Stationarity Conditions Research Articles

Autoregressive moving average model for matrix time series

In the paper, the autoregressive moving average model for matrix time series (MARMA) is investigated. The properties of the MARMA model are investigated by using the conditional least square estimation, the conditional maximum likelihood estimation, the projection theorem in Hilbert space and the decomposition technique of time series, which include necessary and sufficient conditions for stationarity and invertibility, model parameter estimation, model testing and model forecasting.

How trade diversification affects resources sustainability in China: Exploring the role of institutional quality and environmental policies uncertainty

Natural resources are imperative to foster economic progression; however, unsustainable resource allocation elevated natural resource depletion and drastic environmental menaces. Besides, the worldwide COVID-19 pandemic has emerged as a worldwide uncertainty that dampens economic performance, whereas the economic recovery from the pandemic intensifies resource exploitation. Manifestly, China has attained tremendous economic progress in the last few decades, but unremitting and intensive resource consumption triggers the material footprint (MF) in China. Therefore, evaluating the mitigating factors to inhibit MF is essential. This study elucidates the impact of trade diversification (TD), institutional quality (IQ), ecological regulation (ER) and economic uncertainty (EPU) on MF in China from Q1-2000 to Q4-2021. The autoregressive distributed lag (ARDL) approach is employed to investigate the short and long-run model's relationship. The unit root testing affirms the stationarity conditions of variables at the first difference, while the bound ARDL co-integration test endorse the long-run co-integration association. The findings exhibit that TD, IQ and ER reduces MF, whereas the EPU significantly derives MF. Short-run parameters echoed identical outcomes with lower magnitudes, and the co-integrating term significantly converges the model to a long-run equilibrium position with 65.7% per annum. Overall the research offers stringent and resilient-ecological policies to diversify the trade basket and develop a sound institutional structure to decrease the uncertain global shocks and resource inefficiencies.

On Frèchet normal cone for nonsmooth mathematical programming problems with switching constraints

This paper is devoted to the study of a class of nonsmooth programming problems with switching constraints (abbreviated as, (NMPSC)), where all the involved functions in the switching constraints are assumed to be locally Lipschitz. We investigate the properties of Frèchet normal cone of (NMPSC). In particular, we introduce two Guignard type constraint qualifications for (NMPSC) in terms of Michel-Penot subdifferential. Moreover, we derive two estimates for the Frèchet normal cone of (NMPSC) and further establish stationarity conditions at an optimal solution for (NMPSC). To the best of our knowledge, this is for the first time Frèchet normal cone for (NMPSC) have been studied in the setting of Euclidean spaces.

Space-time variational material modeling: a new paradigm demonstrated for thermo-mechanically coupled wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage

We formulate variational material modeling in a space-time context. The starting point is the description of the space-time cylinder and the definition of a thermodynamically consistent Hamilton functional which accounts for all boundary conditions on the cylinder surface. From the mechanical perspective, the Hamilton principle then yields thermo-mechanically coupled models by evaluation of the stationarity conditions for all thermodynamic state variables which are displacements, internal variables, and temperature. Exemplary, we investigate in this contribution elastic wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage. Therein, one key novel aspect are initial and end time velocity conditions for the wave equation, replacing classical initial conditions for the displacements and the velocities. The motivation is intensively discussed and illustrated with the help of a prototype numerical simulation. From the mathematical perspective, the space-time formulations are formulated within suitable function spaces and convex sets. The unified presentation merges engineering and applied mathematics due to their mutual interactions. Specifically, the chosen models are of high interest in many state-of-the art developments in modeling and we show the impact of this holistic physical description on space-time Galerkin finite element discretization schemes. Finally, we study a specific discrete realization and show that the resulting system using initial and end time conditions is well-posed.

Estimation of an Order Book Dependent Hawkes Process for Large Datasets

Abstract A point process for event arrivals in high-frequency trading is presented. The intensity is the product of a Hawkes process and high-dimensional functions of covariates derived from the order book. Conditions for stationarity of the process are stated. An algorithm is presented to estimate the model even in the presence of billions of data points, possibly mapping covariates into a high-dimensional space. Large sample sizes can be common for high-frequency data applications using multiple instruments. Consistency results under weak conditions are established. A test statistic to assess out of sample performance of different model specifications is suggested. The methodology is applied to the study of four stocks that trade on the New York Stock Exchange. The out of sample testing procedure suggests that capturing the nonlinearity of the order book information adds value to the self-exciting nature of high-frequency trading events.

