Commodity futures and market efficiency: A fractional integrated approach

Resources Policy 35 (2010) 276–282

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Commodity futures and market efficiency: A fractional integrated approach Viviana Fernandez n Mining Center, School of Engineering at Pontificia Universidad Catolica de Chile, Trinity College Dublin, Avda. Vicuna Mackenna 4860, Santiago, Chile

a r t i c l e in f o

JEL classification: G13 C22 Keywords: Fractional integration Efficiency market hypothesis

a b s t r a c t In financial time series, persistence or inertia is a feature usually observable in absolute returns, i.e., a proxy for volatility. Moreover, asset return series should be essentially unpredictable according to the efficiency market hypothesis (EMH) in its weak form. Surprisingly, recent literature has found evidence of anti-persistence in technology stocks and commodity futures returns. Anti-persistence would be indicative of an overreaction of asset prices to incoming information. In this article, we concentrate on a sample of 20 DJ-AIG commodity future indices—including broad indices and sub-indices (e.g., energy, grains, industrial metals, and livestock) over the period January 1991–June 2008. We conclude that returns series either over-react or under-react to new market information, which disconfirms the EMH in its weak form. Such disconfirmation would make it possible for market participants to devise non-linear statistical models for improved index forecasting and derivatives valuation. & 2010 Elsevier Ltd. All rights reserved.

Introduction Lagged effects or persistence in time series has been the center of attention of various studies over the past two decades. Seminal work by Granger and Joyeux (1980) and Hosking (1981) characterized fractionally integrated processes, which include highly persistent but stationary processes. Such characterization made it possible to establish a bridge between scaling, a phenomenon extensively studied in physics, hydrology, and other sciences (e.g., Peng et al., 1994; Chen et al., 2002; Chamoli et al., 2007), and the long-memory or persistence feature observable in various economic and financial series. In particular, persistence in asset returns and asset volatility has drawn the attention of several recent studies in the fields of financial econometrics and econophysics (e.g., Pasquini and Serva, 1999; Barkoulas et al., 2000; Carbone et al., 2004; Mills, 2004; Mulligan, 2004; Connor and Rossiter, 2005; Fernandez, forthcoming; Ane´ and Ureche-Rangau, 2008; Los and Yu, 2008; Elder and Jin, 2009).1 In particular, a recent article by Elder and Jin (2009) found evidence of anti-persistence in grain and meat commodity futures returns over the period 1974–2006. The authors conclude that their findings imply that some commodity futures returns, such as

n

Tel.: + 562 686 5895; fax: + 562 686 5805. E-mail address: [email protected] 1 For an analysis of persistence in non-linear time series (e.g., logistic smooth transition autoregressive (LSTAR) models, exponential smooth transition autoregressive (ESTAR) models, and self-exciting threshold autoregressive (SETAR) models), see Kapetanios (2007). 0301-4207/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.resourpol.2010.07.003

soybeans, wheat, and lean hogs, are choppier than white noise, and, therefore, their price dynamics may be such that their returns over-react to incoming information. As a result, commodity futures returns would be subject to considerable periodic highfrequency variation. The anti-persistence phenomenon in financial time series was previously documented by Mulligan (2004) for a sample of 54 technology securities over the period 1993–2001. Mulligan found that various return series in his sample exhibited such a feature, being AT&T, Cisco Systems, Dell Computer, and Time Warner Telecom examples of well-known firms. Mulligan also found evidence of persistence in a few return series, such as Intel, Advance Micro Devices, and Intraware. Persistence in asset returns was also documented by Barkoulas et al. (2000), who analyzed the Greek stock market during the 1980s and the early 1990s. Various statistical methods aimed at quantifying the degree of fractional integration have been devised over time in various fields of knowledge. A thorough discussion on the mean squarederror, size, and power features of some of such methods (e.g., several variants of the rescaled range statistic (R/S), detrended fluctuation analysis (DFA), wavelets, quasi maximum likelihood, among others) can be found in three recent articles by Mielniczuk and Wojyllo (2007), Rea et al. (forthcoming), and Fernandez (forthcoming). The estimation method utilized to gauge fractional integration may be a key factor when drawing conclusions as to the degree of persistence/anti-persistence exhibited by a time series. For instance, in Mulligan’s sample, R/S analysis lends much support to the anti-persistence hypothesis than the variogram method

