Three Essays in Nonstationary Time Series Econometrics

This dissertation comprises three papers that separately study different nonstationary time series models. The first chapter considers the grid bootstrap for constructing confidence intervals for the persistence parameter in a class of continuous-time models driven by a Lévy process. Its asymptotic validity is discussed under the assumption that the sampling interval (h) shrinks to zero, the time span (N) goes to infinity or both. Its improvement over the in-fill asymptotic theory is achieved by expanding the coe¢cient-based statistic around its in-fill asymptotic distribution which is non-pivotal and depends on the initial condition. Monte Carlo studies show that the grid bootstrap method performs better than the in-.ll asymptotic theory and much better than the long-span asymptotic theory. Empirical applications to U.S. interest rate data and volatility data suggest significant differences between the bootstrap con.dence intervals and the con.dence intervals obtained from the in-fill and long-span asymptotic distributions. The second chapter studies a mildly explosive autoregression model with Anti-persistent Errors. An asymptotic distribution is derived for the least squares (LS) estimate of a first-order autoregression with a mildly explosive root and anti-persistent errors. While the sample moments depend on the Hurst parameter asymptotically, the Cauchy limiting distribution theory remains valid for the LS estimates in the model without intercept and a model with an asymptotically negligible intercept. Monte Carlo studies are designed to check the precision of the Cauchy distribution in finite samples. An empirical study based on the monthly NASDAQ index highlights the usefulness of the model and the new limiting distribution. The third chapter considers testing procedures for rational bubbles under strongly dependent errors. A heteroskedasticity and autocorrelation robust (HAR) test statistic is proposed to detect the presence of rational bubbles in financial assets when errors are strongly dependent. The asymptotic theory of the test statistic is developed. Unlike conventional test statistics that lead to a too large type I error under strongly dependent errors, the new test does not su¤er from the same size problem. In addition, it can consistently timestamp the origination and termination dates of a rational bubble. Monte Carlo studies are conducted to check the finite sample performance of the proposed test and estimators. An empirical application to the S&P 500 index highlights the usefulness of the proposed test statistic and estimators.