Non-Gaussian Log-Periodogram Regression

Abstract

We show the consistency of the log-periodogram regression estimate of the long memory parameter for long range dependent linear, not necessarily Gaussian, time series when we make a pooling of periodogram ordinates+ Then, we study the asymptotic behavior of the tapered periodogram of long range dependent time series for frequencies near the origin, and we obtain the asymptotic distribution of the log-periodogram estimate for possibly non-Gaussian observation when the tapered periodogram is used+ For these results we rely on higher order asymptotic properties of a vector of periodogram ordinates of the linear innovations+ Finally, we assess the validity of the asymptotic results for finite samples via Monte Carlo simulation+

Non-stationary log-periodogram regression

“…We will show that tapering allows a reduction of the order of magnitude of the bounds in Theorem 1, so we can estimate bigger values of d. Thus, with the cosine bell taper all the results go through for any d( , since this data taper achieves a reduction of the overall bias from Robinson's (1995) results if f is smooth enough. This was observed by Velasco (1997) for a related problem with non-Gaussian stationary time series. However, as we will see in next section, the full advantage of the tapering improvement in the convergence in the tails of the spectral kernel, only shows up when we use Assumption 2 with *1, increasing the smoothness of the function f near the origin.…”

“…The proof of this theorem follows the lines of the previous one with p"3, though the cosine bell is not of order 3, but shares properties (25) and (A.2) with p"3, except for the integration of the convolutions around the origin of f. Alternatively, we can use the proof of Theorem 2 in Velasco (1997) for the tapered Fourier transform using the cosine bell for stationary processes, taking special care of those intervals.…”

“…The extension of the asymptotic theory given for the log-periodogram estimate to non-Gaussian time series could be tried under related conditions to those used in Velasco (1997).…”

“…This last expression is now equivalent to Assumption 3 in Robinson (1994a) and was used also by Velasco (1997) to study the behaviour of the tapered periodogram for stationary long memory time series. Both assumptions are satis"ed with "2 if f C is the spectral density of a stationary, invertible fractional ARIMA process or fractional Gaussian noise, when d' , so d!13(!…”

Long Memory in Nonlinear Processes

“…standard complex Gaussian have been proposed and theoretically justifed in some cases. The most well known is the GPH estimator of the Hurst index, introduced by [GPH83] and proved consistent and asymptotically Gaussian for Gaussian long memory processes by [Rob95b] and for a restricted class of linear processes by [Vel00]. Another estimator, often referred to as the local Whittle or GSE estimator was introduced by [Kün87] and again proved consistent asymptotically Gaussian by [Rob95a] for linear long memory processes.…”