ARFIMA Model Research Articles

PM10 AIR QUALITY INDEX MODELING USING ARFIMA-GARCH METHOD: BUNDARAN HI AREA OF DKI JAKARTA PROVINCE

Air quality is an essential factor in urban life, and its’ assessment often relies on the concentration of measurable air pollution parameters. One critical parameter is Particulate Matter (PM), particularly PM10, which comprises solid or liquid particles dispersed in the air from various sources. One of the methods employed for predicting stock index prices is ARFIMA. ARFIMA is used to model long memory data characterized by a slowly decreasing Autocorrelation Function (ACF) plot (hyperbolic) or a difference value in the fractional from. This method is widely used due to its ability to handle nonstationarity issues in time series. However, the time series data often contain heteroskedasticity problems. Data with heteroscedasticity are then further addressed using the GARCH model, because it can model volatility changes occurring over longer periods and capture the persistence of volatility. The ARFIMA-GARCH model can explain long-memory patterns in time series data and address heteroscedasticity issues. The data are sourced from the Jakarta open data web, which is integrated with DLH DKI Jakarta Province. The aim of this research was to forecast the PM10 air quality index at the Bundaran HI Area in the Province of DKI Jakarta for the next 14 days, from January 1st to January 14th, 2021, using an ARFIMA model enhanced with GARCH. The analysis reveals that the best model is ARFIMA ([17], d, [1])-GARCH (1,1). Forecasting using this model resulted in a MAPE of 3.47%, indicating that the model is highly capable of forecasting several periods.

Technological progress and economic dynamics: Unveiling the long memory of total factor productivity

This study aims to explore the dynamics of total factor productivity (TFP) in 40 advanced economies and its implications for economic stability, growth patterns, and technological integration. Understanding the concept of long memory in TFP is crucial in comprehending the diverse paths of economic development and technological advancement in different nations. The study endeavors to clarify the concept of long memory in TFP and analyze its impact on economic stability, growth patterns, and technological integration on a global scale. The research employs a sophisticated methodological framework that integrates a combination of unit root tests, non-parametric methods, fractional integration techniques, and ARFIMA models. This multifaceted approach provides a nuanced understanding of the persistence in TFP series and helps distinguish between stable and mean-reverting TFP dynamics across different nations. The analysis highlights the varying trajectories of TFP across these economies, which sheds light on the diverse paths of economic development and technological advancement. The findings of this research are pivotal in comprehending the role of TFP in shaping growth convergence, influencing business cycle fluctuations, and technological progress. The study contributes to policy discourse by proposing strategic policy measures that harness the long-term trends of TFP. These recommendations encompass key areas such as labor, capital, institutional frameworks, and innovation policies. Countries can optimize sustainable economic development and foster effective technological integration in an increasingly interconnected global economy by implementing these measures.

Predictive analysis of stochastic stock pattern utilizing fractional order dynamics and heteroscedastic with a radial neural network framework

Modelling of high-dimension chaotic, noisy, and non-stationary time series of complex fractal dynamics is a big challenge. In this research work, a novel approach of Leverage Convolution LC ARFIMA– GARCH model is presented for sequential learning of irregular, fractal dynamic patterns of stochastic time series dynamics. Different classical methods including the auto-regressive approach indecently are unable to capture fragile patterns and consequently convert some weak signals into random errors by creating white noise patterns. The Convolution Leverage provides an additional degree of freedom for critical noisy points and asymmetrical distribution dynamics to regulate the frequency of imbalance and skewed observation. The designed transformation in the ARFIMA model preserves the loss of information by shifting the skewed data distribution toward a normal pattern. The pattern of population parameters in the proposed leverage paradigm provides out of box approach to track and extract additional information on population variance patterns in the form of an additional degree of freedom to stabilize imbalance signals. The model can provide reliable performance for long-range dependence, particularly for mean reversion chaotic phase variation at critical non-differentiable points. The performance of the dynamic model is verified on chaotic real data of the Ireland Stock Market (ISEQ). The result statistics confirmed the optimized outcome with the addition of the GARCH heteroscedastic multimodal and radial basis neural network. The novel technique can help to address inherited challenges in the imbalance learning and extreme events of the time series modeling by monitoring the chaotic trajectory of fractal physical phenomena, particularly in, finance healthcare, climate, and intelligent sustainability.

