Estimation of Nonparametric Autoregressive Time Series Models Under Dynamical Constraints

Abstract. A method is introduced to estimate nonparametric autoregressive models under the additional constraint that its regression function has a stable cycle. It is based on a penalty approach that chooses a series expansion approximation taking into account both goodness‐of‐fit and fulfillment of the constraint. Consistency of the proposed estimator is obtained under general hypothesis. Feasibility and effective performance of the introduced method are studied through simulated examples and electro‐encephalographic data collected from a subject suffering from epilepsy.     

The Use of Aggregate Time Series in Testing for Gaussianity

Many time series encountered in practice are non-Gaussian. Because of the process of data collection or the practice or researchers, time series used in analysis and modelling are frequently temporal aggregates. In this paper, we study the effects of the use of aggregate time series on testing for Gaussianity. We analyse how the test statistic is affected by aggregation and how that affects the power of the test. The results show that the use of aggregate time series induces Gaussianity and that the degree of inducement increases with the order of aggregation. In fact, the use of aggregate time series reduces the power of the test, although the effect is not significant for low orders of aggregation.     

Robust Hypothesis Testing and Robust Time Series Interpolation And Regression

Abstract. Recent results on minimax robust time series interpolation and regression coefficient estimation are generalized and extended through a relationship with robust hypothesis testing. The spectral uncertainty classes in the time series problems are assumed to be convex and to satisfy an integral constraint such as on the variance of the process. It is shown that robust solutions in such cases can always be obtained from the least-favourable probability density functions for corresponding hypothesis testing problems. A specific class, the bounded spectral densities from the band model, is considered to illustrate the results.     

Higher-Order Accurate Spectral Density Estimation of Functional Time Series

Under the frequency domain framework for weakly dependent functional time series, a key element is the spectral density kernel which encapsulates the second-order dynamics of the process. We propose a class of spectral density kernel estimators based on the notion of a flat-top kernel. The new class of estimators employs the inverse Fourier transform of a flat-top function as the weight function employed to smooth the periodogram. It is shown that using a flat-top kernel yields a bias reduction and results in a higher-order accuracy in terms of optimizing the integrated mean square error (IMSE). Notably, the higher-order accuracy of flat-top estimation comes at the sacrifice of the positive semi-definite property. Nevertheless, we show how a flat-top estimator can be modified to become positive semi-definite (even strictly positive definite) in finite samples while retaining its favorable asymptotic properties. In addition, we introduce a data-driven bandwidth selection procedure realized by an automatic inspection of the estimated correlation structure. Our asymptotic results are complemented by a finite-sample simulation where the higher-order accuracy of flat-top estimators is manifested in practice.     

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.     

Goodness-of-fit Test in Parametric Time Series Models

A goodness-of-fit test is proposed which uses nonparametric curve estimation methods to investigate the fit of parametric models for the spectral density. A test of the null hypothesis that the function has parametric form is considered with a test statistic which compares parametric estimates and nonparametric kernel estimates of the function and its derivatives at a preselected number of points. An important issue for the nonparametric estimator is bandwidth choice, and we propose a data-adaptive method for local bandwidth choice. Under the null hypothesis, asymptotically the test statistic has a χ² distribution. Some practical issues are discussed.     

Testing Separability of Functional Time Series

We derive and study a significance test for determining whether a panel of functional time series is separable. In the context of this paper, separability means that the covariance structure factors into the product of two functions, one depending only on time and the other depending only on the coordinates of the panel. Separability is a property that can dramatically improve computational efficiency by substantially reducing model complexity. It is especially useful for functional data, as it implies that the functional principal components are the same for each member of the panel. However, such an assumption must be verified before proceeding with further inference. Our approach is based on functional norm differences and provides a test with well‐controlled size and high power. We establish our procedure quite generally, allowing one to test separability of autocovariances as well. In addition to an asymptotic justification, our methodology is validated by a simulation study. It is applied to functional panels of particulate pollution and stock market data.     

