We propose herein a factor-augmented semi-varying coefficient model and discuss whether the extracted factors have significant explanatory power. We first use Principal Component Analysis (PCA) to estimate the model and then develop the PCA-based Wald test. We find that the PCA-based Wald test statistic is asymptotically chi-squared distributed with degrees of freedom equal to the unknown number of factors. To avoid estimating the degrees of freedom, we then use Common Correlated Effects (CCE) to estimate the model and develop the CCE-based Wald test. However, as opposed to the PCA-based estimator, the CCE-based estimator of loadings is ambiguous in the sense that the estimation depends on the dimensions of factors and predictors and the estimator can even be inconsistent. If we construct the Wald test based on the CCE-based estimator, the test lacks power. We overcome these difficulties and construct a powerful CCE-based Wald test that is immune to factor-number uncertainty. In addition to the two Wald tests, a new CCE-based goodness-of-fit test is also proposed. The test is irrelevant to the unknown number of factors and spares us the work of estimating asymptotic covariance matrix. Finally, three empirical examples are provided to demonstrate the usefulness of the tests.

中文翻译：

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测试潜在因素的解释力

我们在此提出了一个因子增广的半变系数模型，并讨论了提取的因子是否具有显着的解释力。我们首先使用主成分分析 (PCA) 来估计模型，然后开发基于 PCA 的 Wald 检验。我们发现基于 PCA 的 Wald 检验统计量呈渐近卡方分布，其自由度等于未知因子数。为了避免估计自由度，我们然后使用共同相关效应 (CCE) 来估计模型并开发基于 CCE 的 Wald 检验。然而，与基于 PCA 的估计量相反，基于 CCE 的载荷估计量是模糊的，因为估计量取决于因素和预测变量的维度，而且估计量甚至可能不一致。如果我们基于基于 CCE 的估计量构建 Wald 检验，测试缺乏力量。我们克服了这些困难并构建了一个强大的基于 CCE 的 Wald 检验，该检验不受因子数不确定性的影响。除了两个 Wald 检验之外，还提出了一种新的基于 CCE 的拟合优度检验。该测试与未知数量的因素无关，并且免除了我们估计渐近协方差矩阵的工作。最后，提供了三个实证例子来证明测试的有用性。