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.     

Inference for the Fourth‐Order Innovation Cumulant in Linear Time Series

The rescaled fourth‐order cumulant of the unobserved innovations of linear time series is an important parameter in statistical inference. This article deals with the problem of estimating this parameter. An existing nonparametric estimator is first discussed, and its asymptotic properties are derived. It is shown how the autocorrelation structure of the underlying process affects the behaviour of the estimator. Based on our findings and on an important invariance property of the parameter of interest with respect to linear filtering, a pre‐whitening‐based nonparametric estimator of the same parameter is proposed. The estimator is obtained using the filtered time series only; that is, an inversion of the pre‐whitening procedure is not required. The asymptotic properties of the new estimator are investigated, and its superiority is established for large classes of stochastic processes. It is shown that for the particular estimation problem considered, pre‐whitening can reduce the variance and the bias of the estimator. The finite sample performance of both estimators is investigated by means of simulations. The new estimator allows for a simple modification of the multiplicative frequency domain bootstrap, which extends its considerable range of validity. Furthermore, the problem of testing hypotheses about the rescaled fourth‐order cumulant of the unobserved innovations is also considered. In this context, a simple test for Gaussianity is proposed. Some real‐life data applications are presented.     

Estimating Spatial Changes Over Time of Arctic Sea Ice using Hidden 2×2 Tables

Arctic sea ice extent has been of considerable interest to scientists in recent years, mainly due to its decreasing temporal trend over the past 20 years. In this article, we propose a hierarchical spatio‐temporal generalized linear model for binary Arctic sea‐ice‐extent data, where statistical dependencies in the data are modeled through a latent spatio‐temporal linear mixed effects model. By using a fixed number of spatial basis functions, the resulting model achieves both dimension reduction and non‐stationarity for spatial fields at different time points. An EM algorithm is proposed to estimate model parameters, and an empirical–hierarchical‐modeling approach is applied to obtain the predictive distribution of the latent spatio‐temporal process. We illustrate the accuracy of the parameter estimation through a simulation study. The hierarchical model is applied to spatial Arctic sea‐ice‐extent data in the month of September for 20 years in the recent past, where several posterior summaries are obtained to detect the changes of Arctic sea ice cover. In particular, we consider a time series of latent 2 × 2 tables to infer the spatial changes of Arctic sea ice over time.     

SIGNIFICANT VARIABLE SELECTION AND AUTOREGRESSIVE ORDER DETERMINATION FOR TIME‐SERIES PARTIALLY LINEAR MODELS

This paper is concerned with the regression coefficient and autoregressive order shrinkage and selection via the smoothly clipped absolute deviation (SCAD) penalty for a partially linear model with time‐series errors. By combining the profile semi‐parametric least squares method and SCAD penalty technique, a new penalized estimation for the regression and autoregressive parameters in the model is proposed. We show that the asymptotic property of the resultant estimator is the same as if the order of autoregressive error structure and non‐zero regression coefficients are known in advance, thus achieving the oracle property in the sense of Fan and Li (2001). In addition, based on a prewhitening technique, we construct a two‐stage local linear estimator (TSLLE) for the non‐parametric component. It is shown that the TSLLE is more asymtotically efficient than the one that ignores the autoregressive time‐series error structure. Some simulation studies are conducted to illustrate the finite sample performance of the proposed procedure. An example of application on electricity usage data is also illustrated. Copyright © 2014 Wiley Publishing Ltd     

Robust estimation for the covariance matrix of multi‐variate time series

In this article, we study the robust estimation for the covariance matrix of stationary multi‐variate time series. As a robust estimator, we propose to use a minimum density power divergence estimator (MDPDE) proposed by Basu et al. (1998) . Particularly, the MDPDE is designed to perform properly when the time series is Gaussian. As a special case, we consider the robust estimator for the autocovariance function of univariate stationary time series. It is shown that the MDPDE is strongly consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration.     

Robust estimators under semi‐parametric partly linear autoregression: Asymptotic behaviour and bandwidth selection

Abstract. In this article, under a semi‐parametric partly linear autoregression model, a family of robust estimators for the autoregression parameter and the autoregression function is studied. The proposed estimators are based on a three‐step procedure, in which robust regression estimators and robust smoothing techniques are combined. Asymptotic results on the autoregression estimators are derived. Besides combining robust procedures with M‐smoothers, predicted values for the series and detection residuals, which allow to detect anomalous data, are introduced. Robust cross‐validation methods to select the smoothing parameter are presented as an alternative to the classical ones, which are sensitive to outlying observations. A Monte Carlo study is conducted to compare the performance of the proposed criteria. Finally, the asymptotic distribution of the autoregression parameter estimator is stated uniformly over the smoothing parameter.     

