Fourier Analysis of Serial Dependence Measures

Classical spectral analysis is based on the discrete Fourier transform of the autocovariances. In this article we investigate the asymptotic properties of new frequency‐domain methods where the autocovariances in the spectral density are replaced by alternative dependence measures that can be estimated by U‐statistics. An interesting example is given by Kendall's τ, for which the limiting variance exhibits a surprising behavior.     

An Asymptotic F Test for Uncorrelatedness in the Presence of Time Series Dependence

We propose a simple asymptotically F-distributed Portmanteau test for zero autocorrelations in an otherwise dependent time series. By employing the orthonormal series variance estimator of the variance matrix of sample autocovariances, our test statistic follows an F distribution asymptotically under fixed-smoothing asymptotics. The asymptotic F theory accounts for the estimation error in the underlying variance estimator, which the asymptotic chi-squared theory ignores. Monte Carlo simulations reveal that the F approximation is much more accurate than the corresponding chi-squared approximation in finite samples. The asymptotic F test is as easy to use as the chi-squared test: there is no need to obtain critical values by simulations. Furthermore, it has more accurate empirical sizes and substantial power advantages, comparing to other competitors.     

On Kay's Frequency Estimator

Kay has proposed a technique for estimating the frequency of a complex sinusoid in additive noise, the real and imaginary parts of which are independent and normally distributed with means zero and the same variance. For fixed sample size the estimator achieves the Cramer–Rao lower bound for unbiased estimators of the frequency in the limit as the signal to noise ratio approaches infinity. It has been noted by Lovell and Williamson, however, that the estimator is biased. It is shown in this paper that under Kay's assumptions the estimator is not consistent (for fixed signal to noise ratio the estimator converges almost surely, as the sample size N increases, to a number that is not the true frequency, no matter how large the signal to noise ratio). A class of distributions for the additive noise is proposed under which the technique is strongly consistent and has the correct order of asymptotic variance, namely N ⁻³, for the case where there is some a priori knowledge concerning the range of the frequency. For this class a normal central limit theorem is developed.     

Note on the Asymptotic Efficiency of Sample Covariances in Gaussian Vector Stationary Processes

In this note certain results obtained by Porat ( J. Time Ser. Anal. 8 (1987), 205–20) and Kakizawa and Taniguchi ( J. Time Ser. Anal. 15 (1994), 303–11) concerning the asymptotic efficiency of sample autocovariances of a zero-mean Gaussian stationary process are extended to the case of m -vector processes. It is shown that, for Gaussian vector AR( p ) processes, the sample autocovariance matrix at lag k is asymptotically efficient if 0 ≤ k ≤ p . Further, none of the sample autocovariance matrices is asymptotically efficient for Gaussian vector MA( q ) processes.     

Analysis of X-ray Powder Diffraction Data Using the Maximum Likelihood Estimation Method

The details of substitutional chemistries can be deduced with the associated relaxation of the crystal structure. This paper demonstrates the use of the maximum likelihood estimation method (MLE) for X-ray powder diffraction (XRD) analysis. Detailed calculations are performed for cubic and tetragonal systems. Analysis of yittrium-doped BaTiO₃ prepared under different conditions is shown as an example. The methodology outlined here gives rise to a correct evaluation of the standard deviations of the lattice parameters. In addition, MLE approaches asymptotically the Cramer–Rao lower bound (CRLB) and, therefore, has an advantage over other estimation methods. A link between the output of a commercial software and the standard deviation in the peak position is also suggested.     

Robust Estimation in Vector Autoregressive Moving-Average Models

Bustos and Yohai proposed a class of robust estimates for autoregressive moving-average (ARMA) models based on residual autocovariances (RA estimates). In this paper an affine equivariant generalization of the RA estimates for vector ARMA processes is given. These estimates are asymptotically normal and, when the innovations have an elliptical distribution, their asymptotic covariance matrix differs only by a scalar factor from the covariance matrix corresponding to the maximum likelihood estimate. A Monte Carlo study confirms that the RA estimates are efficient under normal errors and robust when the sample contains outliers. A robust multivariate goodness-of-fit test based on the RA estimates is also obtained.     

Efficient Estimation of Seasonal Long‐Range‐Dependent Processes

Abstract. This paper studies asymptotic properties of the exact maximum likelihood estimates (MLE) for a general class of Gaussian seasonal long‐range‐dependent processes. This class includes the commonly used Gegenbauer and seasonal autoregressive fractionally integrated moving average processes. By means of an approximation of the spectral density, the exact MLE of this class are shown to be consistent, asymptotically normal and efficient. Finite sample performance of these estimates is examined by Monte Carlo simulations and it is shown that the estimates behave very well even for moderate sample sizes. The estimation methodology is illustrated by a real‐life Internet traffic example.     

