A Note on Whittle's Likelihood

The approximate likelihood function introduced by Whittle has been used to estimate the spectral density and certain parameters of a variety of time series models. In this note we attempt to empirically quantify the loss of efficiency of Whittle's method in nonstandard settings. A recently developed representation of some first-order non-Gaussian stationary autoregressive process allows a direct comparison of the true likelihood function with that of Whittle. The conclusion is that Whittle's likelihood can produce unreliable estimates in the non-Gaussian case, even for moderate sample sizes. Moreover, for small samples, and if the autocorrelation of the process is high, Whittle's approximation is not efficient even in the Gaussian case. While these facts are known to some extent, the present study sheds more light on the degree of efficiency loss incurred by using Whittle's likelihood, in both Gaussian and non-Gaussian cases.     

Frequency domain bootstrap for ratio statistics under long-range dependence

A frequency domain bootstrap (FDB) is a common technique to apply Efron’s independent and identically distributed resampling technique (Efron, 1979) to periodogram ordinates – especially normalized periodogram ordinates – by using spectral density estimates. The FDB method is applicable to several classes of statistics, such as estimators of the normalized spectral mean, the autocorrelation (but not autocovariance), the normalized spectral density function, and Whittle parameters. While this FDB method has been extensively studied with respect to short-range dependent time processes, there is a dearth of research on its use with long-range dependent time processes. Therefore, we propose an FDB methodology for ratio statistics under long-range dependence, using semi- and nonparametric spectral density estimates as a normalizing factor. It is shown that the FDB approximation allows for valid distribution estimation for a broad class of stationary, long-range (or short-range) dependent linear processes, without any stringent assumptions on the distribution of the underlying process. The results of a large simulation study show that the FDB approximation using a semi- or nonparametric spectral density estimator is often robust for various values of a long-memory parameter reflecting magnitude of dependence. We apply the proposed procedure to two data examples.     

Local Whittle likelihood approach for generalized divergence

There are many approaches in the estimation of spectral density. With regard to parametric approaches, different divergences are proposed in fitting a certain parametric family of spectral densities. Moreover, nonparametric approaches are also quite common considering the situation when we cannot specify the model of process. In this paper, we develop a local Whittle likelihood approach based on a general score function, with some special cases of which, the approach applies to more applications. This paper highlights the effective asymptotics of our general local Whittle estimator, and presents a comparison with other estimators. Additionally, for a special case, we construct the one-step ahead predictor based on the form of the score function. Subsequently, we show that it has a smaller prediction error than the classical exponentially weighted linear predictor. The provided numerical studies show some interesting features of our local Whittle estimator.     

Semiparametric Whittle estimation of a cyclical long-memory time series based on generalised exponential models

This paper considers a semiparametric estimation of the memory parameter in a cyclical long-memory time series, which exhibits a strong dependence on cyclical behaviour, using the Whittle likelihood based on generalised exponential (GEXP) models. The proposed estimation is included in the so-called broadband or global method and uses information from the spectral density at all frequencies. We establish the consistency and the asymptotic normality of the estimated memory parameter for a linear process and thus do not require Gaussianity. A simulation study conducted using Monte Carlo experiments shows that the proposed estimation works well compared to other existing semiparametric estimations. Moreover, we provide an empirical application of the proposed estimation, applying it to the growth rate of Japan's industrial production index and detecting its cyclical persistence.     

AN EMPIRICAL LIKELIHOOD APPROACH FOR NON‐GAUSSIAN VECTOR STATIONARY PROCESSES AND ITS APPLICATION TO MINIMUM CONTRAST ESTIMATION

We develop the empirical likelihood approach for a class of vector‐valued, not necessarily Gaussian, stationary processes with unknown parameters. In time series analysis, it is known that the Whittle likelihood is one of the most fundamental tools with which to obtain a good estimator of unknown parameters, and that the score functions are asymptotically normal. Motivated by the Whittle likelihood, we apply the empirical likelihood approach to its derivative with respect to unknown parameters. We also consider the empirical likelihood approach to minimum contrast estimation based on a spectral disparity measure, and apply the approach to the derivative of the spectral disparity. This paper provides rigorous proofs on the convergence of our two empirical likelihood ratio statistics to sums of gamma distributions. Because the fitted spectral model may be different from the true spectral structure, the results enable us to construct confidence regions for various important time series parameters without assuming specified spectral structures and the Gaussianity of the process.     

