C51 - Model Construction and EstimationReturn

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.

Volatility of the financial time series belongs to the crucial estimated parameters in finance (e.g. in risk management, derivative pricing). It is well known, that volatility varies in time, so that new approaches of volatility modeling have appeared. In this paper two models of the conditional heteroskedasticity - fractionally integrated GARCH (FIGARCH) and EWMA are presented. These models are illustrated on the daily historical returns of stock index PX and index BUX. Standard tests of normality, autocorrelation and conditional heteroskedasticity are applied to these log-return time series and before estimating the models, which confirm a usability of the conditional heteroskedasticity models. Empirical results of the Rescale Range analysis (R/S) indicate a long memory in the volatility process of PX index and the first 40 autocorrelations of the square log-returns show their hyperbolic decay. The volatility models are estimated by quasi-maximum likelihood method with Student's t-distribution and used to the calculation of the 1-day 95% and 99% Value at Risk values. Finally, the validity of the models is verified by Kupiec's test, TUFF and Christoffersen's test. These tests demonstrate, that the FIGARCH model is a suitable alternative to the EWMA model in the Value at Risk calculation.

We examine whether it is rational to put limit orders in a limit order market. We find that limit orders are not needed and may be even disadvantageous given that the agent trades on-line. Further, we present a numerical study indicating that putting limit orders may be optimal given that the agent trades at discrete times but the benefit from using them in comparison with immediate buying and selling is negligible.