V. Fernandez / Resources Policy 35 (2010) 276–282

does.2 Elder and Jin’s computations in turn are based on Geweke and Porter-Hudak (1983)’s semi-parametric approach (GPH), Jensen (1999)’s wavelet ordinary least-square estimator (WOLS), and Jensen (2000)’s banded wavelet maximum-likelihood estimator (BWMLE). The GPH and BWMLE approaches lend support to the anti-persistence phenomenon in about the same futures commodity series, and they yield numerically similar estimates. The WOLS estimate by contrast tends to predict a smaller degree of anti-persistence, and it detects anti-persistence in a couple of futures commodities, for which neither the GPH nor the BWMLE approach finds evidence of such a feature. The use of fractionally integrated processes has not been limited to in-sample modeling of time series data. Indeed, a strand of the literature has compared the forecast properties of fractional integrated and autoregressive processes, concluding that the former may outperform the latter. For instance, Barkoulas and Baum (2006) focused on US monetary indices and concluded that these series have a fractional order between one and two. When carrying out out-of-sample forecasting, Barkoulas and Baum found that the fitted fractional processes generally yielded consistently more accurate forecasts than benchmark autoregressive processes, as measured by their root-mean squared error. On the other hand, for some of the series analyzed, the authors concluded that the fractional specification encompassed the autoregressive one. An earlier article by Barkoulas et al. (2000) also concluded that a fractional specification may have a better out-of-sample performance than autoregressive and random walk specifications. In this article, we focus on five alternative estimators to gauge fractional integration: a weighted wavelet-based estimator (e.g., Mielniczuk and Wojyllo, 2007; Fernandez, forthcoming), two periodogram-based estimators, obtained by ordinary least squares and least absolute deviations (e.g., Taqqu et al., 1995), GPH’s estimator, and a quasi maximum-likelihood estimator obtained by Haslett–Raftery’s method (1989). Our center of attention is a set of 20 commodity series belonging to the Dow Jones-AIG commodity index family, which includes 5 broad categories—energy, grains, industrial metals, livestock, and precious metals, and 15 sub-indexes. The sample period is January 1991–June 2008. In order to allow for time-variant estimates of fractional integration for each series, we construct rolling estimates based on the above-mentioned methods. In general, our findings show that absolute returns are highly persistent. On the other hand, some returns series may exhibit either anti-persistence or persistence, which challenges the efficiency market hypothesis (EMH) in its weak form. However, the degree of persistence (anti-persistence) detected in the series may be sensitive to the estimation method under consideration. For instance, GPH’s estimator tends to be more conservative than the other four approaches, as regards to the strength of anti-persistence/persistence detected. Nonetheless, the alternative estimation methods considered tend to agree on the sign of fractional integration of some specific broad indexes and sub-indexes. Specifically, four out of the five methods support the existence of anti-persistence in the broad categories of Precious metals. On the other hand, the five methods agree upon the anti-persistence of returns on the Gold and Silver sub-indexes, while they all support the existence of persistence in natural gas, lean hogs, and corn returns.3 P The sample variogram of a time series yt is measured as V(D) ¼ t(yt + D  yt)2/ N, where N is the number of squared differences. The Hurst exponent, H, which equals the persistence parameter d plus 0.5, is computed by a regression (in logs) from the relationship V(D) ¼ D2H. 3 The R/S statistic is not utilized in this article because it becomes very computationally intensive when working with a rolling sample window. 2

277

The contribution of our work is two-fold. Firstly, unlike previous research, we rely upon a dynamic estimator of fractional integration, which allows us to unveil whether one single commodity may adjust too rapidly or too slowly to the arrival of new information, depending upon the time period under consideration. Such a feature cannot certainly be perceived when one relies upon a static estimator. Secondly, by resorting to such a dynamic estimation strategy, our conclusions differ from those found in recent research. For instance, under the three estimation methods that Elder and Jin (2009) use, they find evidence of antipersistence in Lean Hogs returns, whereas we conclude the opposite. In addition, they find some evidence of anti-persistence in corn returns, while our five estimation methods indicate otherwise. This article is organized as follows. Section 2 presents the statistical tools utilized in this study, namely, five alternative methodologies to gauge fractional integration. Section 3 is divided into two subsections: Section 3.1, which describes the data, and Section 3.2, which discusses the empirical findings. Finally, Section 4 presents a summary of the main findings.

Theoretical background Long-memory process A time series yt is a long-memory process or exhibits longrange dependence if its autocovariance function declines hyperbolically to zero. Two independent articles, by Granger and Joyeux (1980) and Hosking (1981), showed that a long-memory process can be parameterized by means of a fractionally integrated process, (1 L)d(yt  Z)¼ et, where L is the lag operator, d is the fractional difference parameter, Z is the expected value of yt and et is a zero-mean and short-memory error term. The fractional difference filter is defined by 1   X d ð1LÞd ¼ ð1Þk Lk k k¼0 where d is a real number, such that d4  1,   d d! Gðd þ1Þ ¼ ¼ k!ðdkÞ! Gðk þ1ÞGðdkþ 1Þ k and G(.) is the gamma function, such that yt admits an AR(N) representation. When 9d941/2, yt is non-stationary; when 0 odo1/2, yt is stationary and it exhibits long-memory; whereas if 1/2od o0, yt is stationary and it displays short memory (i.e., anti-persistence). Alternatives methods to estimate the fractional integration parameter Wavelet-based Veitch and Abry (1999) developed a weighted least-squares estimator of d from a wavelet-based decomposition of the time series of interest. In particular, if u2y ðtj Þ represents the wavelet variance of a time series yt at scale tj  2j  1, a discrete wavelet transform (DWT)-based estimator of it is given by u^ 2y ðtj Þ 

nj 1 X d2 nj k ¼ 1 j,k

ð1Þ

where nj is the number of wavelet coefficients at level j, and dj,k is the kth DWT-wavelet coefficient at level j (see, for instance, Percival and Walden, 2000, Chapter 9). Veitch and Abry derived a heteroscedastic regression model, which relates the wavelet

