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Description:
This dissertation consists of two parts. In the first part, the subject of hypothesis testing is addressed. Here, non-standard formulations of the null hypothesis are discussed, e.g., non-stationarity under the null, and boundary hypotheses. In the second part, stochastic models for financial markets are developed and studied. Particular emphasis is placed on the application of Cox processes. Part one begins with a survey of time-series models which allow for conditional heteroscedasticity and autoregression, AR-GARCH models. These models reduce to a white noise model, when some of the conditional heteroscedasticity parameters take their boundary value at zero, and the autoregressive component is in fact not present. The asymptotic distribution of the pseudo-log-likelihood ratio statistics for testing the presence of conditional heteroscedasticity and the autoregression term is reproduced. For financial market data, the model parameters are estimated and tests for the reduction to white noise are performed. The impact of these results on risk measurement is discussed by comparing several Value-at-Risk calculations assuming the alternative model specifications. Furthermore, the power function of these tests is examined by a simulation study of the ARCH(1) and the AR(1)-ARCH(1) models. First, the simulations are carried out assuming Gaussian innovations and then, the Gaussian distribution is replaced by the heavy tailed t-distribution. This reveals that a substantial loss of power is associated with the use of heavy tailed innovations. A related testing problem arises in the analysis of the Ornstein-Uhlenbeck (OU) model, driven by Levy processes. This model is designed to capture mean reverting behaviour if it exists; but the data may in fact be adequately described by a pure Levy process with no OU (autoregressive) effect. For an appropriate discretized version of the model, likelihood methods are utilized to test for such a reduction of the OU process to Levy motion, deriving the distribution of the relevant ...