ARFIMA Research Articles

Estimating FARIMA models with uncorrelated but non-independent error terms

In this paper we derive the asymptotic properties of the least squares estimator (LSE) of fractionally integrated autoregressive moving-average (FARIMA) models under the assumption that the errors are uncorrelated but not necessarily independent nor martingale differences. We relax the independence and even the martingale difference assumptions on the innovation process to extend considerably the range of application of the FARIMA models. We propose a consistent estimator of the asymptotic covariance matrix of the LSE which may be very different from that obtained in the standard framework. A self-normalized approach to confidence interval construction for weak FARIMA model parameters is also presented. All our results are done under a mixing assumption on the noise. Finally, some simulation studies and an application to the daily returns of stock market indices are presented to corroborate our theoretical work.

Forecasting performance comparison of daily maximum temperature using ARMA based methods

Daily maximum temperature of four different regions in Kerala, India, from 01/01/2019 to 31/12/2020, is recorded and is used for modelling and forecasting. The forecasting methods used are Autoregressive integrated moving average (ARIMA), Seasonal Autoregressive integrated moving average (SARIMA) and Autoregressive fractional integrated moving average (ARFIMA). The comparison of forecasting performance was based on percentage accuracy, mean squared error (MSE) and mean absolute error (MAE). The models used can capture the variations of time series data. All the models exhibit reasonably good performance in predicting the daily maximum temperature. ARFIMA model gives the least forecast errors compared to other models.

Forecasting the Finnish house price returns and volatility: a comparison of time series models

PurposeThe purpose of this paper is to compare different models’ performance in modelling and forecasting the Finnish house price returns and volatility.Design/methodology/approachThe competing models are the autoregressive moving average (ARMA) model and autoregressive fractional integrated moving average (ARFIMA) model for house price returns. For house price volatility, the exponential generalized autoregressive conditional heteroscedasticity (EGARCH) model is competing with the fractional integrated GARCH (FIGARCH) and component GARCH (CGARCH) models.FindingsResults reveal that, for modelling Finnish house price returns, the data set under study drives the performance of ARMA or ARFIMA model. The EGARCH model stands as the leading model for Finnish house price volatility modelling. The long memory models (ARFIMA, CGARCH and FIGARCH) provide superior out-of-sample forecasts for house price returns and volatility; they outperform their short memory counterparts in most regions. Additionally, the models’ in-sample fit performances vary from region to region, while in some areas, the models manifest a geographical pattern in their out-of-sample forecasting performances.Research limitations/implicationsThe research results have vital implications, namely, portfolio allocation, investment risk assessment and decision-making.Originality/valueTo the best of the author’s knowledge, for Finland, there has yet to be empirical forecasting of either house price returns or/and volatility. Therefore, this study aims to bridge that gap by comparing different models’ performance in modelling, as well as forecasting the house price returns and volatility of the studied market.

Modelling and forecasting monthly Brent crude oil prices: a long memory and volatility approach

AbstractThe Standard Generalised Autoregressive Conditionally Heteroskedastic (sGARCH) model and the Functional Generalised Autoregressive Conditionally Heteroskedastic (fGARCH) model were applied to study the volatility of the Autoregressive Fractionally Integrated Moving Average (ARFIMA) model, which is the primary objective of this study. The other goal of this paper is to expand on the researchers’ previous work by examining long memory and volatilities simultaneously, by using the ARFIMA-sGARCH hybrid model and comparing it against the ARFIMA-fGARCH hybrid model. Consequently, the hybrid models were configured with the monthly Brent crude oil price series for the period from January 1979 to July 2019. These datasets were considered as the global economy is currently facing significant challenges resulting from noticeable volatilities, especially in terms of the Brent crude prices, due to the outbreak of COVID-19. To achieve these goals, an R/S analysis was performed and the aggregated variance and the Higuchi methods were applied to test for the presence of long memory in the dataset. Furthermore, four breaks have been detected: in 1986, 1999, 2005, and 2013 using the Bayes information criterion. In the further section of the paper, the Hurst Exponent and Geweke-Porter-Hudak (GPH) methods were used to estimate the values of fractional differences. Thus, some ARFIMA models were identified using AIC (Akaike Information Criterion), BIC (Schwartz Bayesian Information Criterion), AICc (corrected AIC), and the RMSE (Root Mean Squared Error). In result, the following conclusions were reached: the ARFIMA(2,0.3589648,2)-sGARCH(1,1) model and the ARFIMA(2,0.3589648,2)-fGARCH(1,1) model under normal distribution proved to be the best models, demonstrating the smallest values for these criteria. The calculations conducted herein show that the two models are of the same accuracy level in terms of the RMSE value, which equals 0.08808882, and it is this result that distinguishes our study. In conclusion, these models can be used to predict oil prices more accurately than others.

