Alexei Onatski

Let {xi}i∈Z be a sequence of i.i.d. standard normal random variables. We consider two classes of rectangular random matrices: Toeplitz X=(xj−i)1≤i≤p,1≤j≤n and circulant X=(x(j−i)modn)1≤i≤p,1≤j≤n. We let p,n→∞ so that p∕n→c∈(0,1]. Our main result is that, for any c∈(0,1], the smallest eigenvalue of 1nXX⊤ converges to zero both in probability and in expectation. We further establish a lower bound on the rate of this convergence, showing it is faster than any poly-logarithmic rate yet slower than any polynomial rate. In the “rectangular circulant” setting, we additionally derive a simple, explicit upper bound on the convergence rate in terms of c. This upper bound implies that, for sufficiently small c, the decay of the smallest eigenvalue to zero can be made arbitrarily slow on a polynomial scale.

This paper extends the spurious factor analysis of Onatski and Wang (2021, Spurious factor analysis. Econometrica, 89(2), 591–614.) to high-dimensional data with heterogeneous local-to-unit roots. We find a spurious factor phenomenon similar to that observed in the data with unit roots. Namely, the “factors” estimated by the principal components analysis converge to principal eigenfunctions of a weighted average of the covariance kernels of the demeaned Ornstein–Uhlenbeck processes with different decay rates. Thus, such “factors” reflect the structure of the strong temporal correlation of the data and do not correspond to any cross-sectional commonalities, that genuine factors are usually associated with. Furthermore, the principal eigenvalues of the sample covariance matrix are very large relative to the other eigenvalues, creating an illusion of the “factors”capturing much of the data’s common variation. We conjecture that the spurious factor phenomenon holds, more generally, for data obtained from high frequency sampling of heterogeneous continuous time (or spacial) processes, and provide an illustration.