Adaptive Bayesian Procedures Using Random Series Priors

We consider a general class of prior distributions for nonparametric Bayesian estimation which uses finite random series with a random number of terms. A prior is constructed through distributions on the number of basis functions and the associated coefficients. We derive a general result on adaptive posterior contraction rates for all smoothness levels of the target function in the true model by constructing an appropriate ‘sieve’ and applying the general theory of posterior contraction rates. We apply this general result on several statistical problems such as density estimation, various nonparametric regressions, classification, spectral density estimation and functional regression. The prior can be viewed as an alternative to the commonly used Gaussian process prior, but properties of the posterior distribution can be analysed by relatively simpler techniques. An interesting approximation property of B‐spline basis expansion established in this paper allows a canonical choice of prior on coefficients in a random series and allows a simple computational approach without using Markov chain Monte Carlo methods. A simulation study is conducted to show that the accuracy of the Bayesian estimators based on the random series prior and the Gaussian process prior are comparable. We apply the method on Tecator data using functional regression models.     

Automatic Local Smoothing for Spectral Density Estimation

This article uses local polynomial techniques to fit Whittle's likelihood for spectral density estimation. Asymptotic sampling properties of the proposed estimators are derived, and adaptation of the proposed estimator to the boundary effect is demonstrated. We show that the Whittle likelihood-based estimator has advantages over the least-squares based log-periodogram. The bandwidth for the Whittle likelihood-based method is chosen by a simple adjustment of a bandwidth selector proposed in Fan & Gijbels (1995). The effectiveness of the proposed procedure is demonstrated by a few simulated and real numerical examples. Our simulation results support the asymptotic theory that the likelihood based spectral density and log-spectral density estimators are the most appealing among their peers     

Convergence of normalized quadratic forms

The asymptotic behavior of quadratic forms of stationary sequences plays an important role in statistics, for example, in the context of the Whittle approximation to maximum likelihood. The quadratic form, appropriately normalized, may have Gaussian or non-Gaussian limits. Under what circumstances will the limits be of one type or another? And if the limits are non-Gaussian, what are they? The goal of this paper is to describe the historical development of the problem and provide further extensions of recent results.     

Comparing volatility forecasting models during the global financial crisis

Volatility estimation in financial markets has always been a challenge especially in time of crisis. Once asset prices and investment decisions are highly sensitive to such variable, many different models have been proposed in literature. This article estimates the volatility from a new family of stochastic volatility models called non-Gaussian State Space Models, a subclass of state space models where it is possible to compute exact likelihood. Volatilities of important Asian and Oceanian stock market indexes have been estimated and compared to APARCH model estimates. Results showed that non-Gaussian State Space Models outperformed significantly in both in-sample and forecasting cases.     

Consistency of Bernstein polynomial posteriors

A Bernstein prior is a probability measure on the space of all the distribution functions on [0, 1]. Under very general assumptions, it selects absolutely continuous distribution functions, whose densities are mixtures of known beta densities. The Bernstein prior is of interest in Bayesian nonparametric inference with continuous data. We study the consistency of the posterior from a Bernstein prior. We first show that, under mild assumptions, the posterior is weakly consistent for any distribution function P ₀ on [0, 1] with continuous and bounded Lebesgue density. With slightly stronger assumptions on the prior, the posterior is also Hellinger consistent. This implies that the predictive density from a Bernstein prior, which is a Bayesian density estimate, converges in the Hellinger sense to the true density (assuming that it is continuous and bounded). We also study a sieve maximum likelihood version of the density estimator and show that it is also Hellinger consistent under weak assumptions. When the order of the Bernstein polynomial, i.e. the number of components in the beta distribution mixture, is truncated, we show that under mild restrictions the posterior concentrates on the set of pseudotrue densities. Finally, we study the behaviour of the predictive density numerically and we also study a hybrid Bayes–maximum likelihood density estimator.     

