Vector Autoregressive Moving Average Research Articles

Design of an improved process optimization model for enhancing the efficiency of the wastewater treatment process

Abstract Protecting the environment and ensuring the availability of potable water requires efficient wastewater treatment. This paper investigates the need for an advanced process optimization model to enhance the efficacy of wastewater treatment processes. Existing optimization models for wastewater treatment are frequently incapable of effectively analyzing historical data, optimizing dosages, or predicting optimal process parameters. To circumvent these restrictions, a novel method for optimizing various aspects of the treatment procedure using auto encoders (AE), genetic algorithm (GA), and vector autoregressive moving average (VARMA) models is proposed. Extensive experimentation with multiple datasets and samples, including the Melbourne Wastewater Treatment Dataset, Urban Wastewater Treatment, and Global Wastewater Treatment, demonstrates significant improvements in our proposed model. Compared to recently proposed models, our method results in an average improvement of 8.5% in treatment quality, a 4.9% reduction in delay, and a 9.5% increase in water purity. Beyond the datasets mentioned, our model's applications and use cases provide a valuable framework for optimizing wastewater treatment in a variety of settings. The adaptability and effectiveness of the proposed model make it suitable for both small and large treatment plants.

An Interpretable and Efficient Infinite-Order Vector Autoregressive Model for High-Dimensional Time Series

As a special infinite-order vector autoregressive (VAR) model, the vector autoregressive moving average (VARMA) model can capture much richer temporal patterns than the widely used finite-order VAR model. However, its practicality has long been hindered by its non-identifiability, computational intractability, and difficulty of interpretation, especially for high-dimensional time series. This article proposes a novel sparse infinite-order VAR model for high-dimensional time series, which avoids all above drawbacks while inheriting essential temporal patterns of the VARMA model. As another attractive feature, the temporal and cross-sectional structures of the VARMA-type dynamics captured by this model can be interpreted separately, since they are characterized by different sets of parameters. This separation naturally motivates the sparsity assumption on the parameters determining the cross-sectional dependence. As a result, greater statistical efficiency and interpretability can be achieved with little loss of temporal information. We introduce two l 1 -regularized estimation methods for the proposed model, which can be efficiently implemented via block coordinate descent algorithms, and derive the corresponding nonasymptotic error bounds. A consistent model order selection method based on the Bayesian information criteria is also developed. The merit of the proposed approach is supported by simulation studies and a real-world macroeconomic data analysis. Supplementary materials for this article are available online including a standardized description of the materials available for reproducing the work.

Modeling and Analysis Data Production of Oil, and Oil and Gas in Indonesia by Using Threshold Vector Error Correction Model

Data in the fields of finance, business, economics, agriculture, the environment and weather are commonly in the form of time series data. To analyze time series data that involves more than one variable (multivariate), vector autoregressive (VAR) models, vector autoregressive moving average (VARMA) models are generally used. If the variables discussed have cointegration, then the VAR model is modified into a vector error correction model (VECM). The relationship between short-term dynamics and deviation in the VECM model is assumed to be linear. If there is a nonlinear relationship between short-term dynamics and deviation, then a threshold vector error correction model (TVECM) can be used. The variables used in this research consist of oil production and Indonesian oil and gas production from January 2019 to March 2021. The research results show that the best model for data on oil production and oil and gas production is the TVECM 2 Regime model. Based on the TVECM 2 Regime model, further analysis, namely Granger causality and Impulse Response Function are discussed.

EPCMSDB: Design of an ensemble predictive control model for solar PV MPPT deployments via dual bioinspired optimizations

With the increasing demand for renewable energy, solar power has emerged as a promising option for sustainable power generation. However, the effectiveness and efficiency of solar power systems rely on the ability to effectively manage their performance, making it essential to develop efficient control models. This paper proposes a novel ensemble predictive control model for solar deployments using bio-inspired optimizations to improve load-connected solar deployments’ performance. The proposed model integrates multiple control devices, including Maximum Power Point Tracker, Proportional-Integral-Derivative, Proportional-Integral, and Fuzzy Logic Controllers, to selectively control the solar Photovoltaic systems. The proposed model incorporates a predictive control operation utilizing an LSTM-GRU (Long Short-Term Memory-Gated Recurrent Unit) with the VARMA (Vector Auto-Regressive Moving Average) model, which can accurately predict the future power generation of the solar system. This feature can facilitate efficient energy management and increase the system’s performance for different use cases. Implement a SEPIC (Single Ended Primary Inductor Capacitor) converter design to improve the system’s overall efficiency levels. To validate the effectiveness of the proposed approach, the author conducted experiments using real-world data and compared the proposed results with other control strategies. The results demonstrate that the ensemble predictive control model based on bio-inspired optimizations outperforms the existing control models regarding accuracy, efficiency, and stability levels. The proposed model has the potential to significantly improve the performance of load-connected solar deployments, offering a more practical approach to solar power generation. The combination of predictive control operations with bio-inspired optimizations can facilitate the design of sustainable energy systems with higher efficiency and accuracy.

