Measuring the Advantages of Multivariate vs. Univariate Forecasts

Abstract. Suppose we are interested in forecasting a time series and, in addition to the time series data, we have data from many time series related to the one we want to forecast. Since building a dynamic multivariate model for the set of time series can be a complex task, it is important to measure in advance the increase in precision to be attained by using multivariate forecasts with respect to univariate ones. This article presents a simple procedure designed to obtain a consistent estimate of this measure. Its performance is illustrated with Monte Carlo simulations and examples.     

Computationally efficient methods for two multivariate fractionally integrated models

Abstract.  We discuss two distinct multivariate time-series models that extend the univariate ARFIMA (autoregressive fractionally integrated moving average) model. We discuss the different implications of the two models and describe an extension to fractional cointegration. We describe algorithms for computing the covariances of each model, for computing the quadratic form and approximating the determinant for maximum likelihood estimation and for simulating from each model. We compare the speed and accuracy of each algorithm with existing methods individually. Then, we measure the performance of the maximum likelihood estimator and of existing methods in a Monte Carlo. These algorithms are much more computationally efficient than the existing algorithms and are equally accurate, making it feasible to model multivariate long memory time series and to simulate from these models. We use maximum likelihood to fit models to data on goods and services inflation in the United States.     

THE ESTIMATION AND PREDICTION OF SUBSET BILINEAR TIME SERIES MODELS WITH APPLICATIONS

Abstract. In this paper we define subset bilinear time series models, and then describe an algorithm for the estimation of these models. It is also pointed out that for this class of non-linear time series models, it is possible to obtain optimal several step predictors. The estimation technique of these models is illustrated with respect to three time series, and the optimal several steps ahead forecasts of these time series models are calculated. A comparison of these forecasts is made with the forecasts obtained by the best linear autoregressive and threshold autoregressive models. The residuals obtained from the models are tested for independence and Gaussianity using higher order moments.     

Influence diagnostics for multivariate GARCH processes

This article presents diagnostics for identifying influential observations when estimating multivariate generalized autoregressive conditional heteroscedasticity (GARCH) models. We derive influence diagnostics by introducing minor perturbations to the conditional variances and covariances. The derived diagnostics are applied to a bivariate GARCH model of daily returns of the S&P500 and IBM. We find that univariate diagnostic procedures may be unable to identify the influential observations in a multivariate model. Importantly, the proposed curvature‐based diagnostic identified influential observations where the correlation between the two series had a major change. These observations were not identified as influential using the univariate diagnostics for each asset separately. When estimating the bivariate GARCH model allowing for weights at influential observations, we found that the time‐varying correlations behaved differently from that implied by the model ignoring influential observations. The application therefore highlights the importance of extending univariate diagnostic procedures to multivariate settings.     

DIFFERENCING MULTIPLE TIME SERIES: ANOTHER LOOK AT CANADIAN MONEY AND INCOME DATA

Abstract. In multiple time series analysis it is sometimes suggested to remove non-stationarities of the univariate subseries by differencing prior to the multivariate analysis. It is pointed out that, in general, this is not adequate if AR models are built even if stationarity of the univariate subseries can be induced by differencing. Canadian money and income data that were previously analysed by Hsiao are used to illustrate the resulting problems.     

NONPARAMETRIC ESTIMATORS FOR TIME SERIES

Abstract. Kernel multivariate probability density and regression estimators are applied to a univariate strictly stationary time series X _r We consider estimators of the joint probability density of X _t at different t -values, of conditional probability densities, and of the conditional expectation of functionals of X _v given past behaviour. The methods seem of particular relevance in light of recent interest in non-Gaussian time series models. Under a strong mixing condition multivariate central limit theorems for estimators at distinct points are established, the asymptotic distributions being of the same nature as those which would derive from independent multivariate observations.     

Nonlinear modelling of periodic threshold autoregressions using Tsmars

We present new methods for modelling nonlinear threshold-type autoregressive behaviour in periodically correlated time series. The methods are illustrated using a series of average monthly flows of the Fraser River in British Columbia. Commonly used nonlinearity tests of the river flow data in each month indicate nonlinear behaviour in certain months. The periodic nonlinear correlation structure is modelled nonparametrically using TSMARS, a time series version of Friedman's extended multivariate adaptive regression splines (MARS) algorithm, which allows for categorical predictor variables. We discuss two methods of using the computational algorithm in TSMARS for modelling and fitting periodically correlated data. The first method applies the algorithm to data from each period separately. The second method models data from all periods simultaneously by incorporating an additional predictor variable to distinguish different behaviour in different periods, and allows for coalescing of data from periods with similar behaviour. The models obtained using TSMARS provide better short-term forecasts for the Fraser River data than a corresponding linear periodic AR model.     

