An additive -interactive nonlinear volatility model: Its testing and estimation
In this thesis a new separable nonparametric volatility model related to the generalized additive nonlinear ARCH model (GANARCH) of Kim and Linton (2004)  is studied. Unlike the GANARCH model, the proposed additive-interactive nonlinear ARCH model (referred to as AINARCH model) does not assume the known link function but includes second-order interaction terms in both mean and variance functions instead. The assumption of the known link function implies knowing the distribution of the data which is often not easy to verify, especially in the multivariate case; thus, it can be said that the new model imposes less difficulty to verify assumptions. Motivated by the local instrumental variable approach, I propose an instrumental variable-based estimation method for both additive and interactive mean and variance component functions in the AINARCH model. The method is computationally more effective than most other nonparametric estimation methods that can potentially be used to estimate components of such a model. Asymptotic behavior of the resulting estimators is investigated and their asymptotic normality is established. Explicit expressions for asymptotic means and variances of these estimators are also obtained. Simulation experiments provide strong evidence that these estimators are well-behaved in finite samples as well. ^ A procedure that tests the joint significance of the interactive functional components in both mean and variance functions is proposed. The procedure is based on a special grouping of the data that allows for the ANOVA-type analysis. The asymptotic property of the test statistic under the null hypothesis is established and the test performance in finite samples is studied by simulation. The selection of parameters is discussed and a data-driven parameter selection procedure is suggested. ^ The application of the AINARCH model and the testing method is illustrated by a real data example on currency exchange rates. ^
Michael Levine, Purdue University.
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