Vector Autoregressive Moving Average Research Articles

VARMA versus VAR for Macroeconomic Forecasting

In this article, we argue that there is no compelling reason for restricting the class of multivariate models considered for macroeconomic forecasting to vector autoregressive (VAR) models, given the recent advances in vector autoregressive moving average (VARMA) modeling methodology and improvements in computing power. To support this claim, we use real macroeconomic data, and show that VARMA models forecast macroeconomic variables more accurately than VARs.

The asymptotic and exact Fisher information matrices of a vector ARMA process

The exact Fisher information matrix of a Gaussian vector autoregressive-moving average (VARMA) process has been considered for a time series of length N in relation to the exact maximum likelihood estimation method. In this paper it is shown that the Gaussian exact Fisher information matrix converges to the asymptotic Fisher information matrix when N goes to infinity.

Relationship between state-space and ARMAV approaches to modal parameter identification

In this paper we study the relationship between state-space and autoregressive moving average vector (ARMAV) approaches for the problem of estimating the modal parameters of a vibrating system. Using only the singular value decomposition of the block Hankel matrix of covariances, it is shown that these two methods give identical modal parameters in the cases where this block Hankel matrix has full row rank.

Export and production growth: evidence from three major wheat exporters of Australia, Canada and the United States

This article investigates the robustness of the relationship between export and production growth for three major wheat exporters: Australia, Canada and the United States from 1966 to 2000. Combining production, international trade and development theories, a four variable (production, exports, producer price and imports) vector autoregressive moving average (VARMA) model is developed for each country. The causality results show that the hypothesis of export-led development is supported by these three major players in the international wheat trade, though a bi-directional causality is found for Canada and the USA. Variance decomposition and impulse response functions are employed to further investigate the effects of macroeconomic shocks.

Modelling time-varying conditional correlations in the volatility of Tapis oil spot and forward returns

This paper estimates the dynamic conditional correlations in the returns on Tapis oil spot and one-month forward prices for the period 2 June 1992 to 16 January 2004, using recently developed multivariate conditional volatility models, namely the Constant Conditional Correlation Multivariate GARCH (CCC–MGARCH) model of Bollerslev (1990 Bollerslev, T. 1990. Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH approach. Review of Economics and Statistics, 72: 498–505. [Crossref], [Web of Science ®] , [Google Scholar]), Vector Autoregressive Moving Average–GARCH (VARMA–GARCH) model of Ling and McAleer (2003 Ling, S and McAleer, M. 2003. Asymptotic theory for a vector ARMA–GARCH model. Econometric Theory, 19: 278–308. [Crossref] , [Google Scholar]), VARMA–Asymmetric GARCH (VARMA–AGARCH) model of Hoti et al. (2002 Hoti, S, Chan, F and McAleer, M. Structure and asymptotic theory for multivariate asymmetric volatility: empirical evidence for country risk ratings. July2002, Brisbane, Australia. Paper presented to the 2002 Australasian Meeting of the Econometric Society, [Google Scholar]), and the Dynamic Conditional Correlation (DCC) model of Engle (2002 Engle, RF. 2002. Dynamic conditional correlation: a new simple class of multivariate GARCH models. Journal of Business and Economic Statistics, 20: 339–50. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The dynamic correlations are extremely useful in determining whether the spot and forward returns are substitutes or complements, which can be used to hedge against contingencies. Both the univariate ARCH and GARCH estimates are significant for spot and forward returns, whereas the estimates of the asymmetric effect at the univariate level are not statistically significant for either spot or forward returns. Standard diagnostic tests show that the AR(1)–GARCH(1, 1) and AR(1)–GJR(1, 1) specifications are statistically adequate for both the conditional mean and the conditional variance. The multivariate estimates for the VAR(1)–GARCH(1, 1) and VAR(1)–AGARCH(1, 1) models show that the ARCH and GARCH effects for spot (forward) returns are significant in the conditional volatility model for spot (forward) returns. Moreover, there are significant interdependences in the conditional volatilities between the spot and forward markets. The multivariate asymmetric effects are significant for both spot and forward returns. Overall the multivariate VAR(1)–AGARCH(1, 1) dominates its symmetric counterpart. The calculated constant conditional correlations between the conditional volatilities of spot and forward returns using CCC–GARCH(1, 1), VAR(1)–GARCH(1, 1) and VAR(1)–AGARCH(1, 1) are very close to 0.93. Virtually identical results are obtained when the three constant conditional correlation models are extended to include two lags in both the ARCH and GARCH components. Finally, the estimates of the two DCC parameters are statistically significant, which makes it clear that the assumption of constant conditional correlation is not supported empirically. This is highlighted by the dynamic conditional correlations between spot and forward returns, for which its sample mean is virtually identical to the computed constant conditional correlation, regardless of whether a DCC–GARCH(1, 1) or a DCC–GARCH(2, 2) is used. For these models, the dynamic conditional correlations are in the range (0.417, 0.993) and (0.446, 0.993), signifying medium to extreme interdependence. Therefore, the dynamic volatilities in the returns in Tapis oil spot and forward markets are generally interdependent over time. These findings suggest that a sensible hedging strategy would consider spot and forward markets as being characterized by different degrees of substitutability.