An optimized hybrid methodology for short‐term traffic forecasting in telecommunication networks

AbstractWith the rapid development of telecommunication networks, the predictability of network traffic is of significant interest in network analysis and optimization, bandwidth allocation, and load balancing adjustment. Consequently, in recent years, significant research attention has been paid to forecasting telecommunication network traffic. Telecommunication traffic forecasting problems can be considered a time‐series problem, wherein periodic historical data is fed as the input to a model. Time‐series forecasting approaches are broadly categorized as statistical machine learning (ML) methods and their combinations. Statistical approaches forecast linear characteristics of time‐series data, unable to capture nonlinear and complex patterns. ML‐based approaches can model nonlinear characteristics of data. In recent years, hybrid approaches combining statistical and ML‐based approaches have been widely used to model linear and nonlinear data characteristics. However, the performance of these approaches highly depends on feature selection techniques and hyper‐parameter tuning of ML methods. A novel hybrid method is proposed for short‐term traffic forecasting based on feature selection and hyperparameter optimization to address this problem. It combines statistical and ML methods to model linear and nonlinear components of data. First, a novel feature selection technique, modified mutual information based on a linear combination of targets, is proposed to find the candidate input variables. Next, a combination of vector auto regressive moving average (VARMA), long short‐term memory (LSTM), and multilayer perceptron (MLP), called VARMA‐LSTM‐MLP forecaster, is suggested to forecast short‐term traffic. A hybrid metaheuristic algorithm, composed of firefly and BAT, is employed to find the optimal set of hyper‐parameter values. The proposed method is assessed by a real‐world dataset containing Tehran city's daily telecommunication data in IRAN. The evaluation results demonstrate that the proposed method outperforms the existing methods in terms of mean squared error and mean absolute error.

Fault Detection and Localization in Industrial IoT Systems using Deep Learning

In the ever-evolving landscape of Industrial Internet of Things (IoT) systems, the need for robust fault detection and localization has become paramount. Existing methods, while commendable, have faced certain limitations that necessitate the development of more sophisticated solutions. This work embarks on a journey to address these challenges and presents a novel approach to fault detection and localization, leveraging the formidable power of Deep Learning process. The limitations of existing methodologies primarily revolve around their inability to provide the required precision and accuracy in detecting and localizing faults within complex IoT deployments. These methods often struggle to handle the diverse contextual datasets that characterize modern industrial settings. Additionally, their computational efficiency often leaves much to be desired, hindering real-time fault response mechanisms. In response to these limitations, our proposed model introduces a hybrid framework. It combines Vector AutoRegressive Moving Average with Exogenous Variables (VARMAx) for precise fault localization and Convolutional Neural Networks (CNN) for robust fault detection. This fusion harnesses the strengths of both approaches, providing a comprehensive solution for fault management in industrial IoT systems. The advantages of this model are twofold. First, VARMAx offers exceptional accuracy in pinpointing the exact location of faults within the system, facilitating swift corrective actions. Second, the integration of CNN enhances fault detection by effectively capturing patterns and anomalies in the data, resulting in a highly responsive fault detection system. Notably, this model not only outperforms existing methods with a 3.9% boost in precision, 4.5% increase in accuracy, 4.9% higher recall, 8.5% improved AUC, but also boasts a remarkable 5.9% enhancement in computational speed, ensuring real-time responsiveness. The profound impact of this work extends to the realms of industrial automation and IoT system reliability. By addressing the limitations of existing approaches and introducing a robust fault detection and localization model, this paper paves the way for safer and more efficient industrial operations. The improved precision, accuracy, and speed of fault detection will minimize downtime, reduce maintenance costs, and ultimately elevate the performance and reliability of industrial IoT systems to new heights.

