Do Sticky Prices Increase Real Exchange Rate Volatility at the Sector Level?

Transcription

1 Do Sticky Prices Increase Real Exchange Rate Volatility at the Sector Level? Mario J. Crucini y, Mototsugu Shintani z and Takayuki Tsuruga x First Draft: July 2009, This Draft: March 2010 Abstract We introduce the real exchange rate volatility curve as a useful device to understand the role of price stickiness on the deviations from the Law of One Price at the sectoral level. In the presence of both nominal and real shocks in the economy, theory predicts that the real exchange rate volatility curve becomes a U-shaped function of the degree of price stickiness. In general, however, the level of price stickiness which minimizes the volatility depends on the economic structure which varies across country pairs. Using European data on the sectoral real exchange rate and the sectoral frequency of price changes, we estimate the volatility curve and nd that the price stickiness which minimizes the curve is near unity implying that the curve is monotonically decreasing over the range of the distribution of price stickiness across goods. We also conduct the variance decomposition for sectoral goods and nd that contribution of real shocks is at least as large as that of the nominal shocks for most goods. JEL Classi cation: E31; F31; D40 Keywords: Real exchange rates, Law of One Price, Sticky prices, Nonparametric test for monotonicity Preliminary and Incomplete Comments/Corrections Welcome We thank Yasushi Iwamoto for discussion and comments. Mario J. Crucini and Mototsugu Shintani gratefully acknowledge the nancial support of National Science Foundation. Takayuki Tsuruga gratefully acknowledges the nancial support of Grant-in-aid for Scienti c Research. y Department of Economics, Vanderbilt University and NBER. z Department of Economics, Vanderbilt University. x Faculty of Economics, Kansai University. 1

2 1 Introduction Among international macroeconomists, it is widely believed that the variability of real exchange rates is increasing in the degree of local currency price rigidity. Consider the textbook explanation of exchange rate overshooting found in Dornbusch, Fischer, and Startz (2004, p. 534): overshooting results from the rapid response of [nominal] exchange rates to monetary policy and the sluggish adjustment of prices and implies that real exchange rates are highly volatile. Chari, Kehoe, McGrattan (2002) argue that: the most popular story to explain exchange rate uctuations is that they result from the interaction of monetary shocks and sticky prices and they make considerable progress toward advancing this view in a quantitative investigation of the sticky-price mechanism. Sticky prices are discounted somewhat in the border e ect literature pioneered by Engel and Rogers (1996, p. 1113) who estimate that nominal price stickiness appears to account for a large portion of the border e ect, but most of the e ect is left unexplained. Intuitively, during the period of time when both domestic and foreign local currency prices are xed as a consequence of nominal rigidities, the international relative price of the same good expressed in a common currency (i.e., the real exchange rate) moves one-for-one with the nominal exchange rate. The nominal exchange rate, in turn, is presumed to uctuate as a result of current and future monetary shocks. Thus, one can argue that the volatility of real exchange rates is positively related to the degree of price stickiness if nominal shocks are dominant in real exchange rate uctuations. While this mechanism is plausible it falls considerably short of providing a complete empirical accounting of real and nominal exchange rate behavior. Clarida and Gali (1994), for example, nd that nominal shocks account for 45 percent of the forecast error variance for the real German mark rate and 35 percent of the forecast error variance for the real Japanese yen rate. Rogers (1999) nds that monetary shocks account for almost a half of the real dollar-sterling rate in terms of the forecast error variance decomposition. (e.g., Clarida and Galí, 1994 and Rogers, 1999). Because nominal shocks go only part of the way to explaining the unconditional variance and persistence of real exchange rates, economists suspect that real shocks, such as productivity shocks and scal shocks are important additional sources of propagation. An early advocate for the role of real shocks in a general equilibrium setting is Stockman (1980). However, Stockman cast his model in a exible price setting. The di culty, then, in combining the two approaches is that one relies heavily on price stickiness while the other completely abstracts from sticky prices. To nest 2

