C13 - Estimation: GeneralReturn

In the text finite mixtures of lognormal distributions are used for the modelling of net annual per capita income of the Czech households in Czech crowns in 2004-2008. All the households are divided into subgroups with observed group membership according to the attained education of the head of a household (a factor with five levels: basic, secondary, complete secondary, bachelor, magister education), the existence of children in the household (a factor with two levels: children yes or no) and the number of children (a factor with five levels: 0-3, more than 3 children). Then, models with incomplete data (unobserved component membership) were used for 2-5 components. All the estimates in the text are maximum likelihood estimates; explicit formulas exist for the complete data models; an EM algorithm for normal mixtures incorporated in the flexmix package in R was used for the incomplete data models. The models are compared with the use of the Akaike criterion. Maximum likelihood estimates of mixing proportions, expected values and standard deviations of the components are given and their development in the five-year period analysed is discussed. The program R was used for all the computations.

Estimation of volatility of financial time series plays a crucial role in pricing derivatives. Volatility is often estimated from historical data; however, it is well known that volatility varies in time. We propose a method to choose a suitable length of historical data to estimate contemporary volatility. The method is based on adaptation of a procedure used in statistical quality control - a hypothesis, that data contains a changepoint of volatility, is tested and if the test gives a positive answer, the changepoint is estimated. Then, a period of data where no changepoint is statistically significant is used to estimate contemporary volatility. The approach is illustrated on an analysis of CZK/EUR exchange rates.