ARFIMA Research Articles

Detection of long‐range dependence and estimation of fractal exponents through ARFIMA modelling

We evaluate the performance of autoregressive, fractionally integrated, moving average (ARFIMA) modelling for detecting long-range dependence and estimating fractal exponents. More specifically, we test the procedure proposed by Wagenmakers, Farrell, and Ratcliff, and compare the results obtained with the Akaike information criterion (AIC) and the Bayes information criterion (BIC). The present studies show that ARFIMA modelling is able to adequately detect long-range dependence in simulated fractal series. Conversely, this method tends to produce a non-negligible rate of false detections in pure autoregressive and moving average (ARMA) series. Generally, ARFIMA modelling has a bias favouring the detection of long-range dependence. AIC and BIC gave dissimilar results, due to the different weights attributed by the two criteria to accuracy and parsimony. Finally, ARFIMA modelling provides good estimates of fractal exponents, and could adequately complement classical methods, such as spectral analysis, detrended fluctuation analysis or rescaled range analysis.

Dynamic admission control in hybrid QoS networks with WiFi access

Providing quality of service (QoS) guarantees for multimedia applications over Hybrid networks by wired and wireless LANs is rapidly becoming an important issue. In order to protect the high priority traffic flows and improve network performance in a heavy-loaded network, a new dynamic admission control algorithm is proposed in this paper. We firstly propose a method to fit the fractional autoregressive integrated moving average (FARIMA) models to four actual wireless traffic traces and a traffic prediction based on the upper probability limit. Our experiments show that the one-step forecasts based on the FARIMA models are better than those based on autoregressive integrated moving average (ARIMA) models. We also propose a prediction-based admission control algorithm in hybrid QoS networks with WiFi access using the FARIMA models. The com- prehensive simulation experiments are performed to verify the effectiveness of the pro- posed admission control algorithm.

Assessing the forecasting accuracy of alternative nominal exchange rate models: the case of long memory

This paper presents an autoregressive fractionally integrated moving-average (ARFIMA) model of nominal exchange rates and compares its forecasting capability with the monetary structural models and the random walk model. Monthly observations are used for Canada, France, Germany, Italy, Japan and the United Kingdom for the period of April 1973 through December 1998. The estimation method is Sowell's (1992) exact maximum likelihood estimation. The forecasting accuracy of the long-memory model is formally compared to the random walk and the monetary models, using the recently developed Harvey, Leybourne and Newbold (1997) test statistics. The results show that the long-memory model is more efficient than the random walk model in steps-ahead forecasts beyond 1 month for most currencies and more efficient than the monetary models in multi-step-ahead forecasts. This new finding strongly suggests that the long-memory model of nominal exchange rates be studied as a viable alternative to the conventional models. Copyright © 2006 John Wiley & Sons, Ltd.

La prévisibilité des crises de change : essai de modélisation dans le cas de la Turquie

This paper presents a forecasting model of currency crises in the context of Turkey. It attempts to evaluate the relevance of the sensitivity indicators to a currency crisis. We elaborated an exchange market pressure index (IZT) as a warning signal and kept M2/RES ratio and overvaluation of the real rate exchange like explanatory variables. It emerges from our ARF IMA modélisation that most anticipated trajectories have been realized.

Expansions for approximate maximum likelihood estimators of the fractional difference parameter

Summary This paper derives second-order expansions for the distributions of the Whittle and profile plug-in maximum likelihood estimators of the fractional difference parameter in the 'Autoregressive Fractionally Integrated Moving Average of order (0, d, 0)' with unknown mean and variance. Both estimators are shown to be second-order pivotal. This extends earlier findings of Lieberman and Phillips (2004a, Econometric Theory, 20, 464-84), who derived expansions for the Gaussian maximum likelihood estimator under the assumption that the mean and variance are known. One implication of the results is that the parametric bootstrap upper one-sided confidence interval provides an o(n ~1 In η ) improvement over the delta method. For statistics that are not second-order pivotal, the improvement is generally only of the order o{n~l/2 Inn).

A FARIMA-based technique for oil slick and low-wind areas discrimination in sea SAR imagery

This paper introduces a new analysis technique, using the fractionally integrated autoregressive-moving average (FARIMA) model, to distinguish between low-wind and oil slick areas in high-resolution sea synthetic aperture radar (SAR) imagery. The method deals with the estimation of the fractional differencing and autoregressive-moving average parameters of the mean radial power spectral density of sea SAR images. The algorithm is applied and validated on dark areas corresponding to oil slicks, oil spills, and low-wind sea surface anomalies in European Remote Sensing 1 and 2 Precision Images of the Mediterranean Sea, North Sea, and Atlantic Ocean.

