NBER WORKING PAPER SERIESDO STICKY PRICES INCREASE REAL EXCHANGE RATE VOLATILITY AT
THE SECTOR LEVEL?
Mario J. Crucini
Mototsugu Shintani
Takayuki Tsuruga
Working Paper 16081
http://www.nber.org/papers/w16081

NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
June 2010

We thank Yasushi Iwamoto and seminar participants at the Federal Reserve Bank of Atlanta, Clemson
University, the Federal Reserve Bank of Dallas, Otaru University of Commerce for their helpful discussion
and comments. We are grateful to John Taylor for his encouragement at the early stage of the project.
We also thank Virgiliu Midrigan for providing the data set used in the analysis. Mario J. Crucini andMototsugu Shintani gratefully acknowledge the financial support of National Science Foundation (SES0524868). Takayuki Tsuruga gratefully acknowledges the financial support of Grant-in-aid for Scientific
Research. The views expressed herein are those of the authors and do not necessarily reflect the viewsof 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.
© 2010 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.

Do Sticky Prices Increase Real Exchange Rate Volatility at the Sector Level?
Mario J. Crucini, Mototsugu Shintani, and Takayuki Tsuruga
NBER Working Paper No. 16081
June 2010
JEL No. F0,F33,F4,F41
ABSTRACT
We introduce the real exchange rate volatility curve as a useful device to understand the role of pricestickiness in accounting for deviations from the Law of One Price at the sector level. In the presenceof both nominal and real shocks, the theory predicts that the real exchange rate volatility curve is a
U-shaped function of the degree of price stickiness. Using sector-level European real exchange rate
data and frequency of price changes, we estimate the volatility curve. The results are consistent withthe predominance of real effects over nominal effects. Nonparametric analysis suggests the curve isconvex and negatively sloped over the majority of its range. Good-by-good variance decompositions
show that the relative contribution of nominal shocks is smaller at the sector level than what previous
studies have found at the aggregate level. We conjecture that this is due to significant averaging out
of good-specific real microeconomic shocks in the process of aggregation.

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
Department of Economics
Vanderbilt University
Box 1819 Station B
Nashville, TN 37235-1819
mototsugu.shintani@vanderbilt.edu

Takayuki Tsuruga
Graduate School of Economics
Kyoto University
Yoshida Honmachi Sakyo-ku
Kyoto 606-8501
JAPAN
ttsuruga@kansai-u.ac.jp

1. Introduction
Among international macroeconomists, it is widely believed that the variability of realexchange rates is increasing in the degree of local currency price rigidity. The reasoning isfound in passages describing exchange rate overshooting in prominent textbooks:“Exchange rate overshooting results from the rapid response of exchange rates tomonetary policy and the sluggish adjustment of prices. A monetary expansion
will lead to an immediate depreciation but only a gradual increase in prices.
Exchange rate overshooting implies that real exchange rates are highly volatile.”
Dornbusch, Fischer, and Startz (2004, p. 534)
The basic idea is that the nominal exchange rate is an asset price (since currencies areactively traded in the foreign exchange market) and like all asset prices, its value adjusts
instantaneously in response to unexpected changes in exogenous variables in current andfuture periods. In contrast, many goods and services have prices which economists havedocumented to be …xed in local currency for extended periods of time. By de…nition, thismeans nominal and real exchange rates will be highly correlated with each other over time
and will have a comparable level of time series variability, at least at the highest frequencies.
The expectation, then, is a positive correlation between the volatility of real exchange rates
and the degree of price stickiness if nominal shocks dominate the landscape, as they do inmuch theorizing on the topic.
The New Open Economy Macroeconomics (NOEM) framework provides fully articulated
models of the channels through which monetary shocks drive transitory ‡uctuations in realexchange rates. Quantitative investigations of this framework have been undertaken, by
Chari, Kehoe, and McGrattan (2002) who focus on the aggregate real exchange rate in a
one-sector model and by Kehoe and Midrigan (2007) who focus on Law-of-One-Price (LOP)
deviations in a multisector model.

2

The empirical evidence emanating from the VAR literature is mixed on the question of
the relative contribution of real and nominal shocks to real exchange rate variability. Usinga structural VAR model, Clarida and Galí (1994, Table 3) …nd that the relative contributionof nominal shocks to the 1-period-ahead forecast error variance of the quarterly bilateral
U.S. real exchange rate is 47 percent for the German mark, 36 percent for the Japanese yen,
but a mere 2 percent for British pound and less than 1 percent for the Canadian dollar. At
much longer horizons –31 to 36 months –Eichenbaum and Evans (1995, Table 1a) …nd thecontributions of money shocks to the forecast error variance of real exchange rates are 42.9,
38.1, 37.5, 26.2 and 23.0 percent, for Germany, Italy, France, the UK and Japan, respectively(see also Rogers, 1999).
An early advocate for the role of real shocks in the equilibrium determination of realexchange rates is Stockman (1980). Stockman casts his model in a ‡exible price setting, sothat nominal shocks make no contribution to real exchange rate volatility. Crucini, Shintani,
and Tsuruga (2010), on the other hand, neutralize the e¤ect of nominal shocks by focusing
on intranational trade and investigate the role of real shocks on good-level real exchange
rate volatility across cities in the presence of local currency price rigidity. Unlike models
emphasizing the role of the nominal shocks, their model predicts a negative correlation
between price stickiness and real exchange rate variability because only real shocks a¤ectreal exchange rates across locations within a country.
The current paper puts these two views of real exchange rate determination on thesame playing …eld by combining the model of Kehoe and Midrigan (2007) which emphasizes
money shocks and nominal exchange rates with the model of Crucini, Shintani, and Tsuruga(2010) which emphasizes productivity shocks and trade costs. These models rely on thetime dependent pricing assumption of Calvo, but allow the frequencies of price changes
to vary across goods, as measured in the micro-data. Under the synthesized framework,
we theoretically explore the cross-sectional relationship between price stickiness and realexchange rate volatility at the level of individual goods. We refer to this relationship as3

