High-dimensional penalized arch processes

Abstract

We introduce a general methodology to consistently estimate multidimensional ARCH models equation-by-equation, possibly with a very large number of parameters through penalization (Sparse Group Lasso). Some families of multidimensional ARCH models are proposed to tackle homogeneous or heterogeneous portfolios of assets. The corresponding conditions of stationarity and of positive definiteness are studied. We evaluate the relevance of such a strategy by simulation. The relative forecasting performances of our models are compared through the management of financial portfolios.

Keywords:

• Multivariate ARCH
• positive definiteness
• Sparse Group Lasso
• stationarity

JEL CLASSIFICATION:

• C13
• C32
• G17

Notes

1 Alternatively, we can invoke a parametrization of A in the cone of non-negative matrices. The natural basis would be provided by the spectral decomposition of $\mathbb{E}[{\epsilon }_t{\epsilon }_t^{\prime }]$ (or its empirical approximation $[{\widehat{\text{cov}}}_{i,j}]$ instead). Indeed $\exists ({\mathit{v}}_1,\dots ,{\mathit{v}}_N)\in {\mathbb{R}}^N$ s.t. $\mathbb{E}[{\epsilon }_t{\epsilon }_t^{\prime }]\simeq {[{\widehat{\text{cov}}}_{i,j}]}_{1\le i,j\le N}={\sum }_{l=1}^N{\nu }_l{\mathit{v}}_l{\mathit{v}}_l^{\prime },$ where $({\nu }_1,\dots ,{\nu }_N)$ is the associated spectrum, ${\nu }_1\ge {\nu }_2\ge \cdots \ge {\nu }_N\ge 0.$ Then, we could assume that there exist nonnegative real numbers π_l, $l=1,\dots ,N$ s.t. $A={\sum }_{l=1}^N{\pi }_l{\mathit{v}}_l{\mathit{v}}_l^{\prime }.$ This allows to replacing the $N(N+1)/2$ unknown coefficients of A by N parameters $({\pi }_1,\dots ,{\pi }_N).$

2 The homogeneous model has not been compared to the other specifications because it is not able to fairly compete with the others: if the true DGP is actually an homogeneous model, it highly outperforms the other ones; on the other side, this is the opposite under misspecification.

3 An alternative would be to consider a group version of the conservative Lasso. See Caner and Kock (2018) and the references therein, e.g.

Additional information

Funding

This research has been financially supported by the labex Ecodec (“Economics and Decision Sciences”); and the Japan Society for the Promotion of Science.