Publication Date: September 2013
This paper studies nonlinear cointegration models in which the structural coeﬀicients may evolve smoothly over time. These time-varying coeﬀicient functions are well-suited to many practical applications and can be estimated conveniently by nonparametric kernel methods. It is shown that the usual asymptotic methods of kernel estimation completely break down in this setting when the functional coeﬀicients are multivariate. The reason for this breakdown is a kernel-induced degeneracy in the weighted signal matrix associated with the nonstationary regressors, a new phenomenon in the kernel regression literature. Some new techniques are developed to address the degeneracy and resolve the asymptotics, using a path-dependent local coordinate transformation to re-orient coordinates and accommodate the degeneracy. The resulting asymptotic theory is fundamentally diﬀerent from the existing kernel literature, giving two diﬀerent limit distributions with diﬀerent convergence rates in the diﬀerent directions (or combinations) of the (functional) parameter space. Both rates are faster than the usual (vnh) rate for nonlinear models with smoothly changing coeﬀicients and local stationarity. Hence two types of super-consistency apply in nonparametric kernel estimation of time-varying coeﬀicient cointegration models. The higher rate of convergence (nvh) lies in the direction of the nonstationary regressor vector at the local coordinate point. The lower rate (nh) lies in the degenerate directions but is still super-consistent for nonparametric estimators. In addition, local linear methods are used to reduce asymptotic bias and a fully modiﬁed kernel regression method is proposed to deal with the general endogenous nonstationary regressor case. Simulations are conducted to explore the ﬁnite sample properties of the methods and a practical application is given to examine time varying empirical relationships involving consumption, disposable income, investment and real interest rates.
Cointegration; Endogeneity; Kernel degeneracy; Nonparametric regression; Super-consistency; Time varying coeﬀicients
JEL Classification Codes: C13, C14, C23