Posterior consistency for the spectral density of non-Gaussian stationary time series

Various nonparametric approaches for Bayesian spectral density estimation of stationary time series have been suggested in the literature, mostly based on the Whittle likelihood approximation. A generalization of this approximation involving a nonparametric correction of a parametric likelihood has been proposed in the literature with a proof of posterior consistency for spectral density estimation in combination with the Bernstein–Dirichlet process prior for Gaussian time series. In this article, we will extend the posterior consistency result to non-Gaussian time series by employing a general consistency theorem for dependent data and misspecified models. As a special case, posterior consistency for the spectral density under the Whittle likelihood is also extended to non-Gaussian time series. Small sample properties of this approach are illustrated with several examples of non-Gaussian time series.     

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.     

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.     

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.     

Generalized Autoregressive (GAR) Model: A Comparison of Maximum Likelihood and Whittle Estimation Procedures Using a Simulation Study

This article evaluates the performance of two estimators namely, the Maximum Likelihood Estimator (MLE) and Whittle's Estimator (WE), through a simulation study for the Generalised Autoregressive (GAR) model.

As expected, it is found that for the parameters α and σ², the MLE and WE have a better performance than Method of Moments (MOM) estimator. For the parameter δ, MOM sometimes appears to have a slightly better performance than MLE and WE, possibly due to truncation approximations associated with the hypergeometric functions for calculating the autocorrelation function. However, the MLE and WE can be used in practice without loss of efficiency.     

Test de différence de contrastes et somme pondérée de khi-deux

After recalling the framework of minimum-contrast estimation, its consistency and its asymptotic normality, we highlight the fact that these results do not require any stationarity or ergodicity assumptions. The asymptotic distribution of the underlying contrast difference test is a weighted sum of independent chi-square variables having one degree of freedom each. We illustrate these results in three contexts: (1) a nonhomogeneous Markov chain with likelihood contrast; (2) a Markov field with coding, pseudolikelihood or likelihood contrasts; (3) a not necessarily Gaussian time series with Whittle's contrast. In contexts (2) and (3), we compare experimentally the power of the likelihood-ratio test with those of other contrast-difference tests.     

Estimation of the Memory Parameters of the Fractionally Integrated Separable Spatial Autoregressive (FISSAR(1, 1)) Model: A Simulation Study

In this article, we implement the Regression Method for estimating (d ₁, d ₂) of the FISSAR(1, 1) model. It is also possible to estimate d ₁ and d ₂ by Whittle's method. We also compute the estimated bias, standard error, and root mean square error by a simulation study. A comparison was made between the Regression Method of estimating d ₁ and d ₂ to that of the Whittle's method. It was found in this simulation study that the Regression Method of estimation was better when compare with the Whittle's estimator, in the sense that it had smaller root mean square errors (RMSE) values.     

Adaptive multiple importance sampling for Gaussian processes

In applications of Gaussian processes (GPs) where quantification of uncertainty is a strict requirement, it is necessary to accurately characterize the posterior distribution over Gaussian process covariance parameters. This is normally done by means of standard Markov chain Monte Carlo (MCMC) algorithms, which require repeated expensive calculations involving the marginal likelihood. Motivated by the desire to avoid the inefficiencies of MCMC algorithms rejecting a considerable amount of expensive proposals, this paper develops an alternative inference framework based on adaptive multiple importance sampling (AMIS). In particular, this paper studies the application of AMIS for GPs in the case of a Gaussian likelihood, and proposes a novel pseudo-marginal-based AMIS algorithm for non-Gaussian likelihoods, where the marginal likelihood is unbiasedly estimated. The results suggest that the proposed framework outperforms MCMC-based inference of covariance parameters in a wide range of scenarios.     

Quasi-Maximum Likelihood Estimation of GARCH Models With Heavy-Tailed Likelihoods

The non-Gaussian maximum likelihood estimator is frequently used in GARCH models with the intention of capturing heavy-tailed returns. However, unless the parametric likelihood family contains the true likelihood, the estimator is inconsistent due to density misspecification. To correct this bias, we identify an unknown scale parameter η_f that is critical to the identification for consistency and propose a three-step quasi-maximum likelihood procedure with non-Gaussian likelihood functions. This novel approach is consistent and asymptotically normal under weak moment conditions. Moreover, it achieves better efficiency than the Gaussian alternative, particularly when the innovation error has heavy tails. We also summarize and compare the values of the scale parameter and the asymptotic efficiency for estimators based on different choices of likelihood functions with an increasing level of heaviness in the innovation tails. Numerical studies confirm the advantages of the proposed approach.     