Long Memory Time-series Model (ARFIMA) Based Modelling of Jute Prices in the Samsi Market of Malda District, West Bengal

The objective of this paper is modeling and forecasting the weekly jute prices of Samsi market in the Malda district of West Bengal in the presence of long memory process. The long memory behavior of series is investigated by the ACF plot and Hurst R/S analysis. A fractionally integrated autoregressive moving-average (ARssFIMA) model is fitted using 668 weekly data (January 2009-November 2022). This study shows the efficiencies of the Hurst exponent, GPH, Smoothed periodogram, Local Whittle, and Wavelet methods used to estimate the fractional difference parameter in the ARFIMA model. Furthermore, we compared the forecasting abilities of the ARFIMA and ARIMA models. The results show that long memory is present in the jute price series. The models selected according to each method are ARFIMA (3,0.348,0), ARFIMA (3,0.291,1), ARFIMA (2,0.487,0), ARFIMA (3,0.461,0), ARFIMA (2,0.311,0), and ARIMA (2,1,1) on the basis of minimum AIC and BIC using 534 in-sample data. Finally, the wavelet method based ARFIMA (2,0.311,0) model is found to be the best optimal model in terms of MAE, RMSE, and MAPE criteria using 134 out-of-sample data. A comparative study indicates that the forecasting performance of the ARFIMA model is strongly better than that of the ARIMA model in this regard.

Predicting expected idiosyncratic volatility: Empirical evidence from ARFIMA, HAR, and EGARCH models

We investigate the performances of the ARFIMA, HAR, and EGARCH models in capturing the time-varying property of idiosyncratic volatility (IVOL). We find that the expected IVOL predictions by HAR are superior. In diverse portfolio scenarios, a greater degree of judgment is required to assess the pricing ability of expected IVOLs. For the lowest value-weighted quintiles and the expected IVOL estimated by the HAR model, the IVOL-return relationship is negative. Conversely, the IVOL-return relationship is positive for the expected IVOL estimated by the EGARCH model. Further evidence suggests a complicated and mixed relationship between the expected IVOL estimated by the ARFIMA model and stock returns.

Detecting turning points of stock markets in China and the United States

ABSTRACT Choosing the appropriate method to identify turning points is critical for investors and policymakers. We propose a novel hybrid model combining dual long memory with structural breaks in the mean to identify the turning points of the stock market in China and the United States. The results show that models accounting for long memory and structural break perform well in the turning point detection. Notably, the proposed hybrid model generates superior in-sample turning point matching and out-sample forecasts for SSECI over those obtained from the competing ARFIMA models. In contrast, DJIA is better explained by the dual memory model with structural breaks in volatility. Finally, our analysis extends to encompass FTSE, HSI, IBOVESPA, and SENSEX30. The results demonstrate the superior performance of the hybrid model in markets exhibiting fractal characteristics. Such heterogeneity indicates differences in investor risk preference across stock markets and offers elaborate international evidence to the adaptive market hypothesis.

Monitoring the mean of autocorrelated data with long memory from cable production using one‐sided runs rules schemes with ARFIMA (1,d,1) model

AbstractClassical control charts were developed with the assumption that the quality characteristics being monitored are independent. However, observations from several time‐dependent processes, such as industrial or manufacturing, are bound to be autocorrelated such that their autocorrelations decay gradually (long memory) rather than exponentially (short memory). Runs rules monitoring schemes have been proposed in Statistical Process Control literature to monitor processes in the presence of autocorrelation by incorporating the autoregressive integrated moving average (ARIMA) class of models which can only handle autocorrelation with a short memory. This paper therefore incorporates the autoregressive fractionally integrated moving average (ARFIMA (1,d,1)) model into four one‐sided runs rules schemes to monitor the mean of an autocorrelated process with long memory. The performance of these four one‐sided schemes was assessed using average run length (ARL) and expected ARL (EARL) metrics. Based on the ARL and EARL of the improved runs rules (IRR) schemes, the IRR5‐of‐5 and IRR2‐of‐7 were found to issue an out‐of‐control signal faster than the corresponding standard runs rules (RR) schemes, RR5‐of‐5 and RR2‐of‐7, respectively. Also, the four one‐sided schemes IRR5‐of‐5, IRR2‐of‐7, RR5‐of‐5, and RR2‐of‐7 in their steady state condition perform better than their corresponding zero state condition. We further observed that as the fractional parameter d increases, the detection ability of the monitoring schemes reduces. The application of the four one‐sided schemes was demonstrated on the diameter of cable production.