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V. Fernandez / Resources Policy 35 (2010) 276–282

variance at each scale j with the Hurst exponent,4 H, which equals (d  0.5): log2 ðu2y ðtj ÞÞ þ ðnj ln 2Þ1 ¼ c þð2H1Þtj þ ej

ð2Þ

where c is a constant term, E(ej)E0 and Var(ej) E(2nj ln2(2))  1, j ¼jl, y, ju.5 Hence, a wavelet-based estimator of H can be obtained from a regression of (log2 ðu2X ðtj ÞÞ þðnj ln 2Þ1 ) on tj with weights $j ¼nj. Periodogram The periodogram method consists of estimating H from a linear regression of the logarithm of the sample periodogram (i.e., an estimate of the spectral density of a time series) on the logarithm of its corresponding frequency (see Zivot and Wang, 2005, Chapter 8). In order to estimate H from a log–log plot of the periodogram, a given percentage of the frequencies closest to zero must be chosen. Given that observations will be sparse in the neighborhood of the zero frequency, Taqqu et al. (1995) proposed to estimate H by least absolute deviations in addition to ordinary least squares. The former is more robust in the presence of outlying observations. Geweke and Porter-Hudak (GPH) The spectral density of an integrated process, such as (1 L)d(yt  Z)¼ et, is given by  od fe ðoÞ ð3Þ f ðoÞ ¼ 4sin2 2 where o is the Fourier frequency and fe(o) is the spectral density of e. GPH (1983) proposed estimating d from the following linear regression:  o  i þ zi ln f ðoi Þ ¼ b0 d ln 4 sin2 ð4Þ 2 for oi ¼(2pi)/T, i¼1, 2, y, nf(T), where nf(T) is the number of Fourier frequencies utilized in the above regression model. GPH showed that, by utilizing a sample estimate of f(oi), the ordinary least-square estimate of d from (4) is asymptotically normal, provided that nf(T)¼Tg, where 0 o g o1, and T is the sample size.

which Hoskins (1981) obtained by the Levinson–Durbin algorithm.6 In turn, the prediction error variance is given by ut ¼ Varðyt y^ t Þ ¼ s2y k

t1 Y

2

ð1fjj Þ

ð6Þ

j¼1

where s2y ¼ Varðyt Þ, k is the ratio of the innovation variance over the variance of an ARMA(p, q) process.7 In order to make their algorithm more computationally P efficient, Haslett and Raftery approximate t1 j ¼ 1 ftj ytj by t1 X

M X

ftj ytj 

j¼1

ftj ytj 

j¼1

t1 X

fj ytj

ð7Þ

j ¼ Mþ1

where the ji’s are the weights of an ARFIMA(0, d, 0) process in an AR(N)-representation form and ftj E  jj, where jj is O(j  d  1) as t-N, as shown by Hosking (1981). Haslett and Raftery further approximate (7) by setting jj at its approximate average value for M ojrt 1 in the expression Pt1 j ¼ M þ 1 jj ytj :  d ! t1 X M jj ytj  MjM d1 1 yM þ 1,t1M ð8Þ t j ¼ Mþ1 where yM þ 1,t1M ¼

t1M X 1 y t12M j ¼ M þ 1 j

A quasi-maximum likelihood estimator of d ¼(y(L), f(L), d)0 can be obtained by maximizing ^ 2 ðdÞÞ LðdÞp12T lnðs

ð9Þ

where

s^ 2 ðdÞ ¼

T 1X ðyt y^ t Þ2 Tt¼1 ut

conditional on d, for each trial value of d. In practice, an M E100 has found to be a suitable choice (Palma, 2007, Chapter 4).

Empirical results Quasi maximum likelihood In this article, the obtainment of a quasi maximum-likelihood estimate of d is carried out by means of Haslett and Raftery (1989)’s algorithm. This is based on the following results. For an autoregressive fractionally integrated moving average (ARFIMA) (p, d, q) process, yt ¼ ð1LÞd jðLÞ1 yðLÞet , where f(L)¼(1  f1L  ? fpLp), y(L)¼(1  y1L ?  yqLq), such that the roots of f(L) and y(L) lie outside the unit circle, 0 rd r1/2, and {et} is a Gaussian white-noise sequence, an approximate onestep forecast of yt is given by y^ t ¼ fðLÞyðLÞ1

t1 X

ftj ytj

ð5Þ

j¼1

where

ftj ¼ 

t j

!

ðjd1Þ!ðtdjÞ! ðd1Þ!ðtdÞ!

