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 class of optimal tests for contemporaneous non‐causality in VAR models

The aim of this paper was to test for contemporaneous non‐causality defined by Granger (1969) between two groups of variables in a VAR(p) setting. Since contemporaneous correlation of the innovations is a necessary condition for contemporaneous causality (Pierce and Haugh, 1977), we focused on testing some restrictions on the covariance matrix of the noise. The class of the derived tests is locally asymptotically most stringent (in the Le Cam sense), invariant with respect to the group of block affine transformations and asymptotically invariant with respect to the group of continuous monotone radial transformations. Those tests are based on multivariate ranks of distances and multivariate signs of the o bservations and are shown to be asymptotically distribution free under very mild assumptions on the noise.     

Spatio‐temporal models for big multinomial data using the conditional multivariate logit‐beta distribution

We introduce a Bayesian approach for analyzing high‐dimensional multinomial data that are referenced over space and time. In particular, the proportions associated with multinomial data are assumed to have a logit link to a latent spatio‐temporal mixed effects model. This strategy allows for covariances that are nonstationarity in both space and time, asymmetric, and parsimonious. We also introduce the use of the conditional multivariate logit‐beta distribution into the dependent multinomial data setting, which leads to conjugate full‐conditional distributions for use in a collapsed Gibbs sampler. We refer to this model as the multinomial spatio‐temporal mixed effects model (MN‐STM). Additionally, we provide methodological developments including: the derivation of the associated full‐conditional distributions, a relationship with a latent Gaussian process model, and the stability of the non‐stationary vector autoregressive model. We illustrate the MN‐STM through simulations and through a demonstration with public‐use quarterly workforce indicators data from the longitudinal employer household dynamics program of the US Census Bureau.     

An improved dynamic latent variable regression model for fault diagnosis and causal analysis

The advancement of industrial techniques has imposed a high demand for powerful machine learning algorithms to model the increasingly complicated relations in the data. Among them, dynamic models are widely studied to capture the inevitable temporal relations. However, most existing methods only focus on the dynamics between input and output data, failing to exploit other valuable information in the output. In this article, an improved dynamic latent variable regression (LVR) method is proposed to capture both auto-correlations and cross-correlations between input and output with an auto-regressive exogenous model, which is referred to as dynamic regularized LVR with auto-regressive exogenous input (DrLVR-ARX). Further, a DrLVR-ARX-based fault detection and diagnosis framework is designed to identify the root causes of a detected fault. The framework systematically integrates reconstruction-based contribution, time-domain Granger causality, and conditional spectral Granger causality to determine and locate the assignable causes. The effectiveness of the proposed algorithms is demonstrated with two industrial processes.     

Autocovariance Structure of Markov Regime Switching Models and Model Selection

We show that the covariance function of a second-order stationary vector Markov regime switching time series has a vector ARMA( p , q ) representation, where upper bounds for p and q are elementary functions of the number of regimes. These bounds apply to vector Markov regime switching processes with both mean–variance and autoregressive switching. This result yields an easily computed method for setting a lower bound on the number of underlying Markov regimes from an estimated autocovariance function.     

RANDOM AGGREGATION OF UNIVARIATE AND MULTIVARIATE LINEAR PROCESSES

Abstract. The paper is devoted to random aggregation of multivariate autoregressive moving-average (ARMA) processes. We derive second-order characteristics of random aggregate models. We show that random aggregation preserves the ARMA structure. Moreover, we specify a functional relation between the initial model poles and aggregate ones. We then examine the case of univariate ARMA processes. Theorem 4 shows that, if the initial process is ARMA( p, q ), the random aggregate process is an ARMA( p*, q* ) model with p* at most equal to p ; * depends, among other things, on the sampling distribution L . This theorem generalizes the well-known results on the topic of time interval aggregation without overlapping.     

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.     

