Adjustment of Input–Output Tables from Two Initial Matrices

Abstract

The compilation of the information required to construct survey-based input–output (I–O) tables consumes resources and time to statistical agencies. Consequently, a number of non-survey techniques have been developed in the last decades to estimate I–O tables. These techniques usually depart from observable information on the row and column margins, and then the cells of the matrix are adjusted using as a priori information a matrix from a past period (updating) or an I–O table from the same time period (regionalization). This paper proposes the use of a composite cross-entropy approach that allows for introducing both types of a priori information. The suggested methodology is suitable to be applied only to matrices with semi-positive interior cells and margins. Numerical simulations and an empirical application are carried out, where an I–O table for the Euro Area is estimated with this method and the result is compared with the traditional projection techniques.

Keywords:

• Entropy econometrics
• Data-weighted prior estimation
• Cross-entropy
• Non-survey techniques

SUPPLEMENTAL DATA

Supplemental material for this article is available via the supplemental tab on the article's online page at http://dx.doi.org/10.1080/09535314.2015.1007839.

Notes

¹The information of the target matrix is normally assumed to result in non-conflicting constraints. Variants of adjusting procedures that deal with conflicting information can be found in van der Ploeg (1982) or, more recently, in Lenzen et al. (2009).

²The same choice must be made with the RAS algorithm.

³Note that in such a case, these $p_{ij}$ elements can be seen as conditional probabilities for each column.

⁴In other words, the ME solutions are obtained by minimizing the KL divergence $D(\mathbf{\text{P}}||\mathbf{\text{Q}})$ between the unknown $p_{ij}$ and the probabilities $q_{ij}=(1/n);i=1,\dots n;j=1,\dots k$.

⁵See Hewings (1984) for a detailed discussion on the role played by prior information in such estimation problems.

⁶Details about the derivation of the solution can be found in Golan (2001, p. 175).

⁷The Euro Area in 2007 consisted of the following 17 countries belonging to the EU-27: Austria, Belgium, Cyprus, Estonia, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, Malta, the Netherlands, Portugal, Slovakia, Slovenia, and Spain.

⁸The branch 59, corresponding to ‘Private households with employed persons’ has been not considered in the numerical simulation. