fractional integration in commodity futures returns

The Financial Review 44 (2009) 583--602

Fractional Integration in CommodityFutures Returns

John Elder∗Colorado State University

Hyun J. JinChung-Ang University

Abstract

We reexamine commodity futures returns for evidence of fractional integration utilizingtwo estimators based on wavelets. We summarize basic wavelet methods for signal processingand decompose commodity futures returns by wavelet scale. We find the evidence for longmemory is not conclusive based on visual inspection of the wavelet decomposition, but formalstatistical tests suggest evidence of long memory, in the form of antipersistence, in about halfof agricultural commodity futures. We find little evidence of long memory in metal futures.Our results are useful in interpreting previous disparate findings based on frequency domainestimators.

Keywords: futures returns, fractional integration, long memory, wavelets

JEL Classifications: G10, Q14

∗Corresponding author: Department of Finance & Real Estate, 1272 Campus Delivery, Colorado StateUniversity, Fort Collins, CO 80523; Phone: 970-491-2952; E-mail: [email protected].

The authors thank an anonymous referee for very useful comments that improved the paper significantly,as well as Jonathan Dark, Mark Jensen and session participants at the 2006 FMA European meetings inStockholm. We also thank Roman Assilbekov and Arundhati Ghosh for very capable research assistance.

C© 2009, The Eastern Finance Association 583

584 J. Elder and H. J. Jin/The Financial Review 44 (2009) 583–602

1. Introduction

We present two relatively new methods for detecting fractional integration basedon wavelets and apply these methods to a very long sample of commodity futuresreturns. The wavelet-based estimators are appropriate for this analysis because theyare superior to popular frequency domain estimators on the basis of mean-squarederror (MSE), yet they do not impose parametric restrictions on short-term dynamics.

Time series models that incorporate fractional integration can be viewed as a gen-eralization of the usual autoregressive integrated moving average (ARIMA) model.In particular, fractional differencing utilizes series expansion to define the differ-encing operator for noninteger values. An important characteristic of fractionallyintegrated processes is that they can be covariance stationary while also displayingconsiderable persistence—that is, the effects of a shock to such a process decaysvery slowly relative to nonintegrated processes. In this sense, fractional integratedprocesses are correlated at very long lags and are said to exhibit long memory, orlong-range dependence. Hence, fractional differencing can be viewed as a method toparsimoniously model the persistence observed in many financial time series (e.g.,Lo, 1991; Campbell, Lo, and MacKinlay, 1997).

Evidence on fractional integrating dynamics in commodity futures is relevantfor statistical tests of asset pricing models, forecasting, hedging, and related riskmanagement techniques (cf. In and Kim, 2006; Dark, 2007). For example, Dark(2007) shows that, in the presence of long memory, standard approaches to estimatingdynamic minimum variance hedge ratios fails to account for basis convergence, andthat hedging performance can be improved by appropriate modeling of long memorydynamics.

Common methods for estimating the fractional differencing parameter are basedon frequency domain analysis and include the method proposed by Geweke and Porter-Hudak, 1983 (hereafter GPH). The GPH estimator is motivated by the property thatthe spectral density of a long memory processes is infinite at frequency zero and issimply the slope of the sample log periodogram. Comparable estimators of the longmemory parameter can be derived based on wavelet analysis. Wavelet analysis, likefrequency analysis, can be used to decompose the variance of a process into “scales,”separating variation in a process due to small scales (i.e., high frequencies) fromvariation due to large scales (i.e., low frequencies). Wavelet analysis is, however,more flexible in the sense that it can be used to decompose a process by both scaleand time so it is possible, for example, to analyze the large scale variation in a processat a particular point in time. In a more formal context, the variance of the waveletcoefficients at different scales is, for a fractionally integrated process, a function ofthe fractional differencing parameter. The relation allows for a scheme to estimatethe fractional differencing parameter based on wavelet decomposition.

Estimators of the fractional differencing parameter based on wavelets are de-veloped by Wornell and Oppenheim (1992), McCoy and Walden (1996), and Jensen(1999, 2000). Jensen (1999, 2000) shows that wavelet estimators have lower MSE

J. Elder and H. J. Jin/The Financial Review 44 (2009) 583–602 585

than other long memory estimators, including the popular GPH estimator. We utilizetwo wavelet-based estimators to reexamine the evidence of long memory in commod-ity futures, and we provide a wavelet decomposition of commodity futures returns.Our results provide evidence of fractional integration, in the form of antipersistence,in about half of agricultural commodity futures, but little evidence of fractional inte-gration in metal futures.

