Monetary Policy's Role in Exchange Rate Behavior

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 652 November 1999 MONETARY POLICY’S ROLE IN EXCHANGE RATE BEHAVIOR Jon Faust and John H. Rogers NOTE:InternationalFinanceDiscussionPapersarepreliminarymaterialscirculated to stimulate discussion and critical comment. References to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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.bog.frb.fed.us.

MONETARY POLICY’S ROLE IN EXCHANGE RATE BEHAVIOR ∗ Jon Faust and John H. Rogers Abstract: While much empirical work has addressed the role of monetary policy shocksin exchange ratebehavior, conclusions have beenclouded bythelack of plausible identifying assumptions. We apply a recently developed inference procedure allowing us to relax dubious identifying assumptions. This work overturns some earlier results and strengthens others: i) Contrary to earlier (cid:12)ndings of \delayed overshooting," the peak exchange rate e(cid:11)ect of policy shocks may come nearly immediately after the shock; ii) In every otherwise reasonable identi(cid:12)cation, monetary policyshocks lead to large uncovered interest rate parity(UIP) deviations; iii) Monetary policy shocks may account for a smaller portion of the variance of exchange rates than found in earlier estimates. While (i) is consistent with overshooting, (ii) implies that the overshooting cannot be driven by Dornbusch’s mechanism, and (iii) gives reason to doubt whether monetary policy shocks are the main source of exchange rate volatility. Keywords: exchange rates; overshooting; forward premium bias; monetary policy; identi(cid:12)cation. ∗ Division of International Finance, Board of Governors of the Federal Reserve System. Email: faustj@frb.gov and john.h.rogers@frb.gov. The authors thank for useful comments Chris Sims, Eric Leeper, Lars Svensson, Mark Watson, Tao Zha, and seminar participants at Berkeley, Duke, the Federal Reserve Board, IMF, Indiana, Michigan, Michigan State, Ohio State, Penn State, Princeton, U.C. Santa Cruz, and Virginia. Thanks to Michael Sharkey and Molly Wetzel for excellent research assistance. Some computer code for the techniques used in this paper can be found at http://patriot.net/∼faustj. The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of other members of its sta(cid:11).

Exchange rate changes are volatile and di(cid:14)cult to explain. Economists have long suspected that monetary policy shocks might play an important role in accounting for this behavior, and a great deal of theoretical and empirical work has been directed at con(cid:12)rming this suspicion. This paper combines recent developments in international (cid:12)nance and econometrics to assess what (cid:12)rm conclusions can be drawn about the role of monetary policy shocks in exchange rate behavior. An obvious starting point for our study is Dornbusch’s (1976) overshooting 1 model. Having received over 800 citations, this work remains at the core of international (cid:12)nance. Dornbusch’s prediction that the exchange rate should initially overshoot its long-run level in adjusting to a monetary shock owes much of its huge appeal to two factors. First, it provides hope of explaining the empirical regularity thatexchange rates inthepost-Bretton Woods eraaremorevolatile thanmacroeconomic fundamentals such as the money supply, output, and interest rates. Second, theovershootingconclusion follows directlyfromthreefamiliarcomponents: theliquidity e(cid:11)ect of monetary policy shocks on nominal interest rates, uncovered interest rate parity (UIP), and long-run purchasing power parity (PPP). While overshooting is a dominant theory in international (cid:12)nance, its reliance on uncoveredinterestrateparitymeansthatwhenconfrontedwithdata,thetheorywill be enmeshed in a dominant empirical puzzle in international (cid:12)nance|the tendency of the exchange rate to change in the direction opposite to that predicted by UIP. Labelled the forward premium anomaly, this tendency has been extensively documented [Fama (1984), Hodrick (1987), and Engel (1996)]. Because almost all work on this anomaly has been of a reduced form variety, however, it remains an open question whether UIP holds conditionally in response to monetary policy shocks, in particular. Motivated by these facts, we focus on three questions: 1 Does the exchange rate overshoot? More speci(cid:12)cally, at what lag horizon does the exchange rate peak after a monetary policy shock? 1 Social Science Citations Index 1

2 Isthedynamicresponseoftheexchangerateroughlyconsistentwithuncovered interest rate parity? 3 Can monetary policy explain a large share of exchange rate variance under any reasonable theory? The (cid:12)rst two questions are about the relation between overshooting theory and the forward premium anomaly. The third addresses whether|under overshooting or any other theory of international (cid:12)nance|monetary policy shocks can account for a large share of exchange rate variance. Of the papers examining the (cid:12)rst question, one common (cid:12)nding is that the exchange rate overshoots its long-run value in response to policy shocks, but that the peak occurs after one to three years as opposed to happening immediately as 2 predicted by Dornbusch. A typical delayed overshooting result is shown in Figure 1, which gives the estimated dynamic response of the U.S. dollar/U.K. pound and dollar/German mark exchange rates to a stimulative U.S. monetary policy shock in our replication of work by Eichenbaum and Evans (1995). Based on such evidence, aconsensusseemstobeemergingthattheexchangerateshowsdelayed overshooting 3 and theorists are attempting to rationalize this fact. Few papers directly address question two. Eichenbaum and Evans (1995) (cid:12)nd that UIP is violated during the period when the exchange rate is rising toward its 4 delayed peak|UIP predicts that it shouldbefalling. As for question 3, concerning the share of exchange rate variability due to monetary policy shocks, papers report estimates between a few percent to over one-half [Eichenbaum and Evans (1995), 5 Rogers (1999), Clarida and Gali (1994)]. In all of this work, a crucial and highly contentious step is identifying which 2 The results are quite consistent for bilateral rates between the U.S. and Europe and Japan: Eichenbaumand Evans(1995) andClarida andGali (1994) nearly uniformly finddelay; Grilli and Roubini(1996) generally finddelay. Cushman and Zha (1997) findno delay for U.S.-Canada rate. 3 e.g., Gourinchas and Tornell (1996). 4 Cushman and Zha (1997) find that the pointwise confidence intervals for the deviation from UIPgenerally coverzero for theU.S.-Canada exchange rate. 5 In Clarida-Gali and Rogers this is theeffect of therelative money shock in thetwo countries. InEichenbaumandEvans,thisshareisforthemonetarypolicyshockintheU.S.only,aswefocus on in this paper. 2

