Vector Autoregressive Process Research Articles

Multivariate [formula omitted]-stable distributions: VAR(1) processes, measures of dependence and their estimations

Measuring the dependency between lagged random vectors in time series is extremely important to detect appropriate models for fitting real data. Vector autoregressive processes (VAR), in particular, are among the most popular multivariate time series that allow modeling schemes for two or more random variables. However, if the series of innovations process does not have finite second-order moments, the classical analysis that uses the empirical auto-covariance function is not adequate as its theoretical counterpart is not defined. After perceiving the lack of a suitable multivariate version of the codifference function, we propose an adaptation of this tool to measure interdependence in multivariate α-stable processes. We establish such a function based on notions presented in the existing literature. We also show some of its properties and give an estimator based on the empirical characteristic function. Subsequently, we evaluate another measure of dependence with different properties but similar usage for comparison reasons. These two measures are computed for VAR(1) processes with α-stable innovations, proving to be useful in the identification of associated patterns related to them. Finally, we demonstrate how these measures can be applied with real data sets by analyzing the monthly number of patients with community-acquired pneumonia and the registered values of PM10 in the city of São Paulo, Brazil, from January 2008 to December 2021.

Modelling mortality: A bayesian factor-augmented var (favar) approach

AbstractLongevity risk is putting more and more financial pressure on governments and pension plans worldwide due to pensioners’ increasing trend of life expectancy and the growing numbers of people reaching retirement age. Lee and Carter (1992, Journal of the American Statistical Association, 87(419), 659–671.) applied a one-factor dynamic factor model to forecast the trend of mortality improvement, and the model has since become the field’s workhorse. It is, however, well known that their model is subject to the limitation of overlooking cross-dependence between different age groups. We introduce Factor-Augmented Vector Autoregressive (FAVAR) models to the mortality modelling literature. The model, obtained by adding an unobserved factor process to a Vector Autoregressive (VAR) process, nests VAR and Lee–Carter models as special cases and inherits both frameworks’ advantages. A Bayesian estimation approach, adapted from the Minnesota prior, is proposed. The empirical application to the US and French mortality data demonstrates our proposed method’s efficacy in both in-sample and out-of-sample performance.

INSTRUMENTAL VARIABLES INFERENCE IN A SMALL-DIMENSIONAL VAR MODEL WITH DYNAMIC LATENT FACTORS

We study semiparametric inference in a small-dimensional vector autoregressive (VAR) model of order p augmented by unobservable common factors with a dynamic described by a VAR process of order q. This state-space specification is useful to measure separately the direct causality effects and the responses to dynamic common factors. We show that the state-space parameters are identifiable from the autocovariance function of the observed process. We estimate the model by means of a multistep procedure in closed-form, which combines an eigenvalue–eigenvector matrix decomposition and a linear instrumental variable estimation allowing for Hansen–Sargan specification tests. We study the asymptotic and finite-sample properties of the parameter estimators and of rank tests for selecting the number of unobservable factors and VAR orders. In an empirical illustration, we investigate the dynamic common factors and the spillover effects that explain the co-movements among the log daily realized volatilities of four European stock market indices.

Linear Inverse Modeling of Large-Scale Atmospheric Flow Using Optimal Mode Decomposition

Abstract Linear inverse modeling or principal oscillation pattern (POP) analysis is a widely applied tool in climate science for extracting from data dominant spatial patterns together with their dynamics as approximated by a linear Markov model. The system is projected onto a principal linear subspace and the system matrix is estimated from data. The eigenmodes of the system matrix are the POPs, with the eigenvalues providing their decay time scales and oscillation frequencies. Usually, the subspace is spanned by the leading principal components (PCs) and empirical orthogonal functions (EOFs). Outside of climate science, this procedure is now more commonly referred to as dynamic mode decomposition (DMD). Here, we use optimal mode decomposition (OMD) to address the full linear inverse modeling problem of simultaneous optimization of the principal subspace and the linear operator. The method is illustrated on two pedagogical examples and then applied to a three-level quasigeostrophic atmospheric model with realistic mean state and variability. The OMD models significantly outperform the EOF/DMD models in predicting the time evolution of the large-scale flow modes. The advantage of the OMD models stems from finding more persistent modes as well as from better capturing the nonnormality of the linear operator and the associated nonmodal growth. The dynamics of the large-scale flow modes turn out to be markedly non-Markovian and the OMD modes are superior to the EOF/DMD modes also in a modeling setting with a higher-order vector autoregressive process. The OMD modes could also be used as basis functions for a nonlinear dynamical model although they are not optimized for that purpose. Potential applications of the OMD method in weather and climate science include ENSO or MJO prediction, reduced-rank data assimilation, and generation of initial perturbations for ensemble prediction.

