem:max EtPF,t (j,l)

∞
X

∗
]
λhj Qt,t+h [PF,t (j, l) − (1 + τ (j, l))St+h Wt+h

h=0


×

PF,t (j, l)
Pt+h (j, l)−θ Pt+h (j, l)
Pt+h (j)−θ Pt+h (j)
Pt+h

(25)

−θ
ct+h ,

for all cities l ∈ [0, 1]. The optimality condition is of the form similar to (24):∞X
PF,t (j, l) −θhEt
λj Qt,t+h
ct+h
Pt+h
h=0


∞
∗ 
X
(1 + τ (j, l))St+h Wt+h
PF,t (j, l) −θθh=Et
λj Qt,t+h
ct+h .
θ−1
PF,t (j, l)
Pt+h

(26)

h=0

2.3.2

Calvo pricing with infrequent information updating

We now add information stickiness following Mankiw and Reis (2002) to the model. Consider firms
facing two nominal rigidities. First, each firm has a constant probability of price resetting 1 − λjas before. Second, with probability of 1 − ωj , a firm receives an information update in the current
month. The fraction of firms that fail to get updates, ωj , use the information available from themost recent update. For tractability, we assume that the two probabilities are independent eachother.
DKT develop this combined stickiness structure to explain persistent inflation dynamics as
we specified above. In DKT, infrequent price changes arise due to the Calvo assumption of price9changes. However, when firms compute their optimal reset prices, a fraction of firms use the newest
information set and the remaining firms use the stale information set to determine prices. Following
DKT, we employ this structure and refer to it as “dual stickiness” pricing.
All U.S. firms that sell their good j in city l choose different prices according to the vintageof their information set. When firms are allowed to adjust prices, those with the same vintage ofk (j, l) be the optimal price reset by U.S. firms conditional
information choose the same price. Let PH,tk (j, l) for these firms solves
on information of vintage k, its age in months. The price PH,tmax Et−kk (j,l)
PH,t

∞
X

k
λhj Qt,t+h [PH,t(j, l) − Wt+h ]

h=0
k (j, l)
PH,t

×

!−θ 

Pt+h (j, l)

Pt+h (j, l)
Pt+h (j)−θ Pt+h (j)
Pt+h

(27)

−θ
ct+h ,

for k = 0, 1, 2, · · · and for all cities l ∈ [0, 1]. Note the only difference between this problem andthe standard Calvo problem is that the expectation is taken with respect to information of vintage
k and prices that reset are indexed both by the time period they are reset and the vintage of thek (j, l).
information used at the point they are reset, PH,tk (j, l) is
The optimality condition for PH,t

Et−k

∞
X

λhj Qt,t+hh=0

k (j, l)
PH,t

Pt+h

∞

X
θ
=
Et−k
λhj Qt,t+h
θ−1
h=0

!−θ
ct+h

Wt+h
k (j, l)PH,t!

k (j, l)
PH,t

(28)

!−θ
ct+h ,

Pt+h

for k = 0, 1, 2, · · · . Canadian firms that sell their good j by exporting to city l also choose prices
based on their information set that they last updated. They choose prices so as to solve themaximization problem:
max Et−kk (j,l)
PF,t

∞
X

k
∗
λhj Qt,t+h [PF,t(j, l) − (1 + τ (j, l))St+h Wt+h
]

h=0

×

k (j, l)PF,tPt+h (j, l)!−θ 

Pt+h (j, l)
Pt+h (j)−θ Pt+h (j)
Pt+h

(29)

−θ
ct+h ,

for k = 0, 1, 2, · · · . The optimality condition is similar to (28):!
∞k (j, l) −θ
X
PF,t
h
Et−k
λj Qt,t+h
ct+h
Pt+h
h=0

=

θ
Et−k
θ−1

∞
X
h=0

λhj Qt,t+h∗(1 + τ (j, l))St+h Wt+hk (j, l)PF,tfor k = 0, 1, 2, · · · .
10!

k (j, l)
PF,t

Pt+h

(30)

!−θ
ct+h ,

2.3.3Equilibrium

The monetary authority in each country sets the growth rate of the money stock such that it follows
an AR(1):ln µt = ρ ln µt−1 + εt ,(31)

ln µ∗t = ρ ln µ∗t−1 + ε∗t ,(32)∗ . The steady state
where εt and ε∗t are mean-zero i.i.d shock and µt = Mt /Mt−1 and µ∗t = Mt∗ /Mt−1(log) money growth rates is set to zero and the common persistence parameter satisfies ρ ∈ [0, 1).
Total transfers from the government to individuals in each country equal domestic money injections minus the lump sum tax from the government paying interest. For the U.S., we have
Tt = Mt − Mt−1 − (Rt−1 − 1)Wt−1 nt−1 . The total transfer in Canada is of the same form up to∗ − ( St−1 Rt−1 − 1)W ∗ n∗ .
currency conversions: Tt∗ = Mt∗ − Mt−1
t−1 t−1
St

The profits of U.S. firms accrue exclusively to U.S. households. In other words, Πt =R R 12

j z=0 Πt (j, z)dzdj,

where Πt (j, z) is the profit of a U.S. firm. Similarly, the profits of Canadian firms accrue exclusively
R R1
to Canadian households: Π∗t = j z= 1 Π∗t (j, z)dzdj, where Π∗t (j, z) is the profit of a Canadian firm.
2

Recall, market clearing conditions for good markets were given by (21) and (22). The labormarket clearing conditions are
Z Z 1
2

nt =
j

n∗t

Z

z=0
Z 1

j

z= 12

=

nt (j, z)dzdj,n∗t (j, z)dzdj.

Last, but not least, the bond market clears at each date: Bt + Bt∗ = 0 for all t.
An equilibrium of the Calvo pricing economy is a collection of allocations and prices:• {ct (j, l, z)}j,l,z , Mt , Bt+1 , nt for U.S. households;∗ , n∗ for Canadian households;• {c∗t (j, l∗ , z)}j,l,z , Mt∗ , Bt+1t

• {Pt (j, l, z), Pt∗ (j, l∗ , z), nt (j, z), yt (j, z)}j,l,z∈[0,1/2] for U.S. firms;• {Pt (j, l, z), Pt∗ (j, l∗ , z), n∗t (j, z), yt∗ (j, z)}j,l∗ ,z∈(1/2,1] for Canadian firms;• Nominal wages and bond prices satisfy the following conditions:
1. Households’ allocations solve their maximization problem;
11

2. Prices and allocations of firms solve their maximization problem (23) and (25);
3. All markets clear;
4. The money supply process and transfers satisfy the specifications above.
An equilibrium of the dual stickiness pricing economy is not much different from the definition of
the equilibrium of the Calvo pricing economy. Prices and allocations of firms solve the maximization
problems (27) and (29) instead of (23) and (25).

3Model predictions for LOP deviations

We now discuss implications of Kehoe-Midrigan model under Calvo pricing and dual stickinesspricing for the persistence and volatility of deviations from the LOP.

3.1

Calvo pricing

Log-linearization of (24) around the steady state yields the (log) optimal price for U.S. firms that
reset prices in period t:P̂H,t (j, l) = (1 − λj β)∞
X(λj β)h Et M̂t+h ,

(33)

h=0

where P̂H,t (j, l) and M̂t are the log-deviation of PH,t (j, l) and Mt from the steady state, respectively.
Here, we use the proportionality of nominal wages to money supply (i.e., (16)) to replace the logdeviation of Wt with M̂t (i.e., Ŵt = M̂t ). Thus, the firms that adjust prices in period t choose theirprice to equalize it to the weighted average of the current and future path of nominal marginal
costs.
Analogously, we can derive the log-deviation of optimal price for Canadian firms from (26):P̂F,t (j, l) = (1 − λj β)∞
X∗(λj β)h Et (Ŝt+h + M̂t+h).

h=0

Substituting out the equilibrium nominal exchange rate, using (19), gives us
P̂F,t (j, l) = (1 − λj β)∞
X(λj β)h Et M̂t+h .

(34)

h=0

Thus, P̂F,t (j, l) = P̂H,t (j, l), under our specific preference assumption and the log-deviation of priceindex for P̂t (j, l) under Calvo pricing becomesP̂t (j, l) = λj P̂t−1 (j, l) + (1 − λj )P̂H,t (j, l).
12

It is convenient to normalize P̂H,t (j, l) (and P̂t (j, l)) by M̂t to assure stationarity. The deviation
reset prices from their steady-state relative to the movement in the nominal money supply is


∞
X
λj βρ
h
p̂H,t (j, l) = (1 − λj β)
µ̂t ,
(35)
(λj β) Et (M̂t+h − M̂t ) =1 − λj βρh=0where p̂H,t (j, l) = P̂H,t (j, l) − M̂t and µ̂t = M̂t − M̂t−1 . As it turns out p̂F,t (j, l) = P̂F,t (j, l) − M̂t =p̂H,t (j, l) so the short-run dynamics of the optimal prices are the same for home and foreign firms
selling the same good at the same location in spite of the transportation costs which drive a wedge
between the prices in the long-run.
The same normalization for the price deviation for good j in city l yields


λj βρ
p̂t (j, l) = λj p̂t−1 (j, l) − λj µ̂t + (1 − λj )µ̂t ,1 − λj βρ(36)where p̂t (j, l) = P̂t (j, l) − M̂t .
The analogous expression for the Canadian price index for good j and city l∗ is


λj βρ
∗
∗
∗
∗
∗
p̂t (j, l ) = λj p̂t−1 (j, l ) − λj µ̂t + (1 − λj )µ̂∗ ,1 − λj βρ t

(37)

and the log bilateral real exchange rate for good j across cities l and l∗ is q̂t (j, l, l∗ ) = ln qt (j, l, l∗ ) −ln q(j, l, l∗ ), where qt (j, l, l∗ ) is given by
qt (j, l, l∗ ) =St Pt∗ (j, l∗ ),Pt (j, l)(38)

and q(j, l, l∗ ) is its steady state value.
The next proposition characterizes the short-run good-level real exchange rate dynamics underCalvo pricing with a slight generalization of Kehoe and Midrigan (2007).
Proposition 1. Under the preference assumption U (c, n) = ln c − χn, the CIA constraints, theassumption of money growth (31) and (32) and good-specific Calvo pricing, the good-level realexchange rate between any cities l and l∗ follows an AR(2) process of the form:q̂t (j, l, l∗ ) = (λj + ρ)q̂t−1 (j, l, l∗ ) − λj ρq̂t−2 (j, l, l∗ ) + θj ηt ,

(39)

λ βρ

where q̂t (j, l, l∗ ) = Ŝt + P̂t∗ (j, l∗ ) − P̂t (j, l), θj = λj − (1 − λj ) 1−λj j βρ , and ηt (= εt − ε∗t ) is i.i.d.(0, ση2 ).
Proof. From (18) and (19), q̂t (j, l, l∗ ) = p̂∗t (j, l∗ ) − p̂t (j, l). Subtracting (36) from (37) yieldsq̂t (j, l, l∗ ) = λj q̂t−1 (j, l, l∗ ) + θj (µ̂t − µ̂∗t ). Because µ̂t − µ̂∗t follow an AR(1) from (31) and (32),
we obtain (39) and proved Proposition 1.
13

