ARFIMA Research Articles

Identification and validation of stable ARFIMA processes with application to UMTS data

In this paper we present an identification and validation scheme for stable autoregressive fractionally integrated moving average (ARFIMA) time series. The identification part relies on a recently introduced estimator which is a generalization of that of Kokoszka and Taqqu and a new fractional differencing algorithm. It also incorporates a low-variance estimator for the memory parameter based on the sample mean-squared displacement. The validation part includes standard noise diagnostics and backtesting procedure. The scheme is illustrated on Universal Mobile Telecommunications System (UMTS) data collected in an urban area. We show that the stochastic component of the data can be modeled by the long memory ARFIMA. This can help to monitor possible hazards related to the electromagnetic radiation.

Modelling long memory in maximum and minimum temperature series in India

Time series analysis of weather data can be a very valuable tool to investigate its variability pattern and, maybe, even to predict short- and long-term changes in the time series. In this study, the long memory behaviour of monthly minimum and maximum temperature of India for the period 1901 to 2007 by means of fractional integration techniques has been investigated. The results show that the time series can be specified in terms of autoregressive fractionally integrated moving average (ARFIMA) process. Both the series were found to be integrated with orders of integration smaller than 0.5 ensuring the long memory stationarity. Wavelet methodology in frequency domain with Haar wavelet filter was applied in order to see the oscillation at different scale and at different time epochs of the series. Multiresolution analysis (MRA) was carried out to explore the local as well as global variations in both the temperature series over the years. The variability in minimum temperature is found to be more than maximum temperature. Though there is no clear significance trend in the temperature series in the long run, but there are pockets of change in the temperature pattern. The predictive ability of ARFIMA model was investigated in terms of relative mean absolute percentage error.

Using the hybrid Phillips curve with memory to forecast US inflation

AbstractThis paper adopts the Caputo fractional derivative to re-specify the hybrid Phillips curve as a dynamic process of inflation with memory. The Caputo fractional derivative contains a non-integer differencing order, providing the same insight for persistence as emphasized in the Autoregressive Fractionally Integrated Moving Average (ARFIMA) time series models. We utilize the hybrid Phillips curve with memory to forecast US inflation during 1967–2014. The results indicate that our model performs well against a traditional hybrid Phillips curve, an integrated moving average model and a naive random walk model in quasi-in-sample forecasts. In out-of-sample forecasts based on Consumer Price Index (CPI) and Personal Consumption Expenditure (PCE) data, we find that the forecasting performance of Phillips curve models depends on the sample period. Our model with CPI data can outperform others in out-of-sample forecasts during and after the most recent financial crisis (2006–2014).

Average run length of the long-memory autoregressive fractionally integrated moving average process of the exponential weighted moving average control chart

AbstractMeasurement of control chart efficiency by comparison of average run length (ARL) is widely implemented in quality control. The aim of this study is to evaluate the ARL, which is a solution of the integral equation obtained from the Exponential Weighted Moving Average (EWMA) statistic with a long-memory Autoregressive Fractionally Integrated Moving Average (ARFIMA) process. The derivation of the analytical ARL of the EWMA control chart and proof of the existence and uniqueness of the analytical ARL by Fixed Point theory are shown. Moreover, the numerical ARL carried out by the Composite Midpoint Rule technique of the EWMA control chart is demonstrated. A comparison between the analytical and numerical ARL is also illustrated. The findings indicated that analytical ARL of the EWMA control chart is more quickly computational than the numerical ARL. Therefore, the analytical ARL is an alternative method for measuring the efficiency of the EWMA control chart with the long-memory ARFIMA process.

Adjusted empirical likelihood for long-memory time-series models

The empirical likelihood method has been applied to short-memory time-series models by Monti through the Whittle’s estimation method. Yau extended this idea to long-memory time series models. Nordman and Lahiri showed a new formulation of empirical likelihood for inference of dependent data. 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, computing profile empirical likelihood function involving constrained maximization does not always have a solution, which leads to several drawbacks. In this article, we propose an adjusted empirical likelihood procedure to modify the one proposed by Yau for the autoregressive fractionally integrated moving average (ARFIMA) model. It guarantees the existence of a solution to the required maximization problem and maintains an asymptotic chi-squared distribution. Simulations have been carried out to illustrate that the adjusted empirical likelihood method for different long-time series models provides competitive confidence regions and coverage probabilities compared to the unadjusted ones, especially for small sample sizes.