La méthode de vraisemblance pour les processus linéaires spatiaux définis sur un treillis triangulaire

Spatial linear processes {X_s, s ? T} where T is a triangular lattice in R² are considered. Special attention is given to the class of spatial moving-average processes. Precisely, for each site s T, the variable X_s is defined as a linear combination of real-valued random shocks located at the vertices of regular concentric hexagons centered at s. For Gaussian random shocks, the process is also Gaussian, and estimates of its parameters are obtained by maximizing the exact likelihood. For non-Gaussian random shocks, the exact likelihood is difficult to obtain; however, the Gaussian likelihood is still used giving the pseudo-Gaussian likelihood estimates. The behaviour of these estimates is analyzed through the study of asymptotic properties and some simulation experiments based on an isotropic model defined with one coefficient.     

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.     

Monte Carlo Maximum Likelihood Estimation for Generalized Long-Memory Time Series Models

An exact maximum likelihood method is developed for the estimation of parameters in a non-Gaussian nonlinear density function that depends on a latent Gaussian dynamic process with long-memory properties. Our method relies on the method of importance sampling and on a linear Gaussian approximating model from which the latent process can be simulated. Given the presence of a latent long-memory process, we require a modification of the importance sampling technique. In particular, the long-memory process needs to be approximated by a finite dynamic linear process. Two possible approximations are discussed and are compared with each other. We show that an autoregression obtained from minimizing mean squared prediction errors leads to an effective and feasible method. In our empirical study, we analyze ten daily log-return series from the S&P 500 stock index by univariate and multivariate long-memory stochastic volatility models. We compare the in-sample and out-of-sample performance of a number of models within the class of long-memory stochastic volatility models.     

Small sample efficiency gains from a first observation correction for hatanakafs estimator of the lagged dependent variable-serial correlation regression model

Evidence presented by Fomby and Guilkey (1983) suggests that Hatanaka's estimator of the coefficients in the lagged dependent variable-serial correlation regression model performs poorly, not because of poor selection of the estimate of the autocorrelation coefficient, but because of the lack of a first observation correction. This study conducts a Monte Carlo investigationof the small sample efficiency gains obtainable from a first observation correction suggested by Harvey (1981). Results presented here indicate that substantial gains result from the first observation correction. However, in comparing Hatanaka's procedure with first observation correction to maximum likelihood search, it appears that ignoring the determinantal term of the full likelihood function causes some loss of small sample efficiency. Thus, when computer costsand programming constraints are not binding, maximum likelihood search is to be recommended. In contrast, users who have access to only rudimentary least squares programs would be well served when using Hatanaka's two-step procedure with first     

Large Deviation Results for Statistics of Short- and Long-memory Gaussian Processes

This paper discusses the large deviation principle of several important statistics for short- and long-memory Gaussian processes. First, large deviation theorems for the log-likelihood ratio and quadratic forms for a short-memory Gaussian process with mean function are proved. Their asymptotics are described by the large deviation rate functions. Since they are complicated, they are numerically evaluated and illustrated using the Maple V system (Char et al ., 1991a,b). Second, the large deviation theorem of the log-likelihood ratio statistic for a long-memory Gaussian process with constant mean is proved. The asymptotics of the long-memory case differ greatly from those of the short-memory case. The maximum likelihood estimator of a spectral parameter for a short-memory Gaussian stationary process is asymptotically efficient in the sense of Bahadur.     

Bayesian estimation of renewal function for inverse Gaussian renewal process

Two approximation methods are used to obtain the Bayes estimate for the renewal function of inverse Gaussian renewal process. Both approximations use a gamma-type conditional prior for the location parameter, a non-informative marginal prior for the shape parameter, and a squared error loss function. Simulations compare the accuracy of the estimators and indicate that the Tieney and Kadane (T–K)-based estimator out performs Maximum Likelihood (ML)- and Lindley (L)-based estimator. Computations for the T–K-based Bayes estimate employ the generalized Newton's method as well as a recent modified Newton's method with cubic convergence to maximize modified likelihood functions. The program is available from the author.     