The data The Dow Jones-AIG Commodity Indices (DJ-AIGCI) are composed of futures contracts on physical commodities, which are traded on US exchanges, with the exception of aluminum, nickel and zinc, which trade on the London Metal Exchange. The indexes are designed to maintain a long futures position, so that nearby contracts are sold and contracts which have not yet reached the delivery period are purchased. Therefore, the DJ-AIGCI are rolling indices. The indices are calculated on both an excess return and total return basis. The excess return indexes exclusively reflect the return of underlying commodity futures price movements, while the total return indices reflect the return on fully collateralized futures positions. We work with a sample of 20 DJ-AIG indices for the period January 1991–June 2008, measured at a daily frequency and

is a partial linear regression coefficient of an ARFIMA(0, d, 0), 4 Hurst developed the rescale range (R/S) statistic when studying river discharges (see Mills and Markellos, 2008, Chapter 4). 5 Eq. (2) holds only asymptotically for large j. Mielniczuk and Wojyllo (2007) point out that typically ju ¼ log 2(T)  4, where T is the sample size, and that the choice of jl depends on the process under consideration and on the sample size.

6 y^ t ¼ ft1 yt1 þ ft2 yt2 þ    þ ft,t1 y1 , t¼ 1, y, T, represents the best linear predictor of yt based on {y1, y2, y, yt  1}, where fkk ¼ d/(k d) and ftj ¼ ft  1,j  fttft  1,t  j, j¼1, y, t  1. 7 For a stationary and invertible ARMA(p, q) process, yt ¼ f1yt  1 + ?+ fpyt  p + et + y1et  1 + ?+ yqet  q, which admits the MA(N) representation P1 yt ¼ j ¼ 0 cj etj , c(L) ¼ f(L)  1y(L), Var(yt)  g0 is given by g0 ¼ f1g1 + ? + fpgp + s2e (1+ y1c1 +?+ yqcq), where gk ¼ E(ytyt  k). See Box et al. (1994, Chapter 3).

V. Fernandez / Resources Policy 35 (2010) 276–282

computed on an excess return basis. Specifically, we consider 5 broad indices and their sub-indices: Energy (crude oil, heating oil, natural gas, and unleaded gasoline), Grains (corn, soybeans, and wheat), industrial metals (aluminum, copper, nickel, and zinc), livestock (lean hogs and live cattle), and precious metals (gold and silver). All the estimation results reported below were obtained by means of S-Plus 8.0 routines written by this author. Estimation results In order to analyze the degree of fractional integration, we classify the indices into the broad categories of energy, grains, industrial metals, livestock, and precious metals; and, the subindices of energy, livestock, grains, industrial metals, and of precious metals. Our estimation strategy is as follows. We devise rolling estimates of d based on the four methodologies described in Section 2.1. The main reason for doing so is that rolling estimates provide us with a dynamic measurement of fractional integration in the returns and absolute return series. As discussed in the ‘‘Introduction’’ section, the presence of either persistence or antipersistence in returns would challenge the EMH in its weak form. Indeed, the presence of persistence would be indicative of a slow reaction of asset prices to incoming information, whereas the presence of anti-persistence would suggest an overreaction to such new information. On the other hand, persistence in absolute returns, which are a proxy of returns volatility, has been documented in previous studies, and it falls into the well-known phenomenon of volatility clustering. On the other hand, antipersistence in absolute returns is rare but it cannot be completely ruled out (see, for instance, Fernandez, 2007). In order to accommodate for the potential presence of structural breaks in the persistence parameter along the sample period January 1991–June 2008, we utilize an overlapping rolling window. Given the statistical features of the methodologies under consideration—wavelet-based, periodogram-based least squares and least absolute deviations (LAD), Geweke and Porter-Hudak (GPH)’s, and quasi maximum likelihood (QML), we choose a rolling window of 500 observations, i.e., about 2 years of data, which is neither too long or too short. In this regard, it is worth noting that by a choosing a rather large window length, one benefits from greater precision and reliance on a sample size that makes the fulfillment of asymptotic properties more likely. However, one is also forced to impose the same population parameter value along the sample period, which worsens bias.8 Given the dyadic nature of the wavelet filter, the number of wavelet coefficient at each level j is given by T/2j, where T is the sample size.9 Therefore, for a sample of 500 observations, the wavelet variance is computed on the basis of 250 wavelets coefficients at level 1, 125 wavelet coefficients at level 2, and so on, up to 4 wavelet coefficients at level 7. On the other hand, the periodogram-based least square and LAD estimators are computed by using the 25 frequencies closest to zero,10 while GPH’s estimator is obtained by using the 22 frequencies closest to zero 8 A thorough discussion on the properties of these estimators in finite and large samples is contained in Fernandez (forthcoming). Specifically, based on RMSE calculations for simulated ARFIMA(0, d, 0), wavelets and quasi maximum likelihood perform best among the methods utilized in this article. 9 We use a symmlet-8 filter, which is usually the default one, and 7 levels. Such a number of levels enables us to capture 1-year dynamics. Specifically, level j is associated with 2j  2j + 1 day dynamics, e.g., level 7 corresponds with 128–256 day dynamics. 10 This value is determined as 10% of the lowest frequencies out of the sequence of frequencies at which the spectrum is estimated. These correspond to the discrete Fourier frequencies between 0 and p. The spectrum is in turn obtained by means of the S-Plus ‘‘spec.pgram’’ function.

