Scalar correlation functions for model structure selection in high-dimensional time-series modelling

Model structure selection is an important step in high-dimensional time-series modelling. Traditionally AIC and BIC have been used for this purpose, however, only post model estimation. On the other hand, modern approaches use penalized regression methods, but the optimization is in a user-specified model class. In this work, we propose a pre-estimation approach based on two novel correlation functions, namely, the scalar autocorrelation function (SACF) and the scalar inverse autocorrelation function (SIACF) for identifying the appropriate model class among the vector autoregressive (VAR), vector moving average (VMA), and mixed (VARMA) classes. In addition, these scalar functions theoretically provide the exact order of VAR and VMA processes, and are computationally feather light even for high-dimensional series. The proposed functions are obtained through two linear constructs of the given multivariate process with a lagged-correlation equivalence constraint. The key benefit is that only two correlation functions need to be examined as against the standard M² correlation and M² inverse (or partial) correlation plots for an M-dimensional process. This benefit extends to conducting whiteness test in multivariate time-series modelling and is particularly pronounced under small sample conditions, wherein parsimony, structure and class of the identified model is crucial in achieving efficient estimates. Theoretical proofs and case studies are presented to establish the properties and to demonstrate the utility of proposed correlation functions.