why is it so difficult to beat the random walk forecast of exchange rates?

Journal of International Economics 60 (2003) 85–107www.elsevier.com/ locate/econbase

W hy is it so difficult to beat the random walk forecastof exchange rates?

a,c,d , b,c*Lutz Kilian , Mark P. TayloraDepartment of Economics, University of Michigan, Ann Arbor, MI 48109-1220,USA

bDepartment of Economics, University of Warwick, Coventry CV4 7AL, UKcCentre for Economic Policy Research, London, UK

dEuropean Central Bank, 60311 Frankfurt a.M., Germany

Abstract

Recent empirical evidence suggests that the time series behavior of the real exchange rateis well approximated by a nonlinear, exponential smooth transition autoregressive (ESTAR)model. This nonlinearity helps resolve a number of puzzles concerning the persistence andvolatility of real exchange rates. In this paper, we explore whether it may also help resolvethe well-known difficulties of exchange rate forecasting. We develop a bootstrap test of therandom walk hypothesis of the nominal exchange rate, given ESTAR real exchange ratedynamics. We find strong evidence of predictability at horizons of 2 to 3 years, but not atshorter horizons. 2002 Elsevier Science B.V. All rights reserved.

Keywords: Purchasing power parity; Real exchange rate; Random walk; Economic models of exchangerate determination; Long-horizon regression tests

JEL classification: F31; F47; C53

1 . Introduction

After nearly two decades of research since Meese and Rogoff’s pioneering workon exchange rate predictability (see Meese and Rogoff, 1983a,b), the goal of

¨exploiting economic models of exchange rate determination to beat naıve random

*Corresponding author. Tel.:11-734-764-2320; fax:11-734-764-2769.E-mail addresses: [email protected](L. Kilian), [email protected](M.P. Taylor).

0022-1996/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.doi:10.1016/S0022-1996(02)00060-0

mailto:[email protected]

mailto:[email protected]

86 L. Kilian, M.P. Taylor / Journal of International Economics 60 (2003) 85–107

walk forecasts remains as elusive as ever (see Taylor, 1995). One possibleexplanation is simply that standard economic models of exchange rate de-termination are inadequate. Indeed, this appears to be the response of manyprofessional exchange rate forecasters (see e.g. Cheung and Chinn, 1999).

A more charitable interpretation of the dismal forecast performance of economicexchange rate models is that the underlying economic theory is fundamentallysound, but that linear forecasting models of exchange rates fail because there areimportant nonlinearities in the data. In that view, economic models linking thenominal exchange rate to underlying fundamentals such as relative prices orrelative monetary velocity imply long-runequilibrium conditions only, towardwhich the economy adjusts in a nonlinear fashion. Indeed, there has been recentwork documenting various nonlinearities in deviations of the spot exchange ratefrom economic fundamentals (e.g. Balke and Fomby, 1997; Taylor and Peel, 2000;

1Taylor et al., 2001). This evidence of nonlinear mean reversion in the deviationfrom fundamentals has raised expectations that, with the appropriate nonlinearstructure, economic models of the exchange rate will prove useful for forecasting,at least at longer horizons.

The question of forecast accuracy traditionally has been an important test of thecredibility of economic models of nominal exchange rate determination. However,the implications of nonlinear reversion to the long-run exchange rate equilibriumfor nominal exchange rate forecasting have been left largely unexplored. Part ofthe problem relates to technical difficulties in implementing forecast accuracy testsin a nonlinear framework, while another part is related to the small samples of dataavailable for empirical work. In this paper, we focus on the implications ofnonlinear dynamics in the real exchange rate for tests of the hypothesis that thenominal exchange rate follows a random walk. We address these issues in thecontext of a specific nonlinear model that has recently gained popularity and thathas been shown to have empirical support.

We begin, in Section 2, by providing empirical evidence for nonlinear dynamicsin seven real dollar exchange rates in the form of estimates of exponential smoothtransition autoregressive (ESTAR) models. These models link movements in thequarterly nominal exchange rate nonlinearly to movements in relative prices, sothat real (relative price-adjusted) exchange rate dynamics are nonlinear. Ourfindings corroborate the evidence presented in Taylor et al. (2001) that there is

2strong—albeit nonlinear—mean reversion in monthly dollar real exchange rates.This evidence is important for a number of reasons. Firstly, allowing for thisnonlinearity goes some way towards solving Rogoff’s (1996) ‘purchasing power

1See also Obstfeld and Taylor (1997) for empirical evidence on nonlinearities in deviations from thelaw of one price.

2Our results also complement the evidence presented in Taylor and Peel (2000) that the relationshipbetween the nominal exchange rate and monetary fundamentals is nonlinear, given that monetaryfundamentals will clearly be linked to relative prices.

L. Kilian, M.P. Taylor / Journal of International Economics 60 (2003) 85–107 87

parity puzzle’ concerning the apparently very slow speed of adjustment of realexchange rates, since within the context of such a model, the speed of adjustmentof the real exchange rises as the size of the shock impinging on the real exchangerate increases, so that only very small shocks will induce glacially slow speeds of

3adjustment. Secondly, and relatedly, it can be shown that standard univariate testsof non-mean reversion may have very low power againstnonlinear meanreversion, hence explaining why researchers have encountered such difficulty in

4rejecting the null hypothesis of linear unit roots in real exchange rate behavior.Thirdly, as we attempt to demonstrate in this paper, smooth transition dynamicsprovide a plausible source of increased long-horizon nominal exchange rate

5predictability.In Section 3, we discuss the statistical issues involved in bootstrapping

predictability tests in the presence of ESTAR dynamics. We propose an easy-to-use statistical test of the random walk model of nominal exchange rates. Theproposed test not only is highly accurate under the null of no exchange ratepredictability, but it has high power against some plausible nonlinear alternatives,even in relatively small samples.

