ARFIMA Process Research Articles

Short and long memory in stock returns data

The properties of an iterative procedure for the estimation of the parameters of an ARFIMA process are investigated in a Monte Carlo study. The estimation procedure is applied to stock returns data for 15 countries.

A Simple Microstructure Return Model Explaining Microstructure Noise and Epps Effects

We present a simple microstructure model of financial returns that combines (i) the well-known ARFIMA process applied to tick-by-tick returns, (ii) the bid-ask bounce effect, (iii) the fat tail structure of the distribution of returns and (iv) the non-Poissonian statistics of intertrade intervals. This model allows us to explain both qualitatively and quantitatively important stylized facts observed in the statistics of microstructure returns, including the short-ranged correlation of returns, the long-ranged correlations of absolute returns, the microstructure noise and Epps effects. According to the microstructure noise effect, volatility is a decreasing function of the time scale used to estimate it. Paradoxically, the Epps effect states that cross correlations between asset returns are increasing functions of the time scale at which the returns are estimated. The microstructure noise is explained as the result of the negative return correlations inherent in the definition of the bid-ask bounce component (ii). In the presence of a genuine correlation between the returns of two assets, the Epps effect is due to an average statistical overlap of the momentum of the returns of the two assets defined over a finite time scale in the presence of the long memory process (i).

Long Memory Process in Asset Returns with Multivariate GARCH Innovations

The main purpose of this paper is to consider the multivariate GARCH (MGARCH) framework to model the volatility of a multivariate process exhibiting long‐term dependence in stock returns. More precisely, the long‐term dependence is examined in the first conditional moment of US stock returns through multivariate ARFIMA process, and the time‐varying feature of volatility is explained by MGARCH models. An empirical application to the returns series is carried out to illustrate the usefulness of our approach. The main results confirm the presence of long memory property in the conditional mean of all stock returns.

Estimating a change point in the long memory parameter

We propose an estimator of change point in the long memory parameter d of an ARFIMA(p, d, q) process using the sup Wald test. We derive the consistency and the rate of convergence of the estimator for the time of change. The convergence rate of our change point estimator depends on the magnitude of a shift. Furthermore, we obtain the limiting distribution of our change point estimator without depending on the distribution of the process. Therefore, we can construct confidence intervals for the change point. Simulations show the validity of the asymptotic theory of our estimator if the sample size is large enough. We apply our change point estimator to the yearly Nile river minimum water level.

Approximating long-memory DNA sequences by short-memory process

This paper analyzes the approximation of a general long-memory ARFIMA ( p , d , q ) process by a short-memory ARMA(1, 1) process. To validate this approximation, a mean square error forecast criterion is considered, and the calculation of the mean square error between the observation X t + l of an ARFIMA process and the l -step-ahead forecast of the ARMA(1, 1) process is presented. The performance of the ARMA(1, 1) approximation to an ARFIMA model is illustrated by using an application to a DNA sequence of orf virus. This paper gives some theoretical justification on the mean square error forecast criterion based on a selected ARMA(1, 1) model when compared to the general ARFIMA( p , d , q ) process. The paper also provides a DNA sequence of orf virus analysis when the time series is originated from the tangent values of the Chaos Game Representation coordinates for each nucleotide. The conclusions of this paper work well because the estimator value of d is small ( d ˆ = 0.16 ). The paper also gives the other parameter estimate of the fitted ARFIMA (0, d , 1) model and one-step predictions using ARMA(1, 1) model.

A generalized ARFIMA process with Markov-switching fractional differencing parameter

We propose a general class of Markov-switching-ARFIMA (MS-ARFIMA) processes in order to combine strands of long memory and Markov-switching literature. Although the coverage of this class of models is broad, we show that these models can be easily estimated with the Durbin–Levinson–Viterbi algorithm proposed. This algorithm combines the Durbin–Levinson and Viterbi procedures. A Monte Carlo experiment reveals that the finite sample performance of the proposed algorithm for a simple mixture model of Markov-switching mean and ARFIMA(1, d, 1) process is satisfactory. We apply the MS-ARFIMA models to the US real interest rates and the Nile river level data, respectively. The results are all highly consistent with the conjectures made or empirical results found in the literature. Particularly, we confirm the conjecture in Beran and Terrin [J. Beran and N. Terrin, Testing for a change of the long-memory parameter. Biometrika 83 (1996), pp. 627–638.] that the observations 1 to about 100 of the Nile river data seem to be more independent than the subsequent observations, and the value of differencing parameter is lower for the first 100 observations than for the subsequent data.

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.

Estimation Methods of the Long Memory Parameter: Monte Carlo Analysis and Application

Since the seminal paper of Granger & Joyeux (1980), the concept of a long memory has focused the attention of many statisticians and econometricians trying to model and measure the persistence of stationary processes. Many methods for estimating d, the long-range dependence parameter, have been suggested since the work of Hurst (1951). They can be summarized in three classes: the heuristic methods, the semi-parametric methods and the maximum likelihood methods. In this paper, we try by simulation, to verify the two main properties of [dcirc]: the consistency and the asymptotic normality. Hence, it is very important for practitioners to compare the performance of the various classes of estimators. The results indicate that only the semi-parametric and the maximum likelihood methods can give good estimators. They also suggest that the AR component of the ARFIMA (1, d, 0) process has an important impact on the properties of the different estimators and that the Whittle method is the best one, since it has the small mean squared error. We finally carry out an empirical application using the monthly seasonally adjusted US Inflation series, in order to illustrate the usefulness of the different estimation methods in the context of using real data.

