N-Consistent Semiparametric Regression: Partially Linear Models with Unit Roots

We develop unit root tests using additional stationary covariates as suggested in Hansen (1995). However, we allow for the covariates to enter the model in a nonparametric fashion, so that our model is an extension of the semiparametric model analyzed in Robinson (1988). We retain a linear structure for the autoregressive component and show that the parameter is estimated at rate N even though part of the model is estimated nonparametrically. The limiting distribution of the unit root test statistic is a mixture of the standard normal and the Dickey-Fuller distribution. A Monte Carlo experiment is used to evaluate the performance of the tests under various linear and nonlinear specifications for the covariates. We find that the tests are powerful when there is a nonlinear effect and experience a minimal power loss when the covariates have a linear effect or no effect at all.