Exact local Whittle estimation of fractional integration

Nonstationarity-extended Whittle Estimation

For long memory time series models with uncorrelated but dependent errors, we establish the asymptotic normality of the Whittle estimator under mild conditions. Our framework includes the widely used FARIMA models with GARCH-type innovations. To cover nonstationary fractionally integrated processes, we extend the idea of Abadir, Distaso and Giraitis (2007, Journal of Econometrics 141, 1353-1384) and develop the nonstationarity-extended Whittle estimation. The resulting estimator is shown to be asymptotically normal and is more efficient than the tapered Whittle estimator. Finally, the results …

使用（拟）解析小波进行局部 Whittle 估计

Local Whittle estimation with (quasi-)analytic wavelets

针对非高斯非平稳长记忆多元时间序列，提出基于拟解析小波表示的推断方法，利用小波系数协方差和局部 Whittle 近似构建一致估计量，有效估计长记忆、长期协方差及相位参数，并在仿真与神经科学数据中验证了其有效性。

Multiple local whittle estimation in stationary systems

Moving from univariate to bivariate jointly dependent long-memory time series introduces a phase parameter $(γ)$, at the frequency of principal interest, zero; for short-memory series $γ=0$ automatically. The latter case has also been stressed under long memory, along with the ``fractional differencing'' case $γ=(δ_2-δ_1)π/2$, where $δ_1$, $δ_2$ are the memory parameters of the two series. We develop time domain conditions under which these are and are not relevant, and relate the consequent properties of cross-autocovariances to ones of the (possibly bilateral) moving average representation w…

Estimation for seasonal fractional ARIMA with stable innovations via the empirical characteristic function method

Maximum likelihood methods, while widely used, may be non-robust due to disagreement between the assumptions upon which the models are based and the true density probability distribution of observed data. Because the Empirical Characteristic Function (ECF) is the Fourier transform of the empirical distribution function, it retains all the information in the sample but can overcome difficulties arising from the likelihood. This paper discusses an estimation method via the ECF for stable seasonal fractional ARIMA processes. Under some assumptions, we show that the resulting estimators are consis…

Testing the Fractional Integration Parameter Revisited: a Fractional Dickey-Fuller Test

In this paper, in the first step, we show that the fractional Dickey-Fuller test proposed by Dolado et al [10] is useless in practice. In the second step, we propose a new testing procedure for the degree of fractional integration of a time series inspired on the unit root test of Dickey-Fuller [7]. Through a simulation study, we show the good performance of the test in terms of size and power. Finally, in order to show how to use the new testing procedure, the test is applied to the well-known Nelson and Plosser data.

Fractional integration and cointegration

In this chapter we present an overview of the main ideas and methods in the fractional integration and cointegration literature. We do not attempt to give a complete survey of this enormous literature, but rather a more introductory treatment suitable for a researcher or graduate student wishing to learn about this exciting field of research. With this aim, we have surely overlooked many relevant references for which we apologize in advance. Knowledge of standard time series methods, and in particular methods related to nonstationary time series, at the level of a standard graduate course or a…