Autoregressive Fractionally Integrated Moving Average Processes Research Articles

Consistent order selection for ARFIMA processes

Estimating the orders of the autoregressive fractionally integrated moving average (ARFIMA) model has been a long-standing problem in time series analysis. This paper tackles this challenge by establishing the consistency of the Bayesian information criterion (BIC) for ARFIMA models with independent errors. Since the memory parameter of the model can be any real number, this consistency result is valid for short memory, long memory and nonstationary time series. This paper further extends the consistency of the BIC to ARFIMA models with conditional heteroscedastic errors, thereby extending its applications to encompass many real-life situations. Finite-sample implications of the theoretical results are illustrated via numerical examples.

Long-Range Dependence in Financial Markets: A Moving Average Cluster Entropy Approach.

A perspective is taken on the intangible complexity of economic and social systems by investigating the dynamical processes producing, storing and transmitting information in financial time series. An extensive analysis based on the moving average cluster entropy approach has evidenced market and horizon dependence in highest-frequency data of real world financial assets. The behavior is scrutinized by applying the moving average cluster entropy approach to long-range correlated stochastic processes as the Autoregressive Fractionally Integrated Moving Average (ARFIMA) and Fractional Brownian motion (FBM). An extensive set of series is generated with a broad range of values of the Hurst exponent H and of the autoregressive, differencing and moving average parameters . A systematic relation between moving average cluster entropy and long-range correlation parameters H, d is observed. This study shows that the characteristic behaviour exhibited by the horizon dependence of the cluster entropy is related to long-range positive correlation in financial markets. Specifically, long range positively correlated ARFIMA processes with differencing parameter , and are consistent with moving average cluster entropy results obtained in time series of DJIA, S&P500 and NASDAQ. The findings clearly point to a variability of price returns, consistently with a price dynamics involving multiple temporal scales and, thus, short- and long-run volatility components. An important aspect of the proposed approach is the ability to capture detailed horizon dependence over relatively short horizons (one to twelve months) and thus its relevance to define risk analysis indices.

Inhomogeneous membrane receptor diffusion explained by a fractional heteroscedastic time series model.

Single particle tracking experiments have recently uncovered that the motion of cell membrane components can undergo changes of diffusivity as a result of the heterogeneous environment, producing subdiffusion and nonergodic behavior. In this paper, we show that an autoregressive fractionally integrated moving average (ARFIMA) with noise given by generalized autoregressive conditional heteroscedasticity (GARCH) can describe inhomogeneous diffusion in the cell membrane, where the ARFIMA process models anomalous diffusion and the GARCH process explains a fluctuating diffusion parameter.

Estimation of a stationary multivariate ARFIMA process

In this note, we consider an m-dimensional stationary multivariate long memory ARFIMA (AutoRegressive Fractionally Integrated Moving Average) process, which is defined as : A ( L ) D ( L ) ( y 1 ( t ),..., y m ( t ))' = B ( L ) ( ∈ 1 ( t ),..., ∈ m ( t ))', where M ' denotes the transpose of the matrix M . We determine the minimum Hellinger distance estimator (MHDE) of the parameters of a stationary multivariate long memory ARFIMA. This method is based on the minimization of the Hellinger distance between the random function of f n (.) and a theoretical probability density f θ (.). We establish, under some assumptions, the almost sure convergence of the estimator and its asymptotic normality. Keywords: Stationary Multivariate ARFIMA process; Estimation; Long memory; Minimum Hellinger distance AMS 2010 Mathematics Subject Classification: 62F12, 62H12

Mathematical Models for Dynamics of Molecular Processes in Living Biological Cells. A Single Particle Tracking Approach

Abstract In this survey paper we present a systematic methodology of how to identify origins of fractional dynamics. We consider three models leading to it, namely fractional Brownian motion (FBM), fractional Lévy stable motion (FLSM) and autoregressive fractionally integrated moving average (ARFIMA) process. The discrete-time ARFIMA process is stationary, and when aggregated, in the limit, it converges to either FBM or FLSM. In this sense it generalizes both models. We discuss three experimental data sets related to some molecular biology problems described by single particle tracking. They are successfully resolved by means of the universal ARFIMA time series model with various noises. Even if the finer details of the estimation procedures are case specific, we hope that the suggested checklist will still have been of great use as a practical guide. In Appendices A-F we describe useful fractional dynamics identification and validation methods.