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.     

A tobit model with garch errors

In the context of time series regression, we extend the standard Tobit model to allow for the possibility of conditional heteroskedastic error processes of the GARCH type. We discuss the likelihood function of the Tobit model in the presence of conditionally heteroskedastic errors. Expressing the exact likelihood function turns out to be infeasible, and we propose an approximation by treating the model as being conditionally Gaussian. The performance of the estimator is investigated by means of Monte Carlo simulations. We find that, when the error terms follow a GARCH process, the proposed estimator considerably outperforms the standard Tobit quasi maximum likelihood estimator. The efficiency loss due to the approximation of the likelihood is finally evaluated.     

Asymptotic Properties of QML Estimators for VARMA Models with Time‐dependent Coefficients

This paper is about vector autoregressive‐moving average models with time‐dependent coefficients to represent non‐stationary time series. Contrary to other papers in the univariate case, the coefficients depend on time but not on the series' length n. Under appropriate assumptions, it is shown that a Gaussian quasi‐maximum likelihood estimator is almost surely consistent and asymptotically normal. The theoretical results are illustrated by means of two examples of bivariate processes. It is shown that the assumptions underlying the theoretical results apply. In the second example, the innovations are marginally heteroscedastic with a correlation ranging from ?0.8 to 0.8. In the two examples, the asymptotic information matrix is obtained in the Gaussian case. Finally, the finite‐sample behaviour is checked via a Monte Carlo simulation study for n from 25 to 400. The results confirm the validity of the asymptotic properties even for short series and the asymptotic information matrix deduced from the theory.     

Maximum likelihood estimators for ARMA and ARFIMA models: a Monte Carlo study

We analyze by simulation the properties of two time domain and two frequency domain estimators for low-order autoregressive fractionally integrated moving-average Gaussian models, ARFIMA (p,d,q). The estimators considered are the exact maximum likelihood for demeaned data (EML) the associated modified profile likelihood (MPL) and the Whittle estimator with (WLT) and without tapered data (WL). Length of the series is 100. The estimators are compared in terms of pile-up effect, mean square error, bias, and empirical confidence level. The tapered version of the Whittle likelihood turns out to be a reliable estimator for ARMA and ARFIMA models. Its small losses in performance in case of ‘well-behaved’ models are compensated sufficiently in more ‘difficult’ models. The modified profile likelihood is an alternative to the WLT but is computationally more demanding. It is either equivalent to the EML or more favorable than the EML. For fractionally integrated models, particularly, it dominates clearly the EML. The WL has serious deficiencies for large ranges of parameters, and so cannot be recommended in general. The EML, on the other hand, should only be used with care for fractionally integrated models due to its potential large negative bias of the fractional integration parameter. In general, one should proceed with caution for ARMA(1,1) models with almost canceling roots, and, in particular, in case of the EML and the MPL for inference in the vicinity of a moving-average root of +1.     

Approximating hidden Gaussian Markov random fields

Summary.  Gaussian Markov random-field (GMRF) models are frequently used in a wide variety of applications. In most cases parts of the GMRF are observed through mutually independent data; hence the full conditional of the GMRF, a hidden GMRF (HGMRF), is of interest. We are concerned with the case where the likelihood is non-Gaussian, leading to non-Gaussian HGMRF models. Several researchers have constructed block sampling Markov chain Monte Carlo schemes based on approximations of the HGMRF by a GMRF, using a second-order expansion of the log-density at or near the mode. This is possible as the GMRF approximation can be sampled exactly with a known normalizing constant. The Markov property of the GMRF approximation yields computational efficiency.The main contribution in the paper is to go beyond the GMRF approximation and to construct a class of non-Gaussian approximations which adapt automatically to the particular HGMRF that is under study. The accuracy can be tuned by intuitive parameters to nearly any precision. These non-Gaussian approximations share the same computational complexity as those which are based on GMRFs and can be sampled exactly with computable normalizing constants. We apply our approximations in spatial disease mapping and model-based geostatistical models with different likelihoods, obtain procedures for block updating and construct Metropolized independence samplers.