Velasco, Carlos
(1996)
Higher order asymptotic theory for nonparametric time series analysis and related contributions.
PhD thesis, London School of Economics and Political Science.
Abstract
We investigate higher order asymptotic theory in nonparametric time series analysis. The aim of these techniques is to approximate the finite sample distribution of estimates and test statistics. This is specially relevant for smoothed nonparametric estimates in the presence of autocorrelation, which have slow rates of convergence so that inference rules based on first-order asymptotic approximations may not be very precise. First we review the literature on autocorrelation-robust inference and higher order asymptotics in time series. We evaluate the effect of the nonparametric estimation of the variance in the studentization of least squares estimates in linear regression models by means of asymptotic expansions. Then, we obtain an Edgeworth expansion for the distribution of nonparametric estimates of the spectral density and studentized sample mean. Only local smoothness conditions on the spectrum of the time series are assumed, so long range dependence behaviour in the series is allowed at remote frequencies, not necessary only at zero frequency but at possible cyclical and seasonal ones. The nonparametric methods described rely on a bandwidth or smoothing number. We propose a cross-validation algorithm for the choice of the optimal bandwidth, in a mean square sense, at a single point without restrictions on the spectral density at other frequencies. Then, we focus on the performance of the spectral density estimates around a singularity due to long range dependence and we obtain their asymptotic distribution in the Gaussian case. Semiparametric inference procedures about the long memory parameter based on these nonparametric estimates are justified under mild conditions on the distribution of the observed time series. Using a fixed average of periodogram ordinates, we also prove the consistency of the log-periodogram regression estimate of the memory parameter for linear but non-Gaussian time series.
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