Abstract
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 (?nh) 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 (n?h) 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.
Highlights
Cointegration models are one of the most commonly used frameworks for applied research in econometrics, capturing long term relationships among trending macroeconomic time series and present value links between asset prices and fundamentals in finance
Nonlinear cointegrated systems are of particular empirical interest in cases where the data are nonstationary and move together over time yet linear cointegration fails
Time varying coefficient models provide a general mechanism for addressing and capturing such nonlinearities, allowing for smooth structural changes to occur over the sample period
Summary
Cointegration models are one of the most commonly used frameworks for applied research in econometrics, capturing long term relationships among trending macroeconomic time series and present value links between asset prices and fundamentals in finance. One exception is Park and Hahn (1999), who considered the time-varying parameter model (1.1) and used sieve methods to transform the nonlinear cointegrating equation to a linear approximation with a sieve basis of possibly diverging dimension Their asymptotic theory can be seen as an extension of the work by Park and Phillips (1988). The present paper seeks to uncover evolution in the modeling framework for nonstationary time series over a long time horizon by using nonparametric kernel regression methods to estimate f (·), and our asymptotic theory is fundamentally different from that in the paper by Park and Hahn (1999). Some supplementary technical materials and discussions on model specification testing are provided in an online supplement (Phillips, Li and Gao, 2016)
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