VARMA model parameterization using MLLE approach for intraday wind power forecasting application

AbstractAccurate intraday wind power forecasting (WPF) is required by system operators to mitigate challenges arising from small‐scale grid integration of intermittent wind power. This low‐level integration of wind energy into the system urges quality deterministic forecast. Multivariate statistical models considering spatiotemporal correlations are commonly used for such WPF. However, prudent parameterization is a necessity to improve deterministic forecasts even in smaller datasets. This article proposes a computationally efficient vector autoregressive moving average (VARMA) based intraday WPF model that uses the novel maximum log‐likelihood estimation (MLLE) approach to estimate the best‐fit model parameters. MLLE approach reduces the spatiotemporal fitting error by approximately 15% with computational time deduction by more than 25%. Furthermore, an advanced accuracy in parameter estimation (AIPE) technique supports the MLLE approach to obtain the optimal training window length for computing distribution parameters with narrow confidence intervals. Proposed centralized forecast model is implemented on a group of Wind Farms (WFs) in SE Australia. Pre‐processing of wind power generation data is ensured using the Yeo–Johnson transformation and stationarity concept. Proposed model is accurate and computationally efficient as compared to benchmark models in terms of accuracy and computational time. An appreciated forecast quality is achieved using the proposed model having only 0.20 MW and 0.29 MW mean normalized RMSE for 3 and 10 WFs studies, respectively. Possible applicability of the proposed model includes an overall improvement in intraday decision‐making of wind energy producers and system operators.

Information-Criterion-Based Lag Length Selection in Vector Autoregressive Approximations for I(2) Processes

When using vector autoregressive (VAR) models for approximating time series, a key step is the selection of the lag length. Often this is performed using information criteria, even if a theoretical justification is lacking in some cases. For stationary processes, the asymptotic properties of the corresponding estimators are well documented in great generality in the book Hannan and Deistler (1988). If the data-generating process is not a finite-order VAR, the selected lag length typically tends to infinity as a function of the sample size. For invertible vector autoregressive moving average (VARMA) processes, this typically happens roughly proportional to logT. The same approach for lag length selection is also followed in practice for more general processes, for example, unit root processes. In the I(1) case, the literature suggests that the behavior is analogous to the stationary case. For I(2) processes, no such results are currently known. This note closes this gap, concluding that information-criteria-based lag length selection for I(2) processes indeed shows similar properties to in the stationary case.

Applying the Spatial Transmission Network to the Forecast of Infectious Diseases Across Multiple Regions.

ObjectiveTimely and accurate forecast of infectious diseases is essential for achieving precise prevention and control. A good forecasting method of infectious diseases should have the advantages of interpretability, feasibility, and forecasting performance. Since previous research had illustrated that the spatial transmission network (STN) showed good interpretability and feasibility, this study further explored its forecasting performance for infectious diseases across multiple regions. Meanwhile, this study also showed whether the STN could overcome the challenges of model rationality and practical needs.MethodsThe construction of the STN framework involved three major steps: the spatial kluster analysis by tree edge removal (SKATER) algorithm, structure learning by dynamic Bayesian network (DBN), and parameter learning by the vector autoregressive moving average (VARMA) model. Then, we evaluated the forecasting performance of STN by comparing its accuracy with that of the mechanism models like susceptible-exposed-infectious-recovered-susceptible (SEIRS) and machine-learning algorithm like long-short-term memory (LSTM). At the same time, we assessed the robustness of forecasting performance of STN in high and low incidence seasons. The influenza-like illness (ILI) data in the Sichuan Province of China from 2010 to 2017 were used as an example for illustration.ResultsThe STN model revealed that ILI was likely to spread among multiple cities in Sichuan during the study period. During the whole study period, the forecasting accuracy of the STN (mean absolute percentage error [MAPE] = 31.134) was significantly better than that of the LSTM (MAPE = 41.657) and the SEIRS (MAPE = 62.039). In addition, the forecasting performance of STN was also superior to those of the other two methods in either the high incidence season (MAPE = 24.742) or the low incidence season (MAPE = 26.209), and the superiority was more obvious in the high incidence season.ConclusionThis study applied the STN to the forecast of infectious diseases across multiple regions. The results illustrated that the STN not only had good accuracy in forecasting performance but also indicated the spreading directions of infectious diseases among multiple regions to a certain extent. Therefore, the STN is a promising candidate to improve the surveillance work.

