Asymptotics for the Conditional‐Sum‐of‐Squares Estimator in Multivariate Fractional Time‐Series Models

This article proves consistency and asymptotic normality for the conditional‐sum‐of‐squares estimator, which is equivalent to the conditional maximum likelihood estimator, in multivariate fractional time‐series models. The model is parametric and quite general and, in particular, encompasses the multivariate non‐cointegrated fractional autoregressive integrated moving average (ARIMA) model. The novelty of the consistency result, in particular, is that it applies to a multivariate model and to an arbitrarily large set of admissible parameter values, for which the objective function does not converge uniformly in probability, thus making the proof much more challenging than usual. The neighbourhood around the critical point where uniform convergence fails is handled using a truncation argument.     

Cointegrated Linear Processes in Hilbert Space

We extend the notion of cointegration for multivariate time series to a potentially infinite‐dimensional setting in which our time series takes values in a complex separable Hilbert space. In this setting, standard linear processes with nonzero long‐run covariance operator play the role of processes. We show that the cointegrating space for an process may be sensibly defined as the kernel of the long‐run covariance operator of its difference. The inner product of an process with an element of its cointegrating space is a stationary complex‐valued process. Our main result is a version of the Granger–Johansen representation theorem: we obtain a geometric reformulation of the Johansen I(1) condition that extends naturally to a Hilbert space setting, and show that an autoregressive Hilbertian process satisfying this condition, and possibly also a compactness condition, admits an representation.     

The Periodogram of fractional processes1

Abstract. We analyse asymptotic properties of the discrete Fourier transform and the periodogram of time series obtained through (truncated) linear filtering of stationary processes. The class of filters contains the fractional differencing operator and its coefficients decay at an algebraic rate, implying long‐range‐dependent properties for the filtered processes when the degree of integration α is positive. These include fractional time series which are nonstationary for any value of the memory parameter (α ≠ 0) and possibly nonstationary trending (α ≥ 0.5). We consider both fractional differencing or integration of weakly dependent and long‐memory stationary time series. The results obtained for the moments of the Fourier transform and the periodogram at Fourier frequencies in a degenerating band around the origin are weaker compared with the stationary nontruncated case for α > 0, but sufficient for the analysis of parametric and semiparametric memory estimates. They are applied to the study of the properties of the log‐periodogram regression estimate of the memory parameter α for Gaussian processes, for which asymptotic normality could not be showed using previous results. However, only consistency can be showed for the trending cases, 0.5 ≤ α < 1. Several detrending and initialization mechanisms are studied and only local conditions on spectral densities of stationary input series and transfer functions of filters are assumed.     

Density estimation with locally identically distributed data and with locally stationary data

Abstract. We consider multivariate density estimation when the assumptions of identically distributed data or stationary data are relaxed to the assumptions of locally identically distributed data or locally stationary data. We assume that the distribution of the data is changing continuously as function of time. To estimate densities non‐parametrically with these local regularity conditions, we need time localization in addition to the usual space localization. We define a time‐localized kernel estimator that estimates the density non‐parametrically at any given point of time. The consistency of the time‐localized kernel estimator is proved and the rates of convergence of the estimator are derived under conditions on the β‐and α‐mixing coefficients. Both the time‐series setting and spatial setting are covered.     

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.     

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.     

Integration of CARMA processes and spot volatility modelling

Continuous‐time autoregressive moving average (CARMA) processes with a non‐negative kernel and driven by a non‐decreasing Lévy process constitute a useful and very general class of stationary, non‐negative continuous‐time processes which have been used, in particular for the modelling of stochastic volatility. In the celebrated stochastic volatility model of Barndorff‐Nielsen and Shephard (2001) , the spot (or instantaneous) volatility at time t, V(t), is represented by a stationary Lévy‐driven Ornstein‐Uhlenbeck process. This has the shortcoming that its autocorrelation function is necessarily a decreasing exponential function, limiting its ability to generate integrated volatility sequences, , with autocorrelation functions resembling those of observed realized volatility sequences. (A realized volatility sequence is a sequence of estimated integrals of spot volatility over successive intervals of fixed length, typically 1 day.) If instead of the stationary Ornstein–Uhlenbeck process, we use a CARMA process to represent spot volatility, we can overcome the restriction to exponentially decaying autocorrelation function and obtain a more realistic model for the dependence observed in realized volatility. In this article, we show how to use realized volatility data to estimate parameters of a CARMA model for spot volatility and apply the analysis to a daily realized volatility sequence for the Deutsche Mark/ US dollar exchange rate.     

