Methods and Procedures for Element-Wise Reconciliation of Quarterly Macroeconomic Matrices with the Annual Matrix

The element-wise reconciliation of quarterly macroeconomic matrices with the annual matrix in the process of operational balancing of quarterly national accounts is considered in the article as a universal way for quarterly decomposing the annual output matrix. A generalized formulation of the problem of such decomposing is given in terms of mathematical programming. The basis of the proposed approach is the weighted least squares method applied in the linear space of the quarterly coefficients vectors of output distribution by products and industries with weights characterizing a priori relative importance or reliability for every summand of the problem quadratic objective function.
It is shown that the problem of the annual matrix quarterly decomposition in its generalized formulation does not have an optimal solution and is of practical interest only as a source of its two operational versions – the «product» one and the «industry» one. Both versions are quadratic programming problems with linear constraints; their solutions are obtained in analytical form using the Lagrange multiplier method.
The advantages of the developed methods for reconciling quarterly matrices of the products and industries outputs with the annual matrix are the simplicity of practical calculations using compact formulas and a very moderate need for computing resources even with a huge amount of initial data. The proposed optimization approach demonstrates a high degree of flexibility and adaptability in solving problems of reconciling quarterly matrices with annual data on the output of goods and services. High flexibility is provided by the dependence of the considered problems on sets of exogenous parameters that could vary purposefully in the course of performing practical calculations.

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