Abstract
The general theme of this thesis is theoretical properties and evaluation of volatility models. The thesis consists of four papers. In the first chapter the moment structure of the EGARCH model is derived. The second chapter contains new results on the A-PARCH model. The third chapter is about certain stylized facts of financial time series and the idea is to investigate how well the GARCH, EGARCH and ARSV models are able to reproduce these characteristics. The fourth chapter is about evaluating the EGARCH model. A more detailed overview of the chapters follows next. In Chapter 1 we derive the condition for the existence of moments, the expression for the kurtosis and the one for the autocorrelation function of positive powers of the absolute-valued observations for the EGARCH model. The results of the paper are useful, for example, if we want to compare the EGARCH model with the GARCH model. They reveal certain differences in the moment structure between these models. While the autocorrelations of the squared observations decay exponentially in the GARCH model, the decay rate is not exponential in the EGARCH model. While for the GARCH model the conditions for parameters allowing the existence of higher-order moments become more and more stringent for each even moment this is not the case for the EGARCH model. The explicit expressions of the autocorrelation structure of the positive powers of the absolute-valued observations of the model are particularly important in the considerations of Chapter 3 of the thesis.The A-PARCH model contains a particular positive power parameter. By letting the power parameter approach zero, the A-PARCH family of models also includes a family of EGARCH models as a special case. In Chapter 2 we derive the autocorrelation function of squared and logarithmed observations for the A-PARCH family of models and show that it may be obtained as a limiting case of a general power ARCH (GPARCH) model. An interesting thing to notice is that the autocorrelation structure of this GPARCH process, if it exists, is exponential, and that this property is retained at the limit as the power parameter approaches zero, which means that the autocorrelation function of the process of logarithms of squared observations also decay exponentially. While this is true for the logarithmed squared observations of an EGARCH process it cannot simultaneously be true for the untransformed observations defined by these processes as we in Chapter 1 have demonstrated.In order to explain the role of the power parameter we present a detailed analysis of how the autocorrelation functions of the squared observations differ across members of the GPARCH models. In an empirical example we also show that the estimated power parameter considerably improves the correspondence between the estimated autocorrelations on the one hand and the autocorrelation estimates from the model on the other. Financial time series seem to share a number of characteristic features, sometimes called stylized facts. Given a set of stylized facts, one may ask the following question: "Have popular volatility models been parameterized in such a way that they can accommodate and explain the most common stylized facts visible in the data?" Models for which the answer is positive may be viewed as suitable for practical use. In Chapter 3, possible answers to this question for the three popular models of volatility, GARCH, EGARCH and ARSV models are investigated. Model evaluation is an important part of modelling not only for the conditional mean models but for the conditional variance specifications as well. In Chapter 4 we consider misspecification tests for an EGARCH model. We derive two new misspecification tests for an EGARCH model. Because the tests of an EGARCH model against a higher-order EGARCH model and testing parameter constancy are parametric, the alternative may be estimated if the null hypothesis is rejected. This is useful for a model builder who wants to find out possible weakness of estimated specification. Furthermore, we investigate various ways of testing the EGARCH model against GARCH ones as another check of model adequacy. An empirical example shows that there is substantial evidence for parameter nonconstancy in daily return series of the Stockholm Stock Exchange.