First‐order bias correction for fractionally integrated time series

Most of the long memory estimators for stationary fractionally integrated time series models are known to experience non‐negligible bias in small and finite samples. Simple moment estimators are also vulnerable to such bias, but can easily be corrected. In this article, the authors propose bias reduction methods for a lag‐one sample autocorrelation‐based moment estimator. In order to reduce the bias of the moment estimator, the authors explicitly obtain the exact bias of lag‐one sample autocorrelation up to the order n⁻¹. An example where the exact first‐order bias can be noticeably more accurate than its asymptotic counterpart, even for large samples, is presented. The authors show via a simulation study that the proposed methods are promising and effective in reducing the bias of the moment estimator with minimal variance inflation. The proposed methods are applied to the northern hemisphere data. The Canadian Journal of Statistics 37: 476–493; 2009 © 2009 Statistical Society of Canada     

ARFIMA processes and outliers: a weighted likelihood approach

In this paper, we consider the problem of robust estimation of the fractional parameter, d, in long memory autoregressive fractionally integrated moving average processes, when two types of outliers, i.e. additive and innovation, are taken into account without knowing their number, position or intensity. The proposed method is a weighted likelihood estimation (WLE) approach for which needed definitions and algorithm are given. By an extensive Monte Carlo simulation study, we compare the performance of the WLE method with the performance of both the approximated maximum likelihood estimation (MLE) and the robust M-estimator proposed by Beran (Statistics for Long-Memory Processes, Chapman & Hall, London, 1994). We find that robustness against the two types of considered outliers can be achieved without loss of efficiency. Moreover, as a byproduct of the procedure, we can classify the suspicious observations in different kinds of outliers. Finally, we apply the proposed methodology to the Nile River annual minima time series.     

THE EXACT BIAS OF THE LOG-PERIODOGRAM REGRESSION ESTIMATOR

The paper makes two contributions. First, we provide a formula for the exact distribution of the periodogram evaluated at any arbitrary frequency, when the sample is taken from any zero-mean stationary Gaussian process. The inadequacy of the asymptotic distribution is demonstrated through an example in which the observations are generated by a fractional Gaussian noise process. The results are then applied in deriving the exact bias of the log-periodogram regression estimator (Geweke and Porter-Hudak (1983), Robinson (1995)). The formula is computable. Practical bounds on this bias are developed and their arithmetic mean is shown to be accurate and useful.     

Obtaining prediction intervals for FARIMA processes using the sieve bootstrap

The sieve bootstrap (SB) prediction intervals for invertible autoregressive moving average (ARMA) processes are constructed using resamples of residuals obtained by fitting a finite degree autoregressive approximation to the time series. The advantage of this approach is that it does not require the knowledge of the orders, p and q, associated with the ARMA(p, q) model. Up until recently, the application of this method has been limited to ARMA processes whose autoregressive polynomials do not have fractional unit roots. The authors, in a 2012 publication, introduced a version of the SB suitable for fractionally integrated autoregressive moving average (FARIMA (p,d,q)) processes with 0<d<0.5 and established its asymptotic validity. Herein, we study the finite sample properties this new method and compare its performance against an older method introduced by Bisaglia and Grigoletto in 2001. The sieve bootstrap (SB) method is a numerically simpler alternative to the older method which requires the estimation of p, d, and q at every bootstrap step. Monte-Carlo simulation studies, carried out under the assumption of normal, mixture of normals, and exponential distributions for the innovations, show near nominal coverages for short-term and long-term SB prediction intervals under most situations. In addition, the sieve bootstrap method yields better coverage and narrower intervals compared to the Bisaglia–Grigoletto method in some situations, especially when the error distribution is a mixture of normals.     

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 novel auto-regressive fractionally integrated moving average–least-squares support vector machine model for electricity spot prices prediction

In the framework of competitive electricity market, prices forecasting has become a real challenge for all market participants. However, forecasting is a rather complex task since electricity prices involve many features comparably with those in financial markets. Electricity markets are more unpredictable than other commodities referred to as extreme volatile. Therefore, the choice of the forecasting model has become even more important. In this paper, a new hybrid model is proposed. This model exploits the feature and strength of the auto-regressive fractionally integrated moving average model as well as least-squares support vector machine model. The expected prediction combination takes advantage of each model's strength or unique capability. The proposed model is examined by using data from the Nordpool electricity market. Empirical results showed that the proposed method has the best prediction accuracy compared to other methods.     

引入持仓量的沪铜指数长记忆波动性研究

通过协整关系检验、误差修正模型、向量自回归模型、格兰杰因果关系检验、脉冲响应函数证明了在建立模型时引入持仓量序列的必要性。运用修正R/S分析,建立了沪铜指数收益率波动的ARFIMA、FI-GARCH、ARFIMA-FIGARCH模型,并运用此种模型对沪铜指数的收益率序列rt、收益率波动序列｜rt｜及残差序列｜εt｜进行相关研究和分析,结果表明：ARFIMA（0,d1,0）-FIGARCH（1,d2,1）模型的预测效果比较好。     

基于小波变换的长记忆随机波动模型估计方法研究

根据ARFIMA过程的小波分析结果,将小波引入到长记忆随机波动(Long Memory Stochastic Volatility)LMSV模型的估计中,提出了基于小波变换的LMSV模型的参数估计和潜在波动过程的估计方法.用不同参数值和样本容量的数据进行了模拟实验,又用该方法对上海和深圳证券交易所综合指数的收益序列拟合了LMSV模型,结果表明该方法是有效且可行的.     

THE EXACT BIAS OF THE LOG-PERIODOGRAM REGRESSION ESTIMATOR

The paper makes two contributions. First, we provide a formula for the exact distribution of the periodogram evaluated at any arbitrary frequency, when the sample is taken from any zero-mean stationary Gaussian process. The inadequacy of the asymptotic distribution is demonstrated through an example in which the observations are generated by a fractional Gaussian noise process. The results are then applied in deriving the exact bias of the log-periodogram regression estimator (Geweke and Porter-Hudak (1983), Robinson (1995)). The formula is computable. Practical bounds on this bias are developed and their arithmetic mean is shown to be accurate and useful.