Testing the martingale hypothesis for futures prices: Implications for hedgers

The authors thank the referee, Luc Bauwens, Mikael Petitjean, Hans Dewachter, Paul De Grauwe, and theseminar participants at the doctoral workshop and the 5th Corporate Finance Day, both at UCLouvain, fortheir helpful comments.

*Correspondence author, F.W.O. (Flemish Science Foundation), Faculty of Business and Economics,Naamsestraat 69, B-3000 Leuven, Belgium. Tel.: �32-16-326-658, e-mail: [email protected]

Received May 2007; Accepted October 2007

� Cédric de Ville de Goyet is an F.W.O. (Flemish Science Foundation) researcher at the Faculty of Business and Economics, KULeuven, Leuven, Belgium.

� Geert Dhaene is at the Faculty of Business and Economics, KULeuven, Leuven, Belgium.

� Piet Sercu is at the Faculty of Business and Economics, KULeuven, Leuven, Belgium.

The Journal of Futures Markets, Vol. 28, No. 11, 1040–1065 (2008)© 2008 Wiley Periodicals, Inc.Published online in Wiley InterScience (www.interscience.wiley.com).DOI: 10.1002/fut.20343

TESTING THE MARTINGALE

HYPOTHESIS FOR FUTURES

PRICES: IMPLICATIONS

FOR HEDGERS

CÉDRIC de VILLE de GOYET*GEERT DHAENEPIET SERCU

The martingale hypothesis for futures prices is investigated using a nonparametricapproach where it is assumed that the expected futures returns depend (nonparametrically) on a linear combination of predictors. We first collapse thepredictors into a single-index variable where the weights are identified up toscale, using the average derivative estimator proposed by T. Stoker (1986). Wethen use the Nadaraya–Watson kernel estimator to calculate (and visually depict)the relationship between the estimated index and the expected futures returns.An application to four agricultural commodity futures illustrates the technique.Out-of-sample results indicate that for soybeans, wheat, and oats, the estimated

Testing the Martingale Hypothesis for Futures Prices 1041

Journal of Futures Markets DOI: 10.1002/fut

index contains statistically significant information regarding the expected futuresreturns. We discuss implications of this finding for a noninfinitely risk-aversehedger. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:1040–1065, 2008

INTRODUCTION

Under the assumptions that agents are risk neutral, have common and con-stant preferences, and are rational, Samuelson (1965) proved that futuresprices must be martingales with respect to the information set.1 To see the con-sequence for the hedger, consider an agent with a long position in the spotmarket who wishes to place a hedge using a futures contract. The relative sizeof the short position in the futures market, i.e. the hedge ratio, has to be deter-mined in some optimal way. The conventional approach, followed in much ofthe hedging literature, is to solely focus on the risk associated with the randomportfolio return and, accordingly, to minimize the variance of the portfolioreturn (see, for example, Baillie & Myers, 1991; Ederington, 1979; Johnson,1960; Stein, 1961). Some authors, however, have incorporated both risk andreturn in their hedging policy (see, for example, Hsin, Kuo, & Lee, 1994). The optimal hedge ratio in this mean-variance framework is the sum of twocomponents. The first component, a pure hedging term, is the conventionalminimum-variance hedge ratio and the second, the speculative part, is a func-tion of the expected futures return, the risk-aversion parameter, and the condi-tional variance of the futures return. If the hedger is infinitely risk averse orif the futures price is a martingale with respect to the information set, the spec-ulative component vanishes and the optimal mean-variance hedge ratioreduces to the minimum-variance hedge ratio. However, when the agent’sdegree of risk aversion is not too high and when the futures price is not a mar-tingale (implying the futures return has a nonzero conditional or uncondition-al mean), the agent may wish to exploit the bias in an attempt to trade-off riskagainst return. In other words, rejection of Samuelson’s hypothesis indicates apossible risk/return trade-off in hedging.

The purpose of this study is to empirically test whether commodity futuresprices are martingales with respect to the information set, and to discuss theimplication for hedgers if the martingale hypothesis is rejected. Bessembinderand Chan (1992) and Miffre (2002) reported predictability in various futuresmarkets. They use linear models with macroeconomic predictors such as theterm structure of interest rates, inflation, the difference between low- and

1See, for example, LeRoy (1989) or Campbell, Lo, and MacKinley (1996, pp. 23–24) for a review of theproof.

1042 de Ville de Goyet, Dhaene, and Sercu


high-grade bond yields, etc. McCurdy and Morgan (1988) rejected the martin-gale hypothesis for Deutsche Mark futures. The approach adopted here differsfrom earlier research in that it relies on a nonlinear model and uses the laggedunderlying spot return, basis, and futures return as predictors. Besides statisti-cal significance, there is also more focus on the implications for the hedger, inthe sense that the impact of a violation of the martingale hypothesis is meas-ured on the optimal hedge ratio in a mean-variance framework.

To test the martingale hypothesis, a nonparametric approach is usedwhere the expected return depends nonparametrically on a linear index. Moreprecisely, several variables are optimally combined to predict the next periodfutures return. This enables us to judge the importance of each individual vari-able in the index with minimal a priori restrictions on the functional form: itonly requires the expected futures returns to be weakly dependent and to befunctionally related, in a time-invariant way, to a linear index of the predictors.The relationship between the expected return and the index is left unspecifiedand may well be highly nonlinear.

In terms of forecasting methodology, two steps are taken. The predictorsare first collapsed into a single-index variable where the optimal weights areidentified up to scale, using the average derivative estimator (ADE) proposed byStoker (1986). Robust Wald and t-tests are performed on the estimated weightsto assess the significance of the predictors. The Nadaraya–Watson kernel esti-mator is then used to calculate and visually depict the relationship between theestimated index and the expected futures returns.

For wheat, soybeans, and oats, the out-of-sample results show that, thoughthe unconditional mean futures returns are not significantly different fromzero, the constructed index contains information for predicting futures returnsat the daily horizon. The study closes by assessing the relevance of this pre-dictability for the hedger. The findings indicate that, from the hedger’s point ofview, there is a risk/return trade-off in hedging and the optimal mean-variancehedge ratio is substantially affected, even though there is a high uncertaintysurrounding it.

The remainder of this study is organized as follows. The second sectiondescribes the methodology. The use of the nonparametric Nadaraya–Watsonkernel estimator is motivated to test the martingale hypothesis for futuresprices and its applicability when returns are related to several predictors is dis-cussed. The ADE is then used to collapse the set of predictors into a singleindex. How the optimal mean-variance hedge ratio is affected when themartingale hypothesis does not hold is also discussed. The third section appliesthe proposed methodology to U.S. agricultural data covering the period1979–2003. The fourth section concludes.



METHODOLOGY

Nonparametric Analysis of the Martingale Hypothesis

Let st and ft denote the log of the spot price and of the futures price nearest tomaturity, respectively, and let � be the difference operator. The hypothesis istested that ft is a martingale relative to xt�1,

where xt�1 � (x1,t�1, . . ., xq,t�1) is a q � 1 vector of predictors, available to theagent at time t � 1.2 Nonparametric regression is applied to estimate

, where m(�) is an unknown, possibly nonlinear, scalar-valued function, and the hypothesis is tested that m(�) is identically zero.