Forecasting COVID-19 Confirmed Cases in Ghana: A Model Selection Approach

This study seeks to determine an appropriate statistical technique for forecasting the cumulated confirm cases of Coronavirus in Ghana. Cumulated daily data spanning from March 12, 2020, to August 04, 2020, was retrieved from the Center for Systems Science and Engineering at Johns Hopkins University. Four statistical forecasting techniques: Autoregressive Integrated Moving Average, Artificial Neural Network, Exponential smoothing and Autoregressive Fractional Integrated Moving Average were fitted to the COVID-19 series. Their respective forecast accuracy measures were compared to select the appropriate technique for forecasting the COVID-19 cases. Our findings revealed that the ARFIMA technique was a suitable statistical model for predicting COVID-19 cases in Ghana. The "best" model for forecasting is ARFIMA (2, 0.49, 4) which passed all the needed diagnostic tests. An unequal weight was estimated to derive a combined model for all four forecasting techniques. A 149-cumulated daily forecast from the "best" model and the combined model revealed that the number of confirmed COVID-19 cases would increase slightly until the end of this year.

Wind Put Barrier Options Pricing Based on the Nordix Index

Wind power generators face risks derived from fluctuations in market prices and variability in power production, generated by their high dependence on wind speed. These risks could be hedged using weather financial instruments. In this research, we design and price an up-and-in European wind put barrier option using Monte Carlo simulation. Under the existence of a structured weather market, wind producers may purchase an up-and-in European wind barrier put option to hedge wind fluctuations, allowing them to recover their investments and maximise their profits. We use a wind speed index as the underlying index of the barrier option, which captures risk from wind power generation and the Autoregressive Fractionally Integrated Moving Average (ARFIMA) to model the wind speed. This methodology is applied in the Colombian context, an electricity market affected by the El Niño phenomenon. We find that when the El Niño phenomenon occurs, there are incentives for wind generators to sell their energy to the system because their costs, including the put option price, are lower than the power prices. This research aims at encouraging policymakers and governments to promote renewable energy sources and a financial market to trade options to reduce uncertainty in the electrical system due to climate phenomena.

Comparing the performances of symmetric and asymmetric generalized autoregressive conditionally heteroscedasticity models based on long-memory models under different distributions

This research compares the symmetric and asymmetric effects of generalized autoregressive conditional heteroscedasticity (GARCH)-type models to investigate the volatility of the autoregressive fractionally integrated moving average (ARFIMA) model using the monthly Brent crude oil price series for the period of January 1979–July 2019. The best model of volatility is determined by comparing 13 hybrid models of GARCH (sGARCH, fGARCH, EGARCH, TGARCH, IGARCH, AVGARCH, NGARCH, NAGARCH, APARCH, apARCH, GJRGARCH, gjrGARCH, and csGARCH) in terms of symmetric and asymmetric effects at the level of (1,1). R/S analysis is used to achieve this target. The aggregated variance method, the Higuchi method, and the structural break test are performed to determine the presence of long memory in the dataset. Furthermore, the Hurst exponent method and the Geweke and Porter–Hudak method are used to estimate the fractional difference values. The ARFIMA(2,0.3589648,2)–IGARCH(1,1) model under normal distribution is selected as the best model based on the Akaike information criterion, Schwartz Bayesian information criterion, and by the smallest value for root-mean-squared error, in which this model can be used to predict more accurately than other models.