Wavelet analysis is relatively new to financial economics, but there is growinginterest in applications of wavelet analysis, and it has been used to analyze financialdata in several different contexts. For example, In and Kim (2006) utilize waveletanalysis to examine correlation, lead-lag relation, and hedge ratios associated withstock index returns and stock futures, finding that the wavelet correlation and theeffectiveness of hedging strategies varies with a waved-based time-scale decompo-sition. Several papers, including Fernandez (2005, 2006) and Fernandez and Lucey(2007), utilize wavelet analysis to decompose value at risk by both time and scale.Fernandez (2005) applies this method, in the context of an international capital-assetpricing model, to returns in seven emerging markets in Latin America and Asia,finding that value at risk varies with time-scale. Fernandez (2006) and Fernandezand Lucey (2007) also utilize wavelet analysis to decompose value at risk by timeand scale, while the latter study also utilizes wavelets to estimate the degree of frac-tional differencing. Sibbertsen (2004) utilizes wavelet analysis to estimate the degreeof fractional differencing in the volatility of German stock returns. Kyaw, Los, andZong (2006) also utilize wavelet analysis to estimate the degree of fractional differ-encing (or, equivalently, the Hurst exponent) in Latin American stock and currencyreturns, and to conduct a wavelet-based multi-resolution analysis of the sample ofstock returns. As other examples, Los and Yu (2008) utilize wavelets to estimatedependence in Chinese stock markets, finding evidence of greater persistence priorto financial deregulation; and Karuppiah and Los (2005) utilize wavelets to analyzedependence in Asian foreign exchange rates, finding evidence of dependence in theform of antipersistence.

2. Fractional integration in futures

There is an extensive literature on long memory in financial returns (cf. Lo, 1991;Campbell, Lo, and MacKinlay, 1997), and there is growing academic interest in thetime series properties of commodity futures (cf. Adrangi and Chatrath, 2008), but thereis little empirical evidence on long memory in commodity futures. Two exceptionsare Barkoulas, Labys, and Onochie (1999) and Crato and Ray (2000). Barkoulas,Labys, and Onochie (1999) examine 17 agricultural commodity futures return seriesfor evidence of long memory by utilizing the method of GPH to estimate the fractionaldifferencing parameter. These authors find strong evidence of fractional integrationin five of the commodity futures returns and weaker evidence of long memory in anadditional four commodity futures.

In contrast, Crato and Ray (2000) examine 17 commodity futures returns utiliz-ing three different methods: the modified R/S statistic (Lo, 1991), the GPH estimator,


and the nonparametric spectral test of Lobato and Robinson (1998). Using these meth-ods, Crato and Ray (2000) find little or no evidence of long memory in commodityfutures returns. It is difficult to reconcile these disparate results based on the reportedestimates, but it is possible that findings for the GPH estimator may be driven, inpart, by the size of the estimation window. Crato and Ray (2000) use a substantiallyshorter estimation window than Barkoulas, Labys, and Onochie (1999), which tendsto improve bias but also decreases precision. These disparate estimates leave unre-solved, however, the issue of whether commodity futures returns display evidence oflong memory and so a reexamination utilizing more recently developed methods iswarranted.

Barkoulas, Labys, and Onochie (1999) and other studies also examine the prop-erties of metal futures. Panas (2001) tests for evidence of long memory and chaosin six metal futures contracts that trade on the London Metal Exchange and findsevidence of long memory in two of the six contracts. Other related papers includeLabys, Lesourd, and Badillo (1998), who test for evidence of cyclicality in the pricesof nine metals, and Tully and Lucey (2007), who relate gold price volatility to macroe-conomic factors in the form of a power GARCH model. Finally, Souza, Tabak, andCajueiro (2008) examine fed funds futures for evidence of long memory.

3. Methodology

In this section, we review important existing results related to fractional differ-encing and semiparametric estimators of the fractional differencing parameter.

3.1. Fractional integration

Long memory in financial time series is often derived in terms of fractional dif-ferencing. The standard representation of an autoregressive moving-average processintegrated of order d, denoted ARIMA(p, d, q) is

�(L)x(t) = (1 − L)−dφ(L)u(t), (1)

where u(t) is an independently and identically distributed (i.i.d.) random variable withzero mean and variance σ 2

u ; L denotes the lag operator; and �(L) and φ(L) denotefinite polynomials in the lag operator with roots outside the unit circle. For d = 0, theprocess is stationary, and the effect of a shock to u(t) on x(t + j) decays geometrically.For d = 1, the process has a unit root, and the effect of a shock to u(t) on x(t + j)persists indefinitely.

Fractional differencing defines the polynomial in the lag operator (1 − L)−d

for noninteger values of d. Hosking (1981) and Granger and Joyeux (1980) derive adefinition of (1 − L)−d based on power series expansion

(1 − L)−d = 1 + d L + 1

2!d(1 + d)L2 + 1

3!d(1 + d)(2 + d)L3 + · · · , (2)


which is commonly represented in terms of the gamma function as

(1 − L)−d = 1 +∞∑j=1

�( j + d)

�(d)�( j + 1)L j . (3)

For fractional values of d such that |d |≤ 0.5, the process x(t) is both stationaryand invertible, but x(t) also has the important property that the effect of a shocku(t) on x(t+j) tends to decay relatively slowly. The rate of decay is often stated interms of the autocorrelation function, which is said to decay geometrically for zero-integrated processes and hyperbolically for fractionally integrated processes. Sincethe sign of the autocorrelations depends on d, the process x(t) is often described aseither persistent (for 0 < d ≤ 0.5) or antipersistent (for −0.5 ≤ d < 0). For values ofd such that 0.5 ≤ d < 1, the process exhibits long memory, but with infinite variance.

