1. Introduction

The “compass rose” (explained below) is a pervasive, but not universal pattern in financial markets. The pervasiveness of the pattern suggests the value of further study of it. Yet only a few studies of the compass rose have been published. (A reasonably complete list might be Crack and Ledoit, 1996; Kramer and Runde 1997; Chen, 1997; Szpiro, 1998; Lee, Gleason, and Mathur, 1999; Gleason, Lee, and Mathur, 2000; Wang, Hudson, and Keasey, 2000; Koppl and Nardone 2001; Wang and Wang, 2002; and Fang 2002.) We conjecture that this literature has not grown more rapidly because researchers have not had a satisfactory set of statistical tools for testing hypotheses about it.

Crack and Ledoit, who discovered the pattern, say that their explanation of it uses “subjective language” because the statement “the compass rose appears clearly” is itself “subjective” (1996, p. 754). They seem to doubt that “objective” tools can be applied to the compass rose. But “objective” statistical tests can be applied as Koppl and Nardone (2001) show. We extend their work to show how the hypothesis of random walk could be tested in a setting inspired by the compass rose pattern. To exemplify the method we choose the wheat futures contract where the compass rose pattern has been shown to exist (Lee, Gleason, and Mathur 1999).

2. The Compass Rose

Crack and Ledoit (1996) plot daily stock returns against the same values lagged one period. Under certain conditions the result is the compass rose pattern. They report that the pattern is “indisputably present in every stock” on the NYSE. Lee et al. apply this technique to data “from 118 futures contracts traded on 31 futures exchanges in 15 countries” (1999, p. 541). They report mixed results. They could find no general principles to predict when the pattern would appear and when not. Gleason et al. (2000) have found one crucial dimension in determining the appearance of the compass rose, namely, the ratio of tick size to volatility. Wang & Wang (2002) suggest a quantitative measure of the “quality of the [compass rose] pattern” (p. 1101) given by the fraction of points appearing on the 32 most prominent rays of the compass rose (p. 1106).

The results of Lee et al. (1999), Gleason et al. (2000), and of Wang & Wang (2002) highlight one cause of the difficulty in understanding when the compass rose pattern is present, namely, ambiguity in some of the basic concepts. Crack and Ledoit’s conditions for the pattern’s existence include the condition that daily price changes be “small” relative to the price level and that daily price changes are in jumps of a “small” number of ticks. The term “small” is ambiguous. It is also unclear when the pattern is present and when absent. In one and the same paragraph Lee et al. were driven to say of their Figure 2C that “The compass rose pattern is unmistakably observed” and “Because it is reasonable to question whether the pattern observed in Figure 2C is a compass rose pattern, we arbitrarily classify this contract as one without a compass rose pattern” (1999, p. 551).

However, we advance the idea that the presence of the compass rose pattern should not be the focus of the analysis. The focus should be on the statistical properties of financial series. The compass rose can inspire new tests of such properties. The use of “theta histograms” proposed by Koppl and Nardone (2001) would allow researchers to replace the investigation of the compass rose as a pattern with the analysis of the properties of the series under consideration. Specifically the compass rose and its suggested angular distribution are conducive to random walk tests. Recently Fang (2002) investigates how the compass rose pattern would bias standard random walk tests. Our paper proposes a method to directly investigate the random walk hypothesis in a phase plane suggested by the compass rose pattern. Campbell, Lo, and MacKinlay, (1996) discuss many meaning of “random walk” (pp. 27-33). We use one of them, namely, independence of returns. It seems possible to generate tests of other hypotheses by similar methods.

3. Theta Histograms

To construct an empirical theta histogram, one first expresses each point of the compass rose in a modified version of polar coordinates. The point (R_t,R_t+1) becomes (r_t, θ_t) where

Thus, θ_t ranges between –1.0 and 1.0. One then plots the number of points with a given θ_t value against that same θ_tvalue. The θ_t values are sorted into 201 bins of width 0.01. This line of attack is suggested by the compass rose pattern and the importance this pattern accords to the angular distribution of returns.

An empirical theta histogram can be compared to a bootstrapped theta histogram, which shows what the empirical theta histogram would have looked like if 1) returns were independent, 2) the observed distribution of returns were identical to the population distribution, and 3) a much larger sample had been drawn from the population. To construct a bootstrapped theta histogram, sample with replacement from the observed distribution of returns and plot the corresponding theta histogram. Calculating a chi-squared statistic allows one to test whether the differences between the two histograms are attributable to statistical dependence in returns. The null hypothesis is independence of returns. Let n be the number of points on the observed compass rose. Consider a interval ω±δω. Let p be the relative frequency of points in that interval under H₀ . One reads p off of the bootstrapped theta histogram. Given the sample size n and the null hypothesis of independence, the expected number of points in ω is h=np. Let k denote the observed number of points in the interval. Define as follows:

(3)

where . Assuming H₀, in the limit of a large number υ of partitions, the complement cumulative distribution of is Q(χ²|υ) , an incomplete gamma function (Press et al 1992). Selecting the customary confidence level of 0.05, we reject H₀ if

(4)

where Γ(x) is the gamma function.

Use of the Bernoulli distribution allows one to test whether a given ray has too many or too few points. Consider, then, the interval ω±δω. We can easily determine the relative frequency of theta histograms of size n for which the number of hits in that interval, k, is at least equal to the number, h, observed in the bootstrapped sample. Assuming H₀, we have a sequence of Bernoulli trials in which the probability of a hit is p. For every integer k such that 0£ k£n , there are C(k,n) ºn!/[k!(n-k)!] ways you could get k hits. The probability of exactly k hits is C(k,n) p^k (1-p)^(n-k) . Thus, the probability of k³ h is

where B(x;n,p) is the probability function corresponding to the binomial distribution.

