Missing Import Price Changes and Low Exchange Rate Pass-Through

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1040 January 2012 (revised: December 2012) Missing Import Price Changes and Low Exchange Rate Pass-Through Etienne Gagnon, Benjamin R. Mandel, and Robert J. Vigfusson NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from Social Science Research Network electronic library at http://www.ssrn.com/.

Missing Import Price Changes and Low Exchange Rate Pass-Through Etienne Gagnon, Benjamin R. Mandel, and Robert J. Vigfusson (cid:3)y Abstract A large body of empirical work has found that exchange rate movements have only modeste⁄ectsonin(cid:135)ation. However,theresponseofanimportpriceindextoexchange rate movements may be underestimated because some import price changes are missed when constructing the index. We investigate downward biases that arise when items experiencing a price change are especially likely to exit or to enter the index. We show that, in theoretical pricing models, entry and exit have di⁄erent implications for the timing and size of these biases. Using Bureau of Labor Statistics (BLS) microdata, we derive empirical bounds on the magnitude of these biases and construct alternative price indexes that are less subject to selection e⁄ects. Our analysis suggests that the biases induced by selective exits and entries are modest over typical forecast horizons. As such, the empirical evidence continues to support the conclusion that exchange rate pass-through to U.S. import prices is low. JEL Codes: F31, F41, E30, E01, C81 Keywords: exchange rate pass-through, import prices, item replacement ThisresearchwasconductedwithrestrictedaccesstoBureauofLaborStatistics(BLS)data. Theviews (cid:3) in this paper are solely the responsibility of the authors and should not be interpreted as re(cid:135)ecting the views of the Board of Governors of the Federal Reserve System, the Federal Reserve Bank of New York, or any other person associated with the Federal Reserve System or the BLS. We thank Carolyn Evans, Hakan Yilmazkuday, Nicholas Li, Emi Nakamura, J(cid:243)n Steinsson, John Rogers, Kei-Mu Yi, and seminar participants at the Federal Reserve Board, Federal Reserve Bank of New York, Bank of Canada, Banque de France, George Washington University, BLS Price and Index Number Research group, and conference on Microeconomic Sources of Real Exchange Rate Behavior at Vanderbilt University. Kelsey Ayres, Benjamin Hopkins, Gregory Howard, and Robert Sockin provided superb research assistance. E. Gagnon and R. Vigfusson are at the Federal Reserve Board. B. Mandel is at the Federal Rey serve Bank of New York. Comments and suggestions can be directed to etienne.gagnon@frb.gov, benjamin.mandel@ny.frb.org, robert.j.vigfusson@frb.gov.

In conducting monetary policy, central bankers are interested in how much exchange rate movements a⁄ect the prices of imported goods ((cid:147)exchange rate pass-through(cid:148)) as (cid:135)uctuations in these prices can, in turn, a⁄ect domestic prices and output. Commonly, exchange rate pass-through is measured by regressing changes in published import price indexes on changes in trade-weighted exchange rate indexes along with other explanatory variables. Using these regressions, researchers have estimated low rates of exchange rate pass-through for the United States. Recent estimates (for example, Campa and Goldberg [2005], and Marazzi and Sheets [2007]) suggest that, following a 10 percent depreciation of the dollar, U.S. import prices increase about 1 percentage point in the contemporaneous quarter and an additional 2 percentage points over the next year, with little if any subsequent increases. Theselowestimateshaveledseveralauthorstoformulatetheorieswhichgenerateincomplete pass-through.1 In a recent paper, Nakamura and Steinsson (2012) argue that actual long-run passthrough is substantially higher than these standard estimates because published price indexes are missing price changes associated with item replacement. In their words, "[if] the prices of new products entering the index have already adjusted to exchange rate movements ... [then] the response of these prices to movements in exchange rates ... will be (cid:145)lost in transit(cid:146)(i.e., neither picked up by an observed price change of the exiting nor entering product)." They derive an adjustment factor for standard long-run pass-through estimates under speci(cid:133)c assumptions regarding the nature of item substitutions and the way (cid:133)rms set prices. They conclude that measured pass-through is substantially underestimated, implying that exchange rate pass-through to U.S. import prices is much more complete in the long run than was previously thought. Our paper reassesses this conclusion by developing a general framework for understandinghowsampleturnover(cid:151)inconjunctionwithpricingassumptions(cid:151)impactsmeasuredpassthrough, and by proposing and implementing new ways of gauging the biases in standard estimates. Throughout our discussion, we emphasize the implications of sample exits and sample entries over the (cid:133)rst couple years following an exchange rate shock, as such a shortterm horizon is most commonly employed by policymakers and is especially useful for discriminating among theoretical models. Our conceptual framework captures the possibility that an item whose price is about to change is more likely than others to leave the index (resulting in a (cid:147)selective exit(cid:148)) and that an item whose price recently changed is more likely than others to be added the sample (resulting in a (cid:147)selective entry(cid:148)). In either cases, items tend to release their price pressure outside of the period spent in the sample, thus lowering 1These papers include Atkeson and Burstein (2008), Gopinath, Itskhoki, and Rigobon (2010), Gopinath and Itskhoki (2010), and Gust, Leduc, and Vigfusson (2010). 2

the measured response of the price index. We show how the e⁄ects of item replacement on measured pass-through crucially depend on the nature of both exits and entries of items. In turn, the magnitude of these e⁄ects can be sensitive to pricing assumptions and the time horizon of interest. By contrast, the presence of real rigidities is typically of limited consequences. In presenting these (cid:133)ndings, we substantially expand the analysis of Nakamura and Steinsson (2012), who are primarily concerned with the long-run e⁄ects of item substitutions and do not explicitly model sample exits. Ourtheoreticalworkexplorestheimplicationsofselectiveexitsandselectiveentriesunder two popular pricing mechanisms, Calvo and menu costs, which together span a wide range of pricing behavior relevant for understanding pass-through dynamics. The most consequential situation is when price changes are missed both when items exit the sample and when they enter it. In such a case of selective exit and selective entry, standard pass-through estimates omit a roughly constant fraction of the actual price response to an exchange rate movement. This(cid:133)ndingappliestoalltimehorizonsandbothpricingmodels. Whenonlyexitsaresubject to a selection bias, items tend to leave the sample just before releasing their price pressure. The estimation bias as a share of the cumulative price response is largest initially, but then diminishes over time as items that have yet to respond to the exchange rate movement are brought into the sample and eventually release their price pressure. This compensating e⁄ect is largest in the Calvo model due to its relatively slow transmission of shocks. When only entries are subject to a selection bias, measured pass-through is initially unbiased, but a downward bias gradually appears as items entering the sample prove insensitive to past exchangeratemovements. ThebiasgrowslargestintheCalvomodelbecauseofitsrelatively slow pass-through of shocks, which means that item substitutions are likely to take place before items have time to release their price pressure. Item turnover is a potentially important source of bias in standard pass-through regressions. We (cid:133)nd that there is about one item replacement for every 5 price changes in the U.S. import price index. And as reported by Gopinath and Rigobon (2008) and Nakamura and Steinsson (2012), between 30 and 40 percent of items leave the sample without ever displaying a price change. The potential for economically important biases is thus material, providedthat manysampleexits andentriesaresubject toaselectione⁄ect. However, based on a range of empirical exercises, we conclude that the biases induced by selective exits and entries, although a concern and worthy of continued research, do not materially alter the literature(cid:146)s viewthat pass-throughtoU.S. import prices is lowovertypical forecast horizons. To inform our judgement, we (cid:133)rst use BLS microdata to derive empirical bounds on the price level response to an exchange rate shock. To do so, we calibrate our Calvo and menu-cost models to match key features of exchange rate movements, sample turnover, and 3

individual price adjustments, and we then recover the model-implied correction factors. Our empirical analysis focuses on imported (cid:133)nished goods, for which the biases are most likely to be severe. Our baseline empirical estimate of pass-through for these prices after two years is 0.24. After correcting for the most severe case of selective exit and selective entry consistent with the data, this estimate rises only to 0.28. WenextuseBLSmicrodatatoconstructalternativepriceindexesinwhichtheinclusionof newly sampled items in the index is delayed. We formally prove that this delaying procedure can substantially mitigate the selective entry bias over typical forecast horizons in a Calvo model. Theintuitionis that, whenentries are selective, the statistical agencyis addingitems to the index that are too insensitive to past exchange rate movements; delaying their entry in the index gives these items time to be a⁄ected by exchange rate shocks, making them more representative of the population and therefore reducing the selection e⁄ect. As such, theory predicts that if selective entries were a key source of bias, then these alternative price indexes should imply higher pass-through rates. However, when we estimate pass-through using these alternative price indexes, we (cid:133)nd no evidence of bias reduction. GiventhecloseconnectionbetweenourworkandthatofNakamuraandSteinsson(2012), we have attempted throughout to make the similarities and di⁄erences between the two papers clear with respect to both modelling assumptions and empirical conclusions. Section 3.5 compares the theoretical long-run correction factors derived by Nakamura and Steinsson (2012) to the ones obtained in our environment at various horizons. Turning to the empirical results, an important di⁄erence between our (cid:133)ndings and those of Nakamura and Steinsson (2012) re(cid:135)ects our judgment over the incidence of selection biases in item turnover and our treatment of heterogeneity in the observed frequency of price changes across items. As we discussinsection4.3,correctionfactorscandi⁄ergreatlydependingon,amongotherreasons, the assumed mass of observations at very low frequencies of price changes. The remainder of the paper is structured as follows. Section 1 describes the sample of items used by the BLS to compute import price in(cid:135)ation and provides an overview of item exits and entries. Section 2 introduces the baseline Calvo and menu-cost models that we use toillustratethenatureofthevariousbiasesandtogaugetheirquantitativeimportance. Section 3 presents the possible biases associated with selection e⁄ects in sample exit and entry. Section 4 explores the empirical relevance of these biases by computing bounds on standard pass-through estimates and by constructing an alternative price index that mitigates these biases. Section 5 concludes. 4

1 Nature and occurrence of item exits and entries Our study focuses on changes to the BLS import price sample, not on changes to the population (universe) of imported items. For clarity, we reserve the terms (cid:147)exit(cid:148)and (cid:147)entry(cid:148) for changes in the composition of the sample.2 Throughout the presentation of the data and subsequent model-based analysis, we are concerned with the possibility that micro price changes tend to take place just after items exit the sample or shortly before items are added to the sample, so that part of the price response to shocks is censored. We de(cid:133)ne a (cid:147)selective exit(cid:148)as the subtraction of an item from the sample that is triggered by its price being about to change, and a (cid:147)selective entry(cid:148)as a systematic addition to the sample of an item that recently experienced a price change. By contrast, a (cid:147)random exit(cid:148)and a (cid:147)random entry(cid:148) are, respectively, the subtraction from and the addition to the sample of an item without regards to its pricing characteristics. With the above terminology in mind, the remainder of this section provides background informationabouttheconstructionoftheimportpriceindexesusedinstandardpass-through regressions, emphasizing the nature and occurrence of sample exits and entries, their treatment by the BLS, the potential for selection biases, and their relationship to micro price adjustments. 1.1 The International Price Program Given identical data to Gopinath and Rigobon (2008) and Nakamura and Steinsson (2012), we rely on their work to convey the details of the BLS(cid:146)International Price Program (IPP) protocolandsample, aswellasontheBLSHandbookofMethods. Inbrief, importpricesare collected through a monthly survey of U.S. establishments. The sample consists of rolling groups of items, each item having a sampling duration of about three years, on average. The IPP chooses its (cid:133)rms and items based on a proportional-to-size sampling frame with some degree of oversampling of smaller (cid:133)rms and items.3 Respondents must provide prices for actual transactions taking place as closely as possible to the (cid:133)rst day of the month. In total, we observe the price of approximately 13,000 imported items per month from September 2Ouranalysisthusexcludesattheonsetthewell-known(cid:147)newgoodbias(cid:148)and(cid:147)qualitychangebias,(cid:148)which ariseduetodi¢ cultiesinimputingpastreservationpricesforitemsneverobservedbeforebyconsumers. For an introduction to the economics of new goods, see the volume edited by Breshnahan and Gordon (1996). Gordon(2006)andtheconferencesummariesoftheOttawaGroupprovideoverviewsofsubsequentresearch on these biases. 3For instance, if there are two items sampled at a (cid:133)rm, one of which has a 90 percent sales share and the other a 10 percent sales share, allocating weights uniformly would over-weight the smaller item. Whenconstructingitsaggregatepriceindexes,BLScorrectsforthisphenomenonwithitem-levelprobability weights. 5

1993 to July 2007. For the purpose of computing our sample statistics, and consistent with previous studies, we carry forward the last reported price to (cid:133)ll in missing values, e⁄ectively overwriting IPP price imputations and (cid:133)rm estimates of prices in non-traded periods.4 We also restrict our sample to U.S. dollar transactions, which account for about 90 percent of all observations.5 1.2 Nature of exits and entries BLSpricecollectorstakenotewhenanitemexitsthesampleandassigntheretiringitemone of the following codes: (1) regular phaseout, (2) accelerated phaseout, (3) sample dropped, (4) refusal, (5) (cid:133)rm out of business, (6) out of scope, not replaced, and (7) out of scope, replaced. Codes (1) through (3) indicate that item exit is driven primarily by the phaseout schedule of the IPP sampling protocol. Codes (4) and (5) describe situations in which price collectionisimpossiblebecausethesurveyrespondentrefusestorespondorceasestooperate, even though the exiting items may continue to be traded in the universe. Codes (6) and (7), which we collectively refer to as (cid:147)out of scope,(cid:148)are those instances in which price quotes become unavailable because the item ceases to be traded by importers.6 The purpose of item phaseouts is to keep the sample representative of the universe of items; the BLS resamples approximately half of its disaggregated product categories every two years and typically plans to retire items (cid:133)ve years after their entry. An item may retire early if it is insu¢ ciently traded. We see such exits, given their planned nature, as unlikely to be selective. Contrary to phaseouts, refusals and importers going out of business are not foreseenevents. Nevertheless, weviewtheriskthatsuchexitssystematicallymaskindividual price adjustments as relatively modest, as there are several factors unrelated to micro price adjustments that could trigger them. Exits associated with items becoming out of scope likely present the greatest risk of masking price adjustments. For example, an importer could cease to order an item when faced with a price increase eating away its pro(cid:133)t margins. The item could also exit because the foreign producer is adjusting the item(cid:146)s e⁄ective price 4For a given item, reporting (cid:133)rms typically do not provide a transaction price every month. The BLS imputes an item(cid:146)s missing price by either carrying forward the last reported price or by adjusting it by the average price change for the same (cid:133)rm and product category. 5Gopinath,Itskhoki,andRigobon(2010)estimatenearlycompletepass-throughrates(0.95)tonon-dollar importpricesafterasinglepriceadjustment,comparedtoonly0.25fordollarimportprices. Whileexcluding non-dollartransactionsnudgesourestimationresultstoward(cid:133)ndinglowpass-throughrates,ithasthebene(cid:133)t of centering our analysis on transactions for which pass-through is most likely to be underestimated. As such, the correction factors derived in the paper may slightly overstate the extent of the bias when applied to estimates obtained using the full sample of transactions. 6In some instances, the (cid:133)rm can provide an alternative item that meets BLS sampling needs (called (cid:147)replaced(cid:148)), though that new item would still be recorded as a separate entry. About a third of out-outscope exits are (cid:135)agged as (cid:147)replaced.(cid:148) 6

through a change in its characteristics. Other situations leading to out-of-scope items may be unrelated to micro price adjustments. For example, the importer may be curtailing the range of products on o⁄er to streamline its inventory management. It is worthwhile to note that exits are not generally accompanied by the simultaneous entry of a newly sampled item. When an item suddenly becomes out of scope, BLS price analysts ask the reporting (cid:133)rm whether it can provide another item that meets the same sampling criteria. When possible, the BLS may link the price of the entering and exiting items through a one-time quality adjustment, in which case the estimated change in the e⁄ective price is recorded. However, this practice is relatively infrequent. In other instances, the (cid:133)rm may provide an alternative item meeting the BLS sampling needs, though that item is recorded as a separate entry. More often, when no item with similar characteristics is available in the same establishment, or when the planned phaseout date is within the next 18 months, the BLS simply waits until the next biennial sample redrawing. The lag between an unplanned exit and the subsequent item entry can thus be fairly long. Even in the case of planned phaseouts, BLS protocol does not necessitate synchronizing exit and entry.7 Notwithstanding the fact that exits and entries are staggered, the size of the IPP sample hasbeenroughlyconstantsince1993asthegrossnumberofexitshastypicallybeenmatched byacorrespondingnumberofentries. TheBLSusesprobabilitysamplingtechniquestoselect establishments within broad strata of items, and then to select product categories within each stratum-establishment combination. A BLS (cid:133)eld agent next conducts an interview with the establishment to select speci(cid:133)c items. Probability sampling may be used at that stage. In general, special e⁄orts are made to ensure that selected items are traded regularly, which implies that higher-volume items with established price histories are more likely to be selected. In principle, the BLS(cid:146)s decision to sample a given item from within the universe should be unrelated to the timing of that item(cid:146)s price changes. Indeed, our reading of the BLS methodology is that the risk of selective entry is somewhat low, especially for those items entering the sample through planned sample redrawing. The risk of selective entries is arguablylargerforitemsenteringthesampleconcurrentlywithorshortlyafteranunplanned exit when no quality adjustment is made. Our assessment that the risk of a selective entry is somewhat low, especially if occurring in isolation from a selective exit, stands in contrast with Nakamura and Steinsson(cid:146)s (2012) conclusion that selective entries are an empirically important phenomenon. 7For instance, during biennial sample redrawings, some disaggregate product categories may be retired fromfurthersamplingbuttheiritemsmayremainintheindexuntiltheirplannedphaseout. Whereoutgoing and incoming product groups are dissimilar, the bene(cid:133)t to overlapping their items is unclear. 7

