| The dissertation studies three different topics in time series. They are based on a common ground of theories of continuous martingales, Brownian motion, and asymptotic behavior of additive functionals of Brownian motion. In Chapter I, we investigate a t-test for nonlinearity of an integrated I(1) process with a flexible sampling scheme, and develop a theory for asymptotic bootstrap refinements of the test statistic by using the second-order asymptotic expansion. It is found that regardless of postulated nonlinear functional forms and/or sampling schemes, the second-order asymptotic expansions of the test always provide refinements for the asymptotic distributions. The bootstrap refinements, therefore, will inherit the same order of the asymptotic refinements. Chapter II proposes residual-based tests for a longrun relationship among nonstationary time series, i.e. cointegration, in dependent panels. The nonlinear transformations of the adaptively fitted lagged residuals that are obtained from the postulated cointegrating relationships across cross-sections are used as instruments to build the t-ratios for testing the unit root of the panels of the fitted residuals. Such a nonlinear Instrumental Variable (IV) t-ratios are asymptotically normal and cross-sectionally independent under the null hypothesis of no cointegration. The average, minimum, and maximum of the IV t-ratios are considered to test for a fully non-cointegrated panel against a mixed panel, as well as for a mixed panel against a fully cointegrated panel. Finally, Chapter III discusses specification tests for conditional means of asset pricing models in continuous time which cover both stationary and nonstationary stochastic processes. We neither impose any requirement on specifications of the volatility component nor need to estimate an unknown true distribution of the stochastic process. The tests are constructed by combining time change and martingale transformation to obtain asymptotically distribution-free statistics. Under the null of a correct specification, the time change applied on the residuals effectively accounts for various types of volatilities, including but not limited to fat tail, time-varying or stochastic volatilities, and transforms them into a process of iid standard normals. Then, the Khmaladze martingale transformation eliminates the estimators's effect on the fitted time-changed residuals to obtain the asymptotically distribution-free statistics. For each Chapter, simulations are provided to illustrate the corresponding theories. |
