Consistency and asymptotic normality are established for the highly
applied quasi-maximum likelihood estimator in the GARCH(1,1) model.
Contrary to existing literature we allow the parameters to be in the
region where no stationary version of the process exists. This has the
important implication that the likelihood-based estimator for the GARCH
parameters is consistent and asymptotically normal in the entire parameter
region including both stationary and explosive behavior. In particular,
there is no “knife edge result like the unit root case” as
hypothesized in Lumsdaine (1996,
Econometrica 64, 575–596).
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