Using this new test, in Section 4 we provide strong empirical evidence for sevenOECD countries that the predictability of the spot dollar exchange rate improvesdramatically as the forecast horizon is lengthened from one quarter to severalyears. This evidence is based on in-sample predictability tests based on fittedmodels for the entire post-Bretton Woods sample period. For six of seven countrieswe beat the random walk forecast at forecast horizons of 2 or 3 years atconventional significance levels.

If the exchange rate is inherently predictable, why has it been so difficult to beatthe random walk forecast model? We identify two reasons in this paper. Firstly,the ESTAR model implies that near the long-run equilibrium deviations fromeconomic fundamentals will be well approximated by a random walk. This factgoes a long way toward explaining the success of the random walk point forecastsfor OECD exchange rates in earlier work, especially at short horizons (Meese andRogoff, 1983a,b). The view that only occasional, unusually large departures fromfundamentals along the sample path will reveal the inherent tendency toward meanreversion is also consistent with historical evidence that at least during periods ofhyperinflation, the exchange rate does seem to behave as suggested by economictheory (see Frenkel, 1976; Taylor and McMahon, 1998), whereas the evidence ismuch less clear during normal times (see Taylor, 1995; Frankel and Rose, 1995).

Secondly, our evidence of ESTAR dynamics suggests that the power of formal

3See Taylor et al. (2001, pp. 1031–1033).4See Taylor et al. (2001, pp. 1034–1037).5In addition, the existence of ESTAR dynamics has direct implications for the implementation of

bootstrap tests of long-horizon exchange rate predictability (see Kilian, 1999), and these are alsodiscussed in passim below.


statistical tests of the random walk model will be low, unless the sample pathincludes unusually large departures from fundamentals that reveal the presence ofnonlinear mean reversion. This problem is particularly pronounced for recursivereal time (‘out-of-sample’) forecast tests of the random walk model. We show bysimulation that these tests will have much lower power relative to ‘in-sample’t-tests, given the short time span of available post-Bretton Woods data. Indeed, inour empirical analysis, we are unable to establish beyond a reasonable doubt thatthe long-horizon regression forecast is significantly more accurate than the randomwalk forecast in a recursive real-time forecast setting, although we find strongevidence of increased long-horizon predictability consistent with the in-sampleresults.

2 . Econometric evidence of nonlinear mean reversion in real exchangerates

Recently, several papers have investigated the evidence of smooth nonlineartransition dynamics in the deviation of nominal exchange rates from macro-economic fundamentals (e.g. Balke and Fomby, 1997; Taylor and Peel, 2000;

6Taylor et al., 2001). In the present paper, we focus on the example of the realexchange rate, which may be viewed as the deviation of the nominal exchange ratefrom aggregate relative price or ‘purchasing power parity’ (PPP) fundamentals. Aparsimonious parametric model that captures the nature of nonlinear meanreversion in real exchange rates is the exponential smooth transition autoregressive

¨(ESTAR) model of Terasvirta (1994). For example, Taylor et al. (2001) documentevidence of ESTAR dynamics in real (i.e. relative-price adjusted) exchangebehaviour using monthly data for a number of major dollar exchange rates over thepost-Bretton Woods period. We apply the same type of model to quarterly data forCanada, France, Germany, Italy, Japan, Switzerland, and the UK for the period

71973.I–1998.IV. The use of quarterly data allows us to sidestep complications in8bootstrap inference that arise from conditional heteroskedasticity in the error term.

The data source is the International Monetary Fund’s International FinancialStatistics database.

Let e denote the logarithm of the spot nominal exchange rate (foreign price oft

*dollars) and let f ; p 2 p denote the PPP fundamental based on relativet t t

*consumer price indices, wherep is the logarithm of the US CPI andp is thet t

6See Taylor et al. (2001) for further discussion of this literature and further references.7We ended the estimation period at the end of 1998 because two of the currencies under

consideration—the German mark and the French franc—ceased to be traded at that time with theimposition of the first stage of the European Monetary Union.

8Conditional heteroskedasticity in nominal and real exchange rate residuals is a well-known featureof monthly data that tends to vanish at quarterly frequency due to time aggregation.


logarithm of the foreign CPI. Hence, the deviation of the nominal exchange ratefrom the underlying PPP fundamental,z ; e 2 f , is in fact the real exchange ratet t t

(in logarithmic form). For uniformity, we demeanedz for each country prior tot

the empirical analysis. Examination of the partial correlogram forz , as proposedt

in the context of the estimation of nonlinear autoregressive models by Granger and¨ ¨Terasvirta (1993) and Terasvirta (1994), revealed second-order serial correlation

9in the data, suggesting a nonlinear AR(2) model. Specifically, we postulated asmooth transition autoregressive model or STAR model of the form

]dz 2m 5F kz l , m ; G f z 2m 1f z 2m 1 u ,f gs s d s ddt z t2d d51 z 1 t21 z 2 t22 z t

2u | i.i.d.(0,s ). (1)t

]dThe transition functionF kz l , m ; G determines the degree of nonlinearityf gt2d d51 z

in the model and is a function of lagged movements in the real exchange rate,]dkz l , of the equilibrium level of the real exchange rate,m [ (2`, `), and oft2d d51 z]

dthe vector of transition parametersG [ (2`, 0 . Previous work (Taylor and Peel,2000; Taylor et al., 2001) suggests that an exponential form of the transitionfunction is particularly applicable to real exchange rate movements. This func-tional form also makes good economic sense in this application because it implies

10symmetric adjustment of the exchange rate above and below equilibrium.¨Granger and Terasvirta (1993) term STAR models employing exponential

11transition functions exponential STAR or ESTAR models.]

In any empirical application, it is of course necessary to determine the delayd(the dimension ofG and the number of lagged values of the real exchange rateinfluencing the transition function) and whether any of its elements are zero. Ingeneral, applied practice with ESTAR models has favored restrictingG to be a

9There was, however, no evidence of higher-order serial correlation in thenominal exchange rate,suggesting that the standard random walk comparator is still applicable.

