A non‐parametric test for multi‐variate trend functions

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.