The bandwidth-free tests/inferences for a multi-dimensional parameter have attracted considerable attention in econometrics and statistics literature. These tests can be conveniently implemented due to their tuning-parameter free nature and possess more accurate size as compared to the traditional HAC-based approaches, where consistent long run variance estimation was involved. However, when sample size is small/medium, these bandwidth-free tests exhibit large size distortion when both the dimension of the parameter and the magnitude of temporal dependence are moderate, making them unreliable to use in practice. In this paper, we propose a sample splitting based approach to reduce the dimension of the parameter to one for the subsequent bandwidth-free inference. Our SS-SN (sample splitting plus self-normalization) idea is broadly applicable to many testing problems for time series, including mean testing, testing for zero autocorrelation, linear hypotheses testing in a time series regression model and testing for a change point in multivariate mean. Specifically, we propose $L_{\infty}$-type and $L_2$-type SS-SN test statistics and derive their limiting distributions under both the null and alternatives and show their effectiveness in alleviating size distortion via simulations. As an important theoretical contribution, we obtain the limiting distributions for both SS-SN test statistics in the multivariate mean testing problem when the dimension is allowed to diverge as sample size grows to infinity. In addition we show the asymptotic independence of $L_{\infty}$-type and $L_2$-type SS-SN test statistics under the null in the growing dimensional setting.

翻译：多维参数的无带宽检验/推断在计量经济学和统计学文献中引起了广泛关注。这类检验由于无需调整参数而易于实施，且相较于传统基于HAC的方法（需要一致估计长期方差），具有更精确的尺寸。然而，当样本量较小或中等时，若参数维度与时间依赖程度均为中等水平，这些无带宽检验会出现严重的尺寸失真，从而在实践中不可靠。本文提出一种基于样本分割的方法，将参数维度降至一维，以便进行后续的无带宽推断。我们的SS-SN（样本分割加自归一化）思想广泛适用于时间序列的多种检验问题，包括均值检验、零自相关检验、时间序列回归模型中的线性假设检验以及多元均值变点检验。具体地，我们提出了$L_{\infty}$型与$L_2$型SS-SN检验统计量，推导了它们在原假设和备择假设下的极限分布，并通过模拟展示了其在缓解尺寸失真方面的有效性。作为重要的理论贡献，我们在多元均值检验问题中，当维度随样本量增长而发散时，得到了两种SS-SN检验统计量的极限分布。此外，我们证明了在维度增长设定下，$L_{\infty}$型与$L_2$型SS-SN检验统计量在原假设下渐近独立。