What they did not tell you about algebraic (non-)existence, mathematical (ir-)regularity and (non-)asymptotic properties of the full BEKK dynamic conditional covariance model - Michael McAleer (Journal of Risk and Financial Management, 2019)
Persistently high negative covariances between risky assets and hedging instruments are intended to mitigate against risk and subsequent financial losses. In the event of having more than one hedging instrument, multivariate covariances need to be calculated. Optimal hedge ratios are unlikely to remain constant using high frequency data, so it is essential to specify dynamic covariance models. These values can either be determined analytically or numerically on the basis of highly advanced computer simulations. Analytical developments are occasionally promulgated for multivariate conditional volatility models. The primary purpose of the paper is to analyze purported analytical developments for the most widely-used multivariate dynamic conditional covariance model to have been developed to date, namely the Full BEKK model, named for Baba, Engle, Kraft and Kroner. Dynamic models are not straightforward (or even possible) to translate in terms of the algebraic existence, underlying stochastic processes, specification, mathematical regularity conditions, and asymptotic properties of consistency and asymptotic normality, or the lack thereof. The paper presents a critical analysis, discussion, evaluation and presentation of caveats relating to the Full BEKK model, and an emphasis on the numerous dos and don'ts in implementing the Full BEKK and related non-Diagonal BEKK models, such as Triangular BEKK and Hadamard BEKK, in practice.
Keywords: C22; C32; C51; C52; C58; C62; G32; 330; hedging; covariances; existence; mathematical regularity; inevitability; likelihood function; statistical asymptotic properties; caveats; practical implementation