The forward premium anomaly: statistical artefact or economic puzzle? New evidence from robust tests

The forward premium anomaly:statistical artefact or economic puzzle?New evidence from robust tests

Alex Maynard School of Business and Economics, WilfridLaurier University

Abstract. Recent literature has questioned statistical inference in predictive regressionwith persistent regressors, suggesting a possible explanation for puzzles such as the for-ward premium anomaly. We therefore revisit this puzzle using three alternative economet-ric methods known to provide reliable inference in the presence of persistent conditioningvariables. While they provide less evidence against forward rate unbiasedness than tra-ditional predictive regression tests, we still reject using at least one method for all sixcurrencies. Thus, while the econometric problems inherent in predictive regression likelyplay a role in this anomaly, we are left with an economic puzzle even after accounting fortheir influence. JEL classification: F31, C22

L’anomalie de la prime a terme: fabrication statistique ou enigme economique? Nouveauxresultats fondes sur des tests robustes. La litterature specialisee recente a remis en questionl’inference statistique dans des regressions predictives, et suggere une explication possibled’enigmes comme celles de l’anomalie de la prime a terme. L’auteur reexamine cette enigmeen utilisant trois methodes econometriques susceptibles de fournir des inferences fiablesquand il s’agit de variables independantes persistantes. Meme si ces tests produisent desresultats moins probants contre l’hypothese d’un taux a terme qui ne serait pas biaise queles tests traditionnels de regressions predictives, on rejette au moins une methode pourtoutes les six monnaies. Voila qui fait que meme si les problemes econometriques inherentsaux regressions predictives jouent un role dans cette anomalie, on reste avec une enigmeeconomique quand on a pris ces facteurs en compte.

Financial support from SSHRC is gratefully acknowledged. I owe special thanks to KatsumiShimotsu, Jonathan Wright, and two anonymous referees and am grateful to Gordon Anderson,Werner Antweiler, Bryan Campbell, Jon Faust, Silvia Goncalves, Christian Gourieroux,Douglas Hodgson, Angelo Melino, Michael Melvin, and participants at the EEA, CEA, MEG,Georgia Tech IF, and Ryerson international conferences, and seminars at the Federal ReserveBoard, Cornell, Queen’s, Rochester, Toronto, and the University of Western Ontario for usefuldiscussion. Previously entitled ‘Persistence, bias, and the forward premium anomaly: how bad isit?’ Email: [email protected]

Canadian Journal of Economics / Revue canadienne d’Economique, Vol. 39, No. 4November / novembre 2006. Printed in Canada / Imprime au Canada

0008-4085 / 06 / 1244–1281 / C© Canadian Economics Association

The forward premium anomaly 1245

1. Introduction

The forward premium anomaly is a long-standing puzzle in international finance.According to the forward rate unbiasedness hypothesis, the forward exchange rateshould provide an unbiased forecast of the future spot exchange rate. However,not only is this hypothesis rejected by standard regressions of the spot return onthe forward premium (Fama 1984), but puzzling negative coefficients from theseregressions imply that the forward rate is, in fact, a perverse predictor, whichmis-predicts not just the magnitude but even the direction of future exchange ratemovements. Since Covered Interest Parity holds the forward premium equal to thehome/foreign interest rate differential, these results also reverse the direction ofthe classic Uncovered Interest Parity (UIP) condition. Moreover, in conjunctionwith the smooth behaviour of consumption, the estimates imply implausiblylarge risk aversion parameters in the standard international Capital Asset PricingModel (CAPM) (see Mark 1985; Hodrick 1989; Modjtahedi 1991; Kaminskianand Peruga 1990). In this way, the forward premium puzzle poses much the samedifficulty for the international CAPM as the equity premium puzzle does for thedomestic CAPM.

Over the years, several interesting explanations for this puzzle have beensuggested, including a time-varying risk premium (e.g., Hodrick and Srivastava1986), habit persistence (Backus, Gregory, and Telmer, 1993), learning (Lewis,1989), peso effects Evans and Lewis, 1995, central bank feedback rules(McCallum 1994), and models of heterogeneous trading behaviour (e.g., Frankeland Froot 1988). Yet, to this date, there is still no general agreement as to theroot cause of this peculiar empirical finding.

However, the more recent literature has cast some doubt on the econometricvalidity of the stylized facts underlying this puzzle. This is on account of the highlypersistent behaviour of the forward premium. While the exact form of this persis-tence (long-memory, structural breaks, or roots near unity) has been the subjectof some debate,1 a common theme in this literature is that this borderline non-stationarity not only invalidates standard statistical inference, but may potentiallyexplain some of the anomalous empirical findings in tests of forward rate unbi-asedness. For example, in an influential paper Baillie & Bollerslev (2000) showthat by calibrating from a UIP-FIGARCH model, with a long-memory forwardpremium, they are able to recapture some of the salient features of the exchangerate regressions. Maynard & Phillips (2001) come to similar conclusions basedon the large sample theory for regression with long-memory regressors. Similarfindings have been reported under the assumption of structural breaks (Sakoulisand Zivot 2002) or autoregressive roots near unity (see Maynard 2003, Roll andYan 2000; Newbold et al. 1998; Goodheart, McMahon, and Ngama 1997).

1 Crowder (1994, 1995) and Evans and Lewis, 1995 find evidence of a root near unity, whereasBaillie and Bollerslev (1994) and Maynard and Phillips (2001) argue that the behaviour is bettercaptured by a long-memory model, and Sakoulis and Zivot (2002) and Choi and Zivot (2005)argue in favour of a structural break.

1246 A. Maynard

If substantiated, the resolution of this puzzle would have strong implicationsfor both economic modelling and foreign exchange rate practice. Yet, few pre-vious attempts have been made to test this proposition empirically, for exampleby the use of econometric methods that remain reliable in the face of persistentregressors.2 This contrasts with the large financial literature attempting to cor-rect inference on other predictive regressions, such as those dealing with stockreturn predictability (see, e.g., Stambaugh 1999; Torous, Valkanov, and Yan 2005;Valkanov 2003 and Lanne 2002).

This is the goal of the current paper. Instead of relying on regression-basedtests, we provide direct tests of forward rate unbiasedness using three alternativeeconometric methods that are explicitly designed to be robust to the persistentbehaviour of the forward premium. Moreover, the use of these procedures doesnot require us to model explicitly the extent or, in the case of the sign test, eventhe nature of the persistence in the forward premium. In other words, we neednot resolve the debate between near unit roots, long-memory, or structural breakmodels of the forward premium. Nor do we require prior information, estimates,or unit root pre-tests regarding the strength of this persistence.

The three tests we employ are the sign tests of Campbell and Dufour (1995,1997), the covariance-based tests of Maynard and Shimotsu (2005), and theoptimal conditional tests of Jansson and Moreira (2006). All three tests areparticularly well suited to the current application. The sign test has an ex-act finite sample distribution under the null hypothesis, which requires onlymild assumptions on the dependent variable (the excess return) and no as-sumptions whatsoever on the regressor (the forward premium). The exact fi-nite sample nature of the inference rules out any concern whatsoever regardingthe potential for size distortion caused by a persistent or long-memory forwardpremium.

The covariance test is an asymptotic test. However, in contrast to regressiont-statistics, the covariance-based t-statistic has been shown to have a standardnormal limiting distribution even when the conditioning variable has a root nearor equal to unity. Consequently, Monte Carlo studies suggest reliable size insmall samples. It was also found to have power against a more general class ofalternative hypotheses than standard regression tests, including alternatives thatensure a stationary excess return regardless of the persistence in the forwardpremium.

Finally, Jansson and Moreira (2006) have recently introduced an optimal testin the predictive regression context, when the regressor is modelled as a nearunit root.3 Their test eliminates the nuisance parameter problem in standard re-gression by conditioning inference on a set of sufficient statistics for the largestautoregressive root. When the errors are i.i.d. Gaussian the test is exact in finite

2 Two notable exceptions are Bekaert and Hodrick (2000) and Liu and Maynard (2005). However,the corrections they implement are valid only for autoregressive models of the forwardpremium, whereas the literature suggests that the forward premium may well be characterizedby long memory or structural breaks.

3 I thank an anonymous referee for suggesting the use of this test.


samples and is uniformly most powerful among the class of conditionally unbi-ased tests. Under weaker conditions, it maintains correct size and optimal powerunder local-to-unity asymptotics.

Whereas the sign test has been proved to provide reliable inference regardlessof the modelling assumptions on the regressor, the properties of the other twotests have only been studied within the short-memory stationary, unit root, andlocal-to-unity frameworks. Before applying these methods to test unbiasedness,we therefore provide a careful Monte Carlo study demonstrating their reliabilityunder both the long-memory and structural break models that have been con-sidered seriously in the literature as alternative data-generating processes for theforward premium.

Despite the many papers mentioned above that have questioned the standardempirical results based on the persistence of the regressors, few previous attemptshave been made to replace these by tests that are immune to such persistent be-haviour. Notable exceptions are Bekaert and Hodrick (2001), who replace asymp-totic with bootstrapped-based critical values, resampling the residuals from avector autoregressive (VAR) model, and Liu and Maynard (2005), who employthe two-stage bounds procedure of Cavanagh, Elliott, and Stock (1995). Bothprocedures may be expected to work well under the modelling assumptions im-posed in these papers on the forward premium. However, the bootstrap of Bekaertand Hodrick (2001) requires a fixed autoregressive root below unity, and bothprocedures rule out long-memory and structural break models.4

There has also been a related application of sign-testing by Wu and Zhang(1997), who test the sign of the spot return / forward premium relation, but do notexplicitly test forward rate unbiasedness as we do here. It seems that no previousconnection has been made between this work, which had a different motivation,5

and the literature addressed here. Moreover, unlike the finite sample approachof Campbell and Dufour (1997), the tests employed by Wu and Zhang (1997)are based on large sample approximations and do not allow for the possibilityof a non-zero median in the spot return, as would be implied by a constant riskpremium.

Overall, we find less evidence against forward rate unbiasedness using the threerobust tests, than using the standard, but questionable, regression tests. This lendssome support to the contention in the recent literature that the forward premiumpuzzle may be less serious than it appears. On the other hand, for all six dollar-denominated currencies considered here, we are still able to reject using at least oneof the three tests and for some currencies all three tests reject. Moreover, unlike theearlier regression-based evidence, these more robust test results cannot easily beexplained away on the grounds of bias, size distortion, or other purely statistical

4 In ongoing work with Mark Wohar and Aaron Smallwood, we suggest a predictive test in thelong-memory context, where the size distortion is eliminated by fractionally differencing thepredictor. In a recent working paper Rossi (2005) provides inference in a local-to-unity modelfor the long-horizon case, in which the return horizon (e.g., four years) is large relative to thesampling frequency (e.g., one month).

