We propose a test for testing the Kronecker product structure of a factor loading matrix implied by a tensor factor model with Tucker decomposition in the common component. Through defining a Kronecker product structure set, we define if a tensor time series response $\{\mathcal{Y}_t\}$ has a Kronecker product structure, equivalent to the ability to decompose $\{\mathcal{Y}_t\}$ according to a tensor factor model. Our test is built on analysing and comparing the residuals from fitting a full tensor factor model, and the residuals from fitting a (tensor) factor model on a reshaped version of the data. In the most extreme case, the reshaping is the vectorisation of the tensor data, and the factor loading matrix in such a case can be general if there is no Kronecker product structure present. Theoretical results are developed through asymptotic normality results on estimated residuals. Numerical experiments suggest that the size of the tests gets closer to the pre-set nominal value as the sample size or the order of the tensor gets larger, while the power increases with mode dimensions and the number of combined modes. We demonstrate out tests through a NYC taxi traffic data and a Fama-French matrix portfolio of returns.

翻译：本文提出了一种检验方法，用于验证由具有Tucker分解公共分量的张量因子模型所隐含的因子载荷矩阵的Kronecker积结构。通过定义一个Kronecker积结构集，我们判定一个张量时间序列响应$\{\mathcal{Y}_t\}$是否具有Kronecker积结构，这等价于将$\{\mathcal{Y}_t\}$按照张量因子模型进行分解的能力。我们的检验基于分析和比较拟合完整张量因子模型得到的残差，与对数据重塑版本拟合（张量）因子模型得到的残差。在最极端的情况下，重塑操作是对张量数据的向量化，此时若无Kronecker积结构，因子载荷矩阵可以是任意的。理论结果通过估计残差的渐近正态性得以建立。数值实验表明，随着样本量或张量阶数的增大，检验的尺寸更接近预设的名义值，而检验功效则随着模态维度和组合模态数量的增加而提高。我们通过纽约市出租车流量数据和Fama-French矩阵投资组合收益率数据展示了所提检验方法的应用。