Models for Completing the Quarterly Data by Using the Structural Characteristics of a Reference Matrix

Abstract

Completing a partially known vector of quarterly data is considered in the article as a way to identify unknown components of this vector based on structural information contained in its known components as well as in the preselected reference macroeconomic matrix. In other words, the problem of completing a quarterly vector is reduced to «upbuilding» its unknown part according to the known one based on indirect a priori information.The article opens with a formal statement of the general problem of completing vector data together with its economic and statistical interpretation as applying to reconciling the quarterly production accounts. A general analytical solution to the problem of completing the vector of row sums is obtained on the basis of block partitioning of a reference matrix with nonnegative elements in the form of a linear model. It is shown how to use the obtained solution for completing the product output vector in the reporting quarter using the reference matrix associated with the corresponding quarter of the previous year, and how to heuristically estimate the calculation imprecision rate for the reporting quarter by analyzing the deviation of the calculated vector from the homothetic ray determined by the reference matrix. Further generalization of the analytical solution of the completing problem for the joint adjustment of quarterly production results in the reporting year in order to bring them into exact correspondence with the annual data is proposed. Particular attention is paid to heuristic estimating the imprecision rate in calculations for the reporting year based on the analysis of the difference between the system of calculated quarterly vectors and a homothety of their sum (i. e., column vector of the row margin totals for the reference annual matrix). In conclusion, recommendations to improve reliability of the completion results for quarterly vectors in practical situations are given.