In nonparametric regression the conditional expectation is estimated by alocal version of the sample average. When x varies continuously, a general classof nonparametric estimators of can be written as

(1)

where v(�) is a weight function, assigning weights to the observations, and h isa bandwidth or smoothing parameter. The weight function is chosen such thatit gives more importance to observations where is close to x. Followingstandard practice, the Nadaraya–Watson kernel estimator is used, where

and K(�) is a q-variate kernel. The Gaussian kernel is used,3

It is well known that, under general assumptions, is a consistent estimateof the true conditional expectation, .

The choice of the bandwidth, h, is of great importance. It regulates thesize of the neighborhood around x: if h is very large, the estimate is com-puted over a large neighborhood, thus giving importance to observations where

m(x)

m(x) � E[¢ft 0xt�1 � x]m(x)

K(u1, . . . ,uq) � qq

i�1

1

22p e�u2

i�2.

vax�xt�1

hb �

K((x � xt�1)�h)

T�1a

T

t�1K((x � xt�1)�h)

xt�1

m(x) � T�1aT

t�1vax � xt�1

hb¢ft

E[¢ft 0xt�1 � x] � m(x)

E[¢ft 0xt�1] � m(xt�1)

�

H0,x:E[¢ft 0xt�1] � 0

2Following standard practice, continuously compounded futures returns are used, �ft. Simple returns gavenearly identical empirical results.3The choice of the kernel is less important than the choice of the bandwidth (see e.g. Pagan & Ullah, 1999).What makes the Gaussian kernel attractive is its continuity and differentiability to any order.



xt�1 is far away from x, increasing the bias of the estimator. In contrast, if h isvery small, is computed over a small neighborhood, which increasesthe variance of the estimator. The optimal choice of h should balance biasagainst variance. The leave-one-out cross-validation (LOOCV) method is usedto obtain h.4 This is a data-driven bandwidth selection procedure that mini-mizes the mean squared error of the estimates computed in an out-of-samplefashion (see Härdle, 1990, chap. 5).

Testing the null, H0,x, is done by computing 95% pointwise confidencebounds around , for all x. The bounds are derived as follows: the station-ary block-bootstrap of Politis and Romano (1994) is applied to the vector offutures returns and predictors, (�ft, xt�1). The block-bootstrap builds seriesfrom resampled blocks (of random length) of the original data.5 This preservesserial correlation, GARCH effects, and nonnormalities in the data. For eachbootstrap sample, the Nadaraya–Watson estimator is computed, givingthe bootstrap distribution of . The bounds follow as the 2.5th and 97.5thpercentiles of the bootstrap distribution.

In practice, one may wish to relate �ft to many predictors. The laggedfutures and spot returns, the second-order terms, and the lagged level of thebasis were initially considered as predictors:

However, nonparametric regression is not well suited to handle a largenumber of conditioning variables, a problem known as “the curse of dimen-sionality.” One technique for dimension reduction is to impose restrictions onthe type of nonlinearities one is dealing with. One may assume, for instance,that a linear combination of the conditioning variables, a so-called index vari-able , is nonparametrically related to the futures returns. Underthis assumption, the curse of dimensionality problem disappears (as the non-parametric estimator becomes a function of a univariate predictor, the linearindex itself) whereas important nonlinearities in the relationship between and � ft are still allowed. This enables us to track the predictors with the high-est information content and to combine them in an optimal way. The majorissue then becomes the estimation of b as, on that assumption, the null H0,x

can be tested conditional on the one-dimensional estimated index variable zt�1

x�t�1b

zt�1 � x�t�1b

xt�1 � (¢ft�1, ¢st�1, (¢ft�1)2, (¢st�1)2, ¢ft�1¢st�1, ft�1 � st�1)�.

m(x)m(x)

m(x)

m(x)

4We select a single bandwidth h for all q predictors. This is reasonable given that one first standardizes allpredictors.5The length of each block is drawn independently from a geometric distribution. Politis and Romano (1994)further specify that the original data have to be “wrapped” to ensure that whenever a block goes past the lastobservation, it can be filled with observations from the beginning of the series.



by replacing xt�1 with in (1). The estimation of b (up to scale) is the objectof the following subsection.

Estimation of the Index

Estimation of the vector b is first discussed in an i.i.d. context, using the averagederivative technique.6 Then it is indicated how it can be applied to time series.

Average derivative estimatorsConsider a model where the conditional expectation of futures returns may bewritten in the single-index form

(2)

Without specific assumptions on m, Ichimura (1993) searched over a largegrid to identify the value of that minimizes the sum of the squares of theresiduals, . This technique enjoys very good asymptotic proper-ties but is hard to implement because the objective function may not be con-cave or unimodal. Alternatively, Stoker (1986) noted that the q � 1 vector ofcoefficients b can be estimated up to scale through the derivatives of m withrespect to xt�1. This works as follows.

Let d be the vector of average derivatives of �ft with respect to xt�1, i.e. themean of the q � 1 vector of partial derivatives �m/�xt�1 over the distribution ofxt�1. By exploiting the chain rule of differentiation, Stoker (1986) observed that

(3)

where g � E[dm/dzt�1] is a scalar, assumed different from zero. Equation (3)says that d, the vector of average derivatives, is proportional to b, the vector ofcoefficients that define the index . The scaling coefficient g isunimportant as it can be absorbed into m(�). Hence, a consistent estimator ofd measures b up to a scale factor.

Let

and

g(x) � T�1h�q�1RP a

T

t�1K�ax � xt�1

hRPb

g(x) � T�1h�qRPa

T

t�1Kax � xt�1

hRPb

zt�1 � x�t�1b

d � E c 0m0xt�1

d � E c dmdzt�1

db � gb

¢ft � m(x�t�1b)

b

E[¢ft 0xt�1] � m(zt�1), zt�1 � x�t�1b.

zt�1

6An alternative would be to use principal components analysis or factor analysis to determine the weights inthe index. However, such weights would not be directly related to the variables’ predictive power regarding �ft.



be the Rosenblatt–Parzen estimators7 of the marginal density of x and its deriv-ative, respectively. The focus is on the estimate of the negative of the scorevector .

Several consistent ADEs are available in the literature. A simple directestimator of d is given by the sample analog to (3):

(4)

where is an estimate of the q � 1 vector of partial derivatives �m/�xt�1.Two alternative estimators of interest follow from the formula in Stoker

(1986, Theorem 1), which connects the average derivative d to the scores s(x):

(5)

(6)

where the second equality derives from the fact that cov(s(x), x) equals the q �

q identity matrix.8 Each of these two equations induces an estimator of d. Thefirst equality, (5), motivates the indirect estimator. Following Härdle and Stoker(1989),

(7)

where 1[�] is the indicator function and b is chosen so that some chosen frac-tions (between one percent and five percent) of the observations are dropped.This trimming procedure drops observations with a very small estimated density

; the motivation for trimming is that may not behave well when is divided by a very small . The second equality, (6), shows that the averagederivative, d, may be viewed as the vector of slopes in a linear instrumental

g(x)g�(x)s(x)g(x)

dind � T�1aT

t�1s(xt�1)¢ft1[g (xt� 1)b]

� 5cov(s(x), x)6�1cov(s(x), ¢f)

d � cov(s(x), ¢f )

m�(xt�1)

ddir � T�1aT

t�1m�(xt�1)

s(x) � �g�(x)�g(x) � �0log g(x)�0x

7The bandwidth of the Rosenblatt–Parzen estimator, hRP, may be different from the bandwidth of theNadaraya–Watson estimator, h. As the kernel is multivariate standard normal and as the predictors are firststandardized, the popular plug-in estimate of Silverman (1986) is used: hRP � 1.06T�1/5. The same band-width is used for both and .8Assuming that m(x)g(x) S 0 as ||X||S , one has:

Equation (5) follows by noting that E[s(x)] � 0. Similarly, Equation (6) follows from

� ��

�

[0x�0x�]g(x)dx � Iq.cov(s(x), x) � E[s(x)x�] � ��

�

[0g(x)�0x]x�g(x)�g(x)dx

��

�

[0g(x)�0x]m(x)g(x)�g(x)dx � E[m(x)s(x)].