exchange rate movements are due to monetary policy shocks. The paucity of highly credible identifying assumptions forces one to use questionable assumptions and limits the number of variables that can be included in the analysis|since larger modelsrequiremoreassumptions. Thisleads toquestions aboutwhethertheresults are robust to including other arguably relevant variables and to changes in the dubious assumptions. Faust (1998) develops an approach to avoiding these problems. It allows one to impose any highly credible restrictions and then summarize all possible ways of completing identi(cid:12)cation of the model. In this paper, we apply this technique to a standard 7-variable model and a new 14-variable model for both the US-UK and US-German bilateral exchange rates. We (cid:12)nd the following. 1 Thedelayedovershootingresultisquitesensitivetodubiousassumptions. The data are consistent with peak exchange rate e(cid:11)ects that are very early (say, within a month after the shock) or delayed several years. 2 Monetary policy shocks seem to generate large UIP deviations. Even when applying only minimal assumptions about what constitutes a monetary policy shock, a search for a money shock that generates small UIP deviations is fruitless. Thus, if exchange rates do peak early in response to policy shocks, this overshooting is apparently not UIP-driven, Dornbusch overshooting. 3 Consistent with earlier work, we (cid:12)nd in the 7-variable model that the U.S. policy shock might plausibly account for anything between 8 and 56 percent of the forecast error variance of the exchange rate at the 48-month horizon. In the14-variablemodel,however, thisrangeis2toabout30percent. Webelieve that the results for the smaller model may be due to omission of important variables. Theseresultsaredevelopedin5sections. InSections1and2,wediscussrelevant international (cid:12)nance theory and then an example of our approach to identi(cid:12)cation. Section 3 lays out the full approach, Section 4 has results, and Section 5 presents conclusions. 3

1 Overshooting, UIP, and the forward premium anomaly 1.1 Overshooting The Dornbusch overshooting hypothesis predicts that ceteris paribus a one-time permanent increase in the money stock will cause the exchange rate to depreciate on impact beyond its long-run value and then appreciate toward the terminal value. Overshooting is a robust prediction of models exhibiting three standard building blocks: a liquidity e(cid:11)ect of monetary policy shocks, UIP and long-run PPP. By long-run PPP, the exchange rate must ultimately settle at a depreciated value after the money expansion. In the short-run, the liquidity e(cid:11)ect of the money expansion implies that home interest rates to fall relative to foreign rates. UIP requires that Es t+1 −s t = i t −i(cid:3) t , (1) wheresis thelogarithm of thenominalexchange rateandiandi(cid:3) arethehomeand foreign one-periodinterest rates. Ififalls relative to i(cid:3) , thentheexchange rate must be expected to appreciate. Appreciation to a depreciated long-run value implies an 6 initial jump depreciation that overshoots the long-run value. Each of the three building blocks is open to question empirically. Long-run PPP could fail, but any failure present in the data is not signi(cid:12)cant enough to play a large role in our analysis. The liquidity e(cid:11)ect is more problematic. There is still great uncertainty about the size and duration of the liquidity e(cid:11)ect [Leeper and Gordon, 1992; Pagan and Robertson, 1994; Bernanke and Mihov, 1998]. Much of the complication is due to the identi(cid:12)cation problems: the data do not clearly supply us with experiments of unilateral exogenous changes in the money supply, making identi(cid:12)cation of the e(cid:11)ects of a monetary policy shocks controversial [e.g., Rudebusch, 1998; Sims 1998]. 6 The large body of theoretical work related to Dornbusch’s hypothesis and exchange rate dynamicsmoregenerallyincludesAlvarez,Atkeson,andKehoe(1999),Backus,Foresi,andTelmer (1996),Chari,Kehoe,McGrattan(1998),EatonandTurnovsky(1983),Frenkel(1982),Gourinchas and Tornell (1996), Kollmann (1999), Mussa (1982). 4

1.2 UIP and the forward premium anomaly The UIP element of the Dornbusch model is most problematic empirically. Under covered interest parity, i t −i(cid:3) t = f t −s t, where f t is the logarithm of the forward rate. A common test of UIP considers the following regression (or its equivalent under covered interest parity), s t+1 −s t = α+β(i t −i(cid:3) t )+ε t (2) If (1) holds, the population values of the coe(cid:14)cients are α = 0 and β = 1. In practice, for a wide range of currencies and time periods, one (cid:12)nds β signi(cid:12)cantly 7 less zero, with point estimates often below -1. This result is the core of the forward premium anomaly. The deviation from UIP, call it ξ, is the forward premium: ξ t (cid:17) (i t −i(cid:3) t )−(E[s t+1]−s t) = f t −E[s t+1]. (3) So long as capital markets are open and the interest rates are for nominally riskless, highly liquid bonds, then there are two primary explanations for the negative β: ξ t is a time-varying risk premium or the regression does not account for people’s expectations properly. Fama (1984) demonstrated that the negative β implies a negative covariance between ξ t and the expected change in the exchange rates. He argued that this is problematic for the risk premium explanation, as it implies the 8 risk premium is highest when the currency is expected to appreciate. The alternative explanation is that the forward premium regression is mismeasuring expectations. In this view, the β coe(cid:14)cient is a biased estimate of the population value, say, due to learning or peso e(cid:11)ects [see Engel’s (1996) survey]. More recently, Phillips and Maynard (1999) have shown that estimates of β in (2) are biased downward due to the persistence of the interest rate spread. 7 This result is most consistent for short-horizon changes and bilateral dollar exchange rates. SeeFama (1984), Hodrick (1987), Canova and Marrinan (1993), and Engel (1996). 8Backus,ForesiandTelmer(1998)furthercharacterizetheFamapuzzle,bycharacterizingwhat thenegative β implies within a standard class of asset pricing models. 5

