Fast Filtering and Smoothing for Multivariate State Space Models

This paper investigates a new approach to diffuse filtering and smoothing for multivariate state space models. The standard approach treats the observations as vectors, while our approach treats each element of the observational vector individually. This strategy leads to computationally efficient methods for multivariate filtering and smoothing. Also, the treatment of the diffuse initial state vector in multivariate models is much simpler than in existing methods. The paper presents details of relevant algorithms for filtering, prediction and smoothing. Proofs are provided. Three examples of multivariate models in statistics and economics are presented for which the new approach is particularly relevant.     

Time Series Models in Non-Normal Situations: Symmetric Innovations

We consider AR( q ) models in time series with non-normal innovations represented by a member of a wide family of symmetric distributions (Student's t ). Since the ML (maximum likelihood) estimators are intractable, we derive the MML (modified maximum likelihood) estimators of the parameters and show that they are remarkably efficient. We use these estimators for hypothesis testing, and show that the resulting tests are robust and powerful.     

A Simple Test for White Noise in Functional Time Series

We propose a new procedure for white noise testing of a functional time series. Our approach is based on an explicit representation of the L²‐distance between the spectral density operator and its best (L²‐)approximation by a spectral density operator corresponding to a white noise process. The estimation of this distance can be easily accomplished by sums of periodogram kernels, and it is shown that an appropriately standardized version of the estimator is asymptotically normal distributed under the null hypothesis (of functional white noise) and under the alternative. As a consequence, we obtain a very simple test (using the quantiles of the normal distribution) for the hypothesis of a white noise functional process. In particular, the test does not require either the estimation of a long‐run variance (including a fourth order cumulant) or resampling procedures to calculate critical values. Moreover, in contrast to all other methods proposed in the literature, our approach also allows testing for ‘relevant’ deviations from white noise and constructing confidence intervals for a measure that measures the discrepancy of the underlying process from a functional white noise process.     

Adaptive Fourier Series and the Analysis of Periodicities in Time Series Data

A Fourier series decomposes a function x ( t ) into a sum of periodic components that have sinusoidal shapes. This paper describes an adaptive Fourier series where the periodic components of x ( t ) may have a variety of differing shapes. The periodic shapes are adaptive since they depend on the function x ( t ) and the period. The results, which extend both Fourier analysis and Walsh–Fourier analysis, are applied to investigate the shapes of periodic components in time series data sets.     

A State-Space EM Algorithm for Longitudinal Data

An exact EM algorithm is developed for Gaussian longitudinal mixed models in state-space form. These models include between-subject random effects as well as within-subject serial correlation and possibly observational error. Data for each subject may be equally spaced with missing observations, or unequally spaced with different observation times for different subjects. The method uses the Kalman filter and smoothing algorithm to obtain the conditional expectations of the unobserved data given the observations used in the E step of the EM algorithm. Maximum likelihood estimates of the parameters are obtained wtihout the need to use nonlinear optimization routines. Using simulations, the method is shown to give identical results to maximum likelihood methods that use nonlinear optimization.     

Editor's Special Invited Paper: Sequential Estimation for Time Series Models

Abstract

This article revisits sequential estimation of the autoregressive parameter β in a first-order autoregressive (AR(1)) model and construction of a sequential confidence region for a parameter vector θ in a first-order threshold autoregressive (TAR(1)) model. To resolve a theoretical conjecture raised in Sriram (1986 Sriram , T. N. ( 1986 ). Sequential Estimation of Parameters in a First Order Autoregressive Model, Ph.D. diss., Michigan State University, East Lansing.  [Google Scholar]), we provide a comprehensive numerical study that strongly suggests that the regret in using a sequential estimator of β can be significantly negative for many heavy-tailed error distributions and even for normal errors. Secondly, to investigate yet another conjecture about the limiting distribution of a sequential pivotal quantity for θ in a TAR(1) model, we conduct an extensive numerical study that strongly suggests that the sequential confidence region has much better coverage probability than that of a fixed sample counterpart, regardless of whether the θ values are inside or on or near the boundary of the ergodic region of the series. These highlight the usefulness of sequential sampling methods in fitting linear and nonlinear time series models.     

