Transformed contribution ratio test for the number of factors in static approximate factor models

Determining the number of factors (r) is of importance in static approximate factor models. Under some mild conditions, the r largest eigenvalues of the variance matrix of N response variables go to infinity as N increases, while the rest are bounded. Then, “Eigenvalue Ratio” (ER) and “Growth Ratio” (GR) estimators have been well exploited by maximizing the ratio of two adjacent eigenvalues in the literature. As a modification of ER and GR estimators, the new estimator named as “Transformed Contribution Ratio” (TCR) is obtained by maximizing the ratio of two adjacent transformed contribution of the eigenvalues. Under the same conditions of ER and GR estimators, the resulted estimator can be proved to be consistent. It can be further shown that, comparing with the competitors in the existing literature, the new method has desired performance on truly selecting the value of the number of latent common factors, especially when both strong and weak factors or some dominant factors are in static approximate factor models. Monte Carlo simulation experiments and one real data application are carried out for illustration.