Volatility Forecasting for High-Frequency Financial Data Based on Web Search Index and Deep Learning Model

The existing index system for volatility forecasting only focuses on asset return series or historical volatility, and the prediction model cannot effectively describe the highly complex and nonlinear characteristics of the stock market. In this study, we construct an investor attention factor through a Baidu search index of antecedent keywords, and then combine other trading information such as the trading volume, trend indicator, quote change rate, etc., as input indicators, and finally employ the deep learning model via temporal convolutional networks (TCN) to forecast the volatility under high-frequency financial data. We found that the prediction accuracy of the TCN model with investor attention is better than those of the TCN model without investor attention, the traditional econometric model as the generalized autoregressive conditional heteroscedasticity (GARCH), the heterogeneous autoregressive model of realized volatility (HAR-RV), autoregressive fractionally integrated moving average (ARFIMA) models, and the long short-term memory (LSTM) model with investor attention. Compared with the traditional econometric models, the multi-step prediction results for the TCN model remain robust. Our findings provide a more accurate and robust method for volatility forecasting for big data and enrich the index system of volatility forecasting.

Fractional and fractal processes applied to cryptocurrencies price series

IntroductionCryptocurrencies have been attracting the attention from media, investors, regulators and academia during the last years. In spite of some scepticism in the financial area, cryptocurrencies are a relevant subject of academic research. ObjectivesIn this paper, several tools are adopted as an instrument that can help market agents and investors to more clearly assess the cryptocurrencies price dynamics and, thus, guide investment decisions more assertively while mitigating risks. MethodsWe consider three methods, namely the Auto-Regressive Integrated Moving Average (ARIMA), Auto-Regressive Fractionally Integrated Moving Average (ARFIMA) and Detrended Fluctuation Analysis, and three indices given by the Hurst and Lyapunov exponents or the Fractal Dimension. This information allows assessing the behaviour of the time series, such as their persistence, randomness, predictability and chaoticity. ResultsThe results suggest that, except for the Bitcoin, the other cryptocurrencies exhibit the characteristic of mean reverting, showing a lower predictability when compared to the Bitcoin. The results for the Bitcoin also indicate a persistent behavior that is related to the long memory effect. ConclusionsThe ARFIMA reveals better predictive performance than the ARIMA for all cryptocurrencies. Indeed, the obtained residual values for the ARFIMA are smaller for the auto and partial auto correlations functions, as well as for confidence intervals.

Characteristic analysis of coordinate time series of tide gauge station

In this paper, 298 tide gauge stations are used as the research object, and three models of generalized gauss markov (GGM) noise model, autoregressive moving average (ARMA) noise model and Autoregressive fractionally integrated moving average (ARFIMA) noise model are used to estimate the noise model characteristics of the coordinate time series of tide gauge stations and the trend of sea level changes. The influence of time span on the velocity estimation of tide gauge stations analyzed and discussed. The experimental results show that the noise characteristics of coordinate time series of tide gauge stations are mainly ARFIMA(1,0), ARFIMA(2,2), ARMA (1,0); the velocity estimation results of tide gauge stations show that 64.77% of the station velocity values are in the interval of 0 mm/a to 4 mm/a, and the average sea level velocity is 1.25 mm/a, which is on an upward trend. As the time span increases, the velocity uncertainty of the coordinate sequence, of the tide gauge stations, gradually tends to converge from divergence, and a time span more than 110 a helps to obtain a robust estimate of the tide gauge station velocity.

Time Series Modeling of Guinea Fowls Production in Kenya Using the ARIMA and ARFIMA Models

Commercial farming of Guinea Fowls is at its infant stages and is generating a lot of interest for farmers in Kenya. This, coupled with an increased demand for poultry products in the Kenyan market in the recent past, calls for the rearing of the guinea fowls which are birds reared for meat and partly for eggs. In order to have an efficient production of poultry products for this type of poultry farming, there is need for an efficient modeling using sound statistical methodologies. It’s in this regard that the study modeled Guinea Fowl production in Kenya using the Univariate Auto-Regressive Integrated Moving Average (ARIMA) and the Auto-Regressive Fractional Integrated Moving Average (ARFIMA) models. Yearly guinea fowl production data for the period of 2010 to 2019 obtained from Food and Agricultural Organization (FAO-Kenya) was used in the study in which the Augmented Dickey Fuller (ADF) test was used to check for stationarity while the Hurst Exponent was used to test the long-memory property of the series. The ARIMA and ARFIMA models gave a better fit to the data and were used to forecast Guinea Fowl Weights. Fitted model forecast were evaluated via the Random Mean Squared Error (RMSE) in which the ARFIMA model was found to give a better forecast of the Guinea Fowl weights compared to the ARIMA model.