279

Table 1 Weighted wavelet-based least square rolling estimates. H0: d ¼ 0 is rejected in favor of H1 (%)

H0: d¼ 0 is rejected in favor of H1 (%)

do0 (%)

do0 (%)

d 40 (%)

d4 0 (%)

d o0 (%)

do0 (%)

d 40 (%)

d 40 (%)

5

1

5

1

5

1

5

1

(a) Selected DJ-AIGCI sub-indexes Grains rt 1.8 0.5 6.2 1.9 0.0 67.5 28.5 9rt9 0.0 Livestock rt 4.6 1.9 8.5 4.2 0.3 49.0 28.5 9rt9 1.3 Energy

rt 9rt9

Industrial metals 12.1 5.5 10.0 1.3 0.3 28.1 Precious metals 7.6 1.7 3.5 0.4 0.1 39.7

do0 (%)

do 0 (%)

d 40 (%)

d 40 (%)

5 7.2 0.3

1 2.7 0.0

5 3.5 26.3

1 0.8 11.6

3.6 10.5 1.2 20.3

(b) DJ-AIG energy sub-index Natural gas rt 2.8 0.6 7.0 1.5 9rt9 0.0 0.0 58.7 33.0 Heating gas rt 10.9 4.0 1.0 0.2 9rt9 0.5 0.1 29.6 10.1

Crude oil 3.7 0.5 Unleaded 3.3 0.4

0.8 0.1 gas 0.6 0.2

1.1 25.5

0.3 9.6

6.8 42.0

2.2 14.8

(c) DJ-AIG livestock sub-index Live Cattle rt 9.8 3.6 3.4 1.0 9rt9 0.2 0.0 58.8 34.2

Lean hogs 4.5 1.2 3.7 1.2

16.9 15.7

10.2 6.8

(d) DJ-AIG grains sub-index Corn rt 0.2 0.0 13.3 4.5 9rt9 0.0 0.0 74.5 45.0

Soybeans 4.9 1.6 0.1 0.1

3.5 59.5

0.9 29.4

9.4 0.2

0.9 34.8

0.2 14.9

3.3 0.0

4.3 51.0

1.5 33.6%

5.1 0.2

7.3 41.1

3.3 21.5

Wheat

rt 9rt9

do0 (%)

do 0 (%)

d 40 (%)

d 40 (%)

5 1.7 0.2

1 0.1 0.1

5 12.9 34.2

1 5.9 14.8

(e) DJ-AIG industrial metals sub-index Aluminum Copper rt 18.0 9.5 7.7 3.1 18.0 9rt9 0.1 0.0 48.6 29.4 0.5 Nickel Zinc rt 5.4 1.6 12.2 6.8 8.0 9rt9 0.0 0.0 62.6 43.2 0.1 (f) DJ-AIG precious metals sub-index Gold rt 5.8 1.8 5.5 2.1 9rt9 2.5 1.5 46.9 32.4

Silver 14.8 0.5

(i.e., nf(T)¼Tg, T¼500, and g ¼0.5).11 That yields 25 and 22 observations, respectively, to compute the two parameter estimates of the regression model (i.e., a constant term and d). Regarding Haslett–Raftery’s quasi maximum-likelihood estimator, the truncation point is set at 100, which is the default of the ‘‘arima.fracdiff’’ function, built in S-Plus. For each window, we estimate the fractional integration parameter for the return and absolute return series. We next count the number of violations of H0: d ¼0 by relying on the asymptotic normality of the five estimators under consideration. We test H0 against the alternatives H1: d o0 and d40 at the 95

11

g ¼ 0.5 is the default of the gphTest function built in S+ FinMetrics 3.0.

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V. Fernandez / Resources Policy 35 (2010) 276–282

Table 2 Periodogram-based least square rolling estimates. H0: d ¼ 0 is rejected in favor of H1 (%)

Table 3 Periodogram-based LAD rolling estimates.

H0: d ¼0 is rejected in favor of H1 (%)

H0: d ¼0 is rejected in favor of H1 (%)

H0: d¼ 0 is rejected in favor of H1 (%)

do 0 (%)

do0 (%)

d 40 (%)

d 40 (%)

do0 (%)

do0 (%)

d 40 (%)

d 40 (%)

do0 (%)

do0 (%)

d 40 (%)

d 40 (%)

d o0 (%)

d o0 (%)

d 40 (%)

d 40 (%)

5

1

5

1

5

1

5

1

5

1

5

1

5

1

5

1

6.0 57.3

0.0 33.4

0.8 50.6

0.1 32.3

(a) Selected DJ-AIGCI sub-indexes 0.7 0.3 7.7 0.1 0.2 0.1 83.9 61.3 Livestock 5.3 1.2 3.2 0.1 rt 9rt9 0.0 0.0 60.6 37.7

rt 9rt9

Energy

rt 9rt9 (b) DJ-AIG energy Natural gas 0.0 0.0 rt 9rt9 0.0 0.0 Heating gas rt 2.5 0.2 9rt9 0.2 0.0

0.0 0.0 0.2 0.0 Precious metals 18.2 8.4 0.0 0.0

do0 (%)

do0 (%)

d 40 (%)

d 40 (%)