Energy demand forecasting using adaptive ARFIMA based on a novel dynamic structural break detection framework

Forecasting energy demand has become increasingly important due to technological advances, especially new power systems and population growth. Accurate predictions of energy demand and supply are crucial for academics and policymakers. Also, energy consumption is non-stationary and dynamic in time, requiring an adaptive forecasting algorithm. This paper presents a new adaptive hybrid approach for energy time series forecasting using statistical methods. Some of the most popular energy demand time series applications are used to demonstrate the superiority of the proposed algorithm, which covers a wide range of critical energy demand forecasting situations at various scales. The paper's objectives are presented in three phases. In Phase I, ARFIMA is used to forecast the entire original time series as a statistical inspector and later for comparison. Although ARFIMA's fractionality feature helps to capture long and short-term memory behavior in time series, to be more adaptive and become close to a nonlinear algorithm that can handle non-stationarity, and Gaussian and non-Gaussian time series closely, a novel dynamic statistical structural break detection framework is developed, and used in phase II to identify the time series' change points and associated time indexes. Since time series are partitioned based on structural change detections, they approach one in terms of the Hurst exponent and exhibit pure long memory, which is ideal for ARFIMA modeling. Phase III captures non-stationary statistical properties and memory characteristics by applying adaptive ARFIMA modeling to segmented time series. Finally, the predicted partitions are concatenated. Various performance assessment metrics, e.g., Mean Absolute Percentage Error (MAPE), tested on the final achieved results show that the proposed adaptive algorithm outperforms the forecasting capabilities of the existing algorithms on the same energy demand time series used in the current paper. MAPE of the proposed approach for the four utilized real-life cases are 1.72%, 1.35%, 0.22%, and 4.4%, respectively. Therefore, the proposed model is a satisfactory method for energy demand forecasting due to its high accuracy.

Predicting the Long-Term Dependencies in Time Series Using Recurrent Artificial Neural Networks

Long-term dependence is an essential feature for the predictability of time series. Estimating the parameter that describes long memory is essential to describing the behavior of time series models. However, most long memory estimation methods assume that this parameter has a constant value throughout the time series, and do not consider that the parameter may change over time. In this work, we propose an automated methodology that combines the estimation methodologies of the fractional differentiation parameter (and/or Hurst parameter) with its application to Recurrent Neural Networks (RNNs) in order for said networks to learn and predict long memory dependencies from information obtained in nonlinear time series. The proposal combines three methods that allow for better approximation in the prediction of the values of the parameters for each one of the windows obtained, using Recurrent Neural Networks as an adaptive method to learn and predict the dependencies of long memory in Time Series. For the RNNs, we have evaluated four different architectures: the Simple RNN, LSTM, the BiLSTM, and the GRU. These models are built from blocks with gates controlling the cell state and memory. We have evaluated the proposed approach using both synthetic and real-world data sets. We have simulated ARFIMA models for the synthetic data to generate several time series by varying the fractional differentiation parameter. We have evaluated the proposed approach using synthetic and real datasets using Whittle’s estimates of the Hurst parameter classically obtained in each window. We have simulated ARFIMA models in such a way that the synthetic data generate several time series by varying the fractional differentiation parameter. The real-world IPSA stock option index and Tree Ringtime series datasets were evaluated. All of the results show that the proposed approach can predict the Hurst exponent with good performance by selecting the optimal window size and overlap change.

Investigating the Efficacy of ARIMA and ARFIMA Models in Nigeria All Share Index Markets

Investigating the Efficacy of ARIMA and ARFIMA Models in Nigeria All Share Index Markets

Long Memory and Volatility Models in Forecasting Exchange Rate of Nigerian Naira to United State Dollar

Financial time series such as stock prices, inflation rates, interest rates, and exchange rates are known to exhibit upward and downward trend and often possesses long memory and volatility behavior. These behaviors are crucial in the analysis, modeling and forecasting of time series data. Unfortunately, many analysts don’t take into consideration the consequences of long memory and volatility while modeling financial time series data. Therefore, this paper intends to examine the effect of long memory and volatility in forecasting Exchange Rate of Nigerian Naira-United State Dollar. The data used for this study is obtained from Central Bank of Nigeria’s website for the period of January 2002 to Feb 2020. Observations from time series and Autocorrelation function (ACF) plot shows that the data was not stationary. This was confirmed by the Augmented Dickey Fuller (ADF) and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) unit root tests on the datasets. Fractional differencing was used to transform the series and ARFIMA(1,d,1) and ARFIMA(1,d,2) models were selected using AIC criterion. However, the residuals of these models were found to be serially correlated and heteroscedastic. These problems led to combining ARFIMA with GARCH model in order to adequately study long memory and volatility simultaneously. Therefore ARFIMA models were combined with GARCH(1,1) to form ARFIMA(1,d,1)-GARCH(1,1) and ARFIMA(1,d,2)- GARCH(1,1). The results of the forecast performance indicate that the best model is ARFIMA(1,d,2)– GARCH(1,1).

