In this paper, we focus on the BDS test, which is a nonparametric test of independence. Specifically, the null hypothesis $H_{0}$ of it is that $\{u_{t}\}$ is i.i.d. (independent and identically distributed), where $\{u_{t}\}$ is a random sequence. The BDS test is widely used in economics and finance, but it has a weakness that cannot be ignored: over-rejecting $H_{0}$ even if the length $T$ of $\{u_{t}\}$ is as large as $(100,2000)$. To improve the over-rejection problem of BDS test, considering that the correlation integral is the foundation of BDS test, we not only accurately describe the expectation of the correlation integral under $H_{0}$, but also calculate all terms of the asymptotic variance of the correlation integral whose order is $O(T^{-1})$ and $O(T^{-2})$, which is essential to improve the finite sample performance of BDS test. Based on this, we propose a revised BDS (RBDS) test and prove its asymptotic normality under $H_{0}$. The RBDS test not only inherits all the advantages of the BDS test, but also effectively corrects the over-rejection problem of the BDS test, which can be fully confirmed by the simulation results we presented. Moreover, based on the simulation results, we find that similar to BDS test, RBDS test would also be affected by the parameter estimations of the ARCH-type model, resulting in size distortion, but this phenomenon can be alleviated by the logarithmic transformation preprocessing of the estimate residuals of the model. Besides, through some actual datasets that have been demonstrated to fit well with ARCH-type models, we also compared the performance of BDS test and RBDS test in evaluating the goodness-of-fit of the model in empirical problem, and the results reflect that, under the same condition, the performance of the RBDS test is more encouraging.

翻译：本文聚焦于BDS检验，这是一种非参数独立性检验。具体而言，其原假设$H_{0}$为$\{u_{t}\}$是独立同分布（i.i.d.）的随机序列。BDS检验广泛应用于经济学和金融学领域，但存在一个不容忽视的缺陷：即使序列$\{u_{t}\}$的长度$T$高达$(100,2000)$，也会过度拒绝$H_{0}$。为改善BDS检验的过度拒绝问题，考虑到关联积分是BDS检验的基础，我们不仅精确描述了$H_{0}$下关联积分的期望，还计算了关联积分渐近方差中所有阶数为$O(T^{-1})$和$O(T^{-2})$的项，这对提升BDS检验的有限样本性能至关重要。基于此，我们提出修正BDS（RBDS）检验，并证明了其在$H_{0}$下的渐近正态性。RBDS检验不仅继承了BDS检验的所有优势，还有效纠正了BDS检验的过度拒绝问题，本文呈现的仿真结果充分证实了这一点。此外，基于仿真结果我们发现，与BDS检验类似，RBDS检验也会受ARCH型模型参数估计的影响而产生尺寸畸变，但通过对模型估计残差进行对数变换预处理可缓解此现象。同时，通过若干已被证实与ARCH型模型拟合良好的实际数据集，我们在实证问题中比较了BDS检验与RBDS检验在评估模型拟合优度时的表现，结果表明，在相同条件下，RBDS检验的性能更令人满意。