Proposition 1 of Kehoe and Midrigan (2007) is a special case of the one above: when money
growth rates follow an i.i.d. process (ρ = 0) equation (39) reduces to an AR(1) model with its
coefficient λj and θj = λj as Kehoe and Midrigan (2007) prove.
3.1.1

Persistence

Turning to implications for persistence of the good-level real exchange rates we employ the sum ofautoregressive coefficients (SAR) as the persistence metric. This is often the case in applied workwhen moving beyond the AR(1) model (e.g., Andrews and Chen (1994) and Clark (2006)) because
the SAR has a one-to-one relationship to the cumulative long-run impulse response to a shock. We
denote the SAR by αj .
Under Proposition 1, the SAR measure of persistence is αj = λj + ρ(1 − λj ); it simplifies to
αj = λj when ρ = 0. Obviously, SAR is strictly increasing in ρ regardless of the degree of pricestickiness under λj ∈ [0, 1). The left panel of Figure 1 shows the effect of increasing ρ on thepersistence for the two goods: a good with relatively slow price adjustment (λj = 0.95) and a goodwith relatively fast price adjustment (λj = 0.5).
The right panel of Figure 1 plots the SAR against λj . The figure compares the model’s implications for ρ = 0, as calibrated by Kehoe and Midrigan (2007) and ρ = 0.83, the monthly analog to
the CKM calibration.6 The impact of introducing persistence in money growth rates on the SAR
is clear. When ρ = 0, the model predicts that the SAR equals λj , so the two lie on the 45 degreeline in the figure. On the other hand, when ρ > 0, the model predicts a much flatter line. Thus, ahigh persistence of the money growth rates increases the persistence of LOP deviations, regardlessof the frequency of price adjustment, but the quantitative impact is greatest when the frequency
of price adjustment is highest.
To see the intuition behind the persistent dynamics it is instructive to express the current LOP
deviation as a function of its lagged self and the change in the nominal exchange rate:q̂t (j, l, l∗ ) = λj q̂t−1 (j, l, l∗ ) + θj ∆Ŝt ,(40)

where ∆Ŝt = µ̂t − µ̂∗t from (19). When ρ = 0 as in Kehoe and Midrigan (2007), ∆Ŝt is an i.i.d
shock and the good-level real exchange rate follows AR(1) with persistence parameter, λj . When
6

The CKM estimate of the autoregressive coefficient is 0.68 using quarterly U.S. data for M1 growth. We trans-

form this quarterly persistence of M1 growth into the monthly persistence by solving Cov(M̂t − M̂t−3 , M̂t−3 −M̂t−6 )/V ar(M̂t − Mˆt−3 ) = 0.68 for ρ. We obtained the resulting monthly persistence of M1 money growth of 0.83.

14international money growth differential is positively autocorrelated (ρ > 0) so is the change in thenominal exchange rate, which contributes to increased persistence in the real exchange rate.
3.1.2

Volatility

Throughout, real exchange rate volatility will be measured relative to the standard deviation of
the change in the nominal exchange rate: σj = std(qt (j, l, l∗ ))/std(∆St ). When ρ = 0, the modelq
predicts the normalized standard deviation to be σj = σ1 (λj ) = λj / 1 − λ2j and a good with larger
λj will exhibit more variability. When ρ > 0, the normalized standard deviation is predicted to beof the form σj = σ2 (λj , ρ, β) and may be obtained using the variance formula of an AR(2) processpalong with std(∆St ) = std(ηt )/ 1 − ρ2 . Importantly, the volatility function depends not just on
λj , but also on ρ and β.
An implication of this is that increased persistence in money growth, while helpful in resolving
the persistence puzzle, may actually make the volatility puzzle worse because σj = σ2 (λj , ρ, β)turns out not to be monotonic in ρ. Even more disturbing is that the shape of the relationship with
ρ depends on the frequency of price adjustment, which we know differs across goods. The practicalthrust of this is: changes in money growth persistence will have differential impacts across goods.
The left panel of Figure 2 plots the normalized standard deviations σj = σ2 (λj , ρ, β) against
ρ.7 For a good with relatively infrequent price changes (λj = 0.95), volatility of the real exchange
rate rises over most of the range of money growth persistence, before falling sharply as money
growth approaches a random walk. In contrast, for a good with relatively frequent price changes(λj = 0.5), the volatility of the relative price is declining in the money growth rate throughout. The
right panel of Figure 2 shows the ambiguous impact of introducing a positive ρ on the volatility
from another dimension. The normalized standard deviation is smaller for ρ = 0.83 than for ρ = 0
when price adjustment is fast. When the price adjustment is slow, we have a larger normalized
standard deviation for ρ = 0.83 than for ρ = 0.

3.2

Calvo pricing with infrequent information updating

k (j, l) be the log deviation of P k (j, l) from the steady state. Log-linearizing (28) around
Let P̂H,tH,tthe steady state yieldskP̂H,t(j, l) = (1 − λj β)∞
X(λj β)h Et−k M̂t+h , for k = 0, 1, 2, · · · .
h=07

We set the discount factor β to 0.99.

15

The law of iterated expectations implieskP̂H,t(j, l) = Et−k P̂H,t (j, l).
0 (j, l) = P̂
Here, we use P̂H,tH,t (j, l) because of the equivalence between (24) and (28) when k = 0.

Consider the weighted average of newly set prices that U.S. firms choose when they adjust prices
in period t; these firms choose Et−k P̂H,t (j, l) according to their information they last updated.
k (j, l) =Canadian firms choose Et−k P̂F,t (j, l). As before P̂F,t (j, l) = P̂H,t (j, l). Therefore, P̂F,tk (j, l) for k > 0, due to the law of iterated expectations.
P̂H,tDefining X̂t (j, l) as the weighted average for the newly set prices for good j in city l of the U.S.,
based upon different information vintages, we obtainX̂t (j, l) = (1 − ωj )∞X

ωjk Et−k P̂H,t (j, l),

(41)

k=0

which is similar in mathematical formulation to the price index in Mankiw and Reis (2002, p.1300).
Now, using the definition P̂H,t (j, l) = ∆P̂H,t (j, l) + P̂H,t−1 (j, l), (41) can be rewritten asX̂t (j, l) = (1 − ωj )P̂H,t (j, l) + ωj (1 − ωj )∞X

ωjk Et−k−1 ∆P̂H,t (j, l)

k=0+ ωj (1 − ωj )∞X

ωjk Et−k−1 P̂H,t−1 (j, l).

k=0

The second line of the equation is ωj X̂t−1 (j, l) from (41). Hence,X̂t (j, l) = ωj X̂t−1 (j, l) + (1 − ωj )P̂H,t (j, l) + ωj (1 − ωj )∞X

ωjk Et−k−1 ∆P̂H,t (j, l).

k=0

To render the variable stationary, define x̂t (j, l) = X̂t (j, l) − M̂t . Then,x̂t (j, l) = ωj x̂t−1 (j, l) − ωj µ̂t + (1 − ωj )p̂H,t (j, l)+ωj (1 − ωj )∞X

ωjk Et−k−1 [∆p̂H,t (j, l) + µ̂t ].

(42)

k=0

Appendix B shows that we can derive the closed form solution to x̂t (j, l):x̂t (j, l) = ωj x̂t−1 (j, l) + aj µ̂t +λ βρ−ω

jj
where aj = 1−λ, bj =
j βρ

bjµ̂t−1 ,
1 − ωj ρL(43)ωj ρ(1−λj β)(1−ωj ), and L is the lag operator.
1−λj βρThe price index for good j in city l is a Calvo-weighted-average of fixed and reset prices. The
latter being our weighted average of price resets given different vintages of information:P̂t (j, l) = λj P̂t−1 (j, l) + (1 − λj )X̂t (j, l).
16

Again, normalizing by M̂t , givesp̂t (j, l) = λj p̂t−1 (j, l) − λj µ̂t + (1 − λj )x̂t (j, l)(44)

and Canadian versions of these expressions are:bjµ̂∗ ,1 − ωj ρL t−1
p̂∗t (j, l∗ ) = λj p̂∗t−1 (j, l) − λj µ̂∗t + (1 − λj )x̂∗t (j, l∗ ).

x̂∗t (j, l∗ ) = ωj x̂∗t−1 (j, l∗ ) + aj µ̂∗t +The next proposition establishes the rich short-run dynamics of the good-level real exchange
rate emerging from the extended model.
Proposition 2. Under the preference assumption U (c, n) = ln c − χn, the CIA constraints, theassumption of money growth (31) and (32), along with good-specific Calvo pricing and good-specific
Mankiw-Reis information updating, the good-level real exchange rate between any cities l and l∗follows an ARMA(4,2) process of the form:∗q̂t (j, l, l ) =4X∗

φj,r q̂t−r (j, l, l ) +

r=1

2
X

θj,r ηt−r

(45)

r=0

where
φj,1 = φ̃j,1 + ρ,φ̃j,1 = λj + ωj + ωj ρ

φj,2 = φ̃j,2 − φ̃j,1 ρ,φ̃j,2 = −[λj ωj + (λ + ωj )ωj ρ]

φj,3 = φ̃j,3 − φ̃j,2 ρ,φ̃j,3 = λj ωj2 ρ

φj,4 = −φ̃j,3 ρθj,0 = λj − (1 − λj )ajθj,1 = −λj (ωj + ωj ρ) + (1 − λj )(ωj ρaj − bj )θj,2 = λj ωj2 ρ.
Proof. See Appendix C.
When ωj = 0 this proposition reduces to Proposition 1.8 Below, we show that both thepersistence and volatility of good-level real exchange rates predicted by the dual stickiness pricing
can be quite high. Moreover, this is true even if the price adjustment is relatively fast, which isessential in matching the cross-sectional evidence which contains goods with frequent price changes
and, yet, high persistence and variability in their LOP deviations.
8

In particular, we obtain φj,1 = λj + ρ, φj,2 = −λj ρ, and φj,3 = φj,4 = 0 for the AR parameters and θj,0 = θj andθj,1 = θj,2 = 0 for the MA parameters.

17

3.2.1

Persistence

Appendix C shows that the SAR in this generalized case is given by
αj =

4
X

φj,r = 1 − (1 − λj )(1 − ωj )(1 − ωj ρ)(1 − ρ).

r=1

Clearly, the slower the speed of information updating adjustment is (ωj → 1), the larger the SAR
becomes.
For a general ARMA process without parameter restrictions, it is not conventional to use the
SAR as a measure of persistence, because of the presence of MA terms. However, if our model iscorrectly specified, we can show that both the long-run impact of cumulative impulse response ofa unit monetary shock on real exchange rates and the SAR are strictly increasing function of λj ,ωj , and ρ. Furthermore, using the SAR is also convenient in computation and for the purpose of
making comparison with simpler models introduced in the previous subsection. For these reasons,we continue to focus on the SAR as an approximate measure of persistence under the assumptionthat the process (45) is correctly specified.
The extended model works well in generating the persistence of a good-level real exchange rate.
The left panel of Figure 3 shows the SAR among different ωj ’s. The persistence is increasing inωj and is very high regardless of the infrequency of price changes.9 The right panel of Figure 3
plots the persistence against λj . This panel compares cases of two extreme values of ωj . One is thecase in which firms producing good j updates their information every month. (i.e., ωj = 0.) The
other is the case in which firms, on average, update information every 50 months (i.e., ωj = 0.98).
For the former case, the obtained SAR corresponds to the upper straight line in the right panel of
Figure 1 since we set ρ = 0.83 in the computation. In the latter case, the persistence measure isvery close to one whether prices are sticky or flexible.
3.2.2

Volatility

Having improved the potential of the model in accounting for persistence of real exchange rates, we
ask if it helps along the dimension that was more ambiguous in the baseline model, variability. We
calculate the new normalized standard deviation σj = σ3 (ωj , λj , ρ, β), using the fact that the goodlevel real exchange rates now follow the ARMA(4,2) process according to Proposition 2. The left
panel of Figure 4 plots the normalized standard deviations against ωj . It shows that the volatility
9

Even if ωj = 0, q̂t (j, l, l∗ ) is already somewhat persistent, because of the AR(1) money growth.

