一类双函数系数ARCH-M模型的经验似然估计

针对一类双函数系数自回归条件异方差-均值(ARCH-M)模型的估计问题，提出一种基于经验似然的估计方法；在该估计方法中，所提出的模型允许金融时间序列的风险效应和收益效应同时为某一变量的函数结构，可以有效地刻画金融时间序列的风险和平均收益之间的关系，具有较广的适应性；同时，与经典矩估计法和极大似然估计法相比，基于经验似然的估计方法具有独特的优势，可以充分考虑金融序列的异方差性，并且所构造的置信区间不涉及任何渐近方差的估计，因此具有较好的稳健性和有效性；在一些正则条件下，对所构造的经验对数似然比统计量及函数系数估计量的渐近分布进行了理论分析；结果表明：关于风险效应函数系数和收益效应函数系数的经验对数似然比统计量均渐近收敛于中心卡方分布，同时函数系数估计量渐近收敛于正态分布；进而对风险效应函数系数和收益效应函数系数分别构造了相应函数系数的逐点置信区间。

Empirical Likelihood Estimation of a Class of Double-function Coefficient ARCH-M Models

An estimation method based on empirical likelihood was proposed for a class of double-function coefficient autoregressive conditional heteroscedasticity-mean (ARCH-M) model. In this method, the proposed model allows the risk effect and return effect of financial time series to be the functional structure of a certain variable at the same time, which can effectively describe the relationship between the risk and average return of financial time series and has a wide range of adaptability. Meanwhile, compared with the classical moment estimation method and maximum likelihood estimation method, the estimation method based on empirical likelihood has unique advantages. This estimation method can fully consider the heteroscedasticity of financial series, and the constructed confidence interval does not involve any estimation of asymptotic variance, so it has better robustness and effectiveness. Under some regularity conditions, the asymptotic distribution of the constructed empirical log-likelihood ratio statistic and the estimator of function coefficients was analyzed theoretically. The results showed that the empirical log-likelihood ratio statistics about the risk effect function coefficient and the income effect function coefficient asymptotically converged to the central chi-square distribution, while the function coefficient estimator asymptotically converged to the normal distribution. Additionally, the point-by-poin confidence intervals of the corresponding function coefficients were constructed for the risk effect function coefficient and the income effect function coefficient, respectively.