Whittle Estimator Research Articles

Some Improvements of the Bootstrap over the Delta Method Probability Errors for Whittle Estimators

The purpose of this paper is to compare the coverage probability errors of the parametric bootstrap with that of the delta method for the covariance parameters of a regression model with auto-regressive fractionally integrated moving average (ARFIMA) errors. We consider the coverage probability errors of both confidence intervals (CIs) and tests based on the the plug-in Whittle maximum likelihood (PWML) estimators. We first show that, under some sets of conditions on the regression coefficients, the spectral density function, and the parameter values, the bounds on the coverage probability errors of the two-sided delta method and parametric bootstrap confidence intervals on the plug-in Whittle likelihood estimator or the covariance parameter are shown to be $O(n^{-1})$ and $o(n^{-3/2}\ln{n})$, respectively, where n is the sample size. Next, we show that those of the one-sided parametric bootstrap confidence intervals are shown to be $O(n^{-1/2})$ and $o(n^{-1}\ln{n})$, respectively. These results show that for both one-sided and two-sided confidence intervals and tests, the bootstrap provides a significant improvement over that of the delta method.

Frequency domain local bootstrap in short and long memory time series

Bootstrap methods in the frequency domain are effective instruments to approximate the distribution of many statistics of weakly dependent (short memory) series but their validity with long memory remains mostly unsolved. This article proposes a frequency domain local bootstrap (FDLB) based on resampling a locally studentized version of the periodogram in a neighborhood of the frequency of interest. A bound of the Mallows distance between the distributions of the periodogram and its FDLB bootstrap counterpart is offered for anti-persistent, weakly dependent, stationary, and non stationary long memory series. This result is in turn used to justify the use of the FDLB for some statistics that are weighted averages of periodogram ordinates. Finally, the validity of the FDLB to estimate the distribution of the local Whittle estimator is proved and its finite sample behavior analyzed in a Monte Carlo and in an empirical application, comparing its performance with rival alternatives.

Local Whittle estimation in time‐varying long memory series

The memory parameter is usually assumed to be constant in traditional long memory time series. We relax this restriction by considering the memory a time‐varying function that depends on a finite number of parameters. A time‐varying Local Whittle estimator of these parameters, and hence of the memory function, is proposed. Its consistency and asymptotic normality are shown for locally stationary and locally non‐stationary long memory processes, where the spectral behaviour is restricted only at frequencies close to the origin. Its good finite sample performance is shown in a Monte Carlo exercise and in two empirical applications, highlighting its benefits over the fully parametric Whittle estimator proposed by Palma and Olea (2010). Standard inference techniques for the constancy of the memory are also proposed based on this estimator.

A robust M-estimator for Gaussian ARMA time series based on the Whittle approximation

In the frequency domain, the Whittle approach using the classical periodogram is traditionally used to estimate the parameters of a stationary time series. The M-periodogram has recently become an alternative tool for analyzing the dependence of time series. It is particularly useful for dealing with outliers and heavy-tailed noise. This paper proposes an alternative to the standard Whittle approach, the M-Whittle estimator, built from the M-periodogram. We show that the proposed method is a consistent estimator of the true parameters of an ARMA process. A finite sample investigation is carried out to assess the performance of the estimator in the scenarios of contaminated and uncontaminated time series. As expected, for the uncontaminated data, the M-Whittle estimator performs similarly to the classical Whittle approach. However, the superiority of the first method is clear in terms of root mean squared error when the series has additive outliers. Two applications are considered to illustrate the methodologies in real data contexts. Regardless of whether or not the data are contaminated by additive outliers, the results presented here strongly motivate using the M-Whittle estimator in practical problems.

Whittle likelihood estimation in INAR(p) process

The frequency domain analysis is considered in the stationary INAR(p) process under a general innovation. The Whittle likelihood from the demeaned data is not identifiable with respect to the innovation mean and variance. Using an identifiable version of the Whittle likelihood, some asymptotic properties of the Whittle estimator are established. Also, simulation experiments are conducted to illustrate the Whittle likelihood estimation.

On the use of the cumulant generating function for inference on time series

Innovative inference procedures for analyzing time series data are introduced. The methodology covers density approximation and composite hypothesis testing based on Whittle's estimator, which is a widely applied M-estimator in the frequency domain. Its core feature involves the cumulant generating function of Whittle's score obtained using an approximated distribution of the periodogram ordinates. A testing algorithm not only significantly expands the applicability of the state-of-the-art saddlepoint test, but also maintains the numerical accuracy of the saddlepoint approximation. Connections are made with three other prevalent frequency domain techniques: the bootstrap, empirical likelihood, and exponential tilting. Numerical examples using both simulated and real data illustrate the advantages and accuracy of the saddlepoint methods.