E[0m(x)�0x] � ��

�

[0m(x)�0x]g(x)dx � ��

�

[0g(x)�0x]m(x)dx

g�(x)g(x)



variables regression of �ft on xt�1 with the score components as the instrumen-tal variables. This suggests the estimator

(8)

Stoker (1993) showed that the three estimators , , and are -consistent9 and asymptotically normal. Moreover, all the three estimators arefirst-order equivalent10 (Li, 1996). Stoker (1993) favored because it is lesssensitive to inaccuracies in the estimation of cov(s(x),x) and cov(s(x),� f),11 asthe inaccuracy is partially cancelled out owing to the ratio form of the estimator.

Under some general conditions, Härdle and Stoker (1989, Theorem 3.1,Equation (3.6)) showed that a consistent estimator of

var is given by

where

(9)

and Hence, the covariance matrix of is estimated

by . Recalling that cov(s(x), x) � Iq, it is obvious that

2T(dIV � d) ¡d N(0, A©A�)

T�1©Dindr � T�1gT

t�1rt1[g (xt � 1)b].

� Kax � xt�1

hRPb s(xt�1)d ¢ft1[g (x t�1)b]

g(xt�1)

rt � s(xt�1)¢ft1[g(x t�1)b] � T�1h�qRPa

T

t�1ch�1

RPK�ax � xt�1

hRPb

© � T�1aT

t�1rtr

�t1[g(xt � 1)b] � r r �

[2T(dind �d)]© � limTS

dIV

2TdIVdindddir

aT�1aT

t�1s(xt�1)¢ft1[g (xt � 1)b]b.dIV � aT�1

aT

t�1s(x

t�1)x�t�11[g (xt � 1)b]b�1

9Note the (parametric) rate of convergence for the ADEs, whereas and , used in constructingthe ADEs, achieve consistency at a slower rate. The parametric rate is owing to averaging. See Ichimura andTodd (2006, p. 77).10Two estimators, and are first-order equivalent if 11There is a systematic downward bias in the kernel density derivative estimators . This bias carries over to theestimated scores, as ; see Stoker (1993).s(x) � �g�(x)�g(x)

g�(x)

2T(d1 � d2) ¡p

0.d2d1

g�( # )g( # )2T



where . Thus, the covariance matrix of may be esti-mated by with

Time series issues

The theory for average derivative estimation applies to situations where theobservations are i.i.d. However, the kernel density estimators, on which theADEs are built, are still consistent and asymptotically normal under the weak-er assumption that the data are stationary and a-mixing12 (Pagan & Ullah,1999, chap. 2). If so, the ADEs remain consistent13 and asymptotically normalwith weakly dependent observations.

The covariance matrix has to be corrected to account for het-eroskedasticity and autocorrelation in the regression errors, . A het-eroskedasticity and autocorrelation consistent covariance matrix is given by

where is the Newey–West estimator of the long-run covari-ance matrix of rt.

14

Implications for the Optimal Hedge Ratio

Although it can be easily established whether and when a nonzero forecast of�ft is statistically significant, there is no unique measure to assess its rele-vance. The criterion adopted here is the difference between the optimal hedgeratio out; in a mean-variance framework, and the conventional minimum-vari-ance hedge ratio. Suppose the risk-averse hedger maximizes a mean-varianceobjective function of the return.15,16 The time-varying optimal mean-variancehedge ratio that incorporates both risk and return is given by

(10)�

cov(¢st,¢ft 0xt�1) � 1A E[¢ft 0xt�1]

var(¢ft 0xt�1)

Hms2

t�1 � arg minht�1

cE[¢pt 0xt�1] �A

2var[¢pt 0xt�1] d

©NWT�1A©NWA

¢ft � x�t�1d

T�1A©A

A � 5T�1gTt�1s(xt�1)x�

t�11[g(x t�1)b]6�1.T�1A©AdIVA � 5cov(s(x), x)6�1

12An a-mixing series satisfies a condition on the speed at which the dependence between past and futureevents vanishes, as the time between the events increases.13Ghysels and Ng (1998) claim that remains -consistent and asymptotically normal, invoking resultsof Chen and Shen (1998), where -convergence and asymptotic normality of Sieve extremum estimates(SEE) are proved for weakly dependent data. This claim seems incomplete as cannot be written as a SEE. can, at most, be seen as an extremum estimator (as defined, e.g., in Hayashi, 2000, chap. 7), asit is a variant of a GMM estimate with as instrument. However, such an extremum estimator does notbelong to the class of the SEE.14The lag truncation parameter at is set as a common choice.15See e.g. Lien and Tse (2002) and Chen, Lee, and Shrestha (2003) for a review of the hedging literature.16A shortcoming of the model is that, implicitly, it assumes that there is just one source of risk to be hedged,and one hedge. However, in view of the low correlations between returns from commodities and financialassets, a true multirisk solution would come up with essentially the same hedge strategy.

:4(T�100)2�9 ;s(x)

dIVdIV

2T2TdIV



where �pt � �st � ht�1� ft is the portfolio return and A is the relative risk-aver-sion parameter. The optimal mean-variance hedge ratio, , reduces to theminimum-variance hedge ratio,

(11)

if the hedger is infinitely risk averse, i.e. AS , or if the expected return on thefutures contract, , is zero. So, when considering noninfinitely risk-averse hedgers, whether equals zero or not determines the choicebetween the optimal mean-variance and the minimum-variance hedge ratios.

The mean-variance model takes a portfolio point of view, where the pro-ducer identifies payoffs with the assets’ returns and merely tries to combinethem optimally. In reality, corporate hedging affects payoffs also in other ways,for example by reducing the expected costs of financial distress. As this consid-eration cannot be possibly included into the optimal hedge, the optimal hedgecannot be estimated. Still, how much the optimal hedge is changed can becomputed by any nonzero forecast , for a given operational effect: itsdifferential effect on the optimal hedge ratio is measured by

(12)

Thus, the measure is proportional to the ratio of expected return17 to vari-ance. Note that this measure does well even for parts of portfolios, given thatcommodity returns have low correlations with returns from other assets andthat little is gained from cross hedging—say, combining cash oats with wheatfutures next to oats futures. Indeed, to hedge a portfolio of distinct exposures,one just applies the portfolio of separate hedges as determined here; thus, if theindividual hedges are substantially affected by nonzero forecasts, so must bethe hedges of the portfolio.

The impact of a nonzero forecast on the optimal hedge ratio is a very sim-ple, action-related number, easy to explain to users. It takes into account thatthough daily expected returns are small numbers, so are one-day variances;what matters, in a mean-variance framework, is the relative size of the meanversus the variance. In the application, the magnitude of �Ht�1 is investigatedfor measures of risk aversion, A, ranging from two to ten.

