Maximum Likelihood Estimation of the Differencing Parameter for Invertible Short and Long Memory Autoregressive Integrated Moving Average Models
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可逆短记忆与长记忆 ARIMA 模型差分参数的最大似然估计
- 研究问题
- 如何把分数差分参数的估计扩展到非平稳范围，统一拟合传统与分数 ARIMA 模型，并将差分参数的不确定性纳入推断？
- 方法与思路
- 作者采用近似最大似然方法估计大于负二分之一的差分参数，同时估计 ARMA 参数，构造置信区间，并开展平稳性及单位根检验。
- 材料陈述的贡献
- 摘要称，该方法统一处理平稳与非平稳分数 ARIMA，并使 ARMA 参数区间计入差分参数估计的不确定性，另提供算法与应用示例。
- 阅读关注点
- 应核对近似似然、初始观测和非平稳部分的处理，以及可逆性条件、区间有效性和自回归单位根检验的具体构造。
- 阅读边界
- 仅据 Crossref 摘要；摘要未说明近似误差、渐近条件、有限样本覆盖率或实例结论，不能仅凭参数范围认定全面有效。
原文摘要
SUMMARY In practical applications of Box-Jenkins autoregressive integrated moving average (ARIMA) models, the number of times that the observed time series must be differenced to achieve approximate stationarity is usually determined by careful, but mostly informal, analysis of the differenced series. For many time series, some differencing seems appropriate, but taking the first or the second difference may be too strong. As an alternative, Hosking, and Granger and Joyeux proposed the use of fractional differences. For -1/2 < d < 1/2, the resulting fractional ARIMA processes are stationary. For 0 < d < 1/2, the correlations are not summable. The parameter d can be estimated, for instance by maximum likelihood. Unfortunately, estimation methods known so far have been restricted to the stationary range -1/2 < d < 1/2. In this paper, we show how any real d > -1/2 can be estimated by an approximate maximum likelihood method. We thus obtain a unified approach to fitting traditional Box-Jenkins ARIMA processes as well as stationary and non-stationary fractional ARIMA processes. A confidence interval for d can be given. Tests, such as for unit roots in the autoregressive parameter or for stationarity, follow immediately. The resulting confidence intervals for the ARMA parameters take into account the additional uncertainty due to estimation of d. A simple algorithm for calculating the estimate of d and the ARMA parameters is given. Simulations and two data examples illustrate the results.