Frequency Domain Calculation of Seasonal VARMA Autocovariances

In applications of the fitting and forecasting of Seasonal Vector AutoRegressive Moving Average (SVARMA) models, it is important to quickly and accurately compute the autocovariances. Recursive time domain approaches rely on expressing the seasonal AutoRegressive matrix polynomials as a high order non-seasonal AutoRegressive matrix polynomial; initialization of the recursions is therefore costly, because a large-dimensional matrix must be inverted. However, the resulting high-order polynomial has many zeroes that are not intelligently exploited, indicating that this time domain approach is not optimal. It is proposed to compute the autocovariances via integrating the spectral density, taking advantage of analytical expressions for the inverse AutoRegressive matrix polynomials in terms of determinant and adjoint. The R and RCPP code can be hundreds of times faster than the time domain methods when the dimension and seasonal period are large.

A Fast Algorithm for Estimating Parameters of a Multivariate Autoregressive Moving Average Processes

We propose in this paper a fast and iterative algorithm for estimating the parameters of a Gaussian vector autoregressive-moving average (VARMA) model. This algorithm is a multivariate generalization of that suggested by Sabiti (1996) for estimating the parameters of a univariate ARMA(p,q) process. It is proposed, mainly for providing initial estimators for the iterative maximization of a log-likelihood function. Comparisons about the number of computations in terms of multiplication operations are made with a method that uses gradients to locate a maximum of the likelihood function and the fast method suggested by Spliid (1983).

ECM algorithm for estimating vector ARMA model with variance gamma distribution and possible unbounded density

SummaryThe simultaneous analysis of several financial time series is salient in portfolio setting and risk management. This paper proposes a novel alternating expectation conditional maximisation (AECM) algorithm to estimate the vector autoregressive moving average (VARMA) model with variance gamma (VG) error distribution in the multivariate skewed setting. We explain why the VARMA‐VG model is suitable for high‐frequency returns (HFRs) because VG distribution provides thick tails to capture the high kurtosis in the data and unbounded central density further captures the majority of near‐zero HFRs. The distribution can also be expressed in normal‐mean‐variance mixtures to facilitate model implementation using the Bayesian or expectation maximisation (EM) approach. We adopt the EM approach to avoid the time‐consuming Markov chain Monto Carlo sampling and solve the unbounded density problem in the classical maximum likelihood estimation. We conduct extensive simulation studies to evaluate the accuracy of the proposed AECM estimator and apply the models to analyse the dependency between two HFR series from the time zones that only differ by one hour.

The time series regression analysis in evaluating the economic impact of COVID-19 cases in Indonesia

This study aims to determine the impact of COVID-19 cases in Indonesia on the USD/IDR exchange rate using the Transfer Function Model and Vector Autoregressive Moving-Average with Exogenous Regressors (VARMAX) Model. This paper uses daily data on the COVID-19 case in Indonesia, the USD/IDR exchange rate, and the IDX Composite period from 1 March to 29 June 2020. The analysis shows: (1) the higher the increase of the number of COVID-19 cases in Indonesia will significantly weaken the USD/IDR exchange rate, (2) an increase of 1% in the number of COVID-19 cases in Indonesia six days ago will weaken the USD/IDR exchange rate by 0.003%, (3) an increase of 1% in the number of COVID-19 cases in Indonesia seven days ago will weaken the USD/IDR exchange rate by 0.17%, and (4) an increase of 1% in the number of COVID-19 cases in Indonesia eight days ago will weaken the USD/IDR exchange rate by 0.24%.

