Nonparametric Cointegrating Regression With Endogeneity and Long Memory

Abstract

This paper explores nonparametric estimation, inference, and specification testing in a nonlinear cointegrating regression model where the structural equation errors are serially dependent and where the regressor is endogenous and may be driven by long memory innovations. Generalizing earlier results of Wang and Phillips (2009a,b, Econometric Theory 25, 710-738, Econometrica 77, 1901-1948), the conventional nonparametric local level kernel estimator is shown to be consistent and asymptotically (mixed) normal in these cases, thereby opening up inference by conventional nonparametric methods to a wide class of potentially nonlinear cointegrated relations. New results on the consistency of parametric estimates in nonlinear cointegrating regressions are provided, extending earlier research on parametric nonlinear regression and providing primitive conditions for parametric model testing. A model specification test is studied and confirmed to provide a valid mechanism for testing parametric specifications that is robust to endogeneity. But under long memory innovations the test is not pivotal, its convergence rate is parameter dependent, and its limit theory involves the local time of fractional Brownian motion. Simulation results show good performance for the nonparametric kernel estimates in cases of strong endogeneity and long memory, whereas the specification test is shown to be sensitive to the presence of long memory innovations, as predicted by asymptotic theory.

Uniform Convergence Rates Over Maximal Domains in Structural Nonparametric Cointegrating Regression

“…Remark 2.1. The preceding conditions may be compared with those imposed by Phillips (2009b, 2015); but quite unlike those authors, we do not require ε 2 0 < ∞. Although our assumptions are consistent with substantial departures from the standard unit root model -which here coincides with (ii)(a) with α = 2 -{x t } is in all cases a partial sum process, and this feature of the generating mechanism identifies (2.1) as a nonlinear cointegrating regression, in the terminology of Park and Phillips (2001).…”

“…Part (iii) permits the regression disturbance to be serially dependent, and crosscorrelated with the regressor; (2.1) is thus a structural model. Wang and Phillips (2015) allow {u t } to be generated in a slightly more general manner, according to…”

“…Remark 2.3. The requirement that ∞ k=0 |θ k |k 7/6 < ∞ is stronger than is necessary to ensure the pointwise consistency and asymptotic normality of kernel regression estimators in this setting: see for example Wang and Phillips (2015), who merely assume that ∞ k=0 |θ k |k 1/4 < ∞. Were (iii) to be relaxed in this direction, then the arguments used to prove the main results of this paper could still be applied, but the rates of convergence obtained would be complicated by the presence of an additional term, the magnitude of which would depend, in a somewhat complicated manner, on the rate at which θ k → 0 as k → ∞.…”

“…for κ ∈ and B a Brownian motion. Such a process arises as the weak limit of X n , under Wang and Phillips (2009b). (To extend our results to this case would require very little modification to our arguments; indeed, we explicitly considered such a data generating mechanism in an earlier version of this paper.)…”

“…The exogeneity assumption typically -though not universally, see Phillips (2009b, 2015) -imposed in these models is thus unlikely to be satisfied in applications. Moreover, in any application, the bandwidth used to compute a kernel regression estimator will not be determined, a priori, as some function of the sample size, but will instead be chosen with at least some reference to the sample at hand.…”