FITTING SPATIAL CORRELATION MODELS: APPROXIMATING A LIKELIHOOD APPROXIMATION

The paper considers a class of spatial correlation models (stationary Gaussian processes) which includes (spatial) conditional autoregressive, simultaneous autoregressive, moving average and direct covariance models. Given observations on a finite rectangular lattice, a likelihood approximation for estimating the parameters in the spectral density of the model is discussed. The approximation consists of applying the trapezoidal rule, with a her grid of frequencies than the usual Fourier frequencies, to compute the integral in an appraximation due to Whittle (1954) and later modified by Guyon (1984). With this approximation, a Fisher scoring type algorithm has a simple form and in some casea reduces to iteratively reweighted least squares. Methods for computing the unbiased two-dimensional periodogram required by the method are presented and the accuracy of the approximation is discussed. The asymptotic distribution of the parameter estimates computed from the likelihood approximation is also given.     

DATA-DRIVEN NONPARAMETRIC SPECTRAL DENSITY ESTIMATORS FOR ECONOMIC TIME SERIES: A MONTE CARLO STUDY

Spectral analysis at frequencies other than zero plays an increasingly important role in econometrics. A number of alternative automated data-driven procedures for nonparametric spectral density estimation have been suggested in the literature, but little is known about their finite-sample accuracy. We compare five such procedures in terms of their mean-squared percentage error across frequencies. Our data generating processes (DGP) include autoregressive-moving average (ARMA) models, fractionally integrated ARMA models and nonparametric models based on 16 commonly used macroeconomic time series. We find that for both quarterly and monthly data the autoregressive sieve estimator is the most reliable method overall.     

Modeling and forecasting persistent financial durations

This article introduces the Markov-Switching Multifractal Duration (MSMD) model by adapting the MSM stochastic volatility model of Calvet and Fisher (2004) to the duration setting. Although the MSMD process is exponential β-mixing as we show in the article, it is capable of generating highly persistent autocorrelation. We study, analytically and by simulation, how this feature of durations generated by the MSMD process propagates to counts and realized volatility. We employ a quasi-maximum likelihood estimator of the MSMD parameters based on the Whittle approximation and establish its strong consistency and asymptotic normality for general MSMD specifications. We show that the Whittle estimation is a computationally simple and fast alternative to maximum likelihood. Finally, we compare the performance of the MSMD model with competing short- and long-memory duration models in an out-of-sample forecasting exercise based on price durations of three major foreign exchange futures contracts. The results of the comparison show that the MSMD and the Long Memory Stochastic Duration model perform similarly and are superior to the short-memory Autoregressive Conditional Duration models.     

A comparison of nonparametric priors in hierarchical mixture modelling for AFT regression

We will pursue a Bayesian nonparametric approach in the hierarchical mixture modelling of lifetime data in two situations: density estimation, when the distribution is a mixture of parametric densities with a nonparametric mixing measure, and accelerated failure time (AFT) regression modelling, when the same type of mixture is used for the distribution of the error term. The Dirichlet process is a popular choice for the mixing measure, yielding a Dirichlet process mixture model for the error; as an alternative, we also allow the mixing measure to be equal to a normalized inverse-Gaussian prior, built from normalized inverse-Gaussian finite dimensional distributions, as recently proposed in the literature. Markov chain Monte Carlo techniques will be used to estimate the predictive distribution of the survival time, along with the posterior distribution of the regression parameters. A comparison between the two models will be carried out on the grounds of their predictive power and their ability to identify the number of components in a given mixture density.     

Quasi-likelihood Estimation of Non-invertible Moving Average Processes

Classical methods based on Gaussian likelihood or least-squares cannot identify non-invertible moving average processes, while recent non-Gaussian results are based on full likelihood consideration. Since the error distribution is rarely known a quasi-likelihood approach is desirable, but its consistency properties are yet unknown. In this paper we study the quasi-likelihood associated with the Laplacian model, a convenient non-Gaussian model that yields a modified L ₁ procedure. We show that consistency holds for all standard heavy tailed errors, but not for light tailed errors, showing that a quasi-likelihood procedure cannot be applied blindly to estimate non-invertible models. This is an interesting contrast to the standard results of the quasi-likelihood in regression models, where consistency usually holds much more generally. Similar results hold for estimation of non-causal non-invertible ARMA processes. Various simulation studies are presented to validate the theory and to show the effect of the error distribution, and an analysis of the US unemployment series is given as an illustration.     