10In closely related work, Taylor et al. (2001) allowed for asymmetric adjustment in real exchangerates in the form of a logistic smooth transition function, but found no significant evidence ofasymmetry.

11Note that the ESTAR model for the real exchange rate as set out in (1) assumes a constant long-runequilibrium level of the real exchange rate. In practice, there may of course be important real influenceson the equilibrium real exchange rate such as the Harrod–Balassa–Samuelson effect. To that extent, themodel must be regarded as an approximation. We feel that it is reasonable to make such anapproximation in our analysis since the empirical evidence suggests that the effect of productivitydifferentials on the real exchange, at least for real exchange rates between major industrializedcountries over the post-Bretton Woods period, is not very marked. This is the conclusion reached bySarno and Taylor (2002), for example, after an extensive survey of the relevant literature. Such aneffect may, however, be more marked over longer periods (Lothian and Taylor, 2000). Nevertheless,allowance for real effects on the equilibrium level of the real exchange rate within a nonlinearframework remains an important avenue for future research, although we do not expect that this wouldsignificantly alter the main conclusions of the present analysis.


¨singleton (see e.g. Terasvirta, 1994; Taylor et al., 2001), and Granger and¨ ¨Terasvirta (1993) and Terasvirta (1994) suggest a series of nested tests for

determining the appropriate delay in this case. In the present application toquarterly real exchange rate data, however, we found that the model that workedbest for each country—in terms of goodness of fit, statistical significance ofparameters, and adequate diagnostics—set the dimension ofG to 5, with eachelement ofG equal to the same scalar,g, where we expect to findg , 0 if thedegree of mean reversion rises with the magnitude of deviations from equilibrium.This parameterization seems reasonably intuitive since it allows deviations fromequilibrium to affect the nonlinear dynamics with a single lag (rather thansuddenly kicking in at a higher lag) and also allows the effects of persistentdeviations to be cumulative.

In addition, we subsequently found that the restrictionf 1f 5 1 could not be1 2

rejected at standard significance levels for any of the countries. Hence, the modelthat we estimated for each country was of the form:

5

2z 2m 5 exp gO z 2m f z 2m 1 12f z 2ms d s s d s ds ddS H JDt z t2d z 1 t21 z 1 t22 zd51

1 u . (2)t

5 2The exponential transition functionF 5 exp go z 2m , g , 0, wills h jds dd51 t2d z

take the value unity when the last five values of the nominal exchange are equal tothe fundamental equilibrium levelm 1 f , or equivalently the real exchange ratez t2d

z is equal to its equilibrium levelm . Thus, in equilibrium, the real exchanget2d z

rate will follow a unit root process:

z 5f z 1 12f z 1 u . (3)s dt 1 t21 1 t22 t

As departures from the fundamental equilibrium increase, however,F shrinkstowards its limiting value of zero and at any instant the real exchange rate willfollow a mean-reverting AR(2) process with meanm and slope coefficients addingz

up to the instantaneous value ofF , 1.The ESTAR model (2) was estimated by nonlinear least squares for each of our

12seven countries (see Gallant, 1987; Gallant and White, 1988). The modelestimates are reported in Table 1. The models perform well in terms of providinggood fits, statistically significant coefficients and the residual diagnostic statistics

13¨are satisfactory in all cases (see Eitrheim and Terasvirta, 1996). Given the

12Regularity conditions for the consistency and asymptotic normality of this estimator are discussedby Gallant (1987), Gallant and White (1988), Klimko and Nelson (1978) and, in the present context,Tjøstheim (1986).

13For further details of the specification, interpretation and properties of these models the reader isreferred to Taylor et al. (2001) and the references therein.


Table 1ESTAR estimates by country: PPP fundamental

Canada5 2 ˆz 5 (exp h2 0.7060o z j)( 1.1811z 1 (12 1.1811 )z )1 ut d51 t2d t21 t22 t

(16.3074) (12.7201) (12.7201)[0.0000]

s50.0192 AR(1)5[0.09] AR(1–4)5[0.11]

France5 2 ˆ(z 20.0954)5 (exp h2 0.8638o (z 20.0954) j)( 1.3219 (z 20.0954)1 (12 1.3219 )(z 20.0954))1 ut d51 t2d t21 t22 t

(2.8544) (5.75312) (2.8544) (13.8396) (2.8544) (13.8396) (2.8544)[0.0015]

s50.0473 AR(1)5[0.65] AR(1–4)5[0.19]

Germany5 2 ˆ(z 20.0960)5 (exp h2 0.7941o (z 20.0960) j)( 1.2333 (z 20.0960)1 (12 1.2333 )(z 20.0960))1 ut d51 t2d t21 t22 t

(2.4076) (6.3540) (2.4076) (11.6493) (2.4076) (11.6493) (2.4076)[0.0012]

s50.0530 AR(1)5[0.91] AR(1–4)5[0.31]

Italy5 2 ˆz 5 (exp h2 0.9092o z j)( 1.1540z 1 (12 1.1540 )z )1 ut d51 t2d t21 t22 t

(3.6754) (10.2805) (10.2805)[0.0028]

s50.0540 AR(1)5[0.82] AR(1–4)5[0.21]

Japan5 2 ˆz 5 (exp h2 0.7256o z j)( 1.3500z 1 (12 1.3500 )z )1 ut d51 t2d t21 t22 t

(7.7301) (13.5378) (13.5378)[0.0010]

s50.0571 AR(1)5[0.15] AR(1–4)5[0.19]

Switzerland5 2 ˆz 5 (exp h2 0.7242o z j)( 1.2922z 1 (12 1.2922 )z )1 ut d51 t2d t21 t22 t

(5.2450) (13.0796) (13.0796)[0.0019]

s50.0599 AR(1)5[0.45] AR(1–4)5[0.65]