5 They discuss possible bias due to measurement error or a time-varying risk premium.

1248 A. Maynard

reasons. An economic rationale is needed. For several of these currencies we alsoconfirm the significantly negative sign of the spot return / forward premiumrelation. This negative sign underlines the economic significance of the rejectionand constitutes one of the principal stylized facts underlying the forward premiumpuzzle (cf. Engel 1996).

The paper is divided into five sections. Section 2 provides a brief backgrounddiscussion, section 3 outlines the econometric methodology, section 4 presentsand discusses the empirical results, and section 5 concludes. Lemmas are statedin appendix A, proofs are contained in appendix B, and additional details on theeconometric procedures are given in appendix C.

2. Background

2.1. The dataThe empirical results in this paper are based on month-end spot and one-monthforward rates for the USD price of the Australian dollar (AUD), Canadian dollar(CAD), French franc (FFR), German Deutschmark (GDM), Japanese yen (JPY),and British pound (BRP), obtained from Data Resources International. With theexception of the JPY and AUD, for which data start in 79:9 and 86:10, respectively,the series run from June 1973 (73:6) until March 2000 (00:03). The forwardrates are calculated implicitly using the LIBOR one-month interest differential.6

Following Bauer (2001), in order to avoid measurement error due to the timingof maturity dates (Cornell 1989; Bekaert and Hodrick 1993), we used the exactnumber of days between the last business day in each month to translate theannual rates of return into monthly rates.7

2.2. The forward premium puzzleUnder forward rate unbiasedness, the (log) forward exchange rate ( ft) providesan unbiased forecast of the (log) spot rate (s t+1): Etst+1 = ft. This is equivalentto uncovered interest parity (UIP)

Etst+1 − st = ft − st, (1)

where s t+1 − st is the spot return and ft − st the forward premium.8 A morefundamental interpretation is obtained by rewriting (1) as

Et [st+1 − ft] = 0, (2)

6 Covered interest arbitrage holds the forward premium equal to the nominal interest differentialand Maynard and Phillips (2001) report that the nominal interest differential provides a cleanerseries than the forward premium. The forward rate was used for the CAD, since the LIBORseries started much later.

7 The actual number of days between the last business day of each month is divided by the totalnumber of days in the year (365 for the British pound, but 360 by convention for the otherEurocurrency rates – see Bauer (2001)).

8 UIP is often stated as Etst+1 − st = it − i ∗t , where it and i ∗

t are home and foreign interest rates.By covered interest arbitrage it − i ∗

t = ft − st, so that the two definitions are equivalent.


which implies that next period’s excess return (or forecast error) s t+1 − ft mustbe orthogonal to all current and past information in order for unbiasedness tohold. It is this formulation that we work with here.

The range of possible alternatives to (2) is clearly quite rich. However, it iscommon in the literature to parameterize the alternative in terms of a simplelinear model containing only a single lag of the forward premium. With (1) inmind, the most popular formulation is (cf. Fama 1984)

st+1 − st = α + β( ft − st) + et+1, (3)

in which unbiasedness imposes both β = 1 and Etet+1 = 0.9 However, settingγ =β − 1, (3) may also be written in the traditional form of a predictive regressionas

st+1 − ft = α + γ ( ft − st) + et+1, (4)

which for γ �= 0 provides a simple alternative specification to the orthogonalitycondition in (2).

Results from (3) are shown in table 1. These results, which are fairly typical(Engel 1996),10 show strong evidence against unbiasedness. In particular, the twostylized facts emphasized in the literature are that (i) unbiasedness is rejected(β �= 1; alternatively, γ = β − 1 �= 0 in (4)) and (ii) the forward premium appearsto be a perverse predictor of spot return (β < 0). One also notices small R2

values.The perverse sign of the regression coefficients has been particularly puzzling.

In itself, the statistical rejection of unbiasedness, which jointly imposes risk neu-trality and rational expectations, may not seem a complete surprise. For example,a rejection based on β = 0.75 could quite easily be explained away as evidence ofa small time-varying risk premium, a slight deviation from rational expectations,or a modest Peso problem. Such explanations, while still possible, become a bitstrained for a rejection based on, say, β = −2.

In fact, β < 0 has some unattractive implications. Fama (1984) shows thatit requires the risk premium to be both negatively correlated with ft − st andmore variable than the expected spot return. This has proved somewhat difficultto reconcile with reasonable levels of risk aversion (Mark 1985; Hodrick 1989;Modjtahedi 1991; Kaminski and Peruga 1990). On the other hand, if expecta-tional biases are to blame, as suggested by survey data (e.g., Froot and Frankel1989), then β < 0 implies that, on average, the markets misforecast not just the

9 Unbiasedness also requires α = 0. However, α �= 0 could indicate a constant risk premium.10 Recent literature has pointed to some notable exceptions in recent sub-samples (Bekaert and

Hodrick 1993; Baillie and Bollerslev 2000), very long (several years) horizons (Meredith andChinn 1998), and some developing country currencies in recent samples (Bansal and Dahlquist2000). Comprehensive surveys have been provided by Engel (1996), Lewis (1994), and Hodrick(1987).

1250 A. Maynard

TABLE 1UIP regression results

β se β

seβ − 1

se R2 β se β

seβ − 1

se R2

AUD −1.37 0.83 −1.65 −2.86ψ 0.017 GDM −0.72 0.71 −1.01 −2.42ψ 0.003CAD −1.14 0.47 −2.44ψ −4.59ψ 0.018 JPY −2.52 0.95 −2.66ψ −3.72ψ 0.028FFR −0.60 0.53 −1.14 −3.02ψ 0.004 BRP −1.42 0.66 −2.16ψ −3.68ψ 0.014

NOTES: ψ Two-sided rejection at 5%. Entries show the regression coefficient β, its standard error se,the t-statistics β

se and β − 1se for tests of β = 0 and β = 1, respectively, and the coefficient of

determination R2.

magnitude, but even the direction of change. In sum, the finding that β < 0has served to underline the economic, as well as statistical, significance of thisrejection.

2.3. Time series characteristics of the data and their implicationsViewed from the perspective of the time series econometrician, the data also showsome interesting characteristics. The excess return s t+1 − ft is serially uncorre-lated under the null hypothesis and displays little obvious persistence in its firstmoment. On the other hand, the null hypothesis imposes no restriction on thebehaviour of the forward premium, whose persistent behaviour has been doc-umented in the previous literature. However, the exact form of this persistence(root near unity, long memory, or structural break) has nonetheless been some-what controversial. Crowder (1994, 1995) and Evans and Lewis (1995) fail toreject a unit root in the forward premium and Crowder rejects the null of sta-tionarity, whereas Baillie and Bollerslev (1994) and Maynard and Phillips (2001)detect evidence of long memory, and Sakoulis and Zivot (2002) and Choi andZivot (2005) find in favour of structural breaks.

In order to provide robust inference, we explicitly avoid taking a stand onthis debate. Nevertheless, we illustrate the persistent behaviour of the forwardpremium in table 2, columns 2–6. Column 2 shows the Augmented Dickey-Fullertest statistic, using an intercept and k lags, where k (in column 3) is chosenaccording to the Ng and Perron (2001) MIC criterion. Alternatively, columns 4–5shows the Kim and Phillips (1999) Modified Log Periodogram (MLP) estimate ofthe fractional integration parameter (d) and its standard error, using a bandwidthof m = [n3/4] shown in column 6.11 The right half of the table shows the same

11 The MPL modifies the LP regression of the log-periodogram on the log Fourier frequencies, sothat it remains valid for 0 ≤ d ≤ 2. Like the LP estimate, the MLP estimate is

√n/m, consistent

and asymptotically normal with variance π 2/24. We know of no optimal bandwidth rule overthe full range 0 < d < 1 and thus follow Maynard and Phillips (2001) in employing m = n3/4.The overall evidence in favour of long memory in ft − st appears quit robust to this choice andhas also been confirmed in ARFIMA models (Baillie and Bollerslev 1994; Maynard and Phillips2001).


TABLE 2Evidence on the persistence in the forward premium and excess return

ft − st s t+1 − ft

tADF [k] d MLP (sed ) [m] tADF [k] d MLP (sed ) [m]

AUD −2.138 [3] 0.903φ (0.096) [45] −6.02ψ [2] −0.059 (0.096) [45]CAD −2.648 [5] 0.693φ (0.074) [75] −7.52ψ [4] 0.055 (0.074) [75]FFR −3.088ψ [6] 0.529φ (0.074) [75] −6.75ψ [4] 0.231φ (0.074) [75]GDM −1.734 [6] 0.685φ (0.074) [75] −6.18ψ [6] 0.083 (0.074) [75]JPY −2.602 [4] 0.932φ (0.081) [62] −6.94ψ [3] 0.199φ (0.081) [62]BRP −3.632ψ [5] 0.687φ (0.074) [75] −7.42ψ [4] 0.093 (0.074) [75]

NOTES: tADF denotes the augmented Dickey Fuller t-statistic based on an intercept and k lags, wherek is selected by the Ng and Perron (2001) MIC criterion. ψ denotes one-sided rejection of a unit rootat 5% level, using the asymptotic critical value of −2.86. d MLP denotes the Kim and Phillips (1999)Modified Log Periodogram estimate of the fractional integration parameter using a bandwidth ofm = [n0.75], and sed denotes its asymptotic standard error. φ denotes one-sided rejection of d = 0 at a5% level using a standard critical value of 1.645.

set of statistics for the excess return, and it is interesting to note their differences.Whereas a unit root is clearly rejected in all cases for the excess return, just tworejections are observed for the forward premium. Likewise, the MLP estimatesof d are relatively small for the excess return, and we fail to reject d = 0 in four ofsix cases. By contrast, while the estimates of d vary considerably for the forwardpremium, they all lie above one-half. Plots of the excess return and forwardpremium, shown for the DM/USD rate in figures 1 and 2, respectively, convey asimilar impression.

This has some interesting implications for standard regression tests of unbi-asedness, as in the predictive regression in (4).12 First, the near non-stationarityof the right-hand-side variable in regressions of this type are known to lead toboth bias in estimation and size distortion in testing, with actual rejection ratespotentially lying well above their nominal levels (Mankiw and Shapiro 1986;Stambaugh 1999). In other words, standard regression tests are no longer reli-able in this context and may reject with high probability even when unbiasednessholds true. For example, with near unit root regressors, Mankiw and Shapiro(1986) find that actual rejection rates under the null hypothesis may exceed 25%for a nominal 5% level. They may even exceed 50% when a trend is included inthe regression. As we report in the simulations of section 3.5, rejection rates maybe equally unreliable when the regressor follows a long-memory or fractionallyintegrated process.

