CARMICHAEL’S ARCTAN TREND: PRECURSOR OF SMOOTH TRANSITION FUNCTIONS

Abstract

In an almost unreferenced article, Fitzhugh 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 practice 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. This study establishes the precedence of Carmichael’s ideas, re-examines his methods, and solves the problems that he thought would hinder wider applications of his approach, which has since become a central part of contemporary nonlinear econometric methods and for which Carmichael should be given credit.