Long memory conditional random fields on regular lattices

AbstractThis paper draws its motivation from applications in geophysics, agricultural, and environmental sciences where empirical evidence of slow decay of correlations have been found for data observed on a regular lattice. Spatial ARFIMA models represent a widely used class of spatial models for analyzing such data. Here, we consider their generalization to conditional autoregressive fractional integrated moving average (CARFIMA) models, a larger class of long memory models which allows a wider range of correlation behavior. For this class we provide detailed descriptions of important representative models, make the necessary comparison with some other existing models, and discuss some important inferential and computational issues on estimation, simulation and long memory process approximation. Results from model fit comparison and predictive performance of CARFIMA models are also discussed through a statistical analysis of satellite land surface temperature data.

South African inflation modelling using bootstrapped long short-term memory methods.

Inflation is a critical economic series, and proper targeting is required for a stable economy. With the current economic conditions that the world has faced as a result of COVID-19, understanding the effects of this on economies is critical because it will guide policies. Recent research on South African inflation has focused on statistical modelling, specifically the ARFIMA, GARCH, and GJR-GARCH models. In this study, we extend this into deep learning and use the MSE, RMSE, RSMPE, MAE, and MAPE to assess performance. To test which model has better forecasts, we use the Diebold-Mariano test. According to the findings of this study, clustered bootstrap LSTM models outperform the previously used ARFIMA-GARCH and ARFIMA-GJR-GARCH models.

Characterizing Nonlinear Time Series via Sliding-Window Amplitude-Based Dispersion Entropy

In this paper, we propose a hybrid method called sliding-window amplitude-based dispersion entropy, which combines dispersion entropy with sliding-window amplitude, to characterize nonlinear time series. This hybrid method not only inherits the fast calculation speed and the ability to characterize nonlinear time series of dispersion entropy, but also has higher noise resistance than dispersion entropy. We firstly utilize three artificial data (logistic map, Hénon map, ARFIMA model) to qualify the effectiveness of the proposed method, results show that our method can correctly characterize the nonlinear time series, and has stronger robustness to noise. Next, the method is applied to analyze stock market system, the data of stock market are composed of six main indices from different countries, the result shows that the proposed method can easily distinguish the emerging markets and developed markets, and can reveal some features under the financial time series.

Temporal Structure in Sensorimotor Variability: A Stable Trait, But What For?

Human performance shows substantial endogenous variability over time, and this variability is a robust marker of individual differences. Of growing interest to psychologists is the realisation that variability is not fully random, but often exhibits temporal dependencies. However, their measurement and interpretation come with several controversies. Furthermore, their potential benefit for studying individual differences in healthy and clinical populations remains unclear. Here, we gather new and archival datasets featuring 11 sensorimotor and cognitive tasks across 526 participants, to examine individual differences in temporal structures. We first investigate intra-individual repeatability of the most common measures of temporal structures — to test their potential for capturing stable individual differences. Secondly, we examine inter-individual differences in these measures using: (1) task performance assessed from the same data, (2) meta-cognitive ratings of on-taskness from thought probes occasionally presented throughout the task, and (3) self-assessed attention-deficit related traits. Across all datasets, autocorrelation at lag 1 and Power Spectra Density slope showed high intra-individual repeatability across sessions and correlated with task performance. The Detrended Fluctuation Analysis slope showed the same pattern, but less reliably. The long-term component (d) of the ARFIMA(1,d,1) model showed poor repeatability and no correlation to performance. Overall, these measures failed to show external validity when correlated with either mean subjective attentional state or self-assessed traits between participants. Thus, some measures of serial dependencies may be stable individual traits, but their usefulness in capturing individual differences in other constructs typically associated with variability in performance seems limited. We conclude with comprehensive recommendations for researchers.

Investigation of Fractal Market Hypothesis in Emerging Markets: Evidence from the MINT Stock Markets

This study aims to investigate the market efficiency of emerging stock markets, namely the Mexico, Indonesia, Nigeria, and Turkey (MINT) stock markets based on the Fractal Market Hypothesis. For this purpose, the ARFIMA and ARFIMA-FIGARCH type models are used to analyze the MINT stock return series. In this study, the dataset encompasses the daily frequency data of the MINT stock market indices from January 12, 2018, to January 12, 2022. The empirical findings show that long memory is reported for the MINT stock returns. The long memory in the returns implies that the MINT stock prices follow a predictable behavior that is consistent with the Fractal Market Hypothesis. The long memory in the volatility implies that the uncertainty or risk is an important factor in the formation of price movements in the MINT stock prices. Moreover, the MINT stock prices consist of the effect of shocks and news that occurred in the recent past. Thus, this study contributes to investors, academics, and market regulatory authorities. Besides, as far as we know, the current literature on the analysis of the fractal market hypothesis to explore the efficiency of the MINT stock markets has not been found.