Simplified Whittle Estimators for Spectral Parameters of Stationary Linear Models with Tapered Data

Simplified Whittle Estimators for Spectral Parameters of Stationary Linear Models with Tapered Data

Fast and asymptotically-efficient estimation in an autoregressive process with fractional type noise

This paper considers the joint estimation of the parameters of a first-order fractional autoregressive model. A one-step procedure is considered in order to obtain an asymptotically-efficient estimator with an initial guess estimator with convergence speed lower than n and singular asymptotic joint distribution. This estimator is computed faster than the maximum likelihood estimator or the Whittle estimator and therefore allows for faster inference on large samples. The paper also illustrates the performance of this method on finite-size samples via Monte Carlo simulations.

Unraveling the Dynamics of Inflation Persistence: A Long-Memory Analysis of Headline Inflation Convergence and Divergence in European Economies

This comprehensive study delves into the Harmonised Consumer Price Index (HCPI) behavior across European economies from the early 2000s to the post-pandemic period. By analyzing the patterns of convergence and divergence in HCPI persistence, the study aims to uncover the common external and country-specific factors driving these dynamics. Utilizing the local Whittle estimation method, the research examines the long-memory properties of headline inflation through monthly data from January 1996 to January 2023. The study expands upon existing knowledge of inflation persistence by exploring its changes and the various patterns of inflation persistence in Europe. The findings reveal significant differences in the persistence of inflation and the underlying factors behind it. As such, the study emphasizes the need for tailored policy approaches and a deeper understanding of inflation's true nature, expectation formation, price-setting behavior, and monetary policy in shaping inflation persistence. Given its important policy implications, this research is highly relevant to the European Central Bank (ECB) and national governments within the Eurozone.

Local Whittle estimation of high-dimensional long-run variance and precision matrices

Local Whittle estimation of high-dimensional long-run variance and precision matrices

Test of change point versus long‐range dependence in functional time series

In the context of functional time series, we propose a significance test to distinguish between short memory with a change point and long range dependence. The test is based on coefficients of projections onto an optimal direction that captures the dependence structure of the latent stationary functions that are not observable due to a potential change point. The optimal direction must be estimated as well. The test statistic is constructed using the local Whittle estimator applied to these coefficients. It has standard normal distribution under the null hypothesis (change point) and diverges to infinity under the alternative (long range dependence). The article includes asymptotic theory, a simulation study and an application to curve‐valued time series derived from intraday asset prices.

Local Whittle estimation with (quasi‐)analytic wavelets

In the general setting of long‐memory multivariate time series, the long‐memory characteristics are defined by two components. The long‐memory parameters describe the autocorrelation of each time series. And the long‐run covariance measures the coupling between time series, with general phase parameters. It is of interest to estimate the long‐memory, long‐run covariance and general phase parameters of time series generated by this wide class of models although they are not necessarily Gaussian nor stationary. This estimation is thus not directly possible using real wavelets decomposition or Fourier analysis. Our purpose is to define an inference approach based on a representation using quasi‐analytic wavelets. We first show that the covariance of the wavelet coefficients provides an adequate estimator of the covariance structure including the phase term. Consistent estimators based on a local Whittle approximation are then proposed. Simulations highlight a satisfactory behavior of the estimation on finite samples on multivariate fractional Brownian motions. An application on a real neuroscience dataset is presented, where long‐memory and brain connectivity are inferred.

Whittle likelihood estimation in INAR(1) process

The frequency domain analysis is considered in the stationary INAR(1) process under a general innovation. The Whittle likelihood from the demeaned data is shown to be not identifiable with respect to the innovation mean and variance. Using an identifiable version of the Whittle likelihood, the strong consistency and asymptotic normality of the Whittle estimator are established. Furthermore, the Wald-type test about the equidispersion is proposed on the basis of the estimators for the innovation mean and variance.

Efficient Generators of the Generalized Fractional Gaussian Noise and Cauchy Processes

In the last years of the past century, complex correlation structures were empirically observed, both in aggregated and individual traffic traces, including long-range dependence, large-timescale self-similarity and multi-fractality. The use of stochastic processes consistent with these properties has opened new research fields in network performance analysis and in simulation studies, where the efficient synthetic generation of samples is one of the main topics. Nowadays, networks have to support data services for traffic sources that are poorly understood or still insufficiently observed, for which simple, reproducible, and good traffic models are yet to be identified, and it is reasonable to expect that previous generators could be useful. For this reason, as a continuation of our previous work, in this paper, we describe efficient and online generators of the correlation structures of the generalized fractional noise process (gfGn) and the generalized Cauchy (gC) process, proposed recently. Moreover, we explain how we can use the Whittle estimator in order to choose the parameters of each process that give rise to a better adjustment of the empirical traces.

