Fractionally integrated processes and structural changes: theoretical analyses and bootstrap methods
Chang, Seong Yeon
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The first chapter considers the asymptotic validity of bootstrap methods in a linear trend model with a change in slope at an unknown time. Perron and Zhu (2005) analyzed the consistency, rate of convergence, and limiting distributions of the parameter estimates in this model. I provide theoretical results for the asymptotic validity of bootstrap methods related to forming confidence intervals for the break date. I consider two bootstrap schemes, the residual (for white noise errors) and the sieve bootstrap (for correlated errors). Simulation experiments confirm that confidence intervals obtained using bootstrap methods perform well in terms of exact coverage rate. The second chapter extends Perron and Zhu's (2005) analysis to cover more general fractionally integrated errors with memory parameter d in the interval (-0.5,1.5). My theoretical results uncover some interesting features. For example, with a concurrent level shift allowed, the rate of convergence of the estimate of the break date is the same for all values of d in the interval (-0.5,0.5), a feature linked to the contamination induced by allowing a level shift. In all other cases, the rate of convergence is decreasing as d increases. I also provide results about the spurious break issue. The third chapter considers constructing confidence intervals for the break date in linear regressions. I compare the performance of various procedures in terms of the exact coverage rates and lengths: Bai's (1997) based on the asymptotic distribution with shrinking shifts, Elliott and Müller's (EM) (2007) based on inverting a test locally invariant to the magnitude of the change, Eo and Morley's (2013) based on inverting a likelihood ratio test, and various bootstrap procedures. In terms of coverage rates, EM's approach is the best but with a high cost in terms of length. With serially correlated errors and a change in intercept or in the coefficient of a regressor with a high signal-to-noise ratio, or when a lagged dependent variable is present, the length approaches the whole sample as the magnitude of the change increases. This drawback is not present for the other methods. Theoretical results are provided to explain the drawbacks of EM's method.