We propose not to accept H₀ if P(k³h) < 0.05.

4. Wheat Futures

We applied the tools of section 3 to a wheat futures contract. Our data are prices from contracts traded on CBOT. Wheat is traded in 5000 bushel lots, with a tick of 1/4 cent per bushel. Contracts are traded on the open-outcry system. Our data run from January 1970 through June 2000. Figure 1 shows the compass rose for this data. Note that the figure is displaying the pattern poorly enough to inspire some doubt regarding its presence. Figures 2 shows the empirical and bootstrapped theta histograms. Table 1 reports on our χ² test, which indicates that we can reject the hypothesis that the data are statistically independent. Table 2 gives the results of several Bernoulli tests. It shows the rays for which the empirical theta histogram had significantly more points than the bootstrapped histogram, using a 5% significance level. Note that 16 rays have significantly more points than expected under a random walk model. At the same time the four rays defining an X-pattern (-0.745, -0.245, 0.245, 0.745) have an especially high accumulations of points as proven by the significance of the test statistic. This is the X-skewing Koppl and Nardone (2001) found in other data. Our result strengthens their conjecture that X-skewing may be pervasive in financial markets.

5. Conclusions

Lee et al. note “elusiveness of the [compass rose] pattern” (1999, p.544 ). We propose to avoid the question of when a compass rose is a compass rose and to use the angular distributions of returns for random walk hypothesis testing. Section 3 explained a set of tools for such measuring and testing. Applying these tools to wheat futures reveals 16 rays that accumulate more points than under the null of independence. The tests also showed that the wheat futures data exhibits X-skewing. Together with the evidence from Koppl and Nardone (2001), this result points to the possibility that X-skewing may be an important and pervasive pattern in financial markets. This result is only suggestive, but it shows that use of techniques such as those discussed here may lead to the discovery of important new facts about the behavior of financial markets. Further study seems appropriate.

Bibliography

Campbell, John Y., Andrew W. Lo, and A. Craig MacKinlay. 1996. The Econometrics of Financial Markets, Princeton: Princeton University Press.

Chen, An-Sing. 1997. “The Square Compass Rose: the Evidence From Taiwan,” Journal of Multinational Financial Management, 7(2): 127-144.

Crack, Timothy F., and Olivier Ledoit. 1996. “Robust Structure Without Predictability: The ‘Compass Rose’ Pattern of the Stock Market,” Journal of Finance, 51: 751-762.

Fang, Yue. 2002. “The compass rose and random walk tests,” Computational Statistics & Data Analysis, 39: 299-310.

Gleason, Kimberly C., Chun I. Lee, and Ike Mathur. 200. “An explanation for the compass rose pattern,” Economic Letters, 68(2): 127-133.

Koppl, Roger and Carlo Nardone. 2001. “The Angular Distribution of Asset Returns in Delay Space,” Discrete Dynamics in Nature and Society, 6: 101-120.

Kramer, Walter, and Ralf Runde. 1997. “Chaos and the Compass Rose,” Economics Letters, 54(2): 113-118.

Lee, Chun I., Kimberly C. Gleason, and Ike Mathur. 1999. “A Comprehensive Examination of the Compass Rose Pattern in Futures Markets,” Journal of Futures Markets, 19(5): 541-564.

Press, W. H., B. P. Flannery, S. A. Teukolski and W. T. Vetterling. 1992. Numerical Recipes, Cambridge: Cambridge University Press, 2nd ed.

Szpiro, G. G. 1998. “Tick Size, the Compass Rose Pattern and Market Nanostructure,” Journal of Banking and Finance, 22: 1559-1569.

Wang, Eliza, Robert Hudson, and Kevin Keasey. 2000. “Tick size and the compass rose: further insights,” Economic Letters, 68(2): 119-125.

Wang, Huaiqing and Chen Wang. 2002. “Visibility of the compass rose in financial asset returns: A quantitative study,” Journal of Banking and Finance, 26: 1099-1111.

TABLE 1

test on wheat futures contract

The data include n=7,680 observations sorted into 201 bins. is the reduced variable which is asymptotically distributed normally, is the corresponding probability .

x Q(x)

554.26 13.27 1.7*10^-40

TABLE 2

Rays of significant difference between the actual and bootsrapped histograms:

the wheat futures contract

Midpoint	Exp freq %	Act freq %	Bootstrap	Empirical	p-value

-0.875	0.4794	0.6392	37	49	0.0218
-0.745	0.4532	0.8740	35	67	0.0000
-0.735	0.4443	0.6522	34	50	0.0040
-0.685	0.4680	0.6131	36	47	0.0308
-0.675	0.4470	0.6261	34	48	0.0105
-0.665	0.4594	0.6392	35	49	0.0111
-0.245	0.4629	1.1218	36	86	0.0000
-0.145	0.4804	0.6261	37	48	0.0319
0.245	0.4388	1.1218	34	86	0.0000
0.325	0.4776	0.7305	37	56	0.0011
0.355	0.4862	0.8609	37	66	0.0000
0.375	0.5193	0.6653	40	51	0.0366
0.425	0.5307	0.6653	41	51	0.0497
0.745	0.4472	0.9131	34	70	0.0000
0.855	0.4820	0.7305	37	56	0.0013
0.875	0.4817	0.6783	37	52	0.0075

FIGURE 1

Wheat Futures Contract Compass Rose Pattern

FIGURE 2

Histograms for the Wheat Futures Contract