关于张量因子模型中Kronecker积结构的检验
On Testing Kronecker Product Structure in Tensor Factor Models
AI总结：
本文提出一种检验张量因子模型中因子载荷矩阵是否具有Kronecker积结构的方法，通过比较完整模型与重塑数据模型的残差，利用渐近正态性建立理论，实验验证了检验的有效性，并应用于纽约出租车和金融数据。
摘要：
我们提出了一种检验方法，用于检验由张量因子模型隐含的因子载荷矩阵的Kronecker积结构，该模型在共同成分中采用Tucker分解。通过定义一个Kronecker积结构集合，我们判断张量时间序列响应$\{\mathcal{Y}_t\}$是否具有Kronecker积结构，这等价于能否根据张量因子模型对$\{\mathcal{Y}_t\}$进行分解。我们的检验基于分析并比较拟合完整张量因子模型后的残差，以及拟合数据重塑版本上的（张量）因子模型后的残差。在最极端的情况下，重塑是将张量数据向量化，此时若不存在Kronecker积结构，因子载荷矩阵可以是通用的。理论结果通过估计残差的渐近正态性得出。数值实验表明，随着样本量或张量阶数的增大，检验的尺寸（size）逐渐接近预设的名义水平，而功效（power）随模式维度和组合模式数量的增加而提高。我们通过纽约市出租车交通数据和Fama-French矩阵投资组合收益数据展示了我们的检验。
英文摘要：
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.
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