Determining the dimension of factor structures in non-stationary large datasets

We propose a procedure to determine the dimension of the common factor space in a large, possibly non-stationary, dataset. Our procedure is designed to determine whether there are (and how many) common factors (i) with linear trends, (ii) with stochastic trends, (iii) with no trends, i.e. stationary. Our analysis is based on the fact that the largest eigenvalues of a suitably scaled covariance matrix of the data (corresponding to the common factor part) diverge, as the dimension $N$ of the dataset diverges, whilst the others stay bounded. Therefore, we propose a class of randomised test statistics for the null that the $p$-th eigenvalue diverges, based directly on the estimated eigenvalue. The tests only requires minimal assumptions on the data, and no restrictions on the relative rates of divergence of $N$ and $T$ are imposed. Monte Carlo evidence shows that our procedure has very good finite sample properties, clearly dominating competing approaches when no common factors are present. We illustrate our methodology through an application to US bond yields with different maturities observed over the last 30 years. A common linear trend and two common stochastic trends are found and identified as the classical level, slope and curvature factors.