基于Bartlett修正的变换调整经验似然在长记忆时间序列中的应用

Application of the Bartlett-Corrected Transformed Adjusted Empirical Likelihood for Long-Memory Time Series Models

摘要: 本文提出基于Bartlett修正的TAEL_B方法，统一使用标准似然比统计量，并系统比较不同伪观测及其系数对覆盖率的影响。由于经验似然无需计算参数估计的协方差，在卡方近似下容易构造置信区间，并且置信区间的形状由数据进行驱动，因此被广泛应用于包括时间序列等诸多领域。本文在剖面Whittle得分框架下，对四种方法进行蒙特卡洛对比：标准经验似然(EL)、固定权重TAEL(4, a_n)、固定权重TAEL(2, a_n)、以及Bartlett修正TAEL_B。模拟覆盖ARFIMA(0, d, 0)、ARFIMA(1, d, 0)和ARFIMA(0, d, 1)三种模型，样本量n = 50、70、100、150，各重复1000次。模拟结果表明，TAEL(4, a_n)在所有场景下覆盖率系统性偏低，在二维模型中n = 150时降至0.64~0.69，甚至不如未修正的EL。将系数修正为2后，TAEL(2, a_n)和TAEL_B均表现良好，覆盖率接近名义水平。在ARFIMA(1, d, 0)模型(φ, d) = (0.5, 0.4)且n = 150时，TAEL_B的覆盖率达到0.932，比TAEL(2, a_n)的0.902高出三个百分点。在Fisher信息较弱的二维参数区域，Bartlett修正可额外提升2%~3%的覆盖率。本研究为长记忆时间序列的参数区间估计提供了一个更为可靠稳健的方法。

Abstract: This paper proposes a Bartlett-corrected TAEL_B (transformed adjusted empirical likelihood) method, which consistently uses the standard likelihood ratio statistic, and systematically compares the impact of different pseudo-observation strategies and coefficients on coverage probability. Since empirical likelihood does not require estimating the covariance of parameter estimates, confidence intervals can be easily constructed under the chi-squared approximation with data-driven shapes, and it has been widely applied in many fields, including time series. Under the profiled Whittle score framework, Monte Carlo comparisons of four methods are conducted: standard empirical likelihood (EL), fixed-weight TAEL(4, a_n), fixed-weight TAEL(2, a_n), and Bartlett-corrected TAEL_B. Simulations cover three models—ARFIMA(0, d, 0), ARFIMA(1, d, 0), and ARFIMA(0, d, 1)—with sample sizes n = 50, 70, 100, 150, each with 1000 replications. The results show that TAEL(4, a_n) exhibits systematically low coverage across all scenarios, dropping to 0.64~0.69 in two-parameter models at n = 150, which is even worse than the uncorrected EL. After correcting the coefficient to 2, both TAEL(2, a_n) and TAEL_B perform well, with coverage probabilities close to the nominal level. TAEL_B achieves a coverage of 0.932 at (φ, d) = (0.5, 0.4) with n = 150 in ARFIMA(1, d, 0), outperforming TAEL(2, a_n) at 0.902 by three percentage points. The Bartlett correction provides an additional 2%~3% coverage gain in two-parameter regions with weak Fisher information. This study offers a theoretically consistent and computationally robust tool for constructing confidence regions in long-memory time series models.