Central limit theorems for nonparametric estimators with real‐time random variables

In this article, asymptotic theories for nonparametric methods are studied when they are applied to real‐time data. In particular, we derive central limit theorems for nonparametric density and regression estimators. For this we formally introduce a sequence of real‐time random variables indexed by a parameter related to fine gridding of time domain (or fine discretization). Our results show that the impact of fine gridding is greater in the density estimation case in the sense that strong dependence due to fine gridding severely affects the major strength of nonparametric density estimator (or its data‐adaptive property). In addition, we discuss some issues about nonparametric regression model with fine gridding of time domain.     

Spectral estimates for high‐frequency sampled continuous‐time autoregressive moving average processes

In this article, we consider a continuous‐time autoregressive moving average (CARMA) process driven by either a symmetric α‐stable Lévy process with α ∈ (0,2) or a symmetric Lévy process with finite second moments. In the asymptotic framework of high‐frequency data within a long time interval, we establish a consistent estimate for the normalized power transfer function by applying a smoothing filter to the periodogram of the CARMA process. We use this result to propose an estimator for the parameters of the CARMA process and exemplify the estimation procedure by a simulation study.     

THE ESTIMATION AND APPLICATION OF LONG MEMORY TIME SERIES MODELS

Abstract. The definitions of fractional Gaussian noise and integrated (or fractionally differenced) series are generalized, and it is shown that the two concepts are equivalent. A new estimator of the long memory parameter in these models is proposed, based on the simple linear regression of the log periodogram on a deterministic regressor. The estimator is the ordinary least squares estimator of the slope parameter in this regression, formed using only the lowest frequency ordinates of the log periodogram. Its asymptotic distribution is derived, from which it is evident that the conventional interpretation of these least squares statistics is justified in large samples. Using synthetic data the asymptotic theory proves to be reliable in samples of 50 observations or more. For three postwar monthly economic time series, the estimated integrated series model provides more reliable out-of-sample forecasts than do more conventional procedures.     

Separation of Uncorrelated Stationary time series using Autocovariance Matrices

In blind source separation, one assumes that the observed p time series are linear combinations of p latent uncorrelated weakly stationary time series. To estimate the unmixing matrix, which transforms the observed time series back to uncorrelated latent time series, second‐order blind identification (SOBI) uses joint diagonalization of the covariance matrix and autocovariance matrices with several lags. In this article, we find the limiting distribution of the well‐known symmetric SOBI estimator under general conditions and compare its asymptotical efficiencies to those of the recently introduced deflation‐based SOBI estimator. The theory is illustrated by some finite‐sample simulation studies.     

Consistent estimation of the memory parameter for nonlinear time series

Abstract. For linear processes, semiparametric estimation of the memory parameter, based on the log‐periodogram and local Whittle estimators, has been exhaustively examined and their properties well established. However, except for some specific cases, little is known about the estimation of the memory parameter for nonlinear processes. The purpose of this paper is to provide the general conditions under which the local Whittle estimator of the memory parameter of a stationary process is consistent and to examine its rate of convergence. We show that these conditions are satisfied for linear processes and a wide class of nonlinear models, among others, signal plus noise processes, nonlinear transforms of a Gaussian process ξ_t and exponential generalized autoregressive, conditionally heteroscedastic (EGARCH) models. Special cases where the estimator satisfies the central limit theorem are discussed. The finite‐sample performance of the estimator is investigated in a small Monte Carlo study.     

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.     

Asymptotic self‐similarity and wavelet estimation for long‐range dependent fractional autoregressive integrated moving average time series with stable innovations

Abstract. Methods for parameter estimation in the presence of long‐range dependence and heavy tails are scarce. Fractional autoregressive integrated moving average (FARIMA) time series for positive values of the fractional differencing exponent d can be used to model long‐range dependence in the case of heavy‐tailed distributions. In this paper, we focus on the estimation of the Hurst parameter H = d + 1/α for long‐range dependent FARIMA time series with symmetric α‐stable (1 < α < 2) innovations. We establish the consistency and the asymptotic normality of two types of wavelet estimators of the parameter H. We do so by exploiting the fact that the integrated series is asymptotically self‐similar with parameter H. When the parameter α is known, we also obtain consistent and asymptotically normal estimators for the fractional differencing exponent d = H ? 1/α. Our results hold for a larger class of causal linear processes with stable symmetric innovations. As the wavelet‐based estimation method used here is semi‐parametric, it allows for a more robust treatment of long‐range dependent data than parametric methods.     

