Inequality Constraints in the Fractionally Integrated GARCH Model

In this article we derive necessary and sufficient conditionsfor the nonnegativity of the conditional variance in the fractionallyintegrated generalized autoregressive conditional heteroskedastic(p, d, q) (FIGARCH) model of the order p 2 and sufficient conditionsfor the general model. These conditions can be seen as beinganalogous to those derived by Nelson and Cao (1992, Journalof Business & Economic Statistics 10, 229–235) forthe GARCH(p, q) model. However, the inequality constraints whichwe derive for the FIGARCH model illustrate two remarkable propertiesof the FIGARCH model which are in contrast to the GARCH model:(i) even if all parameters are nonnegative, the conditionalvariance can become negative and (ii) even if all parametersare negative (apart from d), the conditional variance can benonnegative almost surely. In particular, the conditions forthe (1, d, 1) model substantially enlarge the sufficient parameterset provided by Bollerslev and Mikkelsen (1996, Journal of Econometrics73, 151–184). The importance of the result is illustratedin an empirical application of the FIGARCH(1, d, 1) model toJapanese yen versus U.S. dollar exchange rate data.