Box-Cox transformations of terms nesting the Trans-Log: the example of rail infrastructure maintenance cost

We explore how the Trans-Log (TL) can be nested in Box-Cox transformed terms and show that a particular specification previously defined, but not fully tested, within the European CATRIN consortium (Gaudry & Quinet, 2010), the Unrestricted Generalized Box-Cox (U-GBC), constitutes a proper incarnation of the Generalized Flexible Quadratic class (Blackorby et al. (1977) and nests a number of more or less known intermediate Box-Cox-inspired partial generalizations of the TL, as well as the target TL itself. After a brief rail cost litterature review, our detailed references to such intermediate model specifications making partial use of Box-Cox transformations are focused on examples developed since 2002 using cross-sectional data, shown to differ profoundly from their ancestor aggregate time-series firm-wide explanations of total or of current maintenance rail cost published before 2002. Notably, the TL, devoid of prices, has since 2002 become a rail engineering degradation cost model under an unchanged econometric terminological form garb on which we dwell. We estimate three main rail maintenance cost model specifications strictly nesting the TL from real 1999 France-wide segment network data and compare their improved log likelihood values under different engineering hypotheses concerning physical interactions among four rail Traffic types and four track Quality characteristics. We find the Trans-Log to be an inadequate model of railway damage because physical interactions among track Quality indicators and train Traffic types are not of log-log form but of other forms better handled by common flexible Box-Cox Transformations, twelve of which are estimated in our most general U-GBC specification, all but one actually differing from the logarithmic case. And, of course, not all physical interactions turn out to matter in the explanation of degradation cost: track Quality-Quality interactions, for instance, are of nugatory importance.