Identifying a Simplifying Structure in Time Series

This article studies how to identify hidden factors in multivariate time series process. This problem is important because, when the series are driven by a set of common factors, (a) a large number of parameters may be needed to obtain an adequate representation of the system and (b) the estimated parameters will be highly correlated. Therefore, a complex and badly defined relationship can appear when, in fact, a simpler and parsimonious model in terms of a few common factors can be operating. This article develops a methodology to identify the number of factors and to build a simplifying transformation to represent the series. It is proved that the number of factors is equal to the rank of the covariance matrices and the parameter matrices of the infinite moving average representation of the process. The eigenvectors of these matrices will provide the canonical transformation. The method is illustrated with one example, using series of the price of wheat in five provinces of Spain in the 19th century. The standard approach to build a vector autore-gressive integrated moving average model showed a complex relationship with all kinds of feedback operating. When the methodology developed in the article was applied, however, two factors were identified and a clearer and simpler representation of the system was achieved.