3.2. Estimators of the fractional differencing parameter

We consider two wavelet-based estimators of the fractional differencing param-eter d and one frequency domain estimator. These wavelet-based estimators are thewavelet ordinary least square (OLS) estimator described by Jensen (1999) and thebanded wavelet maximum likelihood estimator (MLE) of Jensen (2000) and Wornelland Oppenheim (1992).

These estimators are, in some sense, comparable to the popular frequency do-main estimators for the fractional differencing parameter. That is, frequency domainestimators, such as the GPH estimator, exploit the property that fractionally integratedprocesses with 0 < d ≤ 0.5 have very high variance at low frequencies. Similarly, thewavelet estimators exploit the property that fractionally integrated processes with 0 <d ≤ 0.5 have disproportionately high wavelet variance (actually, the metric is relatedto variance) at large scales, where wavelet scale is inversely related to frequency.

For example, an AR(1) process with �1 = 0.9 has positive autocorrelationsthat decline geometrically to zero. In the time domain, a plot of such a process ismuch smoother than white noise, and its spectral density would indicate that a largeproportion of its variance is attributed to low-frequency components, as illustratedin Figure 1. For a fractionally differenced process with 0 < d ≤ 0.5, an extremelylarge portion of the variance is due to very low-frequency components—so much sothat the spectral density is infinite at frequency zero. Similarly, the sample spectraldensity of a fractionally differenced process with—0.5 ≤ d < 0 is zero at frequencyzero. Frequency domain estimators exploit these properties by capturing the slope ofthe (log) sample spectral density in the neighborhood of the origin.

Wavelet analysis captures a somewhat similar property. In particular, waveletestimators exploit the property that the variance of the wavelet coefficients dependson the fractional differencing parameter, and a wavelet decomposition of such aprocess would tend to indicate greater (normalized) energy at larger scales.

A wavelet decomposition by scale, however, has important advantages overa Fourier decomposition by frequency. For example, the Fourier analysis does not


Figure 1

Spectral density for AR(1) process

This figure plots the spectral density of the AR(1) process xt = 0.9 xt−1 + εt−1 where εt ∼ i.i.d. N(0,1).

parsimoniously capture abrupt spikes in a signal, due to the smooth, periodic basisfunctions underlying the Fourier analysis. Wavelet basis functions, however, havefinite oscillations over compact support that are then scaled in size and translated intime to represent the original signal.

For example, let ψ(t) represent a wavelet function as described in the appendix,and let ϕ(t) represent the associated orthogonal scaling function. This formulationdefines the wavelet function as spanning the spaces between that, which is spanned bythe various scales of the scaling function. The wavelet coefficients [am,n] that link thewavelet function to the signal can be obtained by projecting the wavelet and scalingbasis functions onto the signal x(t) in the following manner

x(t) =∑

n

cnϕ(t − n) +∑

m

∑n

am.n2−m/2ψ(2−mt − n), (4)

where m and n are integer indices for the finite or infinite sum. The scaling coefficientis proportional to the mean of the series. The integers m and n scale the wavelet in size(m) and translate the wavelet in time (n), localizing the underlying process in bothtime and frequency. In contrast, the Fourier representation localizes in frequency butnot in time.

An important property of the wavelet transform is that it decorrelates the originaltime series across scale and time. The decorrelation allows the variance of the originaltime series to be decomposed across each scale m so that

var(x) = 1

T

M∑m=1

E(m), (5)


where the energy E(m) = ∑2(M−m)−1n=0 a2

m.n is the sum of the squared wavelet coeffi-cients at scale index m. The decomposition is especially relevant for the analysis oflong memory processes, as it will be shown (below) that for such processes the energyper wavelet coefficient is greater at large scales (low frequencies). The energy perwavelet coefficient at scale m can also be interpreted as the sample variance of thewavelet coefficients at scale m.

A very simple example of a wavelet is the Haar wavelet (cf. Rao and Bopardikar,1998), defined as the step function

ψ(t) =

1 for 0 ≤ t < 0.5

−1 for 0.5 ≤ t < 1

0 otherwise

. (6)

Another wavelet, commonly applied in geophysical applications, is the Mexi-can hat wavelet, which is simply the second derivative of the (normalized) Gaussiandistribution function. A plot of the Mexican hat wavelet is presented in Figure 2.Note, in particular, the finite energy and finite oscillations of the wavelet. As indi-cated above, the wavelet is combined with permutations that are scaled (i.e., eitherstretched or compressed) and translated (i.e., shifted in time) to represent the originalsignal. Other wavelets, however, have more desirable properties for time series data,such as improved frequency localization and the ability to represent continuous sig-nals more parsimoniously. These include the Daubechies (1988) family of wavelets,which cannot be expressed in a closed form algebraic expression. Due to its desirable

Mexican Hat Wavelet

-0.3

Duration

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-5 -4.2

-3.4

-2.6

-1.8 -1 -0

.2 0.6 1.4 2.23

3.8 4.6

Figure 2

Mexican hat wavelet


properties, the Daubechies wavelet is commonly utilized in signal processing as wellas econometrics (cf. Jensen, 1999, 2000; Tkacz, 2001). As such, we also utilize theDaubechies-8 wavelet for our analysis.