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.

Exact maximum likelihood estimation of structured or unit root multivariate time series models

The exact likelihood function of a Gaussian vector autoregressive-moving average (VARMA) model is evaluated in two nonstandard cases: (a) a parsimonious structured form, such as obtained in the echelon form structure or the scalar component model (SCM) structure; (b) a partially nonstationary (integrated of order 1) model in error-correction form. The starting point is any algorithm for computing the exact likelihood of a Gaussian VARMA time series. Our algorithm also provides the parameter estimates and their standard errors. The small sample properties of our algorithm were studied by Monte Carlo methods. Examples with real data are provided.

Algorithms for time synchronization of wireless structural monitoring sensors

Dense networks of wireless structural health monitoring systems can effectively remove the disadvantages associated with current wire-based sparse sensing systems. However, recorded data sets may have relative time-delays due to interference in radio transmission or inherent internal sensor clock errors. For structural system identification and damage detection purposes, sensor data require that they are time synchronized. The need for time synchronization of sensor data is illustrated through a series of tests on asynchronous data sets. Results from the identification of structural modal parameters show that frequencies and damping ratios are not influenced by the asynchronous data; however, the error in identifying structural mode shapes can be significant. The results from these tests are summarized in Appendix A. The objective of this paper is to present algorithms for measurement data synchronization. Two algorithms are proposed for this purpose. The first algorithm is applicable when the input signal to a structure can be measured. The time-delay between an output measurement and the input is identified based on an ARX (auto-regressive model with exogenous input) model for the input–output pair recordings. The second algorithm can be used for a structure subject to ambient excitation, where the excitation cannot be measured. An ARMAV (auto-regressive moving average vector) model is constructed from two output signals and the time-delay between them is evaluated. The proposed algorithms are verified with simulation data and recorded seismic response data from multi-story buildings. The influence of noise on the time-delay estimates is also assessed. Copyright © 2004 John Wiley & Sons, Ltd.

Space–time modeling of traffic flow

This paper discusses the application of space–time autoregressive integrated moving average (STARIMA) methodology for representing traffic flow patterns. Traffic flow data are in the form of spatial time series and are collected at specific locations at constant intervals of time. Important spatial characteristics of the space–time process are incorporated in the STARIMA model through the use of weighting matrices estimated on the basis of the distances among the various locations where data are collected. These matrices distinguish the space–time approach from the vector autoregressive moving average (VARMA) methodology and enable the model builders to control the number of the parameters that have to be estimated. The proposed models can be used for short-term forecasting of space–time stationary traffic-flow processes and for assessing the impact of traffic-flow changes on other parts of the network. The three-stage iterative space–time model building procedure is illustrated using 7.5 min average traffic flow data for a set of 25 loop-detectors located at roads that direct to the centre of the city of Athens, Greece. Data for two months with different traffic-flow characteristics are modelled in order to determine the stability of the parameter estimation.

Rolling regressions and conditional correlations of foreign patents in the USA

Patent registrations have often been used as a proxy of innovation as they reflect a country's technological capability. Recently, some studies have found that the Generalised Autoregressive Conditional Heteroscedasticity (GARCH) model and an asymmetric extension, namely Glosten, Jagannathan and Runkle's (GJR) model, are useful to model the time-varying volatility of the patent ratio, namely the ratio of foreign patents registered in the USA to total patents in the USA. However, this approach assumes that the conditional variance is independent across countries. Furthermore, the time series properties of the patent growth rate, namely the rate of change of foreign patents registered in the USA, have not previously been analysed. This paper examines the conditional variance of the patent growth rate from the leading four foreign countries, namely Canada, France, Germany and Japan, using the Constant Conditional Correlation – Multivariate GARCH (CCC-MGARCH), Vector Autoregressive Moving Average – GARCH (VARMA-GARCH) and VARMA – Asymmetric GARCH (VARMA-AGARCH) models. The results reveal the existence of cross-countries effects in the patent growth rate among the leading four countries, as well as asymmetric effects using monthly data from January 1975 to December 1998. Rolling estimates show that the restrictive assumption of constant conditional correlation is unlikely to hold, and models that accommodate dynamic conditional correlations may provide greater insights for investigating the effects of global factors on changes in innovation for the four leading foreign countries.

