Good-level Real Exchange Rates

Accounting for persistence and volatility of
good-level real exchange rates:
the role of sticky information

Mario J. Crucini,1 Mototsugu Shintani,2Takayuki Tsuruga3
1 Vanderbilt University
2 Vanderbilt University and Bank of Japan
3 Bank of Japan

Macroeconomic Conference, 2007

Good-level Real Exchange Rates
Motivation
Speed of adjustment to LOP = Calvo parameter

Motivation: Law-of-One-Price (LOP) deviation
I Like PPP, the speed of adjustment toward a long-run LOPlevel is measured by estimating αj ofj+ νtj .
qtj = αj qt−1

qtj : (log) real exchange rate for good j
I Kehoe and Midrigan (2007, KM) proved that the Calvo

sticky price model impliesjqtj = λj qt−1+ νtj ,where λj : the probability of no price change (Calvoparameter, degree of price stickiness)

Good-level Real Exchange Rates
Motivation
Speed of adjustment to LOP = Calvo parameter

Motivation: Law-of-One-Price (LOP) deviation
I Like PPP, the speed of adjustment toward a long-run LOPlevel is measured by estimating αj ofj+ νtj .
qtj = αj qt−1

qtj : (log) real exchange rate for good jI Kehoe and Midrigan (2007, KM) proved that the Calvo

sticky price model implies
jqtj = λj qt−1+ νtj ,where λj : the probability of no price change (Calvoparameter, degree of price stickiness)

Good-level Real Exchange Rates
Motivation
Persistence and volatility puzzles

KM’s findings on persistence and volatility
I Using recent micro studies, λj is observable.

KM find the following two puzzles:
1. If the model is correct, αj = λj . However,α̂j  λj (Persistence puzzle)
2. If the model is correct, it will fully explain volatility. However,ˆ j,data )  std(q j,model ) (Volatility puzzle)
std(qtt

Good-level Real Exchange Rates
Motivation
Persistence and volatility puzzles

KM’s findings on persistence and volatility
I Using recent micro studies, λj is observable.

KM find the following two puzzles:
1. If the model is correct, αj = λj . However,α̂j  λj (Persistence puzzle)
2. If the model is correct, it will fully explain volatility. However,ˆ j,data )  std(q j,model ) (Volatility puzzle)
std(qtt

Good-level Real Exchange RatesMotivation
Persistence and volatility puzzles

KM’s findings on persistence and volatility
I Using recent micro studies, λj is observable.

KM find the following two puzzles:
1. If the model is correct, αj = λj . However,α̂j  λj (Persistence puzzle)
2. If the model is correct, it will fully explain volatility. However,ˆ j,data )  std(q j,model ) (Volatility puzzle)
std(qtt

Good-level Real Exchange Rates
Motivation
Persistence and volatility puzzles

KM’s findings on persistence and volatility
I Using recent micro studies, λj is observable.

KM find the following two puzzles:
1. If the model is correct, αj = λj . However,α̂j  λj (Persistence puzzle)
2. If the model is correct, it will fully explain volatility. However,ˆ j,data )  std(q j,model ) (Volatility puzzle)
std(qtt

Good-level Real Exchange Rates
Motivation
Persistence and volatility puzzles

KM’s findings on persistence and volatility
I Using recent micro studies, λj is observable.

KM find the following two puzzles:
1. If the model is correct, αj = λj . However,α̂j  λj (Persistence puzzle)
2. If the model is correct, it will fully explain volatility. However,ˆ j,data )  std(q j,model ) (Volatility puzzle)
std(qtt

Good-level Real Exchange Rates
Contribution of our paper

Contributions of this paper
1. Confirm KM’s findings with highly disaggregated panel
data
Crucini and Shintani’s (2007) data
Highly disaggregated data with 165 goods. (KM: 66 goods)
I Panel data of good-level RER between cities in US andCanada. (KM: time series)II

2. Propose a model to solve the two puzzles.
Integrate sticky information with the standard Calvo sticky
price model
I Add sticky information by Mankiw and Reis (2002)
I We call the model ‘dual stickiness’ model.
IGood-level Real Exchange Rates
Contribution of our paper

Contributions of this paper
1. Confirm KM’s findings with highly disaggregated panel
data
Crucini and Shintani’s (2007) data
Highly disaggregated data with 165 goods. (KM: 66 goods)
I Panel data of good-level RER between cities in US andCanada. (KM: time series)II

2. Propose a model to solve the two puzzles.
Integrate sticky information with the standard Calvo sticky
price model
I Add sticky information by Mankiw and Reis (2002)
I We call the model ‘dual stickiness’ model.
IGood-level Real Exchange Rates
Contribution of our paper
Intuition

Intuition: Why can dual stickiness model solve thepuzzles?
1. Persistence Puzzle
Even if price adjustment is very fast, good-price adjustment
can be slow due to information stickiness (ωj ↑).
I Persistence ↑.