3 nominal and real shocks in an single economic model with sticky prices we follow Crucini, Shintani, and Tsuruga (2009a): a time dependent Calvo pricing model with good-speci c price rigidities augmented with trade costs and productivity shocks. This model predicts a negative correlation between price stickiness and real exchange rate variability conditional on real shocks. Intuitively, a productivity shock reduces the cost of the locally produced variety more than the imported variety, which translates into a relative price change at a given location. Due to positive trade costs individual demands exhibit home bias favoring local varieties. Thus a relative price change at the level of varieties becomes a real exchange rate change at the level of a good due to di erent expenditure weights across varieties, by location. To summarize, we nest two views of real exchange rate determination into a single model. One view emphasizes productivity shocks and trades costs and one view emphasizes money shocks and nominal exchange rates. To put the two views on the same playing eld we assume time dependent pricing in the sense of Calvo, but allow the frequencies to vary across goods as is evident in the micro-data. The nominal (real) shock view predicts a positive (negative) relationship between the frequency of price adjustment and the variability of the real exchange rate. Put di erently: if real shocks predominate, sticky prices make it more di cult to account for real exchange rate variability while if nominal shocks predominate the opposite is true, at least for cross-border pairs where there is a need to convert local currency prices into a common currency when de ning the real exchange rate. Using a standard open economy sticky price model, we quantitatively explore the cross-sectional relationship between price stickiness and real exchange rate volatility at the level of individuals goods. We refer to this relationship as 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 nominal shocks dominate, the volatility curve is upward-sloping: an increasing function of the price stickiness parameter and the good with the least exible price should exhibit the greatest amount of real exchange rate variability. When real shocks dominate, the volatility curve is downward-sloping: a decreasing function of the price stickiness parameter and the good with the most exible price has the greatest amount of real exchange rate variability. We show that when both real and nominal shocks are present, the real exchange rate volatility curve may be U-shaped and therefore produce a zero unconditional correlation between real exchange rate volatility and the frequency of price adjustment. Using real exchanges of Austria, Belgium, France, and Spain vis à vis the US at a very disag- 3

4 gregated level, we estimate the volatility curve and nd that the price stickiness which minimizes the curve is near unity (complete price rigidity). Essentially, the entire estimated volatility curve is monotonically decreasing over the range of observed frequencies of price change in the crosssection of goods for these four bilateral pairs. The nding of a negative correlation between the price stickiness and real exchange rate volatility, suggests that sector-speci c real shocks explain the bulk of short-run volatility of real exchange rates. Our nding on sectoral real exchange rates is in stark contrast with a standard textbook explanation for aggregate real exchange rate, such as Obstfeld and Rogo (1996), that most variabilities of real exchange rates cannot be attributable to real shocks. To reconcile the microeconomic evidence with the macroeconomic evidence, we conduct a variance decomposition of real exchange rate volatility for sectoral goods to evaluate the relative role of each shock at the good level. We nd that contribution of real shocks to sectoral real exchange rates appears to be the largest component for most sectors. We conclude that these sectoral real shocks tend to average out across sectors more than nominal shocks so that when the attribution of nominal and real shocks is assessed at the macroeconomic level, nominal shocks are more on par with real shocks. 1 The fact that nominal and real shocks seem to contribute about equally to real exchange rate variation based on the best available structural VAR analysis suggests that a prototype international macroeconomic model should feature both shocks and their respective propagation mechanisms. However, the nding the real shocks dominate nominal shocks at the microeconomic level suggests that the idiosyncratic shocks experienced by individuals and rms are real, not nominal. To the extent stochastic general equilibrium theory seeks to understand both the macroeconomic cycle and the cross-sectional distribution around the aggregate, our nding suggest that more attention be given to uncovering real microeconomic shocks. The rest of the paper is organized as follows. Section 2 presents the model. Section 3 explores the theoretical relationship between the forecast error variance and price stickiness. Section 4 presents the empirical analysis. Section 5 concludes. 1 This averaging-out conjecture is consistent with a recent nding 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 almost all of the LOP variation unaccounted for and diminishes the role of real shocks. 4

5 2 A Two-country Model of Sectoral Real Exchange Rates The model that we employ to understand sectoral real exchange rate uctuations is a two-country model that incorporates labor productivity variations into New Open Economy Macroeconomics with Calvo (1983) type price stickiness under local currency pricing. The closest model to ours is Kehoe and Midrigan (2007) who assume heterogeneous price stickiness across goods and trade costs to ship goods between countries. 2 In what follows, we will show the key equations of the model and discuss the implication for real exchange rates. The fully edged model is presented in the technical appendix of this paper, not intended for publication as it has been elaborated upon in published papers by the authors. For ease of exposition, we make some simplifying assumptions on the sources of real exchange rate variations. First, the home and foreign money supplies as assumed to follow a random walk and shocks to the money supplies represent the only nominal shock in the model. This is also the assumption made by Kehoe and Midrigan (2007). Second, rms produce goods using a technology that is linear in labor as in much of the New Open Economy Macroeconomics (NOEM) literature. Due to our microeconomic focus, the technology is subject to productivity shocks that, in logs, follow a common stochastic trend: a it = t + " it ; a it = t + " it; where t is a stochastic common trend, t = t 1 + " ct and " s are i.i.d. productivity variations speci c to locations and sectors. These shocks to labor productivity are the real shocks in the model. Variables marked with an asterisk denote foreign analogs of home variables. The central equation of the model describes the forecast error variance of the (log) real exchange rate for sector i, q it, across a bilateral pair of locations. The model implies that the k-period ahead forecast error variance of real exchange rates is given by: V ar t k (q it ) = ik 2 i V ar( t t ) + (1 i ) 2 (1 i ) 2 2 V ar(a it a it) ; (1) 0 1 kx where 2(j 1) A (2) j=1 i where t t denotes the money growth di erential across the home and foreign country and a it 2 Crucini, Shintani and Tsuruga (2009b) introduced sticky information into the Kehoe and Midrigan model to discuss the implications for persistence and volatility of good-level real exchange rates. Though it may be an interesting line of research, the role of sticky information is not focus of this paper. 5

