Memory in Commodity Futures Contracts

Billy P. Helms Fred R. Kaen

Robert E. Rosenman

his article tests the hypothesis that price changes for a selected number of T commodity futures contracts are independent of previous price changes. The statistical procedure used to perform this test is called rescaled range analysis and is capable of identifying persistent or irregular cyclic dependence. I t is the presence of persistent dependence that we choose to label “memory.”

This investigation has been undertaken because there is mounting evidence that a speculative efficiency hypothesis is an inadequate description of auction market price behavior. In the context of commodity futures markets, a speculative effi- ciency hypothesis asserts that consecutive price changes, adjusted for trend, are in- dependent of one another.

Evidence of the absence of speculative efficiency in the foreign exchange market has been reported by Booth, Kaen, and Koveos (1981) in the form of profitable trad- ing rules and nonrandom power spectrums. The trading rule result has recently been confirmed by Dooley and Shafer (1983). Bilson (1981) reaches a similar conclu- sion about an absence of speculative efficiency with respect to the behavior of for- ward and spot exchange rates.

Application of rescaled range (RIS) analysis to the foreign exchange market by Booth, Kaen, and Koveos (1982b); to the gold market by Booth, Kaen, and Koveos (1982a), and to common stocks listed on the New York Stock exchange by Greene and Fielitz (1977) also resulted in rejecting a hypothesis that the respective price changes were independent of previous price changes. Instead, the most frequent finding reported was that long term persistent dependence was present.

Early evidence and conjecture about the persistence of price movements in com- modity future prices may be found in Alexander (1961); Cargill and Rausser (1975),

Billy P. Helms is Associate Professor of Finance at the University of Alabama.

Fred R. Kaen is Professor of Finance at the University of New Hampshire.

Robert E. Rosenman is Assistant Professor of Economics at Washington State University.

The Journal of Futures Markets, Vol. 4, No. 4,559-567 (1984) 01984 by John Wiley & Sons, Inc. CCC 0270-73 141841040559.14S04.00

Houthakker (1961), Larson (1960), and Stevenson and Bear (1970). All of these stud- ies concluded that price trends were apparent in the time series data.

Stevenson and Bear, in particular, performed a battery of tests with respect to soybean futures-the futures contracts that we use in our analysis. These tests con- sisted of an application of trading rules, autocorrelation analysis, and runs analysis. Their overall conclusion was “that the random walk hypothesis does not offer a sat- isfactory explanation of the movement of those [July corn and July soybean] price series” (1970, p. 79).

The results we obtain in this article support and confirm the work of these earlier investigators. Using an entirely different and more powerful type of statistical pro- cedure [see Mandelbrot (1972) and Lawrence and Kottegoda (1977)] we find that there are nonperiodic cycles (persistant dependence) in both daily and intraday commodity futures prices. Hence, we find for our definition of memory.

The remainder of this article is organized in the following manner. Section I ex- plains the research method and describes the data used for the investigation. Sec- tion I1 contains the empirical results and a discussion of some statistical issues associated with their interpretation. Section I11 summarizes our conclusions and speculates about their economic meaning.

I. RESEARCH METHOD AND DATA

A. Research Method

The statistical procedure which we use is called rescaled range analysis and we will refer to it as RlS analysis. It has its origins in the field of hydrology where it was de- veloped to study river flow and dam overflow. Key technical references in the litera- ture are Hurst (1951), Mandelbrot (1972), Mandelbrot and Wallis (1969a,b) and Wallis and Matalas (1970).

The attractive feature of RIS analysis is that it can be used to identify nonperiodic dependence in time series data. This identification is accomplished through the es- timation of a statistic called the Hurst or H coefficient which relates mean RIS val- ues for subsamples of equal length within the modified original time series to the number of consecutive observations within each group of equal length subsamples.

Figure 1 is a convenient chart for characterizing what various values of the Hurst coefficient represent. The three panels contained in Figure 1 are stylized versions of actual data which Mandelbrot (1972) uses in a similar exposition. Panel B in Fig- ure 1 is an independent or “white noise” time series; the values charted were gener- ated by a “no memory” statistical process.

Relative to Panel B, the time series in Panel A appears to be more tightly packed. Furthermore, there are frequently offsetting movements in the sense that large pis- itive deviations from a mean value are quickly offset by equally large negative devi- ations.

The time series in Panel C, on the other hand, exhibits persistent trends in a sin- gle direction. Also, there appear to be short irregular cycles contained within longer irregular cycles.

All three times series in Figure 1, in spite of visual appearances, would be identi- fied as white noise or independent processes by standard autocorrelation tech-

560/ MEMORY IN COMMODITY FUTURES

E 4

0 3! Y

I

- . I ' 0

I U c

8 G

I U c

d z

Ui .- L v1

.- s I

c

4

6 2

HELUS, KAEN, AND ROSENMAN /561

niques such as spectral analysis (1972, p. 274). However, RIS analysis would identify the processes depicted in Figure 1 as being different from one another.'