I

2. Volatility Puzzle
Even if price adjustment is very fast, good-prices can bealmost unaltered due to information stickiness (ωj ↑ ).
I RER keeps track of volatile nominal exchange rate.
I Volatility ↑.
IGood-level Real Exchange Rates
Contribution of our paper
Intuition

Intuition: Why can dual stickiness model solve thepuzzles?
1. Persistence Puzzle
Even if price adjustment is very fast, good-price adjustment
can be slow due to information stickiness (ωj ↑).
I Persistence ↑.

I

2. Volatility Puzzle
Even if price adjustment is very fast, good-prices can bealmost unaltered due to information stickiness (ωj ↑ ).
I RER keeps track of volatile nominal exchange rate.
I Volatility ↑.
IGood-level Real Exchange Rates
Model & Data
Overview of the model

Overview of the model: Two-country generalequilibrium model
I Households
IU(ct , nt ) = log ct − χnt with cash-in-advance constraint.

I Firms producing good jsell goods in monopolistically competitive domestic andforeign local markets.
I set price for good j in each local market(local currency pricing)
I face two constraints:I

1. cannot change price with prob. λj .
2. cannot update info. with prob. ωj .
I Governments
Icontrol money growth rates. AR(1)Good-level Real Exchange Rates
Model & Data
Overview of the model

Sketch of dual stickiness model (Calvo model withinfo. delay)
I Dual stickiness model has two nominal rigidities. (price &information)
I Under sticky prices, the domestic optimal price isjP̂H,t= (1 − βλj )∞X(βλj )h Et (Ŵt+h )h=0∞X∗P̂Fj ,t = (1 − βλj )(βλj )h Et (Ŝt+h + Ŵt+h)
h=0Ŵt (Ŵt∗ ): nominal wages in the home (foreign) country. St :nominal exchange rate.

Good-level Real Exchange Rates
Model & Data
Overview of the modelSketch of dual stickiness model (2)I Sticky information: a fraction of firms cannot have thenewest informationIjProb. ωj : use info. set they last updated ⇒ Et−k P̂H,t,Et−k P̂Fj ,tI

jProb. 1 − ωj : use the newest info. set ⇒ P̂H,t, P̂Fj ,tI

The index for newly set prices X̂tj collects these prices.

I Due to Calvo assumption,jP̂tj = λj P̂t−1+ (1 − λj )X̂tj
I Good-level RER is given by qjj∗
jt = Ŝt + P̂t − P̂t .

Good-level Real Exchange Rates
Model & Data
Overview of Data

Overview of Data: Worldwide Cost of Living Survey
I Prices from 13 US × 4 CAN cities:
I Total of 52 cross-border city pairs
I # of goods = 165. Annual data over 1990-2005.

Good-level Real Exchange Rates
Persistence

Persistence Puzzle

1. KM’s benchmark case (ωj = 0)I

Calvo model without info. delay

2. Our dual stickiness model (ωj ≥ 0)I

Calvo model with info. delay

Good-level Real Exchange Rates
Persistence
Calvo modelCalvo model’s predictions (No information delay)
I We estimate the model with the annual data.
I We have λj : monthly infrequency of price change frommicro studies.
I Panel version of KM’s benchmark case (i.i.d. moneygrowth)
AR(1)

jjj
j
0qi,t= λ12
j qi,t−1 + u D̃t + ζi + νi,tI i: cross-border city pair (e.g., NY and Tronto)
I If the Calvo model is correct,
I our estimate of AR(1) coef. α̂j = λ12 with annual datajGood-level Real Exchange Rates
Persistence
Calvo modelCalvo model’s predictions (No information delay)
I We estimate the model with the annual data.
I We have λj : monthly infrequency of price change frommicro studies.
I Panel version of KM’s benchmark case (i.i.d. moneygrowth)
AR(1)

jjj
j
0qi,t= λ12j qi,t−1 + u D̃t + ζi + νi,tI i: cross-border city pair (e.g., NY and Tronto)
I If the Calvo model is correct,
I our estimate of AR(1) coef. α̂j = λ12 with annual dataj