the real exchange rate volatility curve: the functional relationship between the forecast error
variance of the real exchange rate and the infrequency of price changes at the level of a good.
When real shocks are absent, the volatility curve is upward-sloping: an increasing functionof the price stickiness parameter and the good with the least ‡exible price should exhibit thegreatest amount of real exchange rate variability. When nominal shocks are turned o¤, thevolatility curve is downward-sloping: a decreasing function of the price stickiness parameterand the good with the most ‡exible price has the greatest amount of real exchange ratevariability. When both real and nominal shocks are present, the real exchange rate volatility
curve becomes U-shaped and could result in a zero unconditional correlation between realexchange rate volatility and the frequency of price adjustment.
We estimate the volatility curve using sector-level real exchanges of Austria, Belgium,
France, and Spain vis à vis the US, constructed by Kehoe and Midrigan (2007). We …ndthat the shape of the estimated curve is consistent with the predominance of real shocks
over nominal shocks, in the sense that the curve is minimized at the level of price stickiness
which corresponds to a more than 4 month duration between price changes. Nonparametricanalysis suggests the convexity of the curve as well as a negative relationship between pricerigidity and real exchange rate volatility over the vast majority of empirical frequencies of
price change found in the cross-section of goods. The negative correlation together with thetheoretical prediction of our model suggests that sector-speci…c real shocks explain the bulkof short-run volatility in real exchange rates. We further conduct variance decompositionsof sector-level real exchange rates to evaluate the relative contribution of nominal and realshocks at various horizons. For almost all goods, the contribution of nominal shocks are
smaller than that of real shocks, and real shocks rise in dominance as the forecast horizon
lengthens.
Our …ndings on the role of sector-speci…c real shocks are consistent with recent micro
evidence by Bergin, Glick and Wu (2009), but in stark contrast with the traditional viewthat aggregate real exchange rate variability is attributable mostly to nominal shocks (Rogo¤,41996). To reconcile the microeconomic evidence with the macroeconomic evidence, it seems
necessary to allow for large idiosyncratic real shocks at the sector-level such that thesemicroeconomic sources of variation average out in the move to the CPI-based real exchangerate.

2. The Model
The theory combines the key features of Kehoe and Midrigan (2007) and Crucini, Shintani
and Tsuruga (2010).1 Both of these models assume heterogeneous price stickiness acrossgoods, but the former relies on nominal exchange rate variations whereas the latter focuseson the labor productivity variations in explaining the real exchange rate volatility at sector
level. In what follows, the core implications of the model are discussed focusing on the crosssectional volatility of the (log) real exchange rate for a bilateral pair of countries, de…ned
as:
qit = st + pit

pit ;

(2.1)

where pit (pit ) denotes the (log) sectoral price index in the home (foreign) country and st isthe (log) nominal exchange rate. Throughout the paper, variables marked with an asterisk
denote foreign analogs of home variables.
For ease of exposition, some simplifying assumptions are made on the sources of realexchange rate variation. The …rst assumptions concern nominal shocks and exchange rates.
The nominal shocks in the model are the home and foreign money growth rate,

t and

t,

which are independent and identically distributed (i.i.d.). The semi-log household preferences
over consumption and leisure, combined with a local-currency cash-in-advance constraint,
leads to the equality of the money growth rate di¤erential and the nominal exchange rate
growth rate (i.e.,tt =st ).2 These assumptions are taken from Kehoe and Midrigan

1

The full model is presented in the technical appendix of this paper, which is available from the authors
upon request.
2
To be speci…c, semi-log period utility is given by ln Ct
Lt , where Ct , Lt and (> 0), denote aggregate
consumption, hours worked, and marginal disutility of labor supply, respectively.

5

(2007) and are convenient since the nominal exchange rate st becomes a random walk,consistent with the seminal paper of Meese and Rogo¤ (1983).
The second set of assumptions concern real shocks and trade costs. Monopolisticallycompetitive …rms set prices of their goods, which are produced using a technology thatis linear in labor and subject to productivity shocks. Due to our microeconomic focus,
the productivity shock for each good consists of three components: a global component, a
nation-speci…c component and a good-speci…c component. To be precise, the productivity(in logs) for production of good i at time t, at home ait , and in the foreign country, ait , aregiven by:ait = zt +t + "it ;ait = zt +(2.2)

t + "it ;respectively. The three components, then, are: (i) zt , a common global stochastic trend,
following zt = zt 1 + "zt , "zti.i.d.(0;2z ), (ii)ti.i.d.(0;

2

) and

t

i.i.d.(0;country-speci…c productivity shocks common to all goods, and (iii) "it
i.i.d.(0;"it2), capture

i.i.d.(0;2" ) and32" ) are idiosyncratic shocks to the production of each good in each country.

These labor productivity shocks are the real shocks in the model. Finally, …rms in eachcountry are required to pay an iceberg transportation costto send good across the border.

This transportation cost leads to home bias in consumption because the home variety ofeach good is cheaper than the imported variety.
The focal equation of the model is the k-period-ahead forecast error variance of thesector-level real exchange rate:
V art k (qit ) =
where

ik

=

Pk

j=1

2
ik [ i V ar( t2(j 1),iit ) + (12i ) (1i)2

2

V ar(aitait )](2.3)

is the good-speci…c degree of price stickiness in the senseof Calvo (1983) and the parameter,= 1(1 + )1= 1 + (1 + )1

, captures theasymmetric transmission of productivity to the relative price of good i, across countries due
3

A common stochastic trend zt among two countries can be speci…c to the sector i, zit = zit 1 + "izt ,without changing the substance of our results. Also, for simplicity, we assume that 2" and " 2 are common
across i. In the empirical part of the paper, the possiblity of heterogenous variances of real shocks acrosssectors is considered.

6

to the home bias in expenditure on home and foreign varieties of good i. The veracity of thisasymmetry depends positively on the trade cost, , and the elasticity of substitution amongdi¤erentiated products, .
Equation (2.3) attributes the forecast error variance of the sectoral real exchange rate tott (the nominal shocks) and the varianceof the cross-country productivity di¤erential, ait

ait (the real shocks).4 The …rst term of

the variance of the money growth di¤erential,

Equation (2.3) will be referred to as the nominal e¤ect and the second term as the reale¤ect, alluding to the e¤ects of nominal and real shocks on the volatility of the sectoral realexchange rate.
Under time dependent pricing, monopolistically competitive …rms cannot change prices
with probability

5
i which is assumed to be common across countries but di¤ers across goods.