5 4.3 0.0

1 0.3 0.0

5 12.2 54.6

1 0.4 27.1

Energy

rt 9rt9

sub-index 20.4 74.9

5.2 52.8

6.8 43.8

0.7 15.6

Crude oil 2.4 0.0 Unleaded 3.5 0.0

0.3 0.0 gas 0.1 0.0

(a) Selected DJ-AIGCI sub-indexes Grains rt 11.4 5.5 27.2 12.1 0.1 80.0 69.1 9rt9 0.1 Livestock rt 13.7 6.7 9.8 4.1 0.0 73.3 57.1 9rt9 0.0

6.8 56.2

0.0 26.5

6.1 32.4

0.8 15.3

(b) DJ-AIG energy Natural gas rt 3.8 0.6 9rt9 0.5 0.2 Heating gas rt 0.2 0.0 9rt9 1.7 0.7

Industrial metals 1.8 0.5 18.5 2.8 1.7 70.0 Precious metals 37.0 26.3 2.5 2.3 1.2 56.3

d o0 (%)

d o0 (%)

d 40 (%)

d 40 (%)

5 10.5 1.6

1 5.7 0.4

5 34.0 62.7

1 11.0 45.8

6.4 56.1 1.3 46.0

sub-index 34.0 82.0

15.5 67.4

43.8 64.1

15.6 44.2

Crude oil 10.5 0.5 Unleaded 9.2 1.9

6.1 0.4 gas 5.3 0.7

21.8 68.3

8.3 52.7

17.1 51.0

6.2 37.4

(c) DJ-AIG livestock sub-index Live cattle rt 1.1 0.0 0.5 0.0 9rt9 0.0 0.0 70.8 36.6

Lean hogs 2.5 0.8 0.6 0.0

4.6 67.0

0.3 43.6

(d) DJ-AIG grains sub-index Corn rt 3.6 0.0 20.4 4.6 9rt9 0.0 0.0 77.6 55.0

(c) DJ-AIG livestock sub-index Live cattle rt 9.1 3.5 8.5 1.6 9rt9 0.5 0.0 80.1 66.1

Lean hogs 9.1 5.9 4.0 2.9

12.8 74.3

5.1 66.5

Soybeans 3.2 2.7% 0.0 0.0

6.7 84.2

0.6 65.9

(d) DJ-AIG grains sub-index Corn rt 6.6 3.0 38.6 19.6 9rt9 0.0 0.0 76.1 65.4

Soybeans 4.2 0.8 0.0 0.0

15.5 91.8

6.9 81.2

1.8 0.0

30.1 72.1

17.7 53.0

9.7 0.0

22.0 69.2

15.8 46.6

19.1 1.7

6.6 67.6

2.2 59.0

Wheat

rt 9rt9

do0 (%)

do0 (%)

d 40 (%)

d 40 (%)

5 15.2 0.0

1 9.8 0.0

5 4.7 46.6

1 0.4 31.6

(e) DJ-AIG industrial metals sub-index Aluminum Copper rt 2.7 0.0 6.1 1.0 0.4 9rt9 0.5 0.0 59.6 35.1 0.0 Nickel Zinc rt 2.7 0.9 7.4 1.0 2.7 9rt9 0.3 0.0 56.7 30.5 0.0 (f) DJ-AIG precious metals sub-index Gold rt 10.4 3.9 2.2 0.3 9rt9 0.0 0.0 59.4 42.7

Silver 19.1 0.0

Wheat

rt 9rt9

0.0 0.0

17.4 63.4

4.2 37.0

0.6 0.0

19.0 51.2

10.5 27.1

9.0 0.0

2.7 61.9

0.3 47.3

and 99 confidence levels. We record the number of counts in which the null hypothesis is rejected in favor of the alternative, and the percentage of such a figure with respect to the total. The latter corresponds with the number of observations ( ¼4375) minus the window length ( ¼500), that is 3876 rolling estimates of d. For instance, if the rejection frequency, for a given significance level, is 10%, that means that 388 times, out of 3876, H0 is rejected in favor of H1. It is worth mentioning that we are not taking account of the potential presence of short memory in the data. This should not be a considerable source of bias, however, because, as shown by Fernandez (forthcoming) on the basis of Monte Carlo simulations, except for wavelets, the estimation procedures under consideration are fairly robust in the presence of autoregressive/moving

d o0 (%)

d o0 (%)

d 40 (%)

d 40 (%)

5 21.1 0.2

1 17.1 0.0

5 11.0 57.6

1 4.6 41.3

(e) DJ-AIG industrial metals Aluminum rt 7.5 3.8 19.2 9rt9 0.4 0.1 77.6 Nickel rt 7.6 2.5 13.8 9rt9 1.4 0.9 63.0

sub-index 11.6 60.7 6.8 50.7

(f) DJ-AIG precious metals sub-index Gold rt 26.3 16.0 3.9 2.6 9rt9 0.3 0.2 62.4 52.4

Copper 4.7 0.2 Zinc 16.0 0.3 Silver 25.0 4.3

average components of a moderate magnitude.12 Bearing this in mind, we should take our wavelet-based findings cautiously. In order to make the exposition of our results succinct, we start up by concentrating on the DJ-AIGCI broad indexes. We then discuss and contrast our findings for specific single sub-indexes within the categories of energy, livestock, grains, industrial metals, and precious metals (Tables 1–5). By first focusing on the wavelet-based rolling estimator (Table 1a), we find that the returns on Industrial Metals exhibit

12 The presence of an autoregressive (moving average) component in the data causes an upward (downward) in the estimation of the fractional integration parameter.