Subjected to RIS analysis, the data in Panel A would produce a Hurst coefficient of less than 0.5 and they would be identified as exhibiting negative persistence. The data in Panel C would produce a Hurst coefficient greater than 0.5 and the time se- ries would be said to possess positive persistence. Panel B would produce a Hurst coefficient of 0.5 and would be said to be a "white-noise" time series possessing no memory. In general, the Hurst coefficient may take on values laying between 0 and 1 .O with the aforementioned characterizations holding accordingly.

The procedure for estimating the Hurst coefficient may be explained with the aid of Figure 2. Let X(t) represent a discrete stationary time series. In this article X( t ) is the change in the logs of day-to-day and minute-to-minute prices. Let T represent the total number of observations within each price change time series.

The first step in the procedure is to transform the original series X(t) into a new series X*(t), so that X"(t) represents the cumulative sum of the original price change series X(t). This step is represented by Eq. (1). In Figure 2, this cumulative sum is the solid oscillating line labeled X*(t).

1

X*(t) = C X(u)for 1 5 t I T u = I

I

t t t r

Time

Figurv 2 Illustration ot sample scqucntial range R(l,.s).

'Anticipating tlw results to l x s reported i n this article, I m t h autororri~lation analysis and rutis ana1ysi.i rail- ed to idrnlif! ticiiiindt,pc.ndrnr hr,h;ivior i n I0 (JUI of the 12 tinir wries ~r examined.

562 / MEMORY IN COMMODITY FLI'I'URKS

The second step is to divide the cumulative sum series into smaller unit subseries of equal length s. The length of each group of unit subseries varies from 3 upward; the appropriate sizes may be found in Wallis and Matalas (1970, p. 1583). Once this division has been accomplished, the next task is to measure the trend-adjusted range of cumulative values within each unit subseries. This measurement is repre- sented as R(t,s) in Eq. (2). R(t,s) = max [X*(t + u) - (X*(t) + (u/s)(X*(t + s) - X*(t)))]O u 5 s

- min [X*(t + u) - (X*(t) + (uis) (X*(t + s) - X*(t)))]. o u 5 s (2) In Figure 2 this measurement is the distance between lines UU’ and LL’. Both of

these lines are parallel to line BE which is obtained by connecting the beginning (B) and ending (E) cumulative sum values for the relevant unit subsample size (s) under consideration. Note also that line UU’ is tangent to the highest X* value within the unit subsample and LL’ is tangent to the lowest X* value. By measuring the range (R) in this manner, an adjustment for trend is incorporated into the measurement.

After the range of the unit subseries has been calculated, it is divided by the standard deviation of the observations from the original time series, X(t)-not the cumulative same time series X*(t)-which lie within the same time period. The cal- culation of the standard deviation is represented by Eq. (3).

s 1 o’5 s(t,s)= (Us) c xqt + u) - (lIS2) c X(2 + u)2 . [ .11 o = l (3)

When all the unit R(t,s)/(S(t,s) values have been calculated for the consecutive unit sample sizes of length s which comprise the cumulative time series length T, the mean of these R(t,s)/S(t,s) s is obtained. The sample size s is then raised to its next prescribed number and the process is repeated. Ultimately, the result is a collection of mean RIS values for sample sizes of the prescribed lengths, e.g., 10,11,45, and so forth.

Mandelbrot and Wallis (1969a,b) suggest that these RIS values are asymtotically related to s, the subseries sample size. Specifically,

(4) where c is a constant and h is the measure of long-term dependence. It is this h that is often referred to as the Hurst coefficient and it is this statistic to which we earlier referred.

Estimating the Hurst coefficient is done by recasting Eq. (4) into an equality and transforming it into a log specification. Equation (5) results

(5)

RIS = R(t,s)/S(t,s) - CS”

In (RIS) = 1nC + H In (s)

where C and Hare , respectively, estimators of c and h in Eq. (4). Equation (5) is esti- mated using ordinary least squares regression.

B. Data

Both daily and intraday commodity futures price data are examined for the pres- ence of persistent dependence. All price data are the differences in the logs of se- quential prices.

Day-to-day price changes are examined for six contracts. These are the January 1977 and the March 1976 contracts in soybeans, soybean oil, and soybean meal.

HELMS, KAEN, AND ROSENMAN /563

Intraday price changes are examined for two separate days for the January, March, and May soybean contracts. Intraday price movements have been defined as the difference in the logs of transaction prices every minute on the minute.

All prices were obtained from tapes supplied by the Chicago Board of Trade. The tapes were perused for obvious recording errors.

The number of observations in each time series ranges from 227 to 239. In all cases, this resulted in 26 observations for estimating Eq. (5).

11. EMPIRICAL RESULTS

Table I contains the estimating equations for daily price change analysis. The esti- mated Hurst coefficients range from 0.558 to 0.71 1 with very small standard errors. Consequently, the hypothesis of positive persistent dependence cannot be rejected. Alternatively, the hypothesis of independent white noise price behavior may be re- jected.

Table I1 contains the estimating equations for the intraday minute-to-minute price changes. These, too, are all in excess of 0.5 and also exhibit very small stand- ard errors. Thus, persistent dependence and non-independent behavior describe these series as well as the daily series.