These results are about unconditional UIP|the response of the exchange rate to all shocks on average. The results shed little light on whether monetary policy shocks systematically generate deviations from UIP. Indeed, of all the sources of uncertainty we often speak of, one might suppose that money shocks are least likely to generate large short-run fluctuations in risk premia. To investigate this question, we must identify the response to a monetary policy shock. 2 Conventional identification A linear reduced form model is consistent with in(cid:12)nitely many causal structures of themodel. Theproblemofidenti(cid:12)cation isto chooseamong thesecausalstructures. Take the reduced form dynamic model, B(L)Y t = u t , (4) (cid:80) where Y t is an (n(cid:2)1) vector of data, B(L) = p i=0 B i Li , B 0 = I, LY t = Y t−1, and u t is a vector of shocks. One can premultiply both sides of (4) by any full rank matrix A 0 to arrive at a system A 0 B(L)Y t = A 0 u t, which can be written 9 A(L)Y t = w t . For any A 0, the system has reduced form, (4), and has moving average representation, Y t = A(L) −1w t, or Y t (cid:17) C(L)w t (5) The dynamic response (impulse response) of, say, the ith variable to an impulse to the jth shock, w t, is given by the coe(cid:14)cients of C ij(L). The practical problem is that while each A 0 gives a system with the same reduced form, each gives rise to a di(cid:11)erent impulse response function. As Koopmans and the Cowles commission emphasized [1953], one can only choose among these di(cid:11)erent causal interpretations by bringing to bear a priori identifying restrictions. 9 SinceB 0 =I, it makes sense toname A(L)≡A 0 B(L): thecoefficient of L0 in A(L) is A 0. 6

The standard practice in the recent VAR literature is to identify only the dynamic response to a shock of particular interest, in our case the monetary policy shock. The causal structure of the remainder of the system is left uninterpreted. Conventional VAR identi(cid:12)cation begins with the assumption that the underlying structural shocks are orthogonal. We too will maintain this assumption 10 throughout. After assuming orthogonality of the shocks, identifying the response to the money shock requires N −1 additional assumptions in an N variable system. The identi(cid:12)cation is usually completed using restrictions on contemporaneous interactions: outputdoes notrespondto apolicy shockwithin themonth, or foreign policy 11 does not respond to home policy within the month. One can typically construct plausible arguments for such restrictions [e.g., Leeper, Sims, and Zha (1996)]. 2.1 Example: delayed overshooting in a 7-variable model Take the the seven-variable model of Eichenbaum and Evans (1995). The model contains U.S. and foreign industrial production (Y and Y(cid:3) ), the U.S. consumer price index (P), U.S. and foreign short-term interest rates (i and i(cid:3) ), the ratio of U.S. non-borrowed reserves to total reserves (NBRX), and the exchange rate in dollars per foreign currency (S) (for details, see appendix A). All variables except interest rates are in logs. Data are monthly from 1974:1 to 1997:12. The reduced form is estimated with 6 lags of each variable and a constant. In this example, we focus on the US-UK case. In a preferred identi(cid:12)cation approach, Eichenbaum and Evans identify the responsetoapolicyshockbyimposingtherecursiveordering[Y,P,Y(cid:3),i(cid:3),NBRX,i,S]. The shock to NBRX is interpreted as the money shock. This recursive ordering implies 6 substantive assumptions: Y, P, Y(cid:3) , and i(cid:3) do not respond to U.S. policy shocks within the month that they occur, and policy does not respond to shocks to 10 In data measured at sufficiently high frequency, this assumption is not highly controversial. Even with monthly data, interactions within themonth could cause problems. 11 Somepapersimposerestrictionsimpliedby,e.g.,long-runmonetaryneutrality. SeeBlanchard and Quah (1989) and Faust and Leeper (1997) for a critique. 7

i and s within the month. This basic identi(cid:12)cation scheme has been used in closed-economy settings by Strongin (1995) and others. The impulse responses look familiar from closed- 12 economy applications (See Figure 2, solid lines). The rise in NBRX is associated with a decline in nominal interest rates, a hump-shaped response of output that peaks around 12 to 18 months after the shock, and an initial negative response of prices that eventually turns positive. These sorts of e(cid:11)ects are generally taken as reasonable in the literature. The exchange rate response to money peaks at about three years in these estimates, supporting the delayed overshooting conclusion. 13 Whiletheidenti(cid:12)edmoneyshockpassestheducktest, atleast3ofthe6identifying restrictions are questionable. Fed policymakers are aware of data for exchange rates and domestic interest rates up to the minute when their policy decisions are taken; it is unlikely that surprising movements in those variables are ignored by policymakers. The assumption that the foreign short-term interest rate does not respond to policy within the month is also questionable. The domestic short-term rate and the exchange rate can (and do in the VAR) react contemporaneously to policy. These two variables are tied to the foreign short-term interest rate and the forward exchange rate by covered interest arbitrage. It is di(cid:14)cult to imagine why the foreign short-term rate would not do some of the adjusting to make covered interest parity hold. The use of questionable restrictions is no secret, and the standard response is to present results for a few sets of identifying assumptions. Eichenbaum and Evans assess other recursive orderings of these variables and (cid:12)nd that key results such as delayed overshooting arise systematically. Of course, the arguments against the preferred recursive ordering hold for any recursive ordering. Indeed, identi(cid:12)cations showing simultaneity among money mar- 12 InFigures2and3weprovideerrorbandsaroundtheOLSpointestimates. Thesearecreated usingthetheBayesiansimulationmethodunderthenaturalconjugatepriordescribedintheRATS manualandSimsandZha(1999). Weused1000draws;the68percentcoveragebandsarethe16th and 84th percentile points from thesimulation. 13 Ifit walks like a duckand quackslike a duck,it might actually bea duck. 8