活性污泥法处理染色废水过程模型辨识与参数估计

本文用Ｂｏｘ－Ｊｅｎｋｉｎｓ时间序列分析方法建立了活性污泥法处理染色废水过程中，进、出水高锰酸盐指数（ＯＣｉ，ＯＣｅ）单序列随机模型以及ＯＣｅ对ＯＣｉ序列含量有噪声项的传递函数模型，建立的模型较好地拟合了实际过程，传递函数比噪声项显得重要，模型对ＯＣｉ序列预测平均误差１０．２６％。     

PREDICTION ERROR OF MULTIVARIATE TIME SERIES WITH MIS-SPECIFIED MODELS

We consider the situation in which an incorrectly specified autoregressive moving-average model is used to predict future values of a stationary multivariate time series. The use of an incorrect model for prediction results in an increase in mean-square prediction error over that of the optimal predictor, and an expression for this increase is first given for fixed values of the parameters in the incorrect model. For the case in which the incorrect model is an autoregression, we also take into account parameter estimation error by first deriving the asymptotic distribution and limiting moment properties of the least-squares estimator of the parameters in the mis-specified model. An asymptotic approximation to the increase in mean-square prediction error is then obtained. Numerical examples are provided to demonstrate the accuracy of the asymptotic approximation in finite samples. Our results are consistent with those obtained in the univariate case, indicating that fitted autoregressions of high order can yield substantially sub-optimal forecasts.     

A family of multivariate non-gaussian time series models

In this article, we propose a class of multivariate non-Gaussian time series models which include dynamic versions of many well-known distributions and consider their Bayesian analysis. A key feature of our proposed model is its ability to account for correlations across time as well as across series (contemporary) via a common random environment. The proposed modeling approach yields analytically tractable dynamic marginal likelihoods, a property not typically found outside of linear Gaussian time series models. These dynamic marginal likelihoods can be tied back to known static multivariate distributions such as the Lomax, generalized Lomax, and the multivariate Burr distributions. The availability of the marginal likelihoods allows us to develop efficient estimation methods for various settings using Markov chain Monte Carlo as well as sequential Monte Carlo methods. Our approach can be considered to be a multivariate generalization of commonly used univariate non-Gaussian class of state space models. To illustrate our methodology, we use simulated data examples and a real application of multivariate time series for modeling the joint dynamics of stochastic volatility in financial indexes, the VIX and VXN.     

Testing Non‐Correlation and Non‐Causality between Multivariate ARMA Time Series

Abstract. Haugh [Journal of the American Statistical Association (1976) Vol. 71, pp. 378–85] developed an approach to the problem of testing non‐correlation (at all leads and lags) between two univariate time series. Haugh's tests however have low power against two series which are related over a long distributed lag when individual lag coefficients are relatively small. As a remedy, Koch and Yang [Journal of the American Statistical Association (1986) Vol. 8, pp. 533–44] proposed an alternative method that performs better than Haugh's under such dependencies. A multivariate extension of Haugh's procedure was proposed by El Himdi and Roy [The Canadian Journal of Statistics (1997) Vol. 25, pp. 233–56], but suffers the same weaknesses as the original univariate method. We develop here an asymptotic test generalizing Koch and Yang's method to the multivariate case. Our method includes El Himdi and Roy's as a special case. Based on the same idea, we also suggest a generalization of the El Himdi and Roy procedure for testing causality in the sense of Granger [Econometrica (1969) Vol. 37, pp. 424–38] between two multivariate series. A Monte Carlo study is conducted, which indicates that our approach performs better than El Himdi and Roy's for a wide range of models. Both procedures are applied to the problem of testing the absence of correlation between Canadian and US economic indicators, and to a brief study of causality between money and income in Canada.     

Improved multivariate portmanteau test

A new portmanteau diagnostic test for vector autoregressive moving average (VARMA) models that is based on the determinant of the standardized multivariate residual autocorrelations is derived. The new test statistic may be considered an extension of the univariate portmanteau test statistic suggested by Peňa and Rodríguez (2002) . The asymptotic distribution of the test statistic is derived as well as a chi‐square approximation. However, the Monte–Carlo test is recommended unless the series is very long. Extensive simulation experiments demonstrate the usefulness of this test as well as its improved power performance compared to widely used previous multivariate portmanteau diagnostic check. Two illustrative applications are given.     

Hyper‐spherical and elliptical stochastic cycles

A univariate first‐order stochastic cycle can be represented as an element of a bivariate first‐order vector autoregressive process, or VAR(1), where the transition matrix is associated with a rotation along a circle in the plane, and the reduced form is ARMA(2,1). This paper generalizes this representation in two directions. According to the first, the cyclical dynamics originate from the motion of a point along an ellipse. The reduced form is also ARMA(2,1), but the model can account for certain types of asymmetries. The second deals with the multivariate case: the cyclical dynamics result from the projection along one of the coordinate axis of a point moving in along an hyper‐sphere. This is described by a VAR(1) process whose transition matrix is obtained by a sequence of n‐dimensional Givens rotations. The reduced form of an element of the system is shown to be ARMA(n, n ? 1). The properties of the resulting models are analysed in the frequency domain, and we show that this generalization can account for a multimodal spectral density. The illustrations show that the proposed generalizations can be fitted successfully to some well‐known case studies of the time series literature.     

