Vector ARMA Models Research Articles

Using wavelet analysis to uncover the co-movement behavior of multiple energy commodity prices

This study aims to investigate the dynamic correlations (co-movement) in between energy commodities such as WTI Crude Oil (WOIL), Brent Crude Oil (BOIL), Heating Oil and Electricity prices. To achieve this goal, we employed partial wavelet coherence (PWC) and multiple wavelet coherence (MWC). Wavelet analysis constitutes the core of these methodologies and MWC is essential to determine the dynamic correlation (co-movement) of time intervals and scales between the time series. We have developed a software program to compute PWC and MWC for quadruple data set. Coherent time intervals of the time series are determined. Vector ARMA models are shown to give a good fit due to having low mean squared errors compared to the univariate case. This allowed us to have better forecast performance.

A comparison of multivariate and univariate time series approaches to modelling and forecasting emergency department demand in Western Australia

ObjectiveTo develop multivariate vector-ARMA (VARMA) forecast models for predicting emergency department (ED) demand in Western Australia (WA) and compare them to the benchmark univariate autoregressive moving average (ARMA) and Winters’ models. MethodsSeven-year monthly WA state-wide public hospital ED presentation data from 2006/07 to 2012/13 were modelled. Graphical and VARMA modelling methods were used for descriptive analysis and model fitting. The VARMA models were compared to the benchmark univariate ARMA and Winters’ models to determine their accuracy to predict ED demand. The best models were evaluated by using error correction methods for accuracy. ResultsDescriptive analysis of all the dependent variables showed an increasing pattern of ED use with seasonal trends over time. The VARMA models provided a more precise and accurate forecast with smaller confidence intervals and better measures of accuracy in predicting ED demand in WA than the ARMA and Winters’ method. ConclusionVARMA models are a reliable forecasting method to predict ED demand for strategic planning and resource allocation. While the ARMA models are a closely competing alternative, they under-estimated future ED demand.

Forecasting Density Function: Application in Finance

With increasing availability of data, in many situations it is now possible to reasonably estimate the probability density function (pdf) of a random variable. This is far more informative than using a few summary statistics like mean or variance. In this paper, we propose a method of forecasting the density function based on a time series of estimated density functions. The proposed method uses kernel estimation to pre-process the raw data followed by dimension reduction using functional principal components analysis (FPCA). Then we fit Vector ARMA models to the reduced data to make a prediction of the principal component scores, which can then be used to obtain the forecast for density function. We need to transform and scale the forecasts to ensure non-negativeness and integration to one. We compared our method to [1] for histogram forecasts, on simulated data as well as real data from S&P 500 and the Bombay Stock Exchange. The results showed that our method performed better on both the datasets and the simulation using uniform and Hilbert distance. The time dependence and complexity of density function are different for the two markets, which is captured by our analysis.

On Diagnostic Checking of Vector ARMA-GARCH Models with Gaussian and Student-t Innovations

This paper focuses on the diagnostic checking of vector ARMA (VARMA) models with multivariate GARCH errors. For a fitted VARMA-GARCH model with Gaussian or Student-t innovations, we derive the asymptotic distributions of autocorrelation matrices of the cross-product vector of standardized residuals. This is different from the traditional approach that employs only the squared series of standardized residuals. We then study two portmanteau statistics, called Q1(M) and Q2(M), for model checking. A residual-based bootstrap method is provided and demonstrated as an effective way to approximate the diagnostic checking statistics. Simulations are used to compare the performance of the proposed statistics with other methods available in the literature. In addition, we also investigate the effect of GARCH shocks on checking a fitted VARMA model. Empirical sizes and powers of the proposed statistics are investigated and the results suggest a procedure of using jointly Q1(M) and Q2(M) in diagnostic checking. The bivariate time series of FTSE 100 and DAX index returns is used to illustrate the performance of the proposed portmanteau statistics. The results show that it is important to consider the cross-product series of standardized residuals and GARCH effects in model checking.

Calculating the autocovariances and the likelihood for periodic V ARMA models

This paper concerns the autocovariance calculation and likelihood evaluation for periodic vector ARMA models (PV ARMA). Based on a state space representation of PV ARMA models, we derive an algorithm for computing the PV ARMA autocovariances. The proposed method computes the autocovariances for distinct seasons separately, thereby facilitating efficient calculation for models with a large period. As a result, the obtained autocovariance calculation procedure is exploited in a periodic Chandrasekhar-type filter to evaluate the exact likelihood for Gaussian PV ARMA series. Empirical evidence shows the superiority of the periodic Chandrasekhar algorithm for likelihood evaluation over the Kalman-based one.