1.3 Accounting for exit and entry For a given month t, let exit(t) and entry(t) be the number of items exiting and entering the sample, respectively. These items cannot be used in the computation of in(cid:135)ation at month t because their price in either month t 1 or t is missing. For items whose price is available (cid:0) in both month t 1 and t, let change(t) and no_change(t) be the number of observations (cid:0) with a price change and no price change, respectively. We de(cid:133)ne the exit rate as exit(t) exit_rate(t) = : entry(t 1)+change(t 1)+no_change(t 1) (cid:0) (cid:0) (cid:0) The denominator in the above expression is the number of items whose price was collected in month t 1. The exit rate thus measures the fraction of items present in the sample at (cid:0) the end of month t 1 that leave in the next month. Analogously, the entry rate is measured (cid:0) as entry(t) entry_rate(t) = : entry(t 1)+change(t 1)+no_change(t 1) (cid:0) (cid:0) (cid:0) Table 1 shows summary statistics about exits and entries over the period October 1995 to April 2005.8 For industry groupings, we use the Bureau of Economic Analysis three-digit Enduse classi(cid:133)cation to bring descriptions of the microdata closer to the groups of goods commonly used in aggregate pass-through regressions (for instance, Bergin and Feenstra [2009] and Marazzi, et al. [2005]). In aggregating up from unique items in a given month to industry-level statistics, we weight each measure by its importance to overall U.S. import purchases.9 We aggregate the measures de(cid:133)ned above in two stages: (cid:133)rst, by computing unweighted statistics for each Enduse category in each month. Then, we aggregate across categories and time periods using the 2006 import sales value of each Enduse category.10 The rates of item exits and entries are both approximately 3 percent, indicating that the average size of the IPP sample remained about the same over the sample period. However, 8Incomplete reporting for item discontinuation reasons in the database made available to outside researchers by the IPP truncates our sample at its beginning and end. October 1995 is the (cid:133)rst month for which the discontinuation reason (cid:133)eld is populated, while the months following April 2005 contained incomplete information about exits at the time of our data extraction. 9Doing so assigns the average item frequencies for sampled items and products to those not sampled within the same industry. 10An alternative weighting scheme would be to use the BLS product weights, which are akin to annual importvaluesattheHarmonizedSystem10-digit(HS10)level, spreadevenlyacrossitemswithineachHS10 product. The Enduse weights for a given month would be the sum total of the individual item weights across items and HS10 products within that Enduse. However, due to incomplete weight data for petroleum (Enduse 100), that method tends to under-weight those high-frequency products in the aggregate statistics. Otherwise, at the end-use level, the measures are quite similar. Also, ignoring the BLS probability weights for items and (cid:133)rms within each HS10 product, as we do, does not drastically change the summary statistics. Probability-weighted and unweighted statistics are available upon request. 8

the steadiness of the overall sample size hides a degree of heterogeneity in exit and entry ratesattheEnduseproductlevel. Forinstance, computersandsemiconductors(Enduse213) had an entry rate of 5.0 percent, nearly twice that of agricultural machinery and equipment (Enduse 212). Certain categories (like computers) expanded over the course of the sample as evidenced by higher entry rates relative to exit rates. Di⁄erences in net entries likely re(cid:135)ect the changing trade intensity of certain categories over the course of the sample. Exiting items coded as out of scope, which we see as presenting the highest risk of selective exits, accounted for half (1.5 percentage points) of the total exit rate. Enduse categories with a relatively high share of out-of-scope exits include computers and semiconductors, home entertainment equipment, as well as trucks and buses. By de(cid:133)nition, a selective exit entails a price change concurrent with an item leaving the sample. This pattern suggests that the rate of selective exit should vary over time along with macroeconomic variables triggering price adjustments. Evidence of this phenomenon in scanner data is provided by Broda and Weinstein (2010) in their analysis of barcode creation anddestructionoverthebusinesscycle. Toseeifexitsofimportedgoodssimilarlyrespondto the exchange rate, the top panel of (cid:133)gure 1 presents the time series of the exit rate restricted to out-of-scope items along with an index of the broad nominal dollar.11 This measure is very close to the (cid:147)endogenous exit(cid:148)measure reported in Berger et al. (2009), with the minor di⁄erence that we also exclude exits resulting from refusals and (cid:133)rms out of business. The series is (cid:135)at at about 1 percent throughout most of the early periods with a transient peak at the beginning of 2000. Then, the out-of-scope rate rises by about 50 basis points in 2003 through2005. Thesethreeprominentfeaturesofthetimeseries(i.e., (cid:135)atnessorslightdecline early, peak in 2000, and uptick in 2003-5) correspond inversely to the pattern of the broad nominal dollar index, shown in black. The intuition for this relationship is straightforward: as the dollar depreciates, the pro(cid:133)tability and viability of a higher proportion of imported items is adversely a⁄ected, leading (cid:133)rms to pull the items before the end of their scheduled sample life. We view this evidence as suggestive that exits may, in fact, occur in tandem with price changes. The occurrence of exits related to factors other than items falling out of scope, which we see as presenting a relatively low risk of selection bias, varies far less systematically with the exchange rate. Rather, the random exit series exhibits the fairly normal pattern of peaks every two years (i.e., the end of 1996, 1998, 2000, 2002, and 2004), which is in line with the biennial shu› ing of IPP items. For the most part, the overall entry rate shows a similar pattern with peaks in the middle of the year in 1997, 1999, and so on. Of note, similarly to the out-of-scope exit rate, the rate of overall entry also ticks up towards the end of the 11The exit rates shown in the (cid:133)gure are 12-month moving averages. 9

sample. We also note that the timing of the changes in out-of-scope exit rates and, to a lesser extent, in entry rates, does not seem to account for the decline in measured exchange rate pass-through documented in the literature, which has roughly halved since the 1980(cid:146)s. The decrease in pass-through took place primarily in the 1990(cid:146)s, preceding the upticks in exit rates by quite a few years. 1.4 Micro price adjustments As will be made clear in the next sections, the implications of selective exits and selective entries can be sensitive to underlying pricing frictions. It is convenient for our discussion to de(cid:133)ne the observed frequency of individual price changes as change(t) frequency(t) = . change(t)+no_change(t) The overall weighted incidence of price changes for (cid:133)nished goods is estimated to be 6.2 percent. TheanalogousstatisticfortheentireIPPimportsample(i.e., additionallyincluding industrial supplies, foods, feeds and beverages) is 15.3 percent.12 These levels are consistent with the weighted average of 14.1 percent in Nakamura and Steinsson (2012) and the median of 15 percent in Gopinath and Rigobon (2008). The average absolute (nonzero) price change is6.7percentfor(cid:133)nishedgoodsand8.0percentoverall, inlinewiththemeanoverallestimate of 8.2 percent in Gopinath and Rigobon (2008). Here, again, there is signi(cid:133)cant dispersion across categories, with items belonging to computers and peripherals (Enduse 213) having an average price change of 9.6 percent, compared to 2.0 percent for passenger cars (Enduse 300). 1.5 Other data considerations We conclude the data description by mentioning two additional elements important for the interpretation of the results. First, in any given month, prices are missing for about 40 percent of items in the sample, which could re(cid:135)ect the absence of a transaction or reporting issues. Second,nearlyhalfofallobservationsintheBLSsamplerefertoitemsthataretraded 12Some micro price studies include carried-forward prices in their count of usable observations (i.e., the denominator of the frequency formula), while others do not. We follow the former approach. One bene(cid:133)t is that the number of usable observations from one month to the next is directly determined by the number of entries and exits. If we instead excluded carried-forward prices, then our statistics would need to account for the fact that some quotes are inactive. Our decision makes price changes, exits, and entries somewhat less frequent than if carried-forward prices were excluded. The broad (cid:133)ndings of the paper do not hinge on this methodological choice, however. 10

betweena¢ liatesorentitiesofthesamecompany. AlthoughtheBLSprefersthattheseintracompany transfer prices be market-based or market-in(cid:135)uenced, some have expressed concern over whether these prices play the same allocative role as market transactions. Excluding intra-companytransferpricesfromthesamplehasanegligibleimpactonouranalysisbecause intra-(cid:133)rmandmarkettransactionshaveroughlysimilarentryrates, exitrates, andfrequency of price changes. See Neiman (2010) and Gopinath and Rigobon (2008) for a comparison of intra-(cid:133)rm and market transactions. 2 Pass-through and micro price adjustments: a baseline case ThissectionintroducesthebaselineCalvoandmenu-costmodelsthatwewillusetoillustrate the nature of the selection biases occurring during sample turnover. Although these models are only two of the many pricing mechanisms used in the literature, together they span a wide range of behavior that are relevant for understanding biases in measured pass-through. 2.1 Economic environment We assume that the universe comprises an in(cid:133)nite number of imported items, each produced by a single (cid:133)rm. There are no changes over time to the composition of the universe or to itemcharacteristics. The data-generating process for the change in the price (in logs) of item i at period t is 0 if f = 0 (cid:1)p = Iit : it ( u it +(cid:12)(cid:1)x t +" it if Ii f t = 1 The occurrence of a price change is indicated by the variable f. Given an opportunity (or Iit decision) to change its price, a (cid:133)rm sets (cid:1)p equal to the sum of (a) the amount of price it pressure inherited from previous periods, u , (b) a fraction (cid:12) of the current change in the it exchange rate, (cid:1)x , and (c) the contribution of a (mean-zero) idiosyncratic factor, " . The t it price pressure carried to the beginning of the next period is u +(cid:12)(cid:1)x +" if f = 0 u = it t it Iit : it+1 ( 0 if f = 1 Iit When the (cid:133)rm does not change its price, the shocks occurring in period t are simply added to the price pressure that had already accumulated. If the (cid:133)rm adjusts its price, then the 11

price pressure is fully released.13 The set up so far is quite general and not speci(cid:133)c to import prices. One could, for example, interpret (cid:1)x as the contribution of aggregate shocks, such t as wage in(cid:135)ation, to a (cid:133)rm(cid:146)s reset price. In what follows, we simply assume that (cid:1)x can be t represented by an AR(1) process, (cid:1)x = (cid:11)+(cid:26)(cid:1)x +(cid:24) ; t t 1 t (cid:0) with Gaussian innovations, (cid:24) . t We are ultimately interested in the impact of exchange rate movements on import prices. To this end, we de(cid:133)ne aggregate price in(cid:135)ation (in the universe or in the IPP sample) as the average change in item prices, (cid:1)p = (cid:1)p di: t it Z To recover pass-through dynamics the models, we will estimate linear regressions of the form L (cid:1)p = a+ b (cid:1)x +r ; (1) t l t l t (cid:0) l=0 X where r is an error term. t 2.1.1 Calvo model IntheCalvomodel,thedecisiontochangethepriceisexogenoustothe(cid:133)rm. f equals1with Iit constant probabilityf and0 withprobability1 f. This assumptionhas strong implications (cid:0) for the dynamic responses of import prices to exchange rate movements. It is convenient to consider the case in which innovations to the exchange rate, (cid:1)x , are uncorrelated over time t ((cid:26) = 0), as it allows us to derive analytical expressions for the regression coe¢ cients. As appendix A shows (in a more general environment), the (plim) linear estimate of b l for the baseline Calvo model is b = f (1 f)l(cid:12): (2) l (cid:0) Intuitively, for a movement in the exchange rate l periods earlier to impact an item(cid:146)s price today, the(cid:133)rmmustbegiventheopportunitytoadjustitspricetoday(probabilityf)andno price change must have occurred in each of the previous l periods (probability 1 f in each (cid:0) period). Otherwise, the current price would already re(cid:135)ect (cid:1)x . The Calvo model provides t l (cid:0) a textbook example of a geometric lag model in which the coe¢ cient on the explanatory 13Our pricing rule abstracts from forward-looking concerns, which greatly simpli(cid:133)es the exposition. Monthlyexchangerateinnovationsareonlyweaklycorrelated,soourpricingruleshouldneverthelesscapture central features of micro price adjustments in response to exchange rate movements. 12

variable decays exponentially with the number of lags. Summing up the (plim) coe¢ cients in the regression, we get L b = 1 (1 f)L+1 (cid:12); (3) l (cid:0) (cid:0) X l=0 (cid:16) (cid:17) which converges to (cid:12) as L . Thus, although the e⁄ects of an exchange rate shock are ! 1 neverpassed-throughfullytoimportprices,theeconometriciancanneverthelessapproximate (cid:12) (the (cid:147)long-run(cid:148)pass-through) with an arbitrary degree of precision. 2.1.2 Menu-cost model Inthemenu-costmodel,thedecisiontochangethepriceistheresultofacost-bene(cid:133)tanalysis performed by the (cid:133)rm. As shown by Sheshinski and Weiss (1977), it is optimal for the (cid:133)rm to keep its price unchanged if the deviation from the reset price, u +(cid:12)(cid:1)x +" , falls within it t it a certain range. For simplicity, we assume that this range is symmetric and constant across time and (cid:133)rms. Hence, the price adjustment decision is 0 if u +(cid:12)(cid:1)x +" K f = j it t it j (cid:20) : Iit ( 1 if u it +(cid:12)(cid:1)x t +" it > K j j Unfortunately, deriving analytical results is challenging for the menu-cost model unless one is willing to make stringent assumptions (see Danziger [1999] and Gertler and Leahy [2008] for examples). However, the assumptions required for tractability seem less suitable here. Therefore, we will proceed by simulations to illustrate our main points. Note that the decision to change the price now depends on the value of (cid:12): All else equal, the larger the pass-through coe¢ cient, the more a shock to the exchange rate is likely to trigger a price adjustment. More generally, the more shocks are large and persistent (and thus associated with relatively large bene(cid:133)ts of adjusting the price), the more likely is a (cid:133)rm to change the price immediately. The estimated coe¢ cients in equation 1 are thus sensitive to the particular realization of the shocks in the menu-cost model. 2.2 Calibration of the baseline models To illustrate pass-through in our baseline models for various degrees of price rigidity, we (cid:133)rst set the mean, standard deviation, and autoregressive coe¢ cient of exchange rate innovations to match the corresponding moments of the broad dollar index computed by the Federal Reserve from January 1995 to March 2010. The standard deviation of monthly (end-ofperiod) exchange rate movements was 1:5 percent over that period, with no apparent drift. Exchange rate movements were slightly autocorrelated ((cid:26) = 0:19). We will report results for 13

