DIMENSION REDUCTION IN MULTIVARIATE TIME SERIES VIA ENVELOPE METHODS

Due to the increasing development of information technologies and their applications in many scientific fields, high-dimensional time series data are routinely collected across a wide range of areas, including finance, economics, digital signal processing, neuroscience, and meteorology, among others. The classical vector autoregressive (VAR) models have been widely used to model multivariate time series data, because of their flexibility and ease of use. However, the VAR model suffers from overparameterization particularly when the number of lags and number of time series get large. There are several statistical methods of achieving dimension reduction of the parameter space in VAR models, however, these methods are inefficient to extract relevant information from a complex body of data because it fails to distinguish between information that is useful to the scientific goals. Envelope methods are based on novel parameterizations that use reducing subspaces to link between the mean function and dispersion matrix. The envelope model identifies and removes irrelevant information and is based on material and useful information only and is, therefore, more efficient. We propose new parsimonious VAR models by incorporating the idea of envelope models, and partial envelope models into the reduced-rank VAR models that substantially achieve dimension reduction and efficient parameter estimation simultaneously, and as a result significantly improve the estimation accuracy. Although the envelope VAR models can achieve substantial efficiency gains, they are not invariant or equivariant under the scaling transformations, which restricts their application to time series that are measured in the same or similar units. To overcome this obstacle and achieve efficiency gains by envelope methods in multivariate time series with different scales, we propose scaled envelope VAR models. These models are not only invariant in scale changes but also retain the potential of the standard envelope VAR model to achieve further efficiency gains. For each proposed model, numerical simulations are conducted to demonstrate our theoretical results and to compare the performances of the proposed models with the available models in the literature. The results show that our proposed models are significantly more accurate and efficient than the existing standard models. The proposed methods are applied to some real data