ARMA Processes Research Articles

Jensen-autocorrelation function for weakly stationary processes and applications

The Jensen-variance (JV) information based on Jensen’s inequality and variance has been previously proposed to measure the distance between two random variables. Based on the relationship between JV distance and autocorrelation function of two weakly stationary process, the Jensen-autocovariance and Jensen-autocorrelation functions are proposed in this paper. Furthermore, the distance between two different weakly stationary processes is measured by the Jensen-cross-correlation function. Moreover, autocorrelation function is also considered for ARMA and ARFIMA processes, deriving explicit formulas for Jensen-autocorrelation function that only depends on model parametric space and lag, whose were also illustrated by numeric results. In order to study the usefulness of proposed functions, two real-life applications were considered: the Tree Ring and Southern Humboldt current ecosystem time series.

Mixed orthogonality graphs for continuous‐time state space models and orthogonal projections

In this article, we derive (local) orthogonality graphs for the popular continuous‐time state space models, including in particular multivariate continuous‐time ARMA (MCARMA) processes. In these (local) orthogonality graphs, vertices represent the components of the process, directed edges between the vertices indicate causal influences and undirected edges indicate contemporaneous correlations between the component processes. We present sufficient criteria for state space models to satisfy the assumptions of Fasen‐Hartmann and Schenk (2024a) so that the (local) orthogonality graphs are well‐defined and various Markov properties hold. Both directed and undirected edges in these graphs are characterised by orthogonal projections on well‐defined linear spaces. To compute these orthogonal projections, we use the unique controller canonical form of a state space model, which exists under mild assumptions, to recover the input process from the output process. We are then able to derive some alternative representations of the output process and its highest derivative. Finally, we apply these representations to calculate the necessary orthogonal projections, which culminate in the characterisations of the edges in the (local) orthogonality graph. These characterisations are given by the parameters of the controller canonical form and the covariance matrix of the driving Lévy process.

Rational Expectations Models with Higher-Order Beliefs

Abstract We develop a method of solving rational expectations models with dispersed information and dynamic strategic complementarities. In these types of models, the equilibrium outcome hinges on an infinite number of higher-order expectations which require an increasing number of state variables to keep track of. Despite this complication, we prove that the equilibrium outcome always admits a finite-state representation when the signals follow finite ARMA processes. We also show that such a finite-state result may not hold with endogenous information aggregation. We further illustrate how to use the method to derive comparative statics, characterize equilibrium outcomes in HANK-type network games, reconcile with empirical evidence on expectations, and integrate incomplete information with bounded rationality in general equilibrium.

A robust M-estimator for Gaussian ARMA time series based on the Whittle approximation

In the frequency domain, the Whittle approach using the classical periodogram is traditionally used to estimate the parameters of a stationary time series. The M-periodogram has recently become an alternative tool for analyzing the dependence of time series. It is particularly useful for dealing with outliers and heavy-tailed noise. This paper proposes an alternative to the standard Whittle approach, the M-Whittle estimator, built from the M-periodogram. We show that the proposed method is a consistent estimator of the true parameters of an ARMA process. A finite sample investigation is carried out to assess the performance of the estimator in the scenarios of contaminated and uncontaminated time series. As expected, for the uncontaminated data, the M-Whittle estimator performs similarly to the classical Whittle approach. However, the superiority of the first method is clear in terms of root mean squared error when the series has additive outliers. Two applications are considered to illustrate the methodologies in real data contexts. Regardless of whether or not the data are contaminated by additive outliers, the results presented here strongly motivate using the M-Whittle estimator in practical problems.

Average Run Length Computations of Autoregressive and Moving Average Process using the Extended EWMA Procedure