United Kingdom5 2 ˆz 5 (exp h2 1.0696o z j)( 1.1448z 1 (12 1.1448 ))z )1 ut d51 t2d t21 t22 t

(3.3603) (11.1052) (11.1052)[0.0031]

s50.0520 AR(1)5[0.45] AR(1–4)5[0.21]

Notes: s is the standard error of the regression. AR(1) and AR(1–4) are Lagrange multiplier teststatistics for first-order and up to fourth-order serial correlation in the residuals, respectively,

¨constructed as in Eitrheim and Terasvirta (1996). Figures in parentheses below coefficient estimatesdenote the ratio of the estimated coefficient to the estimated standard error of the coefficient estimate.Figures given in square brackets denote marginal significance levels. The marginal significance levelsfor the estimated transition parameters were calculated by a non-parametric bootstrap under the nullhypothesis of a unit root AR(2) process.

similarity of the estimated parameters across countries, a natural question iswhether the rejections of the linear model are perhaps driven by a single episode inthe data. This is not the case. Plots of the estimated transition functions tend todiffer substantially across countries and reveal that, with the possible exception of


the DM–dollar rate, departures from the linear model are not driven by a singleepisode.

A natural question to ask is whether the ESTAR model in some sense dominatesa linear autoregression for the real exchange rate. We believe that it does. Firstly,the empirical results shown in Table 1 reveal that the estimated transitionparameters are in each case strongly significantly different from zero, which itself

14indicates significant nonlinearity. Secondly, more generally, the ESTAR model isattractive because of its ability to generate endogenously data that exhibit bothhigh short-term volatility and large and persistent deviations from fundamentals ofthe same magnitude as in actual data. Any satisfactory model of real exchangerates must be able to explain: (1) the existence of large deviations of the nominalexchange rate from the PPP fundamentals, (2) the persistence of these deviationsover time, and (3) the short-term volatility of deviations from fundamentals. Ourestimated nonlinear models are consistent with all three of these empiricalregularities. Fig. 1 shows a representative realization from the fitted ESTAR modelfor Germany, plotted against the actual real exchange rate. Not only is there clearevidence of long swings in the real exchange rate data, but the simulated data alsodisplay short-term volatility. Thus, as stressed by Taylor et al. (2001) the ESTARmodel of real exchange rates helps to explain the PPP puzzle of Rogoff (1996)

15concerning the apparently very slow speed of adjustment of real exchange rates.The presence of ESTAR dynamics also helps to explain why researchers haveencountered such difficulty in rejecting the null hypothesis of linear unit roots in

16real exchange rate behavior.Thus, the ESTAR model provides a natural starting point for further analysis. In

this paper we focus on the implications of ESTAR nonlinearities for anotherimportant puzzle in international finance: the difficulty of significantly rejecting asimple random walk forecast of nominal exchange rates in favor of a model based

14A test that the transition parameter,g, is equal to zero, givenf 1f 5 1, is tantamount to a test of1 2

the hypothesis that the real exchange rate follows a linear unit root process against the alternative ofnonlinear mean reversion. For this reason, the ‘P-value’ for the estimated transition parameter in Table1 was estimated using a parametric bootstrap. Note that imposing the restrictionf 1f 5 1 generally1 2

led to an increase in the size and significance of the estimated transition parameter (although thetransition parameter was in any case generally significantly different from zero at the 5% level in theunrestricted estimated model). Intuitively, this is because imposingf 1f 5 1 forces the transition1 2

parameter to pick up any mean reversion that may be in the data. This restriction is consistent withtheoretical models of goods arbitrage with transactions costs (see Taylor et al. (2001) or Sarno andTaylor (2002) for further discussion) and could not be rejected at conventional significance levels.

15In particular, Taylor et al. (2001) derive impulse response functions for their nonlinear ESTARmodels estimated using monthly data, and show that the half-lives of shocks impinging on the realexchange rate are close to 3–5 years (as highlighted in the discussion of Rogoff (1996)) only forrelatively small shocks occurring near equilibrium. Similar results hold for the quarterly data analyzedin this paper (see Kilian and Taylor, 2001).

16For example, Taylor et al. (2001) show that standard univariate tests of non-mean reversion mayhave very low power against a nonlinearly mean-reverting ESTAR alternative.


Fig. 1. Actual and simulated real exchange rates based on a representative random draw from the fittedESTAR model for the Deutsche mark–dollar rate.

Notes: The simulated data were generated by a nonparametric bootstrap approach based on the fittedmodel for Germany in Table 1.

on exchange rate fundamentals. This puzzle was first brought to the profession’sattention by the work of Meese and Rogoff (1983a,b).

If we are interested in forecasting the nominal exchange rate in the presence ofESTAR dynamics in the real exchange rate, it is not enough to modelz . Rathert

we need to model thejoint time series process for fundamentals and nominalexchange rates. The latter task is considerably more demanding than estimating aunivariate model forz and may involve estimation of a large number oft

parameters. Obtaining reliable parameter estimates will be a problem when thesample size is small. An additional problem is that we do not know the functionalform of the bivariate nonlinear process for (e , f )9. We could represent thist t

bivariate process equivalently as a bivariate process in (De , z )9, in which case thet t

functional form for the second equation could be treated as an ESTAR model. Thiswould simplify the problem, but it would leave open the nature of the functionalform of the first equation, which in general might also be nonlinear. Although, in


principle, statistical tools could be used to determine this functional from, suchtools are unlikely to be reliable given the small samples typically available.

In the next section we will propose an easy-to-use econometric test of therandom walk null hypothesis that avoids these difficulties. The proposed procedurerequires us only to specify the model for (De , z )9 under the null hypothesis thatt t

the nominal exchange rate is unpredictable—or in other words follows a randomwalk. This has two advantages. Firstly, under this null hypothesis there is nononlinearity in the first equation and the nature of the nonlinearity in the secondequation is known to be of the ESTAR type. Secondly, imposing the nullhypothesis greatly reduces the number of parameters to be estimated.