A high throughput generative vector autoregression model for stochastic synapses.

By imitating the synaptic connectivity and plasticity of the brain, emerging electronic nanodevices offer new opportunities as the building blocks of neuromorphic systems. One challenge for large-scale simulations of computational architectures based on emerging devices is to accurately capture device response, hysteresis, noise, and the covariance structure in the temporal domain as well as between the different device parameters. We address this challenge with a high throughput generative model for synaptic arrays that is based on a recently available type of electrical measurement data for resistive memory cells. We map this real-world data onto a vector autoregressive stochastic process to accurately reproduce the device parameters and their cross-correlation structure. While closely matching the measured data, our model is still very fast; we provide parallelized implementations for both CPUs and GPUs and demonstrate array sizes above one billion cells and throughputs exceeding one hundred million weight updates per second, above the pixel rate of a 30 frames/s 4K video stream.

Dynamic connectedness and optimal hedging strategy among commodities and financial indices

This study looks at the best portfolio strategy for mitigating the risk associated with the MSCI ACWI & Frontier Markets Index, as well as the volatility spillovers between commodity markets and certain financial markets. Therefore, we empirically explore the connectedness among three financial indicators and five product groups using the framework of Diebold and Yilmaz (2012), which is based on a vector autoregressive process and variance decomposition of prediction errors, between 31 May 2002 and 30 July 2021. We also investigate the best hedging instrument(s) for the MSCI ACWI & Frontier Markets Global Index by combining the Asymmetric Dynamic Conditional Correlation (ADCC) model with the risk reduction index and the hedging ratios. Our empirical findings highlight the importance of volatility spillover effects across financial markets, which is not the case for commodity markets with low volatility externalities. Furthermore, the first markets appear to be net transmitters of volatility, whereas the second markets appear to be net receivers. Using the approach of Kroner and Sultan (1993), we show that the least risk portfolio is a portfolio that combines the MSCI ACWI & Frontier Markets Global Index with financial indices related to socially responsible and irresponsible investing.

Granger Causality Inference in EEG Source Connectivity Analysis: A State-Space Approach.

This article addresses the problem of estimating brain effective connectivity from electroencephalogram (EEG) signals using a Granger causality (GC) characterized on state-space models, extended from the conventional vector autoregressive (VAR) process. The scheme involves two main steps: model estimation and model inference to estimate brain connectivity. The model estimation performs a subspace identification and active source selection based on group-norm regularized least-squares. The model inference relies on the concept of state-space GC that requires solving a Riccati equation for the covariance of estimation error. We verify the performance on simulated datasets that represent realistic human brain activities under several conditions, including percentages and location of active sources, and the number of EEG electrodes. Our model's accuracy in estimating connectivity is compared with a two-stage approach using source reconstructions and a VAR-based Granger analysis. Our method achieved better performances than the two-stage approach under the assumptions that the true source dynamics are sparse and generated from state-space models. When the method was applied to a real EEG SSVEP dataset, the temporal lobe was found to be a mediating connection between the temporal and occipital areas, which agreed with findings in previous studies.

Enforcing Stationarity through the Prior in Vector Autoregressions

Stationarity is a very common assumption in time series analysis. A vector autoregressive process is stable if and only if the roots of its characteristic equation lie outside the unit circle, constraining the autoregressive coefficient matrices to lie in the stationary region. However, the stationary region has a highly complex geometry which impedes specification of a prior distribution. In this work, an unconstrained reparameterization of a stationary vector autoregression is presented. The new parameters are partial autocorrelation matrices, which are interpretable, and can be transformed bijectively to the space of unconstrained square matrices through a simple mapping of their singular values. This transformation preserves various structural forms of the partial autocorrelation matrices and readily facilitates specification of a prior. Properties of this prior are described along with an important special case which is exchangeable with respect to the order of the elements in the observation vector. Posterior inference and computation are described and implemented using Hamiltonian Monte Carlo via Stan. The prior and inferential procedures are illustrated with an application to a macroeconomic time series which highlights the benefits of enforcing stationarity and encouraging shrinkage toward a sensible parametric structure. Supplementary materials for this article are available online.

