Statistical Inference For (Nearly) Unit Root Models With Long Memory Or Possibly Heavy-Tailed Innovations

Since Dickey and Fuller (1979) proposed DF unit root test, its test theory has been further developed. However, the traditional unit-root tests assumed the coefficient to be constant, in which major economic events (such as great economic crisis, oil crisis,etc.) are likely to cause structural break of unit root when it comes to the external environment changes. Therefore, this paper firstly studies unit-root test with structural break points, combined with long-memory properties of financial data. As for the AR(1) model with mean shift and long memory innovations, it is showed that, under some assumptions,the asymptotic distribution of the estimator of the auto-regression coefficient could be expressed as a functional of fractional Brownian motion, whose forms include means μ1and μ2before and after shift and fraction shift τ*. Then, the theorem is verified by using simulated data and empirical data.Between stationary process and unit-root process, there is a kind of process called nearly unit-root (also known as almost non-stationary) process, of which the value of studying its statistical properties is to reveal the special properties of this middle ground, and the statistical properties’differences with smooth process and unit-root process. Therefore, nearly unit-root process occupies an important position in the time series theory. Because of heavy-tailed nature of financial data, error terms with heavy-tailed nearly unit root process has been considered in this paper. For the least square estimation of nearly unit-root process with heavy-tailed error terms, the variance calculation by asymptotic distribution function of the estimation is too troublesome to benefit further interval estimation and statistical inference. The paper has taken estimated parameters based on residual bootstrap as resampling, constructed bootstrap statistics, and theoretically proved that the bootstrap statistics can better approximate the original least squares estimator. In the mean time, by simulation of statistics generated from usual bootstrap resampling and m-out-of-n bootstrap resampling, compared to conventional sample, it is showed that the residual-based m-out-of-n bootstrap resampling is more effective.