Second order asymptotic efficlency in terms of the asymptotic variance of seqrential estimation procedures in the presence of nrisance parameters

In the presence of nuisance parameters, the Bhattacharyya type bound for the asymptotic variance of estimation procedures is obtained. It is shown that the modified maximum likelihood (ML) estimation procedures together with any stopping rule does not attain the bound. Further it is shown that the modified ML estimation procedure with the appropriate stopping rule is second order asymptotically efficient in some class of estimation procedures in the sense that it attains the lower bound for the asymptotic variance in the class.     

Fixed‐b asymptotic approximation of the sampling behaviour of nonparametric spectral density estimators

Abstract. We propose a new asymptotic approximation for the sampling behaviour of nonparametric estimators of the spectral density of a covariance stationary time series. According to the standard approach, the truncation lag grows more slowly than the sample size. We derive first‐order limiting distributions under the alternative assumption that the truncation lag is a fixed proportion of the sample size. Our results extend the approach of Neave (1970) , who derived a formula for the asymptotic variance of spectral density estimators under the same truncation lag assumption. We show that the limiting distribution of zero‐frequency spectral density estimators depends on how the mean is estimated and removed. The implications of our zero‐frequency results are consistent with exact results for bias and variance computed by Ng and Perron (1996) . Finite sample simulations indicate that the new asymptotics provides a better approximation than the standard one.     

Exactly/Nearly Unbiased Estimation of Autocovariances of a Univariate Time Series With Unknown Mean

This article proposes an exactly/nearly unbiased estimator of the autocovariance function of a univariate time series with unknown mean. The estimator is a linear function of the usual sample autocovariances computed using the observed demeaned data. The idea is to stack the usual sample autocovariances into a vector and show that the expectation of this vector is a linear combination of population autocovariances. A matrix that we label, A , collects the weights in these linear combinations. When the population autocovariances of high lags are zero (small), exactly (nearly) unbiased estimators of the remaining autocovariances can be obtained using the inverse of upper blocks of the A matrix. The A ‐matrix estimators are shown to be asymptotically equivalent to the usual sample autocovariance estimators. The A ‐matrix estimators can be used to construct estimators of the autocorrelation function that have less bias than the usual estimators. Simulations show that the A ‐matrix estimators can substantially reduce bias while not necessarily increasing mean square error. More powerful tests for the null hypothesis of white noise are obtained using the A ‐matrix estimators.     

ON RESULTS OF PORAT CONCERNING ASYMPTOTIC EFFICIENCY OF SAMPLE COVARIANCES OF GAUSSIAN ARMA PROCESSES

Abstract. An alternative derivation is given of results first obtained by Porat (1987) concerning the asymptotic efficiencies of sample autocovariances of a stationary Gaussian ARMA process. This is based on an approximation to the likelihood of these autocovariances.     

SOME SIMPLE TESTS OF THE MOVING-AVERAGE UNIT ROOT HYPOTHESIS

Abstract. This paper deals with three test statistics for a moving-average (MA) unit root. The spectral test is based on the estimate of the spectral density at frequency zero. The variance difference statistic compares the sample variance of the integrated series with the estimated variance imposing the MA unit root constraint. Furthermore, Tanaka's score type test statistic is modified to improve the power in higher order models. The asymptotic power of the tests is considered and Monte Carlo experiments are performed to investigate the small sample properties of the tests. Finally, the tests are applied to a number of economic time series to determine the degree of integration.     

On Bartlett's Formula for Non-linear Processes

Bartlett's formula is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocovariances. However, it is derived under precise assumptions (namely linearity of the underlying process and vanishing of its fourth-order cumulants) and effectiv e computations show that the value given by this formula can deviate markedly from the true asymptotic covariance when the requirements on the underlying process are not satisfied. This is the case for a large class of models, for instance bilinear and autoregressive conditionally heteroscedastic processes. For these reasons we investigate the behaviour of smoothed empirical estimates of the covariance between two sample autocovariance s. We prove L ² and strong consistency for strongly mixing stationary processes and define for the covariance matrix of a vector of sample autocovariances a consistent estimate which is a non-negative definite matrix. The choice of the parameters is discussed, applications are given and comparisons are made through a simulation study     

Nonlinear spectral density estimation: thresholding the correlogram

Traditional kernel spectral density estimators are linear as a function of the sample autocovariance sequence. The purpose of this article is to propose and analyse two new spectral estimation methods that are based on the sample autocovariances in a nonlinear way. The rate of convergence of the new estimators is quantified, and practical issues such as bandwidth and/or threshold choice are addressed. The new estimators are also compared with traditional ones using flat‐top lag‐windows in a simulation experiment involving sparse time‐series models.     

