Stable distributions are becoming increasingly popular as appropriate models for stock price changes and other economic phenomena. As a result, there is an expanding body of literature on inferential procedures for this family of distributions. Computationally simple estimators for the parameters of symmetric stable distributions have been provided by Fama and Roll. Little attention, though, has been given to goodness-of-fit tests for members of this family other than the normal.It is the purpose of this paper to discuss simple goodness-of-fit hypothesis tests using kurtosis, b2, to distinguish among members of the stable family. The b2 tests of hypothesis comprise: 1) a null normal versus a nonnormal symmetric stable alternative; 2) a null nonnormal symmetric stable versus a normal alternative; and 3) a null nonnormal stable versus another nonnormal stable alternative. Tables that give the percentage points of b2 and that are necessary for these tests of hypothesis are given. Apart from providing critical values for the tests, the tables allow the researcher to calculate the power. It will be seen that the b2 test exhibits excellent power.It is then hoped that computational convenience will make b2 an important tool for researchers and practitioners in finance. It is also hoped that the procedures we provide will aid these researchers and practitioners in the construction of appropriate financial models.

This paper deals with the theoretical development of some aspects of the trend removal problem. The objective is to show the difference between the two most popular trend removal methods: first differences and linear least squares regression. On the one hand, we show that if first differences are used to eliminate a linear trend, the series of residuals would be stationary but would not be white noises as they contain a first lag autocorrelation of -0.50. Furthermore, the spectral density function (SDF) of these residuals relative to that of a white noise series would be exaggerated at the high frequency portion and attenuated at the low frequency portion. On the other hand, we show that the regression residuals from the linear detrending of a random walk series would contain large positive autocorrelations in the first few lags. Relative to that of white noises, the SDF of the regression residuals would be exaggerated at the low frequency portion and attenuated at the high frequency portion.