6 is a sector-speci c labor productivity di erential in the production function of good i across the home and foreign country, both of which are exogenous and expressed in the logarithms. Equation (1) attributes the forecast error variance of the sectoral real exchange rate to the variances of the money growth di erential and the variance of the productivity di erential across the two countries. When expressed as a function of the frequency of price changes in the crosssection for xed values of the nominal and real shocks, we call this equation the real exchange rate volatility curve equation. Normalizing by the term ik to focus on the role of the frequency of price changes and the relative importance of nominal and real shocks in the shape of the volatility curve, we have: V ar t k (q it ) = V ar( t t ) 2 i + V ar(a it a it) 2 (1 i ) 2 (1 i ) 2 ; (3) ik As is evident, the variance of the nominal and real shocks are coe cients in the volatility curve, with the weights attached to them moving in opposite directions as i increases from 0 to 1. Note that nominal prices are determined in a staggered fashion in the sense of Calvo (1983). Here i captures the degree of price stickiness, common across countries but di ering across goods. 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 across bilateral pairs, this may change. Monopolistically competitive rms indexed by the good they produce, i, are allowed to change prices with probability 1 i. That is, some sectors are allowed to change their prices more frequently than others (we estimate these frequencies in the empirical section below). Consumers in each country have identical preferences, but consume larger shares of the home variety of each good than the imported variety due to iceberg transportation costs. Speci cally, the parameter in the volatility curve captures the economic role of trade costs. Firms in each country are required to pay an iceberg transportation cost to send goods across the border. In our model, the parameter is increasing in the trade cost and in the elasticity of substitution among di erentiated products : = 1 (1 + ) 1 = 1 + (1 + ) 1. To further simplify the argument, consider the case of k = 1. The forecast error variance is: V ar t 1 (q it ) = V ar( t t ) 2 i + 2 V ar(a it a it) (1 i ) 2 (1 i ) 2 : (4) We refer to the rst term of Equation (4) as the nominal e ect and the second term as the real e ect on sectoral real exchange rates. As noted, an increase in i shifts the relative contribution to variance toward the nominal e ect and away from the real e ect. 6

7 To see more explicitly how price stickiness either ampli es or mitigates the impact of a shock on the real exchange rate in (4), recall that real exchange rate for good i is de ned as q it = s t + p it p it ; (5) where p it (p it ) denotes the (log) sectoral price index in the home (foreign) country and s t is the (log) nominal exchange rate. Consider, rst, a positive money growth rate shock in the home country, holding xed foreign money growth. The model predicts an immediate depreciation of the nominal exchange rate (i.e., an increase in s t ). The responses of local currency prices, though, depend on the good-speci c frequencies of price adjustment. Goods that change every period will adjust immediately, completely o setting the impact of the nominal exchange rate depreciation and preserving the original LOP deviation (as determined by the steady-state trade costs). At the other end of the continuum, goods prices that are very sticky will have real exchange rates that tend to follow the path of the nominal exchange rate and exhibit negligible pass-through of the nominal shock. In other words, the nominal e ect on real exchange rate variability is ampli ed by slow local currency price adjustment. Consider, next, a positive shock to home productivity in sector i. In the absence of trade across locations, this would simply lower the price of good i, relative to other goods that individuals consume and this lower price would have symmetric e ects everywhere. Importantly, the relative price of that good would not change across locations. With home bias, this international spillover is less than perfect due to trade costs, and generates a change in the relative price across locations. Basically, trade costs result in price indices at the good level which are disproportionately weighted in home varieties. Thus, an increase in home productivity lowers the price of the home variety which in turn lowers the price index of that good by more at home than abroad. How asymmetric the impact on relative prices is depends on the size of the trade costs and the elasticity of substitution. Because this mechanism requires prices to actually change and thereby induce asymmetric price changes across locations, it is important that prices be exible for this channel to be signi cant. Note: a good-speci c productivity shock is not plausibly going to have an equilibrium e ect on the nominal exchange rate, which is why we are holding it constant here. At the other end of the continuum, goods prices that are very sticky will simply not adjust, so neither will the real exchange rate for those goods. In other words, the real e ect on real exchange rate variability is mitigated by slow local currency price adjustment. 7