Evaluation of European Union carbon and energy networks: Evidence from market efficiency

This article investigates dynamic networks of the informational efficiency between European Union carbon and energy markets in a quantile-based framework. The market (in)efficiency degree is gauged by capturing the potentially fractional integration order of the market price series using the Feasible Exact Local Whittle estimator on a rolling basis. Rather than relying on a mean-based method, we employ the causality-in-quantiles approach, through which the cross-market linkage of the inefficiency degree is uncovered in a full-distributional characterization. The dynamic linkage is identified through the perspectives of data in return and volatility, respectively. We find that both carbon and energy markets are inefficient, while the inefficiency degree of the former is slightly higher. The inefficiency degree of the two types of markets is featured by asymmetry and non-linearity over the data distribution, while its cross-market linkage is found to be bi-directional. The spillover degree of the dynamic linkage remains the highest at around the median quantile of the “information giver,” while such the degree weakens at the extreme quantiles. Our findings possess insightful implications for effective operations of the emission trading toward a low carbon transition.

Testing the law of one-price in the US gasoline market: a long memory approach

This paper aims to shed new light on“law of one-price” in the United States’ gasoline market over the period June-2003–December-2019. Specifically, we test for convergence of the retail prices of gasoline in different US PADDs, states and cities using the Local Whittle estimator (LW) and its variants (Robinson 1995, Shimotsu and Phillips 2005). Mean/trend reversion of the relative price of each unit will imply convergence toward the average price in the US thereby offering support for the law of one-price. LW estimators allow us to consider the case where relative gasoline price are fractionally integrated process and may display long memory implying a slow process of convergence. The results obtained generally offer support for the law of one-price albeit with significant differences in the rate of convergence between PADDs, States and Cities. In detail, a slow convergence (long memory process) is observed for PADD3 (Gulf Coast), PADD5 (West Coast) and then confirmed at the level of States (California, Texas and Washington) and at the level of Cities (Los Angeles, San Francisco and Houston). Among the cities, two of four empirical model employed find a slow convergence process is also in the cities of New York, Cleveland and Miami.

Asymptotic normality of wavelet covariances and multivariate wavelet Whittle estimators

Multivariate processes with long-range dependence properties can be encountered in many fields of application. Two fundamental characteristics in such frameworks are long-range dependence parameters and correlations between component time series. We consider multivariate long-range dependent linear processes, not necessarily Gaussian. We show that the covariances between the wavelet coefficients in this setting are asymptotically Gaussian. We also study the asymptotic distributions of the estimators of the long-range dependence parameter and the long-run covariance by a wavelet-based Whittle procedure. We prove the asymptotic normality of the estimators, and we provide an explicit expression for the asymptotic covariances. An empirical illustration of this result is proposed on a real dataset of rat brain connectivity.

Empirical spectral processes for stationary state space models

In this paper, we consider function-indexed normalized weighted integrated periodograms for equidistantly sampled multivariate continuous-time state space models which are multivariate continuous-time ARMA processes. Thereby, the sampling distance is fixed and the driving Lévy process has at least a finite fourth moment. Under different assumptions on the function space and the moments of the driving Lévy process we derive a central limit theorem for the function-indexed normalized weighted integrated periodogram. Either the assumption on the function space or the assumption on the existence of moments of the Lévy process is weaker. Furthermore, we show the weak convergence in both the space of continuous functions and in the dual space to a Gaussian process and give an explicit representation of the covariance function. The results can be used to derive the asymptotic behavior of the Whittle estimator and to construct goodness-of-fit test statistics as the Grenander–Rosenblatt statistic and the Cramér–von Mises statistic. We present the exact limit distributions of both statistics and show their performance through a simulation study.

Whittle estimation based on the extremal spectral density of a heavy-tailed random field

We consider a strictly stationary random field on the two-dimensional integer lattice with regularly varying marginal and finite-dimensional distributions. Exploiting the regular variation, we define the spatial extremogram which takes into account only the largest values in the random field. This extremogram is a spatial autocovariance function. We define the corresponding extremal spectral density and its estimator, the extremal periodogram. Based on the extremal periodogram, we consider the Whittle estimator for suitable classes of parametric random fields including the Brown–Resnick random field and regularly varying max-moving averages.

Sieve bootstrapping the memory parameter in long-range dependent stationary functional time series

We consider a sieve bootstrap procedure to quantify the estimation uncertainty of long-memory parameters in stationary functional time series. We use a semiparametric local Whittle estimator to estimate the long-memory parameter. In the local Whittle estimator, discrete Fourier transform and periodogram are constructed from the first set of principal component scores via a functional principal component analysis. The sieve bootstrap procedure uses a general vector autoregressive representation of the estimated principal component scores. It generates bootstrap replicates that adequately mimic the dependence structure of the underlying stationary process. We first compute the estimated first set of principal component scores for each bootstrap replicate and then apply the semiparametric local Whittle estimator to estimate the memory parameter. By taking quantiles of the estimated memory parameters from these bootstrap replicates, we can nonparametrically construct confidence intervals of the long-memory parameter. As measured by coverage probability differences between the empirical and nominal coverage probabilities at three levels of significance, we demonstrate the advantage of using the sieve bootstrap compared to the asymptotic confidence intervals based on normality.