Oracle M‐Estimation for Time Series Models

We propose a thresholding M‐estimator for multivariate time series. Our proposed estimator has the oracle property that its large‐sample properties are the same as of the classical M‐estimator obtained under the a priori information that the zero parameters were known. We study the consistency of the standard block bootstrap, the centred block bootstrap and the empirical likelihood block bootstrap distributions of the proposed M‐estimator. We develop automatic selection procedures for the thresholding parameter and for the block length of the bootstrap methods. We present the results of a simulation study of the proposed methods for a sparse vector autoregressive VAR(2) time series model. The analysis of two real‐world data sets illustrate applications of the methods in practice.     

Wiener–Kolmogorov Filtering and Smoothing for Multivariate Series With State–Space Structure

Abstract. Wiener–Kolmogorov filtering and smoothing usually deal with projection problems for stochastic processes that are observed over semi‐infinite and doubly infinite intervals. For multivariate stationary series, there exist closed formulae based on covariance generating functions that were first given independently by N. Wiener and A.N. Kolmogorov around 1940. In this article, we consider multivariate series with a state–space structure and, using a new purely algebraic approach to the problem, we prove the equivalence between Wiener–Kolmogorov filtering and Kalman filtering. Up to now, this equivalence has only been partially shown. In addition, we get some new recursions for smoothing and some new recursions to compute the filter weights and the covariance generating functions of the errors. The results are extended to nonstationary series.     

A Linear Discriminant for Gaussian Time Series

A simple linear discriminant is introduced for Gaussian time series with fixed deterministic component. It is demonstrated that the discriminant is 'efficient' under some general conditions. The proposed procedure is tested using some seismological data.     

Parameter Estimation for Periodically Stationary Time Series

Abstract. The innovations algorithm can be used to obtain parameter estimates for periodically stationary time series models. In this paper, we compute the asymptotic distribution for these estimates in the case, where the innovations have a finite fourth moment. These asymptotic results are useful to determine which model parameters are significant. In the process, we also develop asymptotics for the Yule–Walker estimates.     

A Test of Linearity for Functional Autoregressive Models

We propose a new test for linearity in time series. We consider an asymptotically stationary functional AR( p ) model on ℜ ^d of the form
X _n = f ( X _{n −1}, ..., X _{n − p} ) + ξ _n ( n ∈ N).
The testing procedure is based on a suitably normalized sum of quadratic deviations between two different estimates of the function f evaluated at q distinct points of ℜ ^dp . The estimators are f^ _n , a recursive version of the non-parametric kernel estimator of f , and  _n , a least squares estimator well suited to the linear case. The main result states that the test statistic has a χ² limit distribution under the null hypothesis. A similar result is derived under the alternative hypothesis for the test statistic corrupted by a non-linear term. Our simulations indicate that our asymptotic results hold for moderate sample sizes when the testing procedure is used carefully     

A Powerful Test for Changing Trends in Time Series Models

We propose a non‐parametric test for trend specification with improved properties. Many existing tests in the literature exhibit non‐monotonic power. To deal with this problem, Juhl and Xiao 2005 proposed a non‐parametric test with good power by detrending the data non‐parametrically. However, their test is developed for smooth changing trends and is constructed under the assumption of correct specification in the dynamics. In addition, their test suffers from size distortion in finite samples and imposes restrictive assumptions on the variance structure. The current article tries to address these issues. First, the proposed test allows for both abrupt breaks and smooth structural changes in deterministic trends. Second, the test employs a sieve approach to avoid the misspecification problem. Third, the extended test can be applied to the data with conditional heteroskedasticity and time‐varying variance. Fourth, the power properties under alternatives are also investigated. Finally, a partial plug‐in method is proposed to alleviate size distortion. Monte Carlo simulations show that the new test not only has good size but also has monotonic power in finite samples.     

A Note on the Information Matrix for Multiplicative Seasonal Autoregressive Moving‐Average Models

Abstract. It is shown that there is an invariance property for each of the elements of the information matrix of a multiplicative seasonal autoregressive moving‐average time‐series model, which enables the integral specification of Whittle (1953a,b) to be solved in a straightforward way. The resulting non‐iterative closed procedure shares the property possessed by the piecemeal approach of Godolphin and Bane (2006) of being independent of the seasonal period, but our procedure is preferable if one or more orders of the seasonal components of the model are greater than unity. The procedure is therefore simpler, in general, than the iterative method of Klein and Mélard (1990) that depends necessarily on the seasonal period. In the strictly non‐seasonal case this invariance property prescribes a non‐iterative closed procedure for evaluating the information matrix which improves on the methods of Godolphin and Unwin (1983) , Friedlander (1984) , McLeod (1984) and Klein and Spreij (2003) . Three illustrations of the approach are given.     