We use two wavelet-based estimators of the fractional differencing parameter:the wavelet OLS estimator of Jensen (1999) and the banded wavelet MLE of Jensen(2000) and Wornell and Oppenheim (1992). The wavelet OLS estimator is, in somesense, analogous to the GPH estimator. That is, the GPH estimator is simply a re-gression of the log sample spectrum on the log frequency, while the wavelet OLSestimator is a regression of the (normalized) log wavelet scale spectrum on the logscale.

These estimators are derived from the property that the wavelet coefficientsam,n associated with an ARIMA(p, d, q) process x(t) with |d| ≤ 0.5 are distributedapproximately N(0, σ 22−2d(M−m)). The MLEs exploit the normality of the waveletcoefficients, while the wavelet OLS estimator exploits the log linear relation betweenthe wavelet variance and the fractional differencing parameter. Denote the variance ofthe wavelet coefficients at scale m by var(am,.) = σ 22−2d(M−m). In particular, takinglogs gives

ln[var(am,.)] = ln[σ 2] + d ln[2−2(M−m)]. (7)

The fractional differencing parameter d can be estimated by performing OLSon Equation (7), where var(am, .) is estimated by the sample variance of the waveletcoefficients at scale index m, which is equivalent to the wavelet energy at scale indexm normalized by the number of wavelet coefficients ˆvar(am,.) = 1

2M−m E(m). Referringto Equation (5), this is equivalent to the variance of x at scale index m, multiplied by2m, with T = 2M .

In practice, Percival and Walden (2000) suggest trimming the largest and smallestscales prior to estimating Equation (7) for the wavelet OLS estimator. The rationalefor trimming the lowest scales is that the approximation given by Equation (7) isleast accurate at small scales (high frequencies). Such trimming also ensures thelower bias due to short-run dynamics. The rationale for trimming the highest scalesis that the energy at the highest scale is estimated imprecisely—from a single waveletcoefficient. For the wavelet MLE, some authors suggest trimming only the smallscales.

Jensen (2000) and Wornell and Oppenheim (1992) derive an approximatewavelet MLE of the fractional differencing parameter d that draws on the multi-variate normality of the wavelet coefficients of the data generating process. That is,Jensen (2000) proposes imposing the constraint that cov(am,n aj,k) = 0. He shows thatthe true covariance matrix is relatively sparse so the constraint may be reasonablein practice. Since this estimator approximates the sparse covariance matrix by itsdiagonal elements, Jensen (2000) describes this estimator as a banded wavelet MLE.

If we let σ 2m,n = σ 22−2(M−m)d , then the approximate likelihood of the data as a

function of the parameters and wavelet coefficients can then be written simply as


L(d , σ 2, σ 2

η

) =∏

m

∏n

1√2πσ 2

m,n

exp

{− a2

m,n

2σ 2m,n

}. (8)

We denote the estimate for the fractional differencing parameter obtained bymaximizing the associated log-likelihood as d̂BWMLE.

The wavelet estimators have desirable properties. In particular, Jensen (1999)shows that the wavelet OLS estimator d̂WOLS is consistent, and that the MSE of thewavelet estimator is approximately four to six times smaller than the MSE of theGPH estimator at relevant sample sizes for an ARIMA(0, d, 0) process. The biasof the wavelet OLS estimator, however, tends to be larger than the bias of the GPHestimator. The bias of the wavelet estimator also tends to be negative, while the biasof the GPH estimator is most often positive. The results of Jensen (2000) indicate thatthe wavelet OLS estimator is also valid for more general ARIMA(p, d, q) processes.

With regard to the banded wavelet estimator, Jensen (2000) shows that it hasa lower MSE than the GPH estimator; and that the banded wavelet estimator has alower MSE than an exact MLE for ARIMA(1, d, 0) processes and ARIMA(0, d, 1)processes except for moving average parameters near one. The exact MLE generallyrequires that the short-term dynamics be correctly specified in order for the bias tonot become large, while the bias of the banded wavelet MLE does not depend on thespecification of the short-memory parameters.

The banded wavelet MLE is actually very similar to a wavelet MLE proposedby McCoy and Walden (1996). The likelihood function for the McCoy and Waldenestimator, however, includes the wavelet scaling coefficient and is applicable only toprocesses that are mean zero.