8 This discussion should make clear that it is not necessarily true that real exchange rates become more volatile as price adjustment of goods is slower. It depends on the source of the shock. The next section provides some calibrated examples of the shape of the volatility curve, using (4). We ask: In what type of economic environments is the conventional wisdom on the relationship between real exchange rate volatility and price stickiness plausible? 3 The real exchange rate volatility curve This section shows that the real exchange rate volatility curve is theoretically a U-shaped function of i 2 [0; 1] with the shape varying according to the relative magnitude of the volatility of real and nominal shocks. To compute the volatility curve we need three parameters and two moments from the data. We assume that the unit of time is one month and set the discount factor, = 0:96 1=12, the elasticity of substitution is, = 10, and trade costs are: = 0:5. Our simulations reveal that the salient features of the volatility curve are not very sensitive to these parameters unless is extremely small. More interesting are the role of nominal and real shocks in shaping the curve. We set the standard deviation of the money growth rate di erential to 2.4 percent. The technical appendix of the paper shows assumptions under which the money growth rate di erence is equal to the change in the nominal exchange rate. This is our benchmark case. Thus, we choose 2.4 percent to be consistent with the volatility of the U.S.-European bilateral nominal exchange rates over the period of our data, 1996:1-2006:12. In particular, the standard deviation of the nominal exchange rate growth of the U.S. dollar against Austria, Belgium, France, and Spain relative to the U.S. dollar is 2.36, 2.37, 2.35, 2.36 percent, respectively. While the Euro was o cially introduced part-way through our sample, the nominal exchange rate of these European countries were quite stable against each other and thus the parameterization to a single average nominal shock, seems appropriate. This is why the bilateral real exchange rates tend to have comparable volatilities over the sample. Calibration also requires measures of productivity di erences across the U.S. and Europe, in the sectors spanned by our micro-price data. For the purpose of this section, we illustrate the mechanics of the model by assuming that variances of technological di erence are the same among sectors and consider contrasting ratios of nominal-to-real shocks: (a) Std(a it a it )=Std( t t ) = 5; 8

9 (b) Std(a it a it )=Std( t t ) = 1; and (c) Std(a it a it )=Std( t t ) = 1=5. Each panel of Figure 1 uses (4) to plot the conditional variances against i under the di erent parameterizations of Std(a it a it ) relative to Std( t t ). Each panel shows the variance due to the real e ect expressed by the blue area and variance due to the nominal e ect expressed by the red area. The combined height of blue and red area is the total one-step ahead forecast variance. If goods were uniformly distributed across frequencies, i, this gure would conveniently provide a complete variance of the cross-section into nominal and real e ects under the assumptions of common productivity shock volatility across sectors. The variance due to the real e ect is (1 i ) 2 (1 i ) 2 2 V ar(a it a it ), which is strictly decreasing in i so that the height of the blue area decreases with i. At each frequency of price adjustment, the variance due to the nominal e ect is 2 i V ar(^ t ^ t ), which is strictly increasing in i so the height of the red area increases with i. Note also, the unpleasant sticky-price arithmetic in terms of the fact that, given is close to 1, a little price stickiness goes a long way to mitigating transmission of real shocks due to the power of 4 (1 i ) 2 (1 i ) 2 ' (1 i ) 4 on the frequency of price non-adjustment, (1 of the relative size of real and nominal shocks. i ). Each panel of the gure deals with a di erent parameterization Panel (a) is basically an environment in which only real shocks are of quantitative importance and the volatility curve is downward sloping over almost its entire range. Only as we approach completely rigid local currency prices at i = 1 do we see the curve begin to slope upward, providing the rst hint of a nominal e ect (i.e., the red area becomes visible). The minimum the point at which the slope changes is a good with relatively sluggish adjustment i = 0:75. 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) is the opposite extreme, an environment in which only nominal shocks are of quantitative important and the volatility curve is upward sloping over almost its entire range. Only as we approach goods with completely exible prices do we see the curve begin to slope downward, providing evidence of real e ects (i.e., the blue area becomes visible). The minimum, and therefore the point at which the slope changes is virtually indistinguishable from a completely exible price, i = 0:06. The red area, the nominal e ect, accounts for about 98% of the total area, the blue area (real e ect), a mere 2%. Contrast these cases with the middle panel (b): here the volatility curve is U-shaped. When we parameterize Std(a it a it )=Std( t t ) = 1, the blue area is comparable in size to the red area. 9