(cid:12) = 0:3, which is in-line with recent estimates in the literature (e.g., Marazzi et al. [2005], Gopinath, Itskhoki, and Rigobon [2010]), but somewhat lower than the consensus value for pass-through in the 1980s (e.g., Goldberg and Knetter [1997]). The remaining parameters are calibrated to match salient features of individual import price adjustments. As shown by Gopinath and Itskhoki (2010), the median size of import price changes is insensitive to the frequency of price change changes, hovering between 6 and 7 percent. For the Calvo model, we set the probability of a price change equal to the frequency of interest then calibrate the variance of individual innovations to match a median size of price changes of 6:5 percent. For the menu-cost model, we choose the menu cost K and the standard deviation of " to match both the observed median size and the it average frequency of price changes of interest. We make the additional assumption that " it is normally distributed with mean zero.14 The larger is K, the less frequent and the larger are the individual price changes. Likewise, the larger is the standard deviation of " , the it more frequent and large are individual price changes. 2.3 Impulse response to an exchange rate movement The choice of a particular pricing model can have important consequences for the dynamic response of import prices. The upper, middle, and bottom panels of (cid:133)gure 2 show the response of import price in(cid:135)ation to an exchange rate movement in our baseline Calvo and menu-cost models for (steady-state) frequencies of price changes of 5 percent, 20 percent, and35 percent, respectively. Althoughthemodelssharethesameaveragefrequency, average size of adjustments, and long-run pass-through coe¢ cient, the transmission of an exchange rate shock is markedly faster in the menu-cost model than in the Calvo model.15 In the Calvo model, the frequency of price changes has a direct impact on the speed at which exchange rate disturbances are transmitted to the import price index. For a relatively low frequency (upper panel), the exchange rate movement is still di⁄using by the end of the forecast horizon. For a frequency of 20 percent (middle panel), the shock is almost entirely passed-through by the end of the forecast period. Higher frequencies lead to even faster pass-through. The speed of pass-through is markedly higher in the menu-cost model than in the Calvo model at any given frequency. Under our low-frequency calibration, there is 14We do not posit leptokurtic idiosyncratic shocks as Midrigan (2011) or Gertler and Leahy (2008), an assumption that would make pass-through in the menu-cost model more similar to that in the Calvo model. Whileleptokurticshockshelpimprovethe(cid:133)tofthedistributionofindividualpricechangesinthemenu-cost model, they work against (cid:133)nding fast pass-through, which is a feature that we want to explore later in our discussion. 15In practice, the frequency of price changes and the degree of exchange rate pass-through appear interrelated. Gopinath and Itskhoki (2010) present evidence that items with relatively low frequencies of price changes tend to be associated with relatively low rates of pass-through. 14

negligible amounts of import price in(cid:135)ation left one year after the shock, even for frequencies as low as 5 percent. 3 Selection e⁄ects in sample exits and entries We now alter the baseline models by assuming that the econometrician has only access to a index computed using a large sample of prices from the universe. Whereas the composition of the universe is constant over time, the composition of this index varies as items exit or enter the sample. Prices are collected at the end of the period after nominal adjustments, sample exits, and sample entries have taken place. Entering and exiting items thus cannot be used to compute in(cid:135)ation because either their past or their current price is not observed. Exits occur through two channels. First, items face a probability d of dropping out of the sample every period (the (cid:147)random exit(cid:148)channel). These exits are akin to the sample rotation performed by the BLS in that they do not depend on the amount of price pressure that has cumulated. Second, some exits are triggered by (cid:133)rms changing their prices (the (cid:147)selective exit(cid:148)channel). Conditional on its price being changed in the period, an item faces a probability e of exiting the sample. Such a situation could occur if, for example, a (cid:133)rm changed an items(cid:146)e⁄ective price by altering its characteristics and the price collector, rather than making the required hedonic adjustment, simply dropped the old item from the sample. Selective exits are not (cid:147)endogenous(cid:148)since the decision to exit is exogenous to (cid:133)rms. Nevertheless, they capture in a straightforward way the possibility that some exits partly censor the adjustment of the price index. In total, of all items present in the sample at the beginning of the period, a fraction s = d+(1 d)ef exits by the end of the period, where t t (cid:0) f is the true frequency of price changes among those items. t For convenience, we postulate that exiting items are replaced by an equal number of entering items, which is a rough approximation of the BLS(cid:146)practice over the past two decades. Entries also occur through two channels. A constant fraction 1 n of entering (cid:0) items are drawn at random from the universe of items (the (cid:147)random entry(cid:148)channel). The distributionofdeviationsfromtheoptimalprice, u , isthusthesamefortheseenteringitems it as that of the entire universe, with some fraction of entering items having their price reset duringtheperiod. Anotherfractionnofenteringitemsaresampledsystematically fromprice trajectories with a price change in the current period (the (cid:147)selective entry(cid:148)channel). Their pricealreadyre(cid:135)ectscurrentandpastmovementsintheexchangerateandotherdisturbances (i.e., u = 0). Similarly to the selective exit channel, the selective entry channel has the it feature that items tend to release their price pressure outside of the period spent in the sample. 15

As was the case earlier, it is convenient to (cid:133)rst consider a Calvo model with i:i:d: innovations to the exchange rate. We show in the appendix that the (plim) coe¢ cient on the l-th lag of the exchange rate is 1 e s(1 n) b = (cid:0) (1 d)l + (cid:0) 1 (1 d)l f (1 f)l(cid:12): (4) l 1 fe (cid:0) d (cid:0) (cid:0) (cid:0) (cid:18) (cid:0) (cid:19)(cid:18) (cid:16) (cid:17) (cid:19) The expression in front of (cid:12) is the probability that a price change occurs in period t and the previous price change took place over l periods earlier. Relative to equation 2, the above expression has two new terms , 1 e and (1 d)l + s(1 n) 1 (1 d)l , that capture the 1(cid:0)fe (cid:0) d (cid:0) (cid:0) (cid:0) (cid:0) biases associated with selective exits and selective entries.(cid:16) To gain so(cid:17)me intuition about these biases, it is useful to consider four canonical cases. Following our discussion of these canonical cases, we will then relate these cases with the theoretical work in Nakamura and Steinsson (2012). 3.1 All exits and entries are random When all exits and entries are random (i.e., s = d and n = 0), the (plim) coe¢ cients in the Calvo model with iid exchange rate innovations are identical to equation 2. In short, standard pass-through regressions are unbiased even though, every period, some fraction s = d (with d < 1) of items in the basket is replaced. Intuitively, usable items in the index havethesamedistributionofdeviationsfromtheoptimumpriceasitemsintheuniverse. For thesamereason, biasesareabsentinthemenu-costmodelorwhenexchangerateinnovations are autocorrelated. 3.2 All exits and entries are selective Consider now the case where both exits and entries are selective (i.e., s = fe and n = 1). Thissituationcouldarise, forexample, ifasurveyed(cid:133)rmweretoimplementane⁄ectiveprice change by altering an item(cid:146)s characteristics and the statistical agency, noting this alteration, were to censor the e⁄ective price change by treating the old version as a sample exit and the new version as an unrelated sample entry. When a fraction e of all price changes is censored in this way, the statistical agency is missing a roughly constant share of the index response 16

to shocks.16 In the Calvo model with iid exchange rate innovations, we have 1 e b = (cid:0) f (1 f)l(cid:12): (5) l 1 fe (cid:0) (cid:18) (cid:0) (cid:19) All coe¢ cients are downwardly biased by the same factor (1 e)=(1 fe) relative to the (cid:0) (cid:0) true response. Note that f ^ = (1 e) f is the frequency of price changes observed by the (cid:0) (1 fe) (cid:0) ^ econometrician, so that the estimated coe¢ cients are downwardly biased by a factor f=f. This bias can be large even when the exit rate is low because what crucially matters is the prevalence of exits among price changes rather than among all observations in the index. The left and right panels of (cid:133)gure 3 show the cumulative response of the price index to an exchange rate movement in the Calvo and menu-cost models, respectively, as a share of true long-run pass-through. In addition to n = 1, the (cid:133)gure shows the special case n = 0 (no selective exit), which we will consider shortly. We tentatively assume that a quarter of all price changes are accompanied by an exit, a proportion roughly equal to the median across three-digit Enduse categories of the worse-case probability of selective exit (0:28) that we estimate later in section 4.2. We leave the other parameters of unchanged relative to the baseline calibration described in section 2.2. In both models, the bias is important regardless of how frequently prices are updated. And as noted above, the censoring of price changes through exit reduces the frequency of price changes observed by the econometrician. For underlying frequencies of 5, 20 and 35 percent in the universe of items, the econometrician would observe frequencies of about 4, 16, and 29 percent, respectively, among usable observations in the index. 3.3 All exits are selective and all entries are random It can be challenging for statistical agencies to know if exits are selective or random as they have to press respondents for information about the circumstances in which exits take 16Thereisaninterestinganalogybetweenthiscaseofselectiveexitandselectiveentryandthewell-known (cid:147)quality-change bias(cid:148)by which statistical agencies have di¢ culties accounting for changes in quality from one vintage to the next. In both cases, part or all of an item(cid:146)s e⁄ective price adjustment is censored. The direction of the bias and the e⁄ects on measured pass-through di⁄er between the two situations, however. Mismeasured changes in quality can result in either systematic upward or systematic downward biases in in(cid:135)ation,dependingonwhetherthequalitychangeisunderestimated(e.g.,ignoringimprovementsinacomputer(cid:146)sprocessingpower)oroverestimated(e.g., failingtoaccountfortheuseoflower-qualitycomponents). In practice, the quality change bias is associated with a systematic overestimation of in(cid:135)ation (see BLS [1997b] and Bils [2010]). By contrast, in our canonical case with both selective exits and selective entries, only a fraction of the aggregate price adjustment is recorded, so that in(cid:135)ation is underestimated when it is positive and overestimated when it is negative. As a consequence, pass-through is underestimated, which needsnotbethecaseifin(cid:135)ationissystematicallymismeasuredinaparticulardirectionduetomismeasured quality. 17

place. Statistical agencies have more leeway to avoid selection biases in the entry of items in the basket since, in principle, they can design the sampling procedure to randomly select observations from the universe of items. The special case we now consider assumes that all exits are selective while all entries are random (i.e., s = fe and n = 0). Although we model the sample exit decision as exogenous to the (cid:133)rm, this case captures the essence of (cid:147)endogenous exits(cid:148)problems: Item exits tend to be associated with unobserved price adjustments, so that the price index response to shocks is underestimated.17 When (cid:133)rms choosethetimingofpricechanges,asinourmenu-costmodel,exitsalsotendtobeassociated with relatively large deviations of individual prices from their optimum. Starting again with the Calvo model with uncorrelated exchange rate innovations, we have 1 e b = (cid:0) (1+lfe)f (1 f)l(cid:12): (6) l 1 fe (cid:0) (cid:18) (cid:0) (cid:19) The size of the bias now depends on the relative strength of two opposing forces. On the one hand,selectiveexitscensorpriceadjustments,thusdampeningtheresponseofthepriceindex to past exchange rate movements. This force is represented by the term (1 e)=(1 fe), (cid:0) (cid:0) whichweencounteredearlier. Ontheotherhand, exitsalsocreateopportunitiestointroduce items in the index whose price has not changed for some time. This possibility subsequently makes the price level more responsive to past exchange rate movements. This second force is captured by 1 + lfe. For short lags, the downward bias is the predominant force. In ^ particular, the initial response of the index, b = f(cid:12), is always downwardly biased. As we 0 increase the number of lags, (1+lfe) grows linearly to any arbitrarily large number, so that individual coe¢ cients become upwardly biased at su¢ ciently long lags. Nevertheless, the cumulative index response remains downwardly biased because the coe¢ cients converge more rapidly to zero.18 As shown in the left panels of (cid:133)gure 3, assuming that exiting items are replaced by sampling at random from the population (n = 0) reduces the size of the bias noticeably over the forecast horizon in the Calvo model relative to the case in which entries are selective (n = 1). For frequencies of about 20 percent, the estimated two-year cumulative response is nearly the same as the true one. The randomization of entries mitigates the bias from 17Greenlees and McClelland (2011) o⁄er evidence that exiting food items in the CPI have larger price changes (including zeros) on average than continuing items. They also argue that current CPI techniques may overestimate the extent of quality adjustments for these categories. 18When all exits are selective and all entries are random, the long-run response of the index is given by 1 (cid:0) fe (cid:0) (1 (cid:0) f)e2 (cid:12) < (cid:12). The same conclusion applies to the long-run response in the general case, 1 fe (cid:0) 1 1 (cid:0)f e e f+(d+(1 (cid:0) f d + )f d e) f (1 d (cid:0) n)(1 (cid:0) f) (cid:12). This response is smaller than (cid:12) and increasing in the fraction of en- (cid:0) (cid:0) t(cid:16)ering(cid:17)it(cid:16)ems randomly sampled (cid:17)from the universe. Randomizing entries thus reduces the downward bias without eliminating it entirely. 18

selective exits because some of the entering items have not had a price change in a while, making them responsive to past exchange rate movements. As our (cid:133)gure illustrates, this counterbalancing e⁄ect can be quite large, o⁄setting much of the bias by the end of typical forecast horizons. The gains from resampling at random are more modest in the menu-cost model (right panels) because pass-through is very rapid. As hinted in equation 6, the counterbalancing e⁄ect of randomsubstitutions grows with the number of lags, l, but since coe¢ cients are tiny after a small number of lags, the ultimate impact on cumulative pass-through is modest. 3.4 All exits are random and all entries are selective We next turn our attention to the case in which all exits are random and all entries are selective (s = d and n = 1). This case is similar in spirit to that described by Weber (2012), in which the statistical agency only observes new goods with a lag and hence misses price changesoccurringshortlybeforeanitem(cid:146)sinitiationinthesample. Undertheseassumptions, the (plim) coe¢ cient on the l th lag of exchange innovations in our baseline Calvo model (cid:0) is b = (1 s)lf (1 f)l(cid:12): (7) l (cid:0) (cid:0) This expression has a very intuitive interpretation. For a movement in the exchange rate l periods ago to contribute to in(cid:135)ation in the current period, one must observe a price change in the current period (probability f) and no price change or substitution in the previous l periods (constant probability 1 f and 1 s, respectively, each period). Price changes and (cid:0) (cid:0) substitutionsfromperiodt l tot 1resultinpostedpricesthatalreadyre(cid:135)ectmovementsin (cid:0) (cid:0) the exchange rate at period t l. Relative to equation 2, the above expression is downwardly (cid:0) biased by a factor (1 s)l. (cid:0) A few comments are worth making. First, the nature of the downward bias is that items entering the basket systematically are less sensitive to past movements in the exchange rate than items in general. Including these entering items in pass-through regressions thus lowers estimated pass-through rates. Second, in the special case of l = 0, we have b = f(cid:12); the 0 estimatedinitial impactof anexchangeratemovementonthepriceindexisalwaysunbiased. We also note that the share of the true coe¢ cient correctly measured decays exponentially with the number of lags considered. The importance of the bias as a share of the cumulative response thus grows over time, with estimates of the short-run cumulative response being less biased than estimates of the long-run response. Third, as stressed by Nakamura and Steinsson (2012), the long-run bias under Calvo pricing is most important for product categories with very low frequency of price changes. 19

