On Markov-switching periodic ARMA models

In this work, we propose a generalization of the classical Markov-switching ARMA models to the periodic time-varying case. Specifically, we propose a Markov-switching periodic ARMA (MS-PARMA) model. In addition of capturing regime switching often encountered during the study of many economic time series, this new model also captures the periodicity feature in the autocorrelation structure. We first provide some probabilistic properties of this class of models, namely the strict periodic stationarity and the existence of higher-order moments. We thus propose a procedure for computing the autocovariance function where we show that the autocovariances of the MS-PARMA model satisfy a system of equations similar to the PARMA Yule–Walker equations. We propose also an easily implemented algorithm which can be used to obtain parameter estimates for the MS-PARMA model. Finally, a simulation study of the performance of the proposed estimation method is provided.     

An algorithm for the exact likelihood of periodic autoregressive moving average models

An algorithm to compute the autocovariance functions of periodic autoregressive moving average models is proposed. As a result, an easily implemented algorithm for the exact likelihood of these models is rendered possible.     

Bounded Area Tests For Comparing The Dynamics Between ARMA Processes

This article presents a new test for discerning whether or not two independent autoregressive moving average (ARMA) processes have the same autocovariance structure. This test utilizes a specific geometric feature of a time series plot, namely the area bounded between the line segments that connect adjacent points and the time axis. It will be shown that if you sample two ARMA processes and calculate the magnitudes of the two resulting bounded areas, then a significant difference among these areas tends to imply a significant difference in autocovariances.     

Derivation of the theoretical autocovariance and autocorrelation function of autoregessive moving average processes

Closed form expressions for the theoretical autocovariance and autocorrelation function of mixed autoregressive moving average processes are presented. The results provide insight into the construction of autocovariances and autocorrelatians and are useful in theoretical analysis, model identification as well as in implementing maximum likelihood estimation algorithms.     

Estimators based on ranks for arma models

In this paper we introduce a new family of robust estimators for ARMA models. These estimators are defined by replacing the residual sample autocovariances in the least squares equations by autocovariances based on ranks. The asymptotic normality of the proposed estimators is provided. The efficiency and robustness properties of these estimators are studied. An adequate choice of the score functions gives estimators which have high efficiency under normality and robustness in the presence of outliers. The score functions can also be chosen so that the resulting estimators are asymptotically as efficient as the maximum likelihood estimators for a given distribution.     

A comparison of multivariate signal discrimination techniques

This paper investigates several techniques to discriminate two multivariate stationary signals. The methods considered include Gaussian likelihood ratio tests for variance equality, a chi-squared time-domain test, and a spectral-based test. The latter two tests assess equality of the multivariate autocovariance function of the two signals over many different lags. The Gaussian likelihood ratio test is perhaps best viewed as principal component analyses (PCA) without dimension reduction aspects; it can be modified to consider covariance features other than variances via dimension augmentation tactics. A simulation study is constructed that shows how one can make inappropriate conclusions with PCA tests, even when dimension augmentation techniques are used to incorporate non-zero lag autocovariances into the analysis. The various discrimination methods are first discussed. A simulation study then illuminates the various properties of the methods. In this pursuit, calculations are needed to identify several multivariate time series models with specific autocovariance properties. To demonstrate the applicability of the methods, nine US and Canadian weather stations from three distinct regions are clustered. Here, the spectral clustering perfectly identified distinct regions, the chi-squared test performed marginally, and the PCA/likelihood ratio method did not perform well.     

On the asymptotic distributions of mean,autocovariance, autocorrelation,crossgovariancb and impulse response estimators of a stationary multidimensional random field

Asymptotic properties of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators of a stationary M-dimensionai (M-D) random field are studied. It is shown that only unbiased-type estimators of autocovariances, autocorrelations, crosscovariances and impulse responses have the asymptotic distributions when M≧ 2. Moreover, the asymptotic distributions of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators are presented.     

Improved estimation for the autocovariances of a Gaussian stationary process

For a Gaussian stationary process with mean μ and autocovariance function γ(·), we consider to improve the usual sample autocovariances with respect to the mean squares error (MSE) loss. For the cases μ=0 and μ≠0, we propose sort of empirical Bayes type estimators Γ? and Γ?, respectively. Then their MSE improvements upon the usual sample autocovariances are evaluated in terms of the spectral density of the process. Concrete examples for them are provided. We observe that if the process is near to a unit root process the improvement becomes quite large. Thus, consideration for estimators of this type seems important in many fields, e.g., econometrics.     

EWNA charts for monitoring the mean and the autocovariances of stationary processes

In this paper various types of EWMA control charts are introduced for the simultaneous monitoring of the mean and the autocovariances. The target process is assumed to be a stationary process up to fourth-order or an ARMA process with heavy tailed innovations. The case of a Gaussian process is included in our results as well. The charts are compared within a simulation study. As a measure of the performance the average run length is taken. The target process is an ARMA (1,1) process with Student-t distributed innovations. The behavior of the charts is analyzed with respect to several out-of-control models. The best design parameters are determined for each chart. Our comparisons show that the multivariate EWMA chart applied to the residuals has the best overall performance.     

Sur un test d'égalité des autocovariances de deux séries chronologiques

We propose a test for the equality of the autocovariance functions of two independent and stationary time series. The test statistic is a quadratic form in the vector of differences of the first J + 1 autocovariances. Its asymptotic distribution is derived under the null hypothesis, and the finite-sample properties of the test, namely the bias and the power, are investigated by Monte Carlo methods. A by-product of this study is a new estimator of the covariance between two sample autocovariances which provides a positive definite covariance matrix. We establish the convergence of this estimator in the L₁ norm.     

