Estimation of Models for Stock Returns

Composite distributions where volatility itself is assumed to be a random variable have been used to model stock returns. In this paper, we give details of estimation of these composite distributions when the volatility is assumed to follow an arbitrary distribution and the conditional distribution of stock returns given the volatility follows one of normal, Laplace, uniform, Student's t, Cauchy, logistic of type I, logistic of type II, logistic of type III, logistic of type IV, generalized normal or skew normal distributions. The details given include estimating equations and observed information matrices. An application to Bitcoin exchange rate data is illustrated. Models taking volatility to follow gamma and Weibull distributions are shown to provide excellent fits.