The left panels of (cid:133)gure 4 illustrate the bias over the policy-relevant horizon by plotting the cumulativecontributionofthecoe¢ cients. Asseeninthe(cid:133)gure, thebiasincreasesinseverity with the degree of price stickiness. Only two thirds of the actual cumulative pass-through is correctly estimated at a two-year horizon when the frequency of price changes is 5 percent, and almost one (cid:133)fth is still missing when the frequency is 20 percent. For a frequency of 35 percent, the econometrician captures more than 95 percent of the true response over the forecast horizon. Under Calvo pricing, only (1 s)l of the contribution of lag l to pass- (cid:0) through is correctly estimated. This term typically is decreasing at a slow rate because s is small in practice, meaning that the downward bias kicks in most strongly when much of the exchange rate response occurs at long lags. Under low frequencies of price changes, the coe¢ cients associated with long lags in the Calvo model account for a substantial share of the long-run price response, so that the bias can become large over long horizons. More generally, the size of the bias appears to be related to the speed at which the price index responds to an exchange rate shock. The right panels of (cid:133)gure 4 show the estimated cumulative contribution of the regression coe¢ cients on the various lags of exchange rate movements (the dark-shaded bars), along with the bias left out by the econometrician (the light-shaded bars), under menu-cost pricing. The bias is much less severe than under Calvo pricing. Even for frequencies of price changes as low as 5 percent (upper-left panel), the econometrician captures almost 90 percent of the price index response at the two-year horizon. In the menu-cost model, most of the long-run pass-through occurs in the (cid:133)rst few periods following a shock (cid:150)even at low frequencies (cid:150)so that the bias does not have time to cumulate to something large. Finally, our (cid:133)gure depicts the worst-case assumption that all entries are selective (n = 1). As noted in section 3.1, there would be no bias if price collectors were replacing exiting items by observations randomly selected from the population (n = 0). In the more general case where all exits are random and a fraction n of entries are selective, for the Calvo model, (plim) coe¢ cients are a linear combination of the true coe¢ cient and the biased coe¢ cient under our canonical case, b = 1 n+n(1 s)l f (1 f)l: l (cid:0) (cid:0) (cid:0) (cid:16) (cid:17) Given this expression, we have that long-run pass-through is f (cid:12)(1 n)+n (cid:12): (cid:0) f +s fs (cid:0) Departing from the extreme case of n = 1 can substantially reduce the size of the bias. As 20

a rule of thumb, the bias reduction achieved by the end of our forecast horizon is roughly proportional to 1 n, so that, for example, randomizing half of the entries would roughly (cid:0) halve the area represented by the light bars. 3.5 Comparing our four canonical cases with Nakamura and Steinsson (2012) Having laid out these four canonical cases, we now discuss how the selection biases in our environment relate to the product replacement bias explored by Nakamura and Steinsson (2012). Those authors derive correction factors that apply to pass-through over the in(cid:133)nite horizon under the assumption of Calvo pricing. Our paper extends their analysis in three ways: by emphasizing dynamics, by explicitly considering selection biases in sample exit (Nakamura and Steinsson do not spell out their assumptions about exit), and by exploring both Calvo and menu-cost pricing. Nakamura and Steinsson (2012) proposition 1 relates observed pass-through to true passthrough using the following equation (expressed here using our notation), ~ 1 f b = [(cid:8)(1+(cid:11))(cid:12) +(1 (cid:8))(cid:12)]; l f ~ +s f ~ s (cid:0) X l=0 (cid:0) where (cid:8) denotes the fraction of all price changes that are (cid:133)rst observed price changes. The parameter (cid:11) is a factor governing the extent of overreaction of the (cid:133)rst versus subsequent observedprice changes afteraninnovationinthe exchange rate. This parameteris estimated rather than derived fromthe axioms of their theoretical model. To derive an expression for (cid:11) requires being explicit about the nature of exits and the frequency of random entry. As such, we compare the model-derived cases in our paper to the model-derived cases in Nakamura and Steinsson (2012) (i.e., where (cid:11) equals zero). For the in(cid:133)nite-horizon case, there is a direct connection between Nakamura and Steinsson(cid:146)s (2012) long-run correction factors and those in our environment under Calvo pricing. When all entries are selective (n = 1), we can express measured long-run pass-through as ~ ~ ~ ~ f= f +s fs (cid:12), where f is the observed frequency of price changes in the price index. (cid:0) Th(cid:16)is expression(cid:17)implies the same long-run correction factor, f ~ +s f ~ s =f ~ , derived by (cid:0) Nakamura and Steinsson (2012). Further, over the in(cid:133)nite h(cid:16)orizon, tha(cid:17)t correction factor is independent of whether exits are random or selective.19 The in(cid:133)nite horizon case is 19Theexpressionf~= f~+s f~s (cid:12) forlong-runmeasuredpass-throughisperhapsmosteasilyderivedfor (cid:0) our random exit-selecti(cid:16)ve entry cas(cid:17)e by setting f~=f in equation 7 and summing over all lags. One obtains the same expression in our selective exit-selective entry case by using s=fe and f~=(1 e)f=(1 fe) to (cid:0) (cid:0) 21

special, however. As the parsing of our canonical cases showed, the size of biases over (cid:133)nite horizons can vary greatly with the severity of selection biases in exits. For example, when all entries are selective, the initial response of the price index to a shock is unbiased when exits are random (with b = f(cid:12)) but biased downward when exits are selective (with 0 b = (1 f)=(1 fe)f(cid:12)). 0 (cid:0) (cid:0) Another reason for being explicit about item exit assumptions is that the degree of selective exit a⁄ects the bias to measured pass-through at all horizons when some entries are random. Our framework has the advantage of allowing us to easily reconstitute the bias to short and long-run pass-through with Calvo pricing, by summing up the coe¢ cients given by our general equation 4. Doing so shows that the size of the bias at both short and long horizons is a function of the severity of selection bias in sample exit (e) whenever n < 1. Asshownbyourcomparisonofthemenu-costandCalvomodels, thenatureofthepricing frictions can be quite important in determining the size of the biases. In particular, models with otherwise identical true frequencies of price changes can di⁄er greatly in the severity of the biases. For example, in our random exit-selective entry case, holding constant the observed frequency and exit rate, the menu-cost model had a much smaller bias by the end of our forecast horizon than the Calvo model. What mattered in that case was the speed of pass-through rather than the frequency of price changes per se. In sharp contrast, in our selective exit-selective entry case, the Calvo and menu-cost models with identical exit rates and true frequency of price changes have similar bias. Summing up, our conceptual analysis shows the importance of understanding the nature ofboth exitsandentries,aswellastherelevantpricingfrictions,injudgingoftheimplications for measured pass-through of selection biases in sample turnover. Our section 4 on the empirical relevance of the biases addresses each of those aspects. 3.6 Robustness to the presence of real rigidities Gopinath and Itskhoki (2011) (cid:133)nd that movements in the exchange rate are passed-through to import prices over more than one price adjustment, consistent with the presence of real rigiditiesslowingthedi⁄usionoftheshocktoresetprices. Themodeladoptedsofarabstracts from this possibility. However, we show in appendix B that the inclusion of real rigidities has a negligible impact on the theoretical correction factors one should apply to standard pass-through estimates. In particular, we prove that the correction factors for the initial and the long-run responses are independent of real rigidities in the Calvo model, and then show that the factors are overall insensitive to real rigidities at intermediate horizons. eliminate e and f in equation 5. We thank Emi Nakamura and Jon Steinsson for pointing out this latter link to us. 22

Appendix B also shows that assuming the presence of real rigidities can alter one(cid:146)s judgmentregardingtheempiricalrelevanceoftheCalvoversusthemenu-costmodels. Asweshall see shortly, the empirical impulse response of import prices to an exchange rate movement is consistent with features of both models, in particular the initially rapid response of the index predicted by the menu-cost model, and the continued pass-through over medium-term horizons predicted by the Calvo model. The assumption of real rigidities slows predicted pass-through, which makes it more challenging for the Calvo model to match the initial import price index response to an exchange rate movement. 4 Empirical relevance of selective exits and entries In order to assess the impact of selective exits and selective entries on standard estimates of exchange rate pass-through, one needs to form a view on several objects that are not directly observed, namely the type of pricing frictions giving rise to infrequent nominal adjustments, the extent of price change censoring through exits (e), and the prevalence of entries whose prices are relatively unresponsive to past exchange rate movements (n). In this section, we (cid:133)rst argue that standard estimates of the import price response to exchange rate movements mix features of both the menu-cost and the Calvo models. We next simulate the models to derive bounds on the size of the biases over our forecast horizon. Finally, we present an alternative index construction method to purge standard pass-through estimates of much of the bias induced by selective entry. 4.1 Dynamic transmission of exchange rate shocks: data versus models We focus our empirical analysis on (cid:133)nished goods categories, which account for about 60 percent of the total value of U.S. imports. They comprise automotive products, consumer goods, and capital goods. We leave aside fuel and material-intensive goods because the problems associated with selective exits and selective entries appear relatively benign for those categories given that (i) they are relatively homogeneous products, (ii) they tend to be tradedbetweenalargenumberofbuyersandsellers, and(iii)theirpricescanoftenbereadily observed in electronic trading platforms. In fact, the IPP obtains its crude oil import prices from a source outside of the sampling universe we observe for this paper, which altogether precludes an empirical discussion of exit and entry in that important category. Finally, for an economy as large as the United States, exchange rate movements and the price of fuel and material-intensive categories are arguably simultaneously determined to some degree, 23

which raises additional econometric issues. Our estimation period begins in January 1994 and ends in March 2010. For each threedigit Enduse category (indexed by i), we construct a trade-weighted nominal exchange rate, NEER , and foreign producer price in(cid:135)ation, (cid:25) . We then estimate by ordinary least i;t i(cid:3);t squares the following equation, 24 24 (cid:25) = (cid:11)+ b (cid:1)NEER + c (cid:25) +" : i;t i;l i;t l i;l i(cid:3);t l i;t (cid:0) (cid:0) l=0 l=0 X X The number of lags is greater than is typically used in empirical pass-through literature. However, given the simulation results reported earlier, the additional lags seem to be an appropriate choice for robustness. The estimated impulse responses to a 1-percent depreciation of the U.S. dollar are presented in (cid:133)gure 5. The largest responses are found for machinery and equipment categories (Enduse 210, 211, 212, and 215), and, especially, for computers and semi-conductors (Enduse 213). Incidentally, this last category is also one for which the BLS makes special e⁄orts to hedonically adjust prices. By contrast, some categories show little if any pass-through over our two-year horizon, notably automobiles and other vehicles (Enduse 300 and 301), apparel (Enduse 400), and home entertainment equipment (Enduse 412). Tocomputearesponsefor(cid:133)nishedgoods, weaggregateourthree-digitcategoryresponses using 2006 trade weights. As shown in the lower-left panel, (cid:133)nished goods prices climb more than 0.1 percentage point in the (cid:133)rst two months following a 1-percent exchange rate depreciation, another 0.1 percentage point over the remainder the (cid:133)rst year, and a more modest 0.05 percentage point over the course of the second year. For reference, we also report results for when we regress the price index for (cid:133)nished goods on the exchange rate (the dashed line in the lower-left panel).20 The responses are similar, although as indicated by the 95-percent con(cid:133)dence interval around the estimated responses for the (cid:133)nished goods price index (the gray area), the aggregated response is somewhat faster than the response estimated for the aggregate price index. The shape of the impulse response shares aspects of both the menu-cost and Calvo models. The initially rapid response is qualitatively similar to that in the menu-cost model, whereas the ensuing slow but steady increase is more akin to the protracted response in the Calvo model.21 20Our simple speci(cid:133)cation does not allow for variation in the magnitude of the response over time, a restriction imposed in part due to the short period over which monthly import price data are available. Taking advantage of the longer time coverage of quarterly series, several authors have documented a decline in pass-through rates in recent decades (e.g., Marazzi et al. [2005]), including for (cid:133)nished goods (see Gust et al. [2010]). As mentioned earlier, we (cid:133)nd little evidence that an increase in the occurrence of selective exits and entries could account for that pattern. 21Rather than using a mixture of Calvo and menu-cost models, an alternative approach would be to 24

We next compare the empirical response in each three-digit Enduse category to those generated by our baseline Calvo and menu-cost models. As section 3 showed, the pricing models have di⁄erent implications for the size of the correction factors. As such, in deriving empiricalboundsinthenextsection,wewanttouseourmodel-derivedresponsestomakethe bestapproximationofthericherobservedempiricalresponses. Figure6directlycomparesthe empirical responses in each three-digit Enduse category to those generated by the Calvo and menu-cost models. The models are calibrated to match category-level statistics as outlined in section 2.2, with the minor di⁄erence that we seek to match the observed cumulative rate of pass-though in the last quarter of the forecast horizon rather than some illustrative longrun value. Figure 6 also shows the linear combinations of model responses that minimize the Euclidian distance with the empirical response over the forecast horizon. Again, we (cid:133)nd support for both models, with some Enduse categories clearly preferring one model over the other, and others being best represented by a mixture of the two models. On average, each model is attributed about half of the weight.22 To be clear, this exercise is not an attempt to make a de(cid:133)nitive case of the relative merits of Calvo versus menu-cost models. Rather, the exercise is to estimate the best approximation of the richer observed empirical responses given our model-derived responses. In the next section, we use these approximations in calculating the bounds that are most appropriate given the observed responses. 4.2 Bounding standard pass-through estimates To assess the quantitative importance of selective exits and entries, our next strategy is to derive three sets of bounds on the amount of exchange rate pass-through over the policy horizon. These bounds are related to the canonical cases discussed in sections 3.2 to 3.3, depending on whether we consider, respectively, the largest plausible number of selective exits and entries consistent with the data, the largest plausible number of selective exits in the presence of random entries, or the largest plausible number of selective entries in the presence of random exits. Our worst case of selective exits assumes that all out-of-scope exits mask a price change. We treat exits for other reasons (as de(cid:133)ned in table 1) as random because they typically are planned years in advance by the BLS and thus unlikely to be related to individual pricing decisions. Under these assumptions, we observe the rate of random exit, d, and the rate of selective exits, ef (1 d), as they correspond to the rate of out-of-scope and other exits (cid:0) approximate the responses using heterogeneous Calvo models. However, the empirical evidence in Gagnon, L(cid:243)pez-Salido, and Vincent (2012) favors rules that are not purely time dependent. 22Thoughthemodelresponsesdisplayedassumenoselectione⁄ects,this(cid:133)ndingisrobusttoassumingany degree of selective exits or selective entries in the calibration. 25

shown in table 1. Knowledge of these rates and of the observed frequency of price changes, (1 e)f=(1 fe), is su¢ cient to identify d, e, and f in the model. Our worst case of (cid:0) (cid:0) selective entries occurs when all items added to the sample experience an unobserved price change upon entry (i.e., n = 1). Given the sensitivity of biases to pricing assumptions, we derive our bounds under both Calvo and menu costs. Under Calvo, we compute the correction factors for the estimated cumulative response to an exchange rate movement directly from the analytical expressions for the estimated coe¢ cients, shown in equations 5 to 7. Under menu costs, no such expressions are available, so we compute the corresponding correction factors through simulations. In particular, for each three-digit Enduse category, we select (cid:27) , K, and (cid:12) to match the " observed frequency of price changes, the average absolute size of price changes, and the cumulative amount of pass-through by the last quarter of the forecast horizon following a 1-percent depreciation of the dollar. We then apply the correction factors to the estimated responses and aggregate them using 2006 trade shares to derive our bounds on the response of (cid:133)nished goods. Our worst case of selective exits and entries is shown in the upper-left panel of (cid:133)gure 7. The estimated (cid:133)nished goods price response to a 1-percent depreciation of the dollar is 0:24 percent by the last quarter of the forecast horizon. After correcting for selective exits and selective entries, the (cid:133)gure for the menu-cost and Calvo models are in a similar range at 0:30 percent and 0:32 percent, respectively. If we instead assume that all entries are random (upper-right panel), then the corrected estimate in the last quarter falls to at most 0:26 percent under Calvo, and to at most 0:28 percent under menu costs. The bias is larger under menu costs in this case because the bene(cid:133)ts from randomizing entries are largest when pass-through is slow, as in the Calvo model.23 Under our worst case of selective entry and random exit (lower-left panel), the corrected response is very close to the actual response undermenucosts,butremainssomewhathigher(0:30percent)thantheuncorrectedestimate (0:24 percent) by the last quarter of the forecast horizon. Given our earlier evidence that features of both models are present in the data, we derive afourthsetofboundsunderwhatweviewasmoreplausiblepricingassumptions. Thethreedigit Enduse responses of the Calvo and menu-cost models are (cid:133)rst weighted according to the linear combination that provides the best (cid:133)t of the empirical response, as we did in the previoussection, andthenaggregatedusing2006tradeshares. Wepositthatallout-of-scope exits correspond to selective exits and selective entries, whereas all other exits are associated withrandomexitsandrandomentries, sothatabouthalfofallexitsandentriesareselective. 23We assume that individual (nonzero) price changes leading to exits have the same distribution as those for items remaining in the sample. Whether this is the case in practice remains an open question. 26