Identifying ergodicity breaking for fractional anomalous diffusion: Criteria for minimal trajectory length.

In this paper, we study ergodic properties of α-stable autoregressive fractionally integrated moving average (ARFIMA) processes which form a large class of anomalous diffusions. A crucial practical question is how long trajectories one needs to observe in an experiment in order to claim that the analyzed data are ergodic or not. This will be solved by checking the asymptotic convergence to 0 of the empirical estimator F(n) for the dynamical functional D(n) defined as a Fourier transform of the n-lag increments of the ARFIMA process. Moreover, we introduce more flexible concept of the ε-ergodicity.

Network anomaly detection for railway critical infrastructure based on autoregressive fractional integrated moving average

The article proposes a novel two-stage network traffic anomaly detection method for the railway transportation critical infrastructure monitored using wireless sensor networks (WSN). The first step of the proposed solution is to find and eliminate any outlying observations in the analyzed parameters of the WSN traffic using a simple and fast one-dimensional quartile criterion. In the second step, the remaining data is used to estimate autoregressive fractional integrated moving average (ARFIMA) statistical models describing variability of the tested WSN parameters. The paper also introduces an effective method for the ARFIMA model parameters estimation and identification using Haslett and Raftery estimator and Hyndman and Khandakar technique. The choice of the “economically” parameterized form of the model was based on the compromise between the conciseness of representation and the estimation of the error size. To detect anomalous behavior, i.e., a potential network attack, the proposed detection method uses statistical relations between the estimated traffic model and its actual variability. The obtained experimental results prove the effectiveness of the presented approach and aptness of selection of the statistical models.

Ergodicity testing using an analytical formula for a dynamical functional of alpha-stable autoregressive fractionally integrated moving average processes

In this paper we study asymptotic behavior of a dynamical functional for an α-stable autoregressive fractionally integrated moving average (ARFIMA) process. We find an analytical formula for this important statistics and show its usefulness as a diagnostic tool for ergodic properties. The obtained results point to the very fast convergence of the dynamical functional and show that even for short trajectories one may obtain reliable conclusions on the ergodic properties of the ARFIMA process. Moreover we use the obtained theoretical results to illustrate how the dynamical functional statistics can be used in the verification of the proper model for an analysis of some biophysical experimental data.

Understanding Persistence to Avoid Underestimation of Collective Flood Risk

The assessment of collective risk for flood risk management requires a better understanding of the space-time characteristics of flood magnitude and occurrence. In particular, classic formulation of collective risk implies hypotheses concerning the independence of intensity and number of events over fixed time windows that are unlikely to be tenable in real-world hydroclimatic processes exhibiting persistence. In this study, we investigate the links between the serial correlation properties of 473 daily stream flow time series across the major river basins in Europe, and the characteristics of over-threshold events which are used as proxies for the estimation of collective risk. The aim is to understand if some key features of the daily stream flow data can be used to infer properties of extreme events making a more efficient and effective use of the available data. Using benchmark theoretical processes such as Hurst-Kolmogorov (HK), generalized HK (gHK), autoregressive fractionally integrated moving average (ARFIMA) models, and Fourier surrogate data preserving second order linear moments, our findings confirm and expand some results previously reported in the literature, namely: (1) the interplay between short range dependence (SRD) and long range dependence (LRD) can explain the majority of the serial dependence structure of deseasonalized data, but losing information on nonlinear dynamics; (2) the standardized return intervals between over-threshold values exhibit a sub-exponential Weibull-like distribution, implying a higher frequency of return intervals longer than expected under independence, and expected return intervals depending on the previous return intervals; this results in a tendency to observe short (long) inter-arrival times after short (long) inter-arrival times; (3) as the average intensity and the number of events over one-year time windows are not independent, years with larger events are also the more active in terms of number of events; and (4) persistence influences the distribution of the collective risk producing a spike of probability at zero, which describes the probability of years with no events, and a heavier upper tail, suggesting a probability of more extreme annual losses higher than expected under independence. These results provide new insights into the clustering of stream flow extremes, paving the way for more reliable simulation procedures of flood event sets to be used in flood risk management strategies.