While size corrections are available (see, e.g., Cavanaugh, Elliott, and Stock1995 in the near unit root case), they typically require the researcher to take astand on the nature of persistence. However, as mentioned above there has beena lack of consensus on how to model the forward premium, that is, whether as a

12 Since β = γ + 1, the same issues arise for tests of β = 1 in (3).

1252 A. Maynard

FIGURE 1 Excess exchange rate return s t+1 − ft (DM/USD)

FIGURE 2 Forward premium ft − st (DM/USD)

long-memory series, a structural break process, or an autoregressive series witha root near unity.

Secondly, the persistent nature of ft − st may lead one to question whether thesimple alternative in (4) is the only reasonable one to consider. Note that if ft − st

was truly non-stationary, then (4) allows only two possibilities. Either γ = 0, inwhich case s t+1 − ft is stationary, or γ �= 0, in which case s t+1 − ft inherits thenon-stationarity in ft − st. In other words, (4) is no longer just a predictabilitytest; rather, it is a cointegration test.

As seen above, s t+1 − ft shows relatively little persistence. Therefore, if ft − st

were non-stationary, then one should expect γ = 0 to hold. Yet all that the pa-rameter restriction γ = 0 implies in this context is that s t+1 − ft is stationary.By contrast, the unbiasedness condition in (2) imposes a much stronger restric-tion than stationarity. It is not sufficient that s t+1 − ft be linearly unpredictable


by ft − st, as in the coefficient restriction γ = 0 in (4). Rather, it must not bepredictable using any information available at or before time t. This includes theinformation in past first differences, as well as more complicated functions ofthe past lags, including deviations from recent averages, pre-filtered or stochasti-cally de-trended versions of ft − st. This suggests the consideration of more gen-eral alternatives than those specified in (4). In particular, one would wish to allowan alternative rich enough to detect predictable, but still stationary, behaviour ins t+1 − ft, even when ft − st is non-stationary. It is not clear that this can beachieved in the traditional regression framework described by (4). Thus, con-siderations of both test size and specification motivate us to consider morerobust alternatives to traditional regression tests, as outlined in the followingsection.

3. Econometric methodology

3.1. The broad econometric frameworkAs suggested by the stylized facts discussed above, we consider a dependent vari-able yt, which is serially uncorrelated under the null hypothesis, and a conditioningvariable xt, which may be persistent but where the exact form and strength of thispersistence may be unknown or perhaps difficult to identify in finite sample. Wedefine It as the information set generated by {xs, ys s ≤ t} and Et as the condi-tional expectation with respect to It. Then setting yt+1 = s t+1 − ft and xt = ft −st, the condition Etyt+1 = 0 imposes the unbiasedness hypothesis. In this sectionwe outline three robust tests, the sign, covariance, and optimal conditional teststhat will be employed to test this hypothesis. The mechanism and precise modelassumptions underlying each test are discussed in sections 3.2, 3.3, and 3.4, re-spectively, and some new simulation results are provided in section 3.5. However,we first provide a broad overview of the robustness features important to thepresent application.

Following the discussion of the previous section, we are specifically interestedin two forms of robustness. First, we seek to employ tests that are robust to thedegree and perhaps even the nature of the persistence in xt. For example, if xt ismodelled autoregressively as13

xt = ρ0 + ρ1xt−1 + ux,t t ≥ 0, (5)

then our interest lies in tests which retain the correct significance level regardlessof whether xt is stationary (ρ1 < 1) or unit root non-stationary (ρ1 = 1). The

13 The exact assumptions on u x,t differ across the three tests. The sign test does not require anyassumptions on u x,t; the covariance test allows u x,t to follow a general linear process subject tostandard moment, homoscedasticity, and short-memory conditions. The asymptotic results forthe conditional test allow a similar level of generality, whereas the finite sample results requireu x,t to be i.i.d. Gaussian.

1254 A. Maynard

idea that ρ1 may be close, but not equal, to unity is often formalized by thenear-unit-root/local-to-unity model

ρ1 = 1 + cn

for c ≤ 0, (6)

which is known to yield accurate large sample approximations (Phillips 1987;Chan 1988; Nabeya and Sørensen 1994). All three tests employed here remainvalid across the stationary, near-unit-root, and unit root specifications.

The sign test, which requires no assumption on xt, has the further advantageof retaining its validity under alternative models for the behaviour of xt. Thisincludes the long-memory/fractionally integrated model of the forward premiumadvocated by Baillie and Bollerslev (2000) and Maynard and Phillips (2001),

xt = μx + (1 − L)−dux,t t ≥ 0, xt = 0 t < 0 for 0 ≤ d ≤ 1, (7)

which encompasses both traditional short-memory stationary models, such as theARMA model (d = 0), the unit root model (d = 1), and the empirically relevantlong-memory models (0 < d < 1) that lie in between.14 Likewise, it includesmodels with structural breaks in the forward premium, as suggested by Sakoulisand Zivot (2002) and Choi and Zivot (2005), such as the autoregressive modelwith m historical mean-shifts

xt = ρ0, j + ρ1xt−1 + ux,t for t = n j−1, . . . n j , (8)

where j = 1, . . . m + 1, n0 = 1 and nm+1 = n. Our simulation results suggestthat the conditional and covariance tests may often perform reasonably when xt

follows the type of processes given in (7) or (8). However, unlike the case of thesign test, we know of no analytic results available to establish that this holds moregenerally.

The second aspect in which we seek flexibility is in the specification of thealternative hypothesis. Under the null hypothesis, we simply have

yt+1 = μy + uy,t+1, (9)

where Etuy,t+1 = 0. The alternative hypothesis admits more possibilities. Thelinear alternative,

yt+1 = α + γ xt + uy,t+1 γ �= 0, (10)

as in (4) is the most natural one when xt is short-memory stationary. However,when xt is non-stationary, (10) implies non-stationary behaviour in yt+1. Giventhat excess returns display little apparent persistence, it seems reasonable to allow

14 See Baillie (1996) for an excellent description and survey of long-memory modelling.


also for alternatives under which yt+1 is stationary but predictable. This includesalternatives such as

yt+1 = δ0 + δ(1 − (1 + c/n)L)xt + uy,t+1 = δ0 + δux,t + uy,t+1, δ �= 0, (11)

when xt has a root near unity as in (5), and those of the form

yt+1 = δ0 + δ(1 − L)d xt + uy,t+1 = δ0 + δux,t + uy,t+1, δ �= 0, (12)

when xt has long memory as in (7).In this respect both the sign and the covariance tests are attractive in that

neither is based on parameter restrictions in specific parametric alternatives suchas (10). The sign test imposes only the condition that yt+1 is independent of It

under the null hypothesis, and the covariance test is based on a zero-covariancerestriction on yt+1 and xt, which may be imposed within a wide class of model.In fact, the latter test is specifically designed to provide power against a rangeof alternatives that allow yt to remain stationary regardless of whether xt is sta-tionary. Thus, it maintains consistency against the alternatives in (10) when xt

is stationary and against the alternative in (11) when xt is near non-stationary.The conditional test, which tests γ = 0 in (10) in conjunction with (5), is moreclosely tied to this specific parametric alternative. On the other hand, within thisparametric class, it possesses optimal power properties not shared by other tests.

3.2. Sign testsA fully non-parametric test of forward rate unbiasedness may be obtained usingthe sign statistic tests of Campbell and Dufour (1995, 1997). The sign test isparticularly attractive in the current context, since (i) it allows for exact finitesample critical values, eliminating the previous concern over finite sample biasand size distortion, and (ii) these critical values do not depend in any way on thenature or degree of the persistence in xt. In other words, the test applies equallyunder the autoregressive, long-memory, and structural break models consideredin the literature on the forward premium. Moreover, it is not necessary to knowwhich model applies. In fact, the size remains correct no matter which process xt

follows.Let gt denote any conditioning variable that is a measurable function of It, let

b0 denote the median of yt and define

Sg(b0) =n−1∑t=1

sign((yt+1 − b0)gt

), (13)

as the sign statistic, where sign(u) = 1 for u ≥ 0 and zero otherwise. Underthe null hypothesis that yt+1 has constant median b0 and is independent of It,but without any assumptions on xt, Campbell and Dufour (1995) (proposition 1)

1256 A. Maynard

show that Sg(b0) has an exact binomial finite sample distribution with n trials andprobability of success equal to 0.5. Moreover, it follows immediately from Coudinand Dufour (2003, prop. 3.2) that the independence assumption may be replacedby the much weaker mediangale difference sequence assumption:15

P(yt+1 − b0 > 0 | It) = P(yt+1 − b0 < 0 | It) = 0.5. (14)

Then, for known b0 the quantiles of the binomial distribution may be used toconduct either one- or two-sided tests. To be concrete, denoting the critical valueBi α(n, p) as the 1 − α quantile of the binomial distribution and exploiting itssymmetry about n/2, we would reject for Sg >Bi α(n, p), Sg < Bi 1−α(n, p), orSg �∈ [Bi1−α/2(n, p), Biα/2(n, p)], respectively, in a right-sided, left-sided, or two-sided test of level α.16

In most practical applications the median b0 is unknown. Campbell and Du-four 1997) discuss two methods for handling this. The first is the median sign test,in which b0 is replaced by the sample median of yt. It is no longer exact in finitesample but appears to work well in practice. Employing a first-stage confidenceinterval on b0 in conjunction with a second-stage Bonferroni bound, Campbelland Dufour (1997) also provide a more sophisticated approach, which ensuresa test of level α in finite sample. Below we outline the steps that are required toimplement the two-sided test of level α employed here. One-sided tests may beconstructed analogously.

1. Choose first and second-stage significance levels α1 and α2 such thatα1 + α2 = α.17

2. Construct an exact 1 − α1 first-stage confidence interval J(α1) for themedian b0.

3. Construct second-stage inner bounds (Bi1−(α+α1)/2(n, 0.5), Bi(α+α1)/2(n,

0.5)), and outer bounds (Bi1−α2/2(n, 0.5), Biα2/2(n, 0.5)), using the α +α1 and α2 = α − α1 binomial critical values, respectively.

4. Evaluate Sg(b) over b ∈J(α1), comparing it with the second-stage bounds.(a) Reject if Sg(b) �∈ [Bi1−α2/2(n, 0.5), Biα2/2(n, 0.5)] for all b ∈J(α1) (Sg

lies everywhere outside the outer bounds).(b) Fail to reject if Sg(b) ∈ [Bi1−(α+α1)/2(n, 0.5), Bi(α+α1)/2(n, 0.5)] for all

b ∈J(α1) (Sg lies everywhere inside the inner bounds).(c) Otherwise the test is indeterminate (Sg crosses between bounds).

15 Simply redefine ut in Coudin and Dufour (2003) as ut = (yt − b0)gt−1, omit the conditioning onX , and note that ut inherits the mediangale difference sequence property of yt, since gtε It. Ithank an anonymous referee for noting this.

16 Because the binomial distribution is a discrete distribution, an exact critical value cannot alwaysbe obtained for a test of level α. In this case, we err on the conservative side, choosing the nearestavailable critical value that provides a test with size less than or equal to α.