Cointegration in large VARs

The paper analyzes cointegration in vector autoregressive processes (VARs) for the cases when both the number of coordinates, N, and the number of time periods, T, are large and of the same order. We propose a way to examine a VAR of order 1 for the presence of cointegration based on a modification of the Johansen likelihood ratio test. The advantage of our procedure over the original Johansen test and its finite sample corrections is that our test does not suffer from overrejection. This is achieved through novel asymptotic theorems for eigenvalues of matrices in the test statistic in the regime of proportionally growing N and T. Our theoretical findings are supported by Monte Carlo simulations and an empirical illustration. Moreover, we find a surprising connection with multivariate analysis of variance (MANOVA) and explain why it emerges.

Multivariate Hyper-Rotated GARCH-BEKK

Abstract For large multivariate models of generalized autoregressive conditional heteroskedasticity (GARCH), it is important to reduce the number of parameters to cope with the ‘curse of dimensionality’. Recently, Laurent, Rombouts and Violante (2014 “Multivariate Rotated ARCH Models” Journal of Econometrics 179: 16–30) developed the rotated multivariate GARCH model, which focuses on the parameters for standardized variables. This paper extends the rotated multivariate GARCH model by considering a hyper-rotation, which uses a more flexible structure for the rotation matrix. The paper shows an alternative representation based on a random coefficient vector autoregressive and moving-average (VARMA) process, and provides the regularity conditions for the consistency and asymptotic normality of the quasi-maximum likelihood (QML) estimator for VARMA with hyper-rotated multivariate GARCH. The paper investigates the finite sample properties of the QML estimator for the new model. Empirical results for four exchange rate returns show the new specifications works satisfactory for reducing the number of parameters.

Next-generation reservoir computing based on memristor array

As a kind of brain-inspired computing, reservoir computing (RC) has great potential applications in time sequence signal processing and chaotic dynamics system prediction due to its simple structure and few training parameters. Since in the RC randomly initialized network weights are used, it requires abundant data and calculation time for warm-up and parameter optimization. Recent research results show that an RC with linear activation nodes, combined with a feature vector, is mathematically equivalent to a nonlinear vector autoregression (NVAR) machine, which is named next-generation reservoir computing (NGRC). Although the NGRC can effectively alleviate the problems which traditional RC has, it still needs vast computing resources for multiplication operations. In the present work, a hardware implementation method of using computing-in memory paradigm for NGRC is proposed for the first time. We use memristor array to perform the matrix vector multiplication involved in the nonlinear vector autoregressive process for the improvement of the energy efficiency. The Lorenz63 time series prediction task is performed by simulation experiments with the memristor array, demonstrating the feasibility and robustness of this method, and the influence of the weight precision of the memristor devices on the prediction results is discussed. These results provide a promising way of implementing the hardware NGRC.

Is time a variable like the others in multivariate statistical downscaling and bias correction?

Abstract. Bias correction and statistical downscaling are now regularly applied to climate simulations to make then more usable for impact models and studies. Over the last few years, various methods were developed to account for multivariate – inter-site or inter-variable – properties in addition to more usual univariate ones. Among such methods, temporal properties are either neglected or specifically accounted for, i.e. differently from the other properties. In this study, we propose a new multivariate approach called “time-shifted multivariate bias correction” (TSMBC), which aims to correct the temporal dependency in addition to the other marginal and multivariate aspects. TSMBC relies on considering the initial variables at various times (i.e. lags) as additional variables to be corrected. Hence, temporal dependencies (e.g. auto-correlations) to be corrected are viewed as inter-variable dependencies to be adjusted and an existing multivariate bias correction (MBC) method can then be used to answer this need. This approach is first applied and evaluated on synthetic data from a vector auto-regressive (VAR) process. In a second evaluation, we work in a “perfect model” context where a regional climate model (RCM) plays the role of the (pseudo-)observations, and where its forcing global climate model (GCM) is the model to be downscaled or bias corrected. For both evaluations, the results show a large reduction of the biases in the temporal properties, while inter-variable and spatial dependence structures are still correctly adjusted. However, increasing the number of lags too much does not necessarily improve the temporal properties, and an overly strong increase in the number of dimensions of the dataset to be corrected can even imply some potential instability in the adjusted and/or downscaled results, calling for a reasoned use of this approach for large datasets.