An Overview of FIGARCH and Related Time Series Models

This paper reviews the theory and applications related to fractionally integrated generalized autoregressive conditional heteroscedastic (FIGARCH) models, mainly for describing the observed persistence in the volatility of a time series. The long memory nature of FIGARCH models allows to be a better candidate than other conditional heteroscedastic models for modeling volatility in exchange rates, option prices, stock market returns and inflation rates. We discuss some of the important properties of FIGARCH models inthis review. We also compare the FIGARCH with the autoregressive fractionally integrated moving average (ARFIMA) model. Problems related to parameter estimation and forecasting using a FIGARCH model are presented. The application of a FIGARCH model to exchange rate data is discussed. We briefly introduce some other models, that are closely related to FIGARCH models. The paper ends with some concluding remarks and future directions of research.

Long memory and ARFIMA modelling: The case of CPI inflation rate in Ghana

In macroeconomic theory and economic policy, changes in the general price level or the rate of inflation plays an essential role. Hence, for example, one of the motives behind the adoption of Inflation Targeting policy (IT) by Ghana and the treaty espoused by the European Monetary Union, known as the Maastricht Treaty, was the convergence of inflation rates. On the other hand there is a controversy about which is the order of integration in the inflation rates, some authors arguing that this variable is stationary I(0) Whittle others saying it is nonstationary I(1). In this study we examine the CPI inflation rates of Ghana from a different perspective allowing for fractional degrees of differentiation. Thus, the methodology is based on long memory or long-range dependence processes, using fractional integration and employing techniques based on Whittle parametric and semiparametric methods and autoregressive fractionally integrated moving average (ARFIMA) models. Standard I(0)/I(1) methods were also employed. Our findings indicate that long memory exists in the CPI inflation rate of Ghana. After processing fractional differencing and determining the short memory components, the following two models, ARFIMA(3,0.427,1) and ARFIMA(2,0.499,1) were respectively specified to describe the pre and post introduction of IT policy in May 2007. Consequently, the CPI inflation rate of Ghana is fractionally integrated and mean reverting. Long memory in financial time series has important implications for the critical explanation of financial time series behaviour, as it could provide an opportunity to earn speculative profits in financial markets and cast disbelief on the correctness of the EMH. For instance, when price changes exhibit long memory or long-range dependence, asset pricing models based on the Efficient Market Hypothesis (EMH) may overestimate or underestimate investment risk. Furthermore, the presence of long memory in inflation rates can provide vital information about the likely impact of shocks (e.g. demand/supply) on the economy with respect to time. The results obtained in this study would be very useful in setting up monetary policies or consolidating previous policies such as IT in order to enhance economic growth. Moreover, estimation of long memory in inflation rates can serve as an evaluation tool to assess the performance of monetary policy under different dispensations. Lastly, the presence of long memory can assist in identifying inflationary pressures in the economy.

Efficient Bayesian inference for natural time series using ARFIMA processes

Abstract. Many geophysical quantities, such as atmospheric temperature, water levels in rivers, and wind speeds, have shown evidence of long memory (LM). LM implies that these quantities experience non-trivial temporal memory, which potentially not only enhances their predictability, but also hampers the detection of externally forced trends. Thus, it is important to reliably identify whether or not a system exhibits LM. In this paper we present a modern and systematic approach to the inference of LM. We use the flexible autoregressive fractional integrated moving average (ARFIMA) model, which is widely used in time series analysis, and of increasing interest in climate science. Unlike most previous work on the inference of LM, which is frequentist in nature, we provide a systematic treatment of Bayesian inference. In particular, we provide a new approximate likelihood for efficient parameter inference, and show how nuisance parameters (e.g., short-memory effects) can be integrated over in order to focus on long-memory parameters and hypothesis testing more directly. We illustrate our new methodology on the Nile water level data and the central England temperature (CET) time series, with favorable comparison to the standard estimators. For CET we also extend our method to seasonal long memory.

