# Consistent Multivariate Seasonal Adjustment

A new method makes it possible to compute consistent seasonal corrected series.
Seasonally adjusted series of Gross Domestic Product (GDP) and its breakdown in underlying categories or domains are generally not consistent with each other. Statistical differences between the total GDP and the sum over the underlying domains arise for two reasons. If series are expressed in constant prices, differences arise due to the process of chain linking. These differences increase if in addition a univariate seasonal adjustment, with for instance X-13ARIMA-SEATS, is applied to each series separately. In this paper, it is proposed to model the series for total GDP and its breakdown in underlying domains in one multivariate structural time series model with the restriction that the sum over the different time series components for the domains are equal to the corresponding values for the total GDP. In the proposed procedure this approach is applied as a pre-treatment to remove outliers, level shifts, seasonal breaks and calendar effects, while obeying the aforementioned consistency restrictions. Subsequently, X-13ARIMA-SEATS is used for seasonal adjustment. This reduces inconsistencies remarkably. Remaining inconsistencies due to seasonal and calendar adjustment are removed with a benchmarking procedure.