Empirical Likelihood for a Long Range Dependent Process Subordinated to a Gaussian Process

This article develops empirical likelihood methodology for a class of long range dependent processes driven by a stationary Gaussian process. We consider population parameters that are defined by estimating equations in the time domain. It is shown that the standard block empirical likelihood (BEL) method, with a suitable scaling, has a non‐standard limit distribution based on a multiple Wiener–Itô integral. Unlike the short memory time series case, the scaling constant involves unknown population quantities that may be difficult to estimate. Alternative versions of the empirical likelihood method, involving the expansive BEL (EBEL) methods are considered. It is shown that the EBEL renditions do not require an explicit scaling and, therefore, remove this undesirable feature of the standard BEL. However, the limit law involves the long memory parameter, which may be estimated from the data. Results from a moderately large simulation study on finite sample properties of tests and confidence intervals based on different empirical likelihood methods are also reported.     

Least tail‐trimmed squares for infinite variance autoregressions

We develop a robust least squares estimator for autoregressions with possibly heavy tailed errors. Robustness to heavy tails is ensured by negligibly trimming the squared error according to extreme values of the error and regressors. Tail‐trimming ensures asymptotic normality and super‐‐convergence with a rate comparable to the highest achieved amongst M‐estimators for stationary data. Moreover, tail‐trimming ensures robustness to heavy tails in both small and large samples. By comparison, existing robust estimators are not as robust in small samples, have a slower rate of convergence when the variance is infinite, or are not asymptotically normal. We present a consistent estimator of the covariance matrix and treat classic inference without knowledge of the rate of convergence. A simulation study demonstrates the sharpness and approximate normality of the estimator, and we apply the estimator to financial returns data. Finally, tail‐trimming can be easily extended beyond least squares estimation for a linear stationary AR model. We discuss extensions to quasi‐maximum likelihood for GARCH, weighted least squares for a possibly non‐stationary random coefficient autoregression, and empirical likelihood for robust confidence region estimation, in each case for models with possibly heavy tailed errors.     

Asymptotic Theory and Unified Confidence Region for an Autoregressive Model

Although some unified inferences for the coefficient in an AR(1) model have been proposed in the literature, it remains open as to how to construct a unified confidence region for the intercept and the coefficient jointly without a prior on whether the sequence is stationary or unit root or near unit root or moderate deviations from a unit root or explosive and whether the sequence has a zero or nonzero constant intercept. After deriving the joint limit of the least squares estimator for all of these cases, this article proposes a unified empirical likelihood confidence region by first splitting the data into two parts and then constructing some weighted score equations. The good finite sample performance of the proposed method is demonstrated via a simulation study. Real data applications are provided as well.     

Empirical Likelihood for Outlier Detection and Estimation in Autoregressive Time Series

Identification and estimation of outliers in time series is proposed by using empirical likelihood methods. Theory and applications are developed for stationary autoregressive models with outliers distinguished in the usual additive and innovation types. Some other useful outlier types are considered as well. A simulation experiment is used for studying the behaviour of the empirical likelihood‐based method in finite samples and indicates that the proposed methods are preferable when dealing with the non‐Gaussian data. Our simulations suggest that the usual sequential procedure for multiple outlier detection is suitable also for the methods based on empirical likelihood.     

Empirical likelihood inference for random coefficient INAR(p) process

In this article, we study the empirical likelihood (EL) method for the pth‐order random coefficient integer‐valued autoregressive process. In particular, the limiting distribution of the log EL ratio statistic is established and the confidence regions for the parameter of interest are derived. Also a simulation study is conducted for the evaluation of the developed approach.     

A similarity‐based approach to time‐varying coefficient non‐stationary autoregression

We suggest in this article a similarity‐based approach to time‐varying coefficient non‐stationary autoregression. In a given sample, the model can display characteristics consistent with stationary, unit root and explosive behaviour, depending on the similarity between the dependent variable and its past values. We establish consistency of the quasi‐maximum likelihood estimator of the model, with a general norming factor. Asymptotic score‐based hypothesis tests are derived. The model is applied to a data set comprised of dual stocks traded in NASDAQ and the Tokyo Stock Exchange.     

Time‐series clustering via quasi U‐statistics

The problem of time‐series discrimination and classification is discussed. We propose a novel clustering algorithm based on a class of quasi U‐statistics and subgroup decomposition tests. The decomposition may be applied to any concave time‐series distance. The resulting test statistics are proven to be asymptotically normal for either i.i.d. or non‐identically distributed groups of time‐series under mild conditions. We illustrate its empirical performance on a simulation study and a real data analysis. The simulation setup includes stationary vs. stationary and stationary vs. non‐stationary cases. The performance of the proposed method is favourably compared with some of the most common clustering measures available.     