4. Data

Our data consists of 15 commodity futures price series, including six grains(corn, oats, soybeans, soybean meal, soybean oil, and wheat), three soft commodities(cocoa, coffee, and sugar), three meats (live beef cattle, lean hogs, and pork bellies),and three metals (high-grade copper, platinum, and silver). The grains are tradedon the Chicago Board of Trade, the soft commodities are traded on the New YorkBoard of Trade, the meats are traded on the Chicago Mercantile Exchange, and themetals are traded on the New York Mercantile Exchange. The series consist of dailysettlement prices from March 12, 1974 through December, 29 2006, yielding a verylong series of 8,192 daily observations over more than 30 years. Such a long sampleshould facilitate the detection of long memory dynamics.

The futures prices are nearby futures prices, with the price rolled incrementallyto the successive nearby contract over a five-day period, beginning 30 trading daysprior to expiration. In particular, we roll the price to the next contract over five tradingdays by taking a weighted average of the closing prices on the two contracts, with theweights increasing incrementally over these five days. For example, on the first day of


the roll, we calculate the futures price as 80% from the expiring contract and 20% fromthe next nearby contract. On the second day of the roll, we calculate the futures price as60% from the expiring contract and 40% from the next nearby contract. We continuethis pattern for three more days until 100% weight is shifted to the new contract. Thismethod has several desirable features. First, by beginning the roll 30 days prior toexpiration, we tend to use prices on the contract with the greatest trading volume,as volume typically tapers dramatically in the 30 days prior to expiration, especiallyonce the long position might have to accept delivery. Second, by incrementally rollingto the new contract, we avoid price spikes associated with maturity effects. The rollmethod is similar, for example, to that used in the calculation of the S&P GoldmanSachs Commodity Index.1 Continuously compounded daily returns in basis points,x(t), are then derived from the futures prices, p(t), for all commodities as x(t) ≡10,000 ∗ ln(p(t)/p(t−1)).

Variable names and summary statistics on the daily commodity returns are pre-sented in Table 1. Each series displays considerable volatility, as the standard deviationfor each series is several orders of magnitude larger than the unconditional mean. Sixof the 15 series have significantly negative skewness while each of the series has kur-tosis in excess of Gaussian. Each series also displays some evidence of autocorrelationat long lags, with Ljung-Box Q(20) statistics significant at the usual levels. Finally,we also examine the return series for evidence of a unit root using augmented Dickey-Fuller tests according the procedure advocated by Elder and Kennedy (2001a, 2001b),rejecting the null of a unit root for all commodity returns at conventional significancelevels.

The futures prices are plotted in Figures 3 and 4. Grain prices appear to sharesome common features throughout most of the sample, except for Wheat. Corn, Oats,and Wheat prices each rose in 2006, presumably associated with increased demand forethanol. The soft commodities (Coffee, Cocoa, Sugar) appear not to share commonfactors, with each exhibiting price spikes at different times. Sugar, in particular,spiked in the mid 1970s and early 1980s, with a smaller increase in 2006. Meat prices(FeedCattle, LeanHogs, PorkBellies) were very volatile throughout the sample. Pricesof each of the metals (Platinum, Copper, Silver) have increased considerably since2004, although the price increase for Silver is dwarfed by the crisis in 1979.

5. Results

The statistical properties of futures returns can be analyzed by a wavelet decom-position of the energy of each return series into its component scales, as in Equation(5). Deflating the energy at each scale by the number of wavelet coefficients at that

1 We thank an anonymous referee for suggesting an incremental roll procedure. An alternative roll methodsuggested by Spurgin (1999) continuously averages the two nearby contracts, which facilitates detectionof underlying trends.


Table 1

Summary statistics

This table presents summary statistics on commodity future returns in basis points per day continuouslycompounded. The sample is March 14, 1974 through December 31, 2006. Returns are calculated fromfutures prices on nearby contracts, rolled incrementally to the subsequent contract 30 trading days prior toexpiration. Q(20) represents the Ljung-Box portmanteau test statistics for up to the 20th order autocorre-lation.

Q(20) onSeries Mean Std Dev Max Min Skew Kurtosis returns

GrainsCorn 0.32 131.34 660.36 −703.55 0.04 1.88∗∗ 103.22∗∗Oats 0.79 180.62 1,110.20 −1,193.67 −0.03 1.65∗∗ 88.42∗∗Soybean 0.11 145.12 672.90 −691.60 −0.14∗∗ 1.42∗∗ 48.03∗∗SoyMeal 0.26 161.73 857.27 −853.97 0.08∗∗ 2.80∗∗ 58.97∗∗SoyOil 0.18 160.41 808.04 −829.66 0.07∗∗ 0.97∗∗ 48.05∗∗Wheat −0.05 148.60 793.54 −611.60 0.14∗∗ 1.69∗∗ 40.63∗∗