Temporal Aggregation of Stationary And Nonstationary Discrete‐Time Processes

Abstract. We study the autocorrelation structure and the spectral density function of aggregates from a discrete‐time process. The underlying discrete‐time process is assumed to be a stationary AutoRegressive Fractionally Integrated Moving‐Average (ARFIMA) process, after suitable number of differencing if necessary. We derive closed‐form expressions for the limiting autocorrelation function and the normalized spectral density of the aggregates, as the extent of aggregation increases to infinity. These results are then used to assess the loss of forecasting efficiency due to aggregation.

Power Computations for Intervention Analysis

In many intervention analysis applications, time series data may be expensive or otherwise difficult to collect. In this case the power function is helpful, because it can be used to determine the probability that a proposed intervention analysis application will detect a meaningful change. Assuming that an underlying autoregressive integrated moving average (ARIMA) or fractional ARIMA model is known or can be estimated from the preintervention time series, the methodology for computing the required power function is developed for pulse, step, and ramp interventions with ARIMA and fractional ARIMA errors. Convenient formulas for computing the power function for important special cases are given. Illustrative applications in traffic safety and environmental impact assessment are discussed.

Asymptotic self‐similarity and wavelet estimation for long‐range dependent fractional autoregressive integrated moving average time series with stable innovations

Abstract. Methods for parameter estimation in the presence of long‐range dependence and heavy tails are scarce. Fractional autoregressive integrated moving average (FARIMA) time series for positive values of the fractional differencing exponent d can be used to model long‐range dependence in the case of heavy‐tailed distributions. In this paper, we focus on the estimation of the Hurst parameter H = d + 1/α for long‐range dependent FARIMA time series with symmetric α‐stable (1 < α < 2) innovations. We establish the consistency and the asymptotic normality of two types of wavelet estimators of the parameter H. We do so by exploiting the fact that the integrated series is asymptotically self‐similar with parameter H. When the parameter α is known, we also obtain consistent and asymptotically normal estimators for the fractional differencing exponent d = H − 1/α. Our results hold for a larger class of causal linear processes with stable symmetric innovations. As the wavelet‐based estimation method used here is semi‐parametric, it allows for a more robust treatment of long‐range dependent data than parametric methods.

MPEG-encoded variable bit-rate video traffic modelling

Video streaming and conferencing are expected to be the major services in future communication systems. The design of communication systems to support high-quality video services requires the characterisation and modelling of video traffic. An accurate video traffic model for MPEG encoded video traffic proposed. An analysis of MPEG-encoded video traffic shows that I, P and B frame sizes within the same scene are cross-correlated. However, existing frame-level video traffic models often ignore this correlation. In contrast, a new model utilising the multinomial method (MM) is presented to capture this cross-correlation. The MM is used together with the spatial renewal process (SRP) to model MPEG-encoded VBR video traffic. The model is called SRP-MM. In addition the MM is utilised to enhance the existing model, the nested-autoregressive (nested-AR). The enhanced model is called nested-AR-MM. Simulation results demonstrate that the proposed models, the SRP-MM and nested-AR-MM, predict the empirical frame-size marginal distribution, autocorrelation and queuing performance closely. The nested-AR and fractional autoregressive integrated moving average (FARIMA) models from the existing literature are also implemented for comparison. It is shown that the proposed models outperform both nested-AR and FARIMA.

Modeling and Forecasting S&P 500 Volatility: Long Memory, Structural Breaks and Nonlinearity

textabstractThe sum of squared intraday returns provides an unbiased and almost error-free measure of ex-post volatility. In this paper we develop a nonlinear Autoregressive Fractionally Integrated Moving Average (ARFIMA) model for realized volatility, which accommodates level shifts, day-of-the-week effects, leverage effects and volatility level effects. Applying the model to realized volatilities of the S&P 500 stock index and three exchange rates produces forecasts that clearly improve upon the ones obtained from a linear ARFIMA model and from conventional time-series models based on daily returns, treating volatility as a latent variable.

Monitoring the parameter changes in general ARIMA time series models

We propose methods for monitoring the residuals of a fitted ARIMA or an autoregressive fractionally integrated moving average (ARFIMA) model in order to detect changes of the parameters in that model. We extend the procedures of Box & Ramirez (1992) and Ramirez (1992) and allow the differencing parameter, d to be fractional or integer. Test statistics are approximated by Wiener processes. We carry out simulations and also apply our method to several real time series. The results show that our method is effective for monitoring all parameters in ARFIMA models.