Development and approbation of a software solution for the investment portfolios formation using fractal analysis and predictive models

This paper describes the development and approbation of the software solution, which implements the methodology of forming recommendations for the composition of investment portfolios using fractal analysis and long memory predictive models, which is the result of research conducted over several years at the Department of Information Systems and Mathematical Methods in Economics of PSU. The general algorithm of the program includes four main steps: 1) obtaining and preparing data; 2) sorting assets by the fractal dimension of their price series; 3) forecasting asset returns; 4) forming portfolios (determining asset shares). The described algorithm stages correspond to the structure of the program solution, expressed by the set of its subsystems. The features of the developed program are as follows: possibility to load data of share prices from the FINAM website; calculation of the fractal dimension of asset price series by the DFA and minimum coverage methods; asset returns forecasting using the ARFIMA and ARFIMA-GARCH models; selection of portfolio structures based both on the forecasted and historical data; possibility of multiple generation of portfolios based on random-selected assets with characteristics averaging; evaluation of portfolio characteristics using test data (if any); support for multi-core processors for multiple acceleration of calculations; windowed graphical interface. The results of program approbation on the Russian stock market data under crisis economic conditions are given. On the whole, these results are in line with those obtained earlier on the developed US market data. The developed program can be used by portfolio investors, carrying out investment activities in the international financial markets. Scientific application of the program is important: its users can be scientists, students and other researchers of innovative methods of investment portfolios formation.

Modelling and parameter estimation for discretely observed fractional iterated Ornstein–Uhlenbeck processes

In this work we present how to model an observed time series by a FOU(p) process. We will show that the FOU(p) processes can be used to model a wide range of time series varying from short range dependence to long range dependence, with performance similar to the ARMA or ARFIMA models and in several cases outperforming them. Also, we extend the theoretical results for any FOU(p) processes for the case in which the Hurst parameter is less than 1/2 and we show theoretically and by simulations that under some conditions on T and the sample size n it is possible to obtain consistent estimators of the parameters when the process is observed in a discretized and equispaced interval [0,T]. Lastly, we give a way to obtain explicit formulas for the auto-covariance function for any FOU(p) and we present an application for FOU(2) and FOU(3).

Pricing VIX futures: A framework with random level shifts

We propose a DRLS framework to price VIX futures by modeling the logVIX series dynamics using the ARFIMA and HAR models that introduce the random level shifts component. Compared with other traditional time series models, our model allows the change of the theoretical mean value of the VIX index by time, which is more reasonable since there are different volatility states under different market environments. Using the Kalman filter, we can derive the explicit formula of the VIX futures price without calculating numerical integration that is different from models without random level shifts. The empirical results show that the DRLS framework performs better in both in-sample estimating and out-of-sample forecasting than directly pricing models without random level shifts and is much simpler.

Volatility Puzzle: Long Memory or Antipersistency

The log realized volatility (RV) is often modeled as an autoregressive fractionally integrated moving average model ARFIMA([Formula: see text]). Two conflicting empirical results have been found in the literature. One stream shows that log RV has a long memory (i.e., the fractional parameter d > 0). The other stream suggests that the autoregressive coefficient α is near unity with antipersistent errors (i.e., d < 0). This paper explains how these conflicting empirical findings can coexist in the context of ARFIMA([Formula: see text]) model by examining the finite sample properties of popular estimation methods, including semiparametric methods and parametric maximum likelihood methods. The finite sample results suggest that it is challenging to distinguish [Formula: see text] (ARFIMA([Formula: see text]) with α close to 0 and d close to 0.5) from [Formula: see text] (ARFIMA([Formula: see text]) with α close to unity and d close to –0.5). An intuitive explanation is given. For the 10 financial assets considered, despite that no definitive conclusions can be drawn regarding the data-generating process, we find that the frequency domain maximum likelihood (or Whittle) method can generate the most accurate out-of-sample forecasts.This paper was accepted by Lukas Schmid, finance.Funding: S. Shi acknowledges research support from the Australian Research Council [Project DE190100840]. J. Yu acknowledges financial support from the Ministry of Education–Singapore Tier 2 Academic Research Fund [Project MOE-T2EP402A20-0002] and the Lee Foundation.Supplemental Material: The data files are available at https://doi.org/10.1287/mnsc.2022.4552 .