Variational Bayes for Phase-Type Distribution

This article develops an algorithm for estimating parameters of general phase-type (PH) distribution based on Bayes estimation. The idea of Bayes estimation is to regard parameters as random variables, and the posterior distribution of parameters which is updated by the likelihood function provides estimators of parameters. One of the advantages of Bayes estimation is to evaluate uncertainty of estimators. In this article, we propose a fast algorithm for computing posterior distributions approximately, based on variational approximation. We formulate the optimal variational posterior distributions for PH distributions and develop the efficient computation algorithm for the optimal variational posterior distributions of discrete and continuous PH distributions.     

Cressie–Read Power-Divergence Statistics for Non-Gaussian Vector Stationary Processes

Abstract.  For a class of vector-valued non-Gaussian stationary processes, we develop the Cressie–Read power-divergence (CR) statistic approach which has been proposed for the i.i.d. case. The CR statistic includes empirical likelihood as a special case. Therefore, by adopting this CR statistic approach, the theory of estimation and testing based on empirical likelihood is greatly extended. We use an extended Whittle likelihood as score function and derive the asymptotic distribution of the CR statistic. We apply this result to estimation of autocorrelation and the AR coefficient, and get narrower confidence intervals than those obtained by existing methods. We also consider the power properties of the test based on asymptotic theory. Under a sequence of contiguous local alternatives, we derive the asymptotic distribution of the CR statistic. The problem of testing autocorrelation is discussed and we introduce some interesting properties of the local power.     

Bayes Risk Consistency of Nonparametric Bayes Density Estimates2

Even though the literature on nonparametric density estimation is large, the literature on Bayesian estimation of the density function is relatively small. The reason is the lack of a suitable prior over the space of probability density functions. There have been attempts to define priors over the space of probability measures, but they have not yielded any workable prior for the purpose of density estimation. Dubins & Freedman (1963) have denned random distribution functions which are singular with probability one. Kraft (1964) has denned a class of distribution functions which have derivatives but not continuous derivatives and hence are not suitable for density estimation. The only really convenient prior is the Dirichlet process prior due to Ferguson (1973), but unfortunately this prior concentrates all its mass over the discrete distribution with a dense set of jumps. Recently Lo (1978) has overcome this difficulty by taking convolution of the Dirichlet process with a fixed continuous kernel. In Section 2, the existence of a version of the posterior distribution and the conditional expectation for arbitrary prior over the space of continuous density functions are discussed. The Bayes risk consistency of the Bayes estimator is discussed in Section 3. The Bayes estimator and its properties with respect to two specific prior distributions are discussed in Section 4. In Section 5 some negative results are presented. Finally a numerical example is given in Section 6.     

BAYESIAN ANALYSIS OF VECTOR ARFIMA PROCESSES

A general framework is presented for Bayesian inference of multivariate time series exhibiting long-range dependence. The series are modelled using a vector autoregressive fractionally integrated moving-average (VARFIMA) process, which can capture both short-term correlation structure and long-range dependence characteristics of the individual series, as well as interdependence and feedback relationships between the series. To facilitate a sampling-based Bayesian approach, the exact joint posterior density is derived for the parameters, in a form that is computationally simpler than direct evaluation of the likelihood, and a modified Gibbs sampling algorithm is used to generate samples from the complete conditional distribution associated with each parameter. The paper also shows how an approximate form of the joint posterior density may be used for long time series. The procedure is illustrated using sea surface temperatures measured at three locations along the central California coast. These series are believed to be interdependent due to similarities in local atmospheric conditions at the different locations, and previous studies have found that they exhibit ‘long memory’ when studied individually. The approach adopted here permits investigation of the effects on model estimation of the interdependence and feedback relationships between the series.     

Nonparametric estimation of density derivatives of dependent data

We study here the kernel type, nonparametric estimation of the derivatives of the density function associated with a strongly mixing time series. The consistency and asymptotic normality properties are studied and a method for the selection of the smoothing parameter by means of the modification of the least-squares cross-validation procedure is proposed.