In the past, the control chart served as a statistical tool for detecting process changes. The Exponentially Weighted Moving Average (EWMA) control chart is highly effective for detecting small changes. This research introduces the Extended Exponentially Weighted Moving Average (Extended EWMA) control chart for the Autoregressive and Moving average process with order p = 1 and q = 1 (ARMA(1,1)) The Extended EWMA control chart incorporates two smoothing parameters ( λ1 and λ2 ) derived from the EWMA control chart. A comparative analysis of the performance of the EWMA control chart. The Average Run Length (ARL) value as determined by the explicit formulas in this research, serves as a metric for evaluating the performance of the Extended EWMA control chart and the EWMA control chart. The Numerical Integral Equation (NIE) method is used to verify the accuracy of the ARL for the explicit formulas of the two control charts which has not been before discovered. The effectiveness of control charts can also be evaluated by analyzing SDRL, ARL, MRL, RMI, AEQL, and PCI values as metrics for various design parameter values. After analyzing the results of the ARL and all five performance meters, it was determined that the Extended EWMA control chart is better than the EWMA control chart at all shift sizes of process changes. Finally, the assessment of the ARMA process is being conducted to evaluate the ARL using a dataset on PM2.5 dust levels in Bangkok, Thailand during January and February of 2024.

Efficient Monitoring of Autoregressive and Moving Average Process using HWMA Control Chart

Quality control is an essential process for manufacturing and industry because it enhances product quality, consumer satisfaction, and overall profitability. Among many other statistical process control tools, quality practitioners typically employ control charts to monitor the industrial process and detect production changes. Control charts are widely used to detect flaws in many applications, such as distributed circuits and systems, electronic devices, and systems and signals. In this study, we derived an explicit formula for Average Run Length (ARL) of the Homogenously Weighted Moving Average control chart (HWMA) under the ARMA (p,q) process. The accuracy was checked using the numerical integral equation (NIE) technique. The finding showed that the explicit formulas and numerical solutions presented an outstanding level of agreement. However, the computational time for the explicit formulas was approximately one second, which was less than that required for the NIE. Moreover, the performance efficiency of the HWMA control chart is compared with the cumulative sum control chart for ARMA (p, q) processes including ARMA (2,1), ARMA (2,3), and ARMA (1,1) processes. The results found that the HWMA control chart performance is found to be preferable to the CUSUM control chart performance. Additionally, the explicit formula of the HWMA control chart was implemented in a practical application of the count of nonconformities in printed circuit boards (PCBs).

An abelian way approach to study random extended intervals and their ARMA processes

<abstract><p>An extended interval is a range $ A = [\underline{A}, \overline{A}] $ where $ \underline{A} $ may be bigger than $ \overline{A} $. This is not really natural, but is what has been used as the definition of an extended interval so far. In the present work we introduce a new, natural, and very intuitive way to see an extended interval. From now on, an extended interval is a subset of the Cartesian product $ {\mathbb R}\times {\mathbb Z}_2 $, where $ {\mathbb Z}_2 = \{0, 1\} $ is the set of directions; the direction $ 0 $ is for increasing intervals, and the direction $ 1 $ for decreasing ones. For instance, $ [3, 6]\times\{1\} $ is the decreasing version of $ [6, 3] $. Thereafter, we introduce on the set of extended intervals a family of metrics $ d_\gamma $, depending on a function $ \gamma(t) $, and show that there exists a unique metric $ d_\gamma $ for which $ \gamma(t)dt $ is what we have called an "adapted measure". This unique metric has very good properties, is simple to compute, and has been implemented in the software $ R $. Furthermore, we use this metric to {define variability for random extended intervals. We further study extended interval-valued ARMA} time series and prove the Wold decomposition theorem for stationary extended interval-valued times series.</p></abstract>

Learning Graph ARMA Processes From Time-Vertex Spectra

Learning Graph ARMA Processes From Time-Vertex Spectra

Estimation of prediction error in time series

Summary The accurate estimation of prediction errors in time series is an important problem, which has immediate implications for the accuracy of prediction intervals as well as the quality of a number of widely used time series model selection criteria such as the Akaike information criterion. Except for simple cases, however, it is difficult or even impossible to obtain exact analytical expressions for one-step and multi-step predictions. This may be one of the reasons that, unlike in the independent case (see Efron, 2004), up to now there has been no fully established methodology for time series prediction error estimation. Starting from an approximation to the bias-variance decomposition of the squared prediction error, a method for accurate estimation of prediction errors in both univariate and multivariate stationary time series is developed in this article. In particular, several estimates are derived for a general class of predictors that includes most of the popular linear, nonlinear, parametric and nonparametric time series models used in practice, with causal invertible autoregressive moving average and nonparametric autoregressive processes discussed as lead examples. Simulations demonstrate that the proposed estimators perform quite well in finite samples. The estimates may also be used for model selection when the purpose of modelling is prediction.