Exact maximum likelihood estimation of partially nonstationary vector ARMA models

A useful class of partially nonstationary vector autoregressive moving average (VARMA) models is considered with regard to parameter estimation. An exact maximum likelihood (EML) approach is developed on the basis of a simple transformation applied to the error-correction representation of the models considered. The employed transformation is shown to provide a standard VARMA model with the important property that it is stationary. Parameter estimation can thus be carried out by applying standard EML methods to the stationary VARMA model obtained from the error-correction representation. This approach resolves at least two problems related to the current limited availability of EML estimation methods for partially nonstationary VARMA models. Firstly, it resolves the apparent impossibility of computing the exact log-likelihood for such models using currently available methods. And secondly, it resolves the inadequacy of considering lagged endogenous variables as exogenous variables in the error-correction representation. Theoretical discussion is followed by an example using a popular data set. The example illustrates the feasibility of the EML estimation approach as well as some of its potential benefits in cases of practical interest which are easy to come across. As in the case of stationary models, the proposed EML method provides estimated model structures that are more reliable and accurate than results produced by conditional methods.

MULTICHANNEL IDENTIFICATION OF AIRCRAFT SKELETON STRUCTURES UNDER UNOBSERVABLE EXCITATION: A VECTOR AR/ARMA FRAMEWORK

A comprehensive autoregressive (AR) and linear multistage autoregressive moving average (LMS-ARMA) framework for effective multichannel identification of structures under unobservable excitation is introduced and assessed via the paradigm of aircraft skeleton structure identification. This framework utilises fully parametrised vector AR and ARMA models estimated via linear regression (LR) and a newly developed linear multistage (LMS) method, respectively, statistical model order selection, and structural mode distinction via a dispersion analysis methodology. The framework's effectiveness and limitations are examined via the identification of an experimental and a simulated aircraft skeleton structure, as well as comparisons with non-parametric Welch autospectral density estimation. The results of the study indicate the framework's effectiveness in selecting the necessary model orders, as well as in distinguishing and accurately estimating most structural modes, including closely spaced and ‘local’ modes. The AR an LMS-ARMA methods achieve generally similar accuracy, but the latter leads to lower order models, fewer spurious modes, and less ambiguous ‘stabilisation’ diagrams.

A note on an iterative least-squares estimation method for ARMA and VARMA models

In this note we suggest a new iterative least-squares method for estimating scalar and vector ARMA models. A Monte Carlo study shows that the method has better small sample properties than existing least-squares methods and compares favourably with maximum likelihood estimation as well.

WITHDRAWN: Multichannel identification of aircraft skeleton structures under unobservable excitation: A vector AR/ARMA framework

Abstract A comprehensive autoregressive (AR) and linear multistage autoregressive moving average (LMS-ARMA) framework for effective multichannel identification of structures under unobservable excitation is introduced and assessed via the paradigm of aircraft skeleton structure identification. This framework utilises fully parametrised vector AR and ARMA models estimated via linear regression (LR) and a newly developed linear multistage (LMS) method, respectively, statistical model order selection, and structural mode distinction via a dispersion analysis methodology. The framework's effectiveness and limitations are examined via the identification of an experimental and a simulated aircraft skeleton structure, as well as comparisons with non-parametric Welch autospectral density estimation. The results of the study indicate the framework's effectiveness in selecting the necessary model orders, as well as in distinguishing and accurately estimating most structural modes, including closely spaced and ‘local’ modes. The AR an LMS-ARMA methods achieve generally similar accuracy, but the latter leads to lower order models, fewer spurious modes, and less ambiguous ‘stabilisation’ diagrams.

A Note on an Iterative Least Squares Estimation Method for ARMA and VARMA Models

In this note we suggest a new iterative least squares method for estimating scalar and vector ARMA models. A Monte Carlo study shows that the method has better small sample properties than existing least squares methods and compares favourably with maximum likelihood estimation as well.

The Impact of Seasonal Unit Roots and Vector ARMA Modelling on Forecasting Monthly Tourism Flows

The effect of imposing different numbers of unit roots on forecasting accuracy is examined using univariate ARMA models. To see whether additional information improves forecasting accuracy and increases the informative forecast horizon, the authors cross-relate the time series for inbound tourism in Sweden for different visitor categories and estimate vector ARMA models. The mean-squared forecast error for different filters indicates that models in which unit roots are imposed at all frequencies have the smallest forecast errors. The results from the vector ARMA models with all roots imposed indicate that the informative forecast horizon is greater than for the univariate models. Out-of-sample evaluations indicate, however, that the univariate modelling approach may be preferable.

Elements of Multivariate Time Series Analysis.

