Dynamic Misspeci\u85cation in Nonparametric Cointegrating Regression

Linear cointegration is known to have the important property of invariance un-der 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 (2009a) and is useful for the analysis of nonparametric regression models with a misspeci\u85ed lag structure and in situations where temporal aggregation issues arise. The limit properties of the Nadaraya-Watson (NW) estimator for cointegrating regression under misspeci\u85ed lag structure are de-rived, showing the NW estimator to be inconsistent, in general, with a pseudo-true functionlimit that is a local average of the true regression function. In this respect nonlinear cointegrating regression di¤ers importantly from conventional linear coin-tegration which is invariant to time translation. When centred on the pseudo-true 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 speci cation but the variance of the limit distribution is larger. Moreover, we show that when dynamic misspeci\u85cation is severe, convergence to some pseudo-true func-tion may not hold. The estimator can be divergent or vanishing in this case. Some applications of the limit theory to non-linear distributed lag cointegrating regres-sion are given and the practical import of the results for index models, functional regression models, and temporal aggregation are discussed