18

grows exponentially as ωj increases. The right panel of Figure 4 shows the effect of increasing λj
on the normalized standard deviations under the two extreme cases: ωj = 0 and 0.98. It showsthat real exchange rate volatility becomes substantially greater when the information adjustment isslower. Thus, the introduction of information stickiness enhances the real exchange rate volatility
to a large extent.
The question we pose next is what lengths of information delays are needed to match key
properties of the micro-data, conditional on the model. The key properties are the persistence andvolatility of good-level real exchange rates along with the frequency of price changes observed for
those same goods.

4

Empirical results

4.1

Data

The retail prices come from the Worldwide Cost of Living Survey compiled by the Economist
Intelligence Unit (EIU). It is an extensive annual survey of international retail prices that was
originally designed to help managers to determine compensation levels of their employees residing
in different cities of the world. The coverage of goods and services is broad enough to overlap
significantly with what appears in a typical urban consumption basket (see Rogers (2007), for moredetail on the comparison between EIU data and the CPI data from national statistical agencies).
A notable advantage of the EIU data is the fact that all the individual good prices are listed inabsolute terms with the survey conducted by a single agency in a consistent manner over time.
Because of this convenient panel data format, a number of recent studies on international pricedynamics have used this data, including Crucini and Telmer (2007), Crucini and Shintani (2008),
Engel and Rogers (2004), Parsley and Wei (2007) and Rogers (2007).
For a limited number of countries, the EIU data contains observations from multiple cities. In
our empirical analysis, we focus on U.S.-Canadian city pairs since the assumption of the common
probability of price adjustment for each good seems to be a reasonable approximation between thetwo neighboring countries.10 After removing missing observations to construct a balanced panel forthe period from 1990 to 2005, 3 of the 16 available U.S. cities available in the survey are dropped,
while all 4 Canadian cities remain. This results in a total of 52 unique city pairs. The cities and10

Alternatively, one may use the average of price change frequencies between the two countries, an approach

employed in Kehoe and Midrigan (2007), when data from both countries are available.

19categories of goods included in the analysis are shown in Figure 5 and Table A1, respectively.
For each good j, the log of qt (j, l, l∗ ) for each year t (= 1, ..., 16) is computed using the pricelevel in a U.S. city l (= 1, ..., 13) expressed in U.S. dollars (Pt (j, l)), the price level in a Canadiancity l∗ (= 1, ..., 4) expressed in Canadian dollars (Pt∗ (j, l∗ )), and the spot U.S.–Canadian dollar
exchange rate (St ), all from the EIU data. Since the resulting log real exchange rates represent
the log deviations of the price in a Canadian city relative to that of a U.S. city both expressed in
a common currency, a negative value for the pair of Toronto and New York, for example, implies
that the good is more expensive in New York than in Toronto at year t. Figure 6 plots the log ofqt (j, l, l∗ ), pooling all goods and all city pairs from the end-points of the sample, 1990 and 2005.
Next, for the price stickiness parameter, λj , we utilize the frequency of price changes, fj andtransform it with λj = 1 − fj for good j. Since the EIU data is annual, it is not useful for
constructing estimates of the frequency of price changes. Here we rely on existing studies based onmonthly micro-data from the BLS (Bureau of Labor Statistics). Bils and Klenow (2004) used the
BLS Commodities and Services Substitution Rate Table for 1995-1997 which contains the averagefrequencies of price changes of individual goods and services used in construction of the U.S. CPI.
We took the monthly average frequency of price changes, fj , from Table A1 of their paper andmatched them with the 165 goods in the EIU sample. Since we require persistence and frequency
adjustment parameters good-by-good to evaluate the model, we use only these 165 matched pairs
in our analysis. We assume that the frequency of price changes applies to the entire sample periodof 1990-2005 in our EIU data set.11 In addition, we assume a common frequency of price change
between the U.S. and Canadian cities, good-by-good.
Nakamura and Steinsson (2008) revisited Bils and Klenow’s analysis using more detailed andupdated BLS data. Using the CPI Research Database created by the BLS, they re-estimated thefrequencies of price change after removing temporary price changes associated with sales. They
found that the median duration between regular price changes was 8 - 11 months depending on thetreatment of substitutions, considerably higher than the 4.3 months for the median good, foundby Bils and Klenow (2004). In what follows, we also check the impact on our results of using
the Nakamura and Steinsson’s (2008) data on the frequency of price changes from the period of
11

In some countries which experienced a structural shift in inflation, an assumption of constant frequency of pricechanges over years may not be satisfied. For example, Ahlin and Shintani (2007) use Mexican price data on 44 goods
and report that the average monthly frequency of price changes was 28% in 1994 and as large as 50% in 1995. We
expect that this issue is less serious in our case since both U.S. and Canada had a stable inflation during the period
under consideration.

20

1998-2005.
For the nominal exchange rate changes required for the theoretical volatility calculation, we
use monthly changes in the log of the end-of-month U.S.-Canadian dollar spot rates. While bothprice stickiness parameter (frequency of price changes) and nominal exchange rates are available
in monthly series, real exchange rates are only observed annually. The small number of timeseries observation at the annual frequency is the major limitation of the EIU data. In the next
subsection, we briefly discuss how to reconcile the mixed frequencies of observation in the dynamic
panel estimation and describe the procedure to estimate the time series models.

4.2

Estimation

Table 1 shows how monthly ARMA processes predicted by the model are transformed into theones which have non-zero coefficients for multiples of 12 month lags and finite MA terms. The firstrow of the table shows the easiest transformation. In Calvo pricing with ρ = 0, the equation (39)
directly implies thatq̂t (j, l, l∗ ) = λj q̂t−1 (j, l, l∗ ) + λj ηt .
By repeated substitutions, we get∗q̂t (j, l, l∗ ) = λ12j q̂t−12 (j, l, l ) + λj Λj (L)ηtwhere Λj (L) =

P11

r r
r=0 λj L .

In this equation, the AR term is the 12th lag (in months) and theorder of the MA term is 11. This ARMA(12,11) is equivalent to an AR(1) sampled annually since
λj Λj (L)ηt and q̂t−12 (j, l, l∗ ) are not correlated.
Such a transformation is not necessarily possible with a general ARMA process including AR(2)and ARMA(4,2) processes. However, thanks to a special dynamic feature of the theoretical model,
it is possible that we can make the AR parameters non-zero only if the lags are multiples of 12
and the MA parameters finite under our extended models (39) and (45). Appendix D provides thedetailed derivations of these more elaborate transformations.
Previously, l and l∗ were used for the U.S. and Canadian cities, respectively. Here, they are
replaced by a new single index i (= 1, ..., 52) each representing a city pair spanning a national
border. In addition, the sampling frequency for the model was assumed to be monthly. With someabuse of notation, our new time subscript now represents the time in annual frequency. Namely,
if the true data process is generated for each month t∗ = 1, ..., T ∗ , we now only observe the series
annually at the months of t = 12 × t∗ = 1, ..., T (= T ∗ /12). With this newly introduced index, we
21

jdefine qit
as the log of the real exchange rate for good j between the city pair indexed by i at year

t:
j
qit
= ln qt (j, l, l∗ ).
jThus, the former log deviation from the steady state qbt (j, l, l∗ ) can be rewritten as qit− qij , where

qij is the long-run value which the Appendix E shows to equal:
1[1 + κ1−θ (1 + τ (j, l∗ ))1−θ ] 1−θ
qij = ln q(j, l, l∗ ) = ln.
1
[1 + κ1−θ (1 + τ (j, l))1−θ ] 1−θIntuitively, the relative price of a good in the long-run is higher in the destination market withthe higher shipping cost from the source. Thus if city l∗ is, say, farther from the source of thegood than city l, qij is positive. These heterogeneous long-run deviations justify the presence ofindividual effects (the time invariant city pair-specific effect) in the panel estimation.
Based on the annual transformation shown in Table 1, all the dynamics of the real exchange
rate for good j can be written as
j
qit
=

m
X

j
j
Φj,r qi,t−r+ ζij + ujt + vit,

r=1

where ζij is the time invariant unobserved city pair-specific effect which allows long-run price difference between two cities, ujt is the common time effect which represents the exchange rate shocksjand vitis a good-specific residual term.

This model format nests all the models under consideration: (i) Calvo pricing with ρ = 0 implies
m = 1; (ii) Calvo pricing with ρ 6= 0 implies m = 2; and (iii) dual stickiness pricing implies m = 4.
For the individual specific effect ζij , we can easily see its relationship to the long-run mean andPmjthe persistence from qij = ζij /(1 − αj ) where αj =r=1 Φj,r . For the common time effect ut ,Calvo pricing with ρ 6= 0 predicts a serial correlation of order one, while dual stickiness pricing
predicts a serial correlation of order three. However, in a short panel asymptotic with finite T , thecommon time effects can be treated as unknown parameters to be estimated with time dummies.
In addition, since our main interest is to estimate the persistence expressed in terms of the SAR,
αj , it is convenient to rewrite the model into the augmented Dickey-Fuller (ADF) form. Thus, thenested model is given byjjqit= αj qi,t−1
+

m−1
X

j
j
j
e
γj,r ∆qi,t−r+ u>
j Dt + ζi + vit ,

r=1

22

j
j
j
where ∆qi,t−r= qi,t−r− qi,t−r−1, γj,r =jj >s=r+1 Φj,s for r = 1, ..., k − 1, uj = (um+1 , ..., uT ) is aPme t is a (T − m) × 1 time dummy vector with one in the t-th position and zero
vector of constants, D
otherwise.
To estimate this short dynamic panel model, we employ the generalized method of moments(GMM) estimator in the first differenced form for the purpose of eliminating the individual effectζij . We follow Arellano and Bond (1991) in the choice of instruments and initial weighting matrix.
In particular, the moment condition is given by
!#
"
m−1
X
j
j
j
2 j
>
=0
γj,r ∆ qi,t−r − δj Dt
E qis ∆qit − αj ∆qi,t−1 −
r=1
j
j
j
for s = 1, ..., t−m−1 and t = m+2, ..., T , where ∆2 qi,t−r= ∆qi,t−r−∆qi,t−r−1
δj = (∆ujm+2 , ..., ∆ujT )>is a vector of constants, Dt is a (T − m − 1) × 1 time dummy vector with one in the t-th positionand zero otherwise. The total number of parameters to be estimated is T − 1 with the number ofmoment conditions given by (T − m)(T − m − 1)/2.12 This GMM estimator for αj is consistentunder large N fixed T asymptotics.