That is, some sectors are allowed to change their prices more frequently than others. Viewed
as a function of the sector-speci…c degree of price stickiness,called: the real exchange rate volatility curve. Note that2i ) (1i)2

ik is decreasing in

2ii

2 [0; 1], equation (2.3) is

ik is increasing in

i , while (1i . Fixing the forecast horizon k, an increase ini (‘stickier

prices’) increases the contribution of the nominal e¤ect and decreases the contribution of thereal e¤ect to the total forecast error variance of the sectoral real exchange rate.
To gain some intuition for how price stickiness ampli…es the impact of nominal shocks or
mitigates the impact of real shocks on the real exchange rate volatility represented by (2.3),
recall the de…nition of the real exchange rate given in equation (2.1). To see the impact of
nominal shocks in (2.1), consider a positive money growth rate shock in the home country,
holding …xed foreign money growth. The model predicts an immediate depreciation of thenominal exchange rate, that is, an increase in st in (2.1). The responses of local currency
4Note that we allow shocks to the money supply and labor productivity to have permanent e¤ects on thelevels of prices and outputs which appear to …t the data. However, the relative price (real exchange rate) is
stationary as a consequence of equilibrium adjustment of the nominal exchange rate with respect to nominal
shocks and as a consequence of our cointegration restriction on the productivity across countries.
5
Much of the existing empirical work on the topic has emphasized the dominance of good-dependent frequencies over location-dependent frequencies. However, when in‡ation rates and/or exchange rate properties
di¤er substantially across bilateral pairs, this may change.

7

prices, though, depend on the good-speci…c frequencies of price adjustment. For goods withprices that change every period, their local currency price adjustment completely o¤sets theimpact of the nominal exchange rate depreciation and preserves the original LOP deviation.
At the other end of the continuum, goods with prices that are extremely sticky will have realexchange rates that basically follow the path of the nominal exchange rate with negligible
pass-through of the nominal shock to local currency prices. Simply put: the nominal e¤ect
on real exchange rate variability is ampli…ed by slow local currency price adjustment.
Turning to real shocks, consider a positive shock in home productivity in sector i: Since
this productivity shock is isolated to a single sector, it is assumed to have no equilibrium
consequences for the nominal exchange rate, st in (2.1). What it does is reduce both home andforeign price indexes of sector i, because …rms in the home country sell the goods producedin this sector in both countries. However, due to home bias generated by trade costs, thehome sectoral price index will decrease more than the foreign sectoral price index which
increases the price index di¤erential pit

pit in (2.1). Because this economic channel requires

prices to actually change and thereby induce asymmetric price changes across locations, it ismore quantitatively important when prices are relatively ‡exible. Conversely, the real e¤ect
is mitigated by slow local currency price adjustment. This discussion should make clear thatthe conventional wisdom of a positive relationship between real exchange rate volatility andprice stickiness is predicated on the assumption that the nominal e¤ect dominates the reale¤ect. The next section provides some numerical examples of how di¤erent intensities of realand nominal shocks alter the shape of the real exchange rate volatility curve.

3. Numerical Examples
This section uses numerical examples to show how the shape of the real exchange rate
volatility curve, as function ofi 2 [0; 1], depends upon the relative volatility of real andnominal shocks and a few key structural parameters.
We focus on the one-period-ahead forecast error variance by setting k = 1 in equation

8

(2.3),
V art 1 (qit ) =2
i V ar( tt ) + (12i ) (1i)2

2

V ar(ait(3.1)

ait ):and make note of the fact that the k period ahead forecast is proportional to the one-periodahead forecast: V art k (qit ) =ik

V art 1 (qit ) for any k.

The structural parameters are calibrated as follows: i) the data is monthly, so the discountfactor is set to= 0:961=12 = 0:9966; ii) trade costs, broadly de…ned at the retail level, arein the neighborhood of= 0:5; and iii) the elasticity of substitution is set at

The multiplier on the productivity di¤erential, re‡ecting home bias, then, is

= 10.
2

= 0:9.

While trade costs and elasticities of substitution are expected to di¤er across goods, thesalient features of the volatility curve are not very sensitive to these parameters, leaving
the interesting dimension as the interaction of the frequency of price adjustment

i and thevariances of nominal and real shocks.
For purposes of discussion, the expression is simpli…ed by noting that since the discountfactor and the trade-bias factor are close to 1, the variance decomposition of the real exchange
rate is well approximated by:
V art 1 (qit ) =2
i V ar( tt ) + (14i ) V ar(aitait ) .

(3.2)

Simply put, the variance of the sectoral real exchange rate is a weighted average of nominal
and real shocks. The share of the nominal shock (the money growth di¤erential) in realexchange rate variability rises from zero toward 1 as prices become less ‡exible ( i ! 1)
at a rate equal to the square of the infrequency of price changes. In contrast, the share of
the contribution of the real shock (the productivity di¤erential) toward real exchange ratevariability rises from zero toward 1 as prices become more ‡exible ( i ! 0) at a rate equal to
4th power of the frequency of price changes (1i ). The theory conveniently encompasses

the varied role of nominal and real shocks in the cross-section of goods and the feature thatthese two opposing forces give rise to a real exchange rate volatility curve that is U-shaped
over the support
66
i 2 [0; 1].

To prove this, evaluate the …rst derivative of the total variance with respect to

9

i at

i = 0 and 1.

Turning to numerical examples, Figure 1 uses equation (3.1) to construct conditionalvariances as a function ofwith the Std( tStd(aitiseparately for three distinct stochastic environments (eacht ) normalized to 1 percent):ait )=Std( tt ) = 1;and (c) Std(ait(a) Std(aitait )=Std( tait )=Std( tt ) = 1=5.

t ) = 5; (b)Each panel

shows the variance due to the real e¤ect (in blue) and variance due to the nominal e¤ect(in red). When the distribution of goods across frequencies of price change is uniform on
the interval [0; 1], these areas fully summarize the role of real and nominal shocks across theentire cross-section of goods.
The extent to which the role of real shocks is more quickly mitigated by price rigidity thanis the role of nominal shocks by price ‡exibility is clearly evident in the middle panel where
the two shocks have equal variance. The critical good for which real and nominal shocks
contribute equally to real exchange rate variability has a frequency of price change,i , ofabout 0.4 (rather than 0.5, the midpoint of the unit interval), indicating more price ‡exibility
is needed to handicap the advantage given to nominal shocks by the theory. However, as
evident in panels (a)-(c), the overall size of each area is sensitive to the relative variance ofreal and nominal shocks.
The asymmetries across goods are, of course, greatest at the extremes of price ‡exibility
and in‡exibility. Consider …xing a particular good, indexed by its frequency of price change,i.

