Statistical Inference For Three Types Of Nonlinear Time Series

Nonlinear time series models are widely applied in various fields,such as finance,econometrics,and biological.In this dissertation,we study statistical inference for three types of nonlinear time series models,including ARCH models with trend terms,DAR models,and non-stationary GARCH models.We propose a two-step estimation method and develop an ARCH effect test for an ARCH model with trend.The trend is estimated by the linear B-spline method,and then the least squares estimation(LSE)and a Lagrange multiplier(LM)test based on the resid-uals are presented.Under mild conditions,it is shown that the proposed two-step LSE and the test statistic enjoy the oracle properties;namely,they perform as well as if the true trend function were known and then removed to obtain the ARCH errors.Simulation studies assess the finite-sample performance of the two-step LSE and the test statistic.An empirical example on the US gross private saving series is analyzed to illustrate the usefulness of our procedure.A three-step non-Gaussian quasi-maximum likelihood estimation(TS-NGQMLE)method is proposed for the DAR model.TS-NGQMLE improves efficiency of the GQMLE and circumvents inconsistency of the NGQMLE preferably when the innova-tion is heavy-tailed.Under mild conditions,our estimator not only can achieve consis-tency and asymptotic normality regardless of density misspecification of the innovation,but also outperforms the existing estimators in the literature,such as the weighted least absolute deviations estimator(WLADE),the GQMLE,and the least absolute deviation estimators(LADE),when the innovation is indeed heavy-tailed.Monte Carlo simula-tion studies are conducted to assess the finite-sample performance of our estimator.An empirical example on the trade weighted U.S.dollar index is given.A TS-NGQMLE method is proposed for the non-stationary GARCH(1,1)model(?)ht=?0+a0?t2+?0ht-12.Under mild conditions,the TS-NGQMLE of(?0,?0)is strongly consistent and asymptotically normal.Theoretical and numerical simu-lation studies show that for(?0,?0),the TS-NGQMLE outperforms the existing estimator-GQMLE,when the innovation ?t is heavy-tailed distributed.