Bootstrap Method In ARCH Model

Autoregressive Conditional Heteroskedasticity model is the newly developed time series model, which reflects the special characteristics of stochastic process: the variance changes with the time and the variance is crowed together and fluctuated. ARCH models has been widely applied in modeling and research of economic field, especially of financial markets. For this kind of modeling, people care for not only the estimation of regressive coefficient but also the estimation of unknown parameters in conditional skedasticity. The statistical characters of parameters estimation for ARCH models are valid under asymptotic cases. In fact, the number of sample is finite, so we can test the stability of estimation of parameters by using Bootstrap method repeatedly to enlarge the number of samples. Because standard residual-based bootstrap methods of inference for autoregressions treat the error term as independent and identically distributed and are invalidated by conditional heteroskedasticity. When conditional heteroskedasticity exists, we need to improve the standard residual-based bootstrap method. Usually, recursive-design wild bootstrap, fixed-design wild bootstrap and pairwise bootstrap are used.In this thesis, we prove the validity by using recursive-design wild bootstrap, fixed-design wild bootstrap and pairwise bootstrap twice respectively into autoregressions model with m.d.s. errors subject to possible conditional heteroskedasticity of unknown.The following are main results of this thesis:Theorem 2.1 Under Assumption A , it follows thatwhere P** denotes the probability measure induced by using the recursive-design WB twice.Theorem 2.2 Under Assumption A', it follows thatwhere F**denotes the probability measure induced by using the fixed-design WB twice. Theorem 2.3 Under Assumption A, it follows thatwhereF**denotes the probability measure induced by using the pairwise bootstrap twice.