A random coefficient autoregressive process in which the coefficients are correlated is investigated. First we look at the existence of a strictly stationary causal solution, we give the second-order stationarity conditions and the autocorrelation function of the process. Then we study some asymptotic properties of the empirical mean and the usual estimators of the process, such as convergence, asymptotic normality and rates of convergence, supplied with appropriate assumptions on the driving perturbations. Our objective is to get an overview of the influence of correlated coefficients in the estimation step through a simple model. In particular, the lack of consistency is shown for the estimation of the autoregressive parameter when the independence hypothesis in the random coefficients is violated. Finally, a consistent estimation is given together with a testing procedure for the existence of correlation in the coefficients. While convergence properties rely on ergodicity, we use a martingale approach to reach most of the results.

中文翻译：

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自回归过程的随机系数的相关性检验

研究了系数相关的随机系数自回归过程。首先，我们考察一个严格平稳的因果解的存在，给出二阶平稳条件和过程的自相关函数。然后，我们研究经验均值的渐近性质和该过程的常用估计量，例如收敛性，渐近正态性和收敛速度，并为驱动扰动提供了适当的假设。我们的目标是通过一个简单的模型来概述相关系数在估计步骤中的影响。特别地，当违反随机系数中的独立性假设时，显示出缺乏对自回归参数的估计的一致性。最后，给出了一致的估计值，并给出了系数相关性存在的测试过程。虽然收敛属性依赖于遍历性，但我们使用a方法来获得大部分结果。