4.3

Persistence

In this subsection, we evaluate the Kehoe-Midrigan model and its extension in explaining theobserved persistence of the real exchange rate for each good j. Following the theoretical analysis,
our empirical persistence measure is the SAR αj .
We first revisit the original Kehoe-Midrigan model with an assumption of an i.i.d. money
growth (ρ = 0). In this case, the theory predicts an AR(1) model and thus αj is simply an AR(1)coefficient. A GMM estimation of αj yields a median of 0.56 using annual U.S.-Canadian city pairs
data.13 In terms of monthly frequency, our value corresponds to 0.561/12 = 0.95, which is slightly
less than 0.98, the median value obtained by Kehoe and Midrigan (2007) based on bilateral realexchange rates of 66 goods between the U.S. and four European countries, Austria, Belgium, France
and Spain.
The first panel in Figure 7 plots the estimated persistence measure αj against the (annual)
12 computed based on f from Bils and Klenow’s
infrequency of the price adjustment λ12jj = (1 − fj )12

For the model to be (over-) identified, at least T = 4 is required for m = 1, T = 6 is required for m = 2, andT = 9 is required for m = 4. Since T = 16 is available in our sample, the number of over-identifying restrictions is
51, 76, and 90, respectively, for m = 1, 2, and 4.
13
This value lies between the medians for OECD city pairs (0.65) and LDC city pairs (0.51) obtained by Crucini
and Shintani (2008) based on the same data source.

23

(2004) table. A cross-sectional regression of αj on λ12
j yielded a significantly positive slope coefficientestimate of 0.30 (with a standard error of 0.08) which is consistent with the theoretical prediction
at least in direction: more price stickiness implies higher persistence. However, 160 out of 165
goods lie above the 45 degree line (αj = λ12j ) in the scatter plot with the regression slope being
significantly less than unity. If the model performance is evaluated by computing the ratio of thepredicted persistence (on the 45 degree line) to the observed persistence for each good, the model
can explain merely a 6 percent of the total persistence for the median good. This confirms Kehoeand Midrigan’s claim that a simple model of price stickiness alone is quantitatively insufficient toreproduce the observed persistence in good-level real exchange rates.
We next consider the effect of introducing serially correlated money growth (ρ = 0.83). On thewhole, the persistence estimate αj remains almost unchanged with a median value of 0.57 based on
the AR(2) model. The regression slope shown in the second panel of Figure 7 is 0.35 and is again
significantly positive. Recall that for a given λj , αj is a monotonically increasing function of ρ (seethe left panel of Figure 1). In annual frequency, the predicted SAR from Table 1 is given by
12
12
12 12
αj = 1 − (1 − ρ12 )(1 − λ12j ) = λj + ρ − λj ρ

and the effect of increasing ρ can be seen in the median value of the ratio of prediction and data
provided in the upper panel of Table 2. In terms of the median, the theoretical persistence becomes
the observed persistence when ρ is around 0.95. However, this value is much higher than ρ = 0.83,
the value estimated by CKM. Indeed, when ρ = 0.83 is used, only 31 percent of the persistence
can be explained by the model (the number is provided as the first entry of the lower panel). The
inability of persist money growth to account for the persistence of real exchange rates can be also
seen from the scatter plot. Recall that, from the right panel of Figure 1, increasing ρ shifts thetheoretical line upward with a flatter slope. A similar theoretical prediction line with ρ = 0.83,
expressed in the annual frequency basis, is also drawn in the second panel of Figure 7.14 Comparedto the 45 degree line in the first panel of the same figure (ρ = 0), the predicted line now becomesflatter but is still much steeper than the regression line. Indeed, about 95 percent of data points
are still above the ρ = 0.83 line. Thus, persistence in money growth helps a bit, but the model with
Calvo pricing remains largely unsuccessful in explaining the persistence with a reasonable choiceof money growth process.
Third, we now look at the role of information delay in explaining αj . To simplify the argument,
14

The intercept of the theoretical line is ρ12 = 0.8312 = 0.11.

24

here we assume the information delay parameter to be common across all the goods (namely, ωj = ω
for all j). The persistence estimates based on the AR(4) model become somewhat lower with a
median value of 0.51, but still are much higher than the level predicted by the standard Calvo
pricing without information delay (which corresponds to the ω = 0 line shown in the lower panelof Figure 7). Recall that from the left panel of Figure 3, for a fixed value of λj and ρ(= 0.83), αj isstrictly increasing in ω. This pattern is preserved in the SAR expressed in annual frequency (SeeAppendix D.):1212
αj = 1 − (1 − ρ12 )(1 − λ12j )(1 − ω )(1 − (ωρ) ).

Based on this relationship, the median of the ratio of theoretical value to observed value, provided
in the lower panel of Table 2, increases with ω and reaches one at ω = 0.93 which corresponds to
14 months of average duration between information updates. Therefore, at least in terms of themedian, dual stickiness pricing with a reasonable money growth process is capable of replicating
the observed persistence. In the lower panel of Figure 7, the shaded triangle area shows the range
between the line without information delay (ω = 0) and the line with an enormous information
delay (ω = 0.98 which corresponds to the 50 month average duration of information updates). The
regression line is located almost in the middle of the triangle and its slope is 0.56.
Turning to results based on fj from Nakamura and Steinsson’s (2008) data, Figure 8 shows thescatter plots of the pairs of (αj , λ12j ) for (i) Calvo pricing with ρ = 0; (ii) Calvo pricing with ρ > 0;and (iii) dual stickiness pricing, respectively.
For many goods, the removal of sales results in the lower value of fj . Less frequent price12changes increase the value of λ12
j = (1 − fj ) , and most of the data points in the scatter plot shift

rightward.15 For all the models, the predicted persistence will be higher for the larger values of λ12
j ,and thus excluding the sales from frequency of price changes works in favor of Kehoe-Midrigan’s
model prediction about real exchange rate persistence. The proportion of the data points lie below
the theoretical prediction line increased from 3 percent to 16 percent for Calvo pricing with ρ = 0,
and from 5 percent to 24 percent for Calvo pricing with ρ = 0.83. For all the cases, the regression
slopes shown in the scatter plots are again significant and positive, and the regression fit in termsof the coefficient of determination improves.16 However, because of the rightward shift, more
15

Note that αj for each good j for each AR model remains unchanged between Figures 7 and 8. In addition,

because the sample periods differ between the two data sets, this rightward shift may not occur for some goods.
16
Regression coefficients are 0.36, 0.31, and 0.43 for each panel, respectively. Coefficients of determination increase
from 8 to 26 percent, 10 to 17 percent, and 12 to 15 percent, respectively.

25

data points in the last panel of the figure fall outside the shaded triangle region representing thetheoretical predictions of dual stickiness pricing.
We see improvement in the ratio of predicted to the observed persistence provided in Table 3.
As shown in the first entry of the upper panel of Table 3, even the case with ρ = 0 can accountfor 48 percent of the observed persistence, in comparison to 6 percent based on Bils and Klenow’s
fj . The ratio increases as ρ increases, but because of the higher initial ratio, it becomes one ataround ρ = 0.92 a value lower than the previously selected value of ρ = 0.95. This newly selected
ρ, however, is again higher than the CKM’s reference value of ρ = 0.83. Since the ratio or actual
to predicted persistence is 66 percent at ρ = 0.83, there is still a room for the information delay
structure to fill the gap between the theoretical and observed value. The lower panel of Table
3 shows the effect of increasing ω on the prediction ratio based on Nakamura and Steinsson’sdata. The table shows that 100 percent of the persistence can be explained by setting ω = 0.90
which corresponds to 9.5 months of average duration between information updates. This lengthof months suggests that the role of information stickiness remains important even when Bils andKlenow estimates are replaced with Nakamura and Steinsson estimates.

4.4

Volatility

The second puzzle brought up in Kehoe and Midrigan (2007) is the observation of too much volatility
in good-level real exchange rates which is inexplicable by either a simple sticky price model or amodel with pricing complementarities. In this subsection, we evaluate the role of information
stickiness in terms of explaining the observed volatility.
The performance of the model is evaluated by the ratio of the ‘theoretical’ normalized standard
deviation to the ‘observed’ normalized standard deviation. The procedure of computing each standard deviation is as follows. First, to compute ‘theoretical’ normalized standard deviation, notethat the standard deviation of real exchange rates predicted by the theory has the same implication to both annually sampled data and monthly sampled data. Therefore, unlike the measure of
persistence that required transformations shown in Table 1, using annual data poses no complication. For each good, the theoretical normalized standard deviation σj can be directly obtained bysubstituting λj = 1 − fj into the formula discussed in Section 3.
Second, to compute the ‘observed’ normalized standard deviation, note that using a pooled
sample variance as a volatility measure is not appropriate since it includes the variance component
due to the dispersion of long-run real exchange rate qij among city pairs in our panel data. In
26

addition, the theory predicts volatility caused by the nominal exchange rate fluctuation which iscommon to all the products, but is not designed to incorporate the idiosyncratic variance component
such as the one due to time-varying city specific shocks. For this reason, we conduct a variancedecomposition based on a standard two-way error components model and focus on the extracted
variance component due to a time specific shock. This decomposition seems to be a reasonablechoice in our study because it is consistent with the idea of using time dummies in the dynamic panel
estimation to incorporate the common time specific shocks in our previous analysis of persistence.
We thus use the observed standard deviation of time specific component normalized by the sample
standard deviation of monthly nominal exchange rate growth.
We start with the results presented in Table 4 based on Bils and Klenow’s data. The upper
panel of the table shows the median of the ratio of the theoretical to observed normalized standard
deviation. The original Kehoe-Midrigan setting with ρ = 0 can explain only 13 percent of thevariation in the data. Thus, the evidence of excess volatility discovered by Kehoe and Midrigan(2007) is also confirmed in the EIU panel data of the U.S.-Canadian city pairs. Can we explain thisobserved volatility with an introduction of serially correlated money growth? Unfortunately, unlike
the persistence, the predicted volatility is not a monotonically increasing function of ρ. Examples
presented in the left panel of Figure 2 show that the volatility decreases monotonically for goods
with small λj = 1 − fj and increases only in some range of ρ for goods with a larger λj . As a resultof the combination of the two effects for many goods, none of the median ratios presented in theupper panel of Table 4 is above one and maximum value is only 15 percent at ρ = 0.52.
In contrast to the effect of ρ, the left panel of Figure 4 shows that the volatility increases
monotonically with ω in dual stickiness pricing for any given values of λj and ρ. The lower panelof Table 4 presents the ratio of standard deviations based on dual stickiness pricing with various
ω’s when the CKM’s reference value of ρ = 0.83 is used for the money growth process. With an
introduction of the information delay, the volatility can now be fully explained at ω = 0.94 which
implies 17 months of average duration between information updates. As shown in the previous
section, the observed persistence can be reproduced if any value of ρ is allowed without introducing
information stickiness. For the volatility, however, the observation can be replicated only underdual stickiness pricing. In this sense, the information delay plays an essential role in explaining
volatility.
We now turn to Nakamura and Steinsson’s data with the effect of sales removed from fj . The
median of the ratio of the predicted to observed standard deviation for each pricing is shown in27

Table 5. The performance of the Kehoe-Midrigan model, in terms of explaining volatility, clearly
improves over the results in Table 4 based on Bils and Klenow’s data. Since the removal of sales
results in the lower values of fj , using Nakamura and Steinsson’s data increases the theoreticalvolatility level which is increasing in λj = 1 − fj . For example, when ρ = 0, the ratio increases from13 to 23 percent. However, the degree of increased theoretical volatility is still insufficient to fully
explain the observed volatility under the model without information delay. As shown in the upper
panel of Table 5, the maximum ratio is only 43 percent under the model without information delayat ρ = 0.80. Therefore, the role of sticky information is again crucial in explaining the volatilityof the good-level real exchange rates, even if we use Nakamura and Steinsson’s data. The lower
panel of Table 5 shows that the model with information delay can explain 100 percent of observed
volatility when ω = 0.92 which implies 12 months of average duration between information updates.
In comparison to the result from Bils and Klenow’s data, the reduction of ω reflects the fact thata larger component in the variance is already explained by the reduction of price change frequency
alone in Nakamura and Steinsson’s data.