17 Campbell and Dufour (1997) suggest choosing α1 small. We will use α = 0.05, α1 = 0.01,and α2 = 0.04.


The Bonferroni inequality ensures that false rejections occur with probability nogreater than α = α1 + α2. In order to enhance power, the median of gt is alsoremoved, using past values only. In other words, gt is replaced by g∗

t , where the‘de-median’ operator∗ is defined by

x∗t = xt − mt

1(xt), (15)

where mt1(xt) = sample median (x1, x2, . . . , xt). The reader is referred to Camp-

bell and Dufour (1997), proposition 2(a) for further details.Employing the sign test with yt+1 = s t+1 − ft and gt = ft − st, we test the

hypothesis that the excess return is independent of the available information attime t and this is thus roughly analogous to a test of γ = 0 in (4). In this case thesign statistic is given by Sg = �n−1

t=1 sign ((s t+1 − ft − b0) ( ft − st)), and when b0

= 0 it simply counts the number of times that the forward premium predicts theexcess return with the correct sign. Likewise, 100 ∗(1/n) Sg denotes the percentageof correctly signed predictions. Under the null of unbiasedness, the excess returnis inherently unpredictable, so that ft − st has no predictive content for s t+1 − ft.This implies a roughly equal number of positive and negative yt+1 gt pairs (100 ∗(1/n) Sg ≈ 50%). If, instead, unbiasedness is rejected and ft − st helps to forecasts t+1 − ft, then there ought to be either a significant majority of positive (100 ∗(1/n) Sg > 50%) or negative (100 ∗ (1/n) Sg < 50%) pairs.

To establish the link between the sign tests and the covariance estimation ofsection 3.3, note that, since (1/n) Sg is the sample mean of sign ((yt+1 − b0)gt), itspopulation equivalent is given by E [sign((yt+1 − b0)gt)] = Pr{(yt+1 − b0)gt > 0},the probability that gt predicts yt+1 − b0 with the correct sign. Under someadditional symmetry assumptions, we show in lemma 2 of appendix B that thisprobability is greater (resp., less) than 0.5 precisely when the covariance betweengt and yt+1 is positive (resp., negative).

One of the principal advantages of the sign test in this context is that it makes noassumption on the process for xt. Thus, by employing the sign test, we allow simul-taneously for short-memory stationary, unit root, local-to-unity, long-memory,and structural break models and thus cover the full spectrum of modelling possi-bilities for the forward premium considered in the empirical literature. A secondadvantage of the sign test is that it is robust to oultiers, an important consid-eration when exchange rate data are employed. Finally, as Coudin and Dufour(2003) remark, the mediangale difference assumption in (14) allows for condi-tional heteroscedasticity in yt of a very general nature without altering the exactfinite sample properties of the sign test.18

A related application of sign testing is given in Wu and Zhang (1997), who testthe sign of spot return / forward premium relation. While our work is similar in

18 Note, however, that the results depend on the proper centring of yt+1 about its median.Otherwise, ‘volatility dependence produces sign dependence, so long as expected returns arenonzero’ (Christoffersen, Diebold 2002, abstract). Centring gt improves power, but does notaffect size (Campbell and Dufour 1997).

1258 A. Maynard

spirit to theirs, there are also several significant differences to note. First, our workis directly motivated by a line of literature questioning standard unbiasednessresults on the basis of the persistent features of the forward premium, whereasno clear link between their tests and this branch of the literature has yet beenmade. Secondly, Wu and Zhang (1997) do not actually provide a direct test offorward rate unbiasedness, which involves a non-predictability condition on theexcess return (s t+1 − ft) in (2). They test instead the sign of the relation betweens t+1 − st and ft − st. While both tests are of interest, only the version used hereemploying s t+1 − ft tests forward rate unbiasedness.

The more recent form of the sign tests employed here also offers several ad-vantages from an econometric perspective. Given that we address a question offinite sample size distortion, the exact version of the test that we employ mayhave some additional appeal relative to the asymptotic counterpart used by Wuand Zhang (1997). Perhaps more important, the tests of Wu and Zhang (1997)implicitely assume a zero median for both yt and xt. We improve test power bycentring gt about its median. Likewise, by employing the Campbell and Dufour(1997) bounds procedure described above, we ensure test validity without assum-ing a known median for yt. The median of the excess return may be expected tobe non-zero in the presence of a constant risk premium, which implies a non-zerovalue of α in (4). The consequence of ignoring the median in yt can, in fact, berather severe. For illustration, consider the five (gt,yt+1) pairs, {(1, 5), (2, 4), (3, 3),(4, 2), (5, 1)}, displayed in the left panel of figure 3. They are perfectly negativelycorrelated. Nevertheless, the product yt+1 gt is always positive, so that whenb0 = 0 is imposed, the sign statistic indicates a positive relation between thetwo variables. In the middle panel, the median of yt+1 is removed, leavingequally as many positive as negative pairs. The sign test is no longer mislead-ing, but it has little power. Finally, it is only in the right panel, where the me-dians of both yt+1 and gt are removed, that the sign test detects the negativerelationship.

3.3. Covariance-based testsAs a second robust approach to testing unbiasedness, we employ the recentlyproposed covariance based tests of Maynard and Shimotsu (2005). To explainthe intuition behind this test, it is useful to first consider xt stationary and writethe population coefficient in (4) in its reduced form as γ = cov(yt+1, xt)/var(xt).Given standard assumptions regarding the existence of moments, var(xt) is finite.Therefore, the restriction γ = 0 implied by unbiasedness holds true if and only ifcov(yt+1, xt) = 0. Otherwise, xt contains information that could help to predicts t+1 − ft. This is a true orthogonality restriction.

Now, compare this balanced case with the unbalanced case in which yt+1 isI(0) but xt is I(1). Arguably, the critical difference is that var(xt) is finite in thefirst case but diverges to infinity in the latter. Suppose we were to define thepopulation regression coefficient as γ t = cov(yt+1, xt)/var(xt) and define its limitas γ = lim t→∞ γ t. Then, for xt non-stationary, we would observe that γ = 0 for


FIGURE 3 Sign tests centred and uncentred: a simple illustration

any stationary variable yt+1, not necessarily because cov(yt+1, xt) = 0, but simplybecause var(xt) becomes infinite.

The idea behind the covariance-based approach is to avoid the problemof normalizing by a possibly infinite variance, by testing cov(yt+1, xt) = 0in place of γ = 0. This provides a valid basis for testing unbiasedness sincecov(yt+1, xt) = 0 implies γ = 0. In fact, in the stationary/balanced case, cov(yt+1,xt) = 0 and γ = 0 constitute exactly identical null hypotheses. However, intuitively,in the unbalanced case cov(yt+1, xt) = 0 restricts the null hypothesis to a smallerand more meaningful set of processes, whereas γ = 0 may occur as a result ofeither cov(yt+1, xt) = 0 or var(xt) divergent.

This intuition involves an abuse of notation, since cov(yt+1, xt) is no longerconstant for xt non-stationary. However, in this case, the restriction on the covari-ance may be replaced by a restriction on its limiting value, lim t→∞ cov(yt+1, xt).This limit may be expressed more concretely by writing xt = x0 + ∑t−1

h=0 �xt−h

and cov(yt+1, xt) = cov(yt+1, x0) + ∑t−1h=0 cov(yt+1, �xt−h). Assuming the joint

stationarity of (yt, �xt)′ and the condition lim t→∞ cov(yt, x0) = 0 implied bystandard initializations, such as x0 = 0, Maynard and Shimotsu (2005) define thelimiting covariance as

limt→∞ cov(yt+1, xt) =

∞∑h=1

cov(yt, �xt−h) = λy,�x, (16)

the standard one-sided long-run covariance between yt and �xt (see Maynardand Shimotsu 2005, eq. 8, for the near-unit-root case).

It is apparent that the unbiasedness restriction Etyt+1 = 0 implies thatλy,�x = 0 regardless of the order of integration in xt. Moreover, unlike tests

1260 A. Maynard

based on γ , those based on λy,�x allow us to consider reasonable alternativesunder which yt+1 is still stationary, despite possible non-stationarity in xt. Forexample, in the special case in which ut = (uy,t, ux,t)′ is serially uncorrelated, λy,�x

simplifies to

λy,�x = δvar (ux,t) �= 0 for γ �= 0 (17)

under the alternative in (11) when xt has a near unit root as in (5).19

Expressed in terms of (yt, �xt), λy,�x is just a one-sided version of the long-runcovariance term used for HAC standard errors and may therefore be estimatedby standard kernel estimation techniques (e.g., Newey and West 1987; Andrews1991) of the form

λy,�x =n−1∑h=1

k(

hm

)�y,�x(h), �y,�x(h) = 1

n

n−h∑t=1

�xt yt+h, (18)

where the bandwidth parameter m and kernel weighting function k() satisfythe standard assumptions given in Maynard and Shimotsu (2005, ass. K andM), which are met by common kernels such as the Bartlett kernel.20 FollowingAndrews (1991), m is chosen using a parametric approximating model (here aVARMA(1,1)) to minimize the asymptotic mean squared error of λy,�x given by(Maynard and Shimotsu 2005, 9, eq. 12):

m∗ =(

qk2qα(q)n

/∫ ∞

−∞k2(x) dx

)1/(2q+1)

,

α(q) = 4

⎛⎝ ∞∑h=1

��xy(h)hq

)2/(ωyyω�x,�x), (19)

where q = 1 and k1 = 1 for the Bartlett (Newey-West) kernel used below.Under the assumption that (yt, �xt)′ follows the joint linear process (MA(∞)

representation),

( yt, �xt )′ = A(L)εt =∞∑j=0

Ajεt− j ,

∞∑j=0

j ||Aj || < ∞, εt ∼ i.i.d.(0, I2),(20)

19 See appendix B for the derivation of (17). If unbiasedness instead fails because yt+1 has anon-stationary component cointegrated with xt, then cov(yt+1, xt) is divergent, so that a testbased on λy,�x should again reject.

20 In practice, yt is demeaned prior to estimation, but xt is not, since the difference operationremoves the mean in �xt. Since both yt and �xt are stationary, this has no effect on the limitingdistribution.


which allows for a unit root in xt, Maynard and Shimotsu (2005) show theconsistency and asymptotic normality of λy,�x. Its limit distribution (as n →∞) is given by

√n/m(λy,�x − λy,�x) →d N(0, V ), V = ωyyω�x,�x

∫ ∞

0k2(x) dx, (21)

where ωyy = ∑∞h=−∞ cov(y0, yh) and ω�x,�x = ∑∞

h=−∞ cov(�x0, �xh) respec-tively define the long-run variances of yt and �xt. This then forms the basisfor an asymptotically standard normal t-statistic (Maynard and Shimotsu (2005,13, cor. 6)):

tλ =√ n

m (λy,�x − λy,�x)√V

→d N(0, 1), (22)

where V is a consistent estimator of V .21 tλ may be used in conjunction withstandard normal critical values to test the restriction H0:λy,�x = 0 implied byunbiasedness. Further details and a list of steps required to implement the two-sided tests conducted here are given in appendix C.