A Generalized ARFIMA Process with Markov-Switching Fractional Differencing Parameter

We propose a general class of Markov-switching-ARFIMA processes in order to combine strands of long memory and Markov-switching literature. Although the coverage of this class of models is broad, we show that these models can be easily estimated with the DLV algorithm proposed. This algorithm combines the Durbin-Levinson and Viterbi procedures. A Monte Carlo experiment reveals that the finite sample performance of the proposed algorithm for a simple mixture model of Markov-switching mean and ARFIMA(1, d, 1) process is satisfactory. We apply the Markov-switching-ARFIMA models to the U.S. real interest rates, the Nile river level, and the U.S. unemployment rates, respectively. The results are all highly consistent with the conjectures made or empirical results found in the literature. Particularly, we confirm the conjecture in Beran and Terrin (1996) that the observations 1 to about 100 of the Nile river data seem to be more independent than the subsequent observations, and the value of differencing parameter is lower for the first 100 observations than for the subsequent data.

Aggregation effect and forecasting temporal aggregates of long memory processes

This article studies the parsimonious effect of temporal aggregation on long memory time series with ARFIMA structure, and the efficiency of forecasting temporal aggregates of long memory processes using low order ARFIMA models. It is well known that the aggregates of an ARFIMA(0, d, 0) process (where d is a positive real number) have autocorrelations that follow an ARFIMA (0, d, ∞) structure in general. In this paper, we derive a low order ARFIMA(0, d, d¯1) approximation to the aggregate structure as the level of aggregation tends to infinity (where d¯1 is the greatest integer strictly less than d+1). Numerical evaluation and simulation experiments show that this approximation is close, getting more precise as the value of d increases. For forecasting future aggregates, the efficiency of using the low order approximation for the aggregate series and the efficiency of using the underlying disaggregate model is compared. A simulation study is performed to illustrate the results.

Parametric estimation for ARFIMA models via spectral methods

Given a fractional integrated, autoregressive, moving average,ARFIMA (p, d, q) process, the simultaneous estimation of the short and long memory parameters can be achieved by maximum likelihood estimators. In this paper, following a two-step algorithm, the coefficients are estimated combining the maximum likelihood estimators with the general orthogonal decomposition of stochastic processes. In particular, the principal component analysis of stochastic processes is exploited to estimate the short memory parameters, which are plugged into the maximum likelihood function to obtain the fractional differencingd.

A comparison of estimation methods in non-stationary ARFIMA processes

This paper reports an extensive Monte Carlo simulation study based on six estimators for the long memory fractional parameter when the time series is non-stationary, i.e., ARFIMA(p, d, q) process for d > 0.5. Parametric and semiparametric methods are compared. In addition, the effect of the parameter estimation is investigated for small and large sample sizes and non-Gaussian error innovations. The methodology is applied to a well known data set, the so-called UK short interest rates.

Error and Model Misspecification in ARFIMA Process

In developing the long and short memory estimation, it is usually assumed that the innovations in the ARFIMA model are normally distributed. However, circumstances may occur where this assumption is not true. This paper uses Monte Carlo simulation to evaluate the robustness of different estimators of the fractional parameter in stationary and invertible ARFIMA processes to the misspecification of the error distribution. In particular, we consider misspecification against heavy-tailed, skewed and bimodal distributions. The study is also extended for the incorrect ARFIMA specification.

Using High Frequency Stock Market Index Data to Calculate, Model & Forecast Realized Return Variance

The objective of this paper is to calculate, model and forecast realized volatility, using high frequency stock market index data. The approach taken differs from the existing literature in several aspects. First, it is shown that the decay of the serial dependence of high frequency returns with the sampling frequency, is consistent with an ARMA process under temporal aggregation. This finding has important implications for the modelling of high frequency returns and the optimal choice of sampling frequency when calculating realized volatility. Second, motivated by the outcome of several test statistics for long memory in realized volatility, it is found that the realized volatility series can be modelled as an ARFIMA process. Significant exogenous regressors include lagged returns and contemporaneous trading volume. Finally, the ARFIMA's forecasting performance is assessed in a simulation study. Although it outperforms representative GARCH models, the simplicity and flexibility of the GARCH may outweigh the modest gain in forecasting performance of the more complex and data intensive ARFIMA model.

Identification of the order of a fractionally differenced ARMA model

Long term dependence in time series can be modelled by fractionally integrated ARMA (ARFIMA) models. For an ARFIMA process it is however impossible to identify the order of the short memory polynomials by inspection of the autocorrelation and partial autocorrelation functions. Instead information criteria such as AIC, BIC and HQIC are used to identify the order. This paper investigates the performance of the three information criteria when identifying the order in an ARFIMA model. The impression is that BIC outperforms AIC and HQIC, at least for the ARFIMA models used in this simulation. The overall performance of the information criteria, however, is poor for mixtures of AR and MA processes. Introducing long memory increases the likelihood of identifying the correct orders.

The detection and estimation of long memory in stochastic volatility

We propose a new time series representation of persistence in conditional variance called a long memory stochastic volatility (LMSV) model. The LMSV model is constructed by incorporating an ARFIMA process in a standard stochastic volatility scheme. Strongly consistent estimators of the parameters of the model are obtained by maximizing the spectral approximation to the Gaussian likelihood. The finite sample properties of the spectral likelihood estimator are analyzed by means of a Monte Carlo study. An empirical example with a long time series of stock prices demonstrates the superiority of the LMSV model over existing (short-memory) volatility models.