10 In fact, numerical integration over the entire range of i tells us that real e ects account for 35% of the variation and nominal e ects account for the remaining 65%. The good with the lowest real exchange rate variance is the one with a Calvo parameter equal to As such, total variance of real exchange rates decreases as we increase i from 0 to 0.40, while it increases as we push i above 0.40 and toward 1. 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 real exchange rate. In the intermediate case, again assuming a uniform distribution of goods over the entire range of price adjustment frequencies, the correlation between good-level real exchange rate variability and price stickiness is predicted to be close to zero. This is a textbook example of a situation in which correlation linear dependence is completely misleading about the underlying economic structure. There is obviously a unique functional relationship between the two variables in this model and yet the correlation statistic would not uncover this fact. More generally, the sign of the correlation will depend on the distribution of goods in terms of their frequencies of price adjustment as well as the relative importance of real and nominal shocks. In panel (a), when real shocks dominate, the correlation is robust and negative, while in a panel (c), when nominal shocks dominate, the correlation is robust and positive. Robustness in this context has a number of facets. Consider a researcher with a limited sample of goods prices, which happen to be exible. Using our hybrid model, this researcher will be tempted to conclude that real shocks predominate. For the goods in his sample, this is true, but it need not be true in general given the non-homogeneity of the e ect of price rigidity in the presence of two types of shocks. Consider a researcher who focuses on a period of xed or stable nominal exchange rates using cross-border pairs. This researcher will nd real shocks to be dominant. Consider a research using aggregate price indices. This research will tend to nd nominal shocks dominate if price adjustment frequencies are intermediate between complete rigidity and complete exibility and real shocks average out more across goods than do nominal shocks, which seems plausible. Figure 1 has a number of striking implications for the general cause of identifying the importance of real and nominal shocks. First, because the real exchange rate volatility curve must be U-shaped under a theoretical support of i 2 [0; 1] 3, it is possible that the data suggest negative correlations 3 To prove this, evaluate the rst derivative of the total variance with respect to i at i = 0 and 1. 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 with respect to i is strictly 10

11 between total variance and degree of prices stickiness, contrary to the conventional wisdom. As panels (a) and (b) in Figure 1 suggest, negative correlations can occur when variance in technological di erence substantially contributes to total variance. Second, the degree of price stickiness which minimizes the volatility curve provides information on whether nominal or real shocks are more important in a general sense. This implies that it is important not only to know the slope of the real exchange rate curve over the empirical distribution of i but also to nd the minimum of the curve in evaluating the role of real and nominal e ects on the real exchange rate volatility. It also true that the real and nominal e ects di er considerably across individual sectors even when the shocks are common across sectors because the coe cients on the real and nominal variance are sector-speci c: 2 i or (1 i ) 2 (1 i ) 2 2. To see the practical implications of this, consider i = 0:86, the median value of the degree of price stickiness in the dataset we use below. At this point in the distribution, the nominal shock variance gets a weight of 2 i shock variance gets a coe cient of 0:00036! 4 = 0:74, while the real 4 Empirical Analysis Our analysis focuses on (i) examining the relationship between total variance and the degree of price stickiness; (ii) nding the degree of price stickiness which minimizes the volatility curve; and (iii) assessing the relative importance of the real and nominal e ects at sectoral level. Throughout our analysis, we work with the data constructed by Kehoe and Midrigan (2007). The data consist of sectoral real exchange rates for four European countries (Austria, Belgium, France, and Spain) relative to the U.S. Essentially this involves matching monthly local currency price data from Eurostat and 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 The number of sectors is 66. Kehoe and Midrigan (2007) take the cross-country average monthly infrequencies of price changes within each sector. The country-level frequencies for the U.S. are from the Bils and Klenow (2004) and those in each of the European countries are taken from the following individual country studies: Baumgartner, Glatzer, Rumler, and Stiglbauer (2005) for Austria; Aucremanne 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) that minimizes total variance. 4 Using the formula and the baseline parameterization: (1 0:74) 2 (1 0:74 0:9966) 2 [ 1 (1 + :5) 1 = 1 + (1 + :5) 1 10 ] 2. 11

