Economic Time Series, T. Di Fonzo (Univ Padua) and M. Marini, (ISTAT, Italy) Description of the Sparse Linear System arising from the Reconciliation Problem of a System of Economic Time Series Notes for Professor Tim Davis by Tommaso Di Fonzo and Marco Marini July 3, 2008 Dear Prof. Davis, we have read your message Sparse QR mexFunction on the Matlab Central website, in which you invite the community to provide sparse problems arising from real applications for testing a new sparse QR function. In particular, you seem particularly interested in rank deficient problems. We have a problem with such characteristics, so we have decided to upload in the CISE website the elements of this linear system (coefficient matrix and right- hand side). This problem is generally known as the reconciliation problem in the statistics and economics literature, which we intend to describe you very briefly in this note for your information. Economic statistics are often published in the form of time series, as a collec tion of observations sampled at equally-spaced time periods (months, quar ters). Economic concepts behind such statistics are often linked by a system of linear relationships, deriving from the economic theory. However, these restrictions are rarely met by the original time series for various reasons. Then, data sets of real-world variables generally show discrepancies with respect to prior restrictions on their values. Such discrepancies are rarely accepted because their existence generally causes confusion among users and embarrassment to data producers. The adjustment of a set of data in order to satisfy a number of accounting restrictions -and thus to remove any dis crepancy -is generally known as the reconciliation problem. The reconciliation problem can be seen as the solution of a linear system. The dimension of the system increases rapidly as the number of time pe riods, the number of variables and the number of constraints grow. Since it is convenient to solve the system simultaneously (that is, considering all periods, all variables, all constraints), the use of sparse matrices and algo rithms is strongly suggested for solving practical applications. However, as far as we know, such problems are generally solved by data producers by using dense matrices and/or iterative methods (i.e. the conjugate gradi ent method). Our intention is to encourage the use of direct methods with sparse matrices for solving this kind of problems. A formalization of the problem is helpful to understand the features of this linear system. Let p be an n-dimensional vector containing all the obser- vations of the system to be reconciled (where n is the number of variables multiplied by the number of periods in which such variables are observed). Let H be the matrix defining the relationships between such observations. Supposing there are k constraints, the matrix H has dimension k × n: for each constraint we have a known total that must be satisfied by the obser vations, collected together in the k-dimensional vector w. Notice that (i) k