Limit theory for local polynomial estimation of functional coefficient models with possibly integrated regressors

Limit theory for functional coefficient cointegrating regression was recently found to be considerably more complex than earlier understood. The issues were explained and correct limit theory derived for the kernel weighted local level estimator in Phillips and Wang (2023b). The present paper provides complete limit theory for the general kernel weighted local $p$th order polynomial estimators of the functional coefficient and the coefficient derivatives. Both stationary and nonstationary regressors are allowed. Implications for bandwidth selection are discussed. An adaptive procedure to select the fit order $p$ is proposed and found to work well. A robust $t$-ratio is constructed following the new limit theory, which corrects and improves the usual $t$-ratio in the literature. The robust $t$-ratio is valid and works well regardless of the properties of the regressors, thereby providing a unified procedure to compute the $t$-ratio and facilitating practical inference. Testing constancy of the functional coefficient is also considered. Finite sample studies are provided that corroborate the new asymptotic theory.