12 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). 4.1 Estimating the real exchange rate volatility curve Regression analysis is used to relate real exchange rate volatility to the degree of price stickiness. Let V i be the one-period ahead forecast error variance of real exchange rate for good i. Following Kehoe and Midrigan (2007), when either infrequency of price changes or the real exchange rate is missing or when the forecast error variance can be computed from only small number of samples, we exclude such goods from the sample. 5 In our benchmark analysis, V i is regressed on i using (i) the pooled samples from four country-pairs; and (ii) country-by-country samples. Table 1 reports regression results based on the pooled and country-by-country regressions. In all cases, the sign of the coe cient on i is signi cantly negative based on heteroskedasticity consistent standard errors reported below the point estimates. Even with this simple speci cation, the benchmark pooled regression explains the 70 percent of the cross-sectional di erences in volatility of real exchange rates. 6 The estimated slope coe cients are very close among four nation-speci c and pooled regressions. Regressions t very well especially for Austrian and French data. The regression results suggest that, given the observed degree of price stickiness, real exchange rate volatility appears to decrease with the degree of price stickiness. The negative correlation appears consistent with the dominance of productivity di erentials over money growth rate differentials, as in Panel (a) of Figure 1. Recall, however, that the correlation also depends on the region of the parameter space for frequency of price adjustment for the goods in the micro-sample. In particular, when the observed i are collected around small values, we might have a U-shaped curve which declines with small i but increases with large i outside the range of the observed data. The scatter plot shown in the upper panel of Figure 2 suggests that this is probably not the 5 After excluding the samples, the number of sectors amounts to 57 for Austria, 46 for Belgium, 48 for France, and 31 for Spain. 6 For robustness, we included the country dummy variables into the pooled regression to control for di erences in trade costs paid to carry goods from a country to another country. Also, our benchmark regressions implicitly assume that the volatility of technological di erence is common to all sectors. To relax this assumption, we also included the good dummy variables into the pooled regression as well as the country dummies. These robustness checks revealed that, while the goodness of t increases in both extensions, the magnitudes of the slope coe cient are essentially unaltered. 12

13 case. The 10th percentile of the degrees of price stickiness is 0.71, implying that the stickiness in many sectoral prices is actually distributed around a range of large values. As such, the negative correlation appears to be due to highly volatile technological di erence (or other real shocks) as in Panel (a). Given the theory suggests a possible non-linear unconditional correlation between real exchange rate variability and the i, a squared term is added to the regressions. With this extension, it is possible to locate the degree of price stickiness which minimizes the volatility curve. In particular, we estimate V i = i i + u i ; (6) where k for k = 0; 1; 2 are estimated coe cients. Notice that an expression equivalent to the above regression is V i = b 0 + b 1 ( i b 2 ) 2 + u i ; with the coe cients, b 0 = 0 (1=4) 2 1= 2, b 1 = 2, and b 2 = (1=2) 1 = 2. Since b 2 can be interpreted as the minimizer of this quadratic function, we estimate the minimizer of the curve from the estimates of 1 and 2. Table 2 shows the estimation results when the squared i is added into the benchmark regression. In all cases, the estimated coe cients have the same signs among di erent speci cations and they are all signi cant except for one (i.e., the coe cient on 2 i for Belgium). Overall, the regressions with the squared i fare much better than the regressions without 2 i in terms of the goodness of t. The explanatory power improves signi cantly in most cases. 7 The upper panel of Figure 2 plots the tted curve based on the results of the pooled regression appeared in Table 2. The tted curve qualitatively resembles Panel (a) of Figure 1 in terms of the shape of the curve, which again suggests the importance of real e ects. The last two columns of Table 2 compare the estimated degree of price stickiness which minimizes the curve with the maximum value of empirical i in each regression. The estimates of b 2 are near unity and the technological di erence volatility appears to matter for the real exchange rate volatility curve. Although the point estimates of b 2 exceed a unity in some cases, the 90 percent con dence interval includes a value less than unity, which is consistent with our model s prediction that the minimizer 7 The inclusions of country and good dummies into the pooled regression again did not a ect the signs and magnitudes of the estimated coe cients. 13

14 of the variance must lie between zero and one. In Austrian and French cases estimated precisely, the point estimates are ^b 2 = 0:99 and 0:97, respectively. As a further robustness check, a nonparametric analysis of the real exchange rate volatility curve is conducted by running a regression of the form, V i = m( i ) + u i where m() is an unknown conditional mean function. The lower panel of Figure 2 shows the estimated curve based on two standard kernel-based nonparametric regression estimators, the local constant (Nadaraya-Watson) and local linear estimators. 8 In either case, the shape of the tted curve is somewhat similar to the one based on the quadratic regression (6). The slope of the curve is negative for all the region but seems to become atter as the degree of price stickiness increases. We further investigate the shape of the estimated curve using a formal nonparametric test of monotonicity developed by Ghosal, Sen and van der Vaart (2000). Their test is designed to test the null hypothesis that m() is an increasing (decreasing) function on a certain interval [a; b]. In our context, we are interested in the shape of the curve on empirically observed range [ min ; max ] where min and max are minimum and maximum values of price stickiness in the data. The result of the test is reported in Table 3. Using a conventional signi cance level, the hypothesis of increasing function is signi cantly rejected, and that of decreasing function is not rejected. This result is consistent with the conclusion of parametric analysis that the price stickiness which minimizes the volatility curve is likely to be at the level above all the observed values in the data. 4.2 Variance decomposition Pushing the theory a bit further, the relative importance of the real and nominal e ects at sector level is estimated by examining forecast error variances decomposed using equation (4). The variance due to the nominal e ect is extracted from total variance of the real exchange rate by replacing the money growth rate di erence with the nominal exchange rate growth rate. Recall, this is the appropriate theoretical proxy for the nominal e ect when money growth follows a random walk as in Kehoe and Midrigan (2007). Furthermore, using the observed frequency of price adjustment, the estimated contribution of the nominal shock to the real exchange rate is 2 i V ar(s jk;t ); where s jk;t is the bilateral nominal exchange rate between the United States (k) and country j. Subtracting this term from V i is taken to be the real e ect at the point on the volatility curve indicated by i. 8 For both estimators, Gaussian kernel are used along with the bandwidth selected by the rule of thumb. 14