Good-level Real Exchange Rates
Persistence
Calvo modelCalvo model’s predictions (No information delay)
I We estimate the model with the annual data.
I We have λj : monthly infrequency of price change frommicro studies.
I Panel version of KM’s benchmark case (i.i.d. moneygrowth)
AR(1)

jjj
j
0qi,t= λ12
j qi,t−1 + u D̃t + ζi + νi,tI i: cross-border city pair (e.g., NY and Tronto)
I If the Calvo model is correct,
I our estimate of AR(1) coef. α̂j = λ12 with annual dataj

Good-level Real Exchange Rates
Persistence
Calvo model

Persistence Puzzle: KM’s benchmark caseCalvo model with ρ=0: AR(1)1αj (SAR)

0.8

0.6α = λ12jj

0.4

0.2

000.2

0.4

0.6

λ12
= (1−fj)12j

0.8

Blue line: Calvo model’s prediction

1

Good-level Real Exchange Rates
Persistence
Calvo model

Persistence Puzzle: KM’s benchmark caseCalvo model with ρ=0: AR(1)1αj (SAR)0.8

0.6
Calvo model
OLS
0.4

0.2

000.2

0.4

0.6

λ12
= (1−fj)12
j0.8

1Black dashed line: OLS line from red points.

Good-level Real Exchange Rates
Persistence
Dual Stickness model

Dual stickiness model’s prediction (ωj > 0)
I Panel version of good-level RER under dual stickiness

model
AR(4)

jqi,t=4XjΨr qi,t−r + u 0 D̃t + ζij + νi,tr =1I In a general AR(p) model, a persistence measure is thesum of autoregressive coefficients (SAR).
P4
I If dual stickiness model is correct, αj =r =1 Ψr must be
121212 12α̂j = 1 − (1 − λ12j )(1 − ρ )(1 − ωj )(1 − ωj ρ ).

So, ωj ↑⇒ persistence ↑!!Good-level Real Exchange Rates
Persistence
Dual Stickness model

Dual stickiness model’s prediction (ωj > 0)
I Panel version of good-level RER under dual stickinessmodelAR(4)jqi,t=4XjΨr qi,t−r + u 0 D̃t + ζij + νi,tr =1I In a general AR(p) model, a persistence measure is thesum of autoregressive coefficients (SAR).
P4
I If dual stickiness model is correct, αj =r =1 Ψr must be
121212 12α̂j = 1 − (1 − λ12j )(1 − ρ )(1 − ωj )(1 − ωj ρ ).

So, ωj ↑⇒ persistence ↑!!Good-level Real Exchange Rates
Persistence
Dual Stickness model

Persistence Puzzle: dual stickiness model

Purple line: No info. delay, Green line: 50 month info. delay

Good-level Real Exchange Rates
Persistence
Dual Stickness model

Persistence Puzzle: dual stickiness model

Black dashed line: OLS line from red points.

Good-level Real Exchange Rates
Persistence
Dual Stickness model

How much should be information stickiness needed toexplain persistence?
ωBils andKlenow’s data

00.31

median(αjtheory /αjdata )0.5
0.9
0.95 0.98
0.32

0.79

1.21

1.53

I Our model can explain 100% of the median of persistenceifI ω = 0.93 with Bils and Klenow’s data
I (ω = 0.89 with Nakamura and Steinsson’s data)
I Avg. duration btwn info. updates is 14 and 9.5 months,

resp.

Good-level Real Exchange Rates
Persistence
Dual Stickness model

How much should be information stickiness needed toexplain persistence?
ωBils andKlenow’s data

00.31

median(αjtheory /αjdata )0.5
0.9
0.95 0.98
0.32

0.79

1.21

1.53

I Our model can explain 100% of the median of persistence

ifI ω = 0.93 with Bils and Klenow’s data
I (ω = 0.89 with Nakamura and Steinsson’s data)
I Avg. duration btwn info. updates is 14 and 9.5 months,

resp.