The height of the volatility curve is the model’s prediction for the total one-step ahead

forecast error variance of the real exchange rate for that good. The blue and red segmentsof the vertical line drawn from that point in the

distribution up to the volatility curve

gives the partition of the variance into the contributions of real and nominal shocks. At theextremes, real exchange rate ‡uctuations of goods with fully ‡exible price are driven solely
by real shocks while goods with completely rigid prices are driven solely by nominal shocks.
When evaluated at i = 0, the …rst derivative of the variance due to the nominal e¤ect is zero but that
due to the real e¤ect is negative and …nite, which implies that the …rst derivative of the total variance withrespect to i is strictly negative when i = 0. Analogously, we can also show that the total variance has
a strictly positive slope at i = 1. Because total variance is continuous in i , there exists i 2 (0; 1) thatminimizes total variance.

10Perhaps the closest real-world counterparts of such goods are crude petroleum trading on
the centralized spot market and postage stamps.
Panel (a) calibrates the model such that only real shocks are of consequence and theprediction of the model is that the volatility curve is downward sloping over almost its entire
range. Only as completely rigid local currency prices are approached at

i = 1 do we see thecurve begin to slope upward, providing the …rst hint of a nominal e¤ect (i.e., the red area
becomes visible). The good with the lowest real exchange rate variance in this calibration –the point at which the sign of the slope changes –is

i = 0:75. This corresponds to an average

duration between price changes of 4 months. The average contribution of real and nominal
shocks to the variance of sectoral real exchange rate may be evaluated by integrating the realexchange rate volatility curve over the entire range ofi . The blue area, which corresponds

to real e¤ects, accounts for 93% of the total area, the red area (nominal e¤ects) contributes
a mere 7%.
Panel (c) calibrates the model such that only nominal shocks are of consequence andthe prediction of the model is that the volatility curve is upward sloping over almost its
entire range.

Only as we approach goods with completely ‡exible prices do we see thecurve begin to slope downward, providing evidence of real e¤ects (i.e., the blue area becomesvisible). The point which minimizes variance, and therefore the point at which the sign ofthe slope changes is virtually indistinguishable from complete price ‡exibility,i= 0:06.

This corresponds to an average duration between price changes of 1 month. The red area,
the nominal e¤ect, accounts for about 98% of the total area, the blue area (real e¤ect), a
trivial 2%.
The middle panel (b), with equal variances of real and nominal shocks, displays an
obvious U-shape. The blue area is comparable in size to the red area, with the real e¤ect
accounting for 35% of the variation and nominal e¤ects accounting for the remaining 65%(these are averages across goods). The minimum point on the exchange rate volatility curveisi = 0:40. This corresponds to an average duration between price changes of 1.7 months.

11

Consider, now, what we would expect to …nd in terms of the correlation of price stickiness, as measured by the Calvo parameter,i , and the volatility of the sector-level realexchange rate. Computing the correlation of frequency of price changes and real exchange
rate variability will yield a negative correlation in panel (a) because real shocks dominate;
a positive correlation in panel (c), because nominal shocks dominate; and a near zero correlation in panel (b) due to the U-shape of the volatility curve. This is a textbook example
of a situation in which correlation, a linear measure of dependence, fails to reveal the true
underlying economic structure. The model gives a compelling rationale for investigatinga non-linear relationship between real exchange rate variability and the frequency of priceadjustment.
In practice the sign of the correlation will depend on the distribution of goods in thesample (in terms of their frequencies of price adjustment) as well as the relative importanceof real and nominal shocks. Consider a researcher with a limited sample of goods withprices which happen to be very sticky in local currency units. Even in the case of panel (b),this researcher will …nd positive correlation rather than a zero correlation, and thus will be
tempted to conclude that nominal shocks predominate. For the goods in his sample, this istrue, but it need not be true in general, over the entire distribution ofi across goods in theeconomy.
Note that the blue and red areas in Figure 1 represent the cross-sectional average of thesector-level variance decomposition.7 However, individual goods may have vastly di¤erent
variance decompositions. This arises from the fact that the weights
i

)2

2

2
i and (1appearing in (3.1) are very sensitive to the sector-speci…c value of2i ) (1i . Recall that theblue and red areas in panel (b) account for 35% and 65% (respectively) of the variance of realexchange rates over the entire unit interval. Consider two goods, one withother with

1 = 0:25 and the

2 = 0:56. These two real exchange rates turn out to have the same total forecast

error variance as panel (b). However, due to the manner in which shocks are transmitted to7

It corresponds to the average when the degree of price stickiness is uniformly distributed across goods.

12relative prices, the sources of the variance di¤ers dramatically across the them: the relative
contribution of nominal shocks is only 18% for the …rst good but 90% for the second good.
To summarize our numerical analysis, there are two striking empirical implications ofour model when both real and nominal shocks are allowed to impinge on the economy.
First, it is possible that the data suggest a negative correlation between total real exchange
rate volatility and degree of prices stickiness, contrary to the conventional wisdom. As
panels (a) and (b) in Figure 1 suggest, negative correlations can occur when the variance of
productivity di¤erentials dominate the economic environment for most goods in the crosssection. Second, since the real exchange rate volatility curve is a U-shaped curve, runninga simple linear regression of the sectoral real exchange rate variance on

i may be a poor

method for uncovering the underlying structure. It is useful to consider a ‡exible functional
form in the regression and …nd the degree of price stickiness which minimizes the volatility
curve.