Balanced Bootstrap Joint Confidence Bands for Structural Impulse Response Functions

Constructing joint confidence bands for structural impulse response functions based on a VAR model is a difficult task because of the non‐linear nature of such functions. We propose new joint confidence bands that cover the entire true structural impulse response function up to a chosen maximum horizon with a pre‐specified probability (1?α), at least asymptotically. Such bands are based on a certain bootstrap procedure from the multiple testing literature. We compare the finite‐sample properties of our method with those of existing methods via extensive Monte Carlo simulations. We also investigate the effect of endogenizing the lag order in our bootstrap procedure on the finite‐sample properties. Furthermore, an empirical application to a real dataset is provided.     

Random Walks with Drift – A Sequential Approach

Abstract. In this paper, sequential monitoring schemes to detect nonparametric drifts are studied for the random walk case. The procedure is based on a kernel smoother. As a by‐product we obtain the asymptotics of the Nadaraya–Watson estimator and its associated sequential partial sum process under non‐standard sampling. The asymptotic behaviour differs substantially from the stationary situation, if there is a unit root (random walk component). To obtain meaningful asymptotic results, we consider local nonparametric alternatives for the drift component. It turns out that the rate of convergence at which the drift vanishes determines whether the asymptotic properties of the monitoring procedure are determined by a deterministic or random function. Furthermore, we provide a theoretical result about the optimal kernel for a given alternative.     

Likelihood functions for state space models with diffuse initial conditions

State space models with non‐stationary processes and/or fixed regression effects require a state vector with diffuse initial conditions. Different likelihood functions can be adopted for the estimation of parameters in time‐series models with diffuse initial conditions. In this article, we consider profile, diffuse and marginal likelihood functions. The marginal likelihood function is defined as the likelihood function of a transformation of the data vector. The transformation is not unique. The diffuse likelihood is a marginal likelihood for a data transformation that may depend on parameters. Therefore, the diffuse likelihood cannot be used generally for parameter estimation. The marginal likelihood function is based on an orthonormal data transformation that does not depend on parameters. Here we develop a marginal likelihood function for state space models that can be evaluated by the Kalman filter. The so‐called diffuse Kalman filter is designed for computing the diffuse likelihood function. We show that a minor modification of the diffuse Kalman filter is needed for the evaluation of our marginal likelihood function. Diffuse and marginal likelihood functions have better small sample properties compared with the profile likelihood function for the estimation of parameters in linear time series models. The results in our article confirm the earlier findings and show that the diffuse likelihood function is not appropriate for a range of state space model specifications.     

Stationary bootstrapping for non‐parametric estimator of nonlinear autoregressive model

We consider stationary bootstrap approximation of the non‐parametric kernel estimator in a general kth‐order nonlinear autoregressive model under the conditions ensuring that the nonlinear autoregressive process is a geometrically Harris ergodic stationary Markov process. We show that the stationary bootstrap procedure properly estimates the distribution of the non‐parametric kernel estimator. A simulation study is provided to illustrate the theory and to construct confidence intervals, which compares the proposed method favorably with some other bootstrap methods.     

Frequency domain generalized empirical likelihood method

This paper is concerned with a version of empirical likelihood method for spectral restrictions, which handles stationary time series data via the frequency domain approach. The asymptotic properties of frequency domain generalized empirical likelihood are studied for either strictly stationary processes with vanishing cumulant spectral density function of order 4 or linear processes generated by iid innovations with possibly non‐zero fourth order cumulant. Several statistics for testing parametric restrictions, over‐identified spectral restrictions, and additional spectral restrictions are shown to have the limiting chi‐squared distributions. Some numerical results are presented to investigate the finite sample performance of the proposed procedures. Copyright © 2013 John Wiley & Sons, Ltd.     

Extension of Autocovariance Coefficients Sequence for Periodically Correlated Processes

Abstract. The extension of stationary process autocorrelation coefficient sequence is a classical problem in the field of spectral estimation. In this note, we treat this extension problem for the periodically correlated processes by using the partial autocorrelation function. We show that the theory of the non‐stationary processes can be adapted to the periodically correlated processes. The partial autocorrelation function has a clear advantage for parameterization over the autocovariance function which should be checked for non‐negative definiteness. In this way, we show that contrary to the stationary case, the Yule–Walker equations (for a periodically correlated process) is no longer a tool for extending the first autocovariance coefficients to an autocovariance function. Next, we treat the extension problem and present a maximum entropy method extension through the the partial autocorrelation function. We show that the solution maximizing the entropy is a periodic autoregressive process and compare this approach with others.     