Modelling long‐run trends and cycles in financial time series data

This article proposes a general time series framework to capture the long‐run behaviour of financial series. The suggested approach includes linear and segmented time trends, and stationary and non‐stationary processes based on integer and/or fractional degrees of differentiation. Moreover, the spectrum is allowed to contain more than a single pole or singularity, occurring at both zero but non‐zero (cyclical) frequencies. This framework is used to analyse five annual time series with a long span, namely dividends, earnings, interest rates, stock prices and long‐term government bond yields. The results based on several likelihood criteria indicate that the five series exhibit fractional integration with one or two poles in the spectrum, and are quite stable over the sample period examined.     

Small‐b and Fixed‐b Asymptotics for Weighted Covariance Estimation in Fractional Cointegration

In a standard cointegrating framework, Phillips (1991) introduced the weighted covariance (WC) estimator of cointegrating parameters. Later, Marinucci (2000) applied this estimator to fractional circumstances and, like Phillips (1991), analysed the so‐called small‐b asymptotic approximation to its sampling distribution. Recently, an alternative limiting theory (fixed‐b asymptotics) has been successfully employed to approximate sampling distributions. With the purpose of comparing both approaches, we derive here the fixed‐b limit of WC estimators in a fractional setting, filling also some gaps in the traditional (small‐b) theory. We also provide some Monte Carlo evidence that suggests that the fixed‐b limit is more accurate.     

Frequency Domain Estimation of Continuous Time Cointegrated Models with Mixed Frequency and Mixed Sample Data

Recent work by the author on mixed frequency data analysis has focused on the estimation of cointegrated systems in continuous time based on a fully specified dynamic system of equations, while the estimation of cointegrating vectors in a discrete time system has been approached using a semiparametric frequency domain estimator. We extend the latter approach to cover the continuous time case, establishing the asymptotic properties of the frequency domain estimator and explore, in a simulation study, the effects of misspecifying the continuous time dynamic model in discrete time compared to treating the dynamics non‐parametrically. An empirical illustration is also provided.     

Fractional thermoelectric viscoelastic materials

A new mathematical model of thermoelectro viscoelasticity theory was constructed in the context of a new consideration of heat conduction with fractional order. The state space approach developed earlier by Ezzat was adopted for the solution of a one‐dimensional problem in the presence of heat sources. The Laplace‐transform technique was used. A numerical method was employed for the inversion of the Laplace transforms. According to the numerical results and their graphs, a conclusion about the new theory was constructed. Some comparisons are shown in figures to estimate the effect of the fractional order parameter on all of the studied fields. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2012     

Stability of nonlinear AR‐GARCH models

Abstract. This article studies the stability of nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a nonlinear autoregression of order p [AR(p)] with the conditional variance specified as a nonlinear first‐order generalized autoregressive conditional heteroskedasticity [GARCH(1,1)] model. Conditions under which the model is stable in the sense that its Markov chain representation is geometrically ergodic are provided. This implies the existence of an initial distribution such that the process is strictly stationary and β‐mixing. Conditions under which the stationary distribution has finite moments are also given. The results cover several nonlinear specifications recently proposed for both the conditional mean and conditional variance, and only require mild moment conditions.     

Poisson QMLE of Count Time Series Models

Regularity conditions are given for the consistency of the Poisson quasi‐maximum likelihood estimator of the conditional mean parameter of a count time series model. The asymptotic distribution of the estimator is studied when the parameter belongs to the interior of the parameter space and when it lies at the boundary. Tests for the significance of the parameters and for constant conditional mean are deduced. Applications to specific integer‐valued autoregressive (INAR) and integer‐valued generalized autoregressive conditional heteroscedasticity (INGARCH) models are considered. Numerical illustrations, Monte Carlo simulations and real data series are provided.     