4. Empirical Analysis
The empirical analysis focuses on (i) examining the relationship between total variance andthe degree of price stickiness; (ii) …nding the degree of price stickiness which minimizes thevolatility curve; and (iii) assessing the relative importance of the real and nominal e¤ects atthe sectoral level. The data used here was originally obtained by Kehoe and Midrigan (2007)
and consist of highly disaggregated sectoral real exchange rates for four European countries(Austria, Belgium, France, and Spain) relative to the U.S.. Essentially this involves matchingmonthly local currency micro-price data from Eurostat and the Bureau of Labor Statistics
and converting to a common-currency using spot nominal exchange rates. The sample period
is monthly from January 1996 until December 2006. The number of sectors is 66. Kehoeand Midrigan (2007) take the cross-country average monthly infrequencies of price changes
within each sector. The country-speci…c frequencies for the U.S. are from Bils and Klenow(2004) and those for each of the European countries are taken from the individual country

13

studies by: Baumgartner, Glatzer, Rumler, and Stiglbauer (2005) for Austria; Aucremanne
and Dhyne (2004) for Belgium; Baudry, Le Bihan, Sevestre, and Tarrieu (2007) for France;
Alvarez and Hernando (2004) for Spain. The details of the data construction are found in
the appendix of Kehoe and Midrigan (2007).
While the Euro was o¢ cially introduced part-way through our sample, even before thisthe nominal exchange rate of these European countries were quite stable against each other.
This is evident in the standard deviations of the nominal exchange rate growth of the U.S.
dollar against Austrian Schillings, Belgian Francs, French Francs, and Spanish Pesetas, whichare 2.36, 2.37, 2.35, and 2.36 percent, respectively. The similarity of nominal exchange rate
volatility, e¤ectively the nominal shocks of the theory, rationalize a pooled regression of thefour country-pairs against the dollar as the benchmark in the analysis below. However, theproductivity di¤erentials may di¤er across bilateral pairs, so we estimate the relationship foreach country separately as a robustness check.
4.1. Estimating the real exchange rate volatility curve
Let Vij be the one-period-ahead forecast error variance of the real exchange rate for good
i for country j, vis à vis the United States. The technical appendix of the paper provesthat qijt follows an AR(1) process with an AR coe¢ cientij

under a set of maintained

assumptions. E¤ectively, this means Vij is equal to the sample variance of qijt
using the observed infrequency of price changes,

ij .

When either

ij qijt 1

ij or qijt is missing or

when Vij can be computed from only a short time sample, we exclude such goods from thesample.8
As a preliminary analysis, the …rst set of regression results are simple linear regressions of
Vij onij using (i) the pooled samples of all four country-pairs; and (ii) country-by-country

samples. The results are reported in Table 1. In all cases, the sign of the coe¢ cient on
ij is signi…cantly negative based on heteroskedasticity consistent standard errors reported
8

After excluding the samples, the number of sectors amounts to 57 for Austria, 46 for Belgium, 48 forFrance, and 31 for Spain.

14

below the point estimates. Despite its simplicity, this speci…cation, explains 70 percent of thecross-sectional variation in the volatility of real exchange rates using the pooled regression.9
The estimated slope coe¢ cients are similar across the four nation-speci…c regressions andthe cross-country pooled regression. Regression …t is especially good for Austria and France.
Table 1 also reports the minimum and maximum values of price stickiness in our data(denotedminandmax ).

The linear regression estimates suggest that the real exchange

rate volatility curve is downward sloping within the range of the observedij .

Accordingto the theory, the negative correlation is consistent with the dominance of productivity
di¤erentials over money growth rate di¤erentials, consistent with the stylized numericalexample presented in panel (a) of Figure 1.
Recall, however, that the theory predicts a non-linear relationship between the frequency
of price adjustment and real exchange rate variability when both nominal and real shocks
are present.10 To more adequately address this implication of the theory, the volatility curve
is augmented with a quadratic term and a quartic term following the structural model:
Vij = b1j 2ij + b2j (12ij ) (1ij)2 + uij ;(4.1)

where the b’s are regression coe¢ cients and uij is the regression error term for good i for
country j. The second regressor is constructed by setting= 0:961=12 . According to (3.1),

regression coe¢ cient b1j should capture the nominal e¤ects, due to V ar( tjt ), where

the money growth rate has been replaced by the variance of the bilateral nominal exchange
rate to anticipate the empirical implementation that follows. The regression coe¢ cient,b2j , captures the real e¤ects

2
j V ar(aitaijt ), with the restriction that the variance of

productivity di¤erentials are common across i. Note that since the empirical work involves
more than one bilateral pair (as assumed in the theory), the trade costs, demand elasticities
9

For robustness, we also run the pooled regression with the country dummies for both intercept and slopecoe¢ cient to control for di¤erences in trade costs paid to carry goods from a country to another country.
The results are essentially unchanged.
10
Using Ramsey’s (1969) RESET test, the null hypothesis of linearity is rejected at the one percent
signi…cance level for the pooled case, as well as for the Austrian and French cases. Weaker evidence of
nonlinearity is obtained for Belgian and Spanish cases possibly because the power of the test is lower insmaller samples.

15

and productivity shocks are allowed to vary for across bilateral pairs for each good. For
example, productivity variation may be higher across Spain and the United States (thenumeraire) than between France and the United States. With this nonlinear extension, it
is also possible to locate the degree of price stickiness which minimizes the volatility curveusing the estimates of b1j and b2j .
Table 2 presents the estimation results of regression model (4.1). In all cases, the estimated coe¢ cients have positive signs which is consistent with the theory, all are statisticallysigni…cant. The quartic regression is comparable to the linear regression in terms of thegoodness of …t.11 The role of nominal shocks compared to real shocks, as implied by theestimates b1j and b2j , indicates a much larger role for real shocks. While the regression does
not separately identify the role of home bias

2jand productivity shocks V ar(aitaijt ),

b2j identi…es their combined in‡uence and serves as a lower bound on V ar(ait aijt ) givenp
2
0
1 for j0 and j
1. The same argument establishes that b2j =b1j can bejused as a lower bound for Std(aitaijt )=Std( tjt ).