Gaussian Semiparametric Estimation of Non-stationary Time Series

Generalizing the definition of the memory parameter d in terms of the differentiated series, we showed in Velasco (Non-stationary log-periodogram regression, Forthcoming J. Economet. , 1997) that it is possible to estimate consistently the memory of non-stationary processes using methods designed for stationary long-range-dependent time series. In this paper we consider the Gaussian semiparametric estimate analysed by Robinson (Gaussian semiparametric estimation of long range dependence. Ann. Stat . 23 (1995), 1630–61) for stationary processes. Without a priori knowledge about the possible non-stationarity of the observed process, we obtain that this estimate is consistent for d ∈ (−½, 1) and asymptotically normal for d ∈ (−½,¾) under a similar set of assumptions to those in Robinson's paper. Tapering the observations, we can estimate any degree of non-stationarity, even in the presence of deterministic polynomial trends of time. The semiparametric efficiency of this estimate for stationary sequences also extends to the non-stationary framework.     

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.     

Testing for the Presence of Self-Similarity of Gaussian Time Series Having Stationary Increments

A method for testing for the presence of self-similarity of a Gaussian time series with stationary increments is presented. The test is based on estimation of the distance between the time series and a set of time series containing all the fractional Brownian motions. This distance is constructed from two estimations of multiscale generalized quadratic variations expectations. The second one requires regression estimates of the self-similarity index H . Two estimations of H are then introduced. They present good robustness and computing time properties compared with the Whittle approach, with nearly similar convergence rate. The test is applied on simulated and real data. The self-similarity assumption is notably accepted for the famous Nile River data.     

Testing Normality of Functional Time Series

We develop tests of normality for time series of functions. The tests are related to the commonly used Jarque–Bera test. The assumption of normality has played an important role in many methodological and theoretical developments in the field of functional data analysis. Yet, no inferential procedures to verify it have been proposed so far, even for i.i.d. functions. We propose several approaches which handle two paramount challenges: (i) the unknown temporal dependence structure and (ii) the estimation of the optimal finite‐dimensional projection space. We evaluate the tests via simulations and establish their large sample validity under general conditions. We obtain useful insights by applying them to pollution and intraday price curves. While the pollution curves can be treated as normal, the normality of high‐frequency price curves is rejected.     

A Note on the Behaviour of Nonparametric Density and Spectral Density Estimators at Zero Points of their Support

The asymptotic behaviour of nonparametric estimators of the stationary density and of the spectral density function of a stationary process have been studied in some detail in the last 50–60years. Nevertheless, less is known about the behaviour of these estimators when the target function happens to vanish at the point of interest. In the article at hand, we fill this gap and show that asymptotic normality still holds true but with super‐efficient and different rates of convergence for the density and for the spectral density estimators that are affected also by the dependence structure of the process.     

Marginal Estimation of Parameter Driven Binomial Time Series Models

This article develops asymptotic theory for estimation of parameters in regression models for binomial response time series where serial dependence is present through a latent process. Use of generalized linear model estimating equations leads to asymptotically biased estimates of regression coefficients for binomial responses. An alternative is to use marginal likelihood, in which the variance of the latent process but not the serial dependence is accounted for. In practice, this is equivalent to using generalized linear mixed model estimation procedures treating the observations as independent with a random effect on the intercept term in the regression model. We prove that this method leads to consistent and asymptotically normal estimates even if there is an autocorrelated latent process. Simulations suggest that the use of marginal likelihood can lead to generalized linear model estimates result. This problem reduces rapidly with increasing number of binomial trials at each time point, but for binary data, the chance of it can remain over 45% even in very long time series. We provide a combination of theoretical and heuristic explanations for this phenomenon in terms of the properties of the regression component of the model, and these can be used to guide application of the method in practice.     

A Plug‐in Bandwidth Selection Procedure for Long‐Run Covariance Estimation with Stationary Functional Time Series

In several arenas of application, it is becoming increasingly common to consider time series of curves or functions. Many inferential procedures employed in the analysis of such data involve the long‐run covariance function or operator, which is analogous to the long‐run covariance matrix familiar to finite‐dimensional time‐series analysis and econometrics. This function may be naturally estimated using a smoothed periodogram type estimator evaluated at frequency zero that relies on the choice of a bandwidth parameter. Motivated by a number of prior contributions in the finite‐dimensional setting, in particular Newey and West ( 1994 ), we propose a bandwidth selection method that aims to minimize the estimator's asymptotic mean‐squared normed error (AMSNE) in L²[0,1]². As the AMSNE depends on unknown population quantities including the long‐run covariance function itself, estimates for these are plugged in in an initial step after which the estimated AMSNE can be minimized to produce an empirical optimal bandwidth. We show that the bandwidth produced in this way is asymptotically consistent with the AMSNE optimal bandwidth, with quantifiable rates, under mild stationarity and moment conditions. These results and the efficacy of the proposed methodology are evaluated by means of a comprehensive simulation study, from which we can offer practical advice on how to select the bandwidth parameter in this setting.     