4.5

Good-specific information updating

In the previous subsections, we have shown that an introduction of information stickiness into
Calvo pricing can fully explain the median persistence and volatility by searching for the common
information delay, namely ωj = ω for all goods, j. In this subsection, we will briefly evaluate
the obtained values of common information delay by comparing existing empirical macro studies
on sticky information. Then, we relax our assumption of common information delay and consider
good-specific information delays to account for the individual persistence or volatility.
To evaluate common ω estimates, we first compare them with previous studies’ estimates on
information stickiness based on the aggregate inflation. Using the aggregate data on inflation over
1960:Q1 - 2007:Q2, DKT find that information delay, on average, is 7.1 months with 95 percent
confidence intervals between 5.0 and 16.1 months. Knotek (2006) introduces information stickiness
into the fixed menu cost model and finds the average duration between information updates to be
20.4 months over 1983:Q1 - 2005:Q4.17 Thus, all of our common ω estimates (14 and 17 months
from Bils and Klenow’s data and 9.5 and 12 months from Nakamura and Steinsson’s data) are in
17

Among many empirical studies on the pure sticky information model, Andrés, López-Salido, and Nelson (2005)

estimate the average information duration to be 20 months and Kahn and Zhu (2006) find that the point estimatesof average duration range between 9 and 23 months.

28

line with previous estimates based on aggregate inflation.
So far, our common values of ω was obtained to match the persistence and volatility for the
median good. On the other hand, we can also obtain a good specific ω which matches the individual
persistence or volatility for each good using the following procedure.
First, for each good j(= 1, 2, ..., 165), we obtain good specific information delays from thepersistence using
min [α̂j − α(ωj |λj , ρ)]2 ,ωj ∈[0,1)

where α̂j denotes the SAR estimate of the AR(4) model and α(ωj |λj , ρ) is the theoretical SAR given
12
12
by 1 − (1 − ρ12 )(1 − λ12j )(1 − ωj )(1 − (ωj ρ) ) evaluated at ρ = 0.83 from CKM and λj = 1 − fjfrom the frequency of price changes calculated by either Bils and Klenow (2004) or Nakamura andSteinsson (2008).
Second, we obtain good specific information delays from the volatility using
min [σ̂j − σ(ωj |λj , ρ, β)]2 ,ωj ∈[0,1)jwhere σ̂j is the extracted standard deviation component of qit
due to time specific shocks normalized

by the standard deviation of ∆Ŝt , while σ(ωj |λj , ρ, β) is the predicted normalized standard deviationfrom the model under dual stickiness pricing.18 We take ρ = 0.83, β = 0.99 and λj = 1 − fj fromeither Bils and Klenow (2004) or Nakamura and Steinsson (2008).
We now look at the distribution of good-specific average durations of information updates
1/(1 − ωj ) based on the frequency of price changes from Bils and Klenow (2004). The upper andlower panels of Figure 9 show the relative histogram of information delays implied by the persistence
and volatility, respectively.
The two kernel density estimates shown in the same figure, on the whole, suggest similarity
between the two distributions. The median of the durations implied by persistence is 12.9 months
while that of the durations implied by volatility is 16.6 months.19 These values are close to theaverage durations under the common ω assumption (14 months from persistence and 17 monthfrom volatility).
18

That is, σ(ωj |λj , ρ, β) is the normalized standard deviation σ3 (ωj , λj , ρ, β) evaluated at the fixed values of λj , ρ

and β
19
From the distribution implied by persistence after removing outliers, we obtain the standard deviation of 13.6,
the skewness of 2.0, and the kurtosis of 5.7. On the other hand, we obtain the standard deviation of 15.6, the skewnessof 1.5, and the kurtosis of 3.4 from the distribution implied by volatility.

29With the frequency of price changes from Bils and Klenow (2004), we compute a fraction of
goods in which persistence or volatility can be explained without information stickiness. The
fraction is 11.5 percent from persistence matching and only 6.1 percent from volatility matching.
This computation implies that most goods need to have a positive good specific ωj to fully explain
good-level real exchange rate dynamics.
Next, we turn to Figure 10 which uses Nakamura and Steinsson’s (2008) data on the frequency of
price changes. Once again, the kernel density estimates suggest similarity of the two distributions.
The median duration between information updates implied by persistence is 8.2 months whilethat implied by volatility is 11.9 months.20 The fraction of goods that can match persistence orvolatility without information stickiness has increased to 33.3 percent for persistence matching and21.8 percent from volatility matching, due to the exclusion of sales. However, approximately morethan two-thirds of goods still need to have a positive ωj . Thus, the information stickiness remain
important in explaining persistence and volatility with Nakamura and Steinsson’s (2008) data.
Finally, we ask whether our results are, on the whole, consistent with evidence from micro
studies on prices. No micro studies provide directly comparable distribution of information delay
among goods, but survey results on price reviews done by firms may serve our purpose. Fabiani,
Druant, Hernando, Kwapil, Laudau, Loupias, Martins, Matha, Sabbatini, Stahl, and Stokman(2005) argue that the frequency of price reviews rather than price changes “could be related to thearrival of information.” According to Fabiani et. al. (2005), when additional information on thestate of the economy infrequently arrives, it is sensible for firms to review prices infrequently. In
this sense, we can exploit survey results for price reviews.
Blinder, Canetti, Lebow, and Rudd (1998) surveyed U.S. firms about price setting behavior in
the beginning of 1990s and their results for price reviews allow us to assess our distributions of
average arrival of information. They ask firms what the customary interval (e.g., daily, weekly,monthly, quarterly, and yearly) was between price reviews for their most important product. Table
6 compares our distributions of durations of information updates with Blinder et. al. (1998) survey
results. Overall, our distributions of duration between information updates seem to match the
distribution of price reviews well. In particular, our results are close to their survey results when
the frequency of price changes is taken from Nakamura and Steinsson (2008).
20

Descriptive statistics are as follows. From the distribution implied by persistence, we obtain the standard devia-

tion of 10.8, the skewness of 2.3, and the kurtosis of 9.1. From the distribution implied by volatility, the corresponding
statistics are 16.1, 1.9, and 4.7, respectively.

30

5Conclusion

Using highly disaggregated price data from U.S. and Canadian cities, we have confirmed Kehoeand Midrigan’s main finding that the standard Calvo-type sticky price model fails to explain thepersistence and volatility of good-level real exchange rates. We found that this puzzling but stimulating result remains robust to a change from Bils and Klenow’s data to Nakamura and Steinsson’sdata on the frequency of price changes. The robustness of their finding suggests that the baselinemodel is deficient.
We offer a possible solution to this puzzle by extending the Kehoe-Midrigan model such that
only a fraction of firms have the up-to-date information when resetting prices. Due to the infrequent
arrival of information, real exchange rates become more persistent and track the volatile nominal
exchange rate even if price adjustment is relatively fast. Our model can explain both persistence
and volatility within a reasonable range of average duration of information updates.
We have limited our attention to the implications of our model under many simplifying assumptions. Therefore, there are numerous promising avenues for future research. For example, what
would happen to the prediction of our model if pricing complementarities are included? Whatwould be the impact on good-level real exchange rate dynamics if the non-traded inputs in producing a good are included in the model?21 We believe that answering these questions would help us
further understand the dynamics of price adjustment within and across countries.

A

The real exchange rate
P ∗ c∗P ∗ c∗

t+1
St+1 = Ptt ctt St in each event in
From the intertemporal conditions (12) and (13), we obtain Pt+1
t+1 ct+1

period t + 1. Because qt is defined as

c∗t+1
c∗t−1
St Pt∗c∗t
Pt , it immediately follows that qt+1 ct+1 = qt ct = qt−1 ct−1 =c∗· · · = q0 c00 = κ.

B

The closed form solution to x̂t (j, l)

To derive the closed form solution to x̂t (j, l), we use the closed form solution to p̂H,t (j, l), given µ̂tfollows AR(1). It has been already derived from (35) under an AR µ̂t :


λj βρ
p̂H,t (j, l) =µ̂t ,1 − λj βρ21

See Crucini, Telmer, and Zachariadis (2005) for this line of research.

31

which implies



λj βρ
Et−k−1 (µ̂t − µ̂t−1 )1 − λj βρ


λj βρ
=
(ρk+1 µ̂t−k−1 − ρk µ̂t−k−1 ).
1 − λj βρEt−k−1 [∆p̂H,t (j, l)] =Using this result, we can express x̂t (j, l) as x̂t−1 (j, l) and {µ̂t−k }∞
k=0 :


λj βρ
µ̂t
x̂t (j, l) = ωj x̂t−1 (j, l) − ωj µt + (1 − ωj )1 − λj βρ



∞
X
λj βρ
k+1
k
k+1
k
+ ωj (1 − ωj )(ρ µ̂t−k−1 − ρ µ̂t−k−1 ) + ρ µ̂t−k−1 .
ωj
1 − λj βρ
k=0Using a lag operator L, we can obtain
λj βρ
µ̂t
1 − λj βρ



∞
X
λj βρ
k k k
+ ωj (1 − ωj )
ωj ρ L[ρ − 1] + ρ µ̂t−1 .
1 − λj βρx̂t (j, l) = ωj x̂t−1 (j, l) − ωj µ̂t + (1 − ωj )

k=0

Using

P∞

k k k
−1 and arranging terms yields the closed form solution to x̂ (j, l)t
k=0 ωj ρ L = (1 − ωj ρL)given by (43).

