Vector Autoregressive Process Research Articles

Bayesian Inference of Recurrent Switching Linear Dynamical Systems with Higher-Order Dependence

Many complicated dynamical events may be broken down into simpler pieces and efficiently described by a system that shifts among a variety of conditionally dynamical modes. Building on switching linear dynamical systems, we develop a new model that extends the switching linear dynamical systems for better discovering these dynamical modes. In the proposed model, the linear dynamics of latent variables can be described by a higher-order vector autoregressive process, which makes it feasible to evaluate the higher-order dependency relationships in the dynamics. In addition, the transition of switching states is determined by a stick-breaking logistic regression, overcoming the limitation of a restricted geometric state duration and recovering the symmetric dependency between the switching states and the latent variables from asymmetric relationships. Furthermore, logistic regression evidence potentials can appear as conditionally Gaussian potentials by utilizing the Po´lya-gamma augmentation strategy. Filtering and smoothing algorithms and Bayesian inference for parameter learning in the proposed model are presented. The utility and versatility of the proposed model are demonstrated on synthetic data and public functional magnetic resonance imaging data. Our model improves the current methods for learning the switching linear dynamical modes, which will facilitate the identification and assessment of the dynamics of complex systems.

Medium-term stochastic hydrothermal scheduling with short-term operational effects for large-scale power and water networks

The high integration of variable renewable sources in electric power systems entails a series of challenges inherent to their intrinsic variability. A critical challenge is to correctly value the water available in reservoirs in hydrothermal systems, considering the flexibility that it provides. In this context, this paper proposes a medium-term multistage stochastic optimization model for the hydrothermal scheduling problem solved with the stochastic dual dynamic programming algorithm. The proposed model includes operational constraints and simplified mathematical expressions of relevant operational effects that allow more informed assessment of the water value by considering, among others, the flexibility necessary for the operation of the system. In addition, the hydrological uncertainty in the model is represented by a vector autoregressive process, which allows capturing spatio-temporal correlations between the different hydro inflows. A calibration method for the simplified mathematical expressions of operational effects is also proposed, which allows a detailed short-term operational model to be correctly linked to the proposed medium-term linear model. Through extensive experiments for the Chilean power system, the results show that the difference between the expected operating costs of the proposed medium-term model, and the costs obtained through a detailed short-term operational model was only 0.1%, in contrast to the 9.3% difference obtained when a simpler base model is employed. This shows the effectiveness of the proposed approach. Further, this difference is also reflected in the estimation of the water value, which is critical in water shortage situations.

Confounding Factor Analysis for Vocal Fold Oscillations.

This paper provides a methodology to better understand the relationships between different aspects of vocal fold motion, which are used as features in machine learning-based approaches for detecting respiratory infections from voice recordings. The relationships are derived through a joint multivariate analysis of the vocal fold oscillations of speakers. Specifically, the multivariate setting explores the displacements and velocities of the left and right vocal folds derived from recordings of five extended vowel sounds for each speaker (/aa/, /iy/, /ey/, /uw/, and /ow/). In this multivariate setting, the differences between the bivariate and conditional interactions are analyzed by information-theoretic quantities based on transfer entropy. Incorporation of the conditional quantities reveals information regarding the confounding factors that can influence the statistical interactions among other pairs of variables. This is demonstrated on a vector autoregressive process where the analytical derivations can be carried out. As a proof of concept, the methodology is applied on a clinically curated dataset of COVID-19. The findings suggest that the interaction between the vocal fold oscillations can change according to individuals and presence of any respiratory infection, such as COVID-19. The results are important in the sense that the proposed approach can be utilized to determine the selection of appropriate features as a supplementary or early detection tool in voice-based diagnostics in future studies.

Brazilian macroeconomic dynamics redux: Shocks, frictions, and unemployment in SAMBA model

This paper documents the recent changes in the structure and estimation procedures of the SAMBA model, providing a complete description of the decision problems that each economic agent faces, the first order conditions that solve those problems, and the new techniques employed to estimate the model. This updated version of the model incorporates new features, such as involuntary unemployment, imported goods in the consumption bundle and a new identified vector auto-regressive process for the rest of the world. Reflecting these changes, the set of observables was expanded to include, for instance, participation rates in the labor market and an exogenous measure of output gap. In face of increased complexity and the large number of observables, the model was estimated using Sequential Monte Carlo (SMC) methods, allowing for a smaller sensitivity to the choice of priors.

