The class of observation-driven models (ODMs) includes many models of non-linear time series which, in a fashion similar to, yet different from, hidden Markov models (HMMs), involve hidden variables. Interestingly, in contrast to most HMMs, ODMs enjoy likelihoods that can be computed exactly with computational complexity of the same order as the number of observations, making maximum likelihood estimation the privileged approach for statistical inference for these models. A celebrated example of general order ODMs is the GARCH$(p,q)$ model, for which ergodicity and inference has been studied extensively. However little is known on more general models, in particular integer-valued ones, such as the log-linear Poisson GARCH or the NBIN-GARCH of order $(p,q)$ about which most of the existing results seem restricted to the case $p=q=1$. Here we fill this gap and derive ergodicity conditions for general ODMs. The consistency and the asymptotic normality of the maximum likelihood estimator (MLE) can then be derived using the method already developed for first order ODMs.
翻译:观测驱动模型类别(ODMs)包含许多非线性时间序列模型,这些模型与隐藏的Markov模型(HMMs)类似,但与隐藏的Markov模型(HMMs)不同,涉及隐藏变量。有趣的是,与大多数HMs不同,ODMs享有与观测数量相同的计算复杂性完全可以计算的可能性,因此最有可能估计这些模型的统计推算优异方法。一般顺序ODMs的一个值得称道的例子就是GARCH$(p,q)美元模型,已经广泛研究了该模型的过分性和推断性。但在更普通的模型中,特别是整数估值模型,例如对Poisson GARCH的日志线性模型或NBIN-GARCH的订单$(p,q)美元(美元),其现有结果中的大部分似乎局限于美元=1美元的案例。在这里,我们填补了这一空白,并得出了一般ODMs的惯性条件。然后,对最大可能性测算方法的一致性和相对性常态。