A nice feature of the limit distribution in (22) is that it continues to hold withoutmodification under (5) and (6) for any finite value of the local to unity parameterc ≤ 0 (Maynard and Shimotsu (2005, lemma 3)). On the other hand, when xt

is stationary, �xt is over-differenced and therefore V = ω�x,�x = 0. Thus, thedistribution in (21) is degenerate. Nevertheless, Maynard and Shimotsu (2005,lemma 7) show that the t-statistic in (22) still provides conservative inference inlarge sample. Moreover, the degeneracy was shown to correspond to a faster rateof convergence for the estimator λy,�x. Therefore, while the conservative natureof inference in the stationary case hurts test power, this may be at least partiallyoffset by the increased rate of convergence.

Although Maynard and Shimotsu (2005) do not consider long-memory mod-els, the limit covariance λy,�x in (16) remains well defined under the fractionallyintegrated model (7). Likewise, after establishing the summability condition inlemma 1 (B4) of appendix A, the consistency of λy,�x for 0 ≤ d ≤ 1 followsdirectly from Jansson (2002, theorem 1). However, when d < 1, we again haveV = 0, so that the distribution in (21) is degenerate, just as in the short-memorystationary case. Thus, we conjecture that, as in this case, a faster convergence ratemay apply for the estimator and the t-statistic may again provide conservativeinference. While this conjecture finds support in the simulations of section 3.5,providing a full set of results under the fractionally integrated model (7) wouldlie well beyond the scope of the current paper.

21 The simplest such estimator is V = 1/2ωyyω�x,�x

∫ ∞0 k2(x) dx, where ω is any consistent kernel

estimator of ω. However, the estimator VMDS in (C1) of appendix C is suggested by Maynard andShimotsu (2005, 17) and employed here on account of its better finite sample properties.

1262 A. Maynard

3.4. Optimal conditional testsJansson and Moreira (2006) have recently developed a correctly sized and con-ditionally unbiased test for the parameter γ in the predictive regression model in(10), which is asymptotically independent of the value of c in the local-to-unityspecification for xt in (5) and (6). Their test also has desirable power propertiesfor the alternative in (10). In fact, with Gaussian innovations they show thatit is uniformly most powerful within the class of conditionally unbiased testingprocedures.

An infeasible version of the test obtains exact finite sample size and optimalpower under the assumption that

ut = (uy,t, ux,t)′ ∼ i.i.d. N(0, �) for � =[σyy σxy

σxy σxx

].

The feasible version continues to have correct size and to obtain the same Gaus-sian power envelope in large sample under the local-to-unity model in (6) whenthese assumptions on ut are relaxed in favour of a more general linear process (orMA(∞)) specification, together with standard short-memory, homoscedasticityand moment conditions. The test is unique in simultaneously providing bothcorrect size and optimal power in the local-to-unity framework. On the otherhand, the assumptions on xt are still more restrictive than those of the sign test,and the predictive regression alternative is arguably less general than the implicitalternatives in the previous two tests.

The basic mechanism at work is most easily explained in the finite sampleframework in which ut is i.i.d. Gaussian. In the model given by (5) and (10), γ

is the parameter of interest, while ρ1 acts as a nuisance parameter. The difficultyencountered when applying standard regression tests in this model is that thedistribution of the test statistic typically depends on the value of ρ1. Janssonand Moreira (2006) eliminate this nuisance parameter problem by conductinginference conditional on a set of sufficient statistics for ρ1. In particular, theyshow that S = (Sγ , Sρ , Sγ γ , Sρρ) provides a joint sufficient statistic for the modelparameters, where

Sγ = σ−1yy·x

T∑t=1

(xt−1 − x)(yt − σ−1

xx σxyxt)

Sρ = σ−1xx

T∑t=1

xt−1xt − σ−1xx σxySγ

Sγ γ = σ−1yy·x

T∑t=1

(xt−1 − x)2 Sρρ = σ−1xx

T∑t=1

x2t−1

and σ yy·x = σ yy − σ 2xy/σ xx. Likewise, the last three of these statistics, Sρ = (Sρ ,

Sγ γ , Sρρ), together form a joint sufficient statistic for ρ1, so that the distributionof Sγ conditional on Sρ is a nuisance parameter free in the sense that it doesnot depend on ρ1. Thus correctly sized (infeasible) tests may be conducted by


comparing the test statistic Sγ with critical values based on quantiles of theconditional distribution of Sγ given Sρ . For example, in a two-sided test of level αa rejection occurs if Sγ < qα/2(Sγ | Sρ) or Sγ >q 1−α/2(Sγ | Sρ), where qα(Sγ | Sρ)is the α conditional quantile of Sγ given Sρ . Jansson and Moreira (2006) show thatthis test procedure is uniformly most powerful among all conditionally unbiasedtests, where the conditioning is taken with respect to the specific ancillary (Sγ γ ,Sρρ).

In principle, the test is made feasible by replacing the unknown quantities inS by consistent estimates. However, some additional adjustments are required toproduce a test that works under general conditions on the innovations (Janssonand Moreira 2004, sec. 5 ).22 In implementing the test, the conditional quan-tiles qα/2(Sγ | Sρ) and q 1−α/2(Sγ | Sρ) must also be calculated. While Janssonand Moreira (2004) provide an integral representation of this distribution, ap-proximation of these integrals is highly computer intensive. Instead, we obtainour critical values from the neural network approximation of Polk, Thompson,and Vuolteenaho (2004), who apply the same approach to test for stock returnpredictability.23

The conditional test may also be inverted to provide confidence intervals (Polket al. 2004) and a median unbiased estimator of γ (Eliasz 2005). The 1 − α

confidence interval consists of all values of γ that cannot be rejected at levelα and the median unbiased estimator is the value of γ for which the medianof the conditional distribution of Sγ equals its observed value. This ensures alack of systematic bias, since the estimator is as likely to overestimate γ as tounderestimate it.

3.5. A small Monte Carlo studyAs explained in section 3.1 above, our interest lies in both the robustness ofinference with respect to the process driving xt and the flexibility of the modelrelating yt and xt−1 under the alternative hypothesis. In this section we reportresults from a small simulation study designed to gauge the performance of thetests in these respects.

We focus first on the question of robust inference. All three tests are knownto be robust under the autoregressive (5) and local-to-unity models (6), andsimulations results in the existing literature show them to perform well in finitesample. Thus, we restrict attention to the fractionally integrated and structural

22 The variance/covariance parameters are replaced by their long-run counterparts toaccommodate dependent innovations, and the estimated large-sample sufficient statistics take aslightly different form than their finite sample counterparts described above. The reader isreferred to section 5 of Jansson and Moreira (2006) for details.

23 The neural network approximation is lengthy to describe but is clearly laid out in section 4 ofPolk, Thompson, and Vuolteenaho (2004) to which we refer the reader for further details. Polket al. (2004), also make their Matlab code available for download athttp://kellogg.northwestern.edu/faculty/polk/research/ptv.m. Use of their Matlab code,translated into Gauss for this project, is gratefully acknowledged.

1264 A. Maynard

break models in (7) and (8), respectively. Likewise, since the two-stage Bonferroniversion of the Campbell and Dufour (1997) sign test is provably exact, it seemsmore informative to present the results for the median sign test, in which yt iscentred about its sample median.

We consider tests of the null hypothesis Etyt+1 = 0. Under this null, xt ispredetermined, but need not be strictly exogenous. We therefore specialize theresiduals in (24) to

ut = (uy,tux,t)′ ∼ i.i.d. N(0, �) for � =(

1 σ12

σ12 1

). (23)

Imposing the null hypothesis, yt is generated from (9) with μy,t = 0. Finally, xt

is generated by either (7) (long-memory model) or (8) (structural break model).In the long-memory case, we vary the value of d across the rows and σ 12 acrossthe columns. The structural break model requires the specification of more pa-rameters. Here, we simulate xt from (8), using equally spaced breaks of equalmagnitude, but of opposite sign. We vary the magnitude of the break (|ρ0,j −ρ0,j−1|) across the rows and the fraction of breaks (m/n) across the columns, withρ1 given by (6) with c = −0.1 and σ 12 = 0.95. All simulations are conductedusing 2,000 observations and a sample size of 100.

Table 3 presents rejection rates under the null hypothesis for a two sided test ofthe condition that Etyt+1 = 0 with a nominal level of 5% when xt follows the long-memory process in (7). Four tests are considered: a standard regression t-test (topleft), the median sign test (top right), the covariance test (bottom left), and theconditional test (bottom right). The standard regression test is unreliable and canresult in severe over-rejection, particularly for large values of d and σ 12. Thesesize distortions are qualitatively similar to those known to arise when xt has aroot near unity (Mankiw and Shapiro 1986). By contrast, none of the remainingthree tests shows any tendency to over-reject. Not surprisingly, the sign tests havecorrect level. As conjectured at the end of section 3.3, the covariance test tendsto be conservative for d < 1, but does not over-reject. Finally, the conditionaltest, designed in the autoregressive framework, is surprisingly robust in the long-memory case.

Table 4 presents the equivalent rejection rates for the case in which xt is gen-erated by the autoregression with structural breaks (8). The standard regressiontest again is seen to be unreliable, suffering in many cases from substantial over-rejection. The results for the other three tests again are encouraging. The goodperformance of the sign test is expected from theory. The covariance test alsoperforms well, and although the conditional test slightly over-rejects in one ortwo cases, it is again quite robust.

We turn next to a comparison of test power. Here, we consider only the sign,covariance, and optimal conditional tests, since the standard regression test isunsatisfactory on the basis of size. We generate xt from the long-memory model (7)


TABLE 3Finite sample size comparison (long memory)

d σ 12 = 0 0.25 0.50 0.75 0.95 σ 12 = 0 0.25 0.50 0.75 0.95

Standard regression t-statistic Sign test

1.000 0.050 0.066 0.110 0.175 0.285 0.022 0.026 0.029 0.026 0.0250.800 0.057 0.069 0.085 0.131 0.192 0.026 0.030 0.025 0.028 0.0300.600 0.051 0.057 0.075 0.081 0.093 0.036 0.027 0.036 0.031 0.0320.400 0.061 0.053 0.054 0.064 0.076 0.046 0.038 0.040 0.045 0.045

Covariance test Conditional test

1.000 0.041 0.028 0.036 0.037 0.034 0.051 0.057 0.048 0.053 0.0460.800 0.025 0.024 0.032 0.032 0.022 0.046 0.054 0.053 0.062 0.0650.600 0.018 0.017 0.020 0.019 0.022 0.051 0.049 0.048 0.058 0.0590.400 0.022 0.017 0.013 0.017 0.017 0.042 0.054 0.057 0.054 0.045

NOTES: The table entries show rejection rates under the null hypothesis for nominal 5% two-sidedtests. yt is given by (9) and xt by (7), with ut given by (23). Details are given in the text.

