This paper considers a formulation of the extended
constant or time-varying conditional correlation
GARCH model that allows for volatility feedback of
either the positive or negative sign. In the
previous literature, negative volatility spillovers
were ruled out by the assumption that all the
parameters of the model are nonnegative, which is a
sufficient condition for ensuring the positive
definiteness of the conditional covariance matrix.
In order to allow for negative feedback, we show
that the positive definiteness of the conditional
covariance matrix can be guaranteed even if some of
the parameters are negative. Thus, we extend the
results of Nelson and Cao (1992) and Tsai and Chan
(2008) to a multivariate setting. For the bivariate
case of order one, we look into the consequences of
adopting these less severe restrictions and find
that the flexibility of the process is substantially
increased. Our results are helpful for the
model-builder, who can consider the unrestricted
formulation as a tool for testing various economic
theories.