3 . A bootstrap procedure for generating critical values under the nullhypothesis of a random walk in the nominal exchange rate

The proposed test of the random walk benchmark is based on the well-knowntechnique of long-horizon regression tests. Long-horizon regressions take the form

e 2 e 5 a 1 b z 1´ , k 5 1, 4, 8, 12, 16 (4)t1k t k k t t1k

where the error term in general will be serially correlated. Mean reversion in thenominal exchange rate may be detected by at-test ofH : b 5 0 versusH : b , 00 k 1 k

for a given forecast horizonk, or jointly for all forecast horizons asH : b 5 0;k0 k

versusH : b ,0 for somek. It is well known that asymptotic critical values for1 k

the t-test statistics are severely biased in small samples. In order to mitigate thesesize distortions, critical values are usually calculated based on the bootstrapapproximation of the finite-sample distribution of the test statistic under the nullhypothesis of no exchange rate predictability (see, e.g., Mark, 1995; Kilian, 1999;Mark and Sul, 2001).

Alternatively, the out-of-sample prediction mean-squared error of the twomodels may be evaluated using the DM test of Diebold and Mariano (1995). Aformal test may be based on a sequence of rolling or recursive forecasts andinvolves comparing the null of equal forecast accuracy against the one-sidedalternative that forecasts from the long-horizon regression are more accurate thanrandom walk forecasts. The distribution of the DM test statistic in long-horizonregression problems is not known in general (see McCracken, 1999). In practice, itis common to rely on the bootstrap approach to construct critical values for theDM test.

The bootstrap procedure proposed here applies the principles of nestedhypothesis testing. There are two steps. In step 1 we write down the unrestrictedreduced form representation of the data. This unrestricted process must becompatible with the data. It also must encompass the restricted model under thenull hypothesis as well as the unrestricted model under the alternative hypothesis.In step 2, we generate critical values by estimating this process subject to the


restrictions under the null hypothesis and simulating the distribution of the teststatistics in repeated replications. Note that, by construction, if we relax therestrictions imposed under the null hypothesis, we should obtain the unrestrictedreduced form.

We postulate that the unrestricted data generating process forx 5 (e , f )9 mayt t t

be represented as a bivariate nonlinear model for (De , z )9 such thatz follows ant t t

ESTAR process andDe is possibly predictable based on the past:t

De 2m | g(e , f , e , f . . . )1white noiset e t21 t21 t22 t22 (5)z 2m |ESTARt z

We are deliberately vague about the nature of the functional formg(.) of the firstequation in (5). Model (5) is sufficiently broad to encompass both random walkbehavior of the nominal exchange rate or more complicated linear or nonlinearserially correlated processes governing the nominal rate. Although the nature ofthe unrestricted model affects the power of the test, we never have to estimate thisfully unrestricted model in practice.

We are interested in testing the null hypothesisH that the change in the0

nominal exchange rate is unpredictable based on past realizations ofe and ft t

H : De 2m |white noise. (6)0 t e

Note that this random walk hypothesis puts restrictions only on the first equationof (5), not on thez process in the second equation. Thus, thez process remainst t

nonlinear even under the null hypothesisH . This point is important. We only0

impose and test restrictions on the behavior ofDe , never on the behavior ofz .t t

The parametric structure of the equation forz is the same under the null and undert

the alternative hypotheses and follows from the requirement of finding an17encompassing reduced form model that is compatible with the data.

Thus, by specifying a particular form of the ESTAR process in (5) that isconsistent with the data and by imposingH on model (5), we obtain the restricted0

bootstrap data generating process:

De 2m 5 ut e 1t

5

2z 2m 5 exp gO z 2m f z 2m 1 12f z 2m 1 u .s d s s d s ds ddS H JDt z t2d z 1 t21 z 1 t22 z 2td51

(7)

The innovationsu 5 (u , u )9 in practice will be treated as independent andt 1t 2t

identically distributed (i.i.d.). This system of equations may be estimated by

17The parameter estimates of course may differ, depending on whether we estimate the model underthe null or the alternative hypotheses.


nonlinear least squares. We treat the estimate of this process as the bootstrapdata-generating process (DGP). Bootstrapp-values for the long-horizon regressiontest statistics under the null may be obtained by generating repeated trials from thisbootstrap DGP, re-estimating the long-horizon regression test statistic for each setof bootstrap data, and evaluating the empirical distribution of the resultinglong-horizon regression test statistics. A detailed description of the bootstrapalgorithm can be found in Appendix A.

It is clear that inference based on model (7) will be appropriate only to theextent that (5) is indeed a valid representation of the data. For example, ifzt

followed a Markov switching model instead of an ESTAR model, then theappropriate critical values would be potentially different. Thus, our analysis is bestviewed as an illustration of the implications of nonlinearities inz , conditional on at

particular nonlinear model of the real exchange rate with empirical support.The next section will apply this long-horizon regression test to US dollar

exchange rate data for the UK, Germany, Japan, France, Switzerland, Canada andItaly. An obvious concern is the accuracy of the proposed bootstrap test under thenull hypothesis that the exchange rate is indeed a random walk (possibly with adrift). While a comprehensive Monte Carlo study of the size of the test is beyondthe scope of this paper, we will illustrate the accuracy of the proposed method fora representative DGP of the form:

De 10.0055 ut 1t

5

2z 20.0965 exp 2 0.7941O z 2 0.096s dS H JDt t2dd51

3 1.2333z 2 0.096 1 12 1.2333 z 20.096 1 u (8)s ds s d s ddt21 t22 2t

where the parameter values correspond to the estimates obtained for the US–German exchange rate data and the innovation vectoru is obtained by randomt

sampling with replacement from the actual regression residuals.Fig. 2 shows the effective size of the long-horizon regression test at the nominal

