Tests for the explanatory power of latent factors

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.