A Sparse Loading Full-factor Multivariate GARCH model

Analyzing multivariate time series has been a very important topic in statistics, econometrics and finance. In practice, most time series data arising from finance, biomedicine, meteorology, and social networks are best analyzed by multivariate models. For example, in finance, many popular univariate models such as ARMA, ARCH (GARCH), CAPM and dynamic hedging models are being investigated for possible extensions to their multivariate counterparts, because the multivariate models better represent market dynamics under the conditions of the globalized economy. However, the issue of computation efficiency arises as the number of parameters easily explodes with increasing dimension of the model. To give a practical example, the number of parameters for a typical multivariate GARCH model grows at the rate of $O(n^4)$, where n is the model dimension. So it is almost impossible to use the model when the dimension grows into the order of hundreds, which, however, is the common situation in practice. In this paper, in order to address the problem and make possible the analysis of high-dimensional multivariate time series, we develop a regularization technique for stationary multivariate time series by penalty, which simultaneously selects variables and estimates parameters, and eventually produces a sparse estimator. The idea of penalty is due to several statisticians who in the past decade have been working on regularization problems mainly in the framework of i.i. d. (independently identically distributed) data. The extension to the time series context is not immediate since the associated limit theorems are not valid with dependent structure. In this paper, we reconstruct some of the proofs and develop the consistency results and a central limit theorem for our penalized estimator for stationary time series models. We have proved that under mild conditions our estimator is at least root-T consistent, where T is the length of the observed time series; moreover, the zero-valued parameters are identified with high probability and the non-zero ones are estimated almost as efficiently as if the true sparse model were known in advance. Therefore, if the underlying model is truly sparse, our method provides a great benefit by making efficient use of the limited data. Our method is general in the sense that it can be applied to a large class of stationary multivariate time series models. To demonstrate the advantages of our method, we have applied it to the full-factor multivariate GARCH model (Vrontos, 2003) and developed a sparse-loading version of it. We have carried out a series of simulations and the results are consistent with our theoretical deduction. We have also applied our method to analyze the Dow Jones Industry Average component stocks, and the advantages are noticed to be substantial. Furthermore, we expect our method to be capable in assisting large covariance estimation and generate sparse covariance matrices. This will be very important because it provides a basis for clustering or sub-grouping, by which a large problem can be reasonably split into several smaller ones and so the analysis will be much simpler.