A Sufficient Condition for the Positive Definiteness of the Covariance Matrix of a Multivariate GARCH Model

Basis Convergence and Long Memory in Volatility When Dynamic Hedging with Futures

“…Numerical procedures are used to impose positive definiteness in the diagonal FIGARCH model. 6 Positive definiteness in the diagonal GARCH model is imposed via the conditions in Silberberg and Pafka (2003).…”

“…When M = 2, the GARCH models sometimes failed to achieve strong convergence.27 The diagonal GARCH(1, 1) model ofBollerslev, Engle, and Wooldridge (1988) was also estimated (with and without maturity effects). Positive definiteness was imposed via the conditions inSilberberg and Pafka (2001). The conclusions are insensitive to the use of this model.…”

A Bayesian approach to relaxing parameter restrictions in multivariate GARCH models

“…Silberberg and Pafka (2001) or Chen et al (2005). We must also have positive definite conditional covariance matrices for each time point, however there are no known conditions on parameters that guarantee p.d.…”

Modeling the Conditional Covariance Between Stock and Bond Returns: A Multivariate Garch Approach

“…Since Σ t+1 must be symmetric, so must be the parameter matrices and only the lower portions of these matrices need to be parameterized and estimated. Silberberg and Pafka (2001), for example, prove that a sufficient condition to assure the positive definiteness of the covariance matrix Σ t+1 in 5is that the constant term C, is positive definite and all the other coefficient matrices, A and B, are positive semidefinite.…”

“…In order to derive sufficient conditions to assure positive definiteness of the covariance matrix Σ t+1 in (6), we have to show that the individual matrices in (6) are positive semidefinite as symmetry and positive semi definiteness are preserved by matrix addition (see, e.g., Silberberg and Pafka, 2001). Ding and Engle (2001)…”