The corrected cumulative response under this last set of assumptions is presented in the lower-right panel of (cid:133)gure 7. Following a 1-percent depreciation of the dollar, the corrected cumulative response by the last quarter (0:28 percent) is above the estimated one (0:24 percent). Selective exits and selective entries contribute roughly equally to this di⁄erence, as hinted by the special case with only selective entries that is also displayed in the panel. Summing up, our bound analysis suggests that, even under the stringent assumptions consistent with the microdata, the biases induced by selective exits and selective entries have a limited impact on standard pass-through estimates for (cid:133)nished goods over typical forecast horizons. Given that this conclusion contrasts, at least on the surface, with Nakamura and Steinsson(cid:146)s (2012) assertion that the product replacement bias is quantitatively important, we next discuss how our respective (cid:133)ndings can be reconciled. 4.3 Reconciling our empirical results with those of Nakamura and Steinsson (2012) Wehighlightfourdi⁄erencesbetweenourempiricalworkandthatofNakamuraandSteinsson (2012) informing our conclusion that downward biases due to item replacement are not as severe as asserted by these authors: the price index of interest, the time horizon of interest, theassumeddistributionofheterogeneityinpricingdecisions,andtheextentoftheselectivity in exits and entries. To facilitate comparisons, we begin by estimating an equation for empirical pass-through similar to equation 1 in their paper, 6 (cid:1)pm (cid:1)pcpi = a+ b (cid:1)s +" : (8) t (cid:0) t l t (cid:0) l t l=0 X We regress the di⁄erence between changes in the log of quarterly import prices excluding oil, (cid:1)pm, and changes in the log of the CPI excluding food and energy, (cid:1)pcpi, on a constant as t t well as the contemporaneous and (cid:133)rst 6 lags of the change in the log of the Federal Reserve(cid:146)s trade-weighted major currencies real exchange rate index, (cid:1)s . Our estimation period is t l (cid:0) 1995:Q1 to 2010:Q4 and the import price index is the de(cid:135)ator from the National Economic Accounts published by the U.S. Bureau of Economic Analysis. As table 2 shows, our estimated import price response after 6 quarters is 0:41, a (cid:133)gure nearly identical to that reported by Nakamura and Steinsson (0:43) for the same horizon. Table 2 also presents standard pass-through estimates after 6 quarters for two special groups of products: (cid:133)nished goods (Enduse categories in the 200s, 300s, and 400s) and material-intensive goods excluding oil (Enduse categories below 200, excluding 100). The rate of pass-through after 6 quarters is much lower for (cid:133)nished goods prices (0:26) than for 27

material-intensivegoodsprices(0:93),highlightingthetwogroups(cid:146)sharplydi⁄erentmediumterm responses to exchange rate movements. The high pass-through estimate for materialintensive goods is perhaps not surprising given these items(cid:146)frequent price adjustments, their high commodity content, and the strong comovement between commodity prices and the exchange rate. With pass-through nearly complete after 6 quarters and with their prices being frequently adjusted, it is unlikely that material-intensive goods are plagued by severe downward biases in measured pass-through due to itemturnover, further justifying our focus on (cid:133)nished goods in earlier sections. We momentarily assume that true long-run pass-through, (cid:12), is the same for all items in the IPP sample, leaving only the frequency of price changes free to vary across items. The assumption of identical long-run pass-through rates across product categories is admittedly unappealing given our evidence above for (cid:133)nished goods and material-intensive goods, but it makes it easier to illustrate the importance of other aspects of the data in comparing our (cid:133)ndings with those of Nakamura and Steinsson (2012).24 We set the monthly rate of item substitution for all Enduse categories to 2:5 percent, a (cid:133)gure in the range of rates reported in table 1.25 For now, we also maintain our assumption that items within each three-digit Enduse categories share the same frequency of price changes. Finally, we posit that the data are generated by a Calvo pricing model. We obtain corrected pass-through estimates for each of our canonical cases by applying a correction factor, (cid:3), to the sum of estimated pass-through coe¢ cients from equation 8. Rather than reporting solely long-run correction factors, as Nakamura and Steinsson (2012) do, we also compute correction factors pertaining to a 6-quarter horizon (i.e., an 18-month horizon). The correction factor for measured pass-through after L months is given by L bbiased (cid:3)(L) 1 = w l=0 l;i : (cid:0) i i P L l=0 bu l; n i biased ! X P The right-hand side expression is the weighted share of true pass-through after L months measuredbytheeconometrician,wherew istherelativeweightofEnduseiintheindex. The i termbunbiased is the theoretical unbiased pass-throughcoe¢ cient on the l th lag of exchange l;i (cid:0) rate movements for Enduse i (given by equation 2). The term bbiased is the corresponding l;i biased coe¢ cient in the presence of selective exits and selective entries (given by equation 4 for the general case). The only remaining step for computing (cid:3)(L) is to re-express bunbiased l;i 24We note that Nakamura and Steinsson (2012) also consider the e⁄ect of cross-sectional heterogeneity in pass-through. 25Allowingthesubstitutionratetovaryacross3-digitEndusecategoriesbasedontheentryratesdisplayed in table 1 has only a minor impact on our (cid:133)ndings. 28

and bbiased in terms of observables(cid:151)namely f ^ and s (cid:151)by replacing d, e, and f in equations l;i i i 2 and 4 as needed. See each of our canonical cases for the mapping. Table2reportscorrectedpass-throughratesforourfourcanonicalcasesaftersix-quarters and in the long-run. When all exits and entries are random, both (cid:3)(18) and (cid:3)( ) are equal 1 to one, so that no correction is required. When all exits and entries are selective, (cid:3)(18) and (cid:3)( ) are equal and corrected pass-through for all imported goods is largest at 0:55. For 1 the other two canonical cases, (cid:3)(18) and (cid:3)( ) are not equal. When exits are selective 1 and entries are random, (cid:3)(18) is larger than (cid:3)( ), and as such corrected pass-through 1 declines from 0:48 to 0:45 as the horizon increases from 6 quarters to in(cid:133)nity. In reverse, when exits are random and entries are selective, (cid:3)(18) is less than (cid:3)( ), and corrected 1 pass-through increases from 0:48 to 0:55 as the horizon lengthens to in(cid:133)nity. Hence, one important di⁄erence between our results and those of Nakamura and Steinsson (2012) is that we report correction factors consistent with di⁄erent time horizons and with di⁄erent explicit assumptions about the selectivity of exit and entry. As noted above, because prices for material-intensive goods prices are frequently adjusted, there is little downward bias attributable to item replacement. Indeed, for all biased cases, the theoretical correction factors for material-intensive goods are quite small over the horizons considered. We also reiterate that statistical agencies can signi(cid:133)cantly mitigate downward biases by following sampling procedures that randomize entries. In particular, the long-run correction factor for all goods shrinks from 1:2 under the worst case of selective exits and selective entries to only 1:08 when entries are randomized. Another important di⁄erence between our analysis and that of Nakamura and Steinsson is the treatment of heterogeneity in the frequency of price changes across items. Throughout ourpaper, wehaveassumedthatthefrequencyofpricechangesisconstantwithinthree-digit Endusecategories,otherwiseleavingthefrequencyfreetovaryacrosssectorsofactivity. This approachmayleavesomeheterogeneityunaccountedforwithinthree-digitEndusecategories, which could lead us to underestimate the magnitude of downward biases. Nakamura and Steinsson(2012)insteadcontrolforheterogeneityacross(cid:133)rmsbyassumingthattheobserved frequency of price changes in the index is distributed according to a Beta distribution, which they estimate on observed individual frequencies. To contrast the two approaches, we adopt their parametrization of the Beta distribution (using a = 0:44 and b = 3:50 as parameters) andcomputethelong-runcorrectionfactorsundertheassumptionofselectiveentries. (Recall from our discussion in section 3.5 that long-run pass-through under Calvo pricing does not depend on the prevalence of selective exits when all entries are selective.) As we change our distributional assumptions, the pass-through estimate for imported goods leaps from 0:55 to 0:70. This large correction is driven by a mass of observations at very low frequencies 29

of price changes. One salient feature of the Beta distribution estimated by Nakamura and Steinsson (2012) is the implication that a third of all observations have an observed monthly frequency below 2 percent, with the density of observations becoming arbitrarily large as the observed frequency approaches zero.26 Or, the long-run correction factor under selective ^ ^ ^ entries, f +s fs =f, is quite sensitive to the presence of very low frequencies because it (cid:0) becomes(cid:16)arbitrarily(cid:17)large as f ^ approaches zero.27 To illustrate the sensitivity of the estimated long-run response to the mass of (cid:133)rms updating prices infrequently, we move all observations with a frequency below 2 percent to a mass point at 2 percent. The correction factor for the long-run response drops from 1:70 to 1:47. This decline should caution one against driving strong conclusions that depend on the very low frequencies of price changes, whose density is arguably di¢ cult to estimate. In particular, the Beta distribution has only two parameters to capture the whole range of variation in the density of observed frequencies in the sample. It is thus conceivable that some of the very low frequencies implied by the calibration are o⁄the empirical support. Althoughthecorrectionfactorspresentedintable2areusefulforexploringthesensitivity ofthebiasestoalternativeassumptions, weshallstressinconcludingthiscomparisonthatwe see most of themas implausiblylarge empiricallybecause theyassume rates of selective exits and selective entries that are overly severe. As we argued in section 1, not all exits appear to be selective; the substitution rate of 2:5 percent used our table would halved if we were using only out-of-scope exits as our measure of selective exits. Similarly, we found limited evidence insection1.2thattheBLSmethodologyisconducivetosystematicallyaddingitemsrecently repriced to the IPP sample. Regardless of the nature of exit, the correction factors shrink greatly as we lower the value of n, the fraction of entries that are selective. For example, under the assumption of random exit and assuming that 50 percent of entries are selective, then the correction factor is only 1.26 under Nakamura and Steinsson(cid:146)s Beta distribution. As noted above, the randomization of entries can reduce downward biases much even in the presence of strong selectivity in exits. In short, we see the relatively tight bounds derived in section 4.2 as more indicative of the range of empirically plausible corrected pass-through estimates. 26Whether this parametrization of the distribution of observed individual frequencies also implies that the distribution of actual frequencies of price changes has a large mass near zero depends on one(cid:146)s views about the degree of selectivity in exits. In particular, if all exits are random, then the two distributions are identical, so that many price-setters rarely adjust prices. Nakamura and Steinsson (2012) are not explicit about their assumptions on this point. 27Inanappendix,NakamuraandSteinsson(2012)alsoestimateBetadistributionsseparatelyfor15sectors. While this helps capture sectoral heterogenity, it is still the case in each sector that the slope of the Beta distribution approaches in(cid:133)nity as the frequency of price changes falls to zero, which tends to boost the imputed aggregate correction factor. 30

4.4 Reducing biases through delayed entries If the estimates were subject solely to a bias fromselective entry, then one could use a simple trick to remove much of that bias over the policy-relevant horizon. Recall that selective entry arises because entering items are systematically less responsive to past exchange rate movements than items in the universe. Therefore, simply delaying the entry of substitutes into the index should reduce this bias. We show in the appendix that, when all exits are random and all entries are selective, the estimated (plim) coe¢ cients in a Calvo model with an arbitrary M-period entry delay are given by f (1 f)l(cid:12) if l M b = (cid:0) (cid:20) : l ( (1 s)l (cid:0) M f (1 f)l(cid:12) if l > M (cid:0) (cid:0) Delaying entries thus eliminates the bias due to selective entry for the coe¢ cients associated with the (cid:133)rst M lags of exchange rate movements. The bias on subsequent lags is also reduced, with b representing a fraction (1 s)l M of the true response when entries are l (cid:0) (cid:0) delayed by M periods, compared to only (1 s)l when there is no entry delay. (cid:0) The left panels of (cid:133)gure 8 show that delaying entries by 6 months can go a long way in correcting measured pass-through over policy-relevant horizons in the Calvo model when the only selection bias is in sample entry. The bias at the end of the horizon is negligible when prices are adjusted 20 percent of the time or more. Even at frequencies as low as 5 percent, the prediction over the (cid:133)rst year of the forecast su⁄ers little bias, while the accuracy of the response in the second year is greatly improved. The bias reduction is even larger in the menu-cost model (right panels). Delaying entries by 6 months virtually eliminates the downward bias at all frequencies considered. The consistency gains are especially large in the menu-cost model because delaying entries corrects most e⁄ectively biases associated with short lags of the exchange rate, which account for the bulk of the price level response. The delay can also improve accuracy under our most biased canonical case with both selective exit and selective entry. In such a case, the delay again acts as a mechanism to randomize entry, which, as we illustrated in section 4.3, can go a long way in reducing the bias. Our simulations (not shown) indicate that the bias reduction achieved is sensitive to the true frequency of price changes and the length of the delay. For example, the bias is reduced by about 20 percent when the true frequency of price changes is 5 percent and the delay is 9 months, but by only half as much when the delay is 6 months. If the frequency were instead 15 percent, then a 9-month delay would eliminate over half of the bias. It turns out that our trick of delaying entries can also mitigate biases when the only source of bias is selective exit. We show in the appendix for the general case with arbitrary 31

degrees of selective exit and selective entry that the (plim) regression coe¢ cients under an M-period entry delay in the Calvo model are (1 e) f (1 f)l(cid:12) if l M (cid:0) b (M) = (1 fe)l+1 (cid:0) (cid:20) : l 8 (1 e) (1 d)l M + (cid:0) s(1 n) 1 (1 d)l M f (1 f)l(cid:12) if l > M (cid:0) (cid:0) (cid:0) (cid:0) < (1 fe)M+1 (cid:0) d (cid:0) (cid:0) (cid:0) (cid:0) (cid:16) (cid:16) (cid:17)(cid:17) We also prov:e that the bias diminishes as one increases the entry delay given any forecast horizon. As one delays entries by an arbitrary large number of periods, we have 1 1 (1 e) lim b (M) = (cid:0) f (1 f)l(cid:12) = (cid:12): M l (1 fe)l+1 (cid:0) !1 X l=0 X l=0 (cid:0) In short, the estimated long-run pass-through in the Calvo model is unbiased in the limit, a result that holds whether exits are selective, entries are selective, or both. The short-run response remains downward biased in the presence of selective exit, however. The intuition for why delaying entries can improve pass-through estimates when only exits are selective is somewhat subtle. Remember that, for a movement in the exchange rate l periods ago to have an impact on the index today, there must have been no price change over the past l periods. Delaying entries by M periods eliminates observations incorporated into the index in recent periods, leaving only those present in the index for at least M periods. Or, these surviving observations are less likely than observations in the universe to haveexperiencedapricechangeoverthepastl period(sinceobservationswithapricechange are more likely to have exited), meaning that they are relatively more likely to contribute to measured in(cid:135)ation today. Under Calvo pricing (left panels of (cid:133)gure 9), it turns out that this selection e⁄ect perfectly o⁄sets the downward bias stemming from the censoring of price changesasweconsideranarbitrarilylongentrydelayandforecasthorizon. Undermenu-cost pricing (right panels), the gains are negligible due to the greater mixing of observations. Summing up, our analysis suggests that delaying entries is most e⁄ective at reducing biases associated with selective entry. If estimated pass-through over the forecast horizon increases much after delaying entries, then selective entry may be economically important, giving credence to the canonical cases implying the most severe biases over long horizons. By contrast, if estimated pass-through is insensitive to delaying entries, then selective entries may be unimportant. Using the BLS microdata, we have computed price indexes for the Endusecategoriesbelongingtocapitalgoods,automotiveproducts,andconsumergoods. We have constructed one index using the methodology introduced in section 1 (i.e., there is no entrydelayandmissingpricesarecarriedforward). Wehavealsoconstructedtwoalternative indexesthatimplementa6-monthanda9-monthentrydelay,respectively. Figure10displays 32

the results. As the left-hand column shows, these alternative price indexes are somewhat more volatile than the corresponding published BLS index (the thick black line), especially when entry is delayed. However, estimated pass-through rates are very similar whether we usethepublishedBLSindexorourconstructedindexes. Ifselectiveentrywerequantitatively important, then the estimated pass-through rates for the constructed indexes with delayed entry should be noticeably greater than the pass-through rates for the published index or for the constructed index with no delay. Instead, the estimated pass-through rates are very similar. Thus, the available evidence suggests that selective entry is unlikely to be a large driver of biases in pass-through regressions. 5 Concluding remarks Wehaveinvestigatedselectionbiasesinstandardexchangeratepass-throughregressionsthat arise from missing price changes either due to item exit from or entry into the index. For both Calvo and menu-cost pricing models, we have shown that these selection e⁄ects lower the measured response of an import price index to exchange rate movements over typical policy horizons and that the magnitude of the biases can be sensitive to pricing assumptions. In particular, in the presence of both selective exits and selective entries, the import priceresponseisbiaseddownwardinbothmodels. Assumingthatenteringitemsaresampled randomlyfromtheuniversealleviatessomeofthebias, especiallyunderCalvopricing. When entries are selective and exits occur at random, the downward bias tends to be small in the menu-cost model over any horizon, whereas the bias slowly grows from being negligible at short horizons to quite large over extended horizons in the Calvo model. Assessing the quantitative importance of the biases is inherently challenging because selective exits and selective entries are, by their very nature, not observed. Our review of the BLS methodology suggests a moderate risk of such selection e⁄ects taking place in practice. We also argue that, under plausible assumptions about nominal price stickiness and the incidence of selective exits and selective entries, the presence of downward biases in standard pass-through regressions, although a concern, does not materially alter the literature(cid:146)s view that pass-through to U.S. import prices is low over typical forecast horizons. Even under our worst-case scenario, our estimated empirical bounds imply that at most about a third of an exchange rate shock is passed through to the price of imported (cid:133)nished goods after two years. We note that (cid:133)nished goods categories are precisely those for which one would expect the bias due to selective exit and selective entry to be most pronounced; materialintensive goods have fast and nearly complete pass-through, and hence little scope for large correction factors. Our judgment that biases are likely small empirically is further informed 33

