COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1700 Dynamic Misspecification in Nonparametric Cointegrating Regression Ioannis Kasparis and Peter C.B. Phillips June 2009 Linear cointegration is known to have the important property of invariance under
temporal translation. The same property is shown not to apply for nonlinear cointegration.
The requisite limit theory involves sample covariances of integrable transformations of
non-stationary sequences and time translated sequences, allowing for the presence of a
bandwidth parameter so as to accommodate kernel regression. The theory is an extension of
Wang and Phillips (2008) and is useful for the analysis of nonparametric regression models
with a misspecified lag structure and in situations where temporal aggregation issues
arise. The limit properties of the Nadaraya-Watson (NW) estimator for cointegrating
regression under misspecified lag structure are derived, showing the NW estimator to be
inconsistent with a "pseudo-true function" limit that is a local average of the
true regression function. In this respect nonlinear cointegrating regression differs
importantly from conventional linear cointegration which is invariant to time translation.
When centred on the pseudo-function and appropriately scaled, the NW estimator still has a
mixed Gaussian limit distribution. The convergence rates are the same as those obtained
under correct specification but the variance of the limit distribution is larger. Some
applications of the limit theory to non-linear distributed lag cointegrating regression
are given and the practical import of the results for index models, functional regression
models, and temporal aggregation are discussed. |
