On continuous-time threshold ARMA processes

We review some applications of continuous-time ARMA processes in the modelling of time series observed at discrete times t 1,…, t N . The problem of finding a continuous-time ARMA process in which a given discrete-time ARMA process can be “embedded” is discussed and some of the properties of a family of continuous-time analogues of Tong’s SETARMA models are considered. An approach to inference for such models is described and illustrated with reference to a variety of data sets.KeywordsAutocorrelation FunctionStochastic Differential EquationTime Series AnalysisARMA ModelTransition DistributionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A general approach for modeling the volatility process in continuous-time is based on the convolution of a kernel with a non-decreasing Levy process, which is non-negative if the kernel is non-negative. Within the framework of Continuous-time Auto-Regressive Moving-Average (CARMA) processes, we derive a necessary condition for the kernel to be non-negative, and propose a numerical method for checking the non-negativity of a kernel function. These results can be lifted to solving a similar problem with another approach to modeling volatility via the COntinuous-time Generalized Auto-Regressive Conditional Heteroscedastic (COGARCH) processes.

Continuous‐time autoregressive moving average (CARMA) processes have recently been used widely in the modelling of non‐uniformly spaced data and as a tool for dealing with high‐frequency data of the form ,n = 0, 1, 2,…, where Δ is small and positive. Such data occur in many fields of application, particularly in finance and in the study of turbulence. This article is concerned with the characteristics of the process , when Δ is small and the underlying continuous‐time process is a specified CARMA process.

Considering absolute log returns as a proxy for stochastic volatility, the influence of explanatory variables on absolute log returns of ultra high frequency data is analysed. The irregular time structure and time dependency of the data is captured by utilizing a continuous time ARMA(p,q) process. In particular, we propose a mixed effect model class for the absolute log returns. Explanatory variable information is used to model the fixed effects, whereas the error is decomposed in a non-negative Lévy driven continuous time ARMA(p,q) process and a market microstructure noise component. The parameters are estimated in a state space approach. In a small simulation study the performance of the estimators is investigated. We apply our model to IBM trade data and quantify the influence of bid-ask spread and duration on a daily basis. To verify the correlation in irregularly spaced data we use the variogram, known from spatial statistics. Copyright © 2006 John Wiley & Sons, Ltd.

In this paper we present methods for estimating the parameters of a class of non-Gaussian continuous-time stochastic process, the continuous-time auto regressive moving average (CARMA) model driven by symmetric alpha-stable (SalphaS) Levy processes. In this challenging framework we are not able to evaluate the likelihood function directly, and instead we use a distretized approximation to the likelihood. The parameters are then estimated from this approximating model using a Bayesian Monte Carlo scheme, and employing a Kalman filter to marginalize and sample the trajectory of the state process. An efficient exploration of the parameter space is achieved through a novel reparameterization in terms of an equivalent mechanical system. Simulations demonstrate the potential of the methods

The existing literature on forecasting time series data is primarily based on univariate analysis and techniques such as Univariate Autoregressive (UAR), Univariate Moving Average (UMA), Simple Exponential Smoothing (SES), deep learning models, and, most notably, univariate Long Short-Term Memory (LSTM) built based on univariate variable where the next lag of time series is leveraged for forecasting the next cycle of data. This paper takes this line of research to the next level by focusing on forecasting time series data based on “multivariate” modeling and analysis. To have a better insight of the performance of various deep learning-based models when multivariate analysis is performed, the paper builds and reports the forecasting accuracy for techniques such as the Transformer-based Multi-head Attention network, Long Short-Term Memory (LSTM), Bidirectional Long Short-Term Memory (BI-LSTM), Temporal Convolution Network (TCN), and conventional Vector Autoregressive (VAR) models. The findings revealed that the TCN model achieved the average lowest RMSE values of 0.0589 for stock data and 0.1554 for cryptocurrency data. Notably, the Multi-Head Attention model achieved average <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R^{2}$ </tex-math></inline-formula> values of 0.92 for stock data and −1.98 for cryptocurrency data with respect to five variables (i.e., open, high, low, close and volume). According to the empirical studies conducted and reported in this paper, the transformer-based Multi-head Attention network outperformed other models such as LSTM, BI-LSTM, and more importantly conventional Vector Auto-Regression Models (VAR) in stocks and cryptocurrencies time series data where several variables were leveraged in building these multivariate-based models.

The forecasting of time series data is an integral component for management, planning, and decision making. Following the Big Data trend, large amounts of time series data are available in many application domains. The highly dynamic and often noisy character of these domains in combination with the logistic problems of collecting data from a large number of data sources imposes new requirements on the forecast process. A constantly increasing number of time series has to be forecast over several periods in order to enable long-term planning with high accuracy and short execution time. This is almost impossible, when keeping the traditional focus on creating one forecast model for each individual time series. In addition, often used forecast techniques like ARIMA require complete historical data and fail if time series are intermittent. A method that addresses all these new requirements is the cross-sectional forecasting approach. It utilizes available data from many time series of the same domain in one single model; thus, missing values can be compensated and accurate forecast results are calculated quickly. However, this approach is limited by a rigid data selection and existing forecast methods show that adaptability of the model to the data increases the forecast accuracy. Therefore in this paper, we present CSAR, a model that extends the cross-sectional paradigm by adding more flexibility and allows fine-grained adaptations toward the analyzed data. In this way, we achieve an increased forecast accuracy and thus a wider applicability.

