Sílvia R. C. Lopes

Some simulations and applications of forecasting long-memory time-series models

In this paper, we show some results of forecasting based on the ARFIMA(p,d,q) and ARIMA(p,d,q) models. We show, by simulation, that the technique of forecasting of the ARIMA(p,d,q) model can also be used when d is fractional, i.e., for the ARFIMA(p,d,q) model. We also conduct a simulation study to compare the two estimators of d obtained through regression methods. They are used in the hypothesis test to decide whether or not the series has long memory property and are compared on the basis of their k-step ahead forecast errors. The properties of long-memory models are also investigated using an actual set of data.

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.

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.

Properties of seasonal long memory processes

We consider the fractional ARIMA process with seasonality s, denoted by SARFIMA(p,d,q)x(P,D,Q)s, which describes time series with long memory periodical behavior at finite number of spectrum frequencies. We present the proof of several properties of these processes, such as the spectral density function expression and its behavior near the seasonal frequencies, the stationarity, the intermediate and long memory characteristics, the autocovariance function and its asymptotic expression. We also investigate the ergodicity and we present necessary and sufficient conditions for the causality and the invertibility properties of SARFIMA processes.

Long-Range Dependence in Mean and Volatility: Models, Estimation and Forecasting

In this paper we consider the estimation and forecasting future values of some stochastic processes exhibiting long-range dependence, both in mean and in volatility. We summarize basic definitions, properties and some results considering ARFIMA and SARFIMA processes, which exhibit long memory in mean. We proceed in the same manner considering FIGARCH and Fractionally Integrated Stochastic Volatility (FISV) processes where one can find long memory in volatility.

Statistical properties of detrended fluctuation analysis

The main goal of this work is to consider the detrended fluctuation analysis (DFA), proposed by Peng et al. [Mosaic organization of DNA nucleotides, Phys. Rev. E. 49(5) (1994), 1685–1689]. This is a well-known method for analysing the long-range dependence in non-stationary time series. Here we describe the DFA method and we prove its consistency and its exact distribution, based on the usual i.i.d. assumption, as an estimator for the fractional parameter d. In the literature it is well established that the nucleotide sequences present long-range dependence property. In this work, we analyse the long dependence property in view of the autoregressive moving average fractionally integrated ARFIMA(p, d, q) processes through the analysis of four nucleotide sequences. For estimating the fractional parameter d we consider the semiparametric regression method based on the periodogram function, in both classical and robust versions; the semiparametric R/S(n) method, proposed by Hurst [Long term storage in reservoirs, Trans. Am. Soc. Civil Eng. 116 (1986), 770–779] and the maximum likelihood method (see [R. Fox and M.S. Taqqu, Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series, Ann. Statist. 14 (1986), 517–532]), by considering the approximation suggested by Whittle [Hypothesis Testing in Time Series Analysis (1953), Hafner, New York].

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.

Fixed points in mixed spectrum analysis

Families and sequences of zero-crossing counts generated by parametric time invariant linear filters are called higher order crossings or HOC. Because of the close relationship between zero-crossing counts and first order autocorrelations, families of first order autocorrelations are also referred to as HOC. We investigate the HOC from some particular families of linear filters applied in the problem of multiple frequency detection in noise. Viewing the cosine of each discrete frequency as a fixed point of a certain mapping, it is shown how to construct HOC sequences that converge to the fixed points. A faster convergence rate is achieved by controlling the bandwidth of the parametric filters.

Seasonal FIEGARCH processes

Here we develop the theory of seasonal FIEGARCH processes, denoted by SFIEGARCH, establishing conditions for the existence, the invertibility, the stationarity and the ergodicity of these processes. We analyze their asymptotic dependence structure by means of the autocovariance and autocorrelation functions. We also present some properties regarding their spectral representation. All properties are illustrated through graphical examples and an application of SFIEGARCH models to describe the volatility of the S&P500 US stock index log-return time series in the period from December 13, 2004 to October 10, 2009 is provided.

Mallows Distance in VARFIMA(0, d, 0) Processes

In this work we present an extensive simulation study on Mallows distance in the context of Gaussian and non Gaussian VARFIMA processes. Our main goal is to analyze the dependence among the components of VARFIMA processes through the Mallows distance point of view. A possible relationship between the Mallows distance and the fractional differencing parameter d , the type and level of dependence in the innovation process as well as its marginal behavior is investigated. We study the behavior of the Kendalls τ dependence coefficient under the same framework for comparison purposes. For the Mallows distance, we consider an estimator based on the empirical marginal distribution function. Based on our simulation results, we propose both a semiparametric estimator for the fractional differencing parameter and a testing procedure to assess the presence of strong long range dependence in the components of VARFIMA processes of any (finite) dimension.