V. Fernandez / Resources Policy 35 (2010) 276–282

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an anti-persistence frequency that exceeds 10% (i.e., a 10% rejection rate of d ¼0 in favor of d o0). Such a cutoff, although arbitrary, indicates that the anti-persistence phenomenon is not as rare as one might think. Interestingly, the industrial metals returns also display non-negligible persistence (10% and 12%, respectively, at the 5% significance level). As expected, absolute returns tend to be considerably persistent. In particular, such a feature is evident in grains, for which the persistence frequency exceeds 50%. By contrast, the periodogram-based least square estimate (Table 2a) does not find any evidence of anti-persistence for the industrial metals returns. Instead, the precious metals return series arises as one displaying a relatively large anti-persistence frequency of 18% at the 5% significance level. On the other

hand, under this estimation technique, the presence of persistence in absolute returns is found to be much stronger. For instance, the grains return series display a persistence frequency of 84%. The periodogram-based LAD estimate (Table 3a) lends considerably more support to the existence of anti-persistence in the return series than the other two previous methods. For instance, the anti-persistence frequency is 37% for the precious metals return series. Interestingly, in two return series—energy and a grain; the evidence favors persistence over anti-persistence. For instance, the persistence frequency in the Energy series return is 34%, while that of anti-persistence amounts to 11% approximately. Again, the evidence of persistence in absolute returns is quite strong (e.g., grains and livestock).

Table 4 GPH-based rolling estimates.

Table 5 Quasi maximum likelihood-based rolling estimates.

H0: d ¼ 0 is rejected in favor of H1 (%)

H0: d ¼0 is rejected in favor of H1 (%)

H0: d ¼ 0 is rejected in favor of H1 (%)

H0: d ¼0 is rejected in favor of H1 (%)

d o0 (%)

do 0 (%)

d 40 (%)

d 40 (%)

do0 (%)

do0 (%)

d 40 (%)

d 40 (%)

do0 (%)

do0 (%)

d 40 (%)

d 40 (%)

do0 (%)

do0 (%)

d 40 (%)

d 40 (%)

5

1

5

1

5

1

5

1

5

1

5

1

5

1

5

1

(a) Selected DJ-AIGCI sub-indexes Grains rt 0.5 0.1 10.3 1.1 9rt9 0.2 0.1 81.3 48.7 Livestock rt 0.7 0.0 9.7 0.8 9rt9 0.0 0.0 54.1 27.7 Energy

rt 9rt9 (b) DJ-AIG energy Natural gas rt 0.1 0.0 9rt9 0.0 0.0 Heating gas rt 0.9 0.0 9rt9 0.2 0.0

Industrial metals 0.2 0.0 9.9 0.5 0.0 61.4 Precious metals 12.0 3.0 0.7 0.0 0.0 48.7

do0 (%)

do0 (%)

d 40 (%)

d 40 (%)

5 2.1 0.0

1 0.8 0.0

5 11.4 55.6

1 2.3 24.1

25.1 80.9

8.0 54.0

5.0 34.9

0.4 12.5

Crude oil 2.3 0.0 Unleaded 1.3 0.1

0.3 0.0 gas 0.2 0.0

Lean hogs 0.3 0.0 1.6 0.1

(d) DJ-AIG grains sub-index Corn rt 0.4 0.0 17.8 5.2 9rt9 0.0 0.0 74.7 53.8

Soybeans 2.8 1.5 0.0 0.0

rt 9rt9

0.0 27.7

Energy

rt 9rt9

sub-index

(c) DJ-AIG livestock sub-index Live cattle rt 1.4 0.0 1.6 0.0 9rt9 0.0 0.0 62.2 30.0

Wheat

0.4 28.7

do0 (%)

do0 (%)

d 40 (%)

d 40 (%)

5 15.9 0.0

1 11.0 0.0

5 9.5 44.4

1 1.7 31.7

(e) DJ-AIG industrial metals sub-index Aluminum Copper rt 5.1 0.4 8.9 2.4 0.7 9rt9 0.5 0.2 60.2 34.3 0.0 Nickel Zinc rt 2.3 1.3 3.8 0.7 2.2 9rt9 0.1 0.0 57.7 32.8 0.0 (f) DJ-AIG precious metals sub-index Gold rt 7.9 2.3 2.8 0.3 9rt9 0.0 0.0 62.2 36.5

Silver 12.0 0.0

(a) Selected DJ-AIGCI sub-indexes Grains rt 0.0 0.0 0.0 0.0 9rt9 0.0 0.0 95.3 80.4 Livestock rt 7.2 1.4 0.5 0.0 9rt9 0.0 0.0 61.0 46.3 do0 (%)

do0 (%)

d 40 (%)

d 40 (%)