TABLE 4Finite sample size (autoregression with m breaks in the intercept ρ 0)

m/n: 0.02 0.04 0.06 0.08 0.10 0.02 0.04 0.06 0.08 0.10

|ρ 0,j − ρ 0,j−1| Standard regression t-statistic Sign test

2.0 0.212 0.197 0.182 0.184 0.179 0.029 0.038 0.046 0.053 0.0374.0 0.185 0.151 0.142 0.141 0.133 0.023 0.032 0.043 0.044 0.0508.0 0.130 0.098 0.091 0.097 0.086 0.020 0.037 0.048 0.039 0.049

20.0 0.081 0.069 0.067 0.066 0.052 0.019 0.029 0.044 0.042 0.048

|ρ 0,j − ρ 0,j−1| Covariance test Conditional test

2.0 0.033 0.037 0.032 0.035 0.033 0.048 0.052 0.072 0.053 0.0584.0 0.035 0.033 0.040 0.036 0.036 0.061 0.058 0.056 0.058 0.0548.0 0.039 0.035 0.044 0.040 0.038 0.050 0.050 0.046 0.053 0.046

20.0 0.036 0.040 0.039 0.036 0.043 0.070 0.044 0.056 0.055 0.050

NOTES: The table entries show rejection rates under the null hypothesis for nominal 5% two-sidedtests. yt is given by (9) and xt by (8) with m equally spaced breaks of size |ρ 0,j − ρ 0,j−1| and alternatingsign. ut is given by (23). We set σ 12 = 0.95 and ρ 1 = 0.99.

for d = 0.8 (non-stationary long memory) and d = 0.4 (stationary long memory)and σ 12 = 0.95. In table 5 we consider power against the standard regressionalternative in (10), in which yt+1 is a linear function of xt. The power of thesign and covariance tests seem roughly comparable under this alternative.24 Theconditional test has the best power of the three tests. This is perhaps not surprising,given its known optimal power properties under the alternative in (10).

24 More generally, the comparison between these two tests was found to depend on the parameterspecification, with the sign test showing better power under some specifications and thecovariance test showing better power under others. A more comprehensive comparison isavailable upon request.

1266 A. Maynard

TABLE 5Finite sample power (long-memory, yt+1 = γ xt + u y,t+1)

Test d β = 0.05 0.10 0.15 0.20 0.35 0.50

Sign test 0.80 0.060 0.157 0.273 0.412 0.766 0.903Covariance test 0.80 0.059 0.145 0.301 0.457 0.836 0.939Conditional test 0.80 0.108 0.311 0.594 0.772 0.982 1.000Sign test 0.40 0.060 0.098 0.164 0.259 0.604 0.850Covariance test 0.40 0.036 0.072 0.140 0.286 0.747 0.968Conditional test 0.40 0.061 0.143 0.294 0.512 0.907 0.997

NOTES: The table shows rejection rates under the alternative hypothesis (10) for a nominal 5% test us-ing tλ as defined in (22). xt is given by (7), with ut given by (23) σ 12 = 0.95. Details are given in the text.

TABLE 6Finite sample power (long-memory, yt+1 = δ (1 − L)d xt + u y,t+1)

Test σ 1,2 δ = 0.05 0.10 0.15 0.20 0.35 0.50

Sign test 0.80 0.023 0.024 0.029 0.048 0.086 0.149Covariance test 0.80 0.046 0.068 0.139 0.252 0.639 0.938Conditional test 0.80 0.043 0.034 0.028 0.030 0.047 0.101Sign test 0.40 0.052 0.069 0.116 0.146 0.402 0.716Covariance test 0.40 0.016 0.073 0.134 0.244 0.713 0.974Conditional test 0.40 0.033 0.071 0.146 0.243 0.748 0.981

NOTES: The table shows rejection rates under the alternative hypothesis (12) for a nominal 5% testusing tλ as defined in (22). xt is given by (7), with ut given by (23). n = 100 and σ 12 = 0.95. Detailsare given in the text.

When xt has long memory, the alternative in (10) implies a similar long-memorycomponent in yt. Thus, it may also be of interest to consider the alternative in(12). Under this alternative yt is still predictable based on the past history of xt, sothat Etyt+1 = 0 is violated, but yt does not contain any long-memory components.The power comparisons in this case are presented in table 6, again for d = 0.4and 0.8 and σ 12 = 0.95. All three tests perform well when xt has stationary long-memory (d = 0.4), but only the covariance test, which was explicitly designedwith similar alternatives in mind (Maynard and Shimotsu 2005), has good powerwhen d = 0.8. If this form of the alternative were known a priori, either gt or theregressor could be appropriately transformed to provide a more powerful sign orregression test. In practice, one may not know which alternative is most relevant:(10) or (12). Thus the ability to provide reasonable power against both alternativessimultaneously may prove, in some circumstances, to be a useful property of thecovariance test.

Thus, while the standard regression statistic can suffer substantial size distor-tion, all three robust tests seem especially well suited to applications of the typeconsidered here. Each test has its own particular strength. The sign test remains


the only known test to maintain the correct test level for all possible xt processes.The covariance test shows the greatest flexibility in terms of the alternatives con-sidered above, providing reasonable power under both the linear alternative (10)and the alternative in (12), in which yt maintains short memory. Finally, the con-ditional test is substantially more robust than expected and shows the best poweragainst standard predictive regression alternatives of the type shown in (10). Ac-cordingly, we employ all three to test forward-rate unbiasedness in the followingsection.

4. Empirical results: robust tests of forward rate unbiasedness

The standard regression results shown in table 1 are typical of the literaturein providing strong evidence against unbiasedness. However, as discussed above,their statistical validity has been called into question by the recent literature, basedon the persistent and possible long-memory behaviour in the forward premium.We therefore revisit this hypothesis, using the sign, covariance, and conditionaltests, all three of which remain reliable in the face of persistent conditioningvariables. All tests and p-values reported below are for a two-sided alternativeand all tests are conducted at a 5% significance level.

4.1. Sign testsWe begin with the sign tests of section 3.2. In order to test forward rate unbiased-ness, we set yt+1 = s t+1 − ft and gt = ft − st. This tests for sign predictability ofs t+1 − ft using ft − st, evidence of which would imply a violation of unbiasedness.

Columns 2–4 of table 7 show median sign test results. The second columnfrom the left shows the percentage of correct directional predictions (100 ∗Sg/n).Under the null hypothesis, there is no predictive power, giving a population valueof 50%. This corresponds roughly to γ = 0 or β = 1. Values below 50 % correspondroughly to estimates of γ < 0 (β < 1).

The results may be compared with a standard regression test of γ = 0 in(4). All but one of the point estimates in table 7 lie below 50%, indicating anegative sign to the forward premium/excess return relation. This confirms therobustness of the traditional finding of γ < 0 (i.e. β < 1). On the other hand,the results show less statistical evidence against unbiasedness than is found usingthe conventional, but questionable, regression tests shown in table 1, which sug-gest a strong rejection for all six currencies. In particular, the rejection of un-biasedness is no longer ubiquitous across currencies. We continue to reject forthree currencies: the CAD, GDM, and JPY. On the other hand, the evidenceagainst unbiasedness is marginal for the AUD and BRP and quite weak for theFFR.

The finite sample bounds of Campbell and Dufour (1997), presented graphi-cally in figure 4, provide a more sophisticated median-adjustment that preservesthe exact finite distribution of the sign statistic. The jagged solid lines show the

1268 A. Maynard

TABLE 7Median sign tests

Test of unbiasedness Sign of return/premium relationyt+1 = (s t+1 − ft)∗∗ yt+1 = (s t+1 − st)∗∗

xt = ( ft − st)∗ xt = ( ft − st)∗

100Sgn (se) p-val 100Sg

n (se) p-val

AUD 44.1 (3.94) 0.1558 45.3 (3.94) 0.2700CAD 43.0 (2.79) 0.0139ψ 42.7 (2.79) 0.0101ψ

FFR 51.1 (2.79) 0.3440 52.0 (2.79) 0.5660GDM 43.3 (2.79) 0.0189ψ 43.6 (2.79) 0.0254ψ

JPY 40.2 (3.19) 0.0027ψ 40.2 (3.19) 0.0027ψ

BRP 45.2 (2.79) 0.0938 48.9 (2.79) 0.7380

NOTES: ψ Two-sided rejection at 5%.∗ and ∗∗ are the de-median operators x∗t = xt − mt

1(xt) andx∗∗

t = xt − mn1(xt), respectively, where mt

1(xt) is the sample median of (x1, x2, . . .xt).

FIGURE 4 Sign test on s t+1 − ft: (— Sg(b), — — — outer bound, - - - inner bound)

sign statistic Sg(b) = �nt=1 sign ((yt+1 − b) g∗

t ) evaluated over an exact 99%confidence interval for true median of yt. The dotted and dashed lines provideinner and outer bounds for the test statistic against a two-sided alternative at afive percent level. An unambiguous rejection occurs only when the test statisticlies everywhere outside the outer bound. A failure to reject occurs when it lieseverywhere inside the inner bound and the test is inconclusive if at any point it


TABLE 8Covariance based t-tests (optimal bandwidths )

Test of unbiasedness Return/premium covarianceyt+1 = s t+1 − ft yt+1 = s t+1 − st

xt = ft − st xt = ft − st

105λ 105se(λ) tλ p-val m 105λ 105se(λ) tλ p-val m

AUD −0.197 0.145 −1.353 0.1776 3.1 −0.183 0.143 −1.283 0.1977 2.0CAD −0.398 0.098 −4.065ψ 0.0001 15.0 −0.291 0.091 −3.197ψ 0.0009 12.0FFR −0.829 0.523 −1.584 0.1248 4.7 −0.364 0.489 −0.744 0.4587 3.2GDM −0.465 0.329 −1.411 0.2128 22.0 −0.164 0.192 −0.856 0.3953 1.8JPY −0.155 0.189 −0.824 0.4110 0.6 −0.123 0.186 −0.662 0.5086 0.5BRP −0.673 0.311 −2.164ψ 0.0414 4.2 −0.432 0.291 −1.484 0.1527 3.5

NOTES: ψ Two-sided rejection at 5%. Shown are the covariance estimate λ and its standard errorse(λ), both scaled by 105, the t-statistic tλ, and bandwidth m.

crosses between the two bounds (see sec. 3.2 for further details). The bounds testresults of figure 4 are similar to the median sign test results in table 7, providing anunambiguous rejection of unbiasedness for three of six currencies (CAD, GDM,& JPY) and inconclusive evidence for two (AUD & BRP), and unambiguouslyfailing to reject for one (FFR/USD).