Combining ARFIMA models and fuzzy time series for the forecast of long memory time series

Long memory time series are stationary processes in which there is a statistical long range dependency between the current value and values in different times of the series. Therefore, in this class of series, there is a slow decay of the autocorrelation function as the time difference increases. Many practical forecasting problems fall in this class, for instance, in financial time series, hydrology and earth sciences applications. This research introduces a hybrid method combining Auto Regressive Fractional Integrated Moving Average (ARFIMA) models and Fuzzy Time Series (FTS) for the forecast of long memory (long-range) time series. The proposed method is developed as one algorithm consisting of two phases. The first phase is related to the autoregressive part of the model, while the second phase is related to the Moving Average part. Based on these ideas, the combined ARFIMA and FTS model is introduced. For the parameter estimation of the model, Particle Swarm Optimization (PSO) method is selected, based on its performance on similar optimization problems. In order to illustrate the benefit and potential of the proposed ARFIMA–FTS method, it has been applied to the two stock index databases, namely Taiwan Capitalization Weighted Stock Index (TAIEX)and Dow Jones Industrial Average (DJIA), together with exchange rate data of nine main currencies versus USD. Based on the reported results, it is possible to conclude the superiority of the proposed hybrid method, compared with classical ARFIMA models and other methods in the literature.

Long memory and regime switching: A simulation study on the Markov regime-switching ARFIMA model

Abstract Recent research argues that if the cause of confusion between long memory and regime switching were properly controlled for, they could be effectively distinguished. Motivated by this idea, our study aims to distinguish between them of financial series. We firstly model long memory and regime switching via the Autoregressive Fractionally Integrated Moving Average (ARFIMA) and Markov Regime-Switching (MRS) models, respectively. Their finite-sample properties and the confusion are investigated via simulations. To control for the cause of this confusion, we propose the MRS–ARFIMA model. A Monte Carlo study shows that this framework can effectively distinguish between the pure ARFIMA and pure MRS processes. Furthermore, MRS–ARFIMA outperforms the ordinary ARFIMA model for data simulated from the MRS–ARFIMA process. Finally, empirical studies of hourly and five-minute Garman–Klass and realized volatility of the FTSE index is conducted to demonstrate the advantages and usefulness of the proposed MRS–ARFIMA framework compared with the ARFIMA and MRS models in practice.

Forecasting wholesale price of cluster bean using the Autoregressive Fractionally Integrated Moving-Average Model: the case of Sri Ganganagar of Rajasthan in India

This study explored modelling and forecasting of wholesale cluster beans (guar) monthly prices in Sri Ganganagar district of Rajasthan using Autoregressive Fractionally Integrated Moving-Average Model (ARFIMA). Descriptive statistics of the price series (2003-2015) shows maximum price of cluster beans recorded in Sri Ganganagar market was at price of ₹ 27431.00 in April 2012. Coefficient of variation indicates cluster beans (guar) prices in Sri Ganganagar highly volatile represented by value of 118.82 percent. The results show that, the compound rate growth of prices is about 3.38 percent per month which is higher than instantaneous growth rate of 1.45 percent . Furthermore, prices of cluster beans are are very high in the month of April. Similarly the month of March, May and December records high pric es in the year using the price seasonality indices. Based on the minimum AIC and BIC values, ARFIMA (1/2, 0.309, 1) was selected as the best fit model for forecasting of cluster bean prices. The mean absolute percentage error from the ARFIMA ( ½, 0.309, 1) forecasting from January 2003 to May 2015 is 12.27 percent. On the other hand, the mean absolute percentage error of ARFIMA ( ½, 0.309, 1) based on prediction from December 2014 to May 2015 is 7.049% . Out of sample forecasted cluster bean prices in the coming months of 2015 in Sri Ganganagar (from June to December 2015) has also been presented.

