Transformations and Seasonal Adjustment: Analytic Solutions and Case Studies

We address the problem of seasonal adjustment of a nonlinear transformation of the original time series, such as the Box-Cox transformation of a time series measured on a ratio scale, or the Aranda-Ordaz transformation of proportions, which aims at enforcing two essential features: additivity and orthogonality of the components. The posterior mean and variance of the seasonally adjusted series admit an analytic finite representation only for particular values of the transformation parameter, e.g. for a fractional Box-Cox transformation parameter. Even if available, the analytical derivation can be tedious and difficult. As an alternative we propose to compute the two conditional moments of the seasonally adjusted series by means of numerical and Monte Carlo integration. The former is both fast and reliable in univariate applications. The latter uses the algorithm known as the simulation smoother and it is most useful in multivariate applications. We present several case studies dealing with robust seasonal adjustment under the square root and the fourth root transformation, the seasonal adjustment of the ratio of two series, and the adjustment of time series of proportions. Our overall conclusion is that robust seasonal adjustment under transformations can be carried out routinely and that the possibility of transforming the scale ought to be considered as a further option for improving the quality of seasonal adjustment.