¢Ht�1 � Hms2

t�1 � HMinVart�1 �

�1AE[¢ft 0xt�1]

var(¢ft 0xt�1).

E[¢ft 0xt�1]

E[¢ft 0xt�1]E[¢ft 0xt�1]

�cov(¢st,¢ft 0xt�1)

var(¢ft 0xt�1),

HMinVart�1 � arg min

ht�1

var[¢pt 0xt�1]

Hms2

t�1

17As a futures contract is a zero-investment instrument, its percentage change is similar to an excess returnfor a regular asset.



APPLICATION TO AGRICULTURAL FUTURES

An empirical application of the technique suggested is now considered.The data are described and the potential spurious predictability is discussedarising from the fact that spot and futures prices are not synchronized. Thenthe regression coefficients are estimated, each estimated conditional expecta-tion is graphed given the constructed index, and the implications for hedgersare discussed. This section is ended by checking the robustness of the pre-dictability results out-of-sample.

Data

The data consist of daily spot and nearest-to-maturity futures closing prices ofcorn, wheat, soybeans, and oats. Daytime futures closing prices were extractedfrom the Chicago Board of Trade files, covering the period January 1979,through December 2003 for corn, soybeans, and oats, and the period January1983, through December 2003 for wheat. For the same period, the commodityspot prices were extracted from Datastream. Corn, soybeans, and wheatspot prices (in cents/bushel) are the average of all Central Illinois Elevatorsprices paid by the Country Elevators to the producers, after 2:00 P.M. Oats spotprices are from Minneapolis, Minnesota. The spot prices are for the followingqualities: oats, No.2; wheat, No.2, Soft Red; soybeans, No.1, Yellow; corn,No.2, Yellow.

Following standard practice in the literature (see, for example, Bera,Garcia & Roh, 1997; Harris & Shen, 2003), and to avoid thin trading and expi-ration effects, a contract that expires in month m is replaced with the nextnearest-to-maturity contract on the last day of month m � 1. Specifically, onthe last day of month m � 1, �ft is set equal to the return on the expiring con-tract, whereas on the first day of month m, �ft is set equal to the return on thenew contract. This ensures that at roll-over, i.e. when jumping from the expir-ing contract to the new contract, each return is computed over one contractand does not reflect any price jump due the roll-over of contracts.

Even though both states, Illinois and Minnesota, are in the same timezone, futures returns and the lagged spot return predictors are overlapping.The reason is that the agricultural futures market closes before 1:15 P.M. forthe period under consideration, so that their closing prices miss possible infor-mation available after 2:00 P.M., the afternoon when spot prices are set.Suppose, one wants to predict the Wednesday–Thursday change in the futuresprice, noon to noon. Then, ideally, one should use Tuesday–Wednesdaychanges in the spot price, noon to noon. In reality one only has an afternoon-to-afternoon spot return for Tuesday–Wednesday, which includes information notyet available when the prediction for the futures market is to be made and which,



therefore, possibly overestimates the predictive power of the spot return. Onthe other hand, the alternative of using an afternoon-to-afternoon spot returnfor Monday–Tuesday is likely to underestimate any predictability (if present),because all information that became available Tuesday evening and Wednesdaymorning is ignored. For our purpose, erring on the conservative side is prefer-able to the alternative, so the predictions are based on just price informationthat is available. That is, the following predictors are used:

(13)

giving a “feasible” lower bound on the predictability.Prediction horizons of one day and one week are considered. Similar hori-

zons have been considered in the literature; see, for instance, Baillie and Myers(1991), Bera et al. (1997), Lien, Tse, and Tsui (2002), and Byström (2003).For the weekly prediction horizon, the returns were aggregated to yield weekly(Friday to Friday) returns. Table I gives summary statistics for �st and �ft foreach commodity, during the period covered. It shows the mean, standard devi-ation, skewness, kurtosis, and autocorrelations. The return data are nonnormalas evidenced by the high kurtosis and the significant Jarque-Bera statistics (notreported here).

When testing H0,x a subtle issue arises. As pointed out by a referee, rejec-tion of H0,x may partly (or fully) be the consequence of a nonzero, but constant,conditional mean futures return, instead of being owing to a time-varying con-ditional mean futures return. If this were the case, the null would be rejectedthat the futures returns have an unconditional zero mean, i.e.

would be rejected.Stated differently, jointly rejecting H0,x (the martingale hypothesis) and

not rejecting H0 is a sufficient condition to have a time-varying conditionalmean futures return. A t-test is used to test H0. The associated p-values, report-ed in Table I, are all larger than five percent. Thus, no compelling evidence isfound of a constant nonzero mean futures return, and any rejection of H0,x

must therefore be interpreted as evidence of predictability of the futuresreturns, in the sense that their conditional mean is time varying.

Predicting Futures Returns

�ft is first predicted using xt � 1 as specified in (13). Panels A and B of Table IIreport the ADEs with Newey–West standard errors. As a matter of compari-son, the ordinary least-squares (OLS) estimates are also reported of the linear

dIV

H0:E[¢ft] � 0

xt�1 � (ft�2 � st�2, ¢ft�1, ¢st�2, (¢ft�1)2, (¢st�2)2, ¢ft�2¢st�2)�



regression of �ft on the (standardized) predictors xt�1. The ADEs must yieldsimilar values to the OLS estimates if the relationship between �ft and xt�1 istruly linear. Indeed, both estimators are supposed to measure the same thingbecause, when m(�) is the identity function, g equals unity in (3).

For the daily horizon, the Wald statistics reject the hypothesis that all ADEsare jointly equal to zero. This is less obvious for the OLS estimates. More impor-tantly, in the ADE model, the lagged futures return always carries significant

TABLE I

Spot and Futures Returns

Mean Std. Deviation Skewness Kurtosis

Autocorrelation Coef.

r1 r6 r12

Corn, daily, 1/04/1979–12/31/2003, 6,520 observations�st 5.8072 � 10�6 0.0144 �0.4832 8.9742 0.0189 0.0218 �0.0049�ft �2.8836 � 10�4 0.0122 �0.0060 5.6544 0.0497 �0.0063 0.0098p -Value 0.0571

Corn, weekly, 1/26/1979–12/26/2003, 1,301 observations�st �2.0968 � 10�5 0.0326 �0.4958 7.2140 0.0392 0.0276 �0.0362�ft �0.0014 0.0280 0.3005 6.8211 �0.0166 �0.0101 0.0003p-Value 0.0644

Wheat, daily, 1/04/1983–12/31/2003, 5,477 observations�st �6.5684 � 10�6 0.0172 �0.9733 22.1083 �0.0079 0.0037 0.0205�ft �2.0203 � 10�4 0.0137 0.0811 5.5408 0.0243 0.0034 0.0195p-Value 0.2761

Wheat, weekly, 1/13/1984–12/26/2003, 1,042 observations�st 1.0417 � 10�5 0.0373 �0.6618 10.9648 �0.0203 �0.0206 0.0140�ft �9.9113 � 10�4 0.0301 0.4218 4.6755 0.0014 �0.0357 0.0208p -Value 0.2880

Oats, daily, 1/04/1979–12/31/2003, 6,520 observations�st 2.6532 � 10�5 0.0193 �0.0794 22.9636 �0.0295 0.0108 �0.0008�ft �3.3979 � 10�4 0.0178 �0.0572 5.1270 0.0566 �0.0102 �0.0047p-Value 0.1225

