Abstract
We study a general class of quasi-maximum likelihood estimators for observation-driven time series models. Our main focus is on models related to the exponential family of distributions like Poisson based models for count time series or duration models. However, the proposed approach is more general and covers a variety of time series models including the ordinary GARCH model which has been studied extensively in the literature. We provide general conditions under which quasi-maximum likelihood estimators can be analyzed for this class of time series models and we prove that these estimators are consistent and asymptotically normally distributed regardless of the true data generating process. We illustrate our results using classical examples of quasi-maximum likelihood estimation including standard GARCH models, duration models, Poisson type autoregressions and ARMA models with GARCH errors. Our contribution unifies the existing theory and gives conditions for proving consistency and asymptotic normality in a variety of situations.
Highlights
Our primary aim is to study the properties of the Quasi Maximum Likelihood Estimators (QMLE) for estimating the unknown parameter θ
When a model has been correctly specified, that is when there exists a parameter θ ∈ Θ such that the data are generated according to this specific process, Theorem 3.1 implies consistency of the MLE to θ provided that the set Θ is reduced to the singleton {θ }
We assume for simplicity that X ⊂ R and we initially study the consistency property of the QMLE
Summary
In classical state-space models, referred as parameter-driven models, the observations {Yt , t ∈ N} are modelled hierarchically given a hidden process {Xt , t ∈ N} which has its own (most often Markovian) dynamic structure, see [28] or [12], for instance. For estimating the parameter vector θ, [13] has suggested the use of QMLE by assuming that { t , t ∈ N} is a sequence of i.i.d. exponential random variables with mean one This work includes this specification and gives conditions for obtaining asymptotically normally distributed estimators in the case of model (2.3). We can consider observation-driven models for binary time series by letting Xt = F −1(pt), where F (·) is the cdf of continuous random variable. Where m(Yt−1; α) represents the conditional mean (which depends on an unknown parameter α) and the volatility process is modeled by a non-linear model as discussed above In this example, the complete model depends on the unknown parameter vector θ = (α, λ), whereas the distribution of Yt given σt and Yt−1 depends only on the parameter φ(θ) = α. We proceed to study the asymptotic behavior of the QMLE
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