Vector Autoregressive Process Research Articles

Uniform inference for cointegrated vector autoregressive processes

Uniform inference for cointegrated vector autoregressive processes

Improved Sparse Mean Reverting Portfolio Selection Using Simulated Annealing and Extreme Learning Machine

We study the problem of selecting a sparse, mean reverting portfolio from a universe of assets using simulated annealing (SA). Assuming that assets follow a first order vector autoregressive process (VAR(1)), we make a number of improvements in existing methods. First, we extend the underlying asset dynamics to include a time-independent additive term, thereby enriching the model’s applicability. Second, we introduce Extreme Learning Machine (ELM) to decide whether to apply SA or settle for the much faster greedy solution. Finally, we improve the SA method by better calibration of the initial temperature and by determining the exact value of the weights within a selected dimension using the Rayleigh quotient. On real data, these changes result in more than 90% improvement in run time on average and 4.78% improvement in optimized mean reversion in our simulations. We also test the trading performance of our method on both simulated and real data and managed to achieve positive mean trading results in both cases.

Separating states in astronomical sources using hidden Markov models: with a case study of flaring and quiescence on EV Lac

ABSTRACT We present a new method to distinguish between different states (e.g. high and low, quiescent and flaring) in astronomical sources with count data. The method models the underlying physical process as latent variables following a continuous-space Markov chain that determines the expected Poisson counts in observed light curves in multiple passbands. For the underlying state process, we consider several autoregressive processes, yielding continuous-space hidden Markov models of varying complexity. Under these models, we can infer the state that the object is in at any given time. The continuous state predictions from these models are then dichotomized with the help of a finite mixture model to produce state classifications. We apply these techniques to X-ray data from the active dMe flare star EV Lac, splitting the data into quiescent and flaring states. We find that a first-order vector autoregressive process efficiently separates flaring from quiescence: flaring occurs over 30 per cent–40 per cent of the observation durations, a well-defined persistent quiescent state can be identified, and the flaring state is characterized by higher plasma temperatures and emission measures.

Vector-autoregressive based vibration monitoring of wind energy turbines using laser scanner profile measurements

Structural health monitoring of wind energy turbines (WET) is crucial due to the potential presence of an imbalanced rotor, leading to adverse centrifugal forces. In this research, non-contact vibration monitoring of a WET is conducted using a terrestrial laser scanner (TLS) of the type Zoller+Fröhlich IMAGER 5016 in a two-dimensional (2D) profile mode, and with a rotation speed of 55 revolutions per second. The TLS profile measurements cover the entire pillar up to a height of 100 meter. Therefore, the challenges imposed by conventional measurement systems, such as accelerometers or inductive displacement transducers, are tackled through the high spatial resolution of the TLS measurements and non-contact measurement methods. Additionally, both time and cost are reduced regarding the sensor installation. In general, the WET is measured in two different directions to account for significant movements, both in the direction of the wind and perpendicular to it. To initiate the analysis, time series are generated from the profile measurements for various positions covering the entire pillar. A robust time-domain modal parameter identification approach based on the Vector-autoregressive (VAR) process with multivariate t-distributed random deviations (VAR-RT-MPI) is proposed to estimate modal parameters, including eigenfrequencies, eigenforms, and modal damping. Besides, it allows estimating unknown auto- and cross-correlation coefficients of the VAR process, the cofactor matrix, and the degrees of freedom of the t-distribution. The VAR-RT-MPI algorithm enables to jointly estimate the eigenfrequencies and damping ratio coefficients considering multiple time series. Additionally, the spatio-temporal model of the pillar is characterised based on the estimates of the amplitudes (in a submillimetre range) and phase shifts at different positions. Vibration monitoring was conducted on a WET located in Hannover, Germany, and the results were compared with a well-known covariance driven stochastic subspace identification (SSI-COV) approach.

An efficient multivariate approach to dictionary learning for portfolio selection

Financial market noise greatly limits portfolio optimization by concealing key patterns and resulting in inaccurate asset selection. In this paper, we propose a novel approach that leverages dictionary learning, specifically the modified K-SVD algorithm, for denoising financial time series in the context of multivariate portfolio selection. Our method, which considers highly correlated short datasets, trains the dictionary simultaneously for multiple assets, resulting in more robust and adaptive denoising. We next use a vector auto-regressive process on the denoised data to estimate covariance matrices and build optimal portfolios using the minimum variance approach. Extensive computer simulations are conducted to assess the impact of our denoising method on portfolio performance in terms of several metrics, such as cumulative returns, Sharpe ratio, and model accuracy. The findings indicate how dictionary learning can improve the robustness of investment portfolios in the face of market noise and volatility.

Modelling common bubbles in cryptocurrency prices

Bubbles and spikes in cryptocurrency prices increase considerably the risk on investments in these assets. In the traditional time series literature bubbles are viewed as nonstationary and non-estimable components of a process. In this paper, we adopt a different approach and consider the bubbles as inherent features of a strictly stationary causal-noncausal (mixed) Vector Autoregressive (VAR) process. This approach allows us to model and estimate the common bubbles and spikes in cryptocurrency prices. It also provides us linear combinations of cryptocurrencies that eliminate common bubbles analogously to the cointegrating vectors eliminating common trends in unit root processes. They are used to build cryptocurrency portfolios immune to the risk of common bubbles that ensure stable investment strategies. The mixed VAR model is estimated from the US Dollar prices of Bitcoin, Ethereum, Ripple, and Stellar over the period 2017–2019. We document the common bubbles and illustrate the behaviour of bubble-free portfolios.

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 Pólya-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 l 1 -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.