Asymptotic Theory For Autoregression And Moving Average(1,1) Model Under Heteroskedasticity

This thesis studies the asymptotic theory of the least squares of the ARM A(1,1)model with heteroscedasticity.Although the classic GARCH-type models are successful in capturing many important features in macroeconomic or financial time series,such as volatility clustering and persistent autocorrelation,one obvious disadvantage is its non-robustness to stationary hypotheses,time-varying volatility is exclusively attributed to conditional variance or covariance structure,and the non-conditional variance is assumed to remain constant over time.When this condition is not met,the ARCH or GARCH model can cause serious errors.An approach to modeling time-varying volatility is to use a smooth deterministic nonparametric framework,assuming that the unconditional variance is the main time-changing feature to be captured.In the ARM A(1,1)model,the error is an independent and identically distributed random variable,and the unconditional variance contains an unknown non-parametric time-varying function,namely ?t=g(t/n)ut,where g(.)is a non-parametric time-varying function,{ut} is an independent and identically distributed random variable sequence.And there exist ?>1 and C>0,so that(?).Due to the complexity of the model,the least squares estimator is only given implicit expression,namely(?).When talking about the consistency of estimator,namely(?)by Chebyshev's inequality,Minkowski's inequality and Cauchy-Schwarz's inequality,the consistency of estimator is obtained.And when talking about the asymptotic distribution of estimator,using the Taylor expansion,the(?)is expanded,which converts the proof of asymptotic distribution into the proof of asymptotic distribution of the first-order partial derivatives and the convergence of second-order partial derivatives.By the law of large numbers,the central limit theorem of L1-mixingale and Minkowski's inequality,the asymptotic normal distribution of the first-order partial derivatives and the convergence expression of the second-order partial derivatives are obtained.Finally,we have(?).