Structural breaks and their trace in the memory: Inflation rate series in the long-run

The aim of this paper is to illustrate the risks of neglecting the potential presence of structural changes in economic series when estimating their long memory parameters. To that end, our empirical exercise is carried out for the inflation series of the UK, Italy and Spain during the period of 1874–1998. The persistence of the inflation rates is analysed using an AutoRegressive Fractionally Integrated Moving Average (ARFIMA) model, and we find that the memory parameter is significantly reduced when structural changes are taken into account. Having identified these changes in an endogenous way, the fact that they correspond to war time eras, energy shocks or changes in monetary policy, allows us to think in terms of the probable existence of such economic structural changes. Thus, it seems plausible that these events have altered the mean of the process in the three series, introducing an upward bias in the estimated memory parameters.

A long memory test of the long-run Fisher effect in the G7 countries

The belief that short-term interest rates respond positively to changes in price level, commonly known as the Fisher effect, are currently being investigated extensively by financial researchers. Over the long run the hypothesis implies the presence of an equilibrium relationship between interest rates and inflation. Early evidence favouring the Fisher effect is found not to be consistent in certain time periods and some countries. This paper examines the presence of the effect in the G7 countries. An ARFIMA (Autoregressive Fractionally Integrated Moving Average) model is employed that generalizes the standard ARIMA by allowing fractional differencing. Based on the generalized ARFIMA estimation, the cointegration hypothesis between short-term interest rates and inflation cannot be supported. Interest rates in the G7 countries are not linked to inflation rate in the long run. The puzzling evidence rejecting the Fisher effect remains as the proposed relationship between interest rates and inflation is not r...

Wavelet-based estimators of scaling behavior

Various wavelet-based estimators of self-similarity or long-range dependence scaling exponent are studied extensively. These estimators mainly include the (bi)orthogonal wavelet estimators and the wavelet transform modulus maxima (WTMM) estimator. This study focuses both on short and long time-series. In the framework of fractional autoregressive integrated moving average (FARIMA) processes, we advocate the use of approximately adapted wavelet estimators. For these ideal processes, the scaling behavior actually extends down to the smallest scale, i.e., the sampling period of the time series, if an adapted decomposition is used. But in practical situations, there generally exists a cutoff scale below which the scaling behavior no longer holds. We test the robustness of the set of wavelet-based estimators with respect to that cutoff scale as well as to the specific density of the underlying law of the process. In all situations, the WTMM estimator is shown to be the best or among the best estimators in terms of the mean-squared error (MSE). We also compare the wavelet estimators with the detrended fluctuation analysis (DFA) estimator which was previously proved to be among the best estimators which are not wavelet-based estimators. The WTMM estimator turns out to be a very competitive estimator which can be further generalized to characterize multiscaling behavior.

Purchasing Power Parity under the Gold Standard

1. Introduction Long-run purchasing power parity (LRPPP), mainstay of international economic theory, has proved difficult to back up empirically. Many attempts have been made using various time series of data, yet results continue to be mixed.' A standout exception to these mixed results is presented by Diebold, Husted, and Rush (1991), whose work has been interpreted as very strong evidence of LRPPP because of their across-the-board rejection of the unit root for large sample of real exchange rates. Baillie (1996, p. 51) cites Diebold, Husted, and Rush as reestablishing Purchasing Power Parity as meaningful long run concept despite the substantial mixed results of other studies. These mixed results are usually attributed to the power problems associated with standard unit root tests, especially for small samples. Diebold, Husted, and Rush (1991) overcome these shortcomings by loosening the structure of the model to allow longmemory process. Diebold, Husted, and Rush (1991) model real exchange rate behavior using autoregressive, fractionally integrated moving-average (ARFIMA) models. In doing so, they reject the unit root for 16 real exchange rates from the gold standard era in favor of fractionally integrated alternative hypothesis. A long-memory process is essentially stationary one, where the autocorrelations take far longer to decay than the exponential rate associated with the autoregressive moving average (ARMA) class (Baillie 1996, p. 6). This observation implies that purchasing power parity (PPP) holds, but in some cases only in the extreme long run. We agree with Diebold, Husted, and Rush (1991) that the structure of standard unit root tests is overly restrictive. However, we offer different solution to this problem. From standard unit root tests, one must conclude that either all deviations from PPP are permanent or all deviations dissipate over time. However, there is third possibility that has both theoretical and empirical support. Unit root tests do not allow the possibility that while most deviations dissipate, few remain as permanent shocks. Real exchange rates may show substantial mean reversion, but to changing mean rather than to the constant PPP value. This idea, which we call quasi purchasing power parity (QPPP), draws support from the BalassaSamuelson theory, which asserts that productivity shocks can have permanent effects on real exchange rates. Samuelson (1994, p. 202) criticizes economists who tried to use PPP as guide even when substantive microeconomic changes had taken place between the previous putative equilibrium period and the new status quo. He cites an example from the 1980s when the actual yen/dollar exchange rate was 150 yen per dollar. According to PPP, the equilibrium exchange rate was 200 yen per dollar, yet the rate proceeded to drop to 110 yen per dollar and below. In QPPP terms, this situation would represent not lack of mean reversion, but evidence of an equilibrium that had shifted away from PPP This is form of the Penn effect, whereby a rich country, in comparison with poor one, will be estimated to be richer than it really is if you pretend that the simplified Cassel version of purchasing-power parity ... is correct (Samuelson 1994, p. 201). We assert that this Penn effect can occur not just between rich and poor countries, but between any two countries if one of these countries is affected by permanent external shock. QPPP also supports Engel's (1996, p. 1) proposition that the real exchange rate evolves as the sum of two processes-a stationary but persistent component and non-stationary component. It is generally held that while the exchange rate deviates from its long-run equilibrium value in the short run, it eventually converges to that equilibrium value in the long run. The question, then, is, What is that long-run equilibrium value? Some argue it is the PPP level, while others argue that there is complex set of factors determining the long-run value, including such things as the relative labor productivities at home and abroad (Engel 1996, p. …