Influence of model parameter estimation methods and regression algorithms on curing kinetics and rheological modelling

The accuracy of the phenomenological curing and rheological models are strongly related to the estimated start parameters and selected regression algorithms. Considering the versatile methods for model start parameter estimation (model‐free vs. model‐fitting, dynamic vs. isothermal) and regression analysis algorithm (linear vs. nonlinear, single‐target vs. multi‐target), this paper investigates the theoretical basis and influence of these aspects on the model development process and model quality. The curing kinetics is modelled by model‐free and model‐fitting start parameters and different regression algorithms, followed by cross model validation at the final. The results showed that the different parameter estimation methods and evaluation algorithms have a remarkable influence on the final model parameters and its quality. The study shows the correlation between the different aspects and provides a basis for better selection of model parameter evaluation methods and regression algorithms for model development with improved quality and accuracy. © 2017 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2017 , 134, 45137.     

NON‐PARAMETRIC ESTIMATION OF HIGH‐FREQUENCY SPOT VOLATILITY FOR BROWNIAN SEMIMARTINGALE WITH JUMPS

The availability of high‐frequency financial data has led to substantial improvements in our understanding of financial volatility. Most existing literature focuses on estimating the integrated volatility over a fixed period. This article proposes a non‐parametric threshold kernel method to estimate the time‐dependent spot volatility and jumps when the underlying price process is governed by Brownian semimartingale with finite activity jumps. The threshold kernel estimator combines the threshold estimation for integrated volatility and the kernel filtering approach for spot volatility when the price process is driven only by diffusions without jumps. The estimator proposed is consistent and asymptotically normal and has the same rate of convergence as the estimator studied by Kristensen (2010) in a setting without jumps. The Monte Carlo simulation study shows that the proposed estimator exhibits excellent performance over a wide range of jump sizes and for different sampling frequencies. An empirical example is given to illustrate the potential applications of the proposed method.     

ESTIMATION OF AUTOCOVARIANCE MATRICES FOR INFINITE DIMENSIONAL VECTOR LINEAR PROCESS

Consider an infinite dimensional vector linear process. Under suitable assumptions on the parameter space, we provide consistent estimators of the autocovariance matrices. In particular, under causality, this includes the infinite‐dimensional vector autoregressive (IVAR) process. In that case, we obtain consistent estimators for the parameter matrices. An explicit expression for the estimators is obtained for IVAR(1), under a fairly realistic parameter space. We also show that under some mild restrictions, the consistent estimator of the marginal large dimensional variance–covariance matrix has the same convergence rate as that in case of i.i.d. samples.     

Inference on Multivariate Heteroscedastic Time Varying Random Coefficient Models

In this article, we introduce the general setting of a multivariate time series autoregressive model with stochastic time‐varying coefficients and time‐varying conditional variance of the error process. This allows modelling VAR dynamics for non‐stationary time series and estimation of time‐varying parameter processes by the well‐known rolling regression estimation techniques. We establish consistency, convergence rates, and asymptotic normality for kernel estimators of the paths of coefficient processes and provide pointwise valid standard errors. The method is applied to a popular seven‐variable dataset to analyse evidence of time variation in empirical objects of interest for the DSGE (dynamic stochastic general equilibrium) literature.     

Interventions in INGARCH processes

We study the problem of intervention effects generating various types of outliers in a linear count time‐series model. This model belongs to the class of observation‐driven models and extends the class of Gaussian linear time‐series models within the exponential family framework. Studies about effects of covariates and interventions for count time‐series models have largely fallen behind, because the underlying process, whose behaviour determines the dynamics of the observed process, is not observed. We suggest a computationally feasible approach to these problems, focusing especially on the detection and estimation of sudden shifts and outliers. We consider three different scenarios, namely the detection of an intervention effect of a known type at a known time, the detection of an intervention effect when the type and the time are both unknown and the detection of multiple intervention effects. We develop score tests for the first scenario and a parametric bootstrap procedure based on the maximum of the different score test statistics for the second scenario. The third scenario is treated by a stepwise procedure, where we detect and correct intervention effects iteratively. The usefulness of the proposed methods is illustrated using simulated and real data examples.     

Broadband semi‐parametric estimation of long‐memory time series by fractional exponential models

This article proposes broadband semi‐parametric estimation of a long‐memory parameter by fractional exponential (FEXP) models. We construct the truncated Whittle likelihood based on FEXP models in a semi‐parametric setting to estimate the parameter and show that the proposed estimator is more efficient than the FEXP estimator by Moulines and Soulier (1999) in linear processes. A Monte Carlo simulation suggests that the proposed estimation is more preferable than the existing broadband semi‐parametric estimation.     

A note on the mixture transition distribution and hidden Markov models

We discuss an interpretation of the mixture transition distribution (MTD) for discrete‐valued time series which is based on a sequence of independent latent variables which are occasion‐specific. We show that, by assuming that this latent process follows a first order Markov Chain, MTD can be generalized in a sensible way. A class of models results which also includes the hidden Markov model (HMM). For these models we outline an EM algorithm for the maximum likelihood estimation which exploits recursions developed within the HMM literature. As an illustration, we provide an example based on the analysis of stock market data referred to different American countries.