It is shown that the ML estimates of the popular GARCH(1,1) model are significantly negatively biased in small samples and that in many cases converged estimates are not possible with Bollerslev's non-negativity conditions. Results also indicate that a high level of persistence in GARCH(1,1) models obtained using a large number of observations has autocorrelations lower than these ML estimates suggest in small samples. Considering the size of biases and convergence errors, it is proposed that at least 250 observations are needed for ARCH(1) models and 500 observations for GARCH(1,1) models. A simple measure of how much GARCH conditional volatility explains squared returns is proposed. The measure indicates that for a typical index return volatility whose ARCH parameter is very small, the conditional volatility hardly explains squared returns.Small sample, volatility, GARCH, persistence,

This article analyses the evolution of relative per capita income distribution of Brazilian municipalities over the period 1970-1996. Analyses are based on non-parametric methodologies and do not assume probability distributions or functional forms for the data. Two convergence tests have been carried out - a test for sigma convergence based on the bootstrap principle and a beta convergence test using smoothing splines for the growth regressions. The results obtained demonstrate the need to model the dynamics of income for Brazilian municipalities as a process of convergence clubs, using the methodology of transition matrices and stochastic kernels. The results show the formation of two convergence clubs, a low income club formed by the municipalities of the North and Northeast regions, and another high income club formed by the municipalities of the Center-West, Southeast and South regions. The formation of convergence clubs is confirmed by a bootstrap test for multimodality.

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