Soft CommoditiesCocoa 1.12 194.63 2,172.85 −1,000.59 0.45∗∗ 5.11∗∗ 69.81∗∗Coffee 0.68 221.54 2,323.13 −1,503.09 0.20∗∗ 7.38∗∗ 45.35∗∗Sugar −0.69 245.30 1,531.71 −1,821.83 −0.09∗∗ 2.58∗∗ 52.01∗∗

MeatsLiveCattle 0.82 109.40 417.07 −939.99 −0.17∗∗ 1.39∗∗ 147.81∗∗LeanHogs 0.60 164.97 883.18 −816.57 0.01 1.03∗∗ 192.28∗∗PorkBellies 0.70 224.05 1,566.21 −1,033.09 0.22∗∗ 1.06∗∗ 171.73∗∗

MetalsCopper 1.13 164.12 1,104.49 −1252.03 −0.25∗∗ 3.22∗∗ 46.73∗∗Platinum 1.96 163.92 1047.44 −762.70 −0.14∗∗ 2.68∗∗ 80.64∗∗Silver 1.10 182.17 923.55 −1,394.23 −0.35∗∗ 3.36∗∗ 114.00∗∗

∗∗ and ∗ indicate statistical significance at the 0.05 and 0.10 level, respectively.

scale yields the sample variance of the wavelet coefficients at that scale, that is,var(am,.) at each scale index m = 1, 2, . . . , 13. For a white noise process, the the-oretical wavelet coefficient variance at each scale is constant. For processes withlarge low-frequency variation, such as long memory processes with 0 > d ≥ 0.5, wewould expect high wavelet coefficient variance at large scales. For processes withlarge high-frequency variation, such as long memory processes with −0.5 ≤ d <0, we would expect low wavelet coefficient variation at large scales. The samplewavelet coefficient variances, calculated from the discrete wavelet transform withthe Daubechies wavelet, are plotted versus the scale index in Figures 5 and 6. Recallthat scale is 2m−1, so scale index 5, for example, represents var(a5,.), which is a scaleof 24 = 16 days.

These plots generally indicate very low wavelet coefficient variance at the largestscales. This feature is readily observable in each series, although it is less pronouncedin Cocoa, Platinum, and Silver. For each of the other series, however, the effect is verypronounced. The plots indicate relatively low wavelet coefficient variance at large


1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

100200300400500600

Corn

0

100

200

300

400Oats

400

600

800

1000

1200Soybeans

100150200250300350

Soymeal

10

20

30

40

50Soyoil

200300400500600700

Wheat

0

100

200

300

400Coffee

0

10000

20000

30000Cocoa

Figure 3

Futures prices for grains and soft commodities

This figure plots the U.S. dollar price of the indicated commodity futures.


0

25

50

75Sugar

25

50

75

100FeedCattle

20

40

60

80

100LeanHogs

25

50

75

100

125PorkBellies

0

500

1000

1500Platinum

0

100

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400Copper

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

0

1000

2000

3000

4000

5000Silver

Figure 4

Futures prices for soft commodities, meats, and metals

This figure plots the U.S. dollar price of the indicated commodity futures.


1 2 3 4 5 6 7 8 9 10 11 12 13

05

101520

Corn

1 2 3 4 5 6 7 8 9 10 11 12 13

05

101520

Oats

1 2 3 4 5 6 7 8 9 10 11 12 13

05

101520

Soybean

1 2 3 4 5 6 7 8 9 10 11 12 13

05

101520

SoyMeal

1 2 3 4 5 6 7 8 9 10 11 12 13

05

101520

SoyOil

1 2 3 4 5 6 7 8 9 10 11 12 13

05

101520

Wheat

1 2 3 4 5 6 7 8 9 10 11 12 13

05

101520

Coffee

1 2 3 4 5 6 7 8 9 10 11 12 13

05

101520

Cocoa

Figure 5

Wavelet coefficient variance for grains and soft commodities

This figure plots the proportional wavelet coefficient variance by scale index for the indicated commodityfutures.


1 2 3 4 5 6 7 8 9 10 11 12 13

05

101520

Sugar

1 2 3 4 5 6 7 8 9 10 11 12 13

05

101520

LiveCattle

1 2 3 4 5 6 7 8 9 10 11 12 13

05

101520

Hog

1 2 3 4 5 6 7 8 9 10 11 12 13

05

101520

PorkBellies

1 2 3 4 5 6 7 8 9 10 11 12 13

05

101520

Platinum

1 2 3 4 5 6 7 8 9 10 11 12 13

05

101520

Copper

1 2 3 4 5 6 7 8 9 10 11 12 13

05

101520

Silver

Figure 6

Wavelet coefficient variance for soft commodities, meats, and metals

This figure plots the proportional wavelet coefficient variance by scale index for the indicated commodityfutures.

scales indexes (i.e., low frequencies) and suggest that commodity future returns are“choppier” than white noise. Therefore, if commodity future returns display evidenceof long memory, it is likely in the form antipersistence. Any convincing evidence oflong memory, however, must rely on more formal statistical tests.