Good-level Real Exchange Rates
Volatility

Volatility Puzzle

1. KM’s benchmark case (ωj = 0)I

Calvo model without info. delay

2. Our dual stickiness model (ωj ≥ 0)I

Calvo model with info. delay

Good-level Real Exchange Rates
Volatility
Calvo modelCalvo model’s predictions (No information delay)

We compute std ratio of the theory to the data in terms of atime varying component.
"j,theory #std(qi,t)median= 0.13
from Bils & Klenowj,datastd(qi,t )Good-level Real Exchange RatesVolatility
Calvo model

How much should be information stickiness needed toexplain volatility?

ωBils andKlenow’s data

median[std(q j,theory )/std(q j,data )]00.5
0.9
0.95 0.98
0.13

0.18

0.69

1.13

1.95

I Our model can explain 100% of the median of volatility if
I ω = 0.94 with Bils and Klenow’s data;
I (ω = 0.92 with Nakamura and Steinsson’s data.)
I Avg. duration btwn info. updates is 17 and 12 months.

Good-level Real Exchange Rates
Volatility
Calvo model

How much should be information stickiness needed toexplain volatility?

ωBils andKlenow’s data

median[std(q j,theory )/std(q j,data )]00.5
0.9
0.95 0.98
0.13

0.18

0.69

1.13

1.95

I Our model can explain 100% of the median of volatility if
I ω = 0.94 with Bils and Klenow’s data;
I (ω = 0.92 with Nakamura and Steinsson’s data.)
I Avg. duration btwn info. updates is 17 and 12 months.

Good-level Real Exchange Rates
Volatility
Calvo model

How much should be information stickiness needed toexplain volatility?

ωBils andKlenow’s data

median[std(q j,theory )/std(q j,data )]00.5
0.9
0.95 0.98
0.13

0.18

0.69

1.13

1.95

I Our model can explain 100% of the median of volatility ifI ω = 0.94 with Bils and Klenow’s data;
I (ω = 0.92 with Nakamura and Steinsson’s data.)
I Avg. duration btwn info. updates is 17 and 12 months.

Good-level Real Exchange Rates
Conclusion

Conclusion

1. The Kehoe and Midrigan’s findings are robust to the use ofhighly disaggregate panel data.
ICalvo model fails to explain perisistence and volatility of
good-level RER.

2. One possible explanation is the dual stickiness model.
The dual stickiness model solves persistence and volatility
puzzles.
I Implied durations between info. updates are comparable toestimates of previous studies.
IGood-level Real Exchange Rates
Reconciling monthly models with annual data

Reconciling monthly models with annual data
I e.g., the monthly Calvo model with iid money growth:jqitj = λj qit−1+ λj ηt + ζ̃ij .

where ηt : difference between money growth rates of twocountries.
I Annual transformation
jqitj = λj qit−1+ λj ηt + ζ̃ijj= λ2j qit−2+ λj ηt + λ2j ηt−1 + (1 + λj )ζ̃ij···
j= λ12j qit−12 + λj11Xλj ηt−r +r =0|{zu 0 D̃t11X

λj ζ̃ijr =0}| {z }
jζiGood-level Real Exchange Rates
How much should be information stickiness needed?
Persistence

How much should be information stickiness needed toexplain persistence?Good-specific ωj rather than ω

relative density

Distribution of information delay implied by persistence

0.1

0.05

00102030

4050607080

average duration of information stickiness 1/(1−ωj)90

median = 12.9 months (Bils and Klenow), 8.9 months(Nakamura and Steinsson)

Good-level Real Exchange Rates
How much should be information stickiness needed?
Volatility

How much should be information stickiness needed toexplain volatility?Good-specific ωj rather than ω
Information delay implied by volatility
0.07

relative frequncy

0.06
0.05
0.04
0.03
0.02
0.01
00

102030

40

50

60

70

80

average duration of information stickiness 1/(1−ωj)90

median = 16.6 months (Bils and Klenow), 11.9 months(Nakamura and Steinsson)

Good-level Real Exchange Rates
Strategic complementarities

KM find pricing complementarities do not help forsolving puzzles
I KM also consider pricing complementarities.
I Production function of firms
a 1−ay =m nZ,m=θ−1θmj θZ

θ−1dj,mj =θ−1
θmj,z dz

I Input costs of firms in all good j move together.
I They set an extreme value of a = 0.99. They find....
I The persistence can be explained somewhat better thanotherwise.
I The volatility cannot be explained by pricingcomplementarities.

 θθ−1