4.2. Covariance estimates and testsThe left half of table 8 provides both the covariance estimate λst+1− ft , ft−st

and t-statistic tλ used to test forward rate unbiasedness. Under unbiasedness,λst+1− ft , ft−st = limt→∞ cov(st+1 − ft, ft − st) = 0. Thus, using standard normalcritical values, a significant value of tλ provides a rejection. Moreover, the signof λst+1− ft , ft−st indicates the sign of the covariance between s t+1 − ft and ft − st.Also shown are the standard errors, p-values, and the optimal bandwidth m,details on which are given in appendix C.

The overall results from the covariance tests are qualitatively similar tothose from the sign tests. The point estimates of λst+1− ft , ft−st are uniformlynegative, again confirming the robustness of the traditional finding of γ < 0(i.e., β < 1). Yet there is again less statistical evidence against unbiasedness thanis found in the conventional regression tests. The covariance based t-statisticis significant for the CAD and BRP and shows marginal evidence against un-biasedness for the AUD and FFR, but little evidence for either the GDM orJPY.

Table 9 provides a sensitivity analysis, replacing the optimal bandwidth m bya grid of bandwidths ranging from 2 to 10. Only the t-statistics are shown. In allcases the estimates again are negative. The magnitudes of the test statistic alsoappear relatively insensitive to the choice of bandwidth.

TA

BL

E9

Cov

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base

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test

s(s

ever

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Tes

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unbi

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Ret

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prem

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cova

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cey t

+1=

s t+1

−f t

y t+1

=s t

+1−

s tx t

=f t

−s t

x t=

f t−

s t

m:

2.0

4.0

6.0

8.0

10.0

2.0

4.0

6.0

8.0

10.0

AU

D−1

.328

−1.2

29−0

.925

−0.9

15−0

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51−1

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−0.8

16−0

.783

−0.6

67C

AD

−3.6

78ψ

−3.9

18ψ

−4.2

15ψ

−4.1

15ψ

−4.0

10ψ

−3.1

27ψ

−3.2

02ψ

−3.4

53ψ

−3.3

27ψ

−3.1

76ψ

FF

R−1

.919

−1.6

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.568

−1.6

65−1

.669

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60−0

.701

−0.6

00−0

.669

−0.6

44G

DM

−1.1

47−0

.979

−1.2

31−1

.308

−1.3

15−0

.800

−0.5

39−0

.757

−0.8

34−0

.833

JPY

−0.3

30−0

.521

−0.7

48−1

.047

−1.3

93−0

.089

−0.1

69−0

.339

−0.6

08−0

.931

BR

P−1

.347

−2.1

19ψ

−2.3

40ψ

−2.4

22ψ

−2.5

09ψ

−0.8

76−1

.502

−1.6

39−1

.658

−1.7

05

NO

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5%.T

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TABLE 10Conditional tests

Test of unbiasedness Test of β = 0yt+1 = s t+1 − ft yt+1 = s t+1 − st

xt = ft − st xt = ft − st

γ γmu γ p-val β βmu β p-val

AUD −4.04 −2.37 −0.690 0.0058ψ −3.04 −1.37 0.310 0.1100CAD −3.04 −2.12 −1.200 0.0000ψ −2.04 −1.12 −0.198 0.0173ψ

FFR −2.70 −1.65 −0.594 0.0022ψ −1.70 −0.65 0.406 0.2280GDM −3.16 −1.78 −0.405 0.0112ψ −2.16 −0.78 0.595 0.2650JPY −5.47 −3.59 −1.730 0.0002ψ −4.47 −2.59 −0.729 0.0064ψ

BRP −3.75 −2.45 −1.150 0.0002ψ −2.75 −1.45 −0.153 0.0286ψ

NOTES: ψ two-sided rejection at 5%. Two-sided p-values are shown in columns 5 and 9. Theremaining entries are based on an inversion of the test. Columns 2 and 6 show the lower bound on a95% confidence interval, columns 3 and 7 show the median unbiased estimates, and columns 4 and8 show the upper bound on the confidence interval.

4.3. Conditional testsTests of forward rate unbiasedness employing the optimal conditional approachof Jansson and Moreira (2006) are presented in the left half of table 10. The testis based on the predictive regression model in (4) in which yt+1 = s t+1 − ft isregressed on xt = ft − st and unbiasedness is tested via the coefficient restrictionγ = 0. As explained in section 3.4, the test procedure eliminates the nuisanceparameter problem by conditioning inference on a sufficient statistic for ρ1 in (5)and thus maintains correct size and optimal power against the alternative in (4)regardless of the size of the autoregressive root in xt. The p-value for a two-sidedtest of this restriction is shown in column 4. Interestingly, the restriction that γ

= 0 is rejected at the 5% significance level for all six currencies. Thus, overall, thistest delivers the strongest rejections of the three robust tests and in fact shows asmuch evidence against unbiasedness as do the standard regression tests.

The conditional test may also be inverted to provide confidence intervals andpoint estimates, as suggested by Polk, Thomson, and Vuolteenaho (2004) andEliasz (2005), respectively. The lower and upper bounds on a 95% confidenceinterval for γ are provided in columns 2 and 4. Based on an inversion of the teststatistic, it consists of all values of γ that cannot be rejected at the 5% level. Asimilar inversion yields the Eliasz (2005) estimator of γ , which is median unbiasedwithin the context of the local-to-unity model (6) and thus avoids systematic biasin the sense that it is as likely to overestimate γ as to underestimate it. Theseestimates are displayed in column 3. As in the case of the covariance estimates,these point estimates again are uniformly negative. Thus, overall the conditionaltests both confirm and considerably strengthen the evidence against unbiasednessfound from the sign and covariance tests.

1272 A. Maynard

4.4. The sign of the spot-return/forward premium relationSo far we have focused on the unbiasedness hypothesis itself. Equally relevant tothe forward premium puzzle is the sign of the relationship in (3). In fact, it is thenegative sign of the covariance between the spot-return and forward premium, notjust the rejection of unbiasedness itself, that Engel (1996) highlights in the mostrecent survey of the literature. However, since (3) contains the same persistentregressor as (4), the statistical validity of this finding has also been called intoquestion (e.g., Baillie and Bollerslev 2000; Maynard and Phillips 2001). Usingthese robust tests, we may also revisit this second stylized fact using methods thatare immune to the problems associated with persistent regressors.

Redefining yt+1 = s t+1 − st as the spot return, λy,�x provides a robust es-timate of λst+1−st , ft−st = lim t→∞ cov (st+1 − st, ft − st), the spot return/forwardpremium covariance. The corresponding t-statistic, tλ, may be used to assess thesignificance of the estimated covariance. This may be compared with a test of β =0 in (3). Likewise, we may also use the sign test to re-examine the direction of thespot return/forward premium relation. Setting yt+1 = s t+1 − st and gt = ft − st,we now test the success of the forward premium as a predictor for the sign of thespot return. The percentage of correctly signed predictions (100 ∗Sg/n) equals50 under the null that ft − st has no predictive content for s t+1 − st. This impliesthat st is a martingale and corresponds roughly to β = 0. The two alternativehypotheses are perhaps of greater interest than the null. For 100 ∗Sg/n > 50,ft − st has predictive power for s t+1 − st and tends to forecast at least its sign, ifnot its magnitude, correctly. Alternatively, for 100 ∗Sg/n < 50, the forward rateis perverse predictor, so that st tends to fall rather than rise for ft >st. Finally, byresetting yt+1 = s t+1 − st, the conditional test may be applied to directly test theparameter restriction that β = 0 in (3).

The sign test results are shown in columns 5–7 of table 7 and in figure 5,the covariance estimates and test results are shown in columns 7–11 of Table 8,and the conditional test results are shown in columns 6–9 of table 10. The pointestimates from all three methods clearly confirm the robustness of the nega-tive spot return/forward relation. All of the covariance estimates in table 8,column 6, and all of the median unbiased estimates of β in table 10, column 7,are negative. Likewise, all but one of the sign statistics shown in table 7,column 5, lies below 50%. The sign and conditional test results are also signifi-cant for three of six currencies, whereas just one covariance estimate shows up assignificant.

4.5. DiscussionIn contrast to standard regression tests which may severely over-reject in theface of persistent regressors, all three methods employed above are specificallydesigned to provide robust inference. Thus, to the extent that the sign and co-variance tests show somewhat weaker evidence against unbiasedness, this maybe seen as providing some support for previous contentions that the forward


FIGURE 5 Sign test on s t+1 − st: (— Sg(b), — — outer bound, - - inner bound)

premium puzzle may be less serious than it appears. On the other hand, cautionis always required in interpreting non-rejections, whereas the very robustness ofthese methods lends confidence in those rejections that we do obtain, leaving uslittle room to explain them away as a purely statistical phenomenon. They seemto require instead an economic explanation. Moreover, even stronger evidenceagainst unbiasedness is delivered by the conditional test, which, like the standardregression test, rejects for all six currencies. Likewise, for several currencies we alsoobtained a negative and statistically significant sign on the spot return/forwardpremium relation. This negative relationship has proved particularly difficult toexplain and has come to symbolize the severity and economic significance of therejection of unbiasedness.25

The sign test, which requires no assumptions whatsoever on the behaviourof the forward premium, is the most robust of the three tests. Therefore theresulting p-values may be viewed as constituting a lower bound on the evidenceagainst unbiasedness. On the other hand, the strongest rejections are deliveredby the conditional test. This test enjoys certain optimal power properties not

25 Nevertheless, a brief caveat is in order. While we explicitly test the unbiasedness condition, notall economic models imply that unbiasedness holds exactly. Thus, we have not provided explicittests of economic models that do not imply unbiasedness. For example, unbiasedness does nothold exactly in the UIP-FIGARCH model of Baillie and Bollerslev (2000), so we do notexplicitly test this model. It is possible that models such as this one receive more support afterthe persistent behaviour of the forward premium is accounted for, even in cases where theunbiasedness hypothesis itself does not.

1274 A. Maynard

shared by the other two tests, but it is also more tightly tied to an autoregressivespecification of the regressor. Thus, in principle, if the true model does not followan autoregressive specification, the differences between the two tests could bedue to either size or power differences. However, in this respect the simulationresults of section 3.5, suggesting good size for the conditional test under simplelong-memory and structural break models, may lend some additional credenceto these rejections.

5. Conclusion

The recent literature has highlighted problems in predictive regressions with per-sistent regressors and has suggested them as a possible explanation for empiricalpuzzles, such as the rejection of unbiasedness and the puzzling negative spotreturn/forward premium correlations. Moreover, it has been suggested that theforward premium puzzle may be primarily a statistical artefact rather than a trueeconomic puzzle.

