In the literature, most works of the specification tests focus on the problem with one-dimensional response or fixed multidimensional responses. In this paper, we develop a new specification test for the parametric models with non-stationary regressor under multidimensional setup, where the dimension of responses may tend to infinity, which fills a gap in the literature. The theoretical results about the asymptotic properties of the proposed test are studied and the optimal rate of the local departure under the alternative hypothesis is also given which ensures the models underpinning by the null and alternative hypotheses can be differentiated. Some simulation studies are done to evaluate the performance of the proposed test with the finite sample. Besides, a real data example based on the US aggregate consumers’ consumption data is employed to illustrate the performance. The results of simulation studies and real data analysis both demonstrate the efficiency of our proposed method.

在文献中，规格测试的大多数工作都集中在具有一维响应或固定多维响应的问题上。在本文中，我们为多维设置下具有非平稳回归量的参数模型开发了一种新的规范测试，其中响应的维度可能趋于无穷大，填补了文献中的空白。研究了关于所提出测试的渐近特性的理论结果，并给出了替代假设下局部偏离的最佳率，以确保可以区分原假设和替代假设所支持的模型。进行了一些模拟研究以评估所提出的有限样本测试的性能。除了，以基于美国综合消费者消费数据的真实数据示例来说明性能。模拟研究和真实数据分析的结果都证明了我们提出的方法的有效性。