Long-Memory versus Option-Implied Volatility Predictions

Volatility is a critical parameter in virtually all option pricing models. But the closer we look at volatility, the harder it seems to be to model it correctly. The constant volatility assumption of early pricing models is clearly inadequate. GARCH family models make volatility a function of the asset price process, and stochastic volatility models bring in a second stochastic factor that affects volatility movements. These approaches make sense in theory, but empirically volatility shocks seem to be too persistent to be consistent with them. A further confounding factor is that implied volatilities extracted from option prices in the market are widely thought to give more accurate predictions of future realized volatility, but don9t obey any of these models exactly. In this article, Li examines a volatility model with “long memory,” meaning that it can be made to fit the slow-decay feature of market volatilities. He also introduces a much more extensive data series, with intraday observations every five minutes. Using such a dense price series, realized volatility becomes observable, and the vast number of data points allows precise estimation of model parameters. For exchange rates on the deutsche mark, the yen and the British pound, the ARFIMA (“Autoregressive Fractionally Integrated Moving Average”) model is shown to give a better fit to volatility behavior than the alternative model, and it beats implied volatility substantially in the standard tests of forecasting performance.

ESTIMATION OF PARAMETERS IN ARFIMA PROCESSES: A SIMULATION STUDY

It is known that, in the presence of short memory components, the estimation of the fractional parameter d in an Autoregressive Fractionally Integrated Moving Average, ARFIMA(p, d, q), process has some difficulties (see [1]). In this paper, we continue the efforts made by Smith et al. [1] and Beveridge and Oickle [2] by conducting a simulation study to evaluate the convergence properties of the iterative estimation procedure suggested by Hosking [3]. In this context we consider some semiparametric approaches and a parametric method proposed by Fox-Taqqu[4]. We also investigate the method proposed by Robinson [5] and a modification using the smoothed periodogram function.

Self-similar texture modeling using FARIMA processes with applications to satellite images

A texture model for synthetic aperture radar (SAR) images is presented. Specifically, a sea surface in satellite images is modeled using the two-dimensional (2-D) fractionally integrated autoregressive-moving average (FARIMA) process with a non-Gaussian white driving sequence. The FARIMA process is an ARMA type model which is asymptotically self-similar. It captures the long-range as well as short-range spatial dependence structure of an image with a small number of parameters. To estimate these parameters, an efficient estimation procedure based on a spectral fit is presented. Real-life ocean surveillance radar images collected by the RADARSAT sensor are used to evaluate the practicality of this FARIMA approach. Using the radial power spectral density, the new model is shown to provide a more accurate description of the SAR images than the conventional moving-average (MA), autoregressive (AR), and fractionally differenced (FD) models.

A seasonal fractional ARIMA Model applied to the Nile River monthly flows at Aswan

A seasonal fractional autoregressive integrated moving average (ARIMA) model is introduced, with both short‐ and long‐term persistent periodic components. The estimation of the parameters is carried out by applying the Whittle's approximation to the Gaussian maximum likelihood function, which yields asymptotically consistent estimates. This method is particularly useful for hydrological time series. It is applied here to the Nile River monthly flows at Aswan in order to detect whether long memory is present. The results are compared with ones obtained by applying heuristic procedures, some of which were developed recently, in order to see how these perform on seasonal data.