Oats, weekly, 1/26/1979–12/26/2003, 1,301 observations�st 4.9289 � 10�5 0.0420 �0.0550 7.6995 �0.0720 �0.0335 �0.0278�ft �0.0017 0.0415 0.1307 7.0114 �0.0512 �0.0183 0.0428p-Value 0.1307

Soybeans, daily, 1/04/1979–12/31/2003, 6,520 observations�st 2.4267 � 10�5 0.0135 �0.3723 6.5939 �0.0290 �0.0208 �0.0104�ft �1.6056 � 10�4 0.0129 �0.1511 5.3061 �0.0174 �0.0221 0.0032p-Value 0.3159

Soybeans, weekly, 1/19/1979–12/26/2003, 1,302 observations�st 6.6689 � 1025 0.0307 �0.1426 6.0890 �0.0873 �0.0139 �0.0078�ft �8.2016 � 1024 0.0294 �0.1573 6.3095 �0.0543 �0.0014 �0.0248p-Value 0.3146

Note. p-Values associated with t-test of H0 : E [�ft ] � 0.



information about the next period futures return. In contrast, the two-periodlagged spot return has no significant impact, with the possible exception ofcorn in the ADE model and soybeans in the OLS model. The squared futuresand spot returns do not have significant explanatory power for any commodity

TABLE II

Average Derivative and OLS Estimates for Predicting Futures Return

Corn Wheat Soybeans Oats

ADE OLS ADE OLS ADE OLS ADE OLS

A. Daily

ft�2 � st�2 �0.0023 �0.0001 �0.0079 �0.0102 �0.0052 �0.0270 �0.0295 �0.0450[0.0327] [0.0219] [0.0306] [0.0207] [0.0554] [0.0192] [0.0359] [0.0277]

�st�2 �0.0696 0.0085 �0.0186 �0.0382 0.0060 0.0403 0.1033 0.0040[0.0342] [0.0199] [0.0432] [0.0206] [0.0676] [0.0203] [0.0711] [0.0293]

�ft�1 0.1067 0.0593 0.1391 0.0340 �0.0906 �0.0255 0.1674 0.0983[0.0291] [0.0223] [0.0366] [0.0213] [0.0380] [0.0232] [0.0429] [0.0294]

(�ft�1)2 0.7959 0.0073 �0.0448 �0.0003 1.5929 �0.0274 0.0510 �0.0087

[0.7830] [0.0280] [0.3835] [0.0333] [3.1065] [0.0282] [0.1559] [0.0321](�st�2)

2 1.3375 �0.0557 �0.3539 �0.0315 2.1606 �0.0146 0.0405 0.0575[1.3390] [0.0452] [1.1846] [0.0252] [4.5095] [0.0667] [0.5080] [0.0453]

�ft�2�st�2 0.4007 0.1017 �0.0166 0.0364 0.4939 0.0487 0.2826 0.0448[0.1825] [0.0415] [0.1309] [0.0292] [0.4660] [0.0538] [0.1704] [0.0433]

Wald stat 21.9420 11.5755 15.4056 5.9680 13.0639 11.7784 21.6758 19.5770p -Value 0.0012 0.0721 0.0173 0.4268 0.0420 0.0671 0.0014 0.0033

B. Weekly

ft�2 � st�2 �0.1828 0.1256 0.2253 0.0396 �0.1599 �0.1163 �0.3212 �0.1107[0.1662] [0.1278] [0.1560] [0.0977] [0.1591] [0.0822] [0.1741] [0.1402]

�st�2 0.1687 0.2584 0.2180 0.0222 0.1393 0.2037 0.1302 0.1860[0.2005] [0.0957] [0.1826] [0.1136] [0.1622] [0.0988] [0.2517] [0.1295]

�ft�1 �0.0005 �0.0519 �0.0887 0.0592 0.0647 �0.1569 �0.1498 �0.2329[0.2372] [0.1130] [0.1807] [0.0919] [0.1404] [0.1126] [0.2193] [0.1820]

(�ft�1)2 0.4496 0.0969 �1.6213 �0.1743 �1.9356 �0.2313 �0.7590 0.0055

[3.4684] [0.2231] [0.7933] [0.1476] [2.4883] [0.1393] [3.0644] [0.3014](�st�2)

2 0.0873 0.1877 �2.6008 �0.1955 �4.0273 �0.5643 �0.8692 �0.2551[4.2104] [0.1471] [1.8326] [0.0705] [2.9643] [0.3131] [4.8540] [0.2407]

�ft�2�st�2 0.1823 �0.1096 �0.7113 0.2624 1.2849 0.8855 2.4740 0.5725[0.6581] [0.2461] [0.5008] [0.1387] [0.8733] [0.3599] [0.8414] [0.2331]

Wald stat 3.2242 12.8650 13.2046 20.4691 7.3072 13.3083 16.6477 14.5278p-Value 0.7802 0.0452 0.0599 0.0023 0.2934 0.0384 0.0107 0.0243

C. Daily

ft�2 � st�2 �0.01181 0.00307 �0.01059 �0.0095 �0.00376 �0.02577 �0.05543 �0.04971[0.03177] [0.02085] [0.02919] [0.02046] [0.02835] [0.01993] [0.03358] [0.02919]

�st�2 �0.03735 0.00693 �0.03378 �0.02849 0.00985 0.03855 0.05313 0.00101[0.02661] [0.02096] [0.03059] [0.01934] [0.02669] [0.02092] [0.05814] [0.02864]

�ft�1 0.06404 0.06039 0.08568 0.03418 �0.08398 �0.02439 0.12370 0.09913[0.02403] [0.02290] [0.02906] [0.02144] [0.02522] [0.02354] [0.03406] [0.02918]

Wald stat 8.9393 7.0092 9.9362 4.3164 11.3207 7.5010 17.2495 13.498p-Value 0.0301 0.0716 0.0191 0.2293 0.0101 0.0573 0.0006 0.0037

Note. Newey–West standard errors in brackets; robust Wald statistics; all estimates and standard errors have been multiplied by100; data 1979–2003 (corn, soybeans, oats) or 1983–2003 (wheat). OLS, ordinary least squares; ADE, average derivative estimator.



and the interaction term matters only for corn. Although it is not significant,the estimated effect of the basis is always negative (with the exception of soy-beans in the ADE model), as expected. For the weekly horizon, the uniformityof the results across commodities is much weaker. The predictive power of theADE model disappears for corn, soybeans, and wheat, as indicated by the Waldtests, whereas for the OLS model it remains intact or improves. As can be seenfrom the two last columns of Panel B, there is predictability under both modelsfor oats on a weekly horizon. However, when plotting as afunction of the index together with bootstrap confidence bounds, the pre-dictability area is smaller than in the case of a daily horizon. In the remainderof the analysis, attention is restricted to the daily data.

The ADE and OLS estimates are used as weights to construct two dailyindices, and . It can be shown (not reported here) that all ADEindices based on the full set of (standardized) predictors are mainly driven by(�st�2)

2 and, to a lesser extent, by (�ft�1)2 and/or �ft�2�st�2, even though most

of these regressors are insignificant. To get some insight, take the example ofsoybeans: the ADEs for the standardized predictors (�st�2)

2, (�ft�1)2, and

�ft�2�st�2 are much larger in absolute value than the coefficient estimates ofthe significant predictors ft�2 � st�2, �st�2, �ft�1. This clearly adds noise to theindices. To circumvent that problem and to limit18 potential data snooping cri-tiques by selecting a different set of predictors for each commodity, all indicesare constructed from the constrained regression without (�ft�1)

2, (�st�2)2, and

�ft�2�st�2; that is,

is used as predictors. ADE and OLS estimates of the constrained regression arereported in Panel C of Table II.19 The estimates of the remaining coefficientsare qualitatively similar to those in Panel A. Table III reports summary statisticson the constrained ADE and OLS indices. There is a reassuringly high correla-tion between the two indices.

