The classical ARCH model together with its extensions have been widely applied in the modeling of financial time series. We study a variant of the ARCH model that takes account of liquidity given by a positive stationary process. We provide minimal assumptions that ensure the existence and uniqueness of the stationary solution for this model. Moreover, we give necessary and sufficient conditions for the existence of the autocovariance function. After that, we derive an AR(1) characterization for the stationary solution yielding Yule–Walker type quadratic equations for the model parameters. In order to define a proper estimation method for the model, we first show that the autocovariance estimators of the stationary solution are consistent under relatively mild assumptions. Consequently, we prove that the natural estimators arising out of the quadratic equations inherit consistency from the autocovariance estimators. Finally, we illustrate our results with several examples and a simulation study.

中文翻译：

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具有平稳流动性的 ARCH 模型

经典的 ARCH 模型及其扩展已广泛应用于金融时间序列的建模。我们研究了 ARCH 模型的一个变体，它考虑了正平稳过程给出的流动性。我们提供了最小的假设，以确保该模型的平稳解的存在性和唯一性。此外，我们给出了自协方差函数存在的充要条件。之后，我们推导出固定解的 AR(1) 表征，产生模型参数的 Yule-Walker 型二次方程。为了为模型定义合适的估计方法，我们首先证明在相对温和的假设下，平稳解的自协方差估计量是一致的。最后，我们证明了由二次方程产生的自然估计量继承了自协方差估计量的一致性。最后，我们用几个例子和模拟研究来说明我们的结果。