Fractional Dynamics of PMU Data

Novel dynamics are emerging in the power system due to the new Smart Grid (SG) environment. The high sampling rate of the Phasor Measurement Units (PMUs) enables them to capture the dynamic fluctuations in the power system measurements. Understanding the statistical and dynamic characteristics of the PMU data requires advanced data analytics techniques capable of performing accurate modeling of the power system variables (voltage, frequency, phase angle, and rate of change of frequency (ROCOF)). In this article, we provide evidence of the non-stationarity and fractality of PMU data collected from Europe. We adopt the Autoregressive Fractionally Integrated Moving Average (ARFIMA) models with non-integer differencing parameter to model the short-range and long-range correlations in the PMU data. Furthermore, the goodness-of-fit of the ARFIMA model is confirmed by analyzing the correlation and independence of the model residuals. Anomaly detection is among the promising applications of the PMU ARFIMA models. It is shown that the 2012 Indian blackout is accompanied by a change point in the differencing parameter opening the road to event (anomaly) detection by ARFIMA monitoring.

Forecasting COVID-19 pandemic using optimal singular spectrum analysis.

Coronavirus disease 2019 (COVID-19) is a pandemic that has affected all countries in the world. The aim of this study is to examine the potential advantages of Singular Spectrum Analysis (SSA) for forecasting the number of daily confirmed cases, deaths, and recoveries caused by COVID-19, which are the three main variables of interest. This paper contributes to the literature on forecasting COVID-19 pandemic in several ways. Firstly, an algorithm is proposed to calculate the optimal parameters of SSA including window length and the number of leading components. Secondly, the results of two forecasting approaches in the SSA, namely vector and recurrent forecasting, are compared to those from other commonly used time series forecasting techniques. These include Autoregressive Integrated Moving Average (ARIMA), Fractional ARIMA (ARFIMA), Exponential Smoothing, TBATS, and Neural Network Autoregression (NNAR). Thirdly, the best forecasting model is chosen based on the accuracy measure Root Mean Squared Error (RMSE), and it is applied to forecast 40 days ahead. These forecasts can help us to predict the future behaviour of this disease and make better decisions. The dataset of Center for Systems Science and Engineering (CSSE) at Johns Hopkins University is adopted to forecast the number of daily confirmed cases, deaths, and recoveries for top ten affected countries until October 29, 2020. The findings of this investigation show that no single model can provide the best model for any of the countries and forecasting horizons considered here. However, the SSA technique is found to be viable option for forecasting the number of daily confirmed cases, deaths, and recoveries caused by COVID-19 based on the number of times that it outperforms the competing models.

Stochastic Models for Radon Daily Time Series: Seasonality, Stationarity, and Long-Range Dependence Detection

In this paper, the presence of seasonality, stationarity, and long-range memory structures are detected in daily radon measurements in a permanent monitoring station in central Italy. The transient dynamics and the seasonality structure are identified by power spectral analysis based on the continuous wavelet transformation and a clear 1-year periodicity emerges. The stationarity in the data is assessed with the Dickey Fuller test; the decay of the estimated autocorrelation function and the estimated Hurst exponent indicate the presence of long-range dependence. All the main characteristics of the data have been properly included in a modelling structure. In particular, an autoregressive fractionally integrated moving average (ARFIMA) model is estimated and compared with the classical ARMA and ARIMA models in terms of goodness of fit and, secondarily, of forecast evaluation. An autoregressive model with a non-integer values of the differencing parameter ($d=0.278$) resulted to be the most appropriate on the basis of Akaike Information Criterion, the diagnostic on the residuals, and the Root Mean Squared Error. The results suggest that there is statistically-significant evidence for not rejecting the presence of long memory in the radon concentration. The radon measurements are better characterised as being stationary, but with long memory and so the statistical dependence decays more slowly than an exponential decay.