The next step is to relate the expected futures returns to the estimatedindex . It is worth noting that the Nadaraya–Watson kernel estimator

converges at the optimal nonparametric (univariate) rate T�2/5, even though is replaced by the estimated index (Härdle & Stoker, 1989, Theorem 3.3). The function , plotted in Figures 1–4, is estimated on the interval bounded by the sample mean of

plus and minus twice its standard deviation to diminish the distortingx�t�1d

IV

m(x�t�1d

IV) � E[¢ft 0x�t�1d

IV]x�t�1d

IVx�t�1d

IVm(x�

t�1dIV)

x�t�1d

IV

xt�1 � (¢ft�1, ¢st�2, ft�2 � st�2)�

x�t�1b

OLSx�t�1d

IV

m(x) � E[¢ft 0x�dIV]

18Data snooping is not entirely eliminated, as the set of predictors is still chosen on the basis of pre-testing.19As the exclusion of the second-order terms is based on the results of the ADE model, this may bias subse-quent results in favor of the ADE model. Such a bias is likely to be small, however, because the second-orderterms are nearly always insignificant in the OLS model as well.



TABLE III

ADE and OLS Indices

Corn Wheat

ADE OLS ADE OLS

DailyMean �1.5443 � 10�19 3.1786 � 10�20 �3.0970 � 10�20 7.9722 � 10�20

Variance 5.2549 � 10�7 3.7696 � 10�7 8.3768 � 10�7 1.9894 � 10�7

Skewness 0.0468 �0.0066 0.0414 0.1100Kurtosis 5.2218 5.8683 5.4424 6.9968Correlation 0.7704 0.9429

Soybeans Oats

ADE OLS ADE OLS

DailyMean �3.1728 � 10�20 �2.7022 � 10�19 4.5695 � 10�18 3.9487 � 10�18

Variance 7 1045 � 10�7 2.7087 � 10�7 2.2565 � 10�6 1.2596 � 10�6

Skewness 0.1013 �0.1528 0.2057 0.2596Kurtosis 4.9073 6.4617 5.9414 4.7972Correlation 0.5395 0.9383

Note. indices defined as (ADE) and (OLS) with ; daily data 1979–2003(corn, soybeans, oats) or 1983–2003 (wheat). OLS, ordinary least squares; ADE, average derivative estimator.

xt�1 � (¢ft�1,¢st�2,ft�2 � st�2)�x�t�1d

OLSx�t�1d

IV

�1.5 �1 �0.5 0 0.5 1 1.5x 10�3

�12

�10

�8

�6

�4

�2

0

2

4

6x 10�4

ADE index

Est

imat

ed e

xpec

ted

futu

res

retu

rn

Corn

Expected return95% conf. bound95% conf. bound

FIGURE 1Corn: estimated conditional expected futures return as a function of the ADE index; daily data

1979–2003.



influence of the outliers. A 95% bootstrap pointwise confidence interval is alsoadded.20

The confidence bounds confirm that the ADE index has predictive power,notably when takes on negative values. For corn and wheat, this hap-pens when the lagged futures return takes on large negative values and/or thetwo-period lagged spot return and basis takes on large positive values. For oats,the index is negative when the two-period lagged basis is positive and the two-period lagged spot and one-period lagged futures returns are negative. Forsoybeans, there is predictability when the basis and the lagged futures returnare positive and the two-period lagged spot return is negative. An “asymmetry” inpredictability is also observed: all indices are only able to predict negative expect-ed futures returns with sufficiently high confidence. This is partially explained bythe finding, in Table I, that for all four commodities, the unconditional mean of

x�t�1d

IV

�2 �1.5 �1 �0.5 0 0.5 1 21.5x 10�3

ADE index

�2

�1.5

�1

�0.5

0

0.5

1

1.5

2E

stim

ated

exp

ecte

d re

turn


x 10�3 Wheat

FIGURE 2Wheat: estimated conditional expected futures return as a function of the ADE

index; daily data 1983–2003.

20The optimal cross-validation bandwidths are 0.0007185, 0.0005870, 0.0005670, and 0.0009252 for corn,wheat, soybeans, and oats, respectively. The trimming bound is set such that five percent of the observationsare dropped. Hundred bootstrap replications are used and an average block length of ten periods is chosen.This value seems reasonable given the weak autocorrelation in the daily data.



the futures return is negative, albeit small and insignificant. This makes itharder to predict positive expected futures returns with high confidence.However, even if the curves were shifted upward to offset the negative uncon-ditional mean, the asymmetry in predictability would persist, though to a smallerdegree and with the exception of corn. Note also that, with the exception ofwheat, the relationship between and the expected return is approxi-mately linear over the range considered.

The graphs further illustrate the discussion at the end of the data section,relating H0,x and H0, and how the data support or reject them. H0,x states thatthe conditional expected futures return is identically zero for all values of theADE index; H0 means that the average level of this curve is zero. The data rejectH0,x and support H0, hence, it must be concluded that the curve is not a flat line.That is, the expected futures return effectively varies as the ADE index varies.

Recall that all of the above ignores any information that became availablein the late afternoon and the subsequent morning preceding the predictionperiod, which is noon to noon. In that sense, the observed predictability is infact a lower bound on the “true” predictability.

x�t�1d

IV


�2 �1.5 �1 �0.5 0 0.5 1 21.5x 10�3

ADE index

�2

�1.5

�1

�0.5

0

0.5

1

1.5

Est

imat

ed e

xpec

ted

retu

rn

x 10�3 Soya

FIGURE 3Soybeans: estimated conditional expected futures return as a function of the ADE index; daily

data 1979–2003.



Changes in Optimal Hedge Ratio

In this section the consequences of the observed predictability on the optimalmean-variance hedge ratio are analyzed. The question is whether any pre-dictability has a noticeable impact on the optimal hedge ratio. The (static)minimum-variance hedge ratio is first computed at a one-day horizon,

. Next, the differential impact of the pre-dictability on the hedge ratio is computed, for values of A between two and tenand for a given value of the ADE index, set at the sample mean of the ADEindex minus one standard deviation:

Table IV reports the results. As expected, the static hedge ratios areclose to unity, with the exception of oats. The reason may be that the oats spotmarket is the Minneapolis exchange, whereas the futures contract is traded onthe Chicago Board of Trade. For a degree of risk of aversion A � 3.5, the

HMinVar

¢H � �1�AE[¢ft 0ADE � ADE � s(ADE)]

var(¢ft).