An Overview of Kriging and Cokriging Predictors for Functional Random Fields

This article presents an overview of methodologies for spatial prediction of functional data, focusing on both stationary and non-stationary conditions. A significant aspect of the functional random fields analysis is evaluating stationarity to characterize the stability of statistical properties across the spatial domain. The article explores methodologies from the literature, providing insights into the challenges and advancements in functional geostatistics. This work is relevant from theoretical and practical perspectives, offering an integrated view of methodologies tailored to the specific stationarity conditions of the functional processes under study. The practical implications of our work span across fields like environmental monitoring, geosciences, and biomedical research. This overview encourages advancements in functional geostatistics, paving the way for the development of innovative techniques for analyzing and predicting spatially correlated functional data. It lays the groundwork for future research, enhancing our understanding of spatial statistics and its applications.

Humbert generalized fractional differenced ARMA processes

In this article, we use the generating functions of the Humbert polynomials to define two types of Humbert generalized fractional differenced ARMA processes. We present stationarity and invertibility conditions for the introduced models. The singularities for the spectral densities of the introduced models are investigated. In particular, Pincherle ARMA, Horadam ARMA and Horadam–Pethe ARMA processes are studied. It is shown that the Pincherle ARMA process has long memory property for u=0. Additionally, we employ the Whittle quasi-likelihood technique to estimate the parameters of the introduced processes. Through this estimation method, we attain results regarding the consistency and normality of the parameter estimators. We also conduct a comprehensive simulation study to validate the performance of the estimation technique for Pincherle ARMA process. Moreover, we apply the Pincherle ARMA process to real-world data, specifically to Spain’s 10 years treasury bond yield data, to demonstrate its practical utility in capturing and forecasting market dynamics.

Changes in Thermal Conductivity of the Rocks of the West Siberian Basin Lithosphere in the Vicinity of the Tumenskaya SG-6 Well

Numerical reconstructions of the thermal regime of the lithosphere of the West Siberian basin in the Koltogor-Urengoi graben in the vicinity of the Tyumenskaya SG-6 superdeep well are used to analyze the depth distribution of thermal conductivity of basin rocks. Five depth intervals which differ in the character of changes in thermal conductivity of rocks are distinguished: the permafrost zone, the sedimentary section below this zone, the zone of anomalous rock weakening, the consolidated crust, and the mantle. The algorithms for calculating the thermal conductivity are considered and the main factors affecting its change with depth are determined for each of the five intervals. A sharp decrease in the thermal conductivity of rocks in the bottom part of the sedimentary cover and in the basement top in the vicinity of the SG-6 well is associated with rock weakening due to tectonic fracturing and hydrothermal erosion. The analysis suggests that the stationarity conditions of the process are not observed in optical scanning thermal conductivity measurements and, therefore, this method may overestimate the true thermal conductivity of rocks.

Mathematical properties of optimal fluxes in cellular reaction networks at balanced growth.

The physiology of biological cells evolved under physical and chemical constraints, such as mass conservation across the network of biochemical reactions, nonlinear reaction kinetics, and limits on cell density. For unicellular organisms, the fitness that governs this evolution is mainly determined by the balanced cellular growth rate. We previously introduced growth balance analysis (GBA) as a general framework to model and analyze such nonlinear systems, revealing important analytical properties of optimal balanced growth states. It has been shown that at optimality, only a minimal subset of reactions can have nonzero flux. However, no general principles have been established to determine if a specific reaction is active at optimality. Here, we extend the GBA framework to study the optimality of each biochemical reaction, and we identify the mathematical conditions determining whether a reaction is active or not at optimal growth in a given environment. We reformulate the mathematical problem in terms of a minimal number of dimensionless variables and use the Karush-Kuhn-Tucker (KKT) conditions to identify fundamental principles of optimal resource allocation in GBA models of any size and complexity. Our approach helps to identify from first principles the economic values of biochemical reactions, expressed as marginal changes in cellular growth rate; these economic values can be related to the costs and benefits of proteome allocation into the reactions' catalysts. Our formulation also generalizes the concepts of Metabolic Control Analysis to models of growing cells. We show how the extended GBA framework unifies and extends previous approaches of cellular modeling and analysis, putting forward a program to analyze cellular growth through the stationarity conditions of a Lagrangian function. GBA thereby provides a general theoretical toolbox for the study of fundamental mathematical properties of balanced cellular growth.

Mixed-frequency quantile regressions to forecast value-at-risk and expected shortfall.