Efficient nonparametric estimation of generalised autocovariances

This paper provides a necessary and sufficient condition for asymptotic efficiency of a nonparametric estimator of the generalised autocovariance function of a stationary random process. The generalised autocovariance function is the inverse Fourier transform of a power transformation of the spectral density and encompasses the traditional and inverse autocovariance functions as particular cases. A nonparametric estimator is based on the inverse discrete Fourier transform of the power transformation of the pooled periodogram. We consider two cases: the fixed bandwidth design and the adaptive bandwidth design. The general result on the asymptotic efficiency, established for linear processes, is then applied to the class of stationary ARMA processes and its implications are discussed. Finally, we illustrate that for a class of contrast functionals and spectral densities, the minimum contrast estimator of the spectral density satisfies a Yule–Walker system of equations in the generalised autocovariance estimator.

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.

Scaled Muth–ARMA Process Applied to Finance Market

The analysis of financial market time series is an important source for understanding the economic reality of a country. We introduce a new autoregressive moving average (ARMA) process, the sMuth–ARMA model, which has the sMuth law as the marginal distribution and has one of its parameters as a proportion that can control amodal and unimodal behavior. We propose a procedure for obtaining the maximum likelihood estimators for its parameters and evaluate its performance for various link functions through Monte Carlo simulations. This research also addresses the issue of fluctuations in cryptocurrencies, which has played an increasingly important role in the global economy. An application to the range-based volatility of Tether (USDT) stablecoin prices shows the usefulness of the application of the proposed model over the Gaussian and other models reviewed.

A two-step estimation procedure for locally stationary ARMA processes with tempered stable innovations

A two-step estimation procedure for locally stationary ARMA processes with tempered stable innovations

Count time series with excess zeros: A Bayesian perspective using zero-adjusted distributions

Models for count data which are temporally correlated have been studied using many conditional distributions, such as the Poisson distribution, and the insertion of different dependence structures. Nonetheless, excess of zeros and over dispersion may be observed during the counting process and need to be considered when modelling and choosing a conditional distribution. In this paper, we propose models for counting time series using zero-adjusted distributions by inserting a dependence structure following the ARMA(p, q) process on a Bayesian framework. We perform a simulation study using the proposed Bayesian analysis and analyse the monthly time series of the number of deaths due to dengue haemorrhagic fever (ICD-A91) in Brazil.

Empirical spectral processes for stationary state space models

In this paper, we consider function-indexed normalized weighted integrated periodograms for equidistantly sampled multivariate continuous-time state space models which are multivariate continuous-time ARMA processes. Thereby, the sampling distance is fixed and the driving Lévy process has at least a finite fourth moment. Under different assumptions on the function space and the moments of the driving Lévy process we derive a central limit theorem for the function-indexed normalized weighted integrated periodogram. Either the assumption on the function space or the assumption on the existence of moments of the Lévy process is weaker. Furthermore, we show the weak convergence in both the space of continuous functions and in the dual space to a Gaussian process and give an explicit representation of the covariance function. The results can be used to derive the asymptotic behavior of the Whittle estimator and to construct goodness-of-fit test statistics as the Grenander–Rosenblatt statistic and the Cramér–von Mises statistic. We present the exact limit distributions of both statistics and show their performance through a simulation study.

Control charts for measurement error models

We consider a linear measurement error model (MEM) with AR(1) process in the state equation which is widely used in applied research. This MEM could be equivalently re-written as ARMA(1,1) process, where the MA(1) parameter is related to the variance of measurement errors. As the MA(1) parameter is of essential importance for these linear MEMs, it is of much relevance to provide instruments for online monitoring in order to detect its possible changes. In this paper we develop control charts for online detection of such changes, i.e., from AR(1) to ARMA(1,1) and vice versa, as soon as they occur. For this purpose, we elaborate on both cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts and investigate their performance in a Monte Carlo simulation study. The empirical illustration of our approach is conducted based on time series of daily realized volatilities.

