Abstract
We consider the problem of estimating the variance of the partial sums of a stationary time series that has either long memory, short memory, negative/intermediate memory, or is the first-difference of such a process. The rate of growth of this variance depends crucially on the type of memory, and we present results on the behavior of tapered sums of sample autocovariances in this context when the bandwidth vanishes asymptotically. We also present asymptotic results for the case that the bandwidth is a fixed proportion of sample size, extending known results to the case of flat-top tapers. We adopt the fixed proportion bandwidth perspective in our empirical section, presenting two methods for estimating the limiting critical values—both the subsampling method and a plug-in approach. Simulation studies compare the size and power of both approaches as applied to hypothesis testing for the mean. Both methods perform well–although the subsampling method appears to be better sized–and provide a viable framework for conducting inference for the mean. In summary, we supply a unified asymptotic theory that covers all different types of memory under a single umbrella.
Highlights
Consider a sample Y = {Y1, Y2, · · ·, Yn} from a strictly stationary time series with mean EYt = μ, a∑uthoγchoev−airhiωan. cWe efuanrcetiionnte(raecstfe)dγhin=stCudoyvi(nYgt, Yt+h), and integrable spectral density function f the distribution of the studentized sample mean (ω) = where the normalization involves the summation of sample autocovariances weighted by an arbitrary taper, and when the stochastic process exhibits either short or long memory or even when the process is overdifferenced
The vanishing bandwidth-fraction results of Theorem 2 and Remark 5 indicate a difficulty with using tapers when Long Memory (LM) or Negative Memory (NM) is present, because it is difficult to capture the correct rate for all types of memory
We look at Gaussian processes exhibiting LM, Short Memory (SM), or NM, at a variety of sample sizes and choices of taper
Summary
Our main goal is to develop a practical procedure for conducting inference for the mean using the studentized sample mean, such that the methodology is valid when long memory is present To address this goal we derive novel results regarding the joint distributional properties of Y and WΛ,M for various stochastic processes and various tapers, distinguishing between the case that the bandwidth-fraction b is vanishing and the case that it is a constant proportion. In the former case, we obtain central limit theorems for the sample mean, while the variance estimate tends in probability to a constant when appropriately normalized (Section 3).
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