Inference about long run canonical correlations

This article proposes methods for testing the null hypothesis that a number of so‐called long run canonical correlations (LRCCs) are zero. Two test statistics are proposed and their limiting distributions are derived under the null hypothesis. The finite sample properties of the tests are illustrated via a number of simulation studies that reveal the asymptotic theory provides a good guidance to behaviour in moderate or large sized samples. It is shown that the statistics provide a natural way for testing the asymptotic independence of two standardized sums. The usefulness of the tests is illustrated via the following examples: inference about cointegrating vector in a particular cointegration model; inference about break points in a cointegration model; moment estimation; parameter estimation in Generalized Method of Moments estimation.     

UNIFIED INTERVAL ESTIMATION FOR RANDOM COEFFICIENT AUTOREGRESSIVE MODELS

The consistency of the quasi‐maximum likelihood estimator for random coefficient autoregressive models requires that the coefficient be a non‐degenerate random variable. In this article, we propose empirical likelihood methods based on weighted‐score equations to construct a confidence interval for the coefficient. We do not need to distinguish whether the coefficient is random or deterministic and whether the process is stationary or non‐stationary, and we present two classes of equations depending on whether a constant trend is included in the model. A simulation study confirms the good finite‐sample behaviour of our resulting empirical likelihood‐based confidence intervals. We also apply our methods to study US macroeconomic data.     

Reactions in solid particles—A reappraisal of models

Models such as the Ishida–Wen, or even more commonly the shrinking core models (Ginstling–Brounshtein being an important example), have long been used in the analyses of reactions in particles, particularly in solid–solid systems. There have been a few analyses of the validity of the assumptions made in these models, but to date, no comparison has been undertaken of these models against a general model to delineate their regions of applicability in the parameter space. In this article, we present a general unsteady‐state model that subsumes the earlier models as special cases. Nondimensionalization leads to the identification of two governing parameters in the model, a diffusion‐reaction parameter, and a relative abundance parameter. By solving the general model and comparing the solutions with those of the approximate models in the parameter space, conditions under which the approximate models apply, and the errors that result from their application in other situations, have been identified. © 2012 American Institute of Chemical Engineers AIChE J, 58: 3161–3166, 2012     

GQL Versus Conditional GQL Inferences for Non‐Stationary Time Series of Counts with Overdispersion

Abstract. This article proposes an autoregressive model for time series of counts with non‐stationary means, variances and covariances as functions of certain time‐dependant covariates. For the estimation of the regression, overdispersion and correlation index parameters, a conditional generalized quasilikelihood (CGQL) approach is developed under the assumption that the count responses marginally satisfy the first two moments of a negative binomial distribution. Thus this CGQL approach avoids the use of the likelihood or so‐called partial likelihood of the data which are known to be extremely complicated in the present non‐stationary time series set‐up. It is shown through an extensive simulation study that the proposed CGQL approach performs very well in estimating the parameters of the model. This is also shown that the CGQL approach performs better than an existing GQL approach, especially for the estimation of the overdispersion parameter of the model.     

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.     

Asymptotic results for Fourier‐PARMA time series

Periodically stationary times series are useful to model physical systems whose mean behavior and covariance structure varies with the season. The Periodic Auto‐Regressive Moving Average (PARMA) process provides a powerful tool for modelling periodically stationary series. Since the process is non‐stationary, the innovations algorithm is useful to obtain parameter estimates. Fitting a PARMA model to high‐resolution data, such as weekly or daily time series, is problematic because of the large number of parameters. To obtain a more parsimonious model, the discrete Fourier transform (DFT) can be used to represent the model parameters. This article proves asymptotic results for the DFT coefficients, which allow identification of the statistically significant frequencies to be included in the PARMA model.     

A Generalised Fractional Differencing Bootstrap for Long Memory Processes

A bootstrap methodology suitable for use with stationary and non‐stationary fractionally integrated time series is further developed in this article. The resampling algorithm involves estimating the degree of fractional integration, applying the fractional differencing operator, resampling the resulting approximation to the underlying short memory series and, finally, cumulating to obtain a resample of the original fractionally integrated process. This approach extends existing methods in the literature by allowing for general bootstrap schemes including blockwise bootstraps. Furthermore, we show that it can also be validly used for non‐stationary fractionally integrated processes. We establish asymptotic validity results for the general method and provide simulation evidence which highlights a number of favourable aspects of its finite sample performance, relative to other commonly used bootstrap methods.