Distribution of parallel market premium under stable alternative modeling

This dissertation investigates the tail behavior of stable distributions and uses stable laws as the foundation for modeling random shocks in currency markets in the univariate and multivariate setting. To estimate the characteristic exponent $\alpha$ of stable distributions, convergence of the tails of stable distributions to Pareto is often used. However, while it is known that the tails of stable distributions are asymptotically Paretian the point where the Paretian nature occurs is not known. Using accurate calculations of stable densities and distributions functions, we show that the point where stable distributions are asymptotically Paretian is a complicated function of $\alpha$ and $\beta$. Implications of asymptotic behavior of stable tails on estimates of the characteristic exponent are investigated. In particular, we show that the Hill estimates of $\alpha$ can be misleading, especially when $\alpha$ is near 2. Some analytical properties are derived for modes and densities of stable distributions. Due to exchange controls, most African countries are subject to a dual exchange regime where parallel and official exchange rate coexist. Empirical evidence shows that the marginal distributions of parallel and official exchange rates and parallel market premium are leptokurtic and asymmetric. By fitting a stable law to these distributions and estimating their parameters using maximum likelihood, we found that stable models approximate these distributions better than the normal model which has light tails. The spectral measure $\\Gamma$ for the random vector is estimated using the projection method, and its distribution on the unit circle is used to investigate the dependence structure between official and parallel exchange rate in a dual exchange regime. The location of the measure indicates that official and parallel exchange rates are generally positively associated, and most disturbances in the market seem to originate with the parallel market which has more variability.