Rank-based optimal tests of the adequacy of an elliptic VARMA model

We are deriving optimal rank-based tests for the adequacy of a vector autoregressive-moving average (VARMA) model with elliptically contoured innovation density. These tests are based on the ranks of pseudo-Mahalanobis distances and on normed residuals computed from Tyler’s [Ann. Statist. 15 (1987) 234–251] scatter matrix; they generalize the univariate signed rank procedures proposed by Hallin and Puri [J. Multivariate Anal. 39 (1991) 1–29]. Two types of optimality properties are considered, both in the local and asymptotic sense, a la Le Cam: (a) (fixed-score procedures) local asymptotic minimaxity at selected radial densities, and (b) (estimated-score procedures) local asymptotic minimaxity uniform over a class ℱ of radial densities. Contrary to their classical counterparts, based on cross-covariance matrices, these tests remain valid under arbitrary elliptically symmetric innovation densities, including those with infinite variance and heavy-tails. We show that the AREs of our fixed-score procedures, with respect to traditional (Gaussian) methods, are the same as for the tests of randomness proposed in Hallin and Paindaveine [Bernoulli 8 (2002b) 787–815]. The multivariate serial extensions of the classical Chernoff–Savage and Hodges–Lehmann results obtained there thus also hold here; in particular, the van der Waerden versions of our tests are uniformly more powerful than those based on cross-covariances. As for our estimated-score procedures, they are fully adaptive, hence, uniformly optimal over the class of innovation densities satisfying the required technical assumptions.

A Comparison of Johansen's, Bierens’ and the Subspace Algorithm Method for Cointegration Analysis*

AbstractThe methods listed in the title are compared by means of a simulation study and a real world application. The aspects compared via simulations are the performance of the tests for the cointegrating rank and thequalityof the estimated cointegrating space. The subspace algorithm method, formulated in the state space framework and thus applicable for vector autoregressive moving average (VARMA) processes, performs at least comparably to the Johansen method. Both the Johansen procedure and the subspace algorithm cointegration analysis perform significantly better than Bierens’ method. The real‐world application is an investigation of the long‐run properties of the one‐sector neoclassical growth model for Austria. The results do not fully support the implications of the model with respect to cointegration. Furthermore, the results differ greatly between the different methods. The amount of variability depends strongly upon the number of variables considered and huge differences occur for the full system with six variables. Therefore we conclude that the results of such applications with about five or six variables and 100 observations, which are typical in the applied literature, should possibly be interpreted with more caution than is commonly done.

Forecasting Traffic Flow Conditions in an Urban Network: Comparison of Multivariate and Univariate Approaches

Several univariate and multivariate models have been proposed for performing short-term forecasting of traffic flow. Two different univariate [historical average and ARIMA (autoregressive integrated moving average)] and two multivariate [VARMA (vector autoregressive moving average) and STARIMA (space–time ARIMA)] models are presented and discussed. A comparison of the forecasting performance of these four models is undertaken with data sets from 25 loop detectors located in major arterials in the city of Athens, Greece. The variable under study is the relative velocity, which is the traffic volume divided by the road occupancy. Although the specification of the network’s neighborhood structure for the STARIMA model was relatively simple and can be further refined, the results obtained indicate a comparable forecasting performance for the ARIMA, VARMA, and STARIMA models. The historical average model could not cope with the variability of the data sets at hand.

On the specification of cointegrated autoregressive moving-average forecasting systems

This paper discusses equilibrium correction, echelon canonical form vector autoregressive moving-average, EC-ARMA E, forecasting systems. The echelon canonical form of a vector ARMA model is expanded by the inclusion of an equilibrium correction term to accommodate the possibility of cointegrated variables. A coherent procedure is presented for consistently estimating the Kronecker indices, which characterize the echelon form, and the cointegration rank, which is essential in the specification of the equilibrium correction term. A method of estimation that is fully efficient under Gaussian assumptions is also discussed. The computational burden of these techniques is very moderate because they are based on least squares calculations. The methodology is illustrated by examining a six-equation model of the US economy. An improvement in forecasting performance of the selected EC-ARMA E model over non-equilibrium correction and previously preferred vector AR equilibrium correction models is observed.

