ARFIMA Model Research Articles

A New Variant of ARFIMA Process and Its Predictive Ability

ARFIMA models generated an enormous amount of interest in the literature about three decades ago. However, this interest vaned after Granger (1999) showed that an ARFIMA process might have stochastic properties that do not mimic the properties of the data at all. The empirical results of our research in which we used exchange rate data for the analysis, show that a variant of an ARFIMA process indeed can beat the ARFIMA, the Random Walk and the ARMA process of the order one in out of sample forecasting. This indirectly indicates that our variant of the ARFIMA process can be considered as the data generating process for the long memory time series.

Changing-regime volatility: a fractionally integrated SETAR model

This article presents a 2-regime SETAR model with different long-memory processes in both regimes. We briefly present the memory properties of this model and propose an estimation method. Such a process is applied to the absolute and squared returns of five stock indices. A comparison to simple ARFIMA models is made using some forecastibility criteria. Our empirical results suggest that our model offers an interesting alternative competing framework to describe the persistent dynamics in modelling the returns.

Model Selection and Estimation of Long-Memory Time-Series Models

An exploratory estimation of ARFIMA(p,d,q) models on agricultural spot and futures markets showed us that the estimated d is quite sensitive to the number of lags included in the short-term dynamics. AIC and SIC agreed there were many lags but, familiarly, disagreed on how many. To address this issue, I run a series of Monte Carlo experiments and test the performance (i) of the AIC and the SIC in selecting p and q and (ii) of the AIC, the SIC and the multimodel-inference approach of Burnham and Anderson (2002) in estimating d. I contribute to the literature by studying high-order data generating processes-up to (8,d,8) rather than (2,d,0); by testing also the MMI-approach; and by studying the impact of excluding models close to the data generating process from the set of candidate models. Three findings stand out. First, the familiar result that, in terms of order selection, the SIC outperforms the AIC for low-order models gets reversed for high-order models. Second, for inference on the presence or absence of fractional integration, I find that the SIC still dominates both the AIC and the MMI-approach. Third, set-up snooping (if the true model is also a candidate model) has little impact.

Testing for Threshold Effect in ARFIMA Models: Application to US Unemployment Rate Data

Macroeconomic time series often involve a threshold effect in their ARMA representation, and exhibit long memory features. In this paper we introduce a new class of threshold ARFIMA models to account for this. The threshold effect is introduced in the autoregressive and/or the fractional integration parameters, and can be tested for using LM tests. Monte Carlo experiments show the desirable finite sample size and power of the test with an exact maximum likelihood estimator of the long memory parameter. Simulations also show that a model selection strategy is available to discriminate between the competing threshold ARFIMA models. The methodology is applied to US unemployment rate data where we find a significant threshold effect in the ARFIMA representation and a better forecasting performance over TAR and symmetric ARFIMA models.

Rational Speculative Bubbles: An Empirical Investigation of the Middle East and North African Stock Markets

Despite recent extreme fluctuations of the Middle East and North African (MENA) stock markets, we do not find strong evidence of rational speculative bubbles in the perspective of both domestic and U.S.-based investors. Fractional integration tests built on ARFIMA models do not support the possibility of bubbles in the MENA stock markets. Similarly, duration dependence tests based on nonparametric Nelson-Aalen hazard functions not only reject the existence of bubbles but also support equality of hazard functions between domestic and U.S.-based investors without regard to the rapid financial liberalization and integration in the MENA stock markets.

Forecasting volatility of SSEC in Chinese stock market using multifractal analysis

In this paper, taking about 7 years’ high-frequency data of the Shanghai Stock Exchange Composite Index (SSEC) as an example, we propose a daily volatility measure based on the multifractal spectrum of the high-frequency price variability within a trading day. An ARFIMA model is used to depict the dynamics of this multifractal volatility (MFV) measures. The one-day ahead volatility forecasting performances of the MFV model and some other existing volatility models, such as the realized volatility model, stochastic volatility model and GARCH, are evaluated by the superior prediction ability (SPA) test. The empirical results show that under several loss functions, the MFV model obtains the best forecasting accuracy.

The long memory model of political support: some further results

This article extends the results of Byers et al. (1997) on long memory in support for the Conservative and Labour Parties in the UK using longer samples and additional poll series. It finds continuing support for the ARFIMA(0, d, 0) model, though with somewhat smaller values of the long memory parameter. We find that the move to telephone polling in the mid-1990s had no apparent effect on the estimated value of d for either party. Finally, we find that we cannot reject the hypotheses that the parties share a common long memory parameter which we estimate at around 0.65.

Linkages between the center and periphery stock prices: Evidence from the vector ARFIMA model

This paper provides evidence on the effects of US equity markets on those of selected emerging markets, using the vector fractionally integrated autoregressive moving-average (VARFIMA) model. This model has not so far been employed in examining the interdependence among the equity markets of the world countries. The paper employs four-variate VARFIMA model which allows us to conduct the empirical analysis without transforming the raw data and captures the fractal dynamics. The findings show that while the stock price series of some markets are non-stationary, but mean-reverting, those of some others are non-stationary and non-mean reverting. The more significant finding is that the S&P500 has permanent effects on the stock prices of the emerging markets included in the sample.

