Statistical Inference For Some Financial Time Series Models

This dissertation focuses on the estimation problems of the moderate devi-ations from a unit root model, studies a varying coefficient partially nonlinear model in which the regressors are generated by the multivariate unit root process-es and complements the asymptotic properties of the locally modelled regression for functional data.Firstly, we obtain the asymptotic properties of the LAD estimation for the moderate deviations from a unit root model. We provide the asymptotic nor-mality for the mildly integrated case and the Cauchy limit theory for the mildly explosive case.Secondly, we consider the quantile estimate for the moderate deviations from a unit root model with possibly infinite variance errors. We derive the asymptotic distribution for both the mildly integrated case and mildly explosive case. Numerical simulations illustrate that when the variance of the errors are infinite, the quantile estimate is better than the ordinary least square estimate, and verify our limit results.Thirdly, we investigate a random coefficient AR(1) model with possibly in-finite variance errors and provide a pivot for the estimate. A simple simulation study is presented to illustrate our limit results.Fourthly, we study a varying coefficient partially nonlinear model in which the regressors are generated by the multivariate unit root processes. A profile nonlinear least squares estimation procedure is applied to estimate the parameter vector and the functional coefficients. Under some mild conditions, the asymp-totic distribution theory for the resulting estimators is established. The rate of convergence for the parameter vector estimator depends on the properties of the nonlinear regression function. While, the rate of convergence for the function-al coefficients estimator is the same and enjoys the super-consistency property. Furthermore, a simulation study is conducted to investigate the finite sample performance of the proposed estimating procedures.Finally, we focus on the nonparametric regression of a scalar response on a functional explanatory variable. We establish the asymptotic normality of the locally modelled regression. We also adapt the empirical likelihood method to construct the point-wise confidence intervals for the regression function. Fur-thermore, a simulation study is presented to illustrate our asymptotic normality results and compare the performances of the confidence intervals based on the empirical likelihood inference and the asymptotic normality.