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