10% significance level for the actual sample size of 104. Approximate two-standard error bands for the rejection rates under the null of a 10% significancelevel are indicated by two horizontal lines. The statisticst(20) andt(A) refer toin-samplet-tests for the slope coefficient of the long-horizon regression. The twotests differ only in the computation of the standard error of the slope coefficient.The former test uses a Newey–West standard error based on a fixed truncation lagof 20; the latter uses a truncation lag based on Andrews’ (1991) procedure.DM(20) and DM(A) refer to the corresponding Diebold–Mariano tests of out-of-sample forecast accuracy. The out-of-sample tests are implemented based on asequence of recursive forecasts, starting with a sample size of 32 quarters. Thejoint tests refer to tests of the random walk null against predictability at somehorizon. They are based on the distribution of the maximum value of a given test


Fig. 2. Effective size of bootstrap test procedure under ESTAR null.Notes: Based on 1000 Monte Carlo trials with 2000 bootstrap replications each. The DGP is

described in the text.

statistic across all horizons. A detailed description of these tests can be found inMark (1995).

The first panel of Fig. 2 shows that even for sample sizes as small as 104observations the bootstrap test is remarkably accurate. The effective size of all fourtests is reasonably close to the nominal significance level of 10% and remainsfairly constant across forecast horizons. This result means that any evidence ofincreased long-horizon predictability is unlikely to be caused by size distortions. Ifwe double the sample size, as shown in the second panel of Fig. 2, all sizeestimates are well within the standard error bands for a 10% test.

Next, we will analyze the finite-sample power of the long-horizon regressiontest. The power of the test will in general depend on the alternative model. We willconsider three examples of processes that may be considered empirically plausibleunder the alternative hypothesis of exchange rate predictability. Modeling thepower of the test requires an estimate of the joint DGP of (e , f ) or equivalently oft t

(e , z ) or ( f , z ). We clearly have little hope of correctly identifying the underlyingt t t t

complicated and possibly nonlinear dynamics of the nominal exchange rate from


actual data. Instead, we focus on the easier task of finding a reasonableapproximation to the time series process of the fundamental,f . For expositoryt

purposes we postulate that the DGP forz is the same as in the size study. Givent

the DGP forz , selecting a DGP forf will pin down the implied DGP fore by thet t t

identity e ; z 1 f . Our starting point is once again the US–German data set.t t t

Preliminary tests did not reject the assumption that the German fundamentalsfollow a linear time series process. We selected the following three models as ourDGPs:

DGP 1:Df 5 20.00521 ut 1t

5

2z 20.0965 exp 2 0.7941O z 2 0.096 1.2333z 20.096s d s s dS H JDt t2d t21d51

1 12 1.2333 z 2 0.096 1 u . (9)s ds ddt22 2t

DGP 2:Df 5 20.00302 0.1733Df 10.1419Df 10.1860Dft t21 t22 t23

1 0.2417Df 1 ut24 1t

5

2z 20.0965 exp 2 0.7941O z 2 0.096 1.2333z 20.096s d s s dS H JDi t2d t21d51

1 12 1.2333 z 2 0.096 1 u . (10)s ds ddt22 2t

DGP 3:Df 5 20.00391 0.2022Df 10.0643De 1 ut t22 t22 1t

5

2z 20.0965 exp 2 0.7941O z 2 0.096 1.2333z 20.096s d s s dS H JDt t2d t21d51

1 12 1.2333 z 2 0.096 1 u . (11)s ds ddt22 2t

DGP 1 and DGP 2 were selected by the Schwarz information criterion and theHannan–Quinn criterion, respectively, among the class of linear regressions ofDfton an intercept and up to eight autoregressive lags. DGP 3 was selected among allpossible linear regressions involving up to 4 lags ofDf and De each and ant t

intercept. The innovation vectoru is again obtained by random sampling witht

replacement from the actual regression residuals.The power of the long-horizon regression test against each of these alternatives

is shown in Fig. 3. All power results are based on the nominal 10% bootstrap test.As the actual test size is very close to the nominal size, there is no need for sizecorrections. In Fig. 3a, the sample size isT5104 as in the actual data. Fig. 3asuggests several important conclusions. Firstly, the proposed long-horizon regres-sion test not only is highly accurate under the null of no exchange rate


Fig. 3a. Power of bootstrap test procedure at the 10% significance level, (a)T5104.Notes: Based on 1000 Monte Carlo trials with 2000 bootstrap replications each. The DGP forz ist

the same as for the size analysis, but the DGP forf differs: DGP1:Df regressed on an intercept; DGP2:t t

Df regressed on four lags of itself and an intercept; DGP3:Df regressed on an intercept,Df andt t t21

De .t21

predictability, but it also has high power against empirically plausible alternatives,even in small samples. Secondly, whether the test is conducted in-sample orout-of-sample, Fig. 3a suggests that our ability to predict the exchange rate willimprove at intermediate horizons. The latter point is important because it providesthe rationale for conducting long-horizon regression tests in practice. For example,for the three DGPs considered, the power of the long-horizon regression test tendsto be lowest at the one-quarter horizon. As the forecast horizon is lengthened,power tends to improve initially, but ultimately falls again, resulting in a hump-shaped pattern with a peak at horizons of about 1 or 2 years. Thirdly, power isconsiderably lower for recursive out-of-sample tests than for tests based on the fullsample. The power of the in-sample tests is typically close to 90%, whereas thepower of the out-of-sample tests is closer to 50 or 60%.