Although quantile regression to calculate risk measures is widely established in the financial literature, when considering data observed at mixed-frequency, an extension is needed. In this paper, a model is built on a mixed-frequency quantile regressions to directly estimate the Value-at-Risk (VaR) and the Expected Shortfall (ES) measures. In particular, the low-frequency component incorporates information coming from variables observed at, typically, monthly or lower frequencies, while the high-frequency component can include a variety of daily variables, like market indices or realized volatility measures. The conditions for the weak stationarity of the daily return process are derived and the finite sample properties are investigated in an extensive Monte Carlo exercise. The validity of the proposed model is then explored through a real data application using two energy commodities, namely, Crude Oil and Gasoline futures. Results show that our model outperforms other competing specifications, on the basis of some popular VaR and ES backtesting test procedures.

On Periodic Generalized Poisson INAR(1) Model

In this paper, we introduce a first-order Periodic Generalized Poisson Integer-Valued Autoregressive model which has been shown to be useful to describe overdispersion, equidispersion and underdispersion feature encountered in periodically correlated Integer-Valued time series. Some probabilistic and statistical properties are established, such as the periodically correlated stationarity conditions, in the first and the second moments are provided and the closed-forms of these moments are, under these conditions, derived. Moreover, the structure of the periodic autocovariance is obtained. The estimation problem is addressed through the Yule-Walker the Two-Stage Conditional Least Squares and the Conditional Maximum Likelihood methods. The performance of these methods is done through an intensive simulation study and an application on real data set is accomplished. Keywords and phrases: Periodic Generalized Poisson, Integer-Valued Autoregressive, Periodically correlated process, periodically stationary condition.

Simulation of degradation of pasture ecosystems of arid zone

New approaches to the study of succession dynamics are based on the original concept of system dynamics, which was used in the development of simulation models reflecting the dynamics of pasture ecosystems. The temporal dynamics of the soil and vegetation cover of the natural pastures of the Black Lands of Kalmykia reflects the degradation processes that are studied by simulation modeling. The processes of pasture degradation in the subarid belt of Southern Russia were described by ordinary differential equations. This concept is the basic one in the development of simulation models. The dynamics of changes in effective coatings for phytocenoses of pastures of Black Lands in the process of degradation was traced from the original pastures to complete desertification. Analytical solutions for various conditions of succession dynamics in the degradation of natural pasture lands are a convenient test for assessing the adequacy of simulation modeling. The initial time periods of space observations of effective coatings of phytocenoses were used to parametrize simulation models. The models are implemented in the AnyLogic simulation system. The model development process is based on visual programming using the graphical user interface. Information flows in the Anylogic environment are redistributed over time through a system of storage devices interconnected by an oriented graph of the model. A computational experiment demonstrating simulation results with different parameters was used for an indicative (qualitative) assessment of ecosystem sustainability. The results of analytical modeling using ordinary differential equations and simulation modeling showed a high identity. The study of the dynamics of pasture ecosystems using a variation experiment demonstrates the possibility of visual (cognitive) selection of the parameters of the model that determines the conditions of stationarity of pasture ecosystems. Studying the dynamics of successions of pasture phytocnoses by this method opens up new opportunities for solving problems of optimal use of biological resources and sustainability of pasture ecosystems.

A robust score-driven filter for multivariate time series

A multivariate score-driven filter is developed to extract signals from noisy vector processes. By assuming that the conditional location vector from a multivariate Student’s t distribution changes over time, we construct a robust filter which is able to overcome several issues that naturally arise when modeling heavy-tailed phenomena and, more in general, vectors of dependent non-Gaussian time series. We derive conditions for stationarity and invertibility and estimate the unknown parameters by maximum likelihood. Strong consistency and asymptotic normality of the estimator are derived. Analytical formulae are derived which consent to develop estimation procedures based on a fast and reliable Fisher scoring method. An extensive Monte–Carlo study is designed to assess the finite samples properties of the estimator, the impact of initial conditions on the filtered sequence, the performance when some of the underlying assumptions are violated, such as symmetry of the underlying distribution and homogeneity of the degrees of freedom parameter across marginals. The theory is supported by a novel empirical illustration that shows how the model can be effectively applied to estimate consumer prices from home scanner data.

On the asymmetry in the volatility of financial time series: a buffered transition approach

ABSTRACT In this paper, we propose a new threshold model to deal with the asymmetry and leverage effect in the volatility of financial time series. Our model generalizes the classical threshold stochastic volatility model. By introducing a new flexible and smooth regime-switching mechanism, we avoid the sudden jump in the log-volatility imposed by the classical model. More precisely, we propose a new threshold model that we call Buffered Threshold Stochastic Volatility (BTSV) model. After the definition of our model, we derive a sufficient condition for the strict stationarity of the model. Then, we develop a Sequential Monte Carlo method to estimate the parameters. Next, we analyse the performance of the proposed method through a simulation study. Finally, we apply our new methodology to model the Honeywell International Inc and Standard and Poor Composite 500 stock market indices. This empirical analysis shows that BTSV model is a competitive alternative for the asymmetry modelling in financial returns analysis.