A Note on Calculating Autocovariances of Periodic ARMA Models

An analytically simple and tractable formula for the start-up autocovariances of periodic ARMA (PARMA) models is provided.     

Block bootstrap for periodic characteristics of periodically correlated time series

This research is dedicated to the study of periodic characteristics of periodically correlated time series such as seasonal means, seasonal variances and autocovariance functions. Two bootstrap methods are used: the extension of the usual Moving Block Bootstrap (EMBB) and the Generalised Seasonal Block Bootstrap (GSBB). The first approach is proposed, because the usual Moving Block Bootstrap does not preserve the periodic structure contained in the data and cannot be applied for the considered problems. For the aforementioned periodic characteristics the bootstrap estimators are introduced and consistency of the EMBB in all cases is obtained. Moreover, the GSBB consistency results for seasonal variances and autocovariance function are presented. Additionally, the bootstrap consistency of both considered techniques for smooth functions of the parameters of interest is obtained. Finally, the simultaneous bootstrap confidence intervals are constructed. A simulation study to compare their actual coverage probabilities is provided. A real data example is presented.     

Maximum likelihood estimation of linear continuous time long memory processes with discrete time data

Summary.  We develop a new class of time continuous autoregressive fractionally integrated moving average (CARFIMA) models which are useful for modelling regularly spaced and irregu-larly spaced discrete time long memory data. We derive the autocovariance function of a stationary CARFIMA model and study maximum likelihood estimation of a regression model with CARFIMA errors, based on discrete time data and via the innovations algorithm. It is shown that the maximum likelihood estimator is asymptotically normal, and its finite sample properties are studied through simulation. The efficacy of the approach proposed is demonstrated with a data set from an environmental study.     

Some efficient computational procedures for high order ARMA models

Recursive methods are commonly used to solve Yule—Walker equations for autoregrsssive parameters given an autocovariance function. The reverse procedure can be extended to the efficient solution of various sets of equations which arise in time series analysis. Those presented in this paper include computation of the autocovariance function of an ARMA model, and the Cramer—Wold factorization.     

Statistical Analysis Of Mixture Vector Autoregressive Models

In this paper, we reconsider the mixture vector autoregressive model, which was proposed in the literature for modelling non‐linear time series. We complete and extend the stationarity conditions, derive a matrix formula in closed form for the autocovariance function of the process and prove a result on stable vector autoregressive moving‐average representations of mixture vector autoregressive models. For these results, we apply techniques related to a Markovian representation of vector autoregressive moving‐average processes. Furthermore, we analyse maximum likelihood estimation of model parameters by using the expectation–maximization algorithm and propose a new iterative algorithm for getting the maximum likelihood estimates. Finally, we study the model selection problem and testing procedures. Several examples, simulation experiments and an empirical application based on monthly financial returns illustrate the proposed procedures.     

On Fan's adaptive Neyman tests for comparing two spectral densities

In this paper, we consider tests for assessing whether two stationary and independent time series have the same spectral densities (or same autocovariance functions). Both frequency domain and time domain test statistics for this purpose are reviewed. The adaptive Neyman tests are then introduced and their performances are investigated. Our tests are adaptive, that is, they are constructed completely by the data and do not involve any unknown smoothing parameters. Simulation studies show that our proposed tests are at least comparable to the current tests in most cases. Furthermore, our tests are much more powerful in some cases, such as against the long orders of autoregressive moving average (ARMA) models such as seasonal ARMA series.     

Time Series Properties of the Class of Generalized First-Order Autoregressive Processes with Moving Average Errors

A new class of time series models known as Generalized Autoregressive of order one with first-order moving average errors has been introduced in order to reveal some hidden features of certain time series data. The variance and autocovariance of the process is derived in order to study the behaviour of the process. It is shown that in special cases these new results reduce to the standard ARMA results. Estimation of parameters based on the Whittle procedure is discussed. We illustrate the use of this class of model by using two examples.     

Identification of Periodic Moving-Average Models

Abstract

ARMA models with seasonally-varying parameters and orders, known as periodic ARMA (PARMA) models, have found wide applications in modeling of seasonal processes. This article considers the identification of orders of periodic MA (PMA) models. The identification is based on the cut-off property of the periodic autocorrelation function (PeACF). We derive an explicit expression for the asymptotic variance of the sample PeACF to be used in establishing its bands. A simulated example is also provided which agrees well with the theoretical results.     

Asymptotic Inefficiency of Mean-Correction on Parameter Estimation for a Periodic First-Order Autoregressive Model

A common practice in time series analysis is to fit a centered model to the mean-corrected data set. For stationary autoregressive moving-average (ARMA) processes, as far as the parameter estimation is concerned, fitting an ARMA model without intercepts to the mean-corrected series is asymptotically equivalent to fitting an ARMA model with intercepts to the observed series. We show that, related to the parameter least squares estimation of periodic ARMA models, the second approach can be arbitrarily more efficient than the mean-corrected counterpart. This property is illustrated by means of a periodic first-order autoregressive model. The asymptotic variance of the estimators for both approaches is derived. Moreover, empirical experiments based on simulations investigate the finite sample properties of the estimators.     

Periodic integer-valued GARCH(1, 1) model

This article deals with some probabilistic and statistical properties of a periodic integer-valued GARCH(1,1) model. Necessary and sufficient conditions for the periodical stationary, both in mean and second order, are established. The closed-forms of the mean and the second moment are, under these conditions, obtained. The condition of the existence of higher moment orders and their explicit formula in terms of the parameters are established. The autocovariance structure is studied, while providing the closed-form of the periodic autocorrelation function. The Yule–Walker and the likelihood estimations of the underlying parameters are obtained. A simulation study and an application on real dataset are provided.