The adjustment of prediction intervals to account for errors in parameter estimation

Abstract.  Standard approximate 1 −  α prediction intervals (PIs) need to be adjusted to take account of the error in estimating the parameters. This adjustment may be aimed at setting the (unconditional) probability that the PI includes the value being predicted equal to 1 −  α . Alternatively, this adjustment may be aimed at setting the probability that the PI includes the value being predicted equal to 1 −  α , conditional on an appropriate statistic T . For an autoregressive process of order p , it has been suggested that T consist of the last p observations. We provide a new criterion by which both forms of adjustment can be compared on an equal footing. This new criterion of performance is the closeness of the coverage probability, conditional on all of the data, of the adjusted PI and 1 −  α . In this paper, we measure this closeness by the mean square of the difference between this conditional coverage probability and 1 −  α . We illustrate the application of this new criterion to a Gaussian zero-mean autoregressive process of order 1 and one-step-ahead prediction. For this example, this comparison shows that the adjustment which is aimed at setting the coverage probability equal to 1 −  α conditional on the last observation is the better of the two adjustments.     

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.     

Cointegrated Linear Processes in Hilbert Space

We extend the notion of cointegration for multivariate time series to a potentially infinite‐dimensional setting in which our time series takes values in a complex separable Hilbert space. In this setting, standard linear processes with nonzero long‐run covariance operator play the role of processes. We show that the cointegrating space for an process may be sensibly defined as the kernel of the long‐run covariance operator of its difference. The inner product of an process with an element of its cointegrating space is a stationary complex‐valued process. Our main result is a version of the Granger–Johansen representation theorem: we obtain a geometric reformulation of the Johansen I(1) condition that extends naturally to a Hilbert space setting, and show that an autoregressive Hilbertian process satisfying this condition, and possibly also a compactness condition, admits an representation.     

Prospective surveillance of multivariate spatial disease data

Surveillance systems are often focused on more than one disease within a predefined area. On those occasions when outbreaks of disease are likely to be correlated, the use of multivariate surveillance techniques integrating information from multiple diseases allows us to improve the sensitivity and timeliness of outbreak detection. In this article, we present an extension of the surveillance conditional predictive ordinate to monitor multivariate spatial disease data. The proposed surveillance technique, which is defined for each small area and time period as the conditional predictive distribution of those counts of disease higher than expected given the data observed up to the previous time period, alerts us to both small areas of increased disease incidence and the diseases causing the alarm within each area. We investigate its performance within the framework of Bayesian hierarchical Poisson models using a simulation study. An application to diseases of the respiratory system in South Carolina is finally presented.     

Quadrature-based moment methods for the population balance equation: An algorithm review

The dispersed phase in multiphase flows can be modeled by the population balance model (PBM). A typical population balance equation (PBE) contains terms for spatial transport, loss/growth and breakage/coalescence source terms. The equation is therefore quite complex and difficult to solve analytically or numerically. The quadrature-based moment methods (QBMMs) are a class of methods that solve the PBE by converting the transport equation of the number density function (NDF) into moment transport equations. The unknown source terms are closed by numerical quadrature. Over the years, many QBMMs have been developed for different problems, such as the quadrature method of moments (QMOM), direct quadrature method of moments (DQMOM), extended quadrature method of moments (EQMOM), conditional quadrature method of moments (CQMOM), extended conditional quadrature method of moments (ECQMOM) and hyperbolic quadrature method of moments (HyQMOM). In this paper, we present a comprehensive algorithm review of these QBMMs. The mathematical equations for spatially homogeneous systems with first-order point processes and second-order point processes are derived in detail. The algorithms are further extended to the inhomogeneous system for multiphase flows, in which the computational fluid dynamics (CFD) can be coupled with the PBE. The physical limitations and the challenging numerical problems of these QBMMs are discussed. Possible solutions are also summarized.     