Using the coe¢ cient estimates reported in Table 2 the ratios of the standard deviationof real to nominal shocks are inferred to be: 5.28, 6.17, 4.42, 5.30 and 9.64, for the pooledcase and the Austrian, Belgian, French and Spanish nation-speci…c cases, respectively. Thus,
the quartic regression result is very comparable to the numerical example of case (a) whereStd(ait ait )=Std( tt ) = 5. This is further con…rmed in the left panel of Figure 2 showing

the …tted curve of the pooled quartic regression (the solid line) from Table 2, along withthat of the pooled linear regression (the dashed line) from Table 1. The …tted curve of thequartic regression resembles panel (a) of Figure 1 in terms of the shape of the curve, again
suggesting the importance of real e¤ects.
The last two columns of Table 2 compare the estimated degree of price stickiness which
minimizes the forecast error variance in each regression. For the pooled case, the varianceof the real exchange rate is minimized at

i = 0:76 which is remarkably close to the value of

11

The inclusions of country dummies into the pooled regression again did not a¤ect the signs and magnitudes of the estimated coe¢ cients.

160.75 from our numerical example with dominant real shocks. This frequency of price change
implies that the U-shaped real exchange rate volatility curve is minimized when the duration
between price changes is 4.2 months.
Note that the parametric regression (4.1) imposes a strict theoretical shape restriction on
the real exchange rate volatility curve. As a robustness check, the functional form restriction
is replaced with a general nonparametric regression,
Vij = mj ( ij ) + uij
where mj ( ) is an unknown conditional mean function for country j. The right panel of Figure
2 shows the estimated curve using the nonparametric local linear regression estimator withpooled data.12 The shape of the …tted curve shown as the solid line is very di¤erent from thelinear regression …t shown as the dashed line. This suggests the plausibility of a nonlinear
structure in the real exchange volatility curve.
Turning to a comparison of the quartic regression (4.1) and the non-parametric regression,both similarities and di¤erences are evident. Both estimates imply convexity in the realexchange rate volatility curve. When the …rst derivative of the m function is evaluatednonparametrically, it tends to be increasing in

ij , which is consistent with the theoreticalprediction. The slope of the curve is negative over the empirical range of‡atter as

ij

ij and it becomesincreases. The most notable di¤erence between the quartic regression andnonparametric regression is the location of the bottom of the curve. The value of

ij which

minimizes the forecast error variance in the nonparametric regression is close to unity, avalue larger than the theoretical prediction based on the quartic regression.
To formally investigate the shape of the estimated curve, a nonparametric test of monotonicity developed by Ghosal, Sen and van der Vaart (2000) is employed –a test of the null hypothesis that the m function is an increasing (or decreasing) function over a certain interval.
In the present context, the shape of the curve is examined over the observed range of thedata, [ min ;12max ]. The test is also applied to establish the monotonicity of the …rst derivative

In estimation, Gaussian kernel is used along with the bandwidth selected by the rule of thumb.

17

of the m function. The results are reported in Table 3.
Regarding the m function itself, the hypothesis of an increasing function in

ij is rejected,

and that of decreasing function is not, based on a conventional signi…cance level. For the …rst
derivative, the test fails to reject a monotonically increasing function while a monotonicallydecreasing function is rejected. Simply put: the real exchange rate volatility curve is aconvex function consistent with the U-shape prediction of the theory.
Establishing the in‡ection point on the real exchange rate volatility curve is tenuous, it islikely to be associated with aij larger than estimated from the quartic regression, though

the structurally restricted estimate of 0.76 (a duration of about 4 months between priceadjustments) is our preferred choice. Since the minimum point is largely a function of thestochastic environment and not the underlying structural parameters, the minimum neednot be a crucial focus. However, it would be reassuring in terms of validating the generalityof the theory to explore other samples of goods, cross-sections of countries and historicalperiods such that the nominal shocks play a larger role and the U-shape not be truncatedat the upper boundary of price rigidity.
4.2. Variance decomposition
Let us now turn to the relative importance of the real and nominal e¤ects at sector levelby directly using equation (2.3) at various horizons along with an empirical measure of thevariance of the nominal shock. According to the theory, the appropriate metric ofsjt and the nominal contribution to real exchange rate variance is2
ij

ijk V ar(

t

jt is

sjt ). The

k-period-ahead forecast error variance, V art k (qijt ), is obtained from the quasi-di¤erenceqijtkij qijt k , using observed sectoral infrequency of price changes,

ij .

The relative contribution of nominal shocks to the total forecast error variance of the realexchange is:(i; j; k) =2
ij

ijk V ar(sjt )V art k (qijt )where the indices of the share function, (i; j; k), re‡ect the role of goods (infrequency of price18

changes), country (due to the variability of the bilateral nominal exchange rate of country jvis à vis the U.S. dollar) and horizon. As k ! 1, the sample variance of qijt is used for the2ij )]V ar(

unconditional variance and [ 2ij =(1sjt ) is used to measure the contribution of

the nominal shocks. In addition to its simplicity, this good-by-good variance decomposition
method has the advantage that it does not require the assumption of a common volatility ofreal shocks across sectors, since 1(i; j; k) is unrestricted and may vary due to heterogeneous

variance of real shocks across sectors.
Table 4 reports the summary statistics for the contributions of the nominal shocks to theforecast error variance of sectoral real exchange rates at monthly horizons of k = 1; 3; 6; 12
and 1. Note that unlike the variance decomposition of aggregate real exchange rates often
reported in the literature, the decomposition is calculated for each sectoral good. The …rst
row of the table shows the average contribution of nominal shocks, with the average taken
across all goods and all four bilateral pairs. The numbers in parentheses in the second
row are the standard deviations across goods and countries. The remaining rows reportcorresponding results for each pair of countries.
For the one-period-ahead forecast error decomposition, nominal shocks account for about
40 percent of real exchange rate variation on average and range from a high of 49 percent
for Austria to a low of 35 percent for Spain. The large standard deviations in the table
imply that the contributions of nominal shocks di¤er considerably across goods. This crosssectional dispersion is very similar across countries. For the shortest horizon it seems safe toconclude that the contribution of real shocks is at least as large as that of nominal shocks
for many goods.
The role of nominal shocks becomes smaller as the horizon lengthens. At a horizon of
6 months, the relative contribution is about one half of the 1-month horizon. The longrun contribution of nominal shocks, evaluated at k = 1, is lower than 10 percent for all
countries except for Austria, thus leaving 90 percent to be explained by real shocks. The
cross-sectional variation (across goods) at the longest horizon is much smaller than that at

