Stable Paretian Versus Student's t Stock Market Hypothesis

Abstract

This article investigates the types of probability distributions that can best represent equity returns using a large sample of daily S&P500 index returns. The competing models, Stable Paretian and Pearson families, are compared using Bayesian methods. The evidence against Stable Paretian as a model of S&P500 index returns is overwhelming. The distribution that best fits the data is Pearson Type IV, and Student's t fits almost as well. One implication is that a Bayesian decision maker should strongly shift beliefs in favor of a Pearson distribution with finite means and variances as a model of daily changes in the S&P500 stock index.

Keywords:

• 62F15
• 91G10
• 91G60

Keywords:

• Stable Paretian family
• Pearson family
• Student's t
• Bayesian estimation

Acknowledgments

The authors thank Harry Markowitz for extremely valuable suggestions and the two anonymous referees for improving the exposition of this article.

Notes

1. The evidence against Gaussian distributions for short-term equity returns goes back to Mandelbrot (1963) and Fama (1965). They found that a Stable Paretian distribution with characteristic exponent less than 2 and that is symmetrical better explains the fat tails and higher kurtosis of observed data. Later research, however, challenged the Stable hypotheses in two major, but not mutually exclusive, ways. For one, observed data were better fitted by other distributions, some even with finite variances (see Blattberg and Gonedes 1974). Second, return distributions with thinner tails than those of the Stable, an observation more plausible for distributions that have normal as domain, were more likely to represent the data (see Akgiray and Booth 1988). Markowitz and Usmen launched their research project around this time to find the best fit for daily S&P500 returns, searching through the finite moment members of a large family of distributions known as the Pearson. Their findings appeared in Markowitz and Usmen (1996a; 1996b) and showed that Pearson type IV distribution, which has Student's t as a special case, was the best fit. Around the time the Markowitz and Usmen findings were published, Hurst and Platen (1997) also showed that Student's t fits the data better than the Stable or a compound of normals. Still limited to symmetrical distributions, Fergusson and Platen (2006) found that Student's t fits world stock indices data the best among generalized hyperbolic distributions. This result is further generalized in Platen and Rendek (2008). Although the evidence against the Stable model in favor of Student's t has been building, there has not been a comprehensive study comparing the general families of Stable and Pearson using the most recent data and Bayesian methodologies. The present study intends to do that.

2. From the Bayes formula,

(21)

It follows that

(22)

and the right-hand side is computed as

(23)

3. It is common practice to approximate the integral

(24)

by an integral of a normal function centered on the posterior mode and having variance equal to minus the inverse of the determinant of the matrix of second derivatives of the log posterior density at its mode. This approximation was used in Tierney and Kadane (1986) and Lewis and Raftery (1997) and can be viewed as an application of the Laplace method for integrals, as described in De Bruijn (1981). The approximation works well when the posterior is dominated by a single mode, and we assume the sample size is large enough for this to be the case.

4. The principal of stable estimation of Edwards, Lindman, and Savage (1963) is unrelated to the Stable distribution. It concerns inferences about parameters based upon sufficiently diffuse or gentle prior distributions relative to the likelihood function. The idea used in the article is to approximate the posterior density by the likelihood function in a neighborhood of the posterior mode. Using the principle of stable estimation in a neighborhood . The ratio of posterior to prior odds becomes

(25)

With diffuse priors (e.g. ), is identical to in (6).

5. The Stable likelihood calculations and other Stable methods were made available to us by Nolan via Robust Analysis, Inc.

6. Fmincon uses a gradient search methodology, calculating values and search directions along the way. An optimum is reached when the likelihood does not change by more than since the last iteration.

7. Student's t is a three-parameter model nested within a four-parameter Pearson family. Without any adjustment, the maximum likelihood principle will lead to selecting a model with the highest possible dimension. Schwarz (1978) has a procedure to correct for this when observations come from a Koopman–Darmois family of distributions. This family includes a wide range of distributions such as normal, gamma, , and beta. All Student's t distributions are not in the Koopman–Darmois family.

8. The Bayes factor in general depends on prior distributions, and the computations in Table 4 are based on two assumptions concerning priors. The first is relatively flat priors in the neighborhood of parameter space where the likelihoods are substantial (i.e., the Edwards, Lindman, and Savage (1963) condition) and likelihoods dominate priors. The second assumption is a reasonable bound on the probability of the neighborhoods around likelihood maximizing points (i.e., conditions where is approximately equal to ). Putting aside the problem of whose prior we are talking about, consider the following reasonable assumption on priors, which will yield the results in Table 4 (i.e., ). Consider a uniform prior over neighborhood whose value at the LLH maximizing is defined as as in note 4. Also define for . The prior distribution is as follows:

(26)

where . Integrating over each rectangle gives:

(27) (28) (29) (30)

where is the volume of each rectangle. It follows from that the ratio is given by

(31)

Hence, the ratio is the inverse ratio of volumes. Based on the parameters of sample S in Table 2 for type IV and Stable we calculate:

(32) (33)

Thus, implies the shift in favor of type IV is still massive () and a modicum less than that presented in Table 4.