The GARCH models are widely used to model various financial data with nonlinearity and heteroscedasticity structures. In this article, we propose a functional-coefficient regression model with GARCH(r, s) errors to model these kinds of data. To deal with the effect of heteroscedasticity, we introduce a two-step approach to estimating the unknown coefficient functions and the volatility, which results in unweighted and weighted local linear estimators. Asymptotic properties of the proposed estimators are established. Our results demonstrate that the weighted estimator is more efficient than the unweighted one, and the functional coefficients can be estimated by the weighted estimator as if the volatility was known. Both simulations and real data examples support our theoretical results. In particular, when there are GARCH effects, our two-step estimator mimics the oracle estimator, with the true volatility being known in advance.

中文翻译：

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具有 GARCH 误差的函数系数回归模型

GARCH 模型广泛用于对具有非线性和异方差结构的各种金融数据进行建模。在本文中，我们提出了一个具有 GARCH( r ,  s) 错误来模拟这些类型的数据。为了处理异方差性的影响，我们引入了一种两步法来估计未知系数函数和波动率，从而产生未加权和加权的局部线性估计量。建立了所提出的估计量的渐近特性。我们的结果表明，加权估计量比未加权估计量更有效，并且加权估计量可以估计函数系数，就好像波动率是已知的一样。模拟和真实数据示例都支持我们的理论结果。特别是，当存在 GARCH 效应时，我们的两步估计器会模仿预言机估计器，预先知道真实的波动率。