The long-term linkages between direct and indirect property in Australia

Purpose– Investment in Australia’s property market, whether directly or indirectly through Australian real estate investment trusts (A-REITs), grew remarkably since the 1990s. The degree of segregation between the property market and other financial assets, such as shares and bonds, can influence the diversification benefits within multi-asset portfolios. This raises the question of whether direct and indirect property investments are substitutable. Establishing how information transmits between asset classes and impacts the predictability of returns is of interest to investors. The paper aims to discuss these issues.Design/methodology/approach– The authors study the linkages between direct and indirect Australian property sectors from 1985 to 2013, with shares and bonds. This paper employs an Autoregressive Fractionally Integrated Moving Average (ARFIMA) process to de-smooth a valuation-based direct property index. The authors establish directional lead-lag relationships between markets using bi-variate Granger causality tests. Johansen cointegration tests are carried out to examine how direct and indirect property markets adjust to an equilibrium long-term relationship and short-term deviations from such a relationship with other asset classes.Findings– The authors find the use of appraisal-based property data creates a smoothing bias which masks the extent of how information is transmitted between the indirect property sector, stock and bond markets, and influences returns. The authors demonstrate that an ARFIMA process accounting for a smoothing bias up to lags of four quarters can overcome the overstatement of the smoothing bias from traditional AR models, after individually appraised constituent properties are aggregated into an overall index. The results show that direct property adjusts to information transmitted from market-traded A-REITs and stocks.Practical implications– The study shows direct property investments and A-REITs are substitutible in a multi-asset portfolio in the long and short term.Originality/value– The authors apply an ARFIMA(p,d,q) model to de-smooth Australian property returns, as proposed by Bond and Hwang (2007). The authors expect the findings will contribute to the discussion on whether direct property and REITs are substitutes in a multi-asset portfolio.

Monte Carlo simulation for comparison of different estimators of long memory parameter: An application of ARFIMA model for forecasting commodity price

Time series with long memory or long-range dependence occurs frequently in agricultural commodity prices. For describing long memory, fractional integration is considered. The autoregressive fractionally integrated moving-average (ARFIMA) model along with its different estimation procedures is inve stigated. For the present investigation, the daily spot prices of mustard in Mumbai market are used. Autocorrelation (ACF) and partial autocorrelation (PACF) functions showed a slow hyperbolic decay indicating the presence of long memory. On the basis of minimum AIC values, the best model is identified for each series. Evaluation of forecasting is carried out with root mean squares prediction error (RMSPE), mean absolute prediction error (MAPE) and relative mean absolute prediction error (RMAPE). The residuals of the fitted models were used for diagnostic checking. Long memory parameter of ARFIMA model is computed by Geweke and Porter-Hudak (GPH), Gaussian semiparametric and wavelet method by using Maximal overlap discrete wavelet transform (MODWT). To this end, a comparison in the performance of different estimation procedures is carried out by Monte Carlo simulation technique. The R software package has been used for data analysis.

From physical linear systems to discrete-time series. A guide for analysis of the sampled experimental data.

Modeling physical data with linear discrete-time series, namely, the autoregressive fractionally integrated moving average (ARFIMA) model, is a technique that has attracted attention in recent years. However, this model is used mainly as a statistical tool only, with weak emphasis on the physical background of the model. The main reason for this lack of attention is that the ARFIMA model describes discrete-time measurements, whereas physical models are formulated using continuous-time parameters. In order to eliminate this discrepancy, we show that time series of this type can be regarded as sampled trajectories of the coordinates governed by a system of linear stochastic differential equations with constant coefficients. The observed correspondence provides formulas linking ARFIMA parameters and the coefficients of the underlying physical stochastic system, thus providing a bridge between continuous-time linear dynamical systems and ARFIMA models.

Volatility Behaviors of Financial Time Series by Percolation System on Sierpinski Carpet Lattice

The financial time series is simulated and investigated by the percolation system on the Sierpinski carpet lattice, where percolation is usually employed to describe the behavior of connected clusters in a random graph, and the Sierpinski carpet lattice is a graph which corresponds the fractal — Sierpinski carpet. To study the fluctuation behavior of returns for the financial model and the Shanghai Composite Index, we establish a daily volatility measure — multifractal volatility (MFV) measure to obtain MFV series, which have long-range cross-correlations with squared daily return series. The autoregressive fractionally integrated moving average (ARFIMA) model is used to analyze the MFV series, which performs better when compared to other volatility series. By a comparative study of the multifractality and volatility analysis of the data, the simulation data of the proposed model exhibits very similar behaviors to those of the real stock index, which indicates somewhat rationality of the model to the market application.

Uncertainty, model selection, and the PPP puzzle

Ministerio de Ciencia e Innovacion (ECO2011–30260–C03–02 to M.D.G., ECO2011–25293 to L.M.); Axa research fund