Why are we not able to beat the random walk model more often in real timewhen the null is false by construction? Part of the problem with our real-time


Fig. 3b. Power of bootstrap test procedure at the 10% significance:T5208.Notes: Based on 1000 Monte Carlo trials with 2000 bootstrap replications each. The DGP forz ist

the same as for the size analysis, but the DGP forf differs: DGP1:Df regressed on an intercept; DGP2:t t

Df regressed on four lags of itself and an intercept. DGP3:Df regressed on an intercept,Df andt t t22

De .t22

exercise is the loss of power resulting from the small number of recursive forecasterrors in the sample. Moreover, the small estimation sample underlying theout-of-sample exercise makes it unlikely that we obtain reliable estimates of themean reversion parameterb . For example, at the beginning of the out-of-samplek

forecast exercise we use only 8 years of observations to construct the long-horizonforecast. Clearly, such a small time span may not be enough to capture nonlinearmean reversion. Increasing the initial sample size would seem to be the obvioussolution, except that this increase in turn would further reduce the number ofrecursive forecast errors and thus would further lower the power of the out-of-sample test. Hence, short of obtaining a much larger sample, there is no obvioussolution to the low power of the out-of-sample tests. As both types of tests areequally reliable under the null hypothesis, this evidence suggests that in empiricalwork the in-sample test of the random walk hypothesis will be preferable.

To confirm our interpretation that the much lower power of the out-of-sample


test for T5104 is an artifact of the sample size, we also experimented with asample size ofT5208. The improvement in power is striking. Fig. 3b shows thatthe in-sample tests forT5208 have power of virtually 100% for all horizons. Forthe out-of-sample tests, power is typically in the range of 96% to 99% with a peakat the 8-quarter horizon.

Although our power analysis is limited to three representative DGPs, weconclude that several qualitative implications of our model of nonlinear meanreversion are likely to be robust and can be tested empirically. Firstly, if our modelis supported by the data, the degree of predictability should be highest atintermediate horizons. Secondly, we expect to find less decisive empirical resultsfor the out-of-sample tests than for the in-sample tests in our empirical work. Thisweaker evidence, however, need not indicate a failure of the model. It is fullyexpected given the lower power of out-of-sample tests in small samples. A thirdtestable implication that emerges from the power analysis is that, to the extent thatthe random walk null hypothesis is false, the pattern of predictability for thein-sample and out-of-sample tests ought to be similar, even if the level ofsignificance is much lower out-of-sample than in-sample.

4 . Empirical evidence of long-horizon predictability relative to the randomwalk model

Fig. 4 shows the bootstrapP-values for our four long-horizon regression tests ofthe random walk null. Separate results are shown for horizons ofk51, 4, 8, 12and 16 quarters. As the exchange rate becomes more predictable at longerhorizons, thesep-values should fall. The horizontal bar indicates the nominalsignificance level of 10%. Anyp-value below 0.10 implies a rejection of therandom walk null hypothesis at the 10% significance level. The results in Fig. 4are generally consistent with all three testable implications developed in Section 4.Predictability generally is highest at intermediate horizons. The in-sample evi-dence is much stronger than the out-of-sample evidence, and the pattern ofpredictability across forecast horizons is broadly similar for in-sample and out-of-sample tests.

We will first focus on the results for the in-samplet-tests in columns 1 and 2. Ifour model of exchange rate determination is correct, we would expect to see aclear pattern of increased long-horizon predictability in the form ofp-values thatfall as the horizon grows. This is indeed what we find. There is little differencebetween thet(20) andt(A) test results, suggesting that the results are not sensitiveto the choice of truncation lag. In virtually all cases,p-values fall, as we increasekfrom 1 to 4 and 8. With the exception of France, we also find thatp-values riseagain for very long horizons, resulting in a U-pattern. This result is not surprisinggiven the smaller effective sample size as the forecast horizon is lengthened. It isconsistent with a loss of power at longer horizons, as suggested by Fig. 3. In


Fig. 4. Bootstrapp-values under ESTAR null: PPP fundamental.

addition to the pattern of predictability, in many cases we find that the long-horizon regression is significantly more accurate than the random walk at longerhorizons. For example, fork512, we are able to reject the random walk model atthe 10% significance level for six (five) of the seven countries using thet(A)(t(20)) test. In four (two) cases even the joint test statistic is significant at the 10%level. This number rises to six (four) out of seven if we focus on the 15%significance level. This evidence allows us for the first time to reject conclusivelythe random walk forecast model.

Does this result mean that we can also beat the random walk forecast in realtime? The power study in Fig. 3a suggests that beating the random walk model inreal time will be much more difficult, given the smaller effective sample size. This


Fig. 4. (continued)Notes: Based on 2000 bootstrap replications. Quarterly IFS data for 1973.I–1998.IV.

is indeed what the empirical results suggest. Columns 3 and 4 show thecorrespondingp-values for the DM test of out-of-sample accuracy. These testresults are based on recursive (or real-time) estimation of the forecast model

18starting with a sample size of 32 quarters. Using a conventional significance levelof 10%, with the exception of the UK and of Switzerland at the 3-year horizon,there is no evidence that the long-horizon regression beats the random walk.Moreover, none of the joint tests are significant at even the 15% level. This resultis consistent with the evidence of a drastic loss of power in Fig. 3 for theout-of-sample tests relative to the in-sample tests. There is, however, clearevidence for all seven countries that predictability improves as the forecast horizonis increased from one quarter to 1, 2 and 3 years, before deteriorating at the 4-yearhorizon. This pattern is generally similar to the pattern of the in-samplet-testp-values. The existence of a U-pattern inp-values is consistent with the hump-shaped power pattern we documented in Fig. 3, although the locations and depth

18Qualitative similar results are obtained with an initial sample size of 48 quarters. Note that thelarger the sample size, the smaller is the number of recursive forecasts and the less reliable is the DMtest. This trade-off suggests that our choice of 32 quarters is a reasonable compromise.


of the troughs suggest a somewhat different DGP than those that we considered inthe power study. We conclude that despite clear evidence of nonlinear meanreversion in real exchange rates, the goal of forecasting nominal exchange rates inreal time is likely to remain elusive for the foreseeable future.

