Modeling Compositional Time Series with Vector Autoregressive Models

Multivariate time series describing relative contributions to a total (like proportional data) are called compositional time series. They need to be transformed first to the usual Euclidean geometry before a time series model is fitted. It is shown how an appropriate transformation can be chosen, resulting in coordinates with respect to the Aitchison geometry of compositional data. Using vector autoregressive models, the standard approach based on raw data is compared with the compositional approach based on transformed data. The results from the compositional approach are consistent with the relative nature of the observations, while the analysis of the raw data leads to several inconsistencies and artifacts. The compositional approach is extended to the case when also the total of the compositional parts is of interest. Moreover, a concise methodology for an interpretation of the coordinates in the transformed space together with the corresponding statistical inference (like hypotheses testing) is provided.