TIME–FREQUENCY ANALYSIS OF A SUSPENSION BRIDGE BASED ON GPS

This paper describes the results obtained from full-scale measurements of Humen bridge, which is the second longest suspension bridge in China. A real-time kinematic (RTK) global positioning system (GPS) has been developed and installed on the Humen bridge for on-line monitoring of bridge deck movements. The field wind-induced vibration data were measured by this monitoring system. Three system identification techniques are then adopted in the modal analysis of the wind-induced vibration response: the time–frequency Wigner distribution (WD) technique, the frequency-domain fast Fourier transform (FFT) technique and the time-domain auto-regressive moving average vector (ARMAV) technique. The WD technique can recognize close modal coupling and non-stationary response. The FFT technique can on site verify the quality of the measurements, but its frequency resolution is low and damping estimates are unreliable. The ARMAV method allows for gaining high-frequency resolution. However, it is strictly related to the stationary hypothesis. It is a general conclusion that we can improve the quality of the analysis and get more precise characteristics of the signal by these three methods. In addition, the WD combined with ARMAV seems to be the best case in quantitative analysis of fast-changing vibration signals.

COMPOSITIONAL TIME SERIES ANALYSIS OF MORTALITY PROPORTIONS

Compositional time series are multivariate time series which at each time point are proportions that sum to a constant. Accurate inference for such series which occur in several disciplines such as geology, economics and ecology is important in practice. Usual multivariate statistical procedures ignore the inherent constrained nature of these observations as parts of a whole and may lead to inaccurate estimation and prediction. In this article, a regression model with vector autoregressive moving average (VARMA) errors is fit to the compositional time series after an additive log ratio (ALR) transformation. Inference is carried out in a hierarchical Bayesian framework using Markov chain Monte Carlo techniques. The approach is illustrated on compositional time series of mortality events in Los Angeles in order to investigate dependence of different categories of mortality on air quality.

Specification of Varma Echelon form Models

Vector autoregressive moving average (VARMA) models have certain advantages over pure vector autoregressive (VAR) models because they may permit more parsimonious representations of data generation processes. Moreover for many economic data, for instance, temporal and contemporaneous aggregation leads directly to mixed VARMA models. The major problems in VARMA modelling are related to specification unique representation. To solve these problems, we consider a VARMA model in echelon form, which is automatically identified. To specify the orders of the model, we propose an algorithm which selects a maximal set of linearly independent rows of the Hankel matrix of the estimated covariances. This set is obtained by sequentially testing the smallest singular value of the Hankel matrix and yields estimates of Kronecker indices which characterise the echelon form. Using the matrix perturbation theory, the asymptotic distribution of the test statistic is seen to be chi-squared. An example based on economic data series illustrates the procedure and demonstrates its feasibility in practice. Its performance is compared with other existing methods.

Construction of the exact Fisher information matrix of Gaussian time series models by means of matrix differential rules

The Fisher information matrix is of fundamental importance for the analysis of parameter estimation of time series models. In this paper the exact information matrix of a multivariate Gaussian time series model expressed in state space form is derived. A computationally efficient procedure is used by applying matrix differential rules for the derivatives of a matrix function J= J( θ) with respect to its vector argument. An algorithm is given. It is sketched for the general state space structure without specifying a parametrization. It is then detailed for the vector autoregressive moving average (VARMA) model, with a given parametrization, where explicit recurrent relations are developed.

EXPERIMENTAL MODAL ANALYSIS OF A STRUCTURE EXCITED BY A RANDOM FORCE

A multivariate maximum likelihood procedure for the estimation of modal parameters is presented. The vibrating system is excited by a random force and only output sensors are used to estimate the natural frequencies and damping factors of the system. The method works in time domain and a vector autoregressive moving-average (VARMA) process is used. The information about modal parameters is contained in the multivariate AR part, which is estimated using an iterative maximum likelihood algorithm. This algorithm uses a score technique and output data only. The order of the AR part is obtained via the minimum description length associated with an instrumental variable procedure. Experimental results show the effectiveness of the method for model order selection and modal parameters estimation.

Robust Estimation in Vector Autoregressive Moving‐Average Models

Bustos and Yohai proposed a class of robust estimates for autoregressive moving‐average (ARMA) models based on residual autocovariances (RA estimates). In this paper an affine equivariant generalization of the RA estimates for vector ARMA processes is given. These estimates are asymptotically normal and, when the innovations have an elliptical distribution, their asymptotic covariance matrix differs only by a scalar factor from the covariance matrix corresponding to the maximum likelihood estimate. A Monte Carlo study confirms that the RA estimates are efficient under normal errors and robust when the sample contains outliers. A robust multivariate goodness‐of‐fit test based on the RA estimates is also obtained.