C

The proof of proposition 2

To prove Proposition 2, we use the first order difference equation for p̂t (j, l) and µ̂t and an
ARMA(1,∞) structure for x̂t (j, l). We havep̂t (j, l) = λj p̂t−1 (j, l) − λj µ̂t + (1 − λj )x̂t (j, l)µ̂t = ρµ̂t−1 + εtx̂t (j, l) = ωj x̂t−1 (j, l) + aj µ̂t +

bj
µ̂t−1
1 − ωj ρLfrom (44), (31), and (43), respectively. We can rewrite the first and the third equations as follows:λj
1 − λjµ̂t +x̂t (j, l)
1 − λj L
1 − λj L
bj
aj
µ̂t +
µ̂t−1 .
1 − ωj L(1 − ωj L)(1 − ωj ρL)p̂t (j, l) = −x̂t (j, l) =We eliminate x̂t (j, l) from these equations to get(1 − λj L)(1 − ωj L)(1 − ωj ρL)p̂t (j, l) =(1 − λj )aj (1 − ωj ρL)µ̂t + (1 − λj )bj µ̂t−1− λj (1 − ωj L)(1 − ωj ρL)µ̂t .
32

Arranging terms of the right hand side of the equation yields(1 − λj L)(1 − ωj L)(1 − ωj ρL)p̂t (j, l) = − [λj − (1 − λj )aj ] µ̂t+ [λj (ωj + ωj ρ) − (1 − λj )(ωj ρaj − bj )]µ̂t−1− λj ωj2 ρµ̂t−2 .
Using the definition of θj,0 , θj,1 and θj,2 defined in Proposition 2, we get(1 − λj L)(1 − ωj L)(1 − ωj ρL)p̂t (j, l) = −θj,0 µ̂t − θj,1 µ̂t−1 − θj,2 µ̂t−2 .
The left hand of the equation can be extended so that
(1 − φ̃j,1 L − φ̃j,2 L2 − φ̃j,3 L3 )p̂t (j, l) = −θj,0 µ̂t − θj,1 µ̂t−1 − θj,2 µ̂t−2 .
Since the money growth rate follows an AR(1), µ̂t = (1 − ρL)−1 εt . Then,(1 − ρL)(1 − φ̃j,1 L − φ̃j,2 L2 − φ̃j,3 L3 )p̂t (j, l) = −θj,0 εt − θj,1 εt−1 − θj,2 εt−2 .
Arranging terms the left hand of the equation gives φj,1 , φj,2 , φj,3 , and φj,4 :p̂t (j, l) =4X

φj,r p̂t (j, l) −

r=1

2
X

θj,r εt−r .

r=0

Analogously, we can derive the price index for good j of city l∗ :p̂∗t (j, l∗ ) =4
X

φj,r p̂∗t (j, l∗ ) −

r=1

2
X

θj,r ε∗t−r .

r=0

Because q̂t (j, l, l∗ ) = p̂∗t (j, l∗ ) − p̂t (j, l), we obtain (45).
Finally, note that the coefficient of p̂t (j, l) is(1 − ρL)(1 − φ̃j,1 L − φ̃j,2 L2 − φ̃j,3 L3 ) = (1 − ρL)(1 − λj L)(1 − ωj L)(1 − ωj ρL).
It implies that the SAR

P4

r=1 φj,r is equal to 1 − (1 − ρ)(1 − λj )(1 − ωj )(1 − ωj ρ). Because the AR

coefficients are the same between the same type of good j, it proves Proposition 2.

D

The Detailed Derivation of Transformation from Monthly to
Annual Specification

This appendix shows how we transform a monthly specification into the one which has non-zero AR
coefficients for multiples of 12 month lags and finite MA terms with the remaining AR coefficients33

equal to zero. Table 1 summarizes the obtained results before and after transformations. The
transformations lead us to estimate the model via the annual data.
We have already shown the transformation results of Calvo pricing with ρ = 0 in the main text.
In what follows, we will show the derivation of Calvo pricing with ρ > 0 and dual stickiness pricing.
Calvo pricing (ρ > 0) First, we can rewrite the first order difference equation (40) as∗
q̂t (j, l, l∗ ) = λ12j q̂t−1 (j, l, l ) + θj Λj (L)Ŝt =θj Λj (L)∆Ŝt ,
12
1 − λ12j L(A1)

Second, since ∆Ŝt = µ̂t − µ̂∗t , it immediately follows that∆Ŝt = ρ∆Ŝt−1 + ηt =where R(L) =

R(L)
ηt ,
1 − ρ12 L12

(A2)

P11

r r
r=0 ρ L . Substituting (A2) into (A1) yields:

12
∗
12 12
∗
q̂t (j, l, l∗ ) = (λ12j + ρ )q̂t−12 (j, l, l ) − λj ρ q̂t−24 (j, l, l ) + θj Λj (L)R(L)ηt ,(A3)which produces an ARMA(24,22).22 The AR parameters are non-zero only if the lags are multiples
of 12. Moreover, the length of the MA terms is now finite and of order 22 in this specific ARMA
process. Intuitively, this transformation is made possible because q̂t (j, l, l∗ ) is the first order difference equation and the driving force ∆Ŝt follows an AR(1) process. Remarkably, this monthly
ARMA(24,22) becomes ARMA(2,1) in terms of annually sampled data.
Dual stickiness pricing A similar transformation is possible in dual stickiness pricing. The
next proposition summarizes the transformation result.
Proposition A1. In dual stickiness pricing with ρ, λj , and ωj ∈ (0, 1), the ARMA(4,2) processcharacterized by (45) has an equivalent expression of the following ARMA(48,46) process:∗q̂t (j, l, l ) =4XΦj,r q̂t−12r (j, l, l∗ ) + Θj (L)ηt ,r=1
22

It is because both Λj (L) and R(L) have the power of L of 11 in maximum.

34

(A4)

whereΦj,1 = Φ̃j,1 + ρ12 ,1212Φ̃j,1 = λ12
j + ωj + (ωj ρ)

Φj,2 = Φ̃j,2 − Φ̃j,1 ρ12 ,12
12
12 12 12Φ̃j,2 = −[λ12j ωj + (λj + ωj )ωj ρ ]Φj,3 = Φ̃j,3 − Φ̃j,2 ρ12 ,24 12
Φ̃j,3 = λ12j ωj ρ

Φj,4 = −Φ̃j,3 ρ12(Θj (L) =(1 − ωj12 L12 )(1 − (ωj12 ρ)12 L12 )λj Λj (L)R(L))
−(1 − λj )Λj (L)Ωj (L)R(L) (1 − (ωj ρ)12 L12 )aj + bj L(1 + ΩR
j (L))Ωj (L) =11
X

ωjr Lr ,ΩR
j (L) =

r=0

11X

(ωj ρ)r Lr .

r=1

Proof. To consider the transformation under dual stickiness pricing, note that (43) can be rewrittenasx̂t (j, l) = ωj x̂t−1 (j, l) + aj µ̂t +

bj L
µ̂t ,
1 − ωj ρLusing a lag operator L. This equation has an infinite MA term because the third term of the righthand side has (1 − ωj ρL)−1 µ̂t . We first work on this term.
The infinite MA form (1 − ωj ρL)−1 µ̂t is(1 − ωj ρL)−1 µ̂t= µ̂t +

11X(ωj ρ)r µ̂t−r

r=1+(ωj ρ)12 µ̂t−12 + (ωj ρ)1211X(ωj ρ)r µ̂t−r−12

r=1+(ωj ρ)24 µ̂t−24 + (ωj ρ)24

11X(ωj ρ)r µ̂t−r−24 + · · · .

r=1

Collecting terms by columns yields(1 − ωj ρL)−1 µ̂t = (1 + (ωj ρ)12 L12 + (ωj ρ)24 L24 + · · · )µ̂t
12 12+(1 + (ωj ρ) L

24 24

+ (ωj ρ) L

+ ···)

11X(ωj ρ)r µ̂t−r

r=1
11

=

X
1
1
µ̂
+
(ωj ρ)r Lr µ̂tt
1 − (ωj ρ)12 L12
1 − (ωj ρ)12 L12
r=1

1 + ΩR
j (L)
=
µ̂t ,
1 − (ωj ρ)12 L12
where ΩR
j (L) =P11

r r
r=1 (ωj ρ) L .

35

Using this result, we obtain the first order difference equation for x̂t (j, l):"#bj L(1 + ΩR
j (L))x̂t (j, l) = ωj x̂t−1 (j, l) + aj +µ̂t .
1 − (ωj ρ)12 L12
Equivalently, by repeated substitutions,"x̂t (j, l) = ωj12 x̂t−12 (j, l) +where Ωj (L) =#bj L(1 + ΩR
j (L))Ωj (L)µ̂t ,aj +1 − (ωj ρ)12 L12

(A5)

P11

r r
r=0 ωj L .

Similarly, the equation for the good j price index is the first order difference equation given by(44). It impliesp̂t (j, l) = λ12
j p̂t−12 (j, l) − λj Λj (L)µ̂t + (1 − λj )Λj (L)x̂t (j, l).

(A6)

Substituting (A5) into (A6) yieldsp̂t (j, l) = −+λj Λj (L)µ̂
12 t
1 − λ12j Lhi(1 − λj )Λj (L)Ωj (L) (1 − (ωj ρ)12 L12 )aj + bj L(1 + ΩR
(L))
j
12 12
12
12 12
(1 − λ12
j L )(1 − ωj L )(1 − (ωj ρ) L )

(A7)
µ̂t .

Analogously, we can obtain a similar equation for p̂∗t (j, l∗ ). Then, noting that q̂t (j, l, l∗ ) = p̂∗t (j, l∗ )−p̂t (j, l) and ∆Ŝt = µ̂t − µ̂∗t , we can obtain the following equation for the good-level real exchangerate:q̂t (j, l, l∗ ) =−λj Λj (L)∆Ŝt12
1 − λ12j Lhi(1 − λj )Λj (L)Ωj (L) (1 − (ωj ρ)12 L12 )aj + bj L(1 + ΩR
(L))
j
12 12
12
12 12
(1 − λ12
j L )(1 − ωj L )(1 − (ωj ρ) L )∆Ŝt

Arranging the terms and using ∆Ŝt = (1 − ρ12 L12 )−1 R(L)ηt , we obtain
12
12 12
12 12
∗
(1 − λ12
j L )(1 − ωj L )(1 − (ωj ρ) L )q̂t (j, l, l )(= (1 − ω 12 L12 )(1 − (ωj ρ)12 L12 )λj Λj (L)R(L)


12 12

− (1 − λj )Λj (L)Ωj (L)R(L) (1 − (ωj ρ) L))aj + bj L(1 + ΩR
j (L))

ηt.
1 − ρ12 L12

The terms inside the curly bracket gives Θj (L). Moreover, the first line of the terms has non-zerocoefficient for L46 , because (1 − ω 12 L12 )(1 − (ωj ρ)12 L12 ) have a non-zero coefficient for L24 and36Λj (L)R(L) have a non-zero coefficient for L22 . Since the second line of the terms inside the curly
brackets have L45 , the maximum power for L is 46.
By multiplying both sides of the equation by (1 − ρ12 L12 ) we obtain from (A2) to get(1 − ρ12 L12 )(1 − Φ̃j,1 L12 − Φ̃j,2 L24 − Φ̃j,3 L36 )q̂t (j, l, l∗ ) = Θj (L)ηt ,which gives us Φj,1 , Φj,2 , Φj,3 and Φj,4 .
The implications of Proposition A1 are as follows. First, the number of AR parameters arelimited to four and these four parameters are the coefficients on lags of 12, 24, 36, and 48 months.
Thus, the autoregressive part of the model has a form of autoregression on the past values of thereal exchange rates at annual frequencies. Second, if the AR part has the restriction describedabove and if the maximum order of MA coefficients is 46, dual stickiness pricing with ρ, λj , andωj ∈ (0, 1) can be written only with this representation. Third, this ARMA(48,46) process becomes
ARMA(4,3) in terms of annually sampled data. Finally, under the representation, we can also showthat the SAR is given by
αj =

4
X

12
12
Φj,r = 1 − (1 − ρ12 )(1 − λ12j )(1 − ωj )(1 − (ωj ρ) ),

(A8)

r=1

which is increasing in λj , ωj and ρ.