In particular, we are interested in testing the null hypotheses of no fractionaldifferencing (d = 0) versus the alternative of fractional differencing (d �= 0). Wetherefore estimate the fractional differencing parameter, d, for the realized return


Table 2

Estimates of fractional differencing parameter

The values in parentheses are asymptotic t-statistics for the null hypothesis d = 0. d̂GPH representspoint estimates based on the GPH estimator, d̂WOLS represents point estimates based on the wavelet OLSestimator of Jensen (1999), and d̂BWMLE represents point estimates based on the banded wavelet MLE ofJensen (2000).

d̂GPH d̂WOLS d̂BWMLE

Return series T0.45 Daub-8 Daub-8

GrainsCorn −0.124 −0.111 −0.172∗∗

(−1.77) (−1.45) (−2.01)Oats −0.061 −0.106∗∗ −0.109

(−0.65) (−2.15) (−1.49)Soybean −0.258∗∗ −0.087 −0.247∗∗

(−2.66) (−1.40) (−2.90)SoyMeal −0.309∗∗ −0.098 −0.242∗∗

(−3.87) (−1.59) (−2.86)SoyOil −0.196∗∗ −0.064 −0.142∗

(−1.99) (−1.23) (−1.63)Wheat −0.173 −0.048∗∗ −0.198∗∗

(−0.62) (−2.21) (−2.59)Soft Commodities

Cocoa −0.078 −0.075∗ −0.009(−0.62) (−1.84) (−0.10)

Coffee −0.010 −0.006 −0.035(−0.10) (−0.15) (−0.47)

Sugar 0.007 0.028 −0.068(0.07) (0.75) (−0.86)

MeatsFeedCattle −0.278∗∗ −0.160∗∗ −0.308∗∗

(−2.73) (−3.10) (−5.07)LeanHogs −0.344∗∗ −0.245∗∗ −0.452∗∗

(−3.11) (−2.89) (−5.16)PorkBellies −0.264∗∗ −0.225∗∗ −0.308∗∗

(−2.78) (−3.21) (−3.66)Metals

Copper 0.063 0.083∗∗ −0.046(0.56) (2.29) (−0.63)

Platinum 0.039 0.033 0.024(0.33) (0.74) (0.36)

Silver −0.033 0.004 −0.079(−0.38) (0.07) (−1.24)

∗∗ and ∗ indicate statistical significance at the 0.05 and 0.10 level, respectively.

series by the GPH estimator d̂GPH , the wavelet OLS estimator d̂OLS, and the bandedwavelet MLE d̂BWMLE using the Daubechies wavelet with eight smoothing parameters.The computations are performed in Gauss.

The parameter estimates are reported in Table 2, along with asymptotict-statistics. Consider first the estimates obtained from the GPH estimator with an


estimation window of T0.45. The null hypothesis is rejected for six of the 15 returnseries. These include the three grains related to soy (Soybean, SoyMeal, and Soy-Oil), where the null is rejected at the 0.05 level of significance, and the three meats(FeedCattle, LeanHogs, and PorkBellies). The point estimates in each case are lessthan zero, suggesting antipersistence in each of the return series. This implies thatthe autocorrelations decay very slowly, but are negative in sign. These results are incontrast with, for example, Crato and Ray (2000), who find no evidence of long mem-ory in 17 commodity return series based on the GPH estimator, the rescaled/rangestatistics, and a nonparametric estimator. Our results suggest that the greater powerof the estimators, due to our longer sample, may be substantial.

Consider next the point estimates obtained by the wavelet OLS estimator and thebanded wavelet MLE. As indicated previously, the wavelet estimators do not requirethat the underlying series be covariance stationary, and they are likely to be favoredon the basis of the Monte Carlo results reported by Jensen (1999, 2000). In addition,the wavelet estimators abstract from short-term dynamics due to their semiparametricnature, especially when the appropriate scales are trimmed.2

On the basis of the wavelet OLS estimator, a total of seven return series displaysome evidence of long memory, with the null of no fractional differencing rejectedat the 0.01 or 0.05 level. In addition to the three meats (FeedCattle, LeanHogs, andPorkBellies), these are Oats, Wheat and Copper. Given the lower MSE of the waveletOLS estimator reported by Jensen (1999), possibly it is not surprising to find additionalevidence of the long memory.

Finally, consider the banded wavelet MLE. The wavelet MLE utilizes more in-formation than the wavelet OLS estimator, and Jensen (2000) suggests that it is mostfavored on the basis of MSE. The banded wavelet MLE detects evidence of longmemory in eight futures return series, which is more than half of the observed sam-ple. In each of the eight instances, the estimate of the long memory parameter is againbetween 0 and –0.5, indicating evidence of antipersistence so that the autocorrelationsdecay very slowly, but are negative in sign. In fact, for each of the three estimators,all of the estimates that are significantly different from zero are negative and signif-icant, except the wavelet OLS estimate for Copper. Note in particular that the threemeats (FeedCattle, LeanHogs, and PorkBellies) display evidence of significant an-tipersistence based on all three estimators, while the soy-derived contracts (Soybean,SoyMeal, and SoyOil) display evidence of significant antipersistence based on boththe GPH and wavelet MLE. The wavelet OLS estimates for the soy-derived contractsare negative, but not significant.