15 The rst column of Table 4 reports the cross-sectional average and the standard deviation of contributions of real e ect to the conditional variance of sectoral real exchange rates. Overall, the importance of the real e ect at sector level is comparable to that of the nominal e ect. On average, variances due to real e ect account for 59 percent of total variances when all samples from the four countries are pooled. Although the contributions of the real e ect are somewhat dispersed with the standard deviations of 24 percent, contribution of real shocks is at least as large as that of nominal shocks for most goods. This property does not appear to come from country-speci c factors, because the country-by-country averages and standard deviations do not seem much di erent among countries. In the previous section, the percentage of the blue area relative to the total area was used to summarize the relative importance of the real e ect over the entire range of possible frequencies of price change. In evaluating the relative importance of real e ects from our variance decompositions, the empirical ratio of a blue area to the sum of two areas should put a weight equal to the proportion of goods for which i is the Calvo parameter, instead of putting an equal weight as implicit in our theoretical integration method. As it turns out, the empirical ratio of the blue area to the sum of blue and red areas is actually identical to the sample mean of contributions of real shocks. As such, the estimated ratio of the blue area is simply 59 percent, consistent with the previous ndings that technological di erence is an important source of real exchange rate uctuations. The variance decompositions are valid under the assumption that variance unexplained by the nominal e ect is all attributed to contribution of the real e ect. However, our estimated variance due to the real e ect may include regression errors which cannot be explained by explanatory variables. Hence, if regression errors are large perhaps due to measurement error, the method may over- or under-estimate the contributions of real shocks. To explore such possibilities, the last three columns of Table 4 report contribution of the real shocks after eliminating unexplained part of total variance in the regressions with 2 i and the non-parametric regressions. Overall, while contributions of the real e ect are somewhat larger than the previous estimates, the dominance of the real e ect is robust to this change. It is also worth noting that the nominal exchange rate would naturally vary with national productivity, in which case assigning all of the variation in the nominal exchange rate to the nominal e ect overstates the size of the nominal e ect relative to the real e ect. 15

16 5 Conclusion Using a time-dependent Calvo pricing model, the notion of the real exchange rate volatility curve was developed to advance understanding of the relationship between real exchange rate volatility and price stickiness. In the standard two-country sticky price model with both nominal and real shocks, the real exchange rate volatility curve may be a U-shaped function of 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, it was found that the forecast error variance has negative correlation with the degree of price stickiness and that the level of price stickiness that minimizes the U-shaped volatility curve is near unity. Moreover, the variance of individual real exchange rates at sector level seem to be dominated by the contribution of real shocks, not nominal shocks (though they, too, are signi cant). These results suggest that consideration of both real and nominal shocks are important for understanding the sources of real exchange rate variability. Finally, the cross-sectional patterns of price stickiness and the Calvo parameter are helpful in understanding how di erent sources of shock play out across the distribution of LOP deviations. 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 or nominal shocks and such averaging may be germane to the issue of the source of risks individuals and rms face: are they intertemporal price movements related to interest rate di erentials and aggregate real exchange rates or relative price movements, related to movements of relative price within the distribution (i.e., uctuations in the terms of trade). It is also important to develop an understanding for the reasons the Calvo parameters di er across goods. to be done. We hope to explore these possibilities in future work. Much remains References [1] Alvarez, L. J., and I., Hernando Price setting behavior in Spain: Stylized facts using consumer price micro data. ECB working paper series No [2] Aucremanne, L., and E. Dhyne How frequently do prices change? Evidence based on the micro data underlying the Belgian CPI. National Bank of Belgium Working paper No