by the insensitivity of measured pass-through to delaying the entry of items in the sample. Had the estimates been plagued by the most severe cases of downward bias, then measured pass-through would have risen as we delayed sample entry. Although we have focused on import prices, our (cid:133)ndings are relevant to the study of any price index subject to selection e⁄ects in sample exit and sample entry. The implications of selective exits and selective entries also extend to the measurement of the response of price indexes to aggregate and idiosyncratic shocks other than exchange rate movements. Future research should aim at better identifying the causes of item exits as well as the characteristics of added items. Currently, the information contained in the IPP database provides useful and suggestive, but ultimately limited, guidance on these aspects. References Atkeson, Andrew and Ariel Burstein (2008) (cid:147)Pricing to Market, Trade Costs, and International Relative Prices,(cid:148)American Economic Review, vol. 98(5), pages 1998-2031. Berger, DavidBerger, JonFaust, JohnH.Rogers, andKaiSteverson(2009).(cid:147)BorderPrices and Retail Prices,(cid:148)International Finance Discussion Papers 972, Board of Governors of the Federal Reserve System. Bergin, Paul, and Robert C. Feenstra (2009), (cid:147)Pass-Through of Exchange Rates and Competition between Floaters and Fixers,(cid:148)Journal of Money, Credit and Banking, vol. 41, pages 35-70. Bils, Mark (2009). (cid:147)Do Higher Prices for NewGoods Re(cid:135)ect Quality Growth or In(cid:135)ation?,(cid:148) Quarterly Journal of Economics, Vol. 124(2), pages 637-675. Bresnahan, Timothy F., and Robert J. Gordon (1997). (cid:147)The Economics of New Goods,(cid:148) Studies in Income and Wealth, Volume 58, National Bureau of Economic Research. Broda, Christian, and David E. Weinstein (2010). (cid:147)Product Creation and Destruction: Evidence and Price Implications,(cid:148)American Economic Review, vol. 100(3), pages 691-723. Bureau of Labor Statistics. (cid:147)Handbook of Methods,(cid:148)Washington DC, April 1997. BureauofLaborStatistics.(cid:147)MeasurementIssuesintheConsumerPriceIndex,(cid:148)Washington D.C., June 1997, http://www.bls.gov/cpi/cpigm697.htm (last access: March 2010). Boskin, Michael J., E. Dulberger, R. Gordon, Z. Griliches, and D. Jorgenson (1996). (cid:147)Toward a More Accurate Measure of the Cost of Living,(cid:148)Final Report to the Senate Finance Committee, December 4. Campa, JosØ Manuel and Linda S. Goldberg (2005). (cid:147)Exchange Rate Pass-Through into Import Prices,(cid:148)Review of Economics and Statistics, vol. 87(4), pages 679-690. 34

Danziger, Leif (1999). (cid:147)A Dynamic Economy with Costly Price Adjustments,(cid:148)American Economic Review, vol. 89(4), pages 878-901. Gagnon, Etienne, David L(cid:243)pez-Salido, and Nicolas Vincent (2012). (cid:147)Individual Price Adjustment along the Extensive Margin,(cid:148)NBER Macroeconomics Annual 2012, vol. 27. Gertler, Mark and John Leahy (2008). (cid:147)A Phillips Curve with an Ss Foundation,(cid:148)Journal of Political Economy, vol. 116(3), pages 533-572. Greenlees, John S., and Robert McClelland (2011). (cid:147)Does Quality Adjustment Matter for Technologically Stable Products? An Application to the CPI for Food,(cid:148)American Economic Review, vol. 101(3), pages 200-205. Goldberg, Pinelopi and Michael M. Knetter (1997). (cid:147)Goods Prices and Exchange Rates: What Have We Learned?,(cid:148)Journal of Economic Literature, vol. 35(3), pages 1243-1272. Gopinath, Gita and Oleg Itskhoki (2010). (cid:147)Frequency of Price Adjustment and Pass- Through,(cid:148)Quarterly Journal of Economics, Vol. 125(2), pages 675(cid:150)727. Gopinath, Gita and Oleg Itskhoki (2011). (cid:147)In Search of Real Rigidities,(cid:148)NBER Macroeconomics Annual, 2010, vol. 25, , pages 261-309. Gopinath, Gita and Roberto Rigobon (2008), (cid:147)Sticky Borders,(cid:148)Quarterly Journal of Economics, vol. 123(2), pages 531-75. Gopinath, Gita, Oleg Itskhoki, and Roberto Rigobon (2010). (cid:147)Currency Choice and Exchange Rate Pass-Through,(cid:148)American Economic Review, vol. 100(1), pages 304-36. Gordon, Robert J. (2006). (cid:147)The Boskin Commission Report: A Retrospective One Decade Later,(cid:148)International Productivity Monitor, vol. 12, pages 7-22. Gust, Christopher, Sylvain Leduc, and Robert J. Vigfusson (2010). (cid:147)Trade Integration, Competition, and the Decline in Exchange-Rate Pass-Through,(cid:148)Journal of Monetary Economics, vol. 57(3), pages 309-324. Marazzi, Mario, Nathan Sheets, Robert J. Vigfusson, Jon Faust, Joseph Gagnon, Jaime Marquez, Robert F. Martin, Trevor Reeve, John Rogers (2005). (cid:147)Exchange Rate Pass- Through to U.S. Import Prices: Some New Evidence,(cid:148)International Finance Discussion Papers 833, Board of Governors of the Federal Reserve System. Marazzi, Mario and Nathan Sheets (2007). (cid:147)Declining Exchange Rate Pass-through to U.S. Import Prices: The Potential Role of Global Factors,(cid:148)Journal of International Money and Finance, vol. 26, pages 924-47. Midrigan, Virgiliu(2011).(cid:147)MenuCosts, MultiproductFirms, andAggregateFluctuations,(cid:148) Econometrica, vol. 79, pages 1139(cid:150)1180. Nakamura, Emi and J(cid:243)n Steinsson (2012). (cid:147)Lost in Transit: Product Replacement Bias and Pricing to Market,(cid:148)American Economic Review, vol. 102(7), pages 3277(cid:150)3316. 35

Neiman, Brent (2010). (cid:147)Stickiness, Synchronization, and Passthrough in Intra(cid:133)rm Trade Prices,(cid:148)Journal of Monetary Economics, April 2010. Sheshinski, Eytan and Yoram Weiss (1977). (cid:147)In(cid:135)ation and Costs of Price Adjustment,(cid:148) Review of Economic Studies, vol. 44(2), pages 287-303. Weber, Henning (2012). (cid:147)Product Replacement Bias in In(cid:135)ation and Its Consequences for Monetary Policy,(cid:148)Journal of Money, Credit and Banking, vol. 44(2-3), pages 255-299. 36

A Regression coe¢ cients in the Calvo model This appendix derives analytical expressions for measured pass-through coe¢ cients when the data are generated by a Calvo model with selection biases in sample exit and entry. The environment is as described in section 3 with the extra simplifying assumption that exchange rate innovations are uncorrelated over time. We begin by describing the general case. We then investigate how delaying the entry of items in the index a⁄ect the regression coe¢ cients. We (cid:133)nally prove that the bias on the coe¢ cients declines as one delays the entry of items in the index. A.1 General case Let f, d,and e beindicatorvariablesthatanitemipresentinthesampleatthebeginning Iit Iit Iit of period t has experienced, respectively, a price change, a random exit, and a selective exit (thelatterbeingconditionalonapricechangeandnorandomexit). Foranyexitingitem, we alsode(cid:133)neanindicatorvariable n thatthecorrespondingentryisselective. Forconvenience, Iit let also s = d + 1 d f e be an indicator that an item has exited during the period, Iit Iit (cid:0)Iit IitIit either through a random exit (probability d) or a selective exit (probability (1 d)fe). (cid:0) (cid:1) (cid:0) We (cid:133)rst derive an expression for the contemporaneous impact of an exchange rate movement on the price index. Using the covariance approach, we have cov (cid:1)p di;(cid:1)x s = 0 cov((cid:1)p ;(cid:1)x s = 0)di b = it t jIit = it t jIit 0 var((cid:1)x ) var((cid:1)x ) (cid:0)R t (cid:1) R t cov(u +(cid:12)(cid:1)x +" ;(cid:1)x s = 0; f = 1) = it t it t jIit Iit Pr f = 1 s = 0 var((cid:1)x ) Iit jIit t h i (1 e)f = (cid:0) (cid:12): 1 fe (cid:0) The covariance term is conditioned on s = 0 because, among observations present in the Iit sample at the beginning of the period, only those that do not exit are used to compute in(cid:135)ation. These usable observations either had no price change and no exit (probability (1 d)(1 f)) or a price change and no exit (probability (1 d)f (1 e)). Only the latter (cid:0) (cid:0) (cid:0) (cid:0) observations, which account for a share (1 e)f (1 fe) of usable observations, have a (cid:0) (cid:0) nonzero contribution to in(cid:135)ation. 37

Proceeding similarly with b , 1 cov((cid:1)p ;(cid:1)x s = 0)di b 1 = it t (cid:0) 1 jIit cov((cid:1)x ) R t 1 (cid:0) cov u +(cid:12)(cid:1)x +" ;(cid:1)x s = 0; f = 1 = (1 (cid:0) e)f it t it t (cid:0) 1 jIit Iit : 1 fe (cid:16) cov((cid:1)x ) (cid:17) t (cid:0) Since (cid:1)x and " are assumed to be independent of (cid:1)x , the covariance term is impacted t it t 1 (cid:0) solely through the interaction between (cid:1)x and the cumulated price pressure u . Condit 1 it (cid:0) tioning on past realizations of the indicator variables, there are (cid:133)ve distinct cases: u +(cid:12)(cid:1)x +" if s = 0; f = 0 it (cid:0) 1 t (cid:0) 1 it (cid:0) 1 Iit (cid:0) 1 Iit (cid:0) 1 8 0 if n s = 0; f = 1o > > Iit (cid:0) 1 Iit (cid:0) 1 u = > > u +(cid:12)(cid:1)x +" if s n= 1; n = 0; f o= 0 : (9) it > > > > it (cid:0) 1 t (cid:0) 1 it (cid:0) 1 Iit (cid:0) 1 Iit (cid:0) 1 Iit (cid:0) 1 < 0 if n s = 1; n = 0; f = 1o Iit 1 Iit 1 Iit 1 (cid:0) (cid:0) (cid:0) > > 0 n if s = 1; n = 1 o > > > > Iit (cid:0) 1 Iit (cid:0) 1 > > : (cid:8) (cid:9) Consequently, (1 e)f cov((cid:12)(cid:1)x ;(cid:1)x ) b 1 = 1 (cid:0) fe cov( t (cid:0) (cid:1) 1 x ) t (cid:0) 1 Pr Ii s t 1 = 0; Ii f t 1 = 0 +Pr Ii s t 1 = 1; Ii n t 1 = 0; Ii f t 1 = 0 t (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:16) h i h i(cid:17) (1 e)f = (cid:0) (1 f)(cid:12)((1 d)+(d+(1 d)fe)(1 n)) 1 fe (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) Intuitively,amongallobservationsusabletocomputein(cid:135)ation,onlythosewithapricechange in the current period (marginal probability (1 e)f) and either no price change and no exit (cid:0) 1 fe (cid:0) in the previous period (marginal probability (1 d)(1 f)) or an exit accompanied by an (cid:0) (cid:0) entry with no price change (marginal probability (d+(1 d)fe)(1 n)(1 f)) feature a (cid:0) (cid:0) (cid:0) nonzero contribution of (cid:1)x to in(cid:135)ation. t 1 (cid:0) Thegeneralcasewithb isillustratedintheupperpanelof(cid:133)gure11. The(cid:133)gureshowsthe l various states that are usable in the computation of in(cid:135)ation at each period, along with their associated marginal probability period by period. The arrows indicate the paths through which an exchange rate movement in the period t l is re(cid:135)ected as a nonzero price change in (cid:0) period t. Observations that have not yet responded to an exchange rate movement at period t l can (cid:133)nd their way in the index either by having been present in the sample prior to (cid:0) period t l or by entering the sample through a substitution. The marginal probability from (cid:0) period t l to period t 1 associated with the former event (no price change and no exit) (cid:0) (cid:0) is (1 d)(1 f) for each period. The marginal probability associated with the addition of (cid:0) (cid:0) 38

an item in period t k whose last price change was prior to period t l is the product (cid:0) (cid:0) of the probability of having a substitution (probability d+fr(1 d)), a random selection (cid:0) from the universe (probability 1 n), and no price change for l k+1 periods (probability (cid:0) (cid:0) (1 f)l k+1). Summing up across all usable paths, we have a general expression for the (cid:0) (cid:0) (plim) regression coe¢ cient 1 e s(1 n) b = (cid:0) (1 d)l + (cid:0) 1 (1 d)l f (1 f)l(cid:12): l 1 fe (cid:0) d (cid:0) (cid:0) (cid:0) (cid:18) (cid:0) (cid:19)(cid:18) (cid:16) (cid:17) (cid:19) A.2 Delayed entries Wenextassumethat theeconometricianonlyuses observations thathavebeeninthesample for more than M periods in the computation of in(cid:135)ation. This assumption is made in section 4.4 to argue that delaying the entry of items in the basket can mitigate some of the biases associated with selective exits and entries. We distinguish between two cases: l M and (cid:20) l > M. The (cid:133)rst case is illustrated in the middle panel of (cid:133)gure 11. Because entries are delayed by more periods than the number of lags in the exchange rate movement considered, all observations contributing to in(cid:135)ation and re(cid:135)ecting (cid:1)x must have been in the index t l (cid:0) continuously since before period t l. For this situation to occur, we must have had no (cid:0) price change from period t l to t 1 (marginal probability (1 f)=(1 fe) during l (cid:0) (cid:0) (cid:0) (cid:0) periods), and a price change at t (marginal probability (1 e)f=(1 fe)). The resulting (cid:0) (cid:0) (plim) regression coe¢ cient is (1 e) b (l M) = (cid:0) f (1 f)l(cid:12): (10) l (cid:20) (1 fe)l+1 (cid:0) (cid:0) The case of l > M is illustrated at the bottom of (cid:133)gure 11. It mixes elements of the general case with no delay (upper panel) and the case with l M (middle panel). Prior to period (cid:20) t M, observations that have not yet responded to the exchange rate movement at period (cid:0) t l could have found their way in the index either through a substitution or by having been (cid:0) present in the sample before period t l. From period t M onward, only observations that (cid:0) (cid:0) are continuously present in the index from the end of period t M 1 onward can be used (cid:0) (cid:0) to compute in(cid:135)ation. Summing up the probabilities over all possible paths and simplifying, we get (1 e) s(1 n) b l (l > M) = (1 f (cid:0) e)M+1 (1 (cid:0) d)l (cid:0) M + d (cid:0) 1 (cid:0) (1 (cid:0) d)l (cid:0) M f (1 (cid:0) f)l(cid:12): (11) (cid:0) (cid:18) (cid:16) (cid:17) (cid:19) 39

