Principal component analysis in econometrics: a selective inference perspective

Abstract

We study the long-standing problem of determining the number of principal components in econometric applications from a selective inference perspective. We consider i.i.d. observations from a $p$-dimensional random vector with $p<n$ and define the ``true'' dimensionality as the rank of the population covariance matrix. Building on the sequential testing viewpoint, we propose a data-driven procedure that estimates $\rank(\Sigma_X)$ using a statistic that depends on the eigenvalues of the sample covariance matrix. While the test statistic shares the functional form of its fixed design counterpart Choi et al. (2017), our analysis departs from the non-stochastic setting by treating the design as random and by avoiding parametric Gaussian assumptions. Under a locally defined null hypothesis, we establish asymptotically exact type~I error controls in the sequential testing procedure, with simulation results indicating empirical validity of the proposed method.