Jeffrey's divergence between ARFIMA processes

The symmetric Kullback–Leibler divergence known as Jeffrey's divergence (JD) has found applications in signal and image processing, from radar clutter modeling to texture analysis. Recently, several studies were done on the JD between ergodic wide-sense stationary autoregressive (AR) and/or moving average (MA) processes. It was shown that the so-called asymptotic JD increment can be useful to compare ergodic wide-sense stationary ARMA processes. An interpretation of the asymptotic JD increment was also proposed. It consists in calculating the power of the first process filtered by the inverse filter associated with the second process, and conversely. However, in some biomedical applications, econometrics and other areas, long-memory processes have rather to be studied. Therefore, this paper aims at addressing the JD between ergodic wide-sense stationary autoregressive fractionally integrated moving average (ARFIMA) processes. More particularly, we study the influence of the ARFIMA parameters on the value of the asymptotic JD increment. Then, we analyze if the interpretation of the asymptotic JD increment based on inverse filtering is still valid for this type of process. Finally, some simulation results illustrate the theoretical analysis.

Simulation analysis of multifractal detrended methods based on the ARFIMA process

The two-component autoregressive fractionally integrated moving average (ARFIMA) process and mix-correlated ARFIMA(MC-ARFIMA) are applied in this paper to generate artificial sequence with Hxy = 1/2(Hx + Hy) and Hxy < 1/2(Hx + Hy) respectively and simulate the results of multifractal detrended cross-correlation analysis (MFXDFA), multifractal detrending moving average cross-correlation analysis (MFXDMA), MFDCCA based on maximum overlap wavelet transform (MFDCCA-MODWT), and multifractal detrended partial cross-correlation analysis(MF-DPXA). The advantages and disadvantages of MFXDFA, MFXDMA(θ = 0,0.5,1), and MFDCCA-MODWT are compared to the long-memory of sequences. In the case of Hxy < 1/2(Hx + Hy), these three methods keep around the theoretical value with small fluctuations in a variety of sequence lengths. In the case of Hxy = 1/2(Hx + Hy), the curves are significantly stable and are slightly smaller than the theoretical value. The precision of these estimators may be influenced by the relationship between Hxy and 1/2(Hx + Hy). Multifractal features is detected and the result shows that MFXDMA-0 and MFXDMA-1 is optimal to detect the multifractality. An interesting finding is that MFDCCA-MODWOT performs best in both case of Hxy = 1/2(Hx + Hy) and Hxy < 1/2(Hx + Hy), but it performs worst to detect the multifractality. When Gaussian noise is added to the sequences with different long-memory levels, MFDPXA can eliminate the noise interference compared with MFXDFA, thereby verifying the effectiveness of this method.

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.

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.

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.

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.

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.

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.

MODELLING FOR THE WAVELET COEFFICIENTS OF ARFIMA PROCESSES

AbstractWe consider a model for the discrete nonboundary wavelet coefficients of autoregressive fractionally integrated moving average (ARFIMA) processes in each scale. Because the utility of the wavelet transform for the long‐range dependent processes, which many authors have explained in semi‐parametrical literature, is approximating the transformed processes to white noise processes in each scale, there have been few studies in a parametric setting. In this article, we propose the model from the forms of the (generalized) spectral density functions (SDFs) of these coefficients. Since the discrete wavelet transform has the property of downsampling, we cannot directly represent these (generalized) SDFs. To overcome this problem, we define the discrete non‐decimated nonboundary wavelet coefficients and compute their (generalized) SDFs. Using these functions and restricting the wavelet filters to the Daubechies wavelets and least asymmetric filters, we make the (generalized) SDFs of the discrete nonboundary wavelet coefficients of ARFIMA processes in each scale clear. Additionally, we propose a model for the discrete nonboundary scaling coefficients in each scale.

