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.     

Empirical Characteristic Functions-Based Estimation and Distance Correlation for Locally Stationary Processes

In this article, we propose a kernel-type estimator for the local characteristic function of locally stationary processes. Under weak moment conditions, we prove joint asymptotic normality for local empirical characteristic functions. For time-varying linear processes, we establish a central limit theorem under the assumption of finite absolute first moments of the process. Additionally, we prove weak convergence of the local empirical characteristic process. We apply our asymptotic results to parameter estimation. Furthermore, by extending the notion of distance correlation to locally stationary processes, we are able to provide asymptotic theory for local empirical distance correlations. Finally, we provide a simulation study on minimum distance estimation for α-stable distributions and illustrate the pairwise dependence structure over time of log returns of German stock prices via local empirical distance correlations.     

Bootstrapping the Local Periodogram of Locally Stationary Processes

Abstract. Locally stationary processes are non‐stationary stochastic processes the second‐order structure of which varies smoothly over time. In this paper, we develop a method to bootstrap the local periodogram of a locally stationary process. Our method generates pseudo local periodogram ordinates by combining a parametric time and non‐parametric frequency domain bootstrap approach. We first fit locally a time varying autoregressive model so as to capture the essential characteristics of the underlying process. A locally calculated non‐parametric correction in the frequency domain is then used so as to improve upon the locally parametric autoregressive fit. As an application, we investigate theoretically the asymptotic properties of the bootstrap method proposed applied to the class of local spectral means, local ratio statistics and local spectral density estimators. Some simulations demonstrate the ability of our method to give accurate estimates of the quantities of interest in finite sample situations and an application to a real‐life data‐set is presented.     

Time Deformation,Continuous Euler Processes and Forecasting

Abstract. A continuous Euler model has time‐varying coefficients. Through a logarithmic time transformation, a continuous Euler model can be transformed to a continuous autoregressive (AR) model. By using the continuous Kalman filtering through the Laplace method, this article explores the data application of a continuous Euler process. This time deformation of an Euler process deforms specific time‐variant (non‐stationary) behaviour to time‐invariant (stationary) data on the deformed time scale. With these time‐invariant data on the transformed time scale, one may use traditional tools to conduct parameter estimation and forecasts. The obtained results can then be transformed back to the original time scale. Simulated data and actual data such as bat echolocation and the US residential investment growth are used to demonstrate the usefulness of time deformation in forecasting. The results indicate that fitting a traditional autoregressive moving‐average (ARMA) model on an Euler data set without imposing time transformation leads to forecasts that are out of phase while the forecasts of an Euler model stay mostly in phase.     

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.     

Some Nonparametric Tests for Change-Point Detection Based on the ℙ-ℙ and ℚ-ℚ Plot Processes

Abstract

We propose nonparametric procedures for testing change-point by using the ?-? and ?-? plots processes. The limiting distributions of the proposed statistics are characterized under the null hypothesis of no change and also under contiguous alternatives. We give an estimator of the change-point coefficient and obtain its strong consistency. We introduce the bootstrapped version of ?-? and ?-? processes, requiring the estimation of quantile density, and obtain their limiting laws. Finally, we propose and investigate the exchangeable bootstrap of the empirical ?-? plot and ?-? plot processes which avoids the problem of the estimation of quantile density, which is of its own interest. These results are used for calculating p-values of the proposed test statistics. Emphasis is placed on the explanation of the strong approximation methodology.     

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.     

On geometric ergodicity of CHARME models

In this article we consider a CHARME model, a class of generalized mixture of nonlinear nonparametric AR‐ARCH time series. To provide sets of conditions under which such processes are geometrically ergodic and, therefore, satisfy some mixing conditions, we apply the theory of Markov chains to derive asymptotic stability of this model. These results form the basis for deriving an asymptotic theory for nonparametric estimation. As an illustration, neural network sieve estimates for the autoregressive and volatility functions are considered, and consistency of the parameter estimates is obtained.     

Functional Generalized Autoregressive Conditional Heteroskedasticity

Heteroskedasticity is a common feature of financial time series and is commonly addressed in the model building process through the use of autoregressive conditional heteroskedastic and generalized autoregressive conditional heteroskedastic (GARCH) processes. More recently, multivariate variants of these processes have been the focus of research with attention given to methods seeking an efficient and economic estimation of a large number of model parameters. Because of the need for estimation of many parameters, however, these models may not be suitable for modelling now prevalent high‐frequency volatility data. One potentially useful way to bypass these issues is to take a functional approach. In this article, theory is developed for a new functional version of the GARCH process, termed fGARCH. The main results are concerned with the structure of the fGARCH(1,1) process, providing criteria for the existence of strictly stationary solutions both in the space of square‐integrable and continuous functions. An estimation procedure is introduced, and its consistency and asymptotic normality are verified. A small empirical study highlights potential applications to intraday volatility estimation.     

A note on non‐parametric testing for Gaussian innovations in AR–ARCH models

In this paper, we consider autoregressive models with conditional autoregressive variance, including the case of homoscedastic AR models and the case of ARCH models. Our aim is to test the hypothesis of normality for the innovations in a completely non‐parametric way, that is, without imposing parametric assumptions on the conditional mean and volatility functions. To this end, the Cramér–von Mises test based on the empirical distribution function of non‐parametrically estimated residuals is shown to be asymptotically distribution‐free. We demonstrate its good performance for finite sample sizes in a small simulation study. AMS 2010 Classification: Primary 62 M10, Secondary 62 G10     

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.     

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.     