E

The long-run value of a good-level real exchange rate

This appendix shows the long-run value of qt (j, l, l∗ ). In what follows, we use variables without
time subscript to denote the steady state value.
Consider the steady state value of the price of good j in city l. In the steady state, U.S. firms
set prices such that
PH (j, l) =

θ
θ
W =χ
M.
θ−1
θ−1

Here, we used (16). Canadian firms choose prices such that
PF (j, l) =θθ
(1 + τ (j, l))SW ∗ = χ
κ(1 + τ (j, l))M.
θ−1
θ−1because of (17) and (19). Therefore, the price of good j in city l is
P (j, l) =1χ θ
[1 + κ1−θ (1 + τ (j, l))1−θ ] 1−θ M.
2θ−1

37

(A9)

By similar argument, we can derive P ∗ (j, l∗ ) as follows:P ∗ (j, l∗ ) =1χ θ[κ−(1−θ) + (1 + τ (j, l))1−θ ] 1−θ M ∗ .
2θ−1

(A10)

Given the good-level real exchange rate for good j of a city pair between l and l∗ is given by
q(j, l, l∗ ) = SP (j, l)/P (j, l∗ ), the equations (19), (A9), and (A10) imply
1∗

q(j, l, l ) =[1 + κ1−θ (1 + τ (j, l∗ ))1−θ ] 1−θ
1

.

(A11)

[1 + κ1−θ (1 + τ (j, l))1−θ ] 1−θ
Thus, the long-run value of a good-level of real exchange rate depends on the city pair.

References
Ahlin, C., and M. Shintani (2007): “Menu Costs and Markov Inflation: A Theoretical Revision
with New Evidence,” Journal of Monetary Economics, 54(3), 753–784.
Andrés, J., J. D. López-Salido, and E. Nelson (2005): “Sticky Price Models and the Natural
Rate of Hypothesis,” Journal of Monetary Economics, 52(5), 1025–1053.
Andrews, D. W. K., and H.-Y. Chen (1994): “Approximately Median-Unbiased Estimation ofAutoregressive Models,” Journal of Business and Economic Statistics, 12(2), 187–204.
Arellano, M., and S. Bond (1991): “Some tests of specification for panel data: Monte Carloevidence and an application to employment equations,” Review of Economics Studies, 58(2),277–297.
Bils, M., and P. J. Klenow (2004): “Some Evidence on the Importance of Sticky Prices,”
Journal of Political Economy, 112(5), 947–985.
Blinder, A. S., E. R. D. Canetti, D. E. Lebow, and J. B. Rudd (1998): Asking about Prices:
A New Approach to Understanding Price Stickiness. Russell Sage Foundation, New York.
Calvo, G. (1983): “Staggered Prices in a Utility-Maximizing Framework,” Journal of Monetary
Economics, 12(3), 383–398.
Chari, V. V., P. J. Kehoe, and E. R. McGrattan (2002): “Can Sticky Price Models Generate
Volatile and Persistent Real Exchange Rates?,” Review of Economics Studies, 69(3), 533–563.

38

Clark, T. E. (2006): “Disaggregate Evidence on the Persistence of Consumer Price Inflation,”
Journal of Applied Econometrics, 21(5), 563–587.
Crucini, M. J., and M. Shintani (2008): “Persistence in Law of One Price Deviations: Evidence
from Micro-data,” Journal of Monetary Economics, 55(3), 629–644.
Crucini, M. J., and C. Telmer (2007): “Microeconomic Sources of Real Exchange Rate Variability,” unpublished manuscript.
Crucini, M. J., C. I. Telmer, and M. Zachariadis (2005): “Understanding European RealExchange Rates,” American Economic Review, 95(3), 724–738.
Dupor, B., T. Kitamura, and T. Tsuruga (2008): “Integrating Sticky Prices and Sticky
Information,” Review of Economics and Statistics forthcoming.
Engel, C., and J. H. Rogers (2004): “European Market Integration after Euro,” Economic
Policy, 19, 347–384.
Fabiani, S., M. Druant, I. Hernando, C. Kwapil, B. Laudau, C. Loupias, F. Martins,
T. Y. Matha, R. Sabbatini, H. Stahl, and A. C. J. Stokman (2005): “The Pricing
Behaviour of Firms in the Euro Area New Survey Evidence,” ECB working paper No 535.
Gorodnichenko, Y. (2008): “Endogenous Information, Menu costs, and Inflation Persistence,”
NBER working paper # 14184.
Kahn, H., and Z. Zhu (2006): “Estimates of the Sticky-Information Phillips Curve for the UnitedStates,” Journal of Money, Credit and Banking, 38(1), 195–208.
Kehoe, P. J., and V. Midrigan (2007): “Sticky Prices and Sectoral Real Exchange Rates,”Federal Reserve Bank of Minneapolis Working Paper No 656.
Knotek, E. S. (2006): “A Tale of Two Rigidities: Sticky Prices in a Sticky Information Environment,” Federal Reserve Bank of Kansas City Research Working Paper 06-15.
Mankiw, N. G., and R. Reis (2002): “Sticky Information Versus Sticky Prices: A Proposal toReplace the New Keynesian Phillips Curve,” Quarterly Journal of Economics, 117(4), 1295–1328.
Nakamura, E., and J. Steinsson (2008): “Five Facts About Prices: A Reevaluation of Menu
Cost Models,” Quarterly Journal of Economics forthcoming.
39

Parsley, D. C., and S.-J. Wei (2007): “A Prism into the PPP Puzzles: The Micro-foundationsof Big Mac Real Exchange Rates,” Economic Journal, 117, 1336–1356.
Rogers, J. H. (2007): “Monetary Union, Price Level Convergence, and Inflation: How close isEurope to the USA?,” Journal of Monetary Economics, 54(3), 785–796.
Sims, C. A. (2003): “Implications of Rational Inattention,” Journal of Monetary Economics, 50(3),665–690.
Woodford, M. (2008):“Information-Constrained State-Dependent Pricing,” unpublishedmanuscript.
Yun, T. (1996): “Nominal Price Rigidity, Money Supply Endogeneity, and Business Cycles,”Journal of Monetary Economics, 37(2), 345–370.

40

Figure 1: Persistence without information delay: function of money growth parameter(ρ) and Calvo
parameter (λj )Persistence of AR(2) process

Persistence of AR(2) process

1
0.9

1

λ = 0.95

0.9jρ = 0.83
0.8

0.8

0.7

0.7

λj = 0.5

0.6

SAR

SAR

0.6
0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0
0

0.2

0.4

ρ

0.6

0.8

0
0

1

ρ=0

0.2

0.4

0.6

λ

0.8

1

j

Figure 2: Volatility without information delay: function of money growth parameter(ρ) and Calvo
parameter(λj )Volatility of AR(2) process
8

7

7

6

6

λ = 0.95

5

4

4

3

3

2

2

1

ρ = 0.83

5

jNormalized std

Normalized std

Volatility of AR(2) process8

λ = 0.5

1

0.4

0
0

j

0
0

0.2

ρ=0
ρ

0.6

0.8

1

0.2

0.4

λ

j

NOTES: The discount factor β is 0.99.

41

0.6

0.8

1

Figure 3: Persistence with information delay: function of information stickiness parameter(ωj ) andCalvo parameter(λj )Persistence of ARMA(4,2) process

Persistence of ARMA(4,2) process11

λj = 0.95 & ρ = 0.83
0.98

0.96

0.96

0.94

0.94

λj = 0.5 & ρ = 0.83

0.92

SAR

SAR

0.92

ω = 0.98 & ρ = 0.83
j

0.98

0.9

0.9

0.88

0.88

0.86

0.86

0.84

0.84

0.82

0.82

ω = 0 & ρ = 0.83
j

0.8
0

0.2

0.4

ω

0.6

0.8

0.8
0

1

0.2

0.4

j

λ

0.6

0.8

1

j

Figure 4: Volatility with information delay: function of information stickiness parameter(ωj ) andCalvo parameter(λj )Volatility of ARMA(4,2) process
25

20

20

15

15

Normalized std

Normalized std

Volatility of ARMA(4,2) process25

10ω = 0.98 & ρ = 0.83
j10

λj = 0.95 & ρ = 0.83

55

λ = 0.5 & ρ = 0.83

ωj = 0 & ρ = 0.83

j

0
0

0.2

0.4

ω

0.6

0.8

0
0

1

j

0.2

0.4

λ

j

NOTES: The discount factor β is 0.99.

42

0.6

0.8

1

43

Figure 5: The cross-border city pairs in the U.S. and Canada

Figure 6: Empirical distribution of LOP deviations in 1990 and 2005

1
1990
2005

0.9
0.8
0.7

density

0.6
0.5
0.4
0.3
0.2
0.1
0
−1.5

−1

−0.5

0

0.5

log of q (j,l,l*)t

NOTES: Kernel density estimates.

44

1

1.5

45

Figure 7: Real exchange rate persistence and price stickiness: Bils and Klenow (2004)

46

Figure 8: Real exchange rate persistence and price stickiness: Nakamura and Steinsson (2008)

Figure 9: Empirical distribution of good-specific information delays 1/(1 − ωj ): Bils and Klenow(2004)
Information delay implied by persistence
0.14

0.12

relative frequency

0.1

0.08

0.06

0.04

0.02

0
0

10

20

30

40

50

60

70

80

90

70

80

90

average duration of information stickiness 1/(1−ω )jInformation delay implied by volatility
0.14

0.12

relative frequency

0.1

0.08

0.06

0.04

0.02

0
0

10

20

30

40

50

60

average duration of information stickiness 1/(1−ω )j

NOTES: The upper panel shows the relative histogram of average information delay 1/(1 − ωj ) where ωj ∈ [0, 1) for
each good j is obtained by minimizing the distance between the observed SAR and theoretical prediction from dual
stickiness pricing using λj = 1 − fj from Bils and Klenow (2004) and ρ = 0.83. The smoothed lines are kernel density
estimates. The lower panel shows the distribution when each ωj is obtained by minimizing the distance between theobserved volatility and theoretical prediction using λj = 1 − fj , ρ = 0.83 and β = 0.99.

47

Figure 10: Empirical distribution of good-specific information delays 1/(1 − ωj ): Nakamura andSteinsson (2008)
Information delay implied by persistence

0.3

relative frequency

0.25

0.2

0.15

0.1

0.05

0
0

10

20

30

40

50

60

70

80

90

70

80

90

average duration of information stickiness 1/(1−ω )jInformation delay implied by volatility

0.3

relative frequency

0.25

0.2

0.15

0.1

0.05

0
0

10

20

30

40

50

60

average duration of information stickiness 1/(1−ω )j

NOTES: See the notes of Figure 9. Nakamura and Steinsson’s (2008) frequency of price changes, in stead of Bils andKlenow’s (2004), is used for λj = 1 − fj in the computation of the theoretical value.