ket variables are surely at least as plausible as any recursive ordering. As a result, it is natural to wonder whether results like delayed overshooting are somehow special to recursive formulations or also hold for other plausible formulations. 2.2 Example continued: early peaks in the exchange rate Lacking agreement on a set of credible identifying assumptions, one option is to search all possible identi(cid:12)cations allowing simultaneity among [i(cid:3),NBRX,i,s]. If allthecredibleidenti(cid:12)cations showdelayedovershooting, theissueissettled. Otherwise, one mustadmit to uncertainty about the peaktiming until sharperidentifying restrictions emerge. In the US-UK example, the method described below turns up many identi(cid:12)cations of the 7-variable model that show no delayed overshooting. The exchange rate response to the policy shock in one such identi(cid:12)cation is shown on Fig. 2a (dashed lines). The response to the policy shock is strikingly similar to the recursive identi(cid:12)cation in all respects except that the exchange rate e(cid:11)ect peaks in the (cid:12)rst month after the shock. Indeed, the solid and dashed point estimates typically lie almost entirely within each other’s error bands, except for the (cid:12)rst half-year for 14 the exchange rate and UIP deviation. The dashed line identi(cid:12)cation involves the same recursivity with respect to Y, Y(cid:3) and P as the fully recursive identi(cid:12)cation. Indeed, the only notable di(cid:11)erence is that in the recursive system a policy shock that lowers i (by around 15 to 20 basis points) is restricted to have no impact e(cid:11)ect on i(cid:3) , but such a shock lowers i(cid:3) (by 15 around5to15basispoints)inthedashedlines. Thissortofevidencesuggeststhat 14 Note that the fact that the pointwise error bands do not overlap for the first few months of the exchange rate response does not mean that the peak timings are statistically significantly different. Changing the peak timing on a response involves simultaneously changing several of the point responses and, thus, inference on peak timing requires consideration of both variances and covariances of theresponses. As Kilian and Chang (1998) show, one can simulate the peak timing itself and calculate coverage intervals. We takeup such simulations below. 15 NeitherEichenbaum-Evansnorwereport theresponse of thevariables tothe6uninterpreted shocks in the system. There are, however, no differences in the two systems with respect to the responsetothefirst3uninterpretedshocks(theorthogonalizedshocksinthey,p,andy∗equations). Inthesimultaneous system,i∗,i,NBRX andS shockseach respondtotheorthogonalized shocks to i∗, i and S. Since there is a presumption in favor of simultaneity among these variables, this 9

thedelayed overshooting resultmaybesensitivetodubiousidentifyingassumptions. Thenextsection presents a method to more systematically dothis sortof structural inference when identifying assumptions are questionable. 3 Setting aside dubious identifying assumptions Faust(1998) develops an approachto inferencewhenoneislacking su(cid:14)cientrestrictions to identify the items of interest. One imposes any credible assumptions, but there will generally remain a range of possible answers to questions of interest when the assumptions do not fully identify policy shock. This method allows one to do inference about this range of answers by systematically searching all identi(cid:12)cations 16 consistent with the restrictions. For a more complete description of the approach, see Faust (1998). 3.1 Searching all reasonable identifications Given the reduced form (4) we can always choose an A 0 that transforms the model to have orthogonal errors with unit variance (any recursive ordering will do this): Y t = C(L)w t where Ew t w t 0 = I. The choice of unit variance is merely a normalization. Every money shock in every possible identi(cid:12)cation (that maintains orthogonal, unit variance shocks) can be written, α0w t for some α satisfying α0α = 1. Thus, we can cast our search of reasonable identi(cid:12)cations as a search of the unit vectors α, with each α de(cid:12)ning a shock α0w t. We can limit the search by imposing some identifying restrictions we (cid:12)nd credible. A second useful fact is that for the shock de(cid:12)ned by α, zero restrictions and sign restrictions on the impulse response to a money shock imply linear restrictions on α. Thus, the restriction that a stimulative money shock raises money growth difference is not of much usein distinguishing the credibility of these two formulations. 16 Thisapproachcanbeseenasageneralization oftheapproachinKingandWatson(1992)and Bernankeand Mihov (1998). 10

on impact can be written Rα (cid:21) 0, where the elements of the row vector R depend only on C(L). Each added restriction adds a row to R. 17 Restrictions on linear combinations of impulse responses are also of this form, so one can restrict whether the impulse response function is rising or falling between two points. Once we impose all highly credible assumptions, the problem is that the policy responseis still notidenti(cid:12)ed. We would still like to have away to see whatrange of answerstoourquestionsispossibleafter onlyimposinghighlycredibleassumptions. For some properly structured questions, we can cast the search for this range of answers as a straightforward optimization. Take question 2 in the introduction. One measure of UIP deviations after a policy shock is the root mean square UIP deviation over the (cid:12)rst, say, 4 years after a policy shock. The expected UIP deviation at t+l of a shock at t is given by, 18 c(i,l)−c(i(cid:3),l)−400[c(s,l+3)−c(s,l)]. where c(x,l) is the response of variable x at lag l to the shock de(cid:12)ned by α. The mean square expected UIP deviation (hereafter, UIPD) comes from summing the 19 squared deviations over some horizon. In interpreting the results for UIPD, it is useful to remember that the mean square UIP deviation can be written as the squared mean deviation plus the variance over the chosen horizon. Thus, a large UIPD implies either large absolute deviations or highly variable deviations, or both. Some simple algebra shows that the root mean square expected UIP deviation can be written, (α0Mα) 1/2 , where the elements of M are functions only of C(L). 17 Therestriction thatthemoneyshockhasapositiveeffectonthejth variableatlagk requires puttingthejth row of C k as a row of R. 18Thisisannualized,presumesmonthlydata,andthree-monthinterestratesinannualpercentage rate units. 19 Some trickytiming anddefinition questions arise. Weusemonthly averagedata for exchange rates and interest rates. If the identification is correct, then the calculated UIP deviations should beinterpreted as theexpected path of themonthly-average UIPdeviation in response to a money shock. TheVARtreatsthemonthlyaverageofthemoneymarketvariablessuchasforeign interest rates and the exchange rate as available when policy is made for the month. This is not strictly correctandcouldcontaminatetheidentification. ThisproblemisnodifferentfromotherVARsand isnotsolved,say,bymethodsthatassumethatthepolicymakerseesnoneofthemonthlyaverage. Indeed, the approach of this paper is meant specifically to shed light on this sort of problem by assessing whetherresults are sensitive to different assumptions in this regard. 11