Central Bank Forex Interventions Assessed Using Realized Moments

This paper studies and assesses the impact of G3 Central Bank interventions on the DEM/USD exchange rate properties using daily realized moments of exchange rate returns (obtained from intraday data) for the period 1989-2001. Event studies in terms of the realized moments for the intervention day, the days preceding and following the intervention day illustrate the shape of this impact. Rolling regressions results for an ARFIMA model for realized moments are used to measure the intervention impact and characterize its significance. The analysis confirms previous findings of an increase of volatility after a coordinated Central Bank intervention. It highlights new findings on the timing and the persistence of coordinated interventions on exchange rate volatility, on important volatility spillovers, on the impact on exchange rate covariances and correlations and on skewness coefficients.

Fractal models for event-based and dynamical timers

Some recent papers proposed to distinguish between event-based and emergent timing. Event-based timing is conceived as prescribed by events produced by a central clock, and seems to be used in discrete tasks (e.g., finger tapping). Emergent or dynamical timing refers to the exploitation of the dynamical properties of effectors, and is typically used in continuous tasks (e.g., circle drawing). The analysis of period series suggested that both timing control processes possess fractal properties, characterized by self-similarity and long-range dependence. The aim of this article is to present two models that produce period series presenting the statistical properties previously evidenced in discrete and continuous rhythmic tasks. The first one is an adaptation of the classical activation/threshold models, including a plateau-like evolution of the threshold over time. The second one is a hybrid limit-cycle model, including a time-dependent linear stiffness parameter. Both models reproduced satisfactorily the spectral signatures of event-based and dynamical timing processes, respectively. The models also produced auto-correlation functions similar to those experimentally observed. Using ARFIMA modeling we show that these simulated series possess fractal properties. We suggest in conclusion some possible extensions of this modeling approach, to account for the effects of metronomic pacing, or to analyze bimanual coordination.

Previsão de preços futuros de Commodities agrícolas com diferenciações inteira e fracionária, e erros heteroscedásticos

O presente trabalho tem como objetivo modelar séries temporais para efeito de previsão com diferenciações inteira e fracionária, utilizando dados de preços futuros de commodities agrícolas. Modelos de séries temporais do tipo ARMA/ARIMA (diferenciação inteira) serão estimados como termo de comparação com os modelos do tipo ARFIMA (diferenciação fracionária). Em ambos os casos, os erros dos modelos serão estimados assumindo-se a possibilidade de estimação da volatilidade. O poder de previsão de cada modelo será comparado pelo critério do erro quadrado médio da previsão (EQM). A estimação do termo de diferenciação fracionário (d) também será utilizada para examinar as características de longa dependência das séries. Os resultados indicaram que todas as séries de retornos de preços futuros utilizados são estacionárias. O valor do d fracionário da série de açúcar indicou um comportamento de antipersistência a choques, enquanto que esses valores para as demais commodities apresentaram comportamento persistente. Na maioria dos casos os modelos ARFIMA mostraram um melhor poder de previsão.

A fractionally integrated autoregressive moving average approach to forecasting tourism demand

The primary aim of this paper is to incorporate fractionally integrated ARMA ( p, d, q) ( ARFIMA) models into tourism forecasting, and to compare the accuracy of forecasts with those obtained by previous studies. The models are estimated using the volume of monthly international tourist arrivals in Singapore. Empirical findings demonstrate the evidence that the approach we propose generates relatively lower sample mean absolute percentage errors (MAPEs). This study also deals with the volatile data faced by a forecaster. We use the Asian financial crisis and the September 11 event as examples of economic and political shocks. With respect to the objective of shaving the coefficient MAPE, forecasts based on the selected ARFIMA models dominate convincingly.

Forecasting Exchange Rate Volatility at High Frequency Data: Is the Euro Different?

This paper focuses on forecasting volatility of high frequency Euro exchange rates. Four 15 minute frequency Euro exchange rate series, including Euro/CHF, Euro/GBP, Euro/JPY and Euro/USD, are used to test the forecast performance of six models, including both traditional time series volatility models and the realized volatility model. Besides the normally used regression test and accuracy test, an equal accuracy test, the HLN-DM test, and a superior predictive ability test are also employed in the out-of-sample forecast evaluation. The FIGARCH model is found to be superior in almost all exchange rate series. Although the widely preferred ARFIMA model shows better performance than the traditional daily volatility models, generally speaking, it cannot surpass the FIGARCH model and the intraday GARCH model. Furthermore, the SVX model does not significantly outperform the SV model in the accuracy test, which contradicts the results of some earlier research. The paper confirms the advantage of using high frequency data and modelling the long memory factor. It also analyses the characteristics of Euro exchange rates and compares the test results with the conclusions drawn by previous studies.