We therefore revisit the unbiasedness hypothesis using three alternative econo-metric methods–sign, covariance, and conditional tests–designed to provide in-ference robust to the presence of persistent conditioning variables. All three testsprovide reliable inference in stationary, unit root, and near-unit root models, re-gardless of the size of the largest root. Moreover, the sign test requires no assump-tions whatsoever regarding the underlying process driving the forward premiumand our simulations suggest that the other two tests continue to provide robustinference regardless of whether this persistence is modelled as a near-unit roots,long-memory, or structural break process.

In several instances the robust tests employed here give less evidence againstunbiasedness than do traditional, but questionable, regression tests. This thusprovides some support to the notion that the forward premium puzzle maybe less severe once the long-memory properties of the forward premium areaccounted for. On the other hand, for all currencies considered we reject us-ing at least one test, and for some currencies we reject using all three. Unliketraditional predictive regression tests, the tests we employ do not suffer fromover-rejection in the presence of persistent regressors. This leaves little roomto explain these rejections away as a purely statistical phenomenon. More-over, the negative sign of the spot return/forward premium relation, whichhas underlined the severity of the puzzle, retains its significance for severalcurrencies.

Our results may therefore provide some support for the contention that theeconometric problems inherent in predictive regression likely play a role, andperhaps an important one, in the forward premium anomaly. Nevertheless, asubstantial puzzle is found to remain even after their influence is accounted for,and it cannot be explained as a purely statistical phenomenon. A real economicpuzzle still remains.


Appendix A: Lemmas

LEMMA 1. Let yt = μy + uy,t (short-memory), let xt (long-memory) be given by(7), where

ut = (ux,t, uy,t) = C(L)εt εt ∼ i.i.d. (0, �ε)

C(L) =[

Cyy(L) Cyx(L)

Cxy(L) Cxx(L)

]=

∞∑j=0

Cj Lj , Cj =[

Cyy, j Cyx, j

Cxy, j Cxx, j

], (A1)

∞∑j=1

j2|Cab, j | < ∞ for a, bε{x, y}. Then (A2)

(yt − μy, � xt)′ = D(L)εt =∞∑j=0

Djεt− j , Dj =[

Dyy, j Dyx, j

Dxy, j Dxx, j

](A3)

where∑∞

j=0 ‖Dj‖ < ∞ and where the off-diagonal elements of Dj and the cross-auto-covariances satisfy the short memory summability conditions

∞∑j=0

j 2|Dyx, j | < ∞ and∞∑j=0

j 2|cov(�xt, yt+ j )| < ∞. (A4)

REMARK 1. The lemma shows that the cross-covariances appearing in λy,�x exhibitshort-memory decay rates, despite possible fractional over-differencing in �xt =(1 − L)−(d−1) ux,t for d < 1. Jansson (1999, theorem 1) shows the consistency ofλy,�x under (A4) (his condition (3)).

LEMMA 2. Let (i) Pr{yt > E(yt)} = 0.5 (yt symmetric) and (ii) Pr{(yt −b0)xt−1 > E((yt − b0)xt−1)} = 0.5 (yt − b0) (xt−1 symmetric). Then cov (yt, xt−1) ≥(resp. ≤ ) 0 implies E [sign((yt − b0)xt−1)] ≥ (resp. ≤ ) 0.5.

Appendix B: Proofs

Proof of lemma 1. To simplify notation, define

v ′t = (vy,t, vx,t)′ = (

yt − μy, (1 − L) xt)′ = (

uy,t, (1 − L)−(d−1)ux,t)′. (B1)

1276 A. Maynard

Then, setting q = d − 1, we have26

(1 − L)q = �(L) =∞∑j=0

� j Lj , where �0 = 1 and

� j = �jl=1

l − 1 − ql

≈ j q−1/�(q) for j > 1.

As q ≤ 0, the coefficients are absolutely summable (Baillie 1996, 19)27

∞∑j=0

|� j | < ∞ for d ≤ 1 (q ≤ 0). (B2)

Therefore, from (7) we have28 vt= D(L)εt = ∑∞j=0 Djεt− j where

D(L) =[

Dyy(L) Dyx(L)

Dxy(L) Dxx(L)

]=

[1 0

0 �(L)

] [Cyy(L) Cyx(L)

Cxy(L) Cxx(L)

]. (B3)

By (30) Dyy,j = Cyy,j and Dyx,j = Cyx,j for all j. Therefore Dyy,j

and Dyx,j must also satisfy (25). By (30)∑∞

j=0 Dxy, j = D(1) = �(1)Cxy(1) =(∑∞

j=0 � j )(∑∞

k=0 Cxy,k) and therefore∑∞

j=0 Dxy, j ≤ (∑∞

j=0 |� j |)(∑∞

k=0 |Cxy,k|) <

∞ by (B2) and (25), thus proving the first stated inequality. Note also that

∞∑j=0

j2|Dyy, j |,∞∑j=0

|Dxy, j |,∞∑j=0

|Dxx, j | < ∞, (B4)

since identical arguments apply to∑∞

j=0 |Dxx, j | < ∞, and the remaining inequal-ities follow trivially from the definitions above. For the second inequality oflemma 1, we have from (A3) that

E(v0v ′h) =

∞∑j=0

∞∑k=0

Dj E(ε− jε′h−k)D ′

k =∞∑j=0

Dj�ε D ′j+h, so that

E(vx,0v ′y,h) =

∞∑j=0

(Dxy, jσε,11 + Dxx, jσε,21)Dyy, j+h

+∞∑j=0

(Dxy, jσε,12 + Dxx, jσε,22)Dyx, j+h .

26 This simplifies to �(L) = 1 − L for d = 0 and to �(L) = 1 for d = 1.27 Note that q > 0 would imply d > 1, a possibility that is reasonably ruled out.28 The same result holds under (5) for q = 0 up to an Op(t−1/2) error.


It follows that

∞∑h=0

h2|E(vx,0vy,h)|

=∞∑

h=0

h2

∣∣∣∣∣ ∞∑j=0

(Dxy, jσε,11 + Dxx, jσε,21)Dyy, j+h

+∞∑j=0

(Dxy, jσε,12 + Dxx, jσε,22)Dyx, j+h

∣∣∣∣∣≤

∞∑h=0

h2∞∑j=0

|Dxy, jσε,11 + Dxx, jσε,21||Dyy, j+h |

+∞∑

h=0

h2∞∑j=0

|Dxy, jσε,12 + Dxx, jσε,22||Dyx, j+h |

=∞∑j=0

|Dxy, jσε,11 + Dxx, jσε,21|∞∑

h=0

h2|Dyy, j+h |

+∞∑j=0


h=0

h2|Dyx, j+h |

≤∞∑j=0


h=0

h2|Dyy,h | +∞∑j=0

|Dxy, jσε,12

+ Dxx, jσε,22|∞∑

h=0

h2|Dyx,h |

≤(

|σε,11|∞∑j=0

|Dxy, j | + |σε,21|∞∑j=0

|Dxx, j |) ( ∞∑

h=0

h2|Dyy,h |)

+(

|σε,12|∞∑j=0

|Dxy, j | + |σε,22|∞∑j=0

|Dxx, j |) ( ∞∑

h=0

h2|Dyx,h |)

< ∞,

since∑∞

j=0 |Dxy, j |,∑∞

j=0 |Dxx, j |,∑∞

h=0 h2|Dyy,h |,∑∞

h=0 h2|Dyx,h | < ∞ by (B4).Similar arguments follow for the remaining terms.

Proof of lemma 2.Let cov(yt, gt−1) > 0. (A similar argument holds for cov(yt, gt−1) <

0.) Then E[sign((yt − b0)gt−1)] = Pr{(yt − b0)gt−1 > 0} ≥ Pr{(yt − b0)gt−1 ≥

1278 A. Maynard

cov(yt, gt−1)} = Pr{(yt − b0)gt−1 ≥ E((yt − b0)gt−1)} = 0.5,where the first equal-ity holds by definition, the second since cov(yt, gt−1) > 0, the third since E(yt) =b0 by symmetry of yt, and the fourth by symmetry of ytgt−1.

Derivation of Equation (17). Since ut is serially uncorrelated by as-sumption we may writeλy,�x = ∑∞

h=1 cov (yt, �xt−h) = ∑∞h=1 cov (δ0 + δux,t−1 +

uy,t, �xt−h) = δ∑∞

h=1 cov (ux,t−1, �xt−1) = cov (�xt, ux,t) = var (ux,t), since, ei-ther �xt = �(L)ux,t = ux,t + ∑∞

j=1 � j ux,t− j (long-memory case, see B1 and B2)or �xt = c/nxt−1 + ux,t (near unit root case).

Appendix C: Implementation details for covariance-based test

The Bartlett (Newey-West) kernel is used for k. As in the standard HAC literature(e.g., Andrews 1991) the bandwidth m is chosen according the MSE criterion in(19), using a bivariate ARMA(1,1) as a first stage approximating model for (yt,�xt), estimated using the method of Dufour, and Pelletier (2002). On account ofits small-sample accuracy, Maynard and Shimotsu (2005, 17, line 5) recommendthe variance estimator

VMDS = 1m

n−1∑h′=1

n−1∑h=1

k(

h′ − 1m

)k

(h − 1

m

)k

(h′ − h

m

)��x�x

× (h′ − h)�yy(0)φn(h′ − h, h′, h), (C1)

for V , which is based on a finite sample approximation to the variance of λy,�x

and imposes the null hypothesis Etyt+1 = 0 under which yt is an MDS.29 For h′

>h the step function φn() is defined as

φn(u, h′, h)

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩0, u ≤ −n + h′; = 1 − h′ − u

n, −n + h′ ≤ u ≤ 0

= 1 − h′/n, 0 ≤ u ≤ h′ − h; = 1 − h + un

, h′ − h ≤ u ≤ n − h

= 0, u ≥ n − h.

The estimator V again requires choice of both kernel k and bandwidth m. Weemploy the same Bartlett kernel for k. To ensure conservative inference whenxt ∼ I(0) we require m/m → 0 as n → ∞ and thus choose m = m.0.9 Thus, thefollowing steps are required to implement a two-sided test of level α.

29 This is a specialized version of the more general variance estimator V, defined in Maynard andShimotsu (2005, 13, eq. 16), which remains valid for the null hypothesis in which yt is orthogonalto past xt−j , j > 0 but not to its own past, for example, when yt consists of a long-horizonreturn. Maynard and Shimotsu (2005, 13, lemma 5) shows the consistency of V.


1. Estimate the approximating VARMA(1,1) for (yt, �xt).2. Choose m according to (19) (we use k1 = 1, q = 1).3. Estimate λy,�x according to (18) using the Bartlett kernel k(x) = |1 −

x|1x>1.4. Estimate V according to (C1) (we use k(x) = k(x) and m = m0.9).5. Form tλ as in (22) with λy,�x = 0.6. Reject for |tλ| >zα/2, where zα is the standard normal critical value.

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The forward premium anomaly: statistical artefact or economic puzzle? New evidence from robust tests

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