Granger Causality: A Review and Recent Advances.

Introduced more than a half-century ago, Granger causality has become a popular tool for analyzing time series data in many application domains, from economics and finance to genomics and neuroscience. Despite this popularity, the validity of this framework for inferring causal relationships among time series has remained the topic of continuous debate. Moreover, while the original definition was general, limitations in computational tools have constrained the applications of Granger causality to primarily simple bivariate vector autoregressive processes. Starting with a review of early developments and debates, this article discusses recent advances that address various shortcomings of the earlier approaches, from models for high-dimensional time series to more recent developments that account for nonlinear and non-Gaussian observations and allow for subsampled and mixed-frequency time series.

Lugsail lag windows for estimating time-average covariance matrices

Summary Lag windows are commonly used in time series analysis, econometrics, steady-state simulation and Markov chain Monte Carlo to estimate time-average covariance matrices. In the presence of positive correlation in the underlying process, estimators of this matrix almost always exhibit significant negative bias, leading to undesirable finite-sample properties. We propose a new family of lag windows specifically designed to improve finite-sample performance by offsetting this negative bias. Any existing lag window can be adapted into a lugsail equivalent with no additional assumptions. We use these lag windows in spectral variance estimators and demonstrate their advantages in a linear regression model with autocorrelated and heteroskedastic residuals. We further employ the lugsail lag windows in weighted batch means estimators because of their computational efficiency on large simulation output. We obtain bias and variance results for these multivariate estimators and significantly weaken the mixing condition on the process. Superior finite-sample properties are demonstrated in a vector autoregressive process and a Bayesian logistic regression model.

Generalized binary vector autoregressive processes

Vector‐valued‐60 extensions of univariate generalized binary auto‐regressive (gbAR) processes are proposed that enable the joint modeling of serial and cross‐sectional‐50 dependence of multi‐variate binary data. The resulting class of generalized binary vector auto‐regressive (gbVAR) models is parsimonious, nicely interpretable and allows also to model negative dependence. We provide stationarity conditions and derive moving‐average‐type representations that allow to prove geometric mixing properties. Furthermore, we derive general stochastic properties of gbVAR processes, including formulae for transition probabilities. In particular, classical Yule–Walker equations hold that facilitate parameter estimation in gbVAR models. In simulations, we investigate the estimation performance, and for illustration, we apply gbVAR models to particulate matter (PM10, ‘fine dust’) alarm data observed at six monitoring stations in Stuttgart, Germany.

A tale of two sentiment scales: disentangling short-run and long-run components in multivariate sentiment dynamics

The digitalization of news and social media provides an unprecedented source to investigate the role of information on market dynamics. However, the observed sentiment time-series represent a noisy proxy of the true investor sentiment. Moreover, modeling the joint dynamics of different sentiment series can be beneficial for the assessment of their economic relevance. The main methodological contribution of this paper is twofold: (i) we filter the latent sentiment signals in a genuinely multivariate model; (ii) we propose a decomposition into a long-term random walk component, named long-term sentiment, and a short-term component driven by a stationary Vector Autoregressive process of order one, named short-term sentiment. The proposed framework is a dynamic factor model describing the joint evolution of the observed sentiments of a portfolio of assets. Empirically, we find that the long-term sentiment co-integrates with the market price factor, while the short-term sentiment captures transient and firm-specific swings. By means of quantile regressions, we assess the significance of the explanatory power of filtered present sentiment on future returns. Then, we demonstrate how the lagged relation can be successfully exploited in a portfolio allocation exercise.

