Inequality Constraints in the Univariate GARCH Model Daniel

Since their introduction by Engle (1982) and Bollerslev (1986), respectively, autoregressive conditional heteroscedastic (ARCH) and generalized autoregressive conditional heteroscedastic (GARCH) models have found extraordinarily wide use. The survey article by Bollerslev, Chou, and Kroner (1982) cited more than 300 papers applying ARCH, GARCH, and other closely related models. As they showed, ARCH and GARCH models have been very successful at modeling timevarying volatility in financial time series. One nettlesome feature of GARCH models, however, has been the inequality constraints imposed to keep the conditional variance nonnegative. As we shall see, estimated parameters frequently violate these constraints. This article shows that inequality constraints less severe than commonly imposed are sufficient to keep the conditional variance nonnegative. Is this important in practice? An ARCH model estimated using quasi-maximum likelihood methods will not generate negative conditional variances ot2 in sample, since the log quasi-likelihood involves a term in ln(o-2), which explodes to -oo as or2 approaches 0 and is ill-defined for o-2 0. Nevertheless, an estimated ARCH model may have coefficients that allow ,r2 to become negative out of sample (or, more precisely, assign positive probability to the event that cr2 eventually becomes negative). Such estimated coefficients must either result from sampling error (in which case it may be best to impose the parameter constraints in estimation) or from specification error. In this article, we show that empirically relevant violations of Bollerslev's inequality constraints may be the result neither of sampling error nor of misspecification. The GARCH(p, q) model sets