A local linear estimator of generalized impulse response (GIR) functions is derived for conditional heteroskedastic autoregressive nonlinear processes and shown to be asymptotically normal. We suggest a plug-in bandwidth based on the derived optimal bandwidth. A local linear estimator for the conditional variance function is proposed with simpler bias than the standard estimator. This is achieved by appropriately eliminating the conditional mean. Alternatively to the direct local linear estimators of the k-step prediction functions which enter the GIR estimator we suggest the use of multi-stage prediction techniques. Simulation experiments show the latter estimator to perform best. For quarterly data of the West German real GNP we find that the sizes of generalized impulse response functions vary across different histories, a feature which cannot be captured by linear models.