The Hurst coefficients reported in Tables I and I1 may be compared with those estimated for other markets. Booth, Kaen, and Koveos (1982a) calculated a Hurst coefficient of 0.642 for daily changes in gold prices. In the foreign exchange market they obtained values of 0.670, 0.569, and 0.546 for the pound, mark, and French franc prices of U.S. currency (198213).

Table 1

PRICE CHANGES IN SOYBEAN. SOYBEAN OIL AND SO1 BEAN MEAL FUTURES CONTRACTS

HURST COEFFICIENT ESTIMATING EQUATIONS FOR UAl -TO-DA\

Parameters;'

Contract c H R2

March 1976 Soybeans Soybean Oil

Soybean Meal

January 1977 Soybeans Soybean Oil

Soybean Meal

- 0.444 (0.056)

(0.067)

(0.584)

(0.072)

(0.057)

(0.07 1)

-0.417

- 0.279

- 0.680

- 0.368

-0.613

0.630 0.987 (0.014) 0.63 1 0.982

0.558 0.983 (0.014) 0.71 1 0.984 (0.018) 0.604 0.986 (0.014) 0.696 0.984 (0.0 18)

(0.017)

'Standard errors in parenthesrs.

564 / MEMORY IN COMMODITY FUTURES

Table 11

PRICE: CHANGES Ih SO\ HEAN FUTURLS COhTRACTS

HURST COEFFICIENT ESTIMATING EQUATIONS FOR !kl16UTK-To-M I NLTL

Paranirtrrs"

Soyhran H R' Day Contract c

Mar. 4 Jan. 78 -0.412 (0.082)

(0.609)

(0.066)

(0.059)

(0.081)

(0.068)

Mar. 4 Mar. 77 - 0.34 1

Mar. 4 May 77 - 0.279

Mar. 13 Jan. 78 - 0.493

Mar. 13 Mar. 77 -0.310

Mar. 13 May 77 - 0.367

0.627 0.973 (0.021) 0.609 0.972 (0.021) 0.596 0.980 (0.016) 0.63 1 0.986 (0.01 5) 0.581 0.970 (0.020) 0.568 0.977 (0.017)

'Standard errors in parentheses.

As is the case with virtually all empirical analyses, statistical interpretational is- sues arise because of the assumptions underlying the statistical test and the distri- butional nature of the time series data. RIS analysis is no exception.

Wallis and Matalas (1970) report that first-order autocorrelation may bias the Hurst coefficient. However, the direction and magnitude of the bias depends on the sign and size of the autocorrelation. They found that an upward bias of 0.08 in the estimated Hurst coefficient was present if the first order autocorrelation was 0.3; a downward bias of - 0.03 was obtained for an autocorrelation coefficient of - 0.05. For the daily price changes we have used, the first-order correlation coefficient ranges between 0.05 and -0.08. For the intraday prices, the range is from 0.16 to -0.17. Consequently, very little, if any, bias may be attributed to sample autocorre- lation and that which does exist is more likely to have resulted in our underestimat- ing the Hurst coefficient.

All of our price distributions exhibit some degree of skewness. This, though, may actually be beneficial because Wallis and Matalas also suggest that skewness re- duces the overall estimating bias.

With respect to the estimating procedure itself, Wallis and Matalas (1970) note that unless the true value of the Hurst coefficient is 0.7, a upward bias is encoun- tered. For example, if h = 0.6 (a commonly estimated value in both tables), the up- ward bias is 0.02.

In general, it appears that many of the preceding biases offset one another. What remains is fairly strong statistical evidence of persistent dependence in the selected commodity futures contracts.

HELMS, KAEN, AND ROSENMAN /565

111. DISCUSSION AND CONCLUSIONS

Our acceptance of the presence of irregular cyclic patterns-positive persistent de- pendence-in commodity futures contracts means that we reject the position that even after an adjustment for trend (e.g., the probability of up is 0.6; down, 0.4) price changes are not independent of previous price changes. In other words, we find for memory and against a speculative hypothesis of independence.’ At the present time we have no formal financial theory to explain this outcome. However, we do believe that we have better described and categorized the price change behavior than have previous investigators. Because the modification of existing theory often begins with attempts to explain observed behavior, we believe that our primarily descrip- tive outcome is important contribution.

We also believe that our outcome is more consistent with the observations of practitioners who frequently use such terms as “bandwagon” and “follow-the- leader” to describe market behavior. Such effects are frequently treated with ex- treme skepticism; yet the persistent price patterns implied by these terms are simi- lar to what we have uncovered.

We believe that an interesting economic avenue to explore for explaining such behavior is one where expectations of market participants are not formed indepen- dently of other market participants andlor their actions. Perhaps relaxing assump- tions about independence at the behavioral level would produce expected depen- dence at the empirical level.

The authors would like to thank Clif Horrigan for her programming assistance.

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2Rausser and Carter (1983) have developed an ARIMA model that outperforms the futures market in predicting futures prices for soybeans and soybean meal. Their results imply some price change trends because an ARIMA model implies a degree of serial dependence.

566 / MEMORY IN COMMODITY FUTURES

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HELMS, K A E N , A N D R O S E N M A N /567