Forecasting a long memory process subject to structural breaks

We develop an easy-to-implement method for forecasting a stationary autoregressive fractionally integrated moving average (ARFIMA) process subject to structural breaks with unknown break dates. We show that an ARFIMA process subject to a mean shift and a change in the long memory parameter can be well approximated by an autoregressive (AR) model and suggest using an information criterion (AIC or Mallows’ Cp) to choose the order of the approximate AR model. Our method avoids the issue of estimation inaccuracy of the long memory parameter and the issue of spurious breaks in finite sample. Insights from our theoretical analysis are confirmed by Monte Carlo experiments, through which we also find that our method provides a substantial improvement over existing prediction methods. An empirical application to the realized volatility of three exchange rates illustrates the usefulness of our forecasting procedure. The empirical success of the HAR-RV model can be explained, from an econometric perspective, by our theoretical and simulation results.

Forecasting a Long Memory Process Subject to Structural Breaks

We develop an easy-to-implement method for forecasting a stationary auto-regressive fractionally integrated moving average (ARFIMA) process subject to structural breaks with unknown break dates. We show that an ARFIMA process subject to a mean shift and a change in the long memory parameter can be well approximated by an auto-regressive (AR) model and suggest using an information criterion (AIC or Mallows’ Cp) to choose the order of the approximate AR model. Our method avoids the issue of estimation inaccuracy of the long memory parameter and the issue of spurious breaks in finite sample. Insights from our theoretical analysis are confirmed by Monte Carlo experiments, through which we also find that our method provides a substantial improvement over existing prediction methods. An empirical application to the realized volatility of three exchange rates illustrates the usefulness of our forecasting procedure. The empirical success of the HAR-RV model is explained, from an econometric perspective, by our theoretical and simulation results.

The quasi maximum likelihood approach to statistical inference on a nonstationary multivariate ARFIMA process

We estimate the parameters of a nonstationary multivariate ARFIMA (AutoRegressive Fractionally Integrated Moving Average) process by the quasi likelihood approach. Then, we define the pseudo-spectral density of the process. Under some assumptions, we establish consistency, asymptotic normality.

Estimation par le minimum de distance de Hellinger dʼun processus ARFIMA

In this Note, we determine the minimum Hellinger distance estimate of an ARFIMA (AutoRegressive Fractionally Integrated Moving Average) process. The estimate minimizes the Hellinger distance between the probability density function of the innovation of the process and a parameterized random function. Under some assumptions, we establish the asymptotic properties of this estimate.

Bootstrap Estimation of Long-Range Dependence in Arfima Processes

Long-memory or long-range dependence (LRD) denotes the property of a time series to exhibit a significant dependence between very distant observations. The presence of long-memory is established in diverse fields, for example, in hydrology, finance, network traffic, psychology, etc. From a practical point of view an important issue is the estimation of LRD parameters, especially the estimation of the Hurst parameter proposed in [19]. The Hurst parameter, usually referred as the H parameter, indicates the intensity of LRD.Various methods for estimating the H parameter in a time series data are available some of which are described in [3]. They are validated by appealing to an asymptotic analysis supposing that the sample size of the time series converges to infinity. Taqqu et al. [37] investigated empirical performance of nine methods for estimating the LRD in simulated autoregressive fractionally integrated moving average (ARFIMA) processes for a fairly large sample size of 10000 terms.The bootstrap is a general statistical procedure for statistical inference, originally introduced by Efron [11] that generally yields better estimations in finite sample sizes, than the classical methods. In this paper, we consider a bootstrap technique, which uses blocks composed of cycles, to estimate the H parameter in ARFIMA processes. The bias and the root mean square error (RMSE) of five estimators and their corresponded bootstrap estimators are obtained by a Monte Carlo study.

Modification of autoregressive fractionally integrated moving average models for the estimation of persistence

In this paper, it is proposed to modify autoregressive fractionally integrated moving average (ARFIMA) processes by introducing an additional parameter to comply with the criticism of Hauser et al . (1999) that ARFIMA processes are not appropriate for the estimation of persistence, because of the degenerate behavior of their spectral densities at frequency zero. When fitting these modified ARFIMA processes to the US GNP, it turns out that the estimated spectra are very similar to those obtained with conventional ARFIMA models, indicating that, in this special case, the disadvantage of ARFIMA models cited by Hauser et al. (1999) does not seriously aff ect the estimation of persistence. However, according to the results of a goodness-of-fit test applied to the estimated spectra, both the ARFIMA models and the modified ARFIMA models seem to overfit the data in the neighborhood of frequency zero.