Empirical distributions of stock returns: Paris stock market, 1980–2003

Abstract

The accurate specification of returns distributions has important implications in financial economics. A common practice in financial econometrics is to assume that the logarithms of stock returns are independent and identically distributed and follow a Normal distribution. However, daily stock returns display significant departures from Normality, having fatter tails and more peakedness. This study presents an alternative class of distributions, Levy-stable distributions, which can account for the observed skewness, kurtosis and fat tails, considering a sample of daily returns for nine stocks in Paris Market. Moreover, estimating the Levy-index allows us to determine long-memory behaviour of stock returns. Additionally, this study also tests long-memory hypothesis through an estimation of ARFIMA models. A comparative analysis of both approaches suggests the existence of long-memory in Paris Stock Exchange. The implication of the present work is that Levy-stable distributions are used to better approximate returns distributions and also to explore long-memory effects of stock returns.

Acknowledgements

We would like to thank the editor and an anonymous referee for their constructive comments that helped to improve significantly an earlier version of the article.

Notes

¹ This follows from the fact that, if stock prices follow a random walk, then stock returns should be independent and identically distributed (i.i.d); and if enough i.i.d. returns are collected, the central limit theorem implies that the limiting distributions of these returns should be Normal.

² When the characteristic exponent of a stable Paretian distribution is exactly 2, then Normal distribution is obtained. Hence, the latter is a special case of the former.

³ See Mantengna and Stanley (1995) and McCulloch (1997).

⁴ As in Bidarkota and McCulloch (1998) we relax Normality assumption in favor of stable distributions.

⁵ See Akgiray and Booth (1988); Jansen and de Vries (1991).

⁶ The above mentioned stocks has been chosen as sectoral stocks with an important weight.

⁷ These results support Pagan's (1996) study, in which it is argued that the returns of financial assets have semi-fat tails, or in other words, that the observed kurtosis is higher than the kurtosis of the Normal distribution.

⁸ A distribution is fat-tailed if P[X > s] ∼ s ^−α, as s → α, 0 < α < 2, where b(s) ∼ g(s) which means b(s) / g(s) → 1 as s → α. That is, regardless distribution's behaviour for small values of random variable, if distribution's asymptotic shape is hyperbolic, then distribution is fat-tailed.

⁹ See Levy (1937); Feller (1971); Zolotarev (1986).

¹⁰ See Peters (1994).

¹¹ The Joseph effect denotes the property of certain time series to exhibit persistent behaviour more frequently than would be expected if the series were completely random. The Noah effect refers to the tendency of various time series to exhibit abrupt and discontinuous changes.

¹² See Rachev and Mittnik (2000) and Racheva and Samorodnitsky (2003).

¹³ See Beran(1994).

¹⁴ See Cheung (1993).