Abstract
Invertibility conditions for observation-driven time series models often fail to be guaranteed in empirical applications. As a result, the asymptotic theory of maximum likelihood and quasi-maximum likelihood estimators may be compromised. We derive considerably weaker conditions that can be used in practice to ensure the consistency of the maximum likelihood estimator for a wide class of observation-driven time series models. Our consistency results hold for both correctly specified and misspecified models. The practical relevance of the theory is highlighted in a set of empirical examples. We further obtain an asymptotic test and confidence bounds for the unfeasible true invertibility region of the parameter space.
Highlights
Observation-driven models are widely employed in time series analysis and econometrics
The theory relies on the work of Bougerol (1993) to ensure the invertibility of the filtered time-varying variance and to deliver asymptotic results that are subject to some restrictions on the parameter region where the Quasi Maximum Likelihood (QML) estimator is defined
In general, the typical invertibility conditions needed to ensure the consistency of the maximum likelihood (ML) estimator, which are considered for instance in Straumann and Mikosch (2006), Straumann (2005) and Blasques et al (2014a), lead often to a parameter region that is too small for practical purposes
Summary
Observation-driven models are widely employed in time series analysis and econometrics. This leads researchers to rely on feasible conditions that are typically only satisfied in either degenerate or very small parameter regions, which are unreasonable in practical situations To address this issue and to ensure the asymptotic theory of the QML estimator of the EGARCH(1,1) model of Nelson (1991), Wintenberger (2013) proposed to stabilize the inferential procedure by restricting the optimization of the quasi-likelihood function to a parameter region that satisfies an empirical version of the required invertibility conditions of Straumann and Mikosch (2006). This method provides a consistent QML estimator for the EGARCH(1,1) model.
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