Some properties and a case study of estimation for autoregressive moving-average coefficients in the presence of a regression trend

The residuals of a nonparametric prediction are generally not white-noise distributed. This leads us to consider using a nonparametric model Y_t = r(X_t) + Z_t, where r is an unknown smooth function and {Z_t} is a sequence of causal and invertible autoregressive moving-average error, to improve the nonparametric prediction. We show that under mild assumptions the constructed parametric estimators of error component are asymptotically equivalent to those based on {Z_t}. We also use the well-known Canadian lynx data to compare the performance of nonparametric model fitted to this data with some parametric models.