Pulsar timing array projects measure the pulse arrival times of millisecond pulsars for the primary purpose of detecting nanohertz-frequency gravitational waves. The measurements include contributions from a number of astrophysical and instrumental processes, which can either be deterministic or stochastic. It is necessary to develop robust statistical and physical models for these noise processes because incorrect models diminish sensitivity and may cause a spurious gravitational wave detection. Here we characterise noise processes for the 26 pulsars in the second data release of the Parkes Pulsar Timing Array using Bayesian inference. In addition to well-studied noise sources found previously in pulsar timing array data sets such as achromatic timing noise and dispersion measure variations, we identify new noise sources including time-correlated chromatic noise that we attribute to variations in pulse scattering. We also identify exponential dip events in four pulsars, which we attribute to magnetospheric effects as evidenced by pulse profile shape changes observed for three of the pulsars. This includes an event in PSR J1713$+$0747, which had previously been attributed to interstellar propagation. We present noise models to be used in searches for gravitational waves. We outline a robust methodology to evaluate the performance of noise models and identify unknown signals in the data. The detection of variations in pulse profiles highlights the need to develop efficient profile domain timing methods.
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