Forecasting long range dependent time series with exogenous variable using ARFIMAX model

Time series analysis and forecasting is one of the challenging issues of statistical modelling. Modelling of price and forecasting is a vital matter of concern for both the farming community and policy makers, especially in agriculture. Many practical agricultural data, principally commodity price data shows the typical feature of long memory process or long range dependency. For capturing the long memory behavior of the data Autoregressive Fractionally Integrated Moving Average (ARFIMA) model is generally used. Sometimes, in time series data besides the original series, data on some auxiliary or exogenous variables may be available or can be made available with a lower cost; like besides the market prices of commodities, market arrivals for that commodity may be available and it affects the market price of commodities. This type of exogenous variable may be incorporated in existing model to improve the model performance and forecasting accuracy, like Autoregressive Fractionally Integrated Moving Average with exogenous variables (ARFIMAX) model. In the present study undertaken at ICAR-IASRI, New Delhi during 2019, daily maximum and modal price of potato of Agra market of UP, India are taken along with daily market arrival. Both the ARFIMA and ARFIMAX model with market arrival as exogenous variable are applied for the data under study. Comparative studies of the fitted models are employed by using the Relative Mean Absolute Percentage Error (RMAPE) and Root Mean Square Error (RMSE) criteria. We could establish superiority of the ARFIMAX model over the ARFIMA model in terms of modeling and forecasting efficiency.

Detection of DoS Attacks Using ARFIMA Modeling of GOOSE Communication in IEC 61850 Substations

Integration of Information and Communication Technology (ICT) in modern smart grids (SGs) offers many advantages including the use of renewables and an effective way to protect, control and monitor the energy transmission and distribution. To reach an optimal operation of future energy systems, availability, integrity and confidentiality of data should be guaranteed. Research on the cyber-physical security of electrical substations based on IEC 61850 is still at an early stage. In the present work, we first model the network traffic data in electrical substations, then, we present a statistical Anomaly Detection (AD) method to detect Denial of Service (DoS) attacks against the Generic Object Oriented Substation Event (GOOSE) network communication. According to interpretations on the self-similarity and the Long-Range Dependency (LRD) of the data, an Auto-Regressive Fractionally Integrated Moving Average (ARFIMA) model was shown to describe well the GOOSE communication in the substation process network. Based on this ARFIMA-model and in view of cyber-physical security, an effective model-based AD method is developed and analyzed. Two variants of the statistical AD considering statistical hypothesis testing based on the Generalized Likelihood Ratio Test (GLRT) and the cumulative sum (CUSUM) are presented to detect flooding attacks that might affect the availability of the data. Our work presents a novel AD method, with two different variants, tailored to the specific features of the GOOSE traffic in IEC 61850 substations. The statistical AD is capable of detecting anomalies at unknown change times under the realistic assumption of unknown model parameters. The performance of both variants of the AD method is validated and assessed using data collected from a simulation case study. We perform several Monte-Carlo simulations under different noise variances. The detection delay is provided for each detector and it represents the number of discrete time samples after which an anomaly is detected. In fact, our statistical AD method with both variants (CUSUM and GLRT) has around half the false positive rate and a smaller detection delay when compared with two of the closest works found in the literature. Our AD approach based on the GLRT detector has the smallest false positive rate among all considered approaches. Whereas, our AD approach based on the CUSUM test has the lowest false negative rate thus the best detection rate. Depending on the requirements as well as the costs of false alarms or missed anomalies, both variants of our statistical detection method can be used and are further analyzed using composite detection metrics.

