This paper develops likelihood-based methods for estimation, inference, model selection, and forecasting of continuous-time integer-valued trawl processes. The full likelihood of integer-valued trawl processes is, in general, highly intractable, motivating the use of composite likelihood methods, where we consider the pairwise likelihood in lieu of the full likelihood. Maximizing the pairwise likelihood of the data yields an estimator of the parameter vector of the model, and we prove consistency and, in the short memory case, asymptotic normality of this estimator. When the underlying trawl process has long memory, the asymptotic behaviour of the estimator is more involved; we present some partial results for this case. The pairwise approach further allows us to develop probabilistic forecasting methods, which can be used to construct the predictive distribution of integer-valued time series. In a simulation study, we document the good finite sample performance of the likelihood-based estimator and the associated model selection procedure. Lastly, the methods are illustrated in an application to modelling and forecasting financial bid-ask spread data, where we find that it is beneficial to carefully model both the marginal distribution and the autocorrelation structure of the data.

翻译：本文发展了基于似然的方法，用于连续时间整数值拖网过程的估计、推断、模型选择及预测。整数值拖网过程的完全似然通常高度复杂难以处理，这促使我们采用复合似然方法，即用成对似然替代完全似然。最大化数据的成对似然可得到模型参数向量的估计量，我们证明了该估计量的一致性，并在短记忆情形下证明了其渐近正态性。当底层拖网过程具有长记忆时，估计量的渐近行为更为复杂；针对此情形我们给出了部分结果。成对方法还使我们能够发展概率预测方法，可用于构建整数值时间序列的预测分布。在模拟研究中，我们验证了基于似然的估计量及相关模型选择过程良好的有限样本表现。最后，我们将这些方法应用于金融买卖价差数据的建模与预测，发现仔细建模数据的边际分布和自相关结构是有益的。