19

shorter horizons, implying the dominance of real shocks in real exchange rate ‡uctuations
for most goods. This …nding of a dominant role of sector-speci…c real shocks is consistent
with recent micro evidence by Bergin, Glick and Wu (2009) who claim that idiosyncratic
industry price shocks account for about 80% of variation in LOP deviations in disaggregateddata, with nominal exchange rate shocks playing a very small role.
Let us now compare the variance decompositions of sector-level real exchange rates withprevious studies involving the aggregate real exchange rate. Using a structural VAR model,
Clarida and Galí (1994, Table 3) …nd that the relative contribution of nominal shocks to 1period-ahead forecast error variance of quarterly real exchange rate is 47 percent for Germany
and 36 percent for Japan. In contrast, our three-month (the counterpart to one quarter)
ahead variance decomposition indicates nominal shocks account for between 19 to 31 percent,depending on the country, when results are averaged across sectors (see Table 4). Using over
100 years of annual UK-US real exchange rate data, Rogers (1999) …nds that the contributionof nominal shocks to the 1 year-ahead forecast error variance ranges from 19 percent to
60 percent, with a median value of 41 percent. Our 12-month ahead forecast variancedecomposition estimates indicate nominal shocks only account for about 14 percent when
we average across countries and sectors. The benchmark estimates of Eichenbaum and Evans(1995, Table 1a) show a nominal shock contribution at horizons of 31- to 36-months averaging
38 percent for France, while our estimates imply long-run contributions between 9 and 12
percent for France (again, using averages across sectors). Thus, largely independent of thehorizon or countries examined, nominal shocks play a more important role in accounting for
aggregate real exchange rate ‡uctuations than in accounting for sector-level real exchange
rate ‡uctuations.
What accounts for this di¤erence in the microeconomic and macroeconomic evidence?
Our suspicion is that the sectoral real e¤ects tend to average out across sectors while thenominal e¤ects, almost by de…nition cannot, since there is only one nominal exchange rate
per bilateral pair. When researchers use aggregate level CPI-based real exchange rates,20

the impact of real shocks is attenuated by the aggregation process, while the impact of
nominal shocks is not (since the nominal exchange rate shock is common to all goods, up
to heterogenous price adjustment rates). When using less aggregated data, it is therefore
perhaps not surprising that nominal and real shocks are more on par as contributors to realexchange rate variation.
In terms of the theory, recall productivity shocks found in equation (2.2) are expected
to embody idiosyncratic sector-speci…c shocks "it and "it , at least in part. The aggregationof goods prices across sectors eliminates this idiosyncratic component and one is left with anational productivity di¤erential shock and the money growth di¤erential shock to accountfor the aggregate real exchange rate variance. This averaging-out argument is consistent
with a recent …nding by Crucini and Telmer (2007) who show that only a small fraction of
LOP changes are common to all goods. Assigning all of the common component to nominal
exchange rates in the presence of sticky-prices would, thus, leave most of the LOP variation
unaccounted for. The decomposition performed here …lls this gap with real shocks.

5. Conclusion
We use a time-dependent Calvo pricing model with real and nominal shocks to develop theconcept of a real exchange rate volatility curve. The curve was proved to have a U-shape
as a function of the degree of price stickiness, implying an ambiguous correlation between
the forecast error variance of real exchange rates and price stickiness. Using US-European
real exchange rate data, the correlation between the forecast error variance and the degreeof price stickiness was found to be negative over most of the range of the micro-data. The
good with minimal real exchange rate volatility was estimated to have a duration between
the price changes of about 4.2 months. The downward sloping pro…le suggested that forthis micro-sample of goods and countries, the variance of sectoral real exchange rates isdominated by real shocks, though nominal shocks are important as well.
These results point to the value of examining cross-sectional di¤erences in real exchange

21

rate variability in order to ‡esh out the rich quantitative predictions of models of microprice adjustment currently under development. Di¤erences across goods help us to disentangle heterogeneous responses to common shocks due to di¤erences in economic propagation
mechanisms such as costs of price adjustment and trade costs from heterogeneity in theunderlying shocks themselves. Averaging across goods, as is inevitable in the move to an
aggregate real exchange rate, is not innocuous in terms of the weight given to real and nominal shocks. The same averaging may also lead to an under-appreciation of the sources of therisks that individuals and …rms face. We hope to explore these possibilities in future work.
Much remains to be done.

22References[1] Alvarez, L. J., and I. Hernando. 2004. “Price setting behavior in Spain: Stylized facts
using consumer price micro data.”ECB working paper series No. 416.
[2] Aucremanne, L., and E. Dhyne. 2004. “How frequently do prices change? Evidence
based on the micro data underlying the Belgian CPI.”National Bank of Belgium Working paper No. 44.
[3] Baumgartner, J., E. Glatzer, F. Rumler, and A. Stiglbauer. 2005. “How frequently do
consumer prices change in Austria?”ECB working paper series, No. 523.
[4] Baudry, L., H. Le Bihan, P. Sevestre, and S. Tarrieu. 2007. “What do thirteen million
price records have to say about consumer price rigidity?”Oxford Bulletin of Economics
and Statistics, 69(2), 139-183.
[5] Bergin, P. R., R. Glick, and J.-L. Wu. 2009. “The micro-macro disconnect of purchasing
power parity,”NBER Working Papers 15624, National Bureau of Economic Research.
[6] Bils, M., and P. J. Klenow. 2004. “Some evidence on the importance of sticky prices.”
Journal of Political Economy, 112(5), 947-985.
[7] Calvo, G., 1983. “Staggered prices in a utility-maximizing framework.”Journal of Monetary Economics 12 (3), 383–398.
[8] 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), 533563.
[9] Clarida, R., and J. Galí. 1994. “Sources of real exchange-rate ‡uctuations: How important are nominal shocks?” Carnegie-Rochester Conference Series on Public Policy,
41(1), 1-56.