Reprint of: Generalized Autoregressive Conditional Heteroskedasticity

A natural generalization of the ARCH (Autoregressive Conditional Heteroskedastic) process introduced in Engle (1982) to allow for past conditional variances in the current conditional variance equation is proposed. Stationarity conditions and autocorrelation structure for this new class of parametric models are derived. Maximum likelihood estimation and testing are also considered. Finally an empirical example relating to the uncertainty of the inflation rate is presented.

Geometric ergodicity and conditional self‐weighted M‐estimator of a GRCAR(p) model with heavy‐tailed errors

We establish the geometric ergodicity for general stochastic functional autoregressive (linear and nonlinear) models with heavy‐tailed errors. The stationarity conditions for a generalized random coefficient autoregressive model (GRCAR()) are presented as a corollary. And then, a conditional self‐weighted M‐estimator for parameters in the GRCAR() is proposed. The asymptotic normality of this estimator is discussed by allowing infinite variance innovations. Simulation experiments are carried out to assess the finite‐sample performance of the proposed methodology and theory, and a real heavy‐tailed data example is given as illustration.

Rapid Suboptimal Low-Thrust Space Trajectory Design

The finite Fourier series shape-based approach for fast initial trajectory design satisfies the equations of motion and other problem constraints at discrete points by using a nonlinear programming solver to design the Fourier coefficients. In this paper, the finite Fourier series approach is extended to account for the necessary conditions for optimality during the Fourier coefficient design. The proposed method uses the approximate trajectories to analytically estimate the costates at these discrete points by employing a finite difference technique on the adjoint differential equations. Lagrange multipliers associated with any terminal state constraints are determined using an auxiliary function. Residual errors in the stationarity condition are reduced through the objective function, with the Legendre–Clebsch condition enforced as a nonlinear inequality constraint; applying these two conditions finds suboptimal solutions without an excessive loss of computational efficiency. Test cases use steering angle and thrust acceleration magnitude controls and span Earth-escape spirals, geostationary spacecraft rendezvous, and phasing maneuvers. The presented results demonstrate the proposed method’s ability to achieve trajectories closer to the optimal solution.

Necessary and sufficient conditions for AR vector processes to be stationary: Applications in information theory and in statistical signal processing

As the correlation matrices of stationary vector processes are block Toeplitz, autoregressive (AR) vector processes are non-stationary. However, in the literature, an AR vector process of finite order is said to be “stationary” if it satisfies the so-called stationarity condition (i.e., if the spectral radius of the associated companion matrix is less than one). Since the term “stationary” is used for such an AR vector process, its correlation matrices should “somehow approach” the correlation matrices of a stationary vector process, but the meaning of “somehow approach” has not been mathematically established in the literature. In the present paper we give necessary and sufficient conditions for AR vector processes to be “stationary”. These conditions show in which sense the correlation matrices of an AR “stationary” vector process asymptotically behave like block Toeplitz matrices. Applications in information theory and in statistical signal processing of these necessary and sufficient conditions are also given.

The elliptical Ornstein–Uhlenbeck process

We introduce the elliptical Ornstein-Uhlenbeck (OU) process, which is a generalisation of the well-known univariate OU process to bivariate time series. This process maps out elliptical stochastic oscillations over time in the complex plane, which are observed in many applications of coupled bivariate time series. The appeal of the model is that elliptical oscillations are generated using one simple first order SDE, whereas alternative models require more complicated vectorised or higher order SDE representations. The second useful feature is that parameter estimation can be performed robustly in the frequency domain using only the modelled and observed power spectral density, without having to model and compute cross spectra of individual time series components. We determine properties of the model including the conditions for stationarity, and the geometrical structure of the elliptical oscillations. We demonstrate the utility of the model by measuring periodic and elliptical properties of Earth's polar motion.

On Periodic Generalized Poisson INAR(p) Models

This paper deals with some probabilistic and statistical properties of periodic generalized Poisson integer-valued autoregressive processes of order p, Necessary and sufficient conditions for the periodic stationarity, both in mean and second order, are established. The closed-forms of the mean and the second moment are, under these conditions, obtained. Moreover, the Wold–Cramér expression of the underlying second-order periodically stationary process is then established. The autocovariance structure is studied, while providing the closed-form of the periodic autocorrelation function. The Yule–Walker (YW), the two-stage conditional least squares (CLS) and the conditional maximum likelihood (CML) estimation methods of the underlying parameters are obtained. An intensive simulation study and an application on a real count data consisting of the daily number of daytime road accidents in Schiphol area, in Netherlands are provided.