A.3 Proof that biases are declining in the entry delay We conclude this appendix by proving that delaying the entry of items in the index always improves pass-though estimates. We assume that the number of lags in the regression is at least as large as the forecast horizon, T, a condition typically satis(cid:133)ed in standard passthrough regressions. Let b (M) be the (plim) coe¢ cient associated with the l-th lag of the l exchange rate and an entry delay of M periods. The proof proceeds in two steps. We (cid:133)rst prove that b (M +1) b (M), so that delaying entries by an extra period always (weakly) l l (cid:21) increases the size of the (plim) regression coe¢ cients. We then show that the cumulative response over any forecast horizon remains bounded above by the true response. A.3.1 Step 1: b (M +1) b (M) l l (cid:21) We distinguish between three cases: l < M+1, l = M+1, and l > M+1. When l < M+1, the plim coe¢ cients are given by equation 10 whether the delay is M periods or M + 1 periods, so that b (M) = b (M +1). When l = M + 1, b (M) is given by equation 11 l l M+1 and b (M +1) is given by equation 10. For b (M +1) b (M) to be true in this M+1 M+1 M+1 (cid:21) case, we must have 1 1 d+s(1 n): 1 fe (cid:21) (cid:0) (cid:0) (cid:0) Note that if the above equation holds for n = 0, then it holds for all n [0;1]. Imposing 2 n = 0 and using s = d+(1 d)fe, we have (cid:0) 1 1 d+ef; 1 fe (cid:21) (cid:0) (cid:0) which is always satis(cid:133)ed. Finally, we want to show that b (M +1) b (M) when l > M+1. l l (cid:21) The plim coe¢ cients are given by equation 11. Note that @ 1 (1 d)l M 1 (b l (M +1) b l (M)) = (cid:10) (cid:0) (cid:0) (cid:0) (cid:0) 1 (1 d)l (cid:0) M ; @n (cid:0) (cid:0) 1 fe (cid:0) (cid:0) (cid:0) ! (cid:0) (cid:16) (cid:17) where(cid:10)issomepositiveconstant. Thedi⁄erencebetweenb (M +1)andb (M)isthuslinear l l in n and either always increasing or always decreasing in n. By showing that b (M +1) l (cid:21) b (M) for n = 0 and n = 1, we will have proven that the result holds for the worse scenario l under either case. Consider (cid:133)rst (1 e) b l (M j n = 1) = f (1 (cid:0) f)l (1 f (cid:0) e)M+1 (1 (cid:0) d)l (cid:0) M (cid:12): (cid:0) 40

We have b (M +1 n = 1) b (M n = 1) if and only if (1 d)l M 1 (1 fe)(1 d)l M, l l (cid:0) (cid:0) (cid:0) j (cid:21) j (cid:0) (cid:21) (cid:0) (cid:0) which is always true. Consider next f (1 f)l(1 e) s b l (M j n = 0) = (1 (cid:0) fe)M+ (cid:0) 1 (1 (cid:0) d)l (cid:0) M + d 1 (cid:0) (1 (cid:0) d)l (cid:0) M (cid:12): (cid:0) (cid:16) (cid:16) (cid:17)(cid:17) We have b (M +1 n = 0) b (M n = 0) if and only if l l j (cid:21) j d(1 d)l M 1 +s 1 (1 d)l M 1 (1 fe) d(1 d)l M +s 1 (1 d)l M ; (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:21) (cid:0) (cid:0) (cid:0) (cid:0) (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17) which can be shown to hold if and only if fes 1 (1 d)l M 0; (cid:0) (cid:0) (cid:0) (cid:21) (cid:16) (cid:17) a condition that is always satis(cid:133)ed. Summing up, the individual coe¢ cients are increasing in the entry delay, so that cumulative pass-through over any forecast horizon also is increasing in the entry delay. A.3.2 Step 2: the cumulative estimated response is bounded above by true response To complete the proof, we show that the estimated pass-through under delayed entries never exceeds true pass-through over any forecast horizon. The true pass-through after L periods is L L b = f (1 f)l(cid:12) = 1 (1 f)L+1 (cid:12): l (cid:0) (cid:0) (cid:0) X l=0 X l=0 (cid:16) (cid:17) Because b (M +1) b (M), the estimated pass-through is largest when M L; which is l l (cid:21) (cid:21) associated with L L (1 e) 1 f L+1 b (M L) = (cid:0) f (1 f)l(cid:12) = 1 (cid:0) (cid:12): l (cid:21) (1 fe)l+1 (cid:0) (cid:0) 1 fe ! X l=0 X l=0 (cid:0) (cid:18) (cid:0) (cid:19) Comparing the above expression to equation 3, it is immediate that cumulative pass-through under delayed sample entry is bounded above by the unbiased case. 41

B Real rigidities To explore the incidence of real rigidities on our theoretical correction factors, we follow Gopinath and Itskhoki (2011) in adopting a (cid:135)exible speci(cid:133)cation for the evolution of the (cid:133)rm(cid:146)s reset price, (cid:1)p = (1 (cid:11))((cid:12)(cid:1)x +" )+(cid:11)((cid:1)m ): (cid:3)it (cid:0) t it t The parameter (cid:11) [0;1] controls the extent of real rigidities by capturing the emphasis 2 placed by the (cid:133)rm on matching movements in the average price of competing importers, (cid:1)m , when resetting its own price. As was the case earlier, (cid:133)rms fully release the prest sure accumulated since their last adjustment when updating their price and a fraction (cid:12) of exchange rate movements is passed-through in the long run. Under Calvo pricing, it is possible to derive expressions for actual pass-through to the universe of prices and for measured pass-through in the import price index. Abstracting from idiosyncratic disturbances, which have no in(cid:135)uence on the index, the initial e⁄ect on reset prices of a one-time jump in the exchange rate is the sum of its direct e⁄ect on (cid:1)p (cid:3)i;t and of its indirect e⁄ect through the price of competing importers, (cid:1)p = (1 (cid:11))(cid:12)(cid:1)x +(cid:11)(cid:1)m : (cid:3)i;t (cid:0) t t The impact in subsequent periods depends only on the evolution of the price of competing importers, (cid:1)p = (cid:11)(cid:1)m : (cid:3)i;t+l t+l The price of competing importers evolves according to l (cid:1)m = f (1 f)l k(cid:1)p : (12) t+l (cid:0) (cid:0) (cid:3)i;t+k k=0 X Intuitively, the fraction f of (cid:133)rms updating their price in period t + l takes into account (cid:1)p in setting (cid:1)p . Of these (cid:133)rms, a fraction 1 f did not update in period t+l 1 (cid:3)i;t+l i;t+l (cid:0) (cid:0) andwillalsotakeintoaccount(cid:1)p . Moregenerally,afraction(1 f)s of(cid:133)rmsupdating (cid:3)i;t+l 1 (cid:0) (cid:0) in period t+l will take into account (cid:1)p when setting (cid:1)p . (cid:3)i;t+l s i;t+l (cid:0) The above three equations allow us to solve recursively for (cid:1)p and (cid:1)m . In the (cid:3)i;t+l t+l initial period, (cid:1)p 1 (cid:11) (cid:3)i;t = (cid:0) (cid:12) (13) (cid:1)x 1 (cid:11)f t (cid:18) (cid:0) (cid:19) 42

and (cid:1)m 1 (cid:11) t = (cid:0) f(cid:12): (14) (cid:1)x 1 f(cid:11) t (cid:18) (cid:0) (cid:19) Subsequently, one can show that (cid:1)p (cid:11)(1 (cid:11)) (cid:3)i;t+l = (cid:0) f (1 f)l(cid:12) (15) (cid:1)x (1 f(cid:11))l+1 (cid:0) t (cid:0) and (cid:1)m (1 (cid:11)) t+l = (cid:0) f (1 f)l(cid:12) (cid:1)x (1 f(cid:11))l+1 (cid:0) t (cid:0) for l > 0. Absent real rigidities ((cid:11) = 0), the reset price would initially move by (cid:12)(cid:1)x and t remain unchanged thereafter. With real rigidities ((cid:11) > 0), the reset price moves gradually toward its long-run level so that the response to an exchange rate movement is spread over several priceadjustments. Thecumulativeresponseof thepriceof competingimportersafter L period is given by L (cid:1)m 1 f L+1 t+l = 1 (cid:0) (cid:12); (16) (cid:1)x (cid:0) 1 (cid:11)f l=0 t (cid:18) (cid:0) (cid:19) ! X whichimpliesaslowerrateofpass-throughthanthecasewith(cid:11) = 0. Thecumulativechange in the target price is a weighted sum of the long-run change and the cumulative change in the price of competing importers, L (cid:1)p 1 f L+1 (cid:3)i;t+l = 1 (cid:0) (cid:11)(cid:12) +(1 (cid:11))(cid:12): (cid:1)x (cid:0) 1 (cid:11)f (cid:0) l=0 t (cid:18) (cid:0) (cid:19) ! X Turning to measured pass-through in the sample, using the arguments made in section 3 to derive equation (4), we can express the change in the import price index as l (1 e) s(1 n) (cid:1)p b l = (cid:0) (1 d)k + (cid:0) 1 (1 d)k f (1 f)k (cid:3)i;t+l (cid:0) k : (17) 1 fe (cid:0) d (cid:0) (cid:0) (cid:0) (cid:1)x X k=0 (cid:0) (cid:18) (cid:16) (cid:17) (cid:19) t Thevariableb correspondstothe(plim)regressioncoe¢ cientonthel-thlagoftheexchange l rate. The above equation, along with equations (13) and (15), allow us to compute the b at l all lags. To gain intuition about the role of real rigidities, we shall highlight three results. First, the correction factor on the initial response of the index is independent of the extent of real rigidities. This can be seen by using equations (13), (14), and (17) to write 1 e (cid:1)m t b = (cid:0) : 0 1 fe (cid:1)x (cid:18) (cid:0) (cid:19) t 43

The inverse of (1 e)=(1 fe) is the correction factor applicable to the initial index re- (cid:0) (cid:0) sponse. It does not depend on (cid:11). Second, the correction factor for long-run pass-through is also independent of the extent of real rigidities. This can be proven by considering the cumulative sum of b in equation (17) and taking its limit as the horizon is extended. We l obtain L (1 e) f +s(1 n)(1 f) lim b = (cid:0) (cid:0) (cid:0) (cid:12): l L 1 fe f +d fd !1 l=0 (cid:0) (cid:18) (cid:0) (cid:19) X The inverse of the expression in front of (cid:12) is the correction factor on long-run pass-through, which is again independent of (cid:11). Third, when both exits and entries are selective, the correction factors are independent of real rigidities at all horizons. In such a case, a constant fraction (1 e)=(1 fe) of (cid:1)m =(cid:1)x is missed every period. t+l t (cid:0) (cid:0) In the general case, the correction factors on the cumulative index response may depend on (cid:11) at horizons beyond the initial period but (cid:133)nite. However, we (cid:133)nd that these correction factors are rather insensitive to the extent of real rigidities. This aspect is illustrated in (cid:133)gure 12 for a Calvo model with a frequency of price changes of 15 percent and a rate of substitution of 5 percent. The parameter (cid:11) is set to either 0 (no real rigidities) or 0:6 (with real rigidities). The latter value corresponds to the empirical estimate of Gopinath and Itskhoki (2011). The correction factors are identical for the canonical cases in which all exits and entries are either selective (upper-left panel) or random (upper-right panel), as these cases correspond to the absence of a bias and to the censoring of a constant fraction of the index response, respectively. When all exits are selective and all entries are random (lower-left panel) or all exits are random and all entries are selective (lower-right panel), the presence of real rigidities only has a minor impact on the size of the correction factor. This (cid:133)nding is robust to considering smaller or larger frequency of price changes. Finally, we note a trade-o⁄ between the extent of real rigidities and the degree of state dependence in the calibrated model. For many of the three-digit Enduse (cid:133)nished good categories considered in (cid:133)gure 6, we (cid:133)nd an initially rapid response of the import price index to an exchange rate movement that is inconsistent with the slow response predicted by the Calvo model. For these categories, the addition of real rigidities worsens the (cid:133)t of the Calvo model by further slowing its predicted response. As an experiment, we assumed (cid:11) = 0:6 and recomputedthelinearcombinationsofCalvoandmenu-costimpulseresponsesthatminimize the Euclidian distance withthe empirical response overourforecast horizon. For9 out of the 16 categories, the weight placed on the menu-cost model rose as we added real rigidities. A strong preference for one model over the other was unchanged for the other categories. This (cid:133)nding cautions against concluding that evidence of real rigidities in the data, in the form of pass-through over more than one price adjustment, necessarily implies slow pass-through. 44

Table 1: Exit rate, entry rate, and mean frequency and absolute size of individual price changes in the IPP import price sample Mean Freq. Mean Absolute all out of others of Price Size of Price reasons scope reasons Changes Change 000 Green coffee, cocoa beans, cane sugar 0.3 2.8 1.1 1.7 3.0 47.0 8.7 001 Other agricultural foods 2.8 2.4 0.8 1.5 2.6 21.1 9.4 010 Nonagricultural products 1.1 2.2 0.8 1.4 2.4 20.5 7.1 100 Petroleum & products, excluding gas 16.7 3.6 1.9 1.6 2.5 38.0 11.7 101 Fuels, n.e.s. coal & gas 1.8 3.4 2.1 1.2 4.0 55.9 13.4 110 Paper base stocks 0.2 3.6 1.6 2.0 3.2 36.5 6.1 111 Newsprint & other paper products 0.7 3.3 1.2 2.0 2.5 19.6 5.0 120 Agricultural products 0.5 2.2 0.6 1.5 2.1 28.4 7.4 121 Textile supplies & related materials 0.7 2.6 0.8 1.7 2.5 8.0 6.8 125 Chemicals, excl. meds., food additives 3.4 2.4 0.7 1.7 2.5 11.2 7.3 130 Lumber & unfinished building materials 1.1 2.4 0.9 1.5 2.9 33.2 7.7 131 Building materials, finished 1.0 2.6 0.8 1.8 2.9 10.4 5.7 140 Steelmaking materials unmanufactured 0.4 2.4 0.9 1.5 2.0 21.2 7.1 141 Iron & steel mill products semifinished 1.3 3.5 1.3 2.1 3.9 15.3 21.4 142 Major non Fe metals crude & semifin. 2.7 3.0 1.4 1.5 2.5 43.4 5.8 150 Iron & steel products, ex. advanced mfg. 0.5 2.6 0.8 1.9 2.4 9.5 7.1 151 Iron & steel mfg. advanced 0.4 2.9 0.7 2.2 2.4 13.2 7.2 152 Fin. metal shapes & adv. mfg., ex. steel 0.9 2.7 0.6 2.0 2.4 13.3 5.5 161 Finished 1.5 2.7 1.1 1.6 2.8 7.1 7.9 210 Oil drilling, mining & const. machinery 1.1 2.5 0.9 1.6 2.9 6.9 6.6 211 Industrial & service machinery, n.e.c. 6.2 2.5 0.8 1.7 2.5 6.3 6.7 212 Agricultural machinery & equip. 0.4 2.7 1.2 1.5 3.1 8.9 5.3 213 Computers, periph. & semiconductors 7.5 3.7 2.2 1.5 5.0 9.7 9.6 214 Telecommunications equip. 2.3 3.4 1.8 1.6 3.6 5.8 8.9 215 Business mach. & equip., ex. Computers 0.5 3.3 1.5 1.8 2.5 5.2 6.3 216 Scientific, hospital & medical machinery 1.5 3.1 1.2 1.9 3.2 4.9 6.9 300 Passenger cars, new & used 8.0 2.8 1.6 1.1 3.5 5.3 2.0 301 Trucks, buses, & special purp. vehicles 1.4 2.8 1.9 0.9 3.9 5.8 2.9 302 Parts, engines, bodies, & chassis 5.4 2.8 1.2 1.6 3.0 8.0 7.1 400 Apparel, footwear, & household goods 6.6 3.5 1.7 1.8 3.6 3.9 7.6 401 Other consumer nondurables 5.0 2.4 0.8 1.5 2.7 6.0 7.7 410 Household goods 6.1 2.9 1.2 1.7 3.0 4.6 6.2 411 Recreational equip. & materials 2.3 3.2 1.8 1.5 3.1 4.8 5.7 412 Home entertainment equip. 3.1 3.7 2.2 1.5 4.1 5.6 5.8 413 Coins, gems, jewelry, & collectibles 1.3 3.1 1.1 1.9 3.1 6.9 5.9 500 Imports, N.E.S. 3.5 2.7 0.6 2.1 1.8 5.2 12.5 Total 100.0 3.0 1.4 1.6 3.1 15.3 8.0 ,sdooF & sdeeF segareveB Exit Rate Relative Entry Enduse Category Weight Rate sdooG remusnoC otuA slairetaM dna seilppuS lairtsudnI sdooG latipaC & dooF .xE( evitom ).otuA Notes: These statistics are computed by (cid:133)rst applying uniform weights to items within each Enduse-month combination and then averaging the resulting monthly statistics. The sample period is from October 1995 to April 2005. Missing item prices are imputed by their last observed price and used in the computation of the above statistics. The table also shows the relative 2006 import value shares used to aggregate three-digit Enduse statistics. 45

