Proximity-structured multivariate volatility models

This paper discusses the specification and identification of structured parametrizations for multivariate volatility models, which employ weight matrices induced by economic proximity. Structured parametrizations possess the following four desirable properties: i) they are flexible, allowing for covariance spillover and feedback; ii) they are parsimonious, being characterized by a number of parameters that grows only linearly with the cross-sectional dimension; iii) model parameters have a direct economic interpretation that reflects the chosen notion of economic proximity; iv) model-estimation computations are generally faster than for unstructured specifications. This shows that structured specifications provide a solution to the curse of dimensionality problem for volatility models, which limits feasibility of model-estimation to small cross-sections for unstructured models. We give several examples of structured specifications, discussing how to construct weight matrices and proximity specifications. Identification and estimation of structured specifications is analyzed; we provide rank and order conditions for identification. Asymptotics under identification is discussed. We present an illustrative application of a BEKK specification to 6 asset returns from the NYSE; we compare several specifications both in-sample and out-of-sample. It is found that several of the structured specifications well compare with alternatives for some metrics.