5 . Concluding remarks

The landmark work of Meese and Rogoff (1983a,b), published nearly twodecades ago, launched the profession on a crusade to find the holy grail of beatingthe random walk model of exchange rates. Like the true Holy Grail, the goal of

¨exploiting economic models of exchange rate determination to beat naıveconstantchange forecasts has remained elusive. Recently, empirical evidence has beenforthcoming that the relationship between the nominal exchange rate and theunderlying macroeconomic fundamentals may be inherently nonlinear. In thepresent paper, we have explored the question of whether this evidence ofnonlinearity may help to understand the well-known difficulties in forecasting thenominal exchange rate based on economic fundamentals. Our analysis suggeststhat nonlinear dynamics in the deviation from long-run equilibrium are indeed aplausible explanation of this difficulty when combined with the relatively shorttime span of currently available data.

Throughout the paper we focused on the example of deviations of the exchangerate from so-called PPP fundamentals. We provided empirical support fornonlinear transition dynamics in the real exchange rate in the form of estimates ofexponential smooth transition autoregressive (ESTAR) models fitted to quarterlydata for seven countries over the entire post-Bretton Woods period. Our econo-metric model implies that close to the equilibrium the real exchange rate will bewell approximated by a random walk. This fact helps to explain the apparentsuccess of random walk point forecasts for nominal exchange rates, especially atshort horizons. It also suggests that formal statistical tests of the random walkhypothesis against forecast models based on PPP fundamentals may have lowpower in small samples. The problem is that the nonlinear mean reversion in thereal exchange rate can be detected statistically only following unusually largedepartures from equilibrium and such events may be rare along a given samplepath. Finally, the presence of ESTAR dynamics in the real exchange rate suggeststhat the power of tests of the random walk hypothesis against forecast modelsbased on PPP fundamentals should increase at longer forecast horizons, making iteasier to detect predictability using long-horizon forecasts.

We discussed the implications of ESTAR dynamics in real exchange rates forformal statistical tests of the predictability of nominal exchange rates. We proposeda new bootstrap long-horizon regression test of the random walk benchmarkmodel. This new test was shown to be reliable and powerful against someplausible nonlinear alternatives. The test provided strong empirical evidenceagainst the random walk model at horizons of 2 to 3 years. For example, based on


in-sample tests, at the 3-year horizon we were able to reject the random walkmodel at the 10% level for five or six of the seven countries, depending on thechoice of test statistic. This evidence justifies the continued use of the PPPhypothesis to explain medium-term exchange rate fluctuations.

Our empirical results not only lend support to economists’ beliefs that theexchange rate is inherently predictable, but, at the same time, they also rationalizethe reluctance of foreign exchange traders to rely on economic fundamentals suchas relative prices (see Allen and Taylor, 1990, 1992; Taylor and Allen, 1992;Cheung and Chinn, 1999). Most foreign exchange traders are interested in forecasthorizons of less than 6 months (see Frankel and Froot, 1990). We found noevidence that the use of PPP fundamentals significantly improves forecastaccuracy at such short horizons. Moreover, despite significant rejections of therandom walk based on in-sample tests, we were typically unable to rejectsignificantly the random walk forecast model in real-time recursive forecasts. Ouranalysis suggests that this difficulty of beating the random walk model in real timeneed not reflect an inherent shortcoming of forecasting models based on economicfundamentals. Instead, we showed that this stylized empirical fact appears to be anatural consequence of the small time span of data available for empirical work.

A cknowledgements

This paper has benefited from comments by seminar participants at numerousuniversities, central banks and conferences. We are especially grateful to RobertBarsky, Frank Diebold, Charles Engel, Jon Faust, Gordon Hanson, Nelson Mark,Lucio Sarno, Larry Schembri, Linda Tesar and two anonymous referees forcomments on previous versions of this paper. The views expressed in this paper donot necessarily reflect those of the European Central Bank or its members.

A ppendix A. Bootstrap algorithm for long-horizon regression test

1. Given the sequence of observationshx j wherex 5 (e , f ), definez ; e 2 f ,t t t t t t t

estimate the long-horizon regression

e 2 e 5 a 1 b z 1´ , k 5 1, 4, 8, 12, 16t1k t k k t t1k

ˆand for eachk construct the test statisticu .k

2. Postulate a nonlinear DGP of the form

De 2m 5 ut e 1t5

2z 2m 5 exp gO z 2m f z 2m 1 12f z 2m 1 us d s s d s ds ddS H JDt z t2d z 1 t21 z 1 t22 z 2td51


where the restriction that the exchange rate follows a random walk underH0

has been imposed and the innovationsu 5 (u , u ) are assumed to be zerot 1t 2t

mean, independent and identically distributed. Estimate this process bynonlinear least-squares.

*3. Based on the fitted model generate a sequence of pseudo observationshx j oft

* * *the same length as the original data serieshx j, wherex 5 (e , f ) is obtainedt t t t

from the realizations of the bootstrap DGP:

ˆ* *De 2m 5 ut e 1t

5

2 ˆˆ ˆ ˆ ˆ* * *z 2m 5 exp gO z 2m f z 2mss d s dS H JDt z t2d z 1 t21 zd51

ˆ ˆ* *1 12f z 2m 1 u .s d ds d1 t22 z 2t

* * *The pseudo innovation termu 5 (u , u ) is random and drawn witht 1t 2t21 T˜ ˆ ˆreplacement from the set of recentered residualsu 5 u 2 T o u . Thet t t5p11 t

* *process may be initialized withz 5 0 andDe 50 for j 5 p, . . . , 1. Wet2j t2j

discard the first 500 transients.4. Repeat the preceding step 2000 times. For each of the 2000 bootstrap

*replicationshx j estimate the long-horizon regressiont

* * * *e 2 e 5 a 1 b z 1´ , k 5 1, 4, 8, 12, 16t1k t k k t t1k

ˆ *and construct the test statistics of interest,u .k

5. Use the empirical distribution of the 2000 replications of the bootstrap testˆ ˆ*statisticu to determine thep-value of the test statisticu , wherek 5 1, 4, 8,k k

12, 16.

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