Explicit formulas for the inverses of Toeplitz matrices, with applications

We derive novel explicit formulas for the inverses of truncated block Toeplitz matrices that correspond to a multivariate minimal stationary process. The main ingredients of the formulas are the Fourier coefficients of the phase function attached to the spectral density of the process. The derivation of the formulas is based on a recently developed finite prediction theory applied to the dual process of the stationary process. We illustrate the usefulness of the formulas by two applications. The first one is a strong convergence result for solutions of general block Toeplitz systems for a multivariate short-memory process. The second application is closed-form formulas for the inverses of truncated block Toeplitz matrices corresponding to a multivariate ARMA process. The significance of the latter is that they provide us with a linear-time algorithm to compute the solutions of corresponding block Toeplitz systems.

Putting the Significance of Spectral Peaks on the Level: Implications for the 1470-Yr Peak in Greenland δ18O

Abstract Spectral analysis of the Greenland Ice Sheet Project 2 (GISP2) δ18O record has been interpreted to show a 1/(1470 yr) spectral peak that is highly statistically significant (p < 0.01). The presence of such a peak, if accurate, provides an important clue about the mechanisms controlling glacial climate. As is standard, however, statistical significance was judged relative to a null model, H0, consisting of an autoregressive order one process, AR(1). In this study, H0 is generalized using an autoregressive moving-average process, ARMA(p, q). A rule of thumb is proposed for evaluating the adequacy of H0 involving comparing the expected and observed variances of the logarithm of a spectral estimate, which are generally consistent insomuch as removal of the ARMA structure from a time series results in an approximately level spectral estimate. An AR(1), or ARMA(1, 0), process is shown to be an inadequate representation of the GISP2 δ18O structure, whereas higher-order ARMA processes result in approximately level spectral estimates. After suitably leveling GISP2 δ18O and accounting for multiple hypothesis testing, multitaper spectral estimation indicates that the 1/(1470 yr) peak is insignificant. The seeming prominence of the 1/(1470 yr) peak is explained as the result of evaluating a spectrum involving higher-order ARMA structure and the peak having been selected on the basis of its seeming anomalous. The proposed technique for evaluating the significance of spectral peaks is also applicable to other geophysical records. Significance Statement A suitable null hypothesis is necessary for obtaining accurate test results, but a means for evaluating the adequacy of a null hypothesis for a spectral peak has been lacking. A generalized null model is presented in the form of an autoregressive, moving-average process whose adequacy can be gauged by comparing the observed and expected variance of log spectral density. Application of the method to the GISP2 δ18O record indicates that spectral structure found at 1/(1470 yr) is statistically insignificant.

Time series with infinite-order partial copula dependence

Abstract Stationary and ergodic time series can be constructed using an s-vine decomposition based on sets of bivariate copula functions. The extension of such processes to infinite copula sequences is considered and shown to yield a rich class of models that generalizes Gaussian ARMA and ARFIMA processes to allow both non-Gaussian marginal behaviour and a non-Gaussian description of the serial partial dependence structure. Extensions of classical causal and invertible representations of linear processes to general s-vine processes are proposed and investigated. A practical and parsimonious method for parameterizing s-vine processes using the Kendall partial autocorrelation function is developed. The potential of the resulting models to give improved statistical fits in many applications is indicated with an example using macroeconomic data.

Recursive and Non-Recursive Generalized Least-Squares Methods for Estimation of Time Series Models with Exogenous Variables

We present in this paper two generalized least-squares (GLS) methods for estimating regression coefficients of time series models with exogenous variables. The non-recursive GLS method is a generalization of the GLS method suggested by Cochrane and Orcutt (1949). The proposed GLS method consists of a sequence of four linear regressions. A first regression is fitted and provides residuals. These residuals are modeled as an autoregressive process and are used in a second regression (or autoregression) for obtaining estimators of autoregressive coefficients. These estimators are used to generate transformed endogenous and exogenous variables. A third regression makes use of the lagged values of these transformed variables to estimate the regression coefficients. The estimators of the regression coefficients are used to determine the true residuals which are modeled as an ARMA process which is finally used for obtaining the estimators of autoregressive and moving average parameters. The second GLS method is a recursive version of the first GLS method where the estimators are updated at each time point on receipt of the additional observations. The Simulation results based on different model structures with varying numbers of observations are used to compare the performance of our methods with that of exact maximum likelihood (EML) estimates.