Combined Methodology for Linear Time Series Forecasting

Time series forecasting is an important type of quantitative model used to predict future values given a series of past observations for which the generation process is unknown. Two of the most well‐known methods for the modeling of linear time series are the autoregressive integrated moving average (ARIMA) and the autoregressive fractionally integrated moving average (ARFIMA). For different datasets, the number of past observations necessary for an accurate prediction may vary. Short and long memory dependency problems require different handling, with the ARIMA model being limited to the first, while the ARFIMA model was specifically developed for the latter. Preprocessing techniques and modification on specific components of these models are common approaches used to tackle the memory dependency problem in order to improve their accuracy. However, such solutions are specific to certain datasets. This paper proposes a new method that combines the short and long memory characteristics of the two aforementioned models in order to keep a low accumulative error in several different scenarios. Twelve public time series datasets were used to compare the performance of the proposed method with the original models. The results were also compared with two alternative methods from the literature used to deal with datasets of different memory dependencies. The new approach presented a lower error for the majority of the experiments, failing only for the datasets that contain a large number of features. © 2020 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Forecasting of New Cases of COVID-19 in Nigeria Using Autoregressive Fractionally Integrated Moving Average Models

The emergence of global pandemic known as COVID-19 has impacted significantly on human lives and measures have been taken by government all over the world to minimize the rate of spread of the virus, one of which is by enforcing lockdown. In this study, Autoregressive fractionally integrated moving average (ARFIMA) Models was used to model and forecast what the daily new cases of COVID-19 would have been ten days after the lockdown was eased in Nigeria and compare to the actual new cases for the period when the lockdown was eased. The proposed model ARFIMA model was compared with ARIMA (1, 0, 0), and ARIMA (1, 0, 1) and found to outperform the classical ARIMA models based on AIC and BIC values. The results show that the rate of spread of COVID-19 would have been significantly less if the strict lockdown had continued. ARFIMA model was further used to model what new cases of COVID-19 would be ten days ahead starting from 31st of August 2020. Therefore, this study recommends that government should further enforce measures to reduce the spread of the virus if business must continue as usual.

Should Deep Learning Models be in High Demand, or Should They Simply be a Very Hot Topic? A Comprehensive Study for Exchange Rate Forecasting

Exchange rate movements can significantly impact not only foreign trade, capital flows, and asset portfolio management, but also real economic activity. Therefore, the forecast of exchange rates has always been of great interest among academics, economic agents, and institutions. However, exchange rate series are essentially dynamic and nonlinear in nature and thus, forecasting exchange rates is a difficult task. On the other hand, deep learning models in solving time series forecasting tasks have been proposed in the last half-decade. But the number of formal comparative study in terms of exchange rate forecasting with deep learning models is quite limited. For this purpose, this study applies ten different models (Random Walk, Autoregressive Moving Average, Threshold Autoregression, Autoregressive Fractionally Integrated Moving Average, Support Vector Regression, Multilayer Perceptron, Recurrent Neural Network, Long Short Term Memory, Gated Recurrent Unit and Autoregressive Moving Average-Long Short Term Memory Hybrid Models) and two forecasting modes (recursive and rolling window) to predict three major exchange rate returnsnamely, the Canadian dollar, Australian dollar and British pound against the US Dollar in monthly terms. To evaluate the forecasting performances of the models, we used Model Confidence Set procedure as an advanced test. According to our results, the proposed hybrid model produced the best out-of-sample forecast performance in all samples, without exception.

Inference for Long-memory Time Series Models Based on Modified Empirical Likelihood

Empirical likelihood method has been applied to short-memory time series models by Monti (1997) through the Whittle's estimation method. Yau (2012) extended this idea to long-memory time series models. Asymptotic distributions of the empirical likelihood ratio statistic for short and long-memory time series have been derived to construct confidence regions for the corresponding model parameters. However, it experiences the undercoverage issue which causes the coverage probabilities of parameters lower than the given nominal levels, especially for small sample sizes. In this paper, we propose a modified empirical likelihood which combines the advantages of the adjusted empirical likelihood and the transformed empirical likelihood to modify the one proposed by Yau (2012) for autoregressive fractionally integrated moving average (ARFIMA) model for the purpose of improving coverage probabilities.Asymptotic null distribution of the test statistic has been established as the standard chi-square distribution with the degree of freedom one. Simulations have been conducted to investigate the performance of the proposed method as well as the comparisons of other existing methods to illustrate that the proposed method can provide better coverage probabilities especially for small sample sizes.