Dimensionality determination: A thresholding double ridge ratio approach

Underdetermination of model dimensionality (order) is a longstanding problem when existing eigendecomposition-based criteria are used. To alleviate this difficulty, we propose a thresholding double ridge ratio criterion in this paper. Unlike all existing eigendecomposition-based criteria, the proposed criterion can provide a consistent estimate even when there are several local minima. For illustration, we present the generic strategy with three important applications: dimension reduction in regressions with fixed and divergent dimensions; model checking with local alternative models; and ultra-high dimensional approximate factor models. Numerical studies are conducted to examine the finite sample performance of the proposed method and a real data example is analyzed for illustration.