5 8.6 0.0

1 2.0 0.0

5 0.0 45.4

1 0.0 29.9

8.0 47.4

1.0 15.4

5.0 28.5

0.7 13.9

(b) DJ-AIG energy sub-index Natural gas rt 7.9 0.1 0.0 0.0 9rt9 0.0 0.0 81.9 66.1 Heating gas rt 11.9 0.4 0.0 0.0 9rt9 0.0 0.0 48.6 26.0

1.4 39.4

(c) DJ-AIG livestock sub-index Live cattle rt 12.1 5.6 12.8 10.6 9rt9 0.0 0.0 78.0 70.8

0.1 61.9

(d) DJ-AIG grains sub-index Corn rt 0.0 0.0 11.7 6.1 9rt9 0.0 0.0 97.7 93.0

7.4 67.0

5.8 91.4

Wheat

rt 9rt9

Industrial metals 20.7 9.0 10.3 0.0 0.0 46.9 Precious metals 16.4 4.4 0.0 0.0 0.0 73.0

Crude oil 4.2 0.0 Unleaded 10.1 0.6

0.0 31.6

0.0 55.2

0.0 27.2

Lean hogs 5.4 0.4 3.9 3.0

14.4 36.9

4.5 24.5

Soybeans 2.0 0.5 0.0 0.0

0.0 98.0

0.0 83.4

10.0 0.2

0.0 48.2

0.0 41.1

3.6 0.0

8.2 75.2

0.8 61.3

14.9 0.0

1.9 81.8

0.0 73.6

do0 (%)

do0 (%)

d 40 (%)

d 40 (%)

5 1.4 0.0

1 0.0 0.0

5 5.1 33.0

1 0.1 15.5

0.1 0.0

15.3 48.6

3.9 38.2

0.8 0.0

20.1 51.1

13.5 27.2

0.0 45.2

(f) DJ-AIG precious metals sub-index Gold rt 18.6 4.4 2.3 0.6 9rt9 0.0 0.0 73.9 67.2

1.5 63.8

0.0 58.8

0.0 52.4

(e) DJ-AIG industrial metals sub-index Aluminum Copper rt 31.9 23.5 0.1 0.0 32.8 9rt9 0.0 0.0 63.0 51.1 1.1 Nickel Zinc rt 4.6 0.8 12.1 0.9 17.1 9rt9 0.0 0.0 95.9 74.9 0.0

4.9 0.0

0.3 0.0 gas 1.3 0.0

1.3 21.8

Silver 23.2 0.0

282

V. Fernandez / Resources Policy 35 (2010) 276–282

The most conservative estimates are provided by the GPH method (Table 4a), as anti-persistence in the return series finds little support, except for precious metals (12% frequency). By contrast, persistence seems more plausible in some return series, such as energy, grains, and industrial metals (11.4%, 10.3%, and 9.9% frequency, respectively). In agreement with the other estimation methods, absolute returns display little evidence in favor of anti-persistence, and they are highly persistent in some cases. The degree of support of the anti-persistent returns hypothesis lent by QML (Table 5a) appears to lie somewhere between that of the wavelet-based and periodogram-based LAD methods. Indeed, this method identifies four cases where anti-persistence is rather sizeable, ranging between 16% and 20%—i.e., industrial and precious metals. Moreover, and similarly to the LAD-based method, the evidence of highly persistent absolute returns is substantial, particularly so for grains (95% frequency). Now, by focusing on single sub-indexes, we are able to capture the existence of persistence/anti-persistence, which sometimes may not be observable in the main index categories reported in the panel (a) of each table. For instance, by looking at the precious metals components, we see that silver displays more antipersistence than the precious metals index itself under the wavelet and QML methods. On the other hand, the estimation methods may yield conflicting conclusions for some sub-indices. For instance, the wavelet and QML methods lend support to the anti-persistence of aluminum and copper return series, while the periodogram LAD-based method suggests the opposite. Nevertheless, it is the case that the five methods agree upon some specific sub-indices: gold and silver returns tend to exhibit anti-persistence, while the opposite is observed in natural gas, lean hogs, and corn returns.

Conclusions In this article, we concentrated on five alternative methods to quantify long-memory—wavelets, Geweke and Porter-Hudak’s semi-parametric method, the periodogram regression-based method (ordinary least squares and least absolute deviations), and quasi maximum likelihood. We obtained a rolling estimator of fractional integration for each estimation method, and concentrated on a sample of 20 DJ-AIG commodity indices— including broad indexes and sub-indexes over the period January 1991–June 2008. We concluded that absolute returns tend to be highly persistent over time. However, we found that returns series may exhibit either anti-persistence or persistence over the sample period, which disconfirms the EMH in its weak form. For instance, four out of the five estimation methods support the existence of antipersistence in the broad categories of precious metals. On the other hand, the five methods agree upon the anti-persistence of returns on the gold and silver sub-indexes, while they all support the existence of persistence in natural gas, lean hogs, and corn returns. The above findings indicate that economic agents are incapable of efficiently pricing some commodity indexes. Invalidation of the EMH in its weak form would make it possible for market

participants to come up with non-linear statistical models for improved index forecasting and derivatives valuation.

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