We propose a consistent non-parametric test for the correct specification of parametric trend functions in multi-variate time series. The new test takes the form of the U-statistic and is robust to serial and cross-sectional dependence and time-varying variances in error terms. The test statistic is shown to have a limiting standard normal distribution under the null and diverge to infinity under the alternative. Thus the test is consistent against any fixed alternative. The test is also shown to have non-trivial asymptotic power against two classes of local alternatives approaching the null at different rates. A set of simulations is conducted to evaluate the finite-sample performance of the test.

中文翻译：

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多变量趋势函数的非参数检验

我们提出了一种一致的非参数检验，用于正确指定多变量时间序列中的参数趋势函数。新检验采用U统计量的形式，并且对序列和横截面相关性以及误差项的时变方差具有鲁棒性。检验统计量显示在零下具有限制标准正态分布，而在替代下发散到无穷大。因此，该测试与任何固定的替代方案都是一致的。该检验还显示出对两类以不同速率接近零的局部替代方案具有非平凡的渐近能力。进行一组模拟以评估测试的有限样本性能。