  Factor model is an appealing and effective analytic tool for high-dimensional
time series, with a wide range of applications in economics, finance and
statistics. This paper develops two criteria for the determination of the
number of factors for tensor factor models where the signal part of an observed
tensor time series assumes a Tucker decomposition with the core tensor as the
factor tensor. The task is to determine the dimensions of the core tensor. One
of the proposed criteria is similar to information based criteria of model
selection, and the other is an extension of the approaches based on the ratios
of consecutive eigenvalues often used in factor analysis for panel time series.
Theoretically results, including sufficient conditions and convergence rates,
are established. The results include the vector factor models as special cases,
with an additional convergence rates. Simulation studies provide promising
finite sample performance for the two criteria.