Hˆ MinVar � cov(¢st,¢ft)�var(¢ft)


�4 �2�3 �1 0 1 2 3x 10�3

ADE index

�4

�3

�2

�1

0

1

2

3E

stim

ated

exp

ecte

d fu

ture

s re

turn

x 10�3 Oats

FIGURE 4Oats: estimated conditional expected futures return as a function of the ADE index; daily data

1979–2003.



impact of the mean on is potentially huge, even when using the upperbound, , of the 95% confidence interval forthe expected futures return, instead of the central forecast. But even for A �

10, central-forecast hedge ratios still rise by 30 to 50 percentage points. It isalso clear that the large 95% confidence interval around the estimated expectedmean causes considerable uncertainty regarding the optimal mean-variancehedge ratios, but never to the extent that it would make the minimum-variancehedge ratio a serious candidate for the optimal ratio. Finally, note that thereare many episodes where the forecast is more than one sigma below the mean;for these, the impact would be correspondingly higher than what is shown inthe table. In short, it is concluded that the size of the position may be serious-ly affected for a certain range of negative values of the index, even though theoptimal mean-variance hedge ratio remains determined imprecisely due tothe lack of precision in the predictability.

E[¢ft 0ADE � ADE � s(ADE)]UBHt�1ms2

TABLE IV

Comparison of Optimal Mean-Variance and Minimum-Variance Hedge Ratios

A � A � 10 A � 5 A � 3.5 A � 2

Corn � �0.010% 0.98 0.07 0.13 0.19 0.34Wheat � �0.025% 0.92 0.13 0.27 0.38 0.67Oats � �0.090% 0.46 0.28 0.57 0.81 1.42Soybeans � �0.025% 0.93 0.15 0.30 0.43 0.75

A � A � 10 A � 5 A � 3.5 A � 2


A � A � 10 A � 5 A � 3.5 A � 2


Note. is the lower (upper) bound of the 95% confidence interval for ;

ADE, average derivative estimator.

E[¢ft 0ADE � ADE � s(ADE)]E[¢ft 0ADE � ADE � s(ADE)]LB(UB)

E[¢ft 0ADE � ADE � s(ADE)]LB

¢H �1AE [¢f

t|.]LB

var(¢ft )HMinVar �

cov(¢st,¢ft )var(¢ft )

E[¢ft 0ADE � ADE � s(ADE)]

¢H �

1AE [¢f

t|.]

var(¢ft )HMinVar �cov(¢st,¢ft )

var(¢ft )

ADE � s(ADE)]UBE[¢ft 0ADE �

¢H �1AE [¢ft |.]

UB

var(¢ft )HMinVar �cov(¢st,¢ ft )

var(¢ft )



For the academic economist, the puzzle is why any predictability is confined tonegative forecasts, even if the unconditional mean is slightly negative. It is hardto believe that the commodity futures’ nondiversifiable risk, its covariance with thepricing kernel, goes from zero to negative or vice versa on a day to day basis; in fact,estimated betas for commodity futures have turned slightly positive, recently.21 Oneremaining potential explanation is that there is an unidentified cost of going short.

Out-of-Sample Robustness Check

To check the robustness of the in-sample results, an out-of-sample test is per-formed. The algorithm is first designed in a rolling window framework:

(0) Set i � 1 and j � 0.

(1) The 2,500 � j first observations are used for the in-sample estimation of theADEs and for the identification of the domain of predictability. Specifically,the ADEs are computed for xt�1 � (�ft�1, �st�2, ft�2�st�2)�, theNadaraya–Watson kernel estimator is calculated, relating the constructed index

to the expected futures returns, and 95% confidence bounds areadded. A “promising” domain of predictability, domin, for the index is thenidentified, in two steps. First, domin is defined as the interval delimited by theminimum and the maximum values of where the corresponding 95% upperbound lies below the zero line. Second, domin has to pass two tests to proceedwith it. When domin is very small, it may not be worth speculating on.Therefore, if less than 100 observations are in domin, no speculation is pursuedand the algorithm jumps to step (3). Further, domin may show points where the95% upper bound lies above the zero line, whereas the boundaries of domin byconstruction have a 95% upper bound lying below the zero line. If for morethan 15% of the observations in domin the 95% upper bound lies above the zeroline, no speculation is pursued and the algorithm jumps to step (3).

(2) The next 100 observations (or the number of remaining observations left inthe sample if less than 100) are used to test the out-of-sample predictabili-ty over domin. Using the in-sample ADEs from step (1), an out-of-sampleindex is built. Out-of-sample days for which lies in domin

are identified as days where prediction is “promising.” Let #pred(i) be thenumber of those days, and #predCORRECT(i) the number of those days where,in addition, the sign of the futures return is correctly predicted as negative.If the speculative strategy has predictive power regarding the sign, oneshould find that #predCORRECT(i)/#pred(i) exceeds 1/2 more often than not.Finally, the sum of the (negative and positive) futures returns �ft on days

zoutt�1zoutt�1 � x�t�1d

IV

zint�1

zint�1

zint�1 � x�t�1d

IV

dIV

21The Economist, October 12, 2006.



where lies in domin is computed and denoted by Sum�f(i). If the sum isnegative (positive), this gives a measure of the gain (loss) of using the short-ing speculative strategy on a 100-day basis.

(3) Increase j by 100 observations and i by 1. Return to step (1) unless no out-of-sample observations are left.

Let max(i) denote the number of iterations of the algorithm and Tout thecumulative number of observations in the out-of-sample periods. As the fullsample contains 6,520 (5,477 for wheat) observations, the algorithm isiterated 41(30) times and the last out-of-sample period contains 20 (77)observations. Thus, Tout equals 2,977 for wheat and 4,020 for the threeother commodities.

Alternatively, the algorithm is also run in a moving window frameworkwhere step (1) is replaced by:

(1 ) Observations 1 � j to 2,500 � j are used for the in-sample estimation, etc.

This allows us to figure out whether one wins or loses by discarding thedata that are older than ten years.

As the bandwidth of the Nadaraya–Watson kernel estimator is a functionof the number of observations in the sample under consideration and of thestandard deviation of the index, it must be computed at each iteration of the algorithm. To avoid computationally intensive grid searchs needed by theLOOCV method, the bandwidths are chosen to be proportional to the popularplug-in estimate of Silverman (1986):

where std(.; i) is the standard deviation computed from the sample of size T(i),depending on the moving or rolling window approach, in the ith iteration of thealgorithm. The proportionality constant c is given by the ratio of the LOOCVbandwidth to the plug-in estimate bandwidth when both are computed from thefull sample.22 This renders the results and graphs compatible with those ofthe in-sample analysis.

Table V reports statistics for the whole out-of-sample period.is the total number of out-of-sample days where the agent

should speculate on a negative futures return, according to the methodology.Among these days, is the total number of out-of-sam-ple days where the sign of the futures return is correctly predicted as negative.The ratio indicates how often the agent should go short,or equivalently, how often a “promising” domain of prediction does show up.

amax(i)

i�1# pred(i)�Tout

gmax(i)i�1 � predCORRECT(i)

gmax(i)i�1 � pred(i)

h(i) � c � 1.06 � std(xt�i; i)T(i)�1�5

�

zoutt�1

22For corn, wheat, oats, and soybeans, c equals 5.81, 3.38, 3.37, and 3.68, respectively.



The “promising” domain of prediction does not show up frequently for corn(both windows) and wheat (moving window). In contrast, it shows up frequent-ly for oats and soybeans (both windows) and for wheat (rolling window).

