Three Essays on Non-stationary Time Series

Open Access
Author:
Li, Xiaoye
Graduate Program:
Statistics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
September 16, 2013
Committee Members:
  • Zhibiao Zhao, Dissertation Advisor
  • Runze Li, Committee Chair
  • Fuqing Zhang, Committee Member
  • Jingzhi Huang, Special Member
Keywords:
  • Modulated Stationary
  • Self Normalization
  • Change-point Test
  • Long-run Variance
  • Autocovariance
  • Time-varying exogenous AR model
  • Sieve-wild Bootstrap
Abstract:
We study statistical inference for a class of non-stationary time series with time-dependent variances. Due to non-stationarity and the large number of unknown parameters, existing methods that are developed for stationary or locally stationary time series are not applicable. Based on a self-normalization technique, we address several inference problems, including self-normalized Central Limit Theorem, self-normalized cumulative sum test for change-point problem, long-run variance estimation through blockwise self-normalization, and self-normalization based wild bootstrap for non-stationary time series. Monte Carlo simulation studies show that the proposed self-normalization based methods outperform stationarity based alternatives. We demonstrate the proposed methodology using two real data sets: annual mean precipitation rates in Seoul during 1771–2000, and quarterly U.S. Gross National Product growth rates during 1947–2002. In the literature on change-point analysis, much attention has been paid to detecting changes in certain marginal characteristics, such as mean, variance, and marginal distribution. For time series data with nonparametric time trend, we study the change-point problem for the autocovariance structure of the unobservable error process. To derive the asymptotic distribution of the cumulative sum test statistic, we develop substantial theory for uniform convergence of weighted partial sums and weighted quadratic forms. Our asymptotic results improve upon existing works in several important aspects. The performance of the test statistic is examined through simulations and an application to interest rates data. To model the frequently observed nonstationarity phenomena in social and scientific fields, we propose a class of time-varying exogenous autoregressive models. While the model exhibits nonparametric time-varying dependence structure over a long time span, the model dynamics possess local stationarity within each small time interval. Furthermore, the model can incorporate important external nonstationary inputs to model the main time series of interest. These features allow for theoretical tractability as well as flexible applications. For nonparametric estimation of the coefficient functions, it is shown that the local linear estimation can adapt to the unknown nonstationarity, whereas the local constant estimation is strongly affected by the local stationarity. Some practically important inference and hypothesis testing problems are investigated. To better take into account the nonstationarity and dependence, we propose a sieve-wild bootstrap by combining the ideas from both the sieve and the wild bootstrap. The methodology is illustrated through both Monte Carlo simulations and an application to the stock return-inflation puzzle using the S&P 500 index and Consumer Price Index data during 1982–2012.