Table 2: Reconciliation of our results with those of Nakamura and Steinsson (2011) All goods Material intensive Finished goods excluding oil goods excluding oil Standard pass through estimates after 6 quarters Gagnon, Mandel, and Vigfusson 0.41 0.26 0.93 Nakamura and Steinsson 0.43 n.a. n.a. Item frequencies are constant within 3 digit Enduse categories Case 1: Exits and entries are random Corrected pass through (6 quarters) 0.41 0.26 0.93 Correction factor 1.00 1.00 1.00 Corrected pass through (long run) 0.41 0.26 0.93 Correction factor 1.00 1.00 1.00 Case 2: Exits and entries are selective Corrected pass through (6 quarters) 0.55 0.37 1.04 Correction factor 1.32 1.40 1.12 Corrected pass through (long run) 0.55 0.37 1.04 Correction factor 1.32 1.40 1.12 Case 3: Exits are selective and entries are random Corrected pass through (6 quarters) 0.48 0.32 0.96 Correction factor 1.16 1.21 1.04 Corrected pass through (long run) 0.45 0.29 0.95 Correction factor 1.08 1.10 1.02 Case 4: Exits are random and entries are selective Corrected pass through (6 quarters) 0.48 0.31 1.02 Correction factor 1.16 1.19 1.10 Corrected pass through (long run) 0.55 0.37 1.04 Correction factor 1.32 1.40 1.12 Item frequencies follow Beta distribution All entries are selective Corrected pass through (long run) 0.70 n.a. n.a. Correction factor 1.70 n.a. n.a. Corrected pass through (long run, nof < 0.02) 0.61 n.a. n.a. Correction Factor 1.47 n.a. n.a. Addendum: NS' corrected estimates Corrected pass through (long run) 0.67 n.a. n.a. Correction factor 1.71 n.a. n.a. Notes: Standard pass-through estimates after 6 quarters are obtained from an OLS estimation of equation 8. All correction factors assume Calvo pricing, a monthly frequency of item substitution of 2.5 percent, and uniform long-run pass-through rates across items. Corrected 6-quarter and long-run pass-through estimates are obtained by multiplying standard passthroughestimatesafter6quartersbyacorrectionfactorconsistentwithan18-monthhorizon and an in(cid:133)nite horizon, respectively. The (cid:133)rst set of correction factors assumes a constant frequency of price changes across items within each three-digit Enduse category. The second set uses a Beta distribution with parameters 0.44 and 3.50, as estimated by Nakamura and Steinsson (2012). For reference, the addendum cites the correction factor for the case with (cid:147)local currency pricing imports(cid:148)and (cid:147)no heterogeneity in pass-through(cid:148)from table IV in Nakamura and Steinsson (2012). 46

Figure 1: Exit rate, entry rate, and the dollar Out of scope exits and the dollar Percent Dollar index (Jan. 1997 = 100) 2.5 135 Exit rate, out of scope items (left axis) Federal reserve nominal broad dollar (right axis) 130 125 2.0 120 115 1.5 110 105 100 1.0 95 90 0.5 85 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Exits other than out of scope and entries Percent 5 Exit rate, other than out of scope items Entry rate 4 3 2 1 0 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 47

Figure 2: Coe¢ cients on lags of the exchange rate in pass-through regressions in baseline Calvo and menu-cost models frequency=5% 0.1 Calvo model Menu cost model 0.08 tn 0.06 e c re p 0.04 0.02 0 0 5 10 15 20 25 lag frequency=20% 0.2 0.15 tn e c 0.1 re p 0.05 0 0 5 10 15 20 25 lag frequency=35% 0.25 0.2 tn 0.15 e c re p 0.1 0.05 0 0 5 10 15 20 25 lag 48

Figure 3: Cumulative contribution of coe¢ cients on lagged exchange rate variables under selective exit (e=0.25) Calvo model, frequency=3.8 Menu cost model, frequency=3.7 100 100 Estimated contribution (n=1) Additional contribution (n=0) 80 Missing contribution (n=0) 80 tn 60 tn 60 e e c c re re p p 40 40 20 20 0 0 0 5 10 15 20 25 0 5 10 15 20 25 lag lag Calvo model, frequency=16 Menu cost model, frequency=16 100 100 80 80 tn 60 tn 60 e e c c re re p p 40 40 20 20 0 0 0 5 10 15 20 25 0 5 10 15 20 25 lag lag Calvo model, frequency=29 Menu cost model, frequency=29 100 100 80 80 tn 60 tn 60 e e c c re re p p 40 40 20 20 0 0 0 5 10 15 20 25 0 5 10 15 20 25 lag lag Notes: The (cid:133)gure illustrates the case with selective entries (n = 1) and the case with fullyrandomized entries (n = 0). The observed frequencies of price changes listed in the top, middle, and bottom rows correspond to true frequencies of 5 percent, 20 percent, and 35 49 percent, respectively.

Figure 4: Cumulative contribution of coe¢ cients on lagged exchange rate variables model under random exits and selective entries (n=1, s=0.05) Calvo model, frequency=5 Menu cost model, frequency=5 100 100 Estimated contribution Missing contribution 80 80 tn 60 tn 60 e e c c re re p p 40 40 20 20 0 0 0 5 10 15 20 25 0 5 10 15 20 25 lag lag Calvo model, frequency=20 Menu cost model, frequency=20 100 100 80 80 tn 60 tn 60 e e c c re re p p 40 40 20 20 0 0 0 5 10 15 20 25 0 5 10 15 20 25 lag lag Calvo model, frequency=35 Menu cost model, frequency=35 100 100 80 80 tn 60 tn 60 e e c c re re p p 40 40 20 20 0 0 0 5 10 15 20 25 0 5 10 15 20 25 lag lag 50

Figure 5: Pass-through to imported (cid:133)nished goods prices following a 1-percent depreciation of the dollar (by 3-digit Enduse categories) 210 214 0.6 0.6 211 215 0.5 212 0.5 216 213 300 0.4 0.4 tn tn e e c 0.3 c 0.3 re re p p 0.2 0.2 0.1 0.1 0 0 0.1 0.1 0 5 10 15 20 25 0 5 10 15 20 25 months after shock months after shock 301 410 0.6 0.6 302 411 0.5 400 0.5 412 401 413 0.4 0.4 tn tn e e c 0.3 c 0.3 re re p p 0.2 0.2 0.1 0.1 0 0 0.1 0.1 0 5 10 15 20 25 0 5 10 15 20 25 months after shock months after shock Weighted response 0.6 210 Oil drilling mining and construction machinery and equipment Index 211 Industrial and service machinery 0.5 212 Agricultural machinery and equipment 213 Computers peripherals and semiconductors 0.4 214 Telecommunications equipment 215 Business machinery and equipment except computers 0.3 216 Scientific and medical machinery 300 Passenger cars new and used 0.2 301 Vehicles designed to transport goods 302 Parts engines bodies and chassis 0.1 400 Apparel footwear and household goods 401 Other consumer nondurables 0 410 Household goods 411 Recreational equipment and materials 0.1 412 Home entertainment equipment 0 5 10 15 20 25 413 Coins gems jewelry and collectibles Gray Area: 95 Percent Confidence Interval Around Estimates for Index 51

Figure 6: Pass-through to imported (cid:133)nished goods prices following a 1-percent depreciation of the dollar: models versus data Enduse 210 (l =1.00) Enduse 211 (l =0.04) Enduse 212 (l =1.00) Enduse 213 (l =1.00) 0.4 0.5 0.4 0.3 0.4 0.6 0.3 0.3 tne 0.2 tne tne tne 0.4 cre p 0.1 cre p 0.2 cre p 0.2 cre p 0.1 0 0.1 0.2 0 0.1 0 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 months after shock months after shock months after shock months after shock Enduse 214 (l =0.41) Enduse 215 (l =0.59) Enduse 216 (l =0.00) Enduse 300 (l =0.00) 0.15 0.3 0.2 0.1 0.08 0.1 0.15 0.2 0.06 tne cre p 0.05 tne cre p tne cre p 0.1 tne cre p 0.04 0.1 0 0.05 0.02 0.05 0 0 0 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 months after shock months after shock months after shock months after shock Enduse 301 (l =1.00) Enduse 302 (l =1.00) Enduse 400 (l =1.00) Enduse 401 (l =0.29) 0.05 0.25 0.1 0.3 0.2 0.08 0.2 tne cre p 0 tne cre p 0. 0 1 . 5 1 tne cre p 0 0 . . 0 0 4 6 tne cre p 0.1 0.05 0.02 0 0.05 0 0 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 months after shock months after shock months after shock months after shock Enduse 410 (l =0.76) Enduse 411 (l =0.49) Enduse 412 (l =0.00) Enduse 413 (l =0.59) 0.25 0.15 0.4 0.2 0.2 0.1 0.3 0.15 tne cre p 0. 0 1 . 5 1 tne cre p 0.1 tne cre p 0.05 tne cre p 0.2 0.05 0 0.1 0.05 0 0 0.05 0.05 0 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 months after shock months after shock months after shock months after shock Data Menu costs Calvo Best linear combination Notes: The (cid:133)gure shows the empirical impulse response for three-digit Enduse categories along with calibrated impulse responses in the Calvo and the menu-cost models. The (cid:133)gure alsoshows thecombinationof modelsminimizing (cid:21) IRF +(1 (cid:21)) IRF IRF Calvo MC data k (cid:0) (cid:0) k over (cid:21) [0;1], where is the Euclidian distance and IRF , IRF , and IRF are Calvo MC data 2 kk the impulse responses in the Calvo model, the menu-cost model, and the data. 52

Figure 7: Upper bounds on exchange rate pass-through to imported (cid:133)nished goods Out of scope exits are selective, Out of scope exits are selective, all entries are selective all entries are random 0.4 0.4 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 Empirical Estimate Empirical Estimate 0.05 0.05 Menu Costs Menu Costs Calvo Calvo 0 0 0 5 10 15 20 25 0 5 10 15 20 25 Months after exchange rate movement Months after exchange rate movement All exits are random, all entries are selective Model combination 0.4 0.4 Empirical Estimate Out of scope exits lead to 0.35 0.35 selective exits and entries Out of scope exits lead to selective entries only 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 Empirical Estimate 0.05 0.05 Menu Costs Calvo 0 0 0 5 10 15 20 25 0 5 10 15 20 25 Months after exchange rate movement Months after exchange rate movement 53

Figure 8: Impact of delaying entries on cumulative contribution of coe¢ cients on lagged exchange rate variables under random exits and selective entries (n=1, s=0.05) Calvo model, frequency=5 Menu cost model, frequency=5 100 100 Estimated contribution Extra contribution from delay 80 Remaining bias 80 tn 60 tn 60 e e c c re re p p 40 40 20 20 0 0 0 5 10 15 20 25 0 5 10 15 20 25 lag lag Calvo model, frequency=20 Menu cost model, frequency=20 100 100 80 80 tn 60 tn 60 e e c c re re p p 40 40 20 20 0 0 0 5 10 15 20 25 0 5 10 15 20 25 lag lag Calvo model, frequency=35 Menu cost model, frequency=35 100 100 80 80 tn 60 tn 60 e e c c re re p p 40 40 20 20 0 0 0 5 10 15 20 25 0 5 10 15 20 25 lag lag 54

Figure 9: Impact of delaying entries on cumulative contribution of coe¢ cients on lagged exchange rate variables under selective exits and random entries (n=0, e=0.25) Calvo model, frequency=3.8 Menu cost model, frequency=4 100 100 Estimated contribution Extra contribution from delay 80 Remaining bias 80 tn 60 tn 60 e e c c re re p p 40 40 20 20 0 0 0 5 10 15 20 25 0 5 10 15 20 25 lag lag Calvo model, frequency=16 Menu cost model, frequency=16 100 100 80 80 tn 60 tn 60 e e c c re re p p 40 40 20 20 0 0 0 5 10 15 20 25 0 5 10 15 20 25 lag lag Calvo model, frequency=29 Menu cost model, frequency=29 100 100 80 80 tn 60 tn 60 e e c c re re p p 40 40 20 20 0 0 0 5 10 15 20 25 0 5 10 15 20 25 lag lag Notes: The observed frequencies of price changes listed in the top, middle, and bottom rows correspond to true frequencies of 5 percent, 20 percent, and 35 percent, respectively. 55

Figure 10: Alternative price indexes and measured pass-through Capital Goods Price Indexes Pass through 0.35 100 0.3 95 0.25 90 0.2 0.15 85 Published BLS 0.1 Constructed: No Delay 80 0.05 Constructed: 6 Month Delay Constructed: 9 Month Delay 75 0 1996 1998 2000 2002 2004 2006 0 5 10 15 20 25 Automotive Products Price Indexes Pass through 0.12 108 0.1 0.08 106 0.06 104 0.04 0.02 102 0 100 0.02 1996 1998 2000 2002 2004 2006 0 5 10 15 20 25 Consumer Goods Price Indexes Pass through 0.2 102 101 0.15 100 99 0.1 98 0.05 97 96 0 95 0.05 1996 1998 2000 2002 2004 2006 0 5 10 15 20 25 56

Figure 11: Marginal probabilities of observations usable to computed in(cid:135)ation in period t in the Calvo model with iid exchange rate innovations a) No delay Period t l … t k 1 t k … t 1 t No exit, no price change (1 d)(1 f) … (1 d)(1 f) (1 d)(1 f) … (1 d)(1 f) (1 f)/(1 fe) No exit, price change (1 d)f(1 e) … (1 d)f(1 e) (1 d)f(1 e) … (1 d)f(1 e) f(1 e)/(1 fe) Exit, no price change since t l 1 (d+fe(1 d))(1 n)(1 f) … (d+fe(1 d))(1 n)(1 f) (d+fe(1 d))(1 n)(1 f) … (d+fe(1 d))(1 n)(1 f)^l n.a. Exit, price change since t l 1 (d+fe(1 d))(n+(1 n)f) … (d+fe(1 d))(n+(1 n)f) (d+fe(1 d))(n+(1 n)f) … (d+fe(1 d))(1 (1 n)*(1 f)^l)) n.a. b) Delay ofM periods,M?l Period t l … t k 1 t k … t 1 t No exit, no price change (1 f)/(1 fe) … (1 f)/(1 fe) (1 f)/(1 fe) … (1 f)/(1 fe) (1 f)/(1 fe) No exit, price change f(1 e)/(1 fe) f(1 e)/(1 fe) f(1 e)/(1 fe) f(1 e)/(1 fe) f(1 e)/(1 fe) … … Exit, no price change since t l 1 n.a. n.a. n.a. n.a. n.a. … … Exit, price change since t l 1 n.a. n.a. n.a. n.a. n.a. c) Delay ofM periods, withM<l Period t l … t M 1 t M … t 1 t No exit, no price change (1 d)(1 f) … (1 d)(1 f) (1 f)/(1 fe) … (1 f)/(1 fe) (1 f)/(1 fe) No exit, price change (1 d)f(1 e) … (1 d)f(1 e) f(1 e)/(1 fe) … f(1 e)/(1 fe) f(1 e)/(1 fe) Exit, no price change since t l 1 (d+fe(1 d))(1 n)(1 f) … (d+fe(1 d))(1 n)(1 f)^2 n.a. … n.a. n.a. Exit, price change since t l 1 (d+fe(1 d))(n+(1 n)f) … (d+fe(1 d))(1 (1 n)*(1 f)^2)) n.a. … n.a. n.a. Notes: This (cid:133)gure shows the marginal probabilities in periods t l to t of items whose (cid:0) price can be used to compute in(cid:135)ation in period t. The arrows illustrate the various paths throughwhichamovementintheexchangerateinperiodt l couldbere(cid:135)ectedasanonzero (cid:0) contribution to in(cid:135)ation in period t. The upper, middle, and lower panels show the case in which observations entering the sample are delayed by 0, M l, and M < l period(s), (cid:21) respectively. 57

Figure 12: Correction factors for the cumulative index response to an exchange rate shock in a Calvo model with real rigidities All exits and entries All exits and entries are random are selective 1.5 1.5 1.4 1.4 1.3 alpha=0.00 1.3 alpha=0.60 1.2 1.2 1.1 1.1 1 1 0.9 0.9 0 5 10 15 20 25 0 5 10 15 20 25 period after impulse period after impulse All exits are selective All exits are random and all entries are random and all entries are selective 1.5 1.5 1.4 1.4 1.3 1.3 1.2 1.2 1.1 1.1 1 1 0.9 0.9 0 5 10 15 20 25 0 5 10 15 20 25 period after impulse period after impulse Notes: This(cid:133)gureshowsthetheoreticalcorrectionfactorsonewouldapplytothecumulative import price index response to a one-time jump in the exchange rate. The model assumes Calvo pricing with an exogenous frequency of price changes of 15 percent and a substitution rate of 5 percent. The real rigidity parameter is speci(cid:133)ed as in appendix B. 58