The effect of autocorrelation on the diagnostic procedures

Apart from quick detection of abnormal changes in a multivariate process, it is also critical to accurately identify the factors of abnormal changes after an out-of-control signal in multivariate statistical process control. Some diagnostic methods for fault identification were proposed by scholars on which one of the basic assumptions is that the process data are independent. The independent assumption is reasonable for many applications. However, with the development of industrial automation, the process data usually meet the phenomenon of autocorrelation. As it is well known, the autocorrelation affects detection ability of control chart. It is a natural question that is whether the autocorrelation affects the diagnostic performance of the diagnostic procedures. In the article, we propose the method adopted the analogy residual sequences and compare it with the performance of existing diagnostic methods for different shift sizes and various autocorrelation and cross-correlation structures. To limit the complexity, our discussion employs a first-order vector autoregressive process and focuses mainly on bivariate data in this article. The results of the simulation show the performances of diagnostic procedures were affected by autocorrelation and the impacts become greater as autocorrelation increases. On the other hand, the proposed method in the article can be free from the impacts of autocorrelation. Finally, a real example is also presented to demonstrate the implementation of the proposed method.

Cointegration, Root Functions and Minimal Bases

This paper discusses the notion of cointegrating space for linear processes integrated of any order. It first shows that the notions of (polynomial) cointegrating vectors and of root functions coincide. Second, it discusses how the cointegrating space can be defined (i) as a vector space of polynomial vectors over complex scalars, (ii) as a free module of polynomial vectors over scalar polynomials, or finally (iii) as a vector space of rational vectors over rational scalars. Third, it shows that a canonical set of root functions can be used as a basis of the various notions of cointegrating space. Fourth, it reviews results on how to reduce polynomial bases to minimal order—i.e., minimal bases. The application of these results to Vector AutoRegressive processes integrated of order 2 is found to imply the separation of polynomial cointegrating vectors from non-polynomial ones.

Causality in Reversed Time Series: Reversed or Conserved?

The inference of causal relations between observable phenomena is paramount across scientific disciplines; however, the means for such enterprise without experimental manipulation are limited. A commonly applied principle is that of the cause preceding and predicting the effect, taking into account other circumstances. Intuitively, when the temporal order of events is reverted, one would expect the cause and effect to apparently switch roles. This was previously demonstrated in bivariate linear systems and used in design of improved causal inference scores, while such behaviour in linear systems has been put in contrast with nonlinear chaotic systems where the inferred causal direction appears unchanged under time reversal. The presented work explores the conditions under which the causal reversal happens—either perfectly, approximately, or not at all—using theoretical analysis, low-dimensional examples, and network simulations, focusing on the simplified yet illustrative linear vector autoregressive process of order one. We start with a theoretical analysis that demonstrates that a perfect coupling reversal under time reversal occurs only under very specific conditions, followed up by constructing low-dimensional examples where indeed the dominant causal direction is even conserved rather than reversed. Finally, simulations of random as well as realistically motivated network coupling patterns from brain and climate show that level of coupling reversal and conservation can be well predicted by asymmetry and anormality indices introduced based on the theoretical analysis of the problem. The consequences for causal inference are discussed.

Adjustment models for multivariate geodetic time series with vector-autoregressive errors

Abstract In this contribution, a vector-autoregressive (VAR) process with multivariate t-distributed random deviations is incorporated into the Gauss-Helmert model (GHM), resulting in an innovative adjustment model. This model is versatile since it allows for a wide range of functional models, unknown forms of auto- and cross-correlations, and outlier patterns. Subsequently, a computationally convenient iteratively reweighted least squares method based on an expectation maximization algorithm is derived in order to estimate the parameters of the functional model, the unknown coefficients of the VAR process, the cofactor matrix, and the degree of freedom of the t-distribution. The proposed method is validated in terms of its estimation bias and convergence behavior by means of a Monte Carlo simulation based on a GHM of a circle in two dimensions. The methodology is applied in two different fields of application within engineering geodesy: In the first scenario, the offset and linear drift of a noisy accelerometer are estimated based on a Gauss-Markov model with VAR and multivariate t-distributed errors, as a special case of the proposed GHM. In the second scenario real laser tracker measurements with outliers are adjusted to estimate the parameters of a sphere employing the proposed GHM with VAR and multivariate t-distributed errors. For both scenarios the estimated parameters of the fitted VAR model and multivariate t-distribution are analyzed for evidence of auto- or cross-correlations and deviation from a normal distribution regarding the measurement noise.