FNETS: Factor-Adjusted Network Estimation and Forecasting for High-Dimensional Time Series

We propose FNETS, a methodology for network estimation and forecasting of high-dimensional time series exhibiting strong serial- and cross-sectional correlations. We operate under a factor-adjusted vector autoregressive (VAR) model which, after accounting for pervasive co-movements of the variables by common factors, models the remaining idiosyncratic dynamic dependence between the variables as a sparse VAR process. Network estimation of FNETS consists of three steps: (i) factor-adjustment via dynamic principal component analysis, (ii) estimation of the latent VAR process via -regularized Yule-Walker estimator, and (iii) estimation of partial correlation and long-run partial correlation matrices. In doing so, we learn three networks underpinning the VAR process, namely a directed network representing the Granger causal linkages between the variables, an undirected one embedding their contemporaneous relationships and finally, an undirected network that summarizes both lead-lag and contemporaneous linkages. In addition, FNETS provides a suite of methods for forecasting the factor-driven and the idiosyncratic VAR processes. Under general conditions permitting tails heavier than the Gaussian one, we derive uniform consistency rates for the estimators in both network estimation and forecasting, which hold as the dimension of the panel and the sample size diverge. Simulation studies and real data application confirm the good performance of FNETS.

Towards a dynamical understanding of microstate analysis of M/EEG data

One of the interesting aspects of EEG data is the presence of temporally stable and spatially coherent patterns of activity, known as microstates, which have been linked to various cognitive and clinical phenomena. However, there is still no general agreement on the interpretation of microstate analysis. Various clustering algorithms have been used for microstate computation, and multiple studies suggest that the microstate time series may provide insight into the neural activity of the brain in the resting state. This study addresses two gaps in the literature. Firstly, by applying several state-of-the-art microstate algorithms to a large dataset of EEG recordings, we aim to characterise and describe various microstate algorithms. We demonstrate and discuss why the three “classically” used algorithms ((T)AAHC and modified K-Means) yield virtually the same results, while HMM algorithm generates the most dissimilar results. Secondly, we aim to test the hypothesis that dynamical microstate properties might be, to a large extent, determined by the linear characteristics of the underlying EEG signal, in particular, by the cross-covariance and autocorrelation structure of the EEG data. To this end, we generated a Fourier transform surrogate of the EEG signal to compare microstate properties. Here, we found that these are largely similar, thus hinting that microstate properties depend to a very high degree on the linear covariance and autocorrelation structure of the underlying EEG data. Finally, we treated the EEG data as a vector autoregression process, estimated its parameters, and generated surrogate stationary and linear data from fitted VAR. We observed that such a linear model generates microstates highly comparable to those estimated from real EEG data, supporting the conclusion that a linear EEG model can help with the methodological and clinical interpretation of both static and dynamic human brain microstate properties.

Estimation and inference in a high-dimensional semiparametric Gaussian copula vector autoregressive model

This paper develops simple, robust estimation and inference methods for the transition matrix of a high-dimensional semiparametric Gaussian copula vector autoregressive process. Our estimator is based on rank estimators of the large variance and auto-covariance matrices of a transformed latent high-dimensional Gaussian process. We derive rates of convergence of our estimator based on which we develop de-biased inference for Granger causality. Numerical results demonstrate the efficacy of the proposed methods. Although our focus is on the observable process, by the nature of rank estimators, all the methods developed directly apply to the transformed latent process. In technical terms, our analysis relies heavily on newly developed exponential inequalities for (degenerate) U-statistics under α-mixing condition.

Granger causality tests based on reduced variable information

Granger causality is a classical and important technique for measuring predictability from one group of time series to another by incorporating information of the variables described by a full vector autoregressive (VAR) process. However, in some applications economic forecasts need to be made based on information provided merely by a portion of variates (e.g., removal of a listed stock due to halting, suspension or delisting). This requires a new formulation of forecast based on an embedded subprocess of VAR, whose theoretical properties are often difficult to obtain. To avoid the issue of identifying the VAR subprocess, we propose a computation‐based approach so that sophisticated predictions can be made by utilizing a reduced variable information set estimated from sampled data. Such estimated information set allows us to develop a suitable statistical hypothesis testing procedure for characterizing all designated Granger causal relationships, as well as a useful graphical tool for presenting the causal structure over the prediction horizon. Finally, simulated data and a real example from the stock markets are used to illustrate the proposed method.

Estimation and asymptotics for vector autoregressive models with unit roots and Markov switching trends

We provide a formal definition of an M-state multivariate Markov switching trend, describe its asymptotic distribution, and consider vector autoregressive processes with trends which contain either unit roots or a stationary part. Then, we estimate the coefficients of such models via ordinary least squares , and determine the asymptotic distributions of estimators in terms of functionals on a multivariate Brownian motion.