If

p �a

max(i)

i�1� predCORRECT

(i)

amax(i)

i�1� pred(i)

12

TABLE V

Out-of-Sample Performance of Shorting Speculative Strategy

Moving window Corn Wheat Soybeans Oats

143 8 1636 1588

0.036 0.003 0.407 0.395

73 2 928 854

0.5105 0.2500 0.5672 0.5378

[0.0418] [0.1531] [0.0122] [0.0125]

�0.0885 0.1470 �0.5188 �0.1952

�0.062% 1.8375% �0.032% �0.012%

Rolling window

265 824 847 1849

0.066 0.277 0.211 0.460

126 451 508 982

0.4755 0.5473 0.5998 0.5311

[0.0307] [0.0173] [0.0168] [0.0116]

0.0394 �0.8082 �0.5504 0.0052

0.015% �0.098% �0.065% 0.0003%

Note. is the total number of out-of-sample days where the agent should speculate. Among these days,

is the total number of days where the sign of the futures return is correctly predicted as negative. T out

is the total number of out-of-sample days. Standard errors of are given in brackets.

is the sum of the out-of-sample returns on days where the agent should speculate.amax(i)

i�1Sum¢f(i)

amax(i)

i�1� pred CORRECT(i)�a

max(i)

i�1� pred(i)

amax(i )

i�1� pred CORRECT

(i )

amax(i )

i�1� pred (i)

amax(i)

i�1Sum¢f(i)�a

max(i)

i�1� pred (i)

amax(i)

i�1Sum¢f(i)

amax(i)


(i)�amax(i)

i�1� pred (i)

amax(i)


(i)

amax(i)

i�1� pred (i)�T out

amax(i)

i�1� pred (i)

amax(i)

i�1Sum¢f(i)�a

max(i)

i�1� pred (i)

amax(i)

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the prediction of the sign was correct more than half of the time. The standarderror of , , is also reported. The results show that forsoybeans and oats (moving window) and wheat, soybeans, and oats (rollingwindow), is significantly larger than at the 5% level, meaning that, for thesecommodities, the methodology predicts the correct sign significantly betterthan the toss of a coin.

If one had acted on these predictions for soybeans and oats (both win-dows) and for wheat (rolling window). In four out of five cases, the aggregateout-of-sample return on days that were singled out for the shorting strategy,�i�1

max(i) Sum�f(i), is highly negative: notably returns of �20, �50, �55, and�80%. The single wrong outcome, for oats with rolling windows, is utterlyinsignificant by economic standards. The ratio �i�1

max(i) Sum�f(i)/�i�1max(i) #pred(i)

gives the average daily return conditional on an out-of-sample promising pre-diction. For soybeans (moving window) and wheat and soybeans (rolling win-dow), the daily returns are in line with the in-sample results in Figures 2–4.

It was also checked that the results for wheat (rolling window) and for soybeans and oats (both windows) are not owing to a specific out-of-samplesubperiod. It turns out that the pattern of correct negative sign predictions isstable over time.

In short, it is concluded that the predictability does survive in an out-of-sample setting, even though the performance is uneven and hardlyawesome.

CONCLUSION

This study empirically investigated the martingale hypothesis for agriculturalfutures prices and the practical implications if it is rejected. The question wasaddressed through a nonparametric approach where a linear index combiningseveral predictors is nonparametrically related to the expected futuresreturns.

Statistically strong empirical evidence against Samuelson’s (1965) hypoth-esis was found. The results indicate that the estimated index contains statisti-cally significant information regarding the expected futures returns. The predictibility found has a large effect on the optimal mean-variance hedgeratio, even though it is small and estimated rather imprecisely.

BIBLIOGRAPHY

Baillie, R. T., & Myers, R. J. (1991). Bivariate estimation of the optimal commodityfutures hedge. Journal of Applied Econometrics, 6, 109–124.

12p

2p(1 � p)�©i# pred(i)p



Bera, A. K., Garcia, P., & Roh, J.-S. (1997). Estimation of time-varying hedge ratios forcorn and soybeans: BGARCH and random coefficient approaches. The IndianJournal of Statistics, 59, 346–368.

Bessembinder, H., & Chan, K. (1992). Time-varying risk premia and forecastablereturns in futures markets. Journal of Financial Economics, 32, 169–193.

Byström, H. N. E. (2003). The hedging performance of electricity futures on theNordic power exchange. Applied Economics, 35, 1–11.

Campbell, J. Y., Lo, A. W., & MacKinley, A. C. (1996). The econometrics of financialmarkets. Princeton, New Jersey: Princeton University Press.

Chen, X., & Shen, X. (1998). Sieve extremum estimates for weakly dependent data.Econometrica, 66, 289–314.

Ederington, L. H. (1979). The hedging performance of the new futures markets.Journal of Finance, 34, 157–170.

Ghysels, E., & Ng, S. (1998). A semiparametric factor model of interest rates and testsof the affine term structure. Review of Economics and Statistics, 80, 535–548.

Härdle, W. (1990). Applied nonparametric regression. Cambridge, UK: CambridgeUniversity Press.

Härdle, W., & Stoker, T. (1989). Investigating smooth multiple regression by themethod of average derivatives. Journal of the American Statistical Association, 84,986–995.

Harris, R. D. F., & Shen, J. (2003). Robust estimation of the optimal hedge ratio.Journal of Futures Markets, 23, 799–816.

Hayashi, F. (2000). Econometrics. Princeton, NJ: Princeton University Press.Hsin, C. W., Kuo, J., & Lee, C. F. (1994). A new measure to compare the hedging effec-

tiveness of foreign currency futures. Journal of Futures Markets, 14, 685–707.Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estima-

tion of single index models. Journal of Econometrics, 58, 71–120.Ichimura, H., & Todd, P. E. (2006). Implementing parametric and nonparametric esti-

mators. To appear in the handbook of econometrics, Vol. 5. See http://athena.sas.upenn.edu/�(petra/wbank/curver9.pdf.

Johnson, L. L. (1960). The theory of hedging and speculation in commodity futures.Review of Economic Studies, 27, 139–151.

LeRoy, S. F. (1989). Efficient capital markets and martingales. Journal of EconomicLiterature, 27, 1583–1621.

Li, W. (1996). Asymptotic equivalence of average derivatives. Economics Letters, 52,241–245.

Lien, D.-H. D., Tse, Y. K., & Tsui, A. K. (2002). Evaluating the hedging performance ofthe constant-correlation GARCH model. Applied Financial Economics, 12,791–798.

McCurdy, T. H., & Morgan, I. G . (1988). Testing the martingale hypothesis inDeutsche Mark Futures with models specifying the form of heteroscedasticity.Journal of Applied Econometrics, 3, 187–202.

Miffre, J. (2002). Economic significance of the predictable movements in futuresreturns. Economic Notes, 31, 125–142.

Pagan, A., & Ullah, A. (1999). Nonparametric econometrics. Cambridge: CambridgeUniversity Press.

Politis, D. N., & Romano, J. P. (1994). The stationary bootstrap. Journal of theAmerican Statistical Association, 89, 1303–1313.

Samuelson, P. A. (1965). Proof that properly anticipated prices fluctuate randomly.Industrial Management Review, 6, 41–49.

Silverman, B. (1986). Density estimation for statistics and data analysis. London:Chapman & Hall

Stein, J. L. (1961). The simultaneous determination of spot and futures prices.American Economic Review, 51, 1012–1026.

Stoker, T. (1986). Consistent estimator of scaled coefficients. Econometrica, 54,1461–1481.

Stoker, T. (1993). Smoothing bias in density derivative estimation. Journal of theAmerican Statistical Association, 88, 855–863.



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Testing the martingale hypothesis for futures prices: Implications for hedgers