Robust Estimation For Periodic Autoregressive Time Series

Abstract. A robust estimation procedure for periodic autoregressive (PAR) time series is introduced. The asymptotic properties and the asymptotic relative efficiency are discussed by the estimating equation approach. The performance of the robust estimators for PAR time‐series models with order one is illustrated by a simulation study. The technique is applied to a real data analysis.     

Aggregation and Disaggregation of Structural Time Series Models

The aggregation/disaggregation problem has been widely studied in the time series literature. Some main issues related to this problem are modelling, prediction and robustness to outliers. In this paper we look at the modelling problem with particular interest in the local level and local trend structural time series models together with their corresponding ARIMA(0, 1, 1) and ARIMA(0, 2, 2) representations. Given an observed time series that can be expressed by a structural or autoregressive integrated moving-average (ARIMA) model, we derive the necessary and sufficient conditions under which the aggregate and/or disaggregate series can be expressed by the same class of model. Harvey's cycle and seasonal components models (Harvey, Forecasting, Structural Time Series Models and the Kalman Filter , Cambridge: Cambridge University Press, 1989) are also briefly discussed. Systematic sampling of structural and ARIMA models is also discussed.     

Nonparametric Lag Selection for Time Series

A nonparametric version of the Final Prediction Error (FPE) is analysed for lag selection in nonlinear autoregressive time series under very general conditions including heteroskedasticity. We prove consistency and derive probabilities of incorrect selections that have been previously unavailable. Since it is more likely to overfit (have too many lags) than to underfit (miss some lags), a correction factor is proposed to reduce overfitting and hence increase correct fitting. For the FPE calculation, the local linear estimator is introduced in addition to the Nadaraya-Watson estimator in order to cover a very broad class of processes. To achieve faster computation, a plug-in band-width is suggested for the local linear estimator. Our Monte-Carlo study corroborates that the correction factor generally improves the probability of correct lag selection for both linear and nonlinear processes and that the plug-in bandwidth works at least as well as its commonly used competitor. The proposed methods are applied to the Canadian lynx data and daily returns of DM/US-Dollar exchange rates.     

Functional Coefficient Autoregressive Models: Estimation and Tests of Hypotheses

In this paper, we study nonparametric estimation and hypothesis testing procedures for the functional coefficient AR (FAR) models of the form X_t = f ₁( X _{t − d} ) X _{t − 1}+ ... + f_p ( X _{t − d} ) X _{t − p} +ε _t , first proposed by Chen and Tsay (1993). As a direct generalization of the linear AR model, the FAR model is a rich class of models that includes many useful parametric nonlinear time series models such as the threshold AR models of Tong (1983) and exponential AR models of Haggan and Ozaki (1981). We propose a local linear estimation procedure for estimating the coefficient functions and study its asymptotic properties. In addition, we propose two testing procedures. The first one tests whether all the coefficient functions are constant, i.e. whether the process is linear. The second one tests if all the coefficient functions are continuous, i.e. if any threshold type of nonlinearity presents in the process. The results of some simulation studies as well as a real example are presented.     

Testing Parameter Change in General Integer‐Valued Time Series

We consider the structural change in a class of discrete valued time series, which the conditional distribution belongs to the one‐parameter exponential family. We propose a change point test based on the maximum likelihood estimator of the model's parameter. Under the null hypothesis (of no change), the test statistic converges to a well‐known distribution, allowing the calculation of the critical value of the test. The test statistic diverges to infinity under the alternative, meaning that the test has asymptotic power one. Some simulation results and real data applications are reported to show the effectiveness of the proposed procedure.     

非线性时变系统时滞和参数在线联合估计的SMSA方法

针对输入带纯迟延的一类非线性系统,提出时变时滞和参数在线联合估计的一种SMSA混合算法。该方法有机结合具有概率突跳特性的模拟退火和基于高维欧氏空间中凸多面体结构的单纯形搜索法,在最小化平均平方误差意义下动态寻找时滞和参数的最优估计值。基于实际非线性水箱系统的仿真验证了混合算法对非线性系统时滞和参数在线联合估计的有效性及其一定的抗噪声能力。