Functional coefficient (FC) regressions allow for systematic flexibility in the responsiveness of a dependent variable to movements in the regressors, making them attractive in applications where marginal effects may depend on covariates. Such models are commonly estimated by local kernel regression methods. This paper explores situations where responsiveness to covariates is locally flat or fixed. The paper develops new asymptotics that take account of shape characteristics of the function in the locality of the point of estimation. Both stationary and integrated regressor cases are examined. The limit theory of FC kernel regression is shown to depend intimately on functional shape in ways that affect rates of convergence, optimal bandwidth selection, estimation, and inference. In FC cointegrating regression, flat behavior materially changes the limit distribution by introducing the shape characteristics of the function into the limiting distribution through variance as well as centering. In the boundary case where the number of zero derivatives tends to infinity, near parametric rates of convergence apply in stationary and nonstationary cases. Implications for inference are discussed and a feasible pre-test inference procedure is proposed that takes unknown potential flatness into consideration and provides a practical approach to inference.

函数系数 (FC) 回归允许因变量对回归量运动的响应具有系统灵活性，这使得它们在边际效应可能取决于协变量的应用中具有吸引力。此类模型通常通过局部核回归方法进行估计。本文探讨了对协变量的响应局部平坦或固定的情况。本文开发了新的渐近函数，考虑了估计点局部函数的形状特征。检查静态和积分回归量情况。FC 核回归的极限理论被证明与函数形状密切相关，影响收敛速度、最佳带宽选择、估计和推理。在 FC 协整回归中，平坦行为通过方差和居中将函数的形状特征引入极限分布，从而实质上改变了极限分布。在零导数的数量趋于无穷大的边界情况下，近似参数收敛率适用于平稳和非平稳情况。讨论了推理的含义，并提出了一种可行的预测试推理程序，该程序考虑了未知的潜在平坦度，并提供了一种实用的推理方法。