Detecting Nonlinear Interactions in Complex Systems: Application in Financial Markets

Emerging or diminishing nonlinear interactions in the evolution of a complex system may signal a possible structural change in its underlying mechanism. This type of structural break may exist in many applications, such as in climate and finance, and standard methods for change-point detection may not be sensitive to it. In this article, we present a novel scheme for detecting structural breaks through the occurrence or vanishing of nonlinear causal relationships in a complex system. A significance resampling test was developed for the null hypothesis (H0) of no nonlinear causal relationships using (a) an appropriate Gaussian instantaneous transform and vector autoregressive (VAR) process to generate the resampled multivariate time series consistent with H0; (b) the modelfree Granger causality measure of partial mutual information from mixed embedding (PMIME) to estimate all causal relationships; and (c) a characteristic of the network formed by PMIME as test statistic. The significance test was applied to sliding windows on the observed multivariate time series, and the change from rejection to no-rejection of H0, or the opposite, signaled a non-trivial change of the underlying dynamics of the observed complex system. Different network indices that capture different characteristics of the PMIME networks were used as test statistics. The test was evaluated on multiple synthetic complex and chaotic systems, as well as on linear and nonlinear stochastic systems, demonstrating that the proposed methodology is capable of detecting nonlinear causality. Furthermore, the scheme was applied to different records of financial indices regarding the global financial crisis of 2008, the two commodity crises of 2014 and 2020, the Brexit referendum of 2016, and the outbreak of COVID-19, accurately identifying the structural breaks at the identified times.

Multi-period power utility optimization under stock return predictability

In this paper, we derive an analytical solution to the dynamic optimal portfolio choice problem in the case of an investor equipped with a power utility function of wealth. The results are established by solving the Bellman backward recursion under the assumption that the vector of asset returns follows a vector-autoregressive process with predictable variables. In an empirical study, the performance of the derived solution is compared with the one obtained by applying the numerical method. The comparison is performed in terms of the final wealth and its expected utility. It is documented that the application of the analytical solution to the multi-period portfolio choice problem leads to higher values of both the final wealth and the expected utility.

Analyzing the Stock Volatility Spillovers in Chinese Financial and Economic Sectors

By regarding the Chinese financial and economic sectors as a system, this article studies the stock volatility spillover in the system and explores its effects on the overall performance of the macroeconomy in China. The recent outbreak of COVID-19, U.S.–China trade friction, and three historical financial turbulences are involved to distinguish the changes in the spillover in these distinct crises, which has seldom been unveiled in the literature. By considering that the stock volatility spillover may vary over distinct timescales, the spillovers are disclosed through innovatively constructing the multi-scale spillover networks, followed by connectedness computation, based on variational mode decomposition (VMD) and generalized vector autoregression (GVAR) process. Our empirical analysis first demonstrates the different levels of increases in the total sectoral volatility spillover and changes in the roles of the sectors in the system under the aforementioned crises. Besides, the increases in the sectoral spillover in the long-term are verified to negatively impact the macroeconomy and can thereby act as warning signals.

From Multidimensional Ornstein - Uhlenbeck Process to Bayesian Vector Autoregressive Process

The main purpose of  this paper is to make the connexion between stochastic analysis, the Bayesian Statistics, and time series analysis for policy analysis. This approach solves the problem of mathematical modelling - the presence of uncertainties in the models and parameters -  that reduces the  policy analysis and forecasting   effectiveness. By using the multiple It\^o  integral, the multidimensional Ornstein - Uhlenbeck process can be written as a Vector Autoregressive with lag 1 (VAR(1)) that is the generalization of Vector Autoregressive process. The  limit of this approach is in fact it requires  the strong foundations of stochastic analysis, the Bayesian Statistics, and time series analysis.

Estimation of high-dimensional vector autoregression via sparse precision matrix

Summary We consider the problem of estimating sparse vector autoregression (VAR) via penalized precision matrices. This matrix is the output of the underlying directed acyclic graph of the VAR process, whose zero components correspond to the zero coefficients of the graphical representation of the VAR. The sparsity-based precision matrix estimator is deduced from the D-trace loss with convex and nonconvex penalty functions. We establish the consistency of the penalized estimator and provide the conditions for which all true zero entries of the precision matrix are actually estimated as zero with probability tending to one. The relevance of the method is supported by simulated experiments and a real data application.

Regularized Estimation in High-Dimensional Vector Auto-Regressive Models Using Spatio-Temporal Information

A Vector Auto-Regressive (VAR) model is commonly used to model multivariate time series, and there are many penalized methods to handle high dimensionality. However in terms of spatio-temporal data, most methods do not take the spatial and temporal structure of the data into consideration, which may lead to unreliable network detection and inaccurate forecasts. This paper proposes a data-driven weighted l1 regularized approach for spatio-temporal VAR model. Extensive simulation studies are carried out to compare the proposed method with four existing methods of high-dimensional VAR model, demonstrating improvements of our method over others in parameter estimation, network detection and out-of-sample forecasts. We also apply our method on a traffic data set to evaluate its performance in real application. In addition, we explore the theoretical properties of l1 regularized estimation of VAR model under the weakly sparse scenario, in which the exact sparsity can be viewed as a special case. To the best of our knowledge, this direction has not been considered yet in the literature. For general stationary VAR process, we derive the non-asymptotic upper bounds on l1 regularized estimation errors under the weakly sparse scenario, provide the conditions of estimation consistency, and further simplify these conditions for a special VAR(1) case.

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.