ON BOOTSTRAP PREDICTIVE INFERENCE FOR AUTOREGRESSIVE PROCESSES

Abstract. In this paper we consider bootstrap-based predictive inference for autoregressive processes of order p. We consider both unconditional inference and inference conditional on the last p observed values. We make two contributions. Our first contribution is to point out the best way to apply the bootstrap to unconditional predictive inference when the process is Gaussian. Now, it may be argued that predictive inference for autoregressive processes of order p should be carried out conditional on the last p observed values. When the process is Gaussian, a bootstrap predictive inference conditional on the last p observed values is conveniently computed by 'running' the same autoregressive process backwards in time. This procedure is inappropriate for non-Gaussian autoregressive processes. Our second (and more important) contribution is to present a method (which is not computationally burdensome) for the computation of a bootstrap predictive inference for a non-Gaussian autoregressive process of order p conditional on the last p observed values.     

Capacity Investment Planning for Multiple Vaccines Under Uncertainty: 2: Financial Risk Analysis

Capacity investment planning is a major decision for a vaccine company. Traditionally, due to the inherent flexibility used in almost all vaccine processes and risk-averse decisions, companies always started with limited capacities, thereby reducing the initial capital investment. However, in order to fulfil fast-growing vaccine demands, good and balanced financial risk management for capacity expansion is required to satisfy future demand without over committing capital. To complement the use of financial risk management, known probabilistic definitions of some classical risk measures such as expected downside risk (EDR), opportunity value (OV), value-at-risk (VaR) and conditional value-at-risk (CVaR) are adapted to be used in a scenario-based model for capacity investment planning for manufacture of multiple vaccines. Using these definitions, new models that manage financial risks and aid decisions are developed. Computational results and decision-making analysis methods are also presented and discussed. Numerical results show that this approach enables one to consider and manage the financial risk associated with the different designdjjf\ options, resulting in a set of solutions that can be used for decision-making.     

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.     

Second-order properties of locally stationary processes

Abstract.  In this article, we investigate an optimal property of the maximum likelihood estimator of Gaussian locally stationary processes by the second-order approximation. In the case where the model is correctly specified, it is shown that appropriate modifications of the maximum likelihood estimator for Gaussian locally stationary processes is second-order asymptotically efficient. We also discuss second-order robustness properties.     

MULTIVARIATE LOCAL POLYNOMIAL REGRESSION FOR TIME SERIES:UNIFORM STRONG CONSISTENCY AND RATES

Abstract. Local high-order polynomial fitting is employed for the estimation of the multivariate regression function m ( x₁ ,… x_d ) = E {φ( Y_d )φ X ₁= x ₁,…, X_d = x_d }, and of its partial derivatives, for stationary random processes { Y_i , X_i }. The function φ may be selected to yield estimates of the conditional mean, conditional moments and conditional distributions. Uniform strong consistency over compact subsets of R_d , along with rates, are established for the regression function and its partial derivatives for strongly mixing processes.     

STATIONARITY OF THE SOLUTION OF Xt= AtXt-1+εt AND ANALYSIS OF NON-GAUSSIAN DEPENDENT RANDOM VARIABLES

Abstract. We give general and concrete conditions in terms of the coefficient (stochastic) process {A_t} so that the (doubly) stochastic difference equation X_t= A_tX_t-1+ε_t has a second-order strictly stationary solution. It turns out that by choosing {A_t} and the "innovation" process {ε_t} properly, a host of stationary processes with non-Gaussian marginals and long-range dependence can be generated using this difference equation. Examples of such nowGaussian marginals include exponential, mixed exponential, gamma, geometric, etc. When {A_t} is a binary time series, the conditional least-squares estimator of the parameters of this model is the same as those of the parameters of a Galton-Watson branching process with immigration.     

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.     

DIAGNOSTIC CHECKING ARMA TIME SERIES MODELS USING SQUARED-RESIDUAL AUTOCORRELATIONS

Abstract. Squared-residual autocorrelations have been found useful in detecting nonlinear types of statistical dependence in the residuals of fitted autoregressive-moving average (ARMA) models (Granger and Andersen, 1978; Miller, 1979). In this note it is shown that the normalized squared-residual autocorrelations are asymptotically unit multivariate normal. The results of a simulation experiment confirming the small-sample validity of the proposed tests is reported.