Estimation of the parameters of vector autoregressive moving average (VARMA) time series model with symmetric stable noise

Estimation of the parameters of vector autoregressive moving average (VARMA) time series model with symmetric stable noise

Sparse Identification and Estimation of Large-Scale Vector AutoRegressive Moving Averages

The vector autoregressive moving average (VARMA) model is fundamental to the theory of multivariate time series; however, identifiability issues have led practitioners to abandon it in favor of the simpler but more restrictive vector autoregressive (VAR) model. We narrow this gap with a new optimization-based approach to VARMA identification built upon the principle of parsimony. Among all equivalent data-generating models, we use convex optimization to seek the parameterization that is simplest in a certain sense. A user-specified strongly convex penalty is used to measure model simplicity, and that same penalty is then used to define an estimator that can be efficiently computed. We establish consistency of our estimators in a double-asymptotic regime. Our nonasymptotic error bound analysis accommodates both model specification and parameter estimation steps, a feature that is crucial for studying large-scale VARMA algorithms. Our analysis also provides new results on penalized estimation of infinite-order VAR, and elastic net regression under a singular covariance structure of regressors, which may be of independent interest. We illustrate the advantage of our method over VAR alternatives on three real data examples.

Multivariate time series modelling for urban air quality

We introduce a spatio-temporal model to represent development of atmospheric pollution in an urban area. An important element of this is that recorded measurements are often incomplete which undermines time-series approaches. We identify the multiple imputation by chained equation (MICE) method as effective to complete data sequences synthetically. Following on from this, we develop a vector autoregressive moving average (VARMA) model for the spatio-temporal development of atmospheric pollution in urban areas. This model was fitted to hourly measurements of four pollutants (NO, NO2, NOx and PM10) for the whole of the calendar year 2017 at 30 stations across London, completed by MICE as required. We show by cross-validation that the VARMA model is more effective than other formulations, including the Kriging method of spatial interpolation, and seasonal ARMA models for individual stations with either daily or weekly trends. The resulting model can be used for prediction of air quality in different periods and as the basis for assessment of policy interventions such as increasing vehicle emission standards, and traffic management and control policies such as low and ultra-low emission zones.

Asymptotic Properties of Estimators for Seasonally Cointegrated State Space Models Obtained Using the CVA Subspace Method.

This paper investigates the asymptotic properties of estimators obtained from the so called CVA (canonical variate analysis) subspace algorithm proposed by Larimore (1983) in the case when the data is generated using a minimal state space system containing unit roots at the seasonal frequencies such that the yearly difference is a stationary vector autoregressive moving average (VARMA) process. The empirically most important special cases of such data generating processes are the I(1) case as well as the case of seasonally integrated quarterly or monthly data. However, increasingly also datasets with a higher sampling rate such as hourly, daily or weekly observations are available, for example for electricity consumption. In these cases the vector error correction representation (VECM) of the vector autoregressive (VAR) model is not very helpful as it demands the parameterization of one matrix per seasonal unit root. Even for weekly series this amounts to 52 matrices using yearly periodicity, for hourly data this is prohibitive. For such processes estimation using quasi-maximum likelihood maximization is extremely hard since the Gaussian likelihood typically has many local maxima while the parameter space often is high-dimensional. Additionally estimating a large number of models to test hypotheses on the cointegrating rank at the various unit roots becomes practically impossible for weekly data, for example. This paper shows that in this setting CVA provides consistent estimators of the transfer function generating the data, making it a valuable initial estimator for subsequent quasi-likelihood maximization. Furthermore, the paper proposes new tests for the cointegrating rank at the seasonal frequencies, which are easy to compute and numerically robust, making the method suitable for automatic modeling. A simulation study demonstrates by example that for processes of moderate to large dimension the new tests may outperform traditional tests based on long VAR approximations in sample sizes typically found in quarterly macroeconomic data. Further simulations show that the unit root tests are robust with respect to different distributions for the innovations as well as with respect to GARCH-type conditional heteroskedasticity. Moreover, an application to Kaggle data on hourly electricity consumption by different American providers demonstrates the usefulness of the method for applications. Therefore the CVA algorithm provides a very useful initial guess for subsequent quasi maximum likelihood estimation and also delivers relevant information on the cointegrating ranks at the different unit root frequencies. It is thus a useful tool for example in (but not limited to) automatic modeling applications where a large number of time series involving a substantial number of variables need to be modelled in parallel.

COVID-19 incidence and mortality in Nigeria: gender based analysis.