To (cid:12)nd whether there is any shock satisfying restrictions and leading to small UIP deviations, we can do the optimization, minα0Mα α subject to α0α = 1, R s α (cid:21) 0, R z α = 0, where R s and R z reflect the credible sign and zero restrictions, respectively. Faust (1998) shows how to do this optimization. If the minimum UIPD is large, then we have a robust conclusion that money shocks generate large UIP deviations: small UIP deviations are not mutually consistent with the restrictions. If the minimum UIPD is small, but the analogous maximum UIPD is large, we conclude that UIPD is not sharplyidenti(cid:12)ed. All three 20 questions can be handled in this manner. 3.2 Inference Up to this point, the discussion has focussed only on point estimates, and thus we have not taken account of the fact that the reduced form parameters must be estimated. We propose two inference methods. The (cid:12)rst method is an extension of the conventional simulation method used to 21 produce the error bands on the impulse responses in Figure 2. For any particular value of the reduced form parameters of the VAR, we can calculate the minimum and maximum for the parameter of interest, say, θ, under the chosen restrictions. For example, θ might be the UIPD, and we calculate θ min and θ max. Using the standard simulation method we can obtain coverage intervals for θ min and θ max just as we would for impulse responses. We treat the 5 th percentile of θ min and the 95 th percentile of the θ max as a robust 90 percent coverage band. 20 Question 3 can be handled in the same manner, interpreting the question as asking whether thepolicy shock accountsforalarge shareof theforecast error varianceoftheexchangerate. The forecast errorvarianceshareduetotheshockdefinedbyαcanalso bewrittenasaquadraticform inα. Question1issomewhatdifferent. Forquestion1,wecanimposethattheexchangeratepeak in, say,thefirst or second period after theshock and then usetheoptimization algorithm tosee if thereis any shock that satisfies themoney restrictions and theearly peak restriction. 21 As noted above, these are based method described in the RATS manual and studied recently in Sims and Zha(1999). 12

The intervals are robust in the following sense. Remember that the few restrictionsweimposewillnotbesu(cid:14)cienttoidentifyθ. Thecoverageintervalwecalculate willberobustinthatitwillcontain thecoverage intervalthatwouldbeobtainedun- 22 der any additional restrictions (so long as the restrictions are not overidentifying). Thus, any value outside the coverage interval under the minimal set of restrictions would also be outside under additional restrictions. While robust, these coverage intervals may in practice turn out to be quite large. In computing the minimum and maximum of θ in the simulation, one cannot computationally impose everything one believes about the policy shock, tending to 23 increase the size of the coverage interval. Procedure2ispartialremedyforthisproblem. Wetakethemaximumlikelihood estimate of the reduced form parameters and (cid:12)nd the range, [θ^ min ,θ^ max] , for the parameter of interest. We call this a \nonrejection region" in that it provides a set of points that probably should not be rejected without further evidence. This approach has intuitive appeal, since it basically rests on the assumption that we should not reject any value for θ consistent with the maximum likelihood estimate. We know, for example, that valid classical con(cid:12)dence intervals that always contain [θ^ min ,θ^ max] can be formed. 24 Under procedure 2, we can fully inspect the impulse responses giving rise to the limiting values θ^ min and θ^ max. Thus, we can avoid the problem of procedure 1 by verifying that the validity of any restrictions we believe. Overall, in procedure 1, we impose less than we may believe and reliably learn only about parameter values that are unlikely; in procedure 2 we can impose su(cid:14)cient conditions for a money shock that may be more than is strictly necessary, but 22Thatis,solongasthesupportforthereducedformparametersundertheadditionalidentifying assumptionsisthesameasthatundertheoriginalrestrictions. Therobustnessresultfollowsdirectly from the fact that on each draw θ max must be weakly greater than and θ min weakly less than the θs that would be obtained undermore restrictions. 23 Perhaps20restrictionscanpracticallybeimposedinthe14variablemodel. Oneachdraw,the calculatedminimummustriseandmaximummustfallwhenadditionalrestrictionsareimposed—so longasthereisashock consistent with therestrictions. Inthesimulation, draws inconsistent with therestrictions are thrown out. This is discussed furtherin Faust (1998). 24 Confidence intervals implied, say, by inverting a likelihood ratio test for θ would share this property. While the likelihood ratio test need not have optimality properties in the current case, optimal inference in this case is an open question. 13

we reliably learn only about points that probably should not be rejected without further information. 4 Empirical results In this section we address the three questions posed in the introduction, providing evidence on the (i) timing of the peak exchange rate e(cid:11)ect, (ii) size of UIP deviations following policy shocks, and (iii) maximum share of exchange rate variation that can be explained by money shocks. For each questions, we (cid:12)rst present evidence about the nonrejection region for the parameter of interest|the minimum and maximum values for the parameter consistent with the chosen restrictions at the OLS point estimate of the VAR. We attempt to demonstrate that the impulse responses associated with these minima and maxima are reasonable responses to a money shock. When the nonrejection region is small, we move on to the simulated coverage interval results to see if we can con(cid:12)dently reject any values. We present results for 4 models: 7-variable and 14-variable models for a US- UK system and US-Germany system. The 7-variable model was discussed above. The 14-variable model consists of home and foreign output (Y and Y(cid:3) ), prices (P and P(cid:3) ), money supplies (M and M(cid:3) ), short-term nominal interest rates (i and i(cid:3) ), and long-term nominal interest rates (r and r(cid:3) ). We also include commodity prices (CP), and U.S. non-borrowed reserves (NBR), and total reserves (TR). All variables are in logarithms except the interest rates; the sample period and number of lags are as in the 7-variable model. Itis worth noting thatthe 7-variable modelis large bythestandardsof the VAR literature, but contains no long-term interest rates and has only two foreign variables, Y(cid:3) and i(cid:3) . Clearly, these 7 variables may not contain all variables relevant to sorting out the transmission of monetary shocks at home and abroad. A signi(cid:12)cant bene(cid:12)t of the approach of this paper is the ability to study larger models without having to rely on increasingly questionable identifying assumptions. 14