23

[10] Crucini, M. J., M. Shintani, and T. Tsuruga. 2010. “The law of one price without theborder: the role of distance versus sticky prices.”Economic Journal, 120, 462-480.
[11] Crucini, M. J., and C. Telmer. 2007. “Microeconomic sources of real exchange ratevariability.”unpublished manuscript.
[12] Dornbusch, R., S. Fischer, and R. Startz. 2004. Macroeconomics. McGraw-Hill, NewYork.
[13] Eichenbaum, M., and C. Evans, 1995. “Some empirical evidence on the e¤ects of monetary policy shocks on exchange rates.”Quarterly Journal of Economics, 110, 975–1010.
[14] Ghosal, S., A. Sen, and A. W. van der Vaart. 2000. “Testing monotonicity of regression.”
The Annals of Statistics, 28(4), 1054-1082.
[15] Kehoe, P. J., and V. Midrigan. 2007. “Sticky prices and sectoral real exchange rates."Federal Reserve Bank of Minneapolis Working Paper No. 656.
[16] Meese, R. A. and K. Rogo¤. 1983. “Empirical exchange rate models of the seventies :Do they …t out of sample?”Journal of International Economics, 14(1-2), 3-24.
[17] Ramsey, J. B., 1969. “Tests for speci…cation errors in classical linear least-squares regression analysis.”Journal of the Royal Statistical Society B, 31, 350–371.
[18] Rogers, J. H., 1999. “Monetary shocks and real exchange rates.”Journal of International
Economics, 49, 269-288.
[19] Rogo¤, K., 1996. “The purchasing power parity puzzle.”Journal of Economic Literature,34(2), 647-668.
[20] Stockman, A. C., 1980. “A theory of exchange rate determination.”Journal of Political
Economy, 88(4), 673-698.

24

ConstTable 1: Linear regressions
Adj. R2i

Obs.

[ min ,

max ]Pooled

0.013(0.001)-0.014(0.001)

0.700

182

[0.223, 0.979]-

Austria

0.016(0.002)-0.016(0.002)0.886

57[0.223, 0.979]-

Belgium

0.011(0.001)-0.011(0.001)0.454

46[0.296,0.956]-

France0.013(0.002)-0.014(0.002)0.833

48[0.254, 0.958]
-

Spain

0.014
(0.003)-0.015(0.003)0.589

31[0.524, 0.964]-

NOTES: The heteroskedasticity-consistent standard errors are in parentheses. The“Adj. R2 ”denotes the adjusted R2 . The “Obs.”denotes the number of observations.
The last column shows the empirical range of infrequencies of price changes[ min , max ] in our dataset.

25

2
i

Table 2: Structural regressions
2
2
2(1
Adj. Ruci ) (1i )Pooled0.0016(0.0001)

0.0456(0.0039)

0.751

0.762
(0.008)Austria0.0012(0.0001)

0.0475(0.0066)

0.895

0.784
(0.011)

Belgium

0.0021(0.0004)

0.0419(0.0056)

0.623

0.735
(0.017)

France0.0015(0.0001)

0.0411(0.0040)

0.905

0.763
(0.009)Spain

0.00170.15670.716
0.836(0.0002)(0.0292)(0.011)NOTES:The heteroskedasticity-consistent standard errors are in parentheses.
2
The “Adj. Ruc”denotes the adjusted uncentered R2 . presents the estimatesof which minimize the total variance.

26

Table 3: Tests of monotonicitym( i )
m0 ( i )Null hypothesis
Null hypothesis
Increasing Decreasing Increasing Decreasing

1%

5%

10%

Pooled10.329***-1.831-2.32417.014***5.342 4.386

3.964

Austria

4.927**-1.5461.883

9.362***

5.530 4.488

4.027

Belgium

4.608*-0.6471.866

9.378***

5.737 4.609

4.111France6.369***-1.212

1.397

6.675***

5.645 4.554

4.072

Critical values

Spain
5.098*-1.281-0.0257.731*** 6.924 5.382
4.701
NOTES: The …rst two columns correspond to the hypothesis testing for m( i ) and thesecond two columns correspond to the test for the …rst derivative of m( i ) with respect
to i . Critical values shown in the last three columns are computed from the method by
Ghosal, Sen, and van der Vaart (2000).

27

Table 4: Percentage of forecast error variance accounted for by nominal shocks
k
1
3
6
12
1
Pooled40.6(24.1)

23.6(16.5)

18.7(15.7)

14.2(13.5)

11.4(11.8)

Austria

48.6(24.4)

30.5(16.9)

25.7(17.1)

20.3(15.7)

17.1(16.3)

Belgium

34.9(23.0)

19.9(14.9)

15.3(13.6)

11.4(11.2)

8.9(8.2)

France40.2(22.7)

21.8(15.2)

16.2(13.5)

11.7(10.2)

9.2
(7.6)

Spain

35.2
18.9
14.6
11.1
7.9
(24.5)(16.4)(15.6)(13.3)(8.0)
NOTES: Numbers are in percent. Each column corresponds to the cross-sectional
average of the k-period-ahead forecast error variance of sector-level real exchange rates
accounted for by nominal shocks. Numbers in parentheses are standard deviations.

28

29

0

5

10

15

20

0

0.2

0.4

λi

(a)

0.6

0.8

real effectnominal effect

1

0
0.4

λi

0.6

0.8

1

0

0.1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.2

0.2

real effectnominal effect

0.1

0

(b)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.2

0.4

λi

(c)

0.6

0.8

real effectnominal effect1

Std(ait − a∗it )/Std(µt − µ∗t ) = 5, 1, and 1/5, respectively. Standard deviation of nominal shocks is set to one.

expressed by the heights of blue and red areas. The combined height of the two areas corresponds to the total variance. Panels (a),(b) and (c) show the cases ofof price stickiness. The volatility is measured by the one-period-ahead forecast error variance. For each good, the variances due to real and nominal shocks are

NOTES: Each panel of the figure shows contributions of the nominal and real shocks on the sector-level real exchange rate volatility over the range of the degree

Variance

25

Variance

Figure 1: Simulated real exchange rate volatility curves

Variance

30

λi

0.8

1

0

0.6

0
0.4

0.002

0.002

0.2

0.004

0.008

0.01

0.012

0.014

0.016

0.018

0.004

data
linearquartic

0.02

0.006

0

Quartic regression

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0

0.2

0.4

λi

0.6

0.8

data
linearnonparametricNonparametric regression

1on the nonparametric regression. The dashed line in both panels show the linear regression fit for comparisons.

stickiness. The solid line in the left panel shows the fitted curve based on the quartic regression. The solid line in the right panel represents the fitted curve based

NOTES: The both panels show the scatter plot of the one-period-ahead forecast error variance of the sector-level real exchange rates against the degree of price

Variance

Figure 2: Estimated real exchange rate volatility curves

Variance