On flat-top kernel spectral density estimators for homogeneous random fields

The problem of nonparametric estimation of the spectral density function of a partially observed homogeneous random field is addressed. In particular, a class of estimators with favorable asymptotic performance (bias, variance, rate of convergence) is proposed. The proposed estimators are actually shown to be √N-consistent if the autocovariance function of the random field is supported on a compact set, and close to √N-consistent if the autocovariance function decays to zero sufficiently fast for increasing lags.     

Spectral representation and autocovariance structure of Markov switching DSGE models

Abstract

We investigate the L²-structure of Markov switching Dynamic Stochastic General Equilibrium (MS DSGE) models and derive conditions for strict and second-order stationarity. Then we determine the autocovariance function of the process driven by a stationary MS DSGE model and give a stable VARMA representation of it. It turns out that the autocovariance structure of the process coincides with that of a standard VARMA. Finally, we propose a method to derive the spectral density in a matrix closed-form of MS DSGE models. Our results relate with the works of Francq and Zakoian, Krolzig, Zhang and Stine. Numerical and empirical illustrations complete the article.     

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.     

Improved estimation for the autocovariances of a Gaussian stationary process

For a Gaussian stationary process with mean μ and autocovariance function γ(·), we consider to improve the usual sample autocovariances with respect to the mean squares error (MSE) loss. For the cases μ=0 and μ≠0, we propose sort of empirical Bayes type estimators Γ? and Γ?, respectively. Then their MSE improvements upon the usual sample autocovariances are evaluated in terms of the spectral density of the process. Concrete examples for them are provided. We observe that if the process is near to a unit root process the improvement becomes quite large. Thus, consideration for estimators of this type seems important in many fields, e.g., econometrics.     

Robust estimation in long-memory processes under additive outliers

In this paper, we introduce an alternative semiparametric estimator of the fractional differencing parameter in ARFIMA models which is robust against additive outliers. The proposed estimator is a variant of the GPH estimator [Geweke, J., Porter-Hudak, S., 1983. The estimation and application of long memory time series model. Journal of Time Series Analysis 4, 221–238]. In particular, we use the robust sample autocorrelations of Ma, Y. and Genton, M. [2000. Highly robust estimation of the autocovariance function. Journal of Time Series Analysis 21, 663–684] to obtain an estimator for the spectral density of the process. Numerical results show that the estimator we propose for the differencing parameter is robust when the data contain additive outliers.     

Continuous processes derived from the solution of generalized Langevin equation: theoretical properties and estimation

ABSTRACT

In this paper we present a class of continuous-time processes arising from the solution of the generalized Langevin equation and show some of its properties. We define the theoretical and empirical codifference as a measure of dependence for stochastic processes. As an alternative dependence measure we also consider the spectral covariance. These dependence measures replace the autocovariance function when it is not well defined. Results for the theoretical codifference and theoretical spectral covariance functions for the mentioned process are presented. The maximum likelihood estimation procedure is proposed to estimate the parameters of the process arising from the classical Langevin equation, i.e. the Ornstein–Uhlenbeck process, and of the so-called Cosine process. We also present a simulation study for particular processes arising from this class showing the generation, and the theoretical and empirical counterpart for both codifference and spectral covariance measures.     

Dependence Estimation for High‐frequency Sampled Multivariate CARMA Models

The paper considers high‐frequency sampled multivariate continuous‐time autoregressive moving average (MCARMA) models and derives the asymptotic behaviour of the sample autocovariance function to a normal random matrix. Moreover, we obtain the asymptotic behaviour of the cross‐covariances between different components of the model. We will see that the limit distribution of the sample autocovariance function has a similar structure in the continuous‐time and in the discrete‐time model. As a special case, we consider a CARMA (one‐dimensional MCARMA) process. For a CARMA process, we prove Bartlett's formula for the sample autocorrelation function. Bartlett's formula has the same form in both models; only the sums in the discrete‐time model are exchanged by integrals in the continuous‐time model. Finally, we present limit results for multivariate MA processes as well, which are not known in this generality in the multivariate setting yet.     

A ROLE FOR VARIOGRAMS

In time series texts and journals, variograms are mentioned seldom, if at all. The autocovariance function is preferred. However there are situations where the variogram can be estimated with moderate precision but the autocovariance function cannot, because the variance of the process is not well known. If the problem to be solved does not require the process variance for its solution then it is generally more straightforward to use the variogram rather than the autocovariance function in solving this problem.     

Symbolic cumulant calculations for frequency domain time series

With time series data, there is often the issue of finding accurate approximations for the variance of such quantities as the sample autocovariance function or spectral estimate. Smith and Field (J. Time. Ser. Anal 14: 381–395, 1993) proposed a variance estimate motivated by resampling in the frequency domain. In this paper we present some results on the cumulants of this and other frequency domain estimates obtained via symbolic computation. The statistics of interest are linear combinations of products of discrete Fourier transforms. We describe an operator which calculates the joint cumulants of such statistics, and use the operator to deepen our understanding of the behaviour of the resampling based variance estimate. The operator acts as a filter for a general purpose operator described in Andrews and Stafford (J.R. Statist. Soc. B55, 613–627).     