4.1 When does the exchange rate peak after a monetary policy shock? Return to the 7-variable model discussed in the example above. For the US-UK case, we have already presented a nonrejection region of 1 to 35 months for the peak exchange rate e(cid:11)ect in the example and argued that both values are associated with reasonable shocks. For Germany we (cid:12)nd a range of 1 to 28 months (Table 1). The US-GE impulseresponses are in Figure 2b; once again, the solid line is the recursive identi(cid:12)cation and the dashed line shows an identi(cid:12)cation involving simultaneity among the money market variables. The responses of output, prices, non-borrowed reserves and interest rates in the alternative are remarkably similar to those in the recursive identi(cid:12)cation. This suggests that the alternative is reasonable|at least 25 from the perspective of recent VAR applications. The 14-variable models give very similar nonrejection regions (Table 1). The impulse responses associated with these ranges are reasonable by conventional standards (Figure 3a and 3b). Overall, the results for the 7 and 14 variable models are quite consistent and lead us to conclude that for the peak exchange rate response, a range of one-month to roughly three-years is consistent with the data. 4.2 Monetary policy shocks and UIP We now turn to the question of whether the economy approximately satis(cid:12)es UIP conditionally in response to monetary policy shocks. We know that there are large unconditional UIP deviations in the data. For both countries and both the 7 and 14-variable models, the unconditional UIPD is about 200 basis points. For the 7variable model, we (cid:12)nd a nonrejection region for the conditional UIPD of about 30 25 Althoughhowwefound thesealternativeidentifications doesnotmatterforthepoint,it may be of interest. The dashed lines on Figures 2 were generated by imposing: (1) the impact effects on P, Y, and Y∗ are zero on impact; (2) the impact effect on i is negative, and on NBRX and S positive; (3) the response of P at lag 80 is no larger than at lag 36; (4) (U.K. only) the response of S at lag 23 is nolarger than at lag 12. Figures 2a and 2b represent theidentification consistent with these restrictions that explains the largest share of the forecast error variance of output at a horizonof48months. Ithappensthatthisgivesanearlypeakevenwhenthatisnotimposed. The approach in the 14-variable model is verysimilar. 15

to 90 basis points in the US-UK and US-GE models (Table 2). The lower bounds are associated with the recursive orderingand theupperboundsare associated with the alternative identi(cid:12)cations in Figure 2, which have already been argued to be reasonable. Even the lower bounds are surprisingly large, since they result from short-term interest rate declines that are brief and do not exceed 25 basis points at any time. From Figure 2, it is clear that the UIP deviations are both large at times and quite variable. The peak deviation is much larger than the changes in short-term rates and interest rate di(cid:11)erentials. The non-rejection regions for the UIPD in the 14-variable model are similar (Table 2); impulse responses associated with the minima are shown in the dashed lines on (cid:12)gure 3; maxima are associated with the solid lines. These results suggest that we cannot reject the existence of large and variable UIP deviations, but shed no light on whether there are other reasonable money shocks that produce small deviations. The simulated coverage intervals can help answer this question. Table 2 presents one-sided (left-tail) boundsfor the simulated coverage intervals 26 th th on UIPD. These are the 5 and 10 percentiles of the minimum UIPD from the 27 simulation. On each draw, we calculate minimum UIPD|root mean square UIP deviation at horizon 48|subject to certain restrictions. We consider 2 sets of restrictions. Firstare moneyrestrictions (MR) meant to benecessary fora reasonable monetary policy shock. In the 7-variable model, these are that the responses of: (1) P, Y, Y(cid:3) , NBRX, and S are greater than or equal to zero on impact; (2) i and i(cid:3) are less than or equal to zero on impact; and (3) P at horizon 80 is no larger than at horizon 36, Y(cid:3) is no more than one-half of that of Y on impact, and the decline in i(cid:3) is no larger than one-half of the decline in i on impact. The second type of restrictions are shape restrictions (SR) on the path of the 26 The results are from 1000 simulation draws. 27 Results for thefull distribution are more informative than just thesetwo points. Suchresults areomittedfor brevity,butareavailablefrom theauthors. Ineverycasepresentedintables2and 3, the simulated distributions appear to be single peaked with the peak to the right of the 10th percentile point. 16

exchange rate. Speci(cid:12)cally, we imposethat theexchange rate responsefalls between lags 1{2, 2{3, 3{4, 4{6, 6{12, 12{18, 18{36, and 18{80. In the 14-variable model we use slightly di(cid:11)erent restrictions. This is because the computational burden goes up with the number of sign restrictions we use. In the 14-variable model we use for MR: (1) P, P(cid:3) , and Y(cid:3) are zero on impact; (2) Y, CP, NBR, M, M(cid:3) , S are greater than or equal to zero on impact, as is Y at lag 8; (3) i and i(cid:3) on impact, and i at horizon 4, are less than or equal to zero; (4) P at horizon 80 is no larger than at horizon 36; (5) on impact, the drop in i(cid:3) is no more than one-half of the drop in i; and (6) on impact, the rise in M(cid:3) is no more than one-half of the rise in M. Our shape restrictions on the exchange rate require that it fall between periods 1{2, 2{6, 6-12, 12{18, 18{36, 18{80. We report results under three combinations of these restrictions, (i) MR only; 28 (ii) MR and SR, and (iii) neither MR nor SR, under the columns labelled \none". th For purposes of discussion we focus on the 10 percentile values associated with a 90 percent con(cid:12)dence bound. For both models and countries, we (cid:12)nd that when no restrictions are imposed one cannot rule out UIPDs of less than 10 basis points. Requiring the shock to satisfy the money restrictions raises this total to about 20 basis points. Further requiring that the shock satisfy the shape restrictions|so that the exchange rate th must peak in the (cid:12)rstmonth|raises the 10 percentile about 10 more basis points. Thus, money shocks that generate the sort of modest and short-lived e(cid:11)ects on interest rates seen in the earlier (cid:12)gures, seem to be associated with UIP deviations 29 that are at least 20 basis points (in the root mean square sense) over 4 years. This result is largely una(cid:11)ected by whether one restricts the exchange rate to peak early or not. Thus, even when the exchange rate peaks early, it is not driven by UIP as it would be under Dornbusch overshooting. Given that there are large UIP deviations, we can ask whether these deviations 28By“none”wemeannoimpulseresponserestrictions. Inallcases,weimposethatthestructural shocks are orthogonal with unit variance. 29 WealsoestimatedUIPDsatthe18-monthhorizon. Theresultsweresosimilartothoseatthe 48-month horizon that we omit them for brevity. 17

