We develop a new methodology for the fitting of nonstationary time series that exhibit nonlinearity, asymmetry, local persistence and changes in location scale and shape of the underlying distribution. In order to achieve this goal, we perform model selection in the class of piecewise stationary quantile autoregressive processes. The best model is defined in terms of minimizing a minimum description length criterion derived from an asymmetric Laplace likelihood. Its practical minimization is done with the use of genetic algorithms. If the data generating process follows indeed a piecewise quantile autoregression structure, we show that our method is consistent for estimating the break points and the autoregressive parameters. Empirical work suggests that the proposed method performs well in finite samples.