Power spectra of the timing noise observed in 18 southern pulsars have been derived using a novel technique, based on the CLEAN algorithm. Most of the spectra are well described by a single-or double-component power-law model. Some of these spectra can be interpreted in the context of one or more of the current timing noise models. The results combined with those obtained from the time-domain analyses of the timing activity in these pulsars are used to assess the viability of the various theoretical models of pulsar timing noise.