have the correct correlation patterns to help generate the negative β (in (2)) in the forward premium anomaly. We can decompose this unconditional β into a weighted average of the β associated with each shock: (cid:88)n β = ω j β j j=1 where β j is the β that would emerge if all shocks but shock j were eliminated from thesystemandω j isthevarianceshareofshockj inthetotalvarianceoftheinterest rate di(cid:11)erential (See Appendix B). Thus, each structural shock either contributes to the anomaly|pushing β downward|or tends to o(cid:11)set it some. In our data, the unconditional OLS estimates of βs are -1.71 (t=-3.2) for the UK and -0.54 (t=-0.53) for Germany. For all models except the UK 14-variable model, the nonrejection region for the conditional β is greater than [−1.5,1.5]; for the UK 14-variable model all the conditional βs were above 2. 30 Further, the ωs for the policy shocks are always quite small. Across both countries and both 7-variable 31 and 14-variable models, the non-rejection region is [.02,.07]. Overall, while money shocks generate fairly substantial UIP deviations, these shocks recieve relatively little weight in determining the unconditional β, and the sign on the contribution is not clearly identi(cid:12)ed. Thus, the evidence does not clearly support the view that monetary policy shocks are the source of the forward premium bias. 4.3 How much exchange rate variation is due to monetary policy shocks? Table 3 provides a nonrejection range for the forecast error variance share of the exchange rate explained by the money shock at horizon 48. In the 7-variable model the nonrejection region runs from about 10 to 50 percent for both countries, consistent with earlier estimates for recursive identi(cid:12)cations [Clarida and Gali, 1994; Eichenbaum and Evans, 1995; Rogers, 1999]. 30 Actually, we calculate the contribution of the money shock at the 48-month horizon. This is very close tothe full contribution. 31 In Figure 2a, for example, the implied βs are 0.65 for the solid line and -1.44 for the dashed line, and thecorresponding values for ω are .04 and .06. 18

It is informative to examine the shocks that produce the variance shares at the upper end of the range. These shocks produce the largest deviations from UIP (the UIPDsareover 100basispoints),andareassociated withthelargestnegativevalues of β|values like −11. Further, these shocks explain almost none of the variance of output. Thus, while one can (cid:12)nd money shocks that account for a large part of the variance of the exchange rate, they do so by producing very extreme and, perhaps implausible, exchange rate behavior. The simulated coverage intervals again provide additional evidence on this question. We are interested in whether or not large variance shares are consistent with th theevidence, andhencefocusonthe90 percentile. Inthe7-variablemodel, forthe cases where only money restrictions are imposed, the upper-bound estimate of the exchange rate variance share is over 55 percent for the U.K. and Germany, consistent with earlier results. Onceagain, addingtheshaperestriction that theexchange rate peak early does not change the picture much. Overall, in the 7-variable model, it would be di(cid:14)cult to reject that policy shocks account for over half the variance of exchange rate changes, so long as one is relatively agnostic about the response of the exchange rate. In the 14-variable model, the non-rejection region for the variance share is much smaller than in the 7-variable model: 2 to 6 percent in the case of the U.K. and 2 to 13 percentfor Germany. Thesimulation results for the14-variable modelusingonly themoneyrestrictions alsoproducemuchsmaller upper-boundestimates|about 30 percent for both countries. Theseresultsshouldgivepausetothosewhobelievethatmonetarypolicyshocks are the primary culprit leading to exchange rate variance. In the 7-variable model, policy shocks can account for a large share, but the exchange rate response to the shocks is very odd. In the broader 14-variable model, large shares are far less likely. 19

4.4 Caveats Aswithallworkinthisarea, theseresultsshouldbereadwithcaution. Theyarefor US-UK and US-Germany only and only deal with the U.S. monetary policy shock. While the conclusions are meant to be robust to implausible identifying assumptions, we have imposed some assumptions. If the monetary shocks consistent with theseassumptionsare viewed as too peculiar, perhapstheseidentifying assumptions should be questioned. Further, while our conclusions are robust to some identi(cid:12)cation criticisms, there are ongoing debates about many possible problems with VAR work. For example, Rudebusch [1998] raises many of these arguments; Sims [1998] responds that these problems are not so serious. Continued progress on such issues as seasonal adjustment, structural stability, variable selection, and use of revised data will undoubtedly shed additional light on the questions of this paper. 5 Conclusions Empirical work on the role of monetary policy shocks in explaining exchange rate behaviorisimpededbythelackoffullycredibleidentifyingassumptions. Thispaper applies an inference approach to test the robustness of conclusions to the relaxation of dubious assumptions and changes in the number of variables in the model. We (cid:12)ndthat thedelayed overshooting resultis sensitive to dubiousassumptions. This conclusion comes from loosening the standard assumption of recursiveness in money market variables to allow plausible simultaneity. We also (cid:12)ndthat monetary policy shocks generate large expected root mean square UIP deviations. Even when imposingverylittle onthebehaviorofthemoneyshock, weareunableto(cid:12)ndpolicy shocks that generate interest rate and exchange rate responses roughly consistent with UIP. There is little evidence, however, that these large UIP deviations are the main source of the forward premium anomaly, and, indeed, monetary policy shocks may tend to o(cid:11)set what would otherwise be a larger anomaly. Finally, the results suggest that monetary policy shocks may explain less ex- 20

changeratevariancethanpreviouslybelieved. Inour7-variablemodel,policyshocks that account for much of the variance of the exchange rate also seem to generate veryoddexchangeratebehavior. Inthe14-variable model,we(cid:12)ndithighlyunlikely that U.S. policy shocks account for more than one-third of exchange rate variance. Theseresultshaveimportantimplications forwhatstylized factstheoristsshould be attempting to explain and they present a mixed bag for theorists hoping that relatively conventional theories will do the trick. The results allow for an early peakin the exchange rate, which might give a role for the conventional overshooting model. Unfortunately, the bulk of the variance of the exchange rate after policy shocks is due to large deviations from UIP. This is inconsistent with Dornbusch overshooting, and indeed, no conventional models we are aware of generate large variance in foreign exchange risk premia in response to policy shocks that have the modest e(cid:11)ects on output and interest rates that we (cid:12)nd. Perhaps models in which large ex post UIP deviations arise from information problems o(cid:11)er greater hope. 21