Efficient Bayesian PARCOR approaches for dynamic modeling of multivariate time series

A Bayesian lattice filtering and smoothing approach is proposed for fast and accurate modeling and inference in multivariate non-stationary time series. This approach offers computational feasibility and interpretable time-frequency analysis in the multivariate context. The proposed framework allows us to obtain posterior estimates of the time-varying spectral densities of individual time series components, as well as posterior measurements of the time-frequency relationships across multiple components, such as time-varying coherence and partial coherence. The proposed formulation considers multivariate dynamic linear models (MDLMs) on the forward and backward time-varying partial autocorrelation coefficients (TV-VPARCOR). Computationally expensive schemes for posterior inference on the multivariate dynamic PARCOR model are avoided using approximations in the MDLM context. Approximate inference on the corresponding time-varying vector autoregressive (TV-VAR) coefficients is obtained via Whittle's algorithm. A key aspect of the proposed TV-VPARCOR representations is that they are of lower dimension, and therefore more efficient, than TV-VAR representations. The performance of the TV-VPARCOR models is illustrated in simulation studies and in the analysis of multivariate non-stationary temporal data arising in neuroscience and environmental applications. Model performance is evaluated using goodness-of-fit measurements in the time-frequency domain and also by assessing the quality of short-term forecasting.     

A GENERAL TEST FOR UNIVARIATE SEASONALITY

We propose a general test for univariate seasonality. Starting from a multivariate model for the seasons, some constraints must hold, both on the covariance matrix of the innovations and among coefficients across equations, for a univariate representation of seasonality to be appropriate. Applied to a set of 23 UK macroeconomic variables, our test shows that a multivariate representation of seasonality should be preferred in at least eight cases. This introduces a serious questioning of standard univariate filters to estimate the seasonal component in some economic time series, and suggests the possibility of a more complex but richer way of characterizing relationships among seasonal economic variables.     

ESTIMATION OF MULTIVARIATE TIME SERIES

Abstract. The algorithm proposed here is a multivariate generalization of a procedure discussed by Pearlman (1980) for calculating the exact likelihood of a univariate ARMA model. Ansley and Kohn (1983) have shown how the Kalman filter can be used to calculate the exact likelihood function when not all the observations are known. In Shea (1983) it is shown that this algorithm is much quicker than that of Ansley and Kohn (1983) for all ARMA models except an ARMA (2, 1) and a couple of low-order AR processes and therefore when we have no missing observations this algorithm should be used instead. The Fortran subroutine G13DCF in the NAG (1987) Library fits a vector ARMA model using an adaptation of this algorithm. Experience in the use of this routine suggests that having reasonably good initial estimates of the ARMA parameter matrices, and in particular the residual error covariance matrix, can not only substantially reduce the computing time but more important improve the convergence properties of the minimization procedure. We therefore propose a method of calculating initial estimates of the ARMA parameters which involves using a generalization of the concept of inverse cross covariances from the univariate to the multivariate case. Finally theory is put into practice with the fitting of a bivariate model to a couple of real-life time series.     

Moisture Changes in Grain During Aeration Under Warm Humid Conditions

ABSTRACT

Two mathematical models (an equilibrium model and a combined model) were evaluated, which described the moisture and heat transfer in low temperature drying and aeration. The predicted moisture contents from both models were compared with the experimental data. Comparisons indicated that the combined model was more accurate than the equilibrium model. The combined model is based on the idea that the partial pressure difference is the driving force in moisture transfer. A series of simulation was performed using the combined model to evaluate the effect of air temperature, air relative humidity, and the temperature difference between grain and air on the moisture changes in stored rough rice. The simulation results proved the concept of using the partial pressure difference to describe the moisture transfer in stored grain. A minimum of 5.6°C in     

An Extended Portmanteau Test for VARMA Models With Mixing Nonlinear Constraints

Abstract. The portmanteau test is a widely used diagnostic tool for univariate and multivariate time‐series models. Its asymptotic distribution is known for the unconstrained vector autoregressive moving‐average (VARMA) case and for VAR models with constraints on the autoregressive coefficients. In this article, we give conditions under which the test can be applied to constrained VARMA models. Unfortunately, it cannot generally be applied to models with constraints that simultaneously affect the ARMA polynomial coefficients and the covariance matrix of the innovations (mixing constraints). This happens in latent‐variable models such as dynamic factor models (DFM). In addition, when there are constraints on the covariance matrix it seems convenient to check the goodness of fit using the zero‐lag residual covariances. We propose an extended portmanteau test that not only checks the autocorrelations of the residuals but also whether their covariance matrix is consistent with the constraints. We prove that the statistic is asymptotically distributed as a chi‐square for ARMA models under the assumption that the innovations have Gaussian‐like fourth‐order moments. We also show that the test is appropriate for the DFM, Peña–Box model and factor‐structural vector autoregression (FSVAR).