Transformation to approximate independence for locally stationary Gaussian processes

We provide new approximations for the likelihood of a time series under the locally stationary Gaussian process model. The likelihood approximations are valid even in cases when the evolutionary spectrum is not smooth in the rescaled time domain. We describe a broad class of models for the evolutionary spectrum for which the approximations can be computed particularly efficiently. In developing the approximations, we extend to the locally stationary case the idea that the discrete Fourier transform is a decorrelating transformation for stationary time series. The approximations are applied to fit non‐stationary time‐series models to high‐frequency temperature data. For these data, we fit evolutionary spectra that are piecewise constant in time and use a genetic algorithm to search for the best partition of the time interval.     

A Lagrange Multiplier‐Type Test for Idiosyncratic Unit Roots in the Exact Factor Model

In this article, an exact factor model is considered, and a Lagrange multiplier‐type test is derived for a homogeneous unit root in the idiosyncratic component. It is shown that under sequential asymptotics, its null limiting distribution is standard normal, regardless of whether the factors are integrated, cointegrated or stationary. In a simulation study, the size and local power of the Lagrange multiplier‐type test and some popular non‐likelihood‐based tests are compared. The simulation results show that the Lagrange multiplier‐type test has the highest local power as the panel dimensions tend to infinity, with the actual size tending to the nominal size.     

Nonstationary Cointegration in the Fractionally Cointegrated VAR Model

We consider the fractional cointegrated vector autoregressive (CVAR) model of Johansen and Nielsen (2012a) and make two distinct contributions. First, in their consistency proof, Johansen and Nielsen (2012a) imposed moment conditions on the errors that depend on the parameter space, such that when the parameter space is larger, stronger moment conditions are required. We show that these moment conditions can be relaxed, and for consistency we require just eight moments regardless of the parameter space. Second, Johansen and Nielsen (2012a) assumed that the cointegrating vectors are stationary, and we extend the analysis to include the possibility that the cointegrating vectors are non‐stationary. Both contributions require new analysis and results for the asymptotic properties of the likelihood function of the fractional CVAR model, which we provide. Finally, our analysis follows recent research and applies a parameter space large enough that the usual (non‐fractional) CVAR model constitutes an interior point and hence can be tested against the fractional model using a Chi‐squared‐test.     

Robust empirical likelihood for time series

This article introduces a robust frequency domain empirical likelihood inference procedure for the parametric component in the spectral densities of stationary processes. We construct the empirical likelihood function by using a new spectral estimating function to achieve robustness against contamination in the spectral density. Simulation studies demonstrate the good performance of the proposed robust frequency domain empirical likelihood method, which produces more accurate confidence regions than the ordinary empirical likelihood counterpart.     

Testing for Stationarity in Multivariate Locally Stationary Processes

In this article, we propose a nonparametric procedure for validating the assumption of stationarity in multivariate locally stationary time series models. We develop a bootstrap‐assisted test based on a Kolmogorov–Smirnov‐type statistic, which tracks the deviation of the time‐varying spectral density from its best stationary approximation. In contrast to all other nonparametric approaches, which have been proposed in the literature so far, the test statistic does not depend on any regularization parameters like smoothing bandwidths or a window length, which is usually required in a segmentation of the data. We additionally show how our new procedure can be used to identify the components where non‐stationarities occur and indicate possible extensions of this innovative approach. We conclude with an extensive simulation study, which shows finite‐sample properties of the new method and contains a comparison with existing approaches.     

NORMING RATES AND LIMIT THEORY FOR SOME TIME‐VARYING COEFFICIENT AUTOREGRESSIONS

A time‐varying autoregression is considered with a similarity‐based coefficient and possible drift. It is shown that the random‐walk model has a natural interpretation as the leading term in a small‐sigma expansion of a similarity model with an exponential similarity function as its AR coefficient. Consistency of the quasi‐maximum likelihood estimator of the parameters in this model is established, the behaviours of the score and Hessian functions are analysed and test statistics are suggested. A complete list is provided of the normalization rates required for the consistency proof and for the score and Hessian function standardization. A large family of unit root models with stationary and explosive alternatives is characterized within the similarity class through the asymptotic negligibility of a certain quadratic form that appears in the score function. A variant of the stochastic unit root model within the class is studied, and a large‐sample limit theory provided, which leads to a new nonlinear diffusion process limit showing the form of the drift and conditional volatility induced by sustained stochastic departures from unity. The findings provide a composite case for time‐varying coefficient dynamic modelling. Some simulations and a brief empirical application to data on international Exchange Traded Funds are included. Copyright © 2014 Wiley Publishing Ltd     

Modelling Count Data Time Series with Markov Processes Based on Binomial Thinning

Abstract. We obtain new models and results for count data time series based on binomial thinning. Count data time series may have non‐stationarity from trends or covariates, so we propose an extension of stationary time series based on binomial thinning such that the univariate marginal distributions are always in the same parametric family, such as negative binomial. We propose a recursive algorithm to calculate the probability mass functions for the innovation random variable associated with binomial thinning. This simplifies numerical calculations and estimation for the classes of time series models that we consider. An application with real data is used to illustrate the models.