Appendix A: Data The data were acquired through the Federal Reserve Board’s database and the IMFs International Financial Statistics database. All series are expressed in natural logarithms except interest rates, which are expressed in percentage points. The series de(cid:12)nitions and sources are listed as follows: Source: Federal Reserve Board Y (Y(cid:3) ) = index of U.S. (foreign) industrial production - total, 1992 base; P = U.S. CPI - all urban, all items; NBR = non-borrowed reserves plus extended credit, seasonally adjusted, monthly average; TR = total reserves, seasonally adjusted, monthly average; NBRX = NBR/TR; S = spot exchange rate; monthly average; US$/foreign currency; CP = commodity prices - materials component of the U.S. producer price index. M (M(cid:3) ) = U.S. (foreign) money supply, seasonally adjusted; M1 for U.S. and Germany, M0 for the U.K. r (r(cid:3) ) = U.S. (foreign) ten-year Treasury bond rate. Source: IMFs International Financial Statistics i(cid:3) = foreign t-bill rate, percent per annum (line 60c); i = U.S. t-bill rate, percent per annum (line 60c); P(cid:3) = foreign consumer price index, (line 64). Appendix B: Decomposing β The population β is, β = cov(s t+3 −s t ,i t −i(cid:3) t ) var(i t −i(cid:3) t ) (cid:88)n covj(s t+3 −s t ,i t −i(cid:3) t ) = j=1 var(i t −i(cid:3) t ) (cid:88)n varj(i t −i(cid:3) t )covj(s t+3 −s t ,i t −i(cid:3) t ) = j=1 var(i t −i(cid:3) t ) varj(i t −i(cid:3) t ) (cid:88)n = ω j β j , j=1 where the sums are across all shocks j and covj and varj are the covariances and variances of the argument variables if shock j were the only shock with positive variance. The weight ω j is the variance share of shock j in interest di(cid:11)erential and β j is the β that would result if shock j were the only shock. 22

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Table 1: Nonrejection ranges for timing of peak exchange rate e(cid:11)ect in months Country nvar Min. Max. US-UK 7 1 35 US-GE 7 1 28 US-UK 14 0 47 US-GE 14 0 30 Notes: Reading from the top to bottom row, the impulse response functions associated with these peaks are shown in Figures 2a, 2b, 3a, and 3b. In each case the minimum is from the dashed line; the maximum is from the solid line. Table 2: Nonrejection range and one-sided con(cid:12)dence interval for UIPD (root mean square UIP deviation in percent) Rejection Nonrejection MR MR+SR none th th th th th th country nvar min. max. 5 10 5 10 5 10 US-UK 7 0.37 0.82 0.19 0.21 0.30 0.34 0.08 0.09 US-GE 7 0.31 0.92 0.16 0.18 0.24 0.27 0.08 0.09 US-UK 14 0.28 0.70 0.20 0.23 0.23 0.27 0.07 0.07 US-GE 14 0.40 0.92 0.19 0.22 0.22 0.24 0.07 0.07 Notes: The impulse responses giving the nonrejection ranges are as in Table 1. In the 7-variable models (Figs. 2a and 2b), the minimum is from the solid line and the maximum is from the dashed line; in the 14-variable models (Figs. 3a and 3b), this is reversed. From top left to bottom right, the posterior odds in favor of the restrictions are 61.5, 4.1, 1, 999, 2.9,1,1, 25.3, 1,1,199,1. Table 3: Nonrejection range and one-sided con(cid:12)dence interval for exchange rate forecast error variance share Rejection Nonrejection MR MR+SR none th th th th th th country nvar min. max. 90 95 90 95 90 95 US-UK 7 0.08 0.52 0.57 0.63 0.54 0.59 0.78 0.83 US-GE 7 0.10 0.56 0.56 0.61 0.48 0.53 0.83 0.87 US-UK 14 0.02 0.06 0.28 0.32 0.24 0.27 0.74 0.80 US-GE 14 0.02 0.13 0.32 0.35 0.25 0.28 0.77 0.81 Notes: The impulse responses giving the nonrejection ranges are as in Table 2 with the exception of the 7-variable maxima. These are common values in the literature and are omitted for brevity. The posterior odds are as in Table 2. 26

UK GE 0.75 0.25 -0.25 0 12 24 36 48 60 72 84 0 12 24 36 48 60 72 84 Figure 1: Responses of the $/pound (UK) and $/DM (GE) nominal exchange rates to a stimulative U.S. monetary policy shock in our replication of a 7-variable model of Eichenbaum and Evans (1995). The response horizon in months is given on the horizontal axis. The units on the vertical axis are in approximate percent and are the same for each panel.

y y* p 0.75 0.5 0.25 0 -0.25 i i* 0.2 0.1 0 -0.1 -0.2 -0.3 NBRX S UIP 3 2 1 0 -1 -2 0 12 24 36 48 60 72 84 0 12 24 36 48 60 72 84 0 12 24 36 48 60 72 84 Figure 2a: Response to monetary policy shock in two identifications of the 7-variable US- UK model. The scale on the vertical axis is the same for each panel in any row. The units are approximate percent(in annualpercentage rates for i, i∗ , andthedeviations from UIP). The horizontal axis is lag horizon in months. Error bands (as defined in the text) for the solid and dashed lines are provided by the gray solid and dashed lines, respectively.

y y* p 0.75 0.5 0.25 0 -0.25 i i* 0.2 0.1 0 -0.1 -0.2 -0.3 NBRX S UIP 3 2 1 0 -1 -2 0 12 24 36 48 60 72 84 0 12 24 36 48 60 72 84 0 12 24 36 48 60 72 84 Figure2b: Responsetomonetarypolicyshockintwoidentificationsofthe7-variableUS-GE model. See notes to Figure 2a.

y y* p p* 0.4 0.2 0 -0.2 -0.4 i i* R R* 0.2 0.1 0 -0.1 -0.2 1 NBR TR M M* 0.5 0 -0.5 -1 1.5 CP S Q UIP/3 1 0.5 0 -0.5 -1 0 12 24 36 48 60 72 84 0 12 24 36 48 60 72 84 0 12 24 36 48 60 72 84 0 12 24 36 48 60 72 84 Figure 3a: Response to monetary policy shock in two identifications of the 14-variable US- UK model. In this figure only, UIP is scaled by 1/3 for readability. See the notes to Figure 2a.

y y* p p* 0.4 0.2 0 -0.2 -0.4 i i* R R* 0.2 0 -0.2 -0.4 NBR TR M M* 1.5 1 0.5 0 -0.5 3 CP S Q UIP 1.5 0 -1.5 -3 0 12 24 36 48 60 72 84 0 12 24 36 48 60 72 84 0 12 24 36 48 60 72 84 0 12 24 36 48 60 72 84 Figure 3b: Response to monetary policy shock in two identifications of the 14-variable US-GE model. See the notes to Figures 2a.