Published online by Cambridge University Press: 26 October 2009
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.
For insightful suggestions and constructive comments we thank Bruce E. Hansen, and two anonymous referees. We would also like to thank Luc Bauwens, Tim Bollerslev, Charles Bos, Michael Burmeister, James Davidson, Russell Davidson, Jurgen A. Doornik, Martin Gassebner, Siem Jan Koopman, Michael J. Lamla, Bent Nielsen, Marius Ooms, Ruey Tsay, and Giovanni Urga for their valuable suggestions. We have also benefited from the comments received from participants at the seminar held in the Department of Economics at the University of Exeter in November 2007, the Inaugural Conference of the Society for Financial Econometrics (New York, 2008), the Annual Conference of the ESRC Econometric Study Group (Bristol, 2008), the 63rd European Meeting of the Econometric Society (Milan, 2008), the Oxford-Man Institute Financial Econometrics and Vast Data Conference (Oxford, 2008), and the 7th OxMetrics User Conference (London, 2008).