Carmichael's Arctan Trend: Precursor of Smooth Transition Functions

In an almost unreferenced article Carmichael (1928), writing of the period around the First World War, noted that “During the past twelve years many economic series have undergone what appears to be a permanent change in level.” These are prescient words that are widely applicable today. Carmichael noted that the then standard method of linear detrending was inappropriate in the presence of what we would now call structural breaks; as a result he proposed a method that would not only model a nonlinear trend, but would be suitable for situations where the transition from one regime to another was smooth in the sense that we now associate with LSTAR transition functions. Moreover, in an even greater understanding of the underlying processes, he extended the possibility of transition to a double transition, a clear but unacknowledged precursor of recent work in that area. This study establishes the precedence of Carmichael’s work, re-examines his methods and solves the problems that he thought would hinder the then wider applications of his approach. Carmichael shows considerable skill in assessing the complex practical problems of determining the switch points and strength of adjustment of the proposed transition functions.