Non-normal Stable Distribution Research Articles

Can fat-tail create the momentum and reversal?

ABSTRACT We suggest that fat-tail variations can cause both short-term momentum and long-term reversal simultaneously, in both the time series and cross-sectional returns of securities. The fat-tail of the distribution is known to explain many anomalies in the financial market, but not momentum. To support our argument, we adopt widely accepted models in the literature, which generate reversal only, and revise a single assumption: Each random variable follows a non-normal stable distribution rather than a normal distribution. This single difference generates additional short-term return momentum. This finding shows that 1) investor irrationality is not essential to generate both phenomena, and 2) we must be cautious not to overuse normal distributions in the models.

Least Absolute Deviation Estimation of Linear Econometric Models: A Literature Review

Econometricians generally take for granted that the error terms in the econometric models are generated by distributions having a finite variance. However, since the time of Pareto the existence of error distributions with infinite variance is known. Works of many econometricians, namely, Meyer & Glauber (1964), Fama (1965) and Mandlebroth (1967), on economic data series like prices in financial and commodity markets confirm that infinite variance distributions exist abundantly. The distribution of firms by size, behaviour of speculative prices and various other recent economic phenomena also display similar trends. Further, econometricians generally assume that the disturbance term, which is an influence of innumerably many factors not accounted for in the model, approaches normality according to the Central Limit Theorem. But Bartels (1977) is of the opinion that there are limit theorems, which are just likely to be relevant when considering the sum of number of components in a regression disturbance that leads to non-normal stable distribution characterized by infinite variance. Thus, the possibility of the error term following a non-normal distribution exists. The Least Squares method of estimation of parameters of linear (regression) models performs well provided that the residuals (disturbances or errors) are well behaved (preferably normally or near-normally distributed and not infested with large size outliers) and follow Gauss-Markov assumptions. However, models with the disturbances that are prominently non-normally distributed and contain sizeable outliers fail estimation by the Least Squares method. An intensive research has established that in such cases estimation by the Least Absolute Deviation (LAD) method performs well. This paper is an attempt to survey the literature on LAD estimation of single as well as multi-equation linear econometric models.

Gauss-Newton and M-estimation for ARMA processes with infinite variance

We consider two estimation procedures, Gauss-Newton and M-estimation, for the parameters of an ARMA ( p, q) process when the innovations belong to the domain of attraction of a nonnormal stable distribution. The Gauss-Newton or iterative least squares estimate is shown to have the same limiting distribution as the maximum likelihood and Whittle estimates. The latter was derived recently by Mikosch et al. (1995). We also establish the weak convergence for a class of M-estimates, including the case of least absolute deviation, and show that, asymptotically, the M-estimate dominates both the Gauss-Newton and Whittle estimates. A brief simulation is carried out comparing the performance of M-estimation with iterative and ordinary least squares. As suggested by the asymptotic theory, M-estimation, using least absolute deviation for the loss function, outperforms the other two procedures.

Convergence to a Stable Distribution Via Order Statistics

Let $X_1, X_2, \cdots$ be i.i.d. random variables whose common distribution function $F$ is in the domain of attraction of a nonnormal stable distribution. A simple, probabilistic proof of the convergence of the normalized partial sums to the stable distribution is given. The proof makes use of an elementary property of order statistics and clarifies the manner in which the largest few summands determine the limiting distribution. The method is applied to determine the limiting distribution of self-norming sums and deduce a representation for the limiting distribution. The representation affords an explanation of the infinite discontinuities of the limiting densities which occur in some cases. Application of the technique to prove weak convergence in a separable Hilbert space is explored.

On the Use of Limit Theorem Arguments in Economic Statistics

Abstract The heavy reliance on the normal distribution in the teaching of economic statistics is usually justified by appealing to the central limit theorem. Limit theorem arguments can, however, also lead to the nonnormal stable distributions. It is argued that the conditions required for nonnormal stable limits are, in fact, frequently satisfied by economic variables and in particular by disturbance terms in regressions. Normal theory results should, therefore, be used with caution, and it might be wise for economic statisticians to devote more attention to robust statistical methods.

A Closer Look at the Implications of the Stable Paretian Hypotheses

T HE form of the distribution of security returns has important implications for financial theory and for the testing and implementation of theory. Because of this, a number of studies have dealt with estimating the form of the distribution of investment relatives. Almost all of this work has examined the distribution of the log of price relatives and concluded that this distribution is Gaussian (see Moore (1962) and Osborne (1954) or non-normal stable (see Mandelbrot (1963) and (1967), Fama (1965), and Roll (1970)). Many authors who have accepted the non-normal Stable Paretian Hypothesis have treated stock price returns as if they followed a non-normal stable distribution.