A New Recursive Estimation Method for Single Input Single Output Models

This article is devoted to a new recursive estimation method for dynamic time series models, more precisely for single input single output models. In that method, the recurrence for updating the Hessian is avoided, but the recurrence for updating the estimator makes use of the Fisher information matrix. The asymptotic properties, consistency and asymptotic normality, of the new estimator are obtained under weak assumptions. Monte Carlo experiments and examples indicate that the estimates converge well, comparatively with alternative methods.     

Broadband Semiparametric Estimation of the Memory Parameter of a Long-Memory Time Series Using Fractional Exponential Models

We consider a fractional exponential, or FEXP estimator of the memory parameter of a stationary Gaussian long-memory time series. The estimator is constructed by fitting a FEXP model of slowly increasing dimension to the log periodogram at all Fourier frequencies by ordinary least squares, and retaining the corresponding estimated memory parameter. We do not assume that the data were necessarily generated by a FEXP model, or by any other finite-parameter model. We do, however, impose a global differentiability assumption on the spectral density except at the origin. Because of this, and its use of all Fourier frequencies, we refer to the FEXP estimator as a broadband semiparametric estimator. We demonstrate the consistency of the FEXP estimator, and obtain expressions for its asymptotic bias and variance. If the true spectral density is sufficiently smooth, the FEXP estimator can strongly outperform existing semiparametric estimators, such as the Geweke–Porter-Hudak (GPH) and Gaussian semiparametric estimators (GSE), attaining an asymptotic mean squared error proportional to (log n )/ n , where n is the sample size. In a simulation study, we demonstrate the merits of using a finite-sample correction to the asymptotic variance, and we also explore the possibility of automatically selecting the dimension of the exponential model using Mallows' C_L criterion.     

A Stochastic Approximation Algorithm for the Adaptive Control of Time Series Following Generalized Linear Models

A recursive estimation method for time series models following generalized linear models is developed in two ways. The estimation procedure, suitably modified, gives rise to a stochastic approximation scheme. We use the modified estima-tion procedure to illustrate a connection between control theory and generalized linear models by employing a logistic regression model.     

Tests for Harmonic Components in the Spectra of Categorical Time Series

A test for categorical time series is developed which is based on Fisher's test for continuous-parameter time series. Instead of using a test based on the Fourier periodog ram for spectral analysis, we utilize the Walsh–Fourier periodogram for testing purposes. We briefly explain the theory behind Walsh–Fourier analysis and some of its recent applications. Asymptotic results for the distribution of the new test statistic for Walsh–Fourier spectra are presented and compared with a simulated distribution. We also perform power studies in order to assess the detection capability of the test. In the presence of multiple peaks in the spectrum, this test tends to lose power. Therefore, we also explore several alternatives to the test for Walsh–Fourier spectra and apply all of the alternative methods to a realization of geomagnetic reversals     

A Structural‐Factor Approach to Modeling High‐Dimensional Time Series and Space‐Time Data

This article considers a structural‐factor approach to modeling high‐dimensional time series and space‐time data by decomposing individual series into trend, seasonal, and irregular components. For ease in analyzing many time series, we employ a time polynomial for the trend, a linear combination of trigonometric series for the seasonal component, and a new factor model for the irregular components. The new factor model simplifies the modeling process and achieves parsimony in parameterization. We propose a Bayesian information criterion to consistently select the order of the polynomial trend and the number of trigonometric functions, and use a test statistic to determine the number of common factors. The convergence rates for the estimators of the trend and seasonal components and the limiting distribution of the test statistic are established under the setting that the number of time series tends to infinity with the sample size, but at a slower rate. We study the finite‐sample performance of the proposed analysis via simulation, and analyze two real examples. The first example considers modeling weekly PM_2.5 data of 15 monitoring stations in the southern region of Taiwan and the second example consists of monthly value‐weighted returns of 12 industrial portfolios.