17 [3] Baumgartner, J., E. Glatzer, F. Rumler, and A. Stiglbauer How frequently do consumer prices change in Austria? ECB working paper series, No [4] Baudry, L., H. Le Bihan, P. Sevestre, and S. Tarrieu What do thirteen million price records have to say about consumer price rigidity? Oxford Bulletin of Economics and Statistics, 69(2), [5] Bils, M., and P.J. Klenow Some evidence on the importance of sticky prices. Journal of Political Economy, 112(5), [6] Chari, V. V., P. J. Kehoe, and E. R. McGrattan Can sticky price models generate volatile and persistent real exchange rates?," Review of Economics Studies, 69(3), [7] Clarida, R., and J. Galí Sources of real exchange-rate uctuations: How important are nominal shocks? Carnegie-Rochester Conference Series on Public Policy, 41(1), [8] Crucini, M. J., M. Shintani, and T. Tsuruga. 2009a. The law of one price without the border: the role of distance versus sticky prices." NBER working paper [9] Crucini, M. J., M. Shintani, and T. Tsuruga. 2009b. Accounting for persistence and volatility of good-level real exchange rates: the role of sticky information." unpublished manuscript, Vanderbilt University. [10] Dornbusch, R., S. Fischer, and R. Startz Macroeconomics. McGraw-Hill, New York. [11] Ghosal, S., A. Sen, and A. W. van der Vaart "Testing monotonicity of regression." The Annals of Statistics, 28(4) [12] Engel, C., and J. H. Rogers, "How wide is the border?" American Economic Review, 86(5), [13] Kehoe, P. J., and V. Midrigan Sticky Prices and Sectoral Real Exchange Rates." Federal Reserve Bank of Minneapolis Working Paper No [14] Obstfeld, M., and K. Rogo, Foundations of International Macroeconomics, MIT press. [15] Rogers, J. H, Monetary shocks and real exchange rates, Journal of International Economics, 49,

18 Table 1: The benchmark regressions Const. i Adj. R 2 Pooled Reg (0.001) (0.001) Austria (0.002) (0.002) Belgium (0.001) (0.001) France (0.002) (0.002) Spain (0.003) (0.003) NOTES: The standard errors shown in parenthesis are computed with heteroskedasticity consistent estimator. Pooled Reg. denotes the regression pooling four countries. Table 2: The regression results with 2 i Const. i 2 i Adj. R 2 b 2 Empirical max Pooled Reg (0.002) (0.005) (0.003) (0.071) - Austria (0.001) (0.003) (0.002) (0.037) - Belgium (0.003) (0.009) (0.007) (0.879) - France (0.0004) (0.002) (0.002) (0.026) - Spain (0.003) (0.010) (0.008) (0.163) - NOTES: The standard errors shown in parenthesis are computed with heteroskedasticity consistent estimator. The value of b 2 presents the estimates of the degree of price stickiness which minimizes the total variance, which can be compared with the maximum value of empirical max in our dataset. 18

19 Table 3: Test for monotonicity Critical values H 0: Increasing H 0:Decreasing 1% 5% 10% Pooled *** Austria 4.927** Belgium 4.608* France 6.369*** Spain 5.098* NOTES: Critical values are computed from the method by Ghosal, Sen, and van der Vaart (2000). * signi cant at 10 percent level; ** signi cant at 5 percent level; *** signi cant at 1 percent level. Table 4: Cross-sectional average of contributions of real shocks to total variance Actual Estimated (1) Estimated (2) Estimated (3) All countries (24.1) (20.4) (16.5) (20.6) Austria (24.4) (19.6) (16.0) (19.6) Belgium (23.0) (16.0) (11.3) (21.0) France (22.7) (18.2) (14.7) (22.0) Spain (24.5) (28.2) (15.0) (24.5) NOTES: The unit of numbers are all percent. Numbers in parenthesis are standard deviations of contributions of the real e ect. Estimated (1) denotes contributions of the real e ect using parametric regressions in Table 2: Estimated (2) and Estimated (3) denote contributions of the real e ect using nonparametric regressions based on local constant estimator and local linear estimator, respectively. 19

20 Figure 1: Simulated variance of real exchange rates λ λ λ NOTES: Each panel of the figure shows the nominal and real effects on the real exchange rate volatility against the degree of price stickiness. The height of blue area in each panel measures variances due to the real effect while the height of red area placed above the blue area measures total variance. The variance due to the nominal effect can be seen from the height of red area alone. Each panel calibrates variances over different degree of price stickiness based on different values of standard deviation due to technological difference:(a) Std(ait a it)/std(µt µ t ) = 5; (b) Std(ait a it)/std(µt µ t ) = 1; (c) Std(ait a it)/std(µt µ t ) = 1/5. 17

21 Figure 2: Real exchange rate volatility and degrees of price stickiness: data and fitted curves λ λ NOTES: Both panels show the scatter plot of the degree of price stickiness and the conditional variance of real exchange rates. The upper panel of the figure shows the fitted curves based on parametric estimations where the dotted line is the fitted curve from the regression of volatility on λ i and the solid line is the fitted curve from the regression of volatility on λ i and λ 2 i. The lower panel of the figure shows the fitted curves based on nonparametric estimation. The dotted line is based on local constant estimator and the solid line is based on local linear estimator. 18