Forecasting volatility and volume in the Tokyo Stock Market: Long memory, fractality and regime switching

We investigate the predictability of both volatility and volume for a large sample of Japanese stocks. The particular emphasis of this paper is on assessing the performance of long memory time series models in comparison to their short-memory counterparts. Since long memory models should have a particular advantage over long forecasting horizons, we consider predictions of up to 100 days ahead. In most respects, the long memory models (ARFIMA, FIGARCH and the recently introduced multifractal model) dominate over GARCH and ARMA models. However, while FIGARCH and ARFIMA also have quite a number of cases with dramatic failures of their forecasts, the multifractal model does not suffer from this shortcoming and its performance practically always improves upon the naïve forecast provided by historical volatility. As a somewhat surprising result, we also find that, for FIGARCH and ARFIMA models, pooled estimates (i.e. averages of parameter estimates from a sample of time series) give much better results than individually estimated models.

Optimal prediction with nonstationary ARFIMA model

AbstractWe propose two methods to predict nonstationary long‐memory time series. In the first one we estimate the long‐range dependent parameter d by using tapered data; we then take the nonstationary fractional filter to obtain stationary and short‐memory time series. In the second method, we take successive differences to obtain a stationary but possibly long‐memory time series. For the two methods the forecasts are based on those obtained from the stationary components. Copyright © 2007 John Wiley & Sons, Ltd.

Empirical study of ARFIMA model based on fractional differencing

In this paper, we studied the long-term memory of Hong Kong Hang Sheng index using MRS analysis, established ARFIMA model for it, and detailed the procedure of fractional differencing. Furthermore, we compared the ARFIMA model built by this means with the one that took first-order differencing as an alternative. The result showed that, if doing so, much useful information of time series would be lost. The forecast formula of ARFIMA model was corrected according to the method of fractional differencing, and was employed in the empirical study. It was illustrated that the forecast performance of ARFIMA model was not as good as we expected since the ARFIMA model was ineffective in forecasting Hang Sheng index. The certainty of this conclusion was proposed from two different aspects.

Likelihood‐based Analysis of a Class of Generalized Long‐Memory Time Series Models

Abstract. This article introduces a family of ‘generalized long‐memory time series models’, in which observations have a specified conditional distribution, given a latent Gaussian fractionally integrated autoregressive moving‐average (ARFIMA) process. The observations may have discrete or continuous distributions (or a mixture of both). The family includes existing models such as ARFIMA models themselves, long‐memory stochastic volatility models, long‐memory censored Gaussian models and others. Although the family of models is flexible, the latent long‐memory process poses problems for analysis. Therefore, we introduce a Markov chain Monte Carlo sampling algorithm and develop a set of recursions which makes it feasible. This makes it possible, among other things, to carry out exact likelihood‐based analysis of a wide range of non‐Gaussian long‐memory models without resorting to the use of likelihood approximations. The procedure also yields predictive distributions that take into account model parameter uncertainty. The approach is demonstrated in two case studies.

Invariance of the first difference in ARFIMA models

A desirable property for an estimator of the fractional ARFIMA parameter is to be first difference invariant. This paper investigates the effects on the fractional parameter estimator in nonstationary ARFIMA(p,d,q) processes before and after applying a first difference. We consider semiparametric and parametric approaches for estimating d. The study is based on a Monte Carlo simulation for different sample sizes. The Brazilian exchange rate series is given as an application of the methodology.

Bootstrap approaches and confidence intervals for stationary and non-stationary long-range dependence processes

This paper deals with different bootstrap approaches and bootstrap confidence intervals in the fractionally autoregressive moving average ( ARFIMA ( p , d , q ) ) process [J. Hosking, Fractional differencing, Biometrika 68(1) (1981) 165–175] using parametric and semi-parametric estimation techniques for the memory parameter d. The bootstrap procedures considered are: the classical bootstrap in the residuals of the fitted model [B. Efron, R. Tibshirani, An Introduction to the Bootstrap, Chapman and Hall, New York, 1993], the bootstrap in the spectral density function [E. Paparoditis, D.N Politis, The local bootstrap for periodogram statistics. J. Time Ser. Anal. 20(2) (1999) 193–222], the bootstrap in the residuals resulting from the regression equation of the semi-parametric estimators [G.C Franco, V.A Reisen, Bootstrap techniques in semiparametric estimation methods for ARFIMA models: a comparison study, Comput. Statist. 19 (2004) 243–259] and the Sieve bootstrap [P. Bühlmann, Sieve bootstrap for time series, Bernoulli 3 (1997) 123–148]. The performance of these procedures and confidence intervals for d in the stationary and non-stationary ranges are empirically obtained through Monte Carlo experiments. The bootstrap confidence intervals here proposed are alternative procedures with some accuracy to obtain confidence intervals for d.

Correlated Errors in the Parameters Estimation of the ARFIMA Model: A Simulated Study

Processes with correlated errors have been widely used in economic time series. The fractionally integrated autoregressive moving-average processes—ARFIMA(p, d, q)—(Hosking, 1981) have been explored to model stationary and non stationary time series with long-memory property. This work uses the Monte Carlo simulation method to evaluate the performance of some parametric and semiparametric estimators for long and short-memory parameters of the ARFIMA model with conditional heteroskedastic (ARFIMA-GARCH model). The comparison is based on the empirical bias and the mean squared error of each estimator.