Inequality Constraints in the Fractionally Integrated GARCH Model

In this article we derive necessary and sufficient conditions for the nonnegativity of the conditional variance in the fractionally integrated generalized autoregressive conditional heteroskedastic (p, d, q) (FIGARCH) model of the order p ≤ 2 and sufficient conditions for the general model. These conditions can be seen as being analogous to those derived by Nelson and Cao (1992, Journal of Business & Economic Statistics 10, 229--235) for the GARCH(p, q) model. However, the inequality constraints which we derive for the FIGARCH model illustrate two remarkable properties of the FIGARCH model which are in contrast to the GARCH model: (i) even if all parameters are nonnegative, the conditional variance can become negative and (ii) even if all parameters are negative (apart from d), the conditional variance can be nonnegative almost surely. In particular, the conditions for the (1, d, 1) model substantially enlarge the sufficient parameter set provided by Bollerslev and Mikkelsen (1996, Journal of Econometrics 73, 151--184). The importance of the result is illustrated in an empirical application of the FIGARCH(1, d, 1) model to Japanese yen versus U.S. dollar exchange rate data. Copyright 2006, Oxford University Press.