On consistency of minimum description length model selection for piecewise autoregressions

The Auto-PARM (Automatic Piecewise AutoRegressive Modeling) procedure, developed by Davis et al. (2006), uses the minimum description length (MDL) principle to estimate the number and locations of structural breaks in a non-stationary time series. Consistency of this model selection procedure has been established when using conditional maximum (Gaussian) likelihood variance estimates. In contrast, the estimate of the number of change-points is inconsistent in general if Yule-Walker variance estimates are used instead. This surprising result is due to an exact cancellation of first-order terms in a Taylor series expansion in the conditional maximum likelihood case, which does not occur in the Yule-Walker case. In order to simplify notation and make the arguments more transparent, we only treat in detail the simple case where the time series follows an AR(p) model with no change-points.