BackgroundCoronavirus Disease 2019 (COVID-19) has been surging globally. Risk strata in medical attention are of dynamic significance for apposite assessment and supply distribution. Presently, no known cultured contrivance is available to fill this gap of this pandemic. The aim of this study is to develop a predictive model based on vector autoregressive moving average (VARMA) model of various orders for gender based daily COVID-19 incidence in Nigeria. This study also aims to proffer empirical evidence that compares incidence between male and female for COVID-19 risk factors.MethodsWilcoxon signed-rank test is employed to investigate the significance of the difference in the gender distributions of the daily incidence. A VARMA model of various orders is formulated for the gender based daily COVID-19 incidence in Nigeria. The optimal VARMA model is identified using Bayesian information criterion. Also, a predictive model based on univariate autoregressive moving average model is formulated for the daily death cases in Nigeria. Fold change is estimated based on crude case-fatality risk to investigate whether there is massive underreporting and under-testing of COVID-19 cases in Nigeria.ResultsDaily incidence is higher in males on most days from 11 April 2020 to 12 September 2020. Result of Wilcoxon signed-rank test shows that incidence among male is significantly higher than female (p-value < 2.22 × 10−16). White neural network test shows that daily female incidence is not linear in mean (p-value = 0.00058746) while daily male incidence is linear in mean (p-value = 0.4257). McLeod-Li test shows that there is autoregressive conditional heteroscedasticity in the female incidence (Maximum p-value = 1.4277 × 10−5) and male incidence (Maximum p-value = 9.0816 × 10−14) at 5% level of significance. Ljung-Box test (Tsay, 2014) shows that the daily incidence cases are not random (p-value=0.0000). The optimal VARMA model for male and female daily incidence is VARMA (0,1). The optimal model for the Nigeria’s daily COVID-19 death cases is identified to be ARIMA (0,1,1). There is no evidence of massive underreporting and under-testing of COVID-19 cases in Nigeria.ConclusionsComparison of the observed incidence with fitted data by gender shows that the optimal VARMA and ARIMA models fit the data well. Findings highlight the significant roles of gender on daily COVID-19 incidence in Nigeria.

Spatial transmission network construction of influenza-like illness using dynamic Bayesian network and vector-autoregressive moving average model

BackgroundAlthough vaccination is one of the main countermeasures against influenza epidemic, it is highly essential to make informed prevention decisions to guarantee that limited vaccination resources are allocated to the places where they are most needed. Hence, one of the fundamental steps for decision making in influenza prevention is to characterize its spatio-temporal trend, especially on the key problem about how influenza transmits among adjacent places and how much impact the influenza of one place could have on its neighbors. To solve this problem while avoiding too much additional time-consuming work on data collection, this study proposed a new concept of spatio-temporal route as well as its estimation methods to construct the influenza transmission network.MethodsThe influenza-like illness (ILI) data of Sichuan province in 21 cities was collected from 2010 to 2016. A joint pattern based on the dynamic Bayesian network (DBN) model and the vector autoregressive moving average (VARMA) model was utilized to estimate the spatio-temporal routes, which were applied to the two stages of learning process respectively, namely structure learning and parameter learning. In structure learning, the first-order conditional dependencies approximation algorithm was used to generate the DBN, which could visualize the spatio-temporal routes of influenza among adjacent cities and infer which cities have impacts on others in influenza transmission. In parameter learning, the VARMA model was adopted to estimate the strength of these impacts. Finally, all the estimated spatio-temporal routes were put together to form the final influenza transmission network.ResultsThe results showed that the period of influenza transmission cycle was longer in Western Sichuan and Chengdu Plain than that in Northeastern Sichuan, and there would be potential spatio-temporal routes of influenza from bordering provinces or municipalities into Sichuan province. Furthermore, this study also pointed out several estimated spatio-temporal routes with relatively high strength of associations, which could serve as clues of hot spot areas detection for influenza surveillance.ConclusionsThis study proposed a new framework for exploring the potentially stable spatio-temporal routes between different places and measuring specific the sizes of transmission effects. It could help making timely and reliable prediction of the spatio-temporal trend of infectious diseases, and further determining the possible key areas of the next epidemic by considering their neighbors’ incidence and the transmission relationships.

An indirect proof for the asymptotic properties of VARMA model estimators

Strong consistency and asymptotic normality of a Gaussian quasi-maximum likelihood estimator for the parameters of a causal, invertible, and identifiable vector autoregressive-moving average (VARMA) model are established in an indirect way. The proof is based on similar results for a much wider class of VARMA models with time-dependent coefficients, hence in the context of non-stationary and heteroscedastic time series. For that reason, the proof avoids spectral analysis arguments and does not make use of ergodicity. The results presented are also applicable to ARMA models.