## Fachbereich Mathematik

We work in the setting of time series of financial returns. Our starting point are the GARCH models, which are very common in practice. We introduce the possibility of having crashes in such GARCH models. A crash will be modeled by drawing innovations from a distribution with much mass on extremely negative events, while in ''normal'' times the innovations will be drawn from a normal distribution. The probability of a crash is modeled to be time dependent, depending on the past of the observed time series and/or exogenous variables. The aim is a splitting of risk into ''normal'' risk coming mainly from the GARCH dynamic and extreme event risk coming from the modeled crashes. We will present several incarnations of this modeling idea and give some basic properties like the conditional first and second moments. For the special case that we just have an ARCH dynamic we can establish geometric ergodicity and, thus, stationarity and mixing conditions. Also in the ARCH case we formulate (quasi) maximum likelihood estimators and can derive conditions for consistency and asymptotic normality of the parameter estimates. In a special case of genuine GARCH dynamic we are able to establish L_1-approximability and hence laws of large numbers for the processes itself. We can formulate a conditional maximum likelihood estimator in this case, but cannot completely establish consistency for them. On the practical side we look for the outcome of estimating models with genuine GARCH dynamic and compare the result to classical GARCH models. We apply the models to Value at Risk estimation and see that in comparison to the classical models many of ours seem to work better although we chose the crash distributions quite heuristically.

Competing Neural Networks as Models for Non Stationary Financial Time Series -Changepoint Analysis-
(2005)

The problem of structural changes (variations) play a central role in many scientific fields. One of the most current debates is about climatic changes. Further, politicians, environmentalists, scientists, etc. are involved in this debate and almost everyone is concerned with the consequences of climatic changes. However, in this thesis we will not move into the latter direction, i.e. the study of climatic changes. Instead, we consider models for analyzing changes in the dynamics of observed time series assuming these changes are driven by a non-observable stochastic process. To this end, we consider a first order stationary Markov Chain as hidden process and define the Generalized Mixture of AR-ARCH model(GMAR-ARCH) which is an extension of the classical ARCH model to suit to model with dynamical changes. For this model we provide sufficient conditions that ensure its geometric ergodic property. Further, we define a conditional likelihood given the hidden process and a pseudo conditional likelihood in turn. For the pseudo conditional likelihood we assume that at each time instant the autoregressive and volatility functions can be suitably approximated by given Feedfoward Networks. Under this setting the consistency of the parameter estimates is derived and versions of the well-known Expectation Maximization algorithm and Viterbi Algorithm are designed to solve the problem numerically. Moreover, considering the volatility functions to be constants, we establish the consistency of the autoregressive functions estimates given some parametric classes of functions in general and some classes of single layer Feedfoward Networks in particular. Beside this hidden Markov Driven model, we define as alternative a Weighted Least Squares for estimating the time of change and the autoregressive functions. For the latter formulation, we consider a mixture of independent nonlinear autoregressive processes and assume once more that the autoregressive functions can be approximated by given single layer Feedfoward Networks. We derive the consistency and asymptotic normality of the parameter estimates. Further, we prove the convergence of Backpropagation for this setting under some regularity assumptions. Last but not least, we consider a Mixture of Nonlinear autoregressive processes with only one abrupt unknown changepoint and design a statistical test that can validate such changes.