Phillips averaging procedure as a "crude" version of the Haar filter

The aim of this study is to investigate the exact nature of Phillips’ (1958) findings. We show that the application of the simplest type of wavelet basis function developed by Haar in 1910 allows to replicate the output of Phillips’ data transformation procedure, i.e. the six mean coordinates. Specifically, the resemblance between the coarsest scale level coefficients from the Haar wavelet filter and the six crosses suggests the long-term nature of Phillips’ (wage-unemployment) relationship. The application of the Haar wavelet filter allows us to examine the effects of two main features of Phillips’ ‘unorthodox’ averaging procedure: the arbitrarily choice of variable-width intervals and the choice of sorting observations in ascending order of unemployment rate values. Our results show that the arbitrary selection of intervals affects only the smoothness (regularity) of the nonlinear pattern of the wage- unemployment relationship, but not its shape which is determined by sorting and grouping unemployment rate values in ascending order. Indeed, when observations are ordered according to a chronological sequence a simple linear relationship is evident. These findings are robust to different samples, 1861-1913 and 1861-1958.