48

Table 1: Summary of transformations from monthly to annual specification
Monthly specification

Annual specification

Calvo (ρ=0)

q̂t (j, l, l∗ ) = λj q̂t−1 (j, l, l∗ ) − λj ηt∗q̂t (j, l, l∗ ) = λ12j q̂t−12 (j, l, l ) − λj Λj (L)ηt

Calvo (ρ > 0)q̂t (j, l, l∗ ) = (λj + ρ)q̂t−1 (j, l, l∗ )12∗q̂t (j, l, l∗ ) = (λ12j + ρ )q̂t−12 (j, l, l )Dual stickiness−λj ρq̂t−2 (j, l, l∗ ) − θj ηtP4q̂t (j, l, l∗ ) = r=1 φj,r q̂t−r (j, l, l∗ )P2
+ r=0 θj,r ηt−r

12
∗
−λ12
j ρ q̂t−24 (j, l, l ) − λj Λj (L)R(L)ηtP4q̂t (j, l, l∗ ) = r=1 Φj,r q̂t−12r (j, l, l∗ )+Θj (L)ηt

NOTES: The left panel shows the original monthly ARMA processes which are in the main text. The right panel
shows corresponding conversions such that autoregressive coefficients are non-zero only if the lags are multiples of
12 and that moving average terms are finite. These conversions allow us to estimate the original monthly ARMA
process with annually sampled data. The autoregressive parameters Φj,r and moving average polynomials, Λj (L),
R(L) and Θj (L) are given in Appendix D.

Table 2: Proportions of explained persistence of good-level real exchange rates: Bils and Klenow(2004)
Calvo pricing with various ρρTheory/Data

0

0.3

0.5

0.7

0.9

0.95

0.98

0.946

0.059

0.058

0.058

0.088

0.634

1.043

1.425

1.000

Dual stickiness pricing with ρ = 0.83
ωTheory/Data

0

0.3

0.5

0.7

0.9

0.95

0.98

0.930

0.306

0.323

0.323

0.350

0.792

1.209

1.529

1.000

NOTES: Numbers are median ratios of the theoretical persistence, predicted by Bils and Klenow (2004), to observed
persistence measured by SAR estimated from real exchange rate data. Theoretical persistence for the upper panel isthe SAR for various ρ when Calvo pricing is used. Theoretical persistence for the lower panel is the SAR for various
common ω with ρ = 0.83 when dual stickiness pricing is used. Median SAR estimates for AR(1), AR(2) and AR(4)models are 0.563, 0.568, and 0.508, respectively. The last column of each panel shows the value of ρ and ω, giving
the median ratio closest to one.

49

Table 3: Proportions of explained persistence of good-level real exchange rates: Nakamura andSteinsson (2008)
Calvo pricing with various ρρTheory/Data

0

0.3

0.5

0.7

0.9

0.95

0.98

0.923

0.484

0.505

0.506

0.549

0.922

1.226

1.522

1.000

Dual stickiness pricing with ρ = 0.83
ωTheory/Data

0

0.3

0.5

0.7

0.9

0.95

0.98

0.895

0.664

0.659

0.660

0.681

1.015

1.329

1.619

1.000

NOTES: See the notes of Table 2. Nakamura and Steinsson’s (2008) data, instead of Bils and Klenow’s (2004), isused for the computation of the theoretical value.

Table 4: Proportions of explained volatility of good-level real exchange rates: Bils and Klenow(2004)
Calvo pricing with various ρρTheory/Data

0

0.3

0.5

0.7

0.9

0.95

0.98

0.521

0.130

0.143

0.153

0.148

0.096

0.064

0.036

0.153

Dual stickiness pricing with ρ = 0.83
ωTheory/Data

0

0.3

0.5

0.7

0.9

0.95

0.98

0.940

0.125

0.149

0.181

0.276

0.691

1.129

1.950

1.000

NOTES: Numbers are median ratios of the theoretical volatility, predicted by Bils and Klenow (2004), to observed
volatility measured by normalized standard deviation of real exchange rate data. Theoretical volatility for the upper
panel is the normalized standard deviation for various ρ when Calvo pricing is used. Theoretical volatility for the
lower panel is the normalized standard deviation for various common ω with ρ = 0.83 when dual stickiness pricing isused. The normalized sample standard deviation of real exchange rate is the extracted standard deviation component
due to time specific shocks in the two-way error component model. The last column of each panel shows the value ofρ and ω, giving the median ratio closest to one.

50

Table 5: Proportions of explained volatility of good-level real exchange rates: Nakamura andSteinsson (2008)
Calvo pricing with various ρρTheory/Data

0

0.3

0.5

0.7

0.9

0.95

0.99

0.801

0.234

0.298

0.351

0.403

0.398

0.312

0.212

0.426

Dual stickiness pricing with ρ = 0.83
ωTheory/Data

0

0.3

0.5

0.7

0.9

0.95

0.98

0.916

0.423

0.449

0.478

0.562

0.882

1.311

2.082

1.000

NOTES: See the notes of Table 4. Nakamura and Steinsson’s (2008) data, instead of Bils and Klenow’s (2004), isused for the computation of the theoretical value.

Table 6: Intervals between information update
one month1.01-5.99

6-11.99

12 months

or lessmonthsmonthsor above

25.6

13.2

16.5

44.5

Persistence

11.5

8.5

26.7

53.3

Volatility

6.1

4.2

18.2

71.5

Nakamura

Persistence

33.3

12.7

18.2

35.8

and Steinsson

Volatility

21.8

13.9

14.5

49.7

Blinder et. al.’s (1998)
survey data
Bils and Klenow

NOTES: The numbers in the first row represent the distribution, in percentages, of the frequency of price reviews
reported in Blinder, Canetti, Lebow, and Rudd (1998, Table 4.7 in p. 90). The second and third rows show the
distribution of information delay implied by the observed persistence and volatility of real exchange rates based on
Bils and Klenow’s (2004) data on the frequency of price changes. The fourth and fifth rows show the distribution of
information delay when Nakamura and Steinsson’s (2008) data on the frequency of regular price changes is used.

51

Table A1: Frequency of price changes and information updates by category
ELI

Category name

Price

Information

Price

Bils &

implied byNakamura &KlenowPer.

Vol.

Information

Steinsson

implied by

# of

Per.

Vol.

goods

FA

Cereals and cereals products

26.5

11.1

4.9

11.5

100.0

6.9

7

FB

Bakery products

25.7

5.5

4.4

9.8

8.7

6.8

1

FC

Beef and veal

47.2

12.2

4.8

25.5

13.8

5.5

8

FD

Pork

47.9

10.2

3.8

23.2

12.3

4.4

6

FF

Poultry

39.4

53.0

2.7

16.6

53.6

3.1

2

FG

Fish and seafood

42.4

8.7

10.6

20.4

9.7

15.2

1

FH

Eggs

61.8

7.5

6.5

47.6

7.5

6.8

1

FJ

Dairy and related products

33.7

6.7

4.4

24.9

7.2

5.3

4

FK

Fresh fruits

36.4

7.5

5.6

16.6

17.3

6.9

8

FL

Fresh vegetables

62.4

24.4

3.4

40.8

25.6

3.6

6

FM

Processed fruits and vegetables

24.9

5.2

4.1

10.5

7.7

6.0

6

FN

Juices and nonalcoholic drinks

35.6

6.1

2.4

13.1

8.2

2.9

4

FP

Beverage incl. coffee and tea

21.1

8.8

7.3

8.9

18.1

13.2

11

FR

Sugar and sweets

22.9

4.8

7.0

9.9

7.1

12.7

2

FS

Fats and oils

29.5

14.5

6.1

18.1

16.0

6.7

8

FV

Food away from home

9.0

3.6

12.9

5.0

5.9

88.5

3

FW

Alcoholic beverages at home

19.3

6.1

6.8

10.6

7.5

10.0

7

FX

Alcoholic beverages away from home

6.4

2.4

14.1

5.0

3.0

25.1

1

HB

Lodging away from home

38.1

11.2

4.9

41.7

11.2

4.8

2

HF

Gas and electricity

43.4

3.6

5.3

38.1

3.6

5.4

1

HK

Appliances

19.0

2.7

3.6

3.6

15.0

25.6

2

HL

Other equipment and furnishings

16.1

10.2

6.6

2.8

100.0

100.0

1

HN

Housekeeping supplies

19.2

9.1

3.2

9.4

60.0

5.7

8

HP

Household operations

6.5

6.7

38.6

4.3

10.8

100.0

1

AA

Men’s apparel

26.0

3.1

7.5

4.5

11.3

100.0

5

AB

Boy’s apparel

25.9

2.4

11.5

4.3

6.9

100.0

1

AC

Women’s apparel

45.0

6.3

6.8

2.5

100.0

100.0

6

AE

Footwear

28.0

4.8

7.1

3.5

60.0

100.0

2

AF

Infants’ and toddlers’ apparel

36.3

7.6

7.8

3.5

100.0

100.0

2

TA

New and used motor vehicles

39.1

7.5

5.7

31.3

7.6

6.0

7

TB

Motor fuel

78.9

11.3

6.3

88.6

11.3

6.2

1

TD

Motor vehicle maintenance and repair

11.6

6.7

6.1

10.7

7.1

6.4

2

TE

Motor vehicle insurance

15.5

3.2

11.8

8.2

4.6

27.7

1

TG

Public transportation

5.0

4.3

19.8

4.4

4.9

31.2

3

MB

Nonprescription drugs and medical supplies

13.7

5.8

14.8

7.9

8.7

42.6

2

RA

Video and audio

22.0

10.3

10.2

9.4

55.7

24.8

2

RD

Photography

8.6

9.6

16.2

8.8

12.0

30.5

2

RF

Recreation services

8.8

6.7

13.3

9.0

6.6

12.9

1

RG

Recreational reading materials

12.4

15.1

34.5

5.4

100.0

100.0

3

GA

Tobacco and smoking products

21.6

4.3

1.3

23.2

4.3

1.3

4

GB

Personal care products

11.1

4.7

10.8

3.9

14.7

100.0

10

GC

Personal care services

4.1

78.7

100.0

3.1

100.0

100.0

2

GD

Miscellaneous personal services

5.1

13.8

100.0

3.0

100.0

100.0

8

NOTES: ELI in the first column stands for the entry level item in the CPI. EIU price series for good and service used in
the analysis are matched to BLS’s ELI codes. The third column shows the median value of average monthly frequencies of
price changes from Bils and Klenow (2004), among the goods included in each category. The fourth and fifth columns show
the median value of the estimated average monthly frequencies of information updates implied by the persistence (Per.) andvolatility (Vol.) of good-level real exchange rates, when Bils and Klenow (2004) is used to compute the theoretical prediction.
The sixth column is the median of the frequencies of regular price changes from Nakamura and Steinsson (2008). The seventhand eighth columns show the median of frequencies of information updates when Nakamura and Steinsson’s (2008) data is used.
The last column shows the total numbers of goods and services included in each category of ELI codes.

52