Our results suggest that the price dynamics of two groups of commodity futures,meats and soy, may each be driven by a common factor, and the evidence of anti-persistence suggests that the price dynamics may be such that returns tend to overreact

2 We trim the two smallest scales and three largest scales from the wavelet OLS estimator, and the fivesmallest scales from the banded wavelet MLE (cf. Jensen, 2000; Percival and Walden, 2000).


to new information, which induces substantive periodic high-frequency variation. Asshown by Dark (2007) and Lien and Shrestha (2007), such results can be useful, forexample, in designing optimal hedges over long horizons. In particular, the standardapproaches to estimating dynamic minimum variance hedge ratios will fail to accountfor basis convergence under long memory dynamics.

In sum, three of 12 agricultural commodity futures (the meats) display evidenceof long memory by all three estimators, and another three (the soy-derived contracts)display evidence of long memory by two estimators. The three metal futures displaylittle evidence of long memory, which is generally consistent with Panas (2001). Toensure that our results are not sensitive to the particular wavelet chosen, we recal-culate the wavelet OLS and banded wavelet MLE estimates for the Daubechies(4),Daubechies(10), and Coiflet(4) wavelets. The estimates (not reported, but availablefrom the authors) are remarkably robust across the various wavelets. Our results there-fore provide additional evidence of long memory in commodity futures, somewhatsimilar to those obtained by Barkoulas, Labys, and Onochie (1999) although theyfind evidence of long memory in different contracts. Their result may be due, in part,to their long estimation window for the GPH estimator, which increases precision butalso increases bias. We find evidence of long memory with the GPH estimator basedon a shorter estimation window and a very long sample.

6. Conclusion

Previous empirical investigations of fractional differencing in commodity futureshas been inconclusive. Barkoulas, Labys, and Onochie (1999) find evidence of frac-tional differencing in about one-third of commodity futures returns, but Crato and Ray(2000) find no evidence of fractional differencing. Both studies utilize the commonGPH estimator, but with different estimation windows. We reexamine commodityfutures returns for evidence of long memory by applying relatively new wavelet-based estimators to a very long sample of over 30 years. In particular, we employa wavelet OLS estimator and a wavelet MLE, which are generally superior to thepopular GPH estimator and exact MLEs on the basis of MSE. We find little evidenceof fractional differencing in metal futures, but we find that about half of agriculturalcommodity futures exhibit some evidence of fractional differencing in the form ofantipersistence.

Appendix

The wavelet function ψ(t) in Equation (4) must satisfy a few relatively weakconditions. In particular, let ψ(t) be a real valued, square integrable function with finiteoscillations that decrease to zero as t → ±∞. That is, let ψ(t) satisfy

∫ ∞−∞ ψ(t)dt = 0

and∫ ∞−∞ ψ(t)2dt = 1. Then ψ(t) is a wavelet if it also satisfies the admissibility

condition described, for example, by Percival and Walden (2000). The wavelet is


scaled by integer m and translated in time by integer n

ψ(t)m,n = 2−m/2ψ(2−mt − n), (A1)

to form an orthonormal basis for a general set of function spaces that includes, forexample, the set of square integrable functions L2(�). The term 2−m/2 maintains thenormalization at different scales.

The wavelet transform given by Equation (4) applied to a discrete time se-ries can also be represented by a sequence of matrix transformations that producesT/2m (=2M−m) wavelet coefficients at scale 2m−1 for m = 1, . . . , M, where m is some-times referred to as the scale index. Each wavelet coefficient reflects the differencebetween a generalized average of 2m−1 observations sampled before and after timeintervals separated by length 2m. For example, for the scale index m = 1, the scale is21−1 = 1 and there are T/2 wavelet coefficients {a1,0, a1,1, . . . , a1,T/2−1}, with eachcoefficient reflecting the difference between a generalized average of one observationbefore and after t = 0.5, 2.5, 4.5, . . . . These small scale wavelet coefficients thereforereflect information at high frequencies. For scale index m = 2, the scale is 22−1 = 2and there are T/4 wavelet coefficients {a2,0, a2,1, . . . , a2,T/4−1}, with each coefficientreflecting the difference between a generalized average of two observations beforeand after t = 1.5, 5.5, 9.5, . . . . At the largest scale index M, the scale is 2M−1, and thereis one wavelet coefficient {aM, 0} reflecting the difference between a generalized av-erage of 2M−1 = T/2 observations before and after the middle observation. This largescale wavelet coefficient reflects information at the lowest frequency. The discretewavelet transform also produces one scaling coefficient c that reflects information atscales greater than 2M−1. In this case, the scaling coefficient is simply proportionalto the mean of x(t).

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