Benchmarking systems of seasonally adjusted time series according.

When a system of time series is seasonally adjusted, generally the accounting constraints originally linking the series are not fulfilled. To overcome this problem, we discuss an extension to a system of series linked by an accounting constraint of the classical univariate benchmarking procedure due to Denton (1971), which is founded on a movement preservation principle very appreciable in this case. The presence of linear dependence between the variables makes it necessary to conveniently deal with the whole set of contemporaneous and temporal aggregation relationships. The case of the one-way classified (e.g. by regions or by industries) and of two-ways classified (e.g., by regions and by industries) systems of series are studied. Moreover, simplified expressions of the matrices involved in the calculations are presented, which turn out useful in practical implementation of benchmatking for large systems of series. An empirical application of the Canadian retail trade series by province (12 series) and trade groups (18 series) is finally considered to show the capability of the proposed procedures