A comparison of multivariate signal discrimination techniques

This paper investigates several techniques to discriminate two multivariate stationary signals. The methods considered include Gaussian likelihood ratio tests for variance equality, a chi-squared time-domain test, and a spectral-based test. The latter two tests assess equality of the multivariate autocovariance function of the two signals over many different lags. The Gaussian likelihood ratio test is perhaps best viewed as principal component analyses (PCA) without dimension reduction aspects; it can be modified to consider covariance features other than variances via dimension augmentation tactics. A simulation study is constructed that shows how one can make inappropriate conclusions with PCA tests, even when dimension augmentation techniques are used to incorporate non-zero lag autocovariances into the analysis. The various discrimination methods are first discussed. A simulation study then illuminates the various properties of the methods. In this pursuit, calculations are needed to identify several multivariate time series models with specific autocovariance properties. To demonstrate the applicability of the methods, nine US and Canadian weather stations from three distinct regions are clustered. Here, the spectral clustering perfectly identified distinct regions, the chi-squared test performed marginally, and the PCA/likelihood ratio method did not perform well.     

Some efficient computational procedures for high order ARMA models

Recursive methods are commonly used to solve Yule—Walker equations for autoregrsssive parameters given an autocovariance function. The reverse procedure can be extended to the efficient solution of various sets of equations which arise in time series analysis. Those presented in this paper include computation of the autocovariance function of an ARMA model, and the Cramer—Wold factorization.     

On Fan's adaptive Neyman tests for comparing two spectral densities

In this paper, we consider tests for assessing whether two stationary and independent time series have the same spectral densities (or same autocovariance functions). Both frequency domain and time domain test statistics for this purpose are reviewed. The adaptive Neyman tests are then introduced and their performances are investigated. Our tests are adaptive, that is, they are constructed completely by the data and do not involve any unknown smoothing parameters. Simulation studies show that our proposed tests are at least comparable to the current tests in most cases. Furthermore, our tests are much more powerful in some cases, such as against the long orders of autoregressive moving average (ARMA) models such as seasonal ARMA series.     

Derivation of the theoretical autocovariance and autocorrelation function of autoregessive moving average processes

Closed form expressions for the theoretical autocovariance and autocorrelation function of mixed autoregressive moving average processes are presented. The results provide insight into the construction of autocovariances and autocorrelatians and are useful in theoretical analysis, model identification as well as in implementing maximum likelihood estimation algorithms.     

Autocovariance Estimation in Regression with a Discontinuous Signal and m‐Dependent Errors: A Difference‐Based Approach

We discuss a class of difference‐based estimators for the autocovariance in nonparametric regression when the signal is discontinuous and the errors form a stationary m‐dependent process. These estimators circumvent the particularly challenging task of pre‐estimating such an unknown regression function. We provide finite‐sample expressions of their mean squared errors for piecewise constant signals and Gaussian errors. Based on this, we derive biased‐optimized estimates that do not depend on the unknown autocovariance structure. Notably, for positively correlated errors, that part of the variance of our estimators that depend on the signal is minimal as well. Further, we provide sufficient conditions for ‐consistency; this result is extended to piecewise Hölder regression with non‐Gaussian errors. We combine our biased‐optimized autocovariance estimates with a projection‐based approach and derive covariance matrix estimates, a method that is of independent interest. An R package, several simulations and an application to biophysical measurements complement this paper.     

Block bootstrap for periodic characteristics of periodically correlated time series

This research is dedicated to the study of periodic characteristics of periodically correlated time series such as seasonal means, seasonal variances and autocovariance functions. Two bootstrap methods are used: the extension of the usual Moving Block Bootstrap (EMBB) and the Generalised Seasonal Block Bootstrap (GSBB). The first approach is proposed, because the usual Moving Block Bootstrap does not preserve the periodic structure contained in the data and cannot be applied for the considered problems. For the aforementioned periodic characteristics the bootstrap estimators are introduced and consistency of the EMBB in all cases is obtained. Moreover, the GSBB consistency results for seasonal variances and autocovariance function are presented. Additionally, the bootstrap consistency of both considered techniques for smooth functions of the parameters of interest is obtained. Finally, the simultaneous bootstrap confidence intervals are constructed. A simulation study to compare their actual coverage probabilities is provided. A real data example is presented.     

Spatial discrimination and classification maps

A method of constructing maps through spatial discrimination is given. The discrimination depends basically on the assumption of local spatial continuity, and a factorized covariance matrix. Given an autocovariance function, this formulation in particular, leads to a deeper insight into the pioneering work of Switzer (1980). Certain windows for the maps are examined, and choice of window size is discussed in relation to the classification error when the variables are dependent versus independent. When a training data is given, we give a method of estimating the parameters in the model. Some numerical examples are also given.     

Nonnegative Definiteness of the Sample Autocovariance Function

Various textbooks on time series analysis assert that the usual version of the sample autocovariance function (1) is nonnegative definite. Two simple proofs of this result are presented.     

Continuous-Time Stochastic Processes with Cyclical Long-Range Dependence

This paper introduces continuous‐time random processes whose spectral density is unbounded at some non‐zero frequencies. The discretized versions of these processes have asymptotic properties similar to those of discrete‐time Gegenbauer processes. The paper presents some properties of the covariance function and spectral density as well as a theory of statistical estimation of the mean and covariance function of such processes. Some directions for further generalizations of the results are indicated.     

A note on maximum entropy spectrum estimation

This article studies the maximum entropy spectrum estimation. After a bnei discussion on iiow co select ciic appropriate constraints and tiie objec¬tive functions, we decide to choose the constraints containing only the first four sample moments and, consequently, to employ the second order spectral entropy as the objective function. The resulting (maximum entropy) spectral estimate is the power spectral density of an ARMA sequence. Examples for comparing our proposal with the traditional maximum entropy spectral estimate follow at the end.     

A method for computing the autocovariance of renewal processes

In this paper we derive formulae for the autocovariance functions of renewal and renewal reward processes. The derivation is based on a Poissonization technique of a renewal process. The formulae are expressed in the form of Laplace transforms. In some cases we may invert the Laplace transforms analytically, but in general we have to invert them numerically.