Two-Step Reconciliation of Time Series New Formulation and Validation

Two-step reconciliation methods solve the temporal constraint in the first step, while in the second step the contemporaneous constraint is satisfied without altering the temporal constraint. Both in Quenneville and Rancourt and in Di Fonzo and Marini methods, the methodology used applies the Denton benchmarking technique in the first step. The work done in this study is based on an alternative two-step procedure for the reconciliation of systems of time series, proposing an algorithm which allows to choose one of the two different solutions for the second step, and introduces the possibility of using well-known established techniques in the first step, such a Chow and Lin, Fernandez and Litterman. Furthermore, a way of dealing with the reconciliation of hierarchical systems of time series is presented. An innovative test for detecting common seasonal patterns in time series is also presented. Such test could be used for deciding at which level to seasonally adjust an aggregated time series before applying reconciliation. Moreover, together with a simulation study, several aspects of the validation of a reconciliation technique are shown, including a new methodology for detecting whether the outliers at the end of series are consistent. Two real examples using the European industrial production index and the euro area quarterly sector accounts data will also be presented.