This study is devoted to the analysis of multivariate time series data. Such data might arise in business and economics, engineering, geophysical sciences, agriculture, and many other fields. The emphasis is on providing an account of the basic concepts and methods which are useful in analyzing such data. The book presupposes a familiarity with univariate time series as might be gained from one term of a graduate course, but it is otherwise self-contained. It covers the basic topics such as autocovariance matrices of stationary processes, vector ARMA models and their properties, forecasting ARMA processes, least squares and maximum likelihood estimation techniques for vector AR and ARMA models, and associated likelihood ratio testing procedures for model building. In addition, it presents more advanced topics and techniques including reduced rank structure, structural indices, scalar component models, canonical correlation analyses for vector time series, multivariate nonstationary unit root models and co-integration structure, and state-space models and Kalman flltering techniques.

Cointegration test of the monetary theory of inflation and forecasting accuracy of the univariate and vector ARMA models of inflation

Conducts both the cointegration test of the monetary theory of inflation and the Granger‐causality test between the variables in the system, and also develops univariate and multivariate time series models to forecast inflation rates. Quarterly time series data for Pakistan, from 1972‐2 to 1993‐4 is used for empirical investigation. Results suggest no cointegrating or long‐run relationship between the variables in the monetary model. Observes that there is some evidence of Granger‐causality running from inflation to output growth. Comparison of out‐of‐sample quarterly forecasts for the 1988‐1 to 1993‐4 period are made for univariate and vector ARMA models of inflation. States that the forecasting accuracy of the multivariate ARMA model is not statistically different from that of the univariate ARMA model.

A simulation study on vector arma processes with nonstationary innovation:a new approach to identification

This paper considers the problem of identification of vector autoregressive-moving average (VARMA) processes with nonstationary innovations and suggests a new approach to the problem. This approach has successfully been applied recently by Singh and Tampubolon (1995) to the recordings of the Kobe earthquake which occurred on 16 January, 1995, in the univariate case. The same techniques have been extended in this paper for the identification of vector ARMA processes with nonstationary white-noise in the sense that its covariance structure is assumed to be time-dependent; a few simulated examples are discussed for illustration. Furthermore, they have been applied to the identification of the vector ARMA models that were fitted to the Kobe earthquake vibrations recorded at two stations, viz. (i) Charters Towers, Queensland and (ii) Hobart, Tasmania, both in Australia. The theoretical results have been supported by simulation studies.

Exact Maximum Likelihood Estimation of Stationary Vector ARMA Models

The problems of evaluating and maximizing the exact likelihood function of vector ARMA models are considered separately. A new and efficient procedure for evaluating the exact likelihood function is presented. This method puts together a set of useful features which can only be found separately in currently available algorithms. A procedure for maximizing the exact likelihood function, which takes full advantage of the properties offered by the evaluation algorithm, is also considered. Combining these two procedures, a new algorithm for exact maximum likelihood estimation of vector ARMA models is obtained. Comparisons with existing procedures, in terms of both analytical arguments and a numerical example, are given in order to show that the new estimation algorithm performs at least as well as existing ones, and that relevant real situations occur in which it do es better.

On the Asymptotic Relative Efficiency of Gaussian and Least Squares Estimators for Vector ARMA Models

This paper is concerned with the asymptotic relative efficiency of the Gaussian and least squares estimators when employed to estimate the parameters of vector ARMA models presented in echelon canonical form. The relative efficiency is assessed via the variance-covariance matrices of the limiting normal distributions of the two estimators. Situations under which substantial loss or gain in efficiency could be exprected are discussed and illustrated with some numerical examples.

The approximation of the one-step ahead forecast error covariance for vector ARMA models

A vector case parallel to that described by Ledolter and Abraham ( Technometrics, 23 (1981) 411-414) is extended for an approximation of one-step ahead forecast error covariance for vector ARMA models.

Simultaneous transfer functions versus vector arma models

This work compares two classes of multiple time series models which have been developed in past decades and are usually believed to be equivalent: the vector ARMA model and the system of simultaneous transfer functions (STF). The first part analyzes the mathematical structure of the two schemes; their properties of stability, structural identification and realization. In the second, algorithms of order identification and parameter estimation are derived, following the approach of stochastic approximation. The proposed solutions are easily implementable on standard statistical software and in an extended empirical example their performance is checked. The superiority of the STF model will be well established.

MTS: A review

Journal of Applied EconometricsVolume 4, Issue 2 p. 205-206 A Review MTS: A review Mark W. Watson, Mark W. Watson Depatment of Economics, Northwestern University, 2003 Sheridan Road, Evanston, IL 60201, USASearch for more papers by this author Mark W. Watson, Mark W. Watson Depatment of Economics, Northwestern University, 2003 Sheridan Road, Evanston, IL 60201, USASearch for more papers by this author First published: April/June 1989 https://doi.org/10.1002/jae.3950040208AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume4, Issue2April/June 1989Pages 205-206 RelatedInformation

Testing causality using efficiently parametrized vector ARMA models

We outline a practical methodology for fitting to data vector autoregressive moving-average models, and employ these models to test for Granger causality. The procedures are applied to data on employment and hourly wages in manufacturing in the United States.