This article considers goodness-of-fit tests for bivariate INAR and bivariate Poisson autoregression models. The test statistics are based on an L2-type distance between two estimators of the probability generating function of the observations: one being entirely nonparametric and the second one being semiparametric computed under the corresponding null hypothesis. The asymptotic distribution of the proposed tests statistics both under the null hypotheses as well as under alternatives is derived and consistency is proved. The case of testing bivariate generalized Poisson autoregression and extension of the methods to dimension higher than two are also discussed. The finite-sample performance of a parametric bootstrap version of the tests is illustrated via a series of Monte Carlo experiments. The article concludes with applications on real data sets and discussion.

中文翻译：

---

双变量时间序列的拟合优度检验

本文考虑了双变量INAR和双变量泊松自回归模型的拟合优度检验。检验统计基于观察值的概率生成函数的两个估计量之间的L2型距离：一个是完全非参数的，第二个是在相应的无效假设下计算的半参数。推导了在零假设下以及在备选方案下所提出的检验统计量的渐近分布，并证明了一致性。还讨论了测试二元广义Poisson自回归和将方法扩展到大于2维的情况。通过一系列的蒙特卡洛实验说明了测试的参数引导程序版本的有限样本性能。