Factor modelling for tensor time series

Abstract

High dimensional tensor time series data is increasingly prevalent across various fields. In the analysis of such data, factor modelling plays a crucial role as a dimension reduction tool. While traditional factor models primarily handle vector time series, the exploration of matrix or tensor factor models under various assumptions is still in its early stages and has attracted increasing interest in recent years. In this thesis, we develop a tensor factor model under the presence of both serial and cross-correlations in the idiosyncratic components, assuming only bounded fourth order moments for the time series variables. Moreover, we incorporate a spectrum of different factor strengths into the model, in contrast to the prevalent assumption in many literature that considers only pervasive factors. The inclusion of serial dependence noise and weak factors makes our model more compatible with real data, especially in economics and finance. With the relaxed assumptions in our model, we propose a pre-averaging procedure to initially estimate the factor loading spaces, which achieves signal accumulation through the random projection of tensor fibres. Furthermore, we develop an iterative projection algorithm to improve the re-estimation of factor loadings by projecting the data onto the strongest estimated factor directions. To estimate the number of factors, we introduce a new core tensor rank estimation method through correlation analysis on the projected data. Theoretical guarantees are provided for all estimators, and extensive simulations, as well as analyses of real datasets, are conducted to compare our methods with other state-of-the-art or traditional alternatives. Finally, we present a new method for estimating factor strengths with empirical results provided and introduce a novel matrix convergence criterion for specific covariance matrix estimators, offering valuable insights into directions for future research