Limit Theory for Locally Flat Functional Coefficient Regression

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. In such cases, 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. 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. Locally flat behavior in the coefficient function has, as expected, a major effect on bias and thereby on the trade-off between bias and variance, and on optimal bandwidth choice. 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. Both bias and variance depend on the number of zero derivatives in the coefficient function. In the boundary case where the number of zero derivatives tends to infinity, near parametric rates of convergence apply for both stationary and nonstationary cases. Implications for inference are discussed and simulations characterizing finite sample behavior are reported.