A Wavelet Characterization of Continuous‐Time Periodically Correlated Processes with Application to Simulation

We introduce a wavelet characterization of continuous‐time periodically correlated processes based on a linear combination of infinite‐dimensional stationary processes. The finite version of this linear combination converges to the main process. The first‐order and second‐order estimators based on the wavelets are presented. Under a simple and easy algorithm, the periodically correlated process is simulated for a given autocovariance function. The proposed algorithm has two main advantages: first, it is fast, and second, it is distribution free. We indicate through four examples that the simulated data are periodically correlated with the desired period.     

On the Distribution Estimation of Power Threshold Garch Processes

The aim of this article is to estimate the probability distribution of power threshold generalized autoregressive conditional heteroskedasticity processes by establishing bounds for their finite dimensional laws. These bounds only depend on the parameters of the model and on the distribution function of its independent generating process. The application of this study to some particular models allows us to conjecture that this procedure is an adequate alternative to the corresponding estimation using the empirical distribution functions, particularly useful in the development of control charts for this kind of models.     

Using Difference‐Based Methods for Inference in Regression with Fractionally Integrated Processes

Abstract. This paper suggests a difference‐based method for inference in the regression model involving fractionally integrated processes. Under suitable regularity conditions, our method can effectively deal with the inference problems associated with the regression model consisting of nonstationary, stationary and intermediate memory regressors, simultaneously. Although the difference‐based method provides a very flexible modelling framework for empirical studies, the implementation of this method is extremely easy, because it completely avoids the difficult problems of choosing a kernel function, a bandwidth parameter, or an autoregressive lag length for the long‐run variance estimation. The asymptotic local power of our method is investigated with a sequence of local data‐generating processes (DGP) in what Davidson and MacKinnon [Canadian Journal of Economics. (1985) Vol. 18, pp. 38–57] call ‘regression direction’. The simulation results indicate that the size control of our method is excellent even when the sample size is only 100, and the pattern of power performance is highly consistent with the theoretical finding from the asymptotic local power analysis conducted in this paper.     

A superharmonic prior for the autoregressive process of the second‐order

Abstract. The Bayesian estimation of the spectral density of the AR(2) process is considered. We propose a superharmonic prior on the model as a non‐informative prior rather than the Jeffreys prior. Theoretically, the Bayesian spectral density estimator based on it dominates asymptotically the one based on the Jeffreys prior under the Kullback–Leibler divergence. In the present article, an explicit form of a superharmonic prior for the AR(2) process is presented and compared with the Jeffreys prior in computer simulation.     

High Moment Partial Sum Processes of Residuals in ARMA Models and their Applications

Abstract. In this article, we study high moment partial sum processes based on residuals of a stationary autoregressive moving average (ARMA) model with known or unknown mean parameter. We show that they can be approximated in probability by the analogous processes which are obtained from the i.i.d. errors of the ARMA model. However, if a unknown mean parameter is used, there will be an additional term that depends on model parameters and a mean estimator. When properly normalized, this additional term will vanish. Thus the processes converge weakly to the same Gaussian processes as if the residuals were i.i.d. Applications to change‐point problems and goodness‐of‐fit are considered, in particular, cumulative sum statistics for testing ARMA model structure changes and the Jarque–Bera omnibus statistic for testing normality of the unobservable error distribution of an ARMA model.     

Consistent Estimation for Non-Gaussian Non-Causal Autoregessive Processes

Traditional estimation based on least squares or Gaussian likelihood cannot distinguish between causal and non-causal representation of a stationary autoregressive (AR) process. Breidt et al . (Maximum likelihood estimation for non-causal autoregressive processes. J. Multivariate Anal. 36 (1991), 175–98) proved the existence of a consistent likelihood estimation of possibly non-causal AR processes; however, in this case an existence result is not very useful since the likelihood function generally exhibits multiple maxima. Moreover the method assumes full knowledge of the distribution of the innovation process. This paper shows a constructive proof that a modified L ₁ estimate is consistent if the innovation process has a stable law distribution with index α∈ (1, 2). It is also shown that neither non-Gaussianity nor infinite variance is sufficient to ensure consistency.     

A Note on Non‐Negative Arma Processes

Abstract. Recently, there has been much research on developing models suitable for analysing the volatility of a discrete‐time process. Since the volatility process, like many others, is necessarily non‐negative, there is a need to construct models for stationary processes which are non‐negative with probability one. Such models can be obtained by driving autoregressive moving average (ARMA) processes with non‐negative kernel by non‐negative white noise. This raises the problem of finding simple conditions under which an ARMA process with given coefficients has a non‐negative kernel. In this article, we derive a necessary and sufficient condition. This condition is in terms of the generating function of the ARMA kernel which has a simple form. Moreover, we derive some readily verifiable necessary and sufficient conditions for some ARMA processes to be non‐negative almost surely.     

Filling the gap between Continuous and Discrete Time Dynamics of Autoregressive Processes

Discretization of continuous time autoregressive (AR) processes driven by a Brownian motion and embedding of discrete time AR sequences driven by a Gaussian white noise are classical issues. The article aims at establishing and using such discretization and embedding formulae between extended AR continuous time processes and discrete time sequences. The continuous-time processes are driven by either Brownian or jump processes, and may have random coefficients depending on time; Lévy-driven processes are also considered. The innovation of the discrete time processes may be of many types – including Gaussian. In one way, observing the continuous time AR process at discrete times leads the AR dynamics of the discretized process to be characterized. The other way round, AR sequences can be embedded, in the almost sure sense, into continuous time AR processes with the same dynamics. Illustration is provided through many examples and simulation.