Abstract. Embedding a discrete‐time autoregressive moving average (DARMA) process in a continuous‐time ARMA (CARMA) process has been discussed by many authors. These authors have considered the relationship between the autocovariance structures of continuous‐time and related discrete‐time processes. In this article, we treat the problem from a slightly different point of view. We define embedding in a more rigid way by taking account of the probability structure. We consider Gaussian processes. First we summarize the necessary and sufficient condition for a DARMA process to be able to be embedded in a CARMA process. Secondly, we show a concrete condition such that a DARMA process can be embeddable in a CARMA process. This condition is new and general. Thirdly, we show some special cases including new examples. We show how we can examine embeddability for these special cases.

Continuous-time threshold autoregressive (CTAR) processes have been developed in the past few years for modelling non-linear time series observed at irregular intervals. Several approximating processes are given here which are useful for simulation and inference. Each of the approximating processes implicitly defines conditions on the thresholds, thus providing greater understanding of the way in which boundary conditions arise.

This paper examines the existence of limit cycles in Japanese macroeconomic variables using a threshold autoregressive (TAR) model. Recent business cycle theories are grouped into two main categories: (1) real business cycle and (2) endogenous business cycle. Real business cycle theory, which is modelled by an autoregressive (AR) model, has a linear dynamic system with consecutive exogenous shocks that cause cyclical deviations from a growth trend. On the other hand, endogenous business cycle theory, which includes limit cycles characterized by a TAR model, is based on a nonlinear dynamic system that has a mechanism that induces complicated economic fluctuations endogenously. Which theory is appropriate has significant implications for policy makers. Accordingly, it is necessary to investigate whether economic fluctuations depend on an endogenous mechanism or on exogenous shocks. To investigate this, we test for linearity in Japanese macroeconomic variables. The linearity test in the paper distinguishes the TAR model from the AR model. We find that most fluctuations have TAR processes. The results indicate that most Japanese macroeconomic data have limit cycles, and imply that the government could control the business cycle in Japan, even though the cyclical behaviour is extremely complicated.

We introduce a simple stochastic system able to generate anomalous diffusion for both position and velocity. The model represents a viable description of the Fermi's acceleration mechanism and it is amenable to analytical treatment through a linear Boltzmann equation. The asymptotic probability distribution functions for velocity and position are explicitly derived. The diffusion process is highly non-Gaussian and the time growth of moments is characterized by only two exponents nu(x) and nu(v). The diffusion process is anomalous (non-Gaussian) but with a defined scaling property, i.e., P(|r|,t)=1/t(nu(x))Fx(|r|/t(nu(x))) and similarly for velocity.

In this essay we review some of the emerging trends in modeling repeated measures data. Three general forms of longitudinal models are discussed: panel model designs, growth curve models, and intensive within-person assessments. Each section discusses design elements that should be considered when using each of these types of longitudinal models, and introduces some emerging trends. In the section on panel designs, continuous time models and planned missing data models are introduced; these ideas will revolutionize the modeling and collection of panel data. In the section on growth curve models, the necessity of separately evaluating mean and covariance model fit is discussed. This section also introduces methods being used to carefully consider the time of measurements in temporal designs. Finally, the budding analysis of intensive within individual observations is considered, including recent work from mathematics that limits the generalizability of interindividual studies to individual outcomes. Keywords: longitudinal; structural equation modeling; latent variable modeling; cross-lagged and auto-regressive panel modeling; growth curve modeling; continuous time modeling; dynamical systems modeling

USING CONTINUOUS-TIME MODELING FOR ERRORS-IN-VARIABLES IDENTIFICATION

Semiparametric estimation for partially linear models with [formula omitted]-weak dependent errors

We calculated the probability distribution function (PDF) from simulations for an initially spherical wave propagated through homogeneous atmospheric turbulence. The onset of strong scintillation was calculated. By onset of strong scintillation, we mean conditions of weak scintillation changing to the condition of strong focusing. In addition, one case in the saturation regime was calculated. The simulations' PDF's are compared with several heuristic models of the PDF and are seen to progress from close to log normal for the cases of weakest scintillation to close to the log normally modulated exponential PDF (LNME PDF) for the cases of strong scintillation. The simulations' PDF's are not in agreement with the K PDF for any of the calculated cases. The best agreement was obtained in comparison with Beckmann's PDF [P. Beckmann, Probability in Communication Engineering (Harcourt, Brace, & World, 1967)]. Beckmann's PDF varies from being the log-normal PDF for weak scintillation to being the LNME PDF for strong scintillation and progresses further to the theoretically expected exponential PDF in the limit of saturated scintillation. We recommend that simulation be used to predict the irradiance PDF for plane and diverged waves in homogeneous turbulence in preference to using heuristic models. More complicated propagation cases remain in the domain of heuristic argumentation.