SOME ASYMPTOTIC PROPERTIES OF THE SAMPLE COVARIANCES OF GAUSSIAN AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. The paper deals with the asymptotic variances of the sample covariances of autoregressive moving average processes. Using state-space representations and some matrix Lyapunov equation theory, closed-form expressions are derived for the asymptotic variances of the sample covariances and for the Cramer-Rao bounds on the process covariances. The main results obtained from these expressions are as follows: For ARMA ( p, q ) processes with p ≥ q , the sample covariance of order n is asymptotically efficient if and only if 0 ≤ n ≤ p – q .
For ARMA ( p, q ) processes with p < q , none of the sample covariances is asymptotically efficient.     

A refined efficiency rate for ordinary least squares and generalized least squares estimators for a linear trend with autoregressive errors

When a straight line is fitted to time series data, generalized least squares (GLS) estimators of the trend slope and intercept are attractive as they are unbiased and of minimum variance. However, computing GLS estimators is laborious as their form depends on the autocovariances of the regression errors. On the other hand, ordinary least squares (OLS) estimators are easy to compute and do not involve the error autocovariance structure. It has been known for 50 years that OLS and GLS estimators have the same asymptotic variance when the errors are second‐order stationary. Hence, little precision is gained by using GLS estimators in stationary error settings. This article revisits this classical issue, deriving explicit expressions for the GLS estimators and their variances when the regression errors are drawn from an autoregressive process. These expressions are used to show that OLS methods are even more efficient than previously thought. Specifically, we show that the convergence rate of variance differences is one polynomial degree higher than that of least squares estimator variances. We also refine Grenander's (1954) variance ratio. An example is presented where our new rates cannot be improved upon. Simulations show that the results change little when the autoregressive parameters are estimated.     

An Estimating Method for Parametric Spectral Densities of Gaussian Time Series

An estimating method for spectral densities of Gaussian time series that belong to a parametric model is proposed. Spectral density estimators are evaluated by using average Kullback–Leibler divergence from the true spectral density to estimated spectral densities. In the classical approach, unknown spectral densities are estimated by replacing the unknown parameters by asymptotically efficient estimates. In the alternative method introduced in the present paper, spectral density estimates usually do not belong to the model. The alternative spectral density estimators asymptotically dominate the classical ones. The difference in average Kullback–Leibler divergence between them can be regarded as the mixture mean curvature of the model in the space of all spectral densities. The explicit expression for the proposed estimators of spectral densities of autoregressive processes is obtained. The accuracy of prediction can be improved by using predictors that correspond to the alternative spectral density estimators.     

Subsampling inference for the autocovariances and autocorrelations of long‐memory heavy‐ tailed linear time series

We provide a self‐normalization for the sample autocovariances and autocorrelations of a linear, long‐memory time series with innovations that have either finite fourth moment or are heavy‐tailed with tail index 2 < α < 4. In the asymptotic distribution of the sample autocovariance there are three rates of convergence that depend on the interplay between the memory parameter d and α, and which consequently lead to three different limit distributions; for the sample autocorrelation the limit distribution only depends on d. We introduce a self‐normalized sample autocovariance statistic, which is computable without knowledge of α or d (or their relationship), and which converges to a non‐degenerate distribution. We also treat self‐normalization of the autocorrelations. The sampling distributions can then be approximated non‐parametrically by subsampling, as the corresponding asymptotic distribution is still parameter‐dependent. The subsampling‐based confidence intervals for the process autocovariances and autocorrelations are shown to have satisfactory empirical coverage rates in a simulation study. The impact of subsampling block size on the coverage is assessed. The methodology is further applied to the log‐squared returns of Merck stock.     

Bootstrap-assisted Goodness-of-fit Tests in the Frequency Domain

The validity of a stationary time series model may be measured by the goodness of fit of the spectral distribution function. Anderson (Technical Report 27, 1991; Technical Report 309, 1995; Stanford University) has worked out the closed-form characteristic functions for the Cramer–von Mises criterion for general linear processes, under the condition that all values of the parameters are specified. The asymptotic approach is not easily implemented and usually requires a case by case analysis. In this paper we propose a bootstrap goodness-of-fit test in the frequency domain. By properly resampling the residuals, we can consistently estimate the p values for many weakly dependent semiparametric models with unspecified parameter values. This is the content of the main theorem that we try to explain. A group of simulations is conducted, showing consistent significance level and good power. The special tests are applied to the lynx data and reveal structure unexplained by the AR(1) model fitted by Tong ( J. R. Stat. Soc. A 140 (1977), 432–36). A possible generalization with application to financial data analysis is also discussed.     

A CENTRAL LIMIT THEOREM FOR m(n) AUTOCOVARIANCES

Many of the fundamental results in time series analysis depend on the joint asymptotic normality of a fixed number m of the sample autocovariances. However, in practice, the m is often chosen after the number of observations, n , is known, with m then treated as fixed. In this paper a Berry-Esseen type result is proved for m ( n ) autocovariances for m growing at a certain rate.