Inferences in volatility models

Tagore, Vickneswary (2010) Inferences in volatility models. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

In some real life time series, especially in financial time series, the variance of the responses over time appear to be non-stationary. The changes in the variances of such data are usually modeled through a dynamic relationship among these variances, and subsequently the responses are modeled in terms of the non-stationary variances. This type of time series model is referred to as the stochastic volatility model. However, obtaining the consistent and efficient estimators for the parameters of such a model has been proven to be difficult. Among the existing estimation approaches, the so-called generalized method of moments (GMM) and the quasi-maximum likelihood (QML) estimation techniques are widely used. In this thesis, we introduce a simpler method of moments (SMM), which, unlike the existing GMM approach, does not require an arbitrarily large number of unbiased moment functions to construct moment estimating equations for the parameters involved. We also demonstrate numerically that the proposed SMM approach is asymptotically more efficient than the existing QML approach. We also provide another simpler 'working' generalized quasi likelihood (WGQL) approach which is similar but different than the SMM approach. Furthermore, the small and large sample behavior of the SMM and WGQL approaches are examined through a simulation study. The effect of the SMM estimation approach is also examined for kurtosis estimation. -- In volatility models mentioned above, the responses are assumed to be uncorrected. However, in some situations, it may happen that the responses become influenced by certain time dependent covariates, and as opposed to the standard stochastic volatility models, the responses become correlated. In the later part of thesis, we introduce an observation-driven dynamic (ODD) regression model with non-stationary error variances, these variances being modeled as in the standard stochastic volatility models. We refer to such a model as the observation driven dynamic-dynamic (dynamic²) (ODDD) volatility model. The parameters of this wider model are estimated by using a hybrid estimation technique by combining the GQL and SMM approaches.

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