A Comparative Note about Estimation of the Fractional Parameter under Additive Outliers

In recent articles, Fajardo et al. (2009 Fajardo Molinares, F., Reisen, V.A., Cribari-Neto, F. (2009). Robust estimation in long-memory processes under additive outliers. Journal of Statistical Planning and Inference 139:2511–2525.[Crossref], [Web of Science ®] , [Google Scholar]) and Reisen and Fajardo (2012) propose an alternative semiparametric estimator of the fractional parameter in ARFIMA models which is robust to the presence of additive outliers. The results are very interesting, however, they use samples of 300 or 800 observations which are rarely found in macroeconomics. In order to perform a comparison, I estimate the fractional parameter using the procedure of Geweke and Porter-Hudak (1983 Geweke, J., Porter-Hudak, S. (1983). The estimation and application of long memory time series model. Journal of Time Series Analysis 4:221–238.[Crossref] , [Google Scholar]) augmented with dummy variables associated with the (previously) detected outliers using the statistic τ_d suggested by Perron and Rodríguez (2003 Perron, P., Rodríguez, G. (2003). Searching for additive outliers in nonstationary time series. Journal of Time Series Analysis 24(2):193–220.[Crossref], [Web of Science ®] , [Google Scholar]). Comparing with Fajardo et al. (2009 Fajardo Molinares, F., Reisen, V.A., Cribari-Neto, F. (2009). Robust estimation in long-memory processes under additive outliers. Journal of Statistical Planning and Inference 139:2511–2525.[Crossref], [Web of Science ®] , [Google Scholar]) and Reisen and Fajardo (2012), I found better results for the mean and bias of the fractional parameter when T = 100 and the results in terms of the standard deviation and the MSE are very similar. However, for higher sample sizes such as 300 or 800, the robust procedure performs better. Empirical applications for seven monthly Latin-American inflation series with very small sample sizes contaminated by additive outliers are discussed.     

双长记忆GARCH族模型的预测能力比较研究——基于沪深股市数据的实证分析

GARCH族模型在金融风险的度量中有着广泛的应用。在考虑股市收益率和波动率序列双长记忆性的基础上,基于上证综合指数和深圳成份指数的日收盘价序列,从证券投资风险量化的角度,引入受险值VaR和相对正确符号指标PCS作为模型预测误差衡量指标,比较分析了双长记忆GARCH族模型在不同分布假设情况下的的拟合与预测精度。结果显示:偏t分布能较好描述沪深股市的厚尾特征;在较小的VaR水平下ARFIMA(2,d₁,0)-FIAPARCH(1,d₂,1)-skt模型对股市波动风险具有较强的预测能力,而ARFIMA(2,d₁,0)-HYGARCH(1,d₂,1)-skt对股市的涨跌趋势具有较强的预测能力。     

A Fractional Dickey–Fuller Test for Unit Roots

This paper presents a new test for fractionally integrated (FI) processes. In particular, we propose a testing procedure in the time domain that extends the well–known Dickey–Fuller approach, originally designed for the I(1) versus I(0) case, to the more general setup of FI(d₀) versus FI(d₁), with d₁<d₀. When d₀=1, the proposed test statistics are based on the OLS estimator, or its t–ratio, of the coefficient on Δ^d1y_t−1 in a regression of Δy_t on Δ^d1y_t−1 and, possibly, some lags of Δy_t. When d₁ is not taken to be known a priori, a pre–estimation of d₁ is needed to implement the test. We show that the choice of any T^1/2–consistent estimator of d₁∈[0 ,1) suffices to make the test feasible, while achieving asymptotic normality. Monte–Carlo simulations support the analytical results derived in the paper and show that proposed tests fare very well, both in terms of power and size, when compared with others available in the literature. The paper ends with two empirical applications.     

Maximum likelihood estimation of ARFIMA models with a Box-Cox transformation

In this paper we study the interaction between the estimation of the fractional differencing parameter d of ARFIMA models and the common practice of instantaneous transformation of the observed time series. At this aim, we first discuss the effect of a nonlinear transformation of the data on the identification of the process and on the estimate of d. Thus, we propose a joint estimation of the Box-Cox parameter and d by means of a modified normalized version of the Whittle likelihood. Then, the variance and covariance matrix of the parameters estimates is obtained. Finally, a Monte Carlo study is performed in order to check the behaviour of the proposed estimators in finite samples.The paper is the result of a joint research of the two authors. As far as it concerns this version of the work, A. DElia wrote Sects. 2, 3, 4, while D. Piccolo wrote Sects. 1, 5, 6.     

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.     

石油价格的ARFIMA模型预测研究

经检验，石油价格波动具有长记忆性，而通常用于定量预测的ARMA模型是不考虑长记忆性的．应用考虑长记忆性的ARFIMA模型对石油价格进行了预测研究，预测结果表明，ARFIMA模型的预测结果要好于不考虑长记忆性的ARMA模型．     

沪深股票市场长程记忆相关性研究

针对上海和深圳的日收益序列,采用重标级差(R/S分析)对其进行了实证研究.从统计结果来看,样本序列呈现出尖峰、胖尾等有偏特征,明显不满足正态分布的假设,表明收益序列可能具有长程相关或记忆性.采用ARFIMA模型对沪深股市收益率的长期记忆性进行了检验,根据分段检验的结果,得出了一些我国证券市场有效性的结论.     

Robust estimation in long-memory processes under additive outliers

In this paper, we introduce an alternative semiparametric estimator of the fractional differencing parameter in ARFIMA models which is robust against additive outliers. The proposed estimator is a variant of the GPH estimator [Geweke, J., Porter-Hudak, S., 1983. The estimation and application of long memory time series model. Journal of Time Series Analysis 4, 221–238]. In particular, we use the robust sample autocorrelations of Ma, Y. and Genton, M. [2000. Highly robust estimation of the autocovariance function. Journal of Time Series Analysis 21, 663–684] to obtain an estimator for the spectral density of the process. Numerical results show that the estimator we propose for the differencing parameter is robust when the data contain additive outliers.     

Bayesian estimation and the application of long memory stochastic volatility models

A new sampling-based Bayesian approach to the long memory stochastic volatility (LMSV) process is presented; the method is motivated by the GPH-estimator in fractionally integrated autoregressive moving average (ARFIMA) processes, which was originally proposed by J. Geweke and S. Porter-Hudak [The estimation and application of long memory time series models, Journal of Time Series Analysis, 4 (1983) 221–238]. In this work, we perform an estimation of the memory parameter in the Bayesian framework; an estimator is obtained by maximizing the posterior density of the memory parameter. Finally, we compare the GPH-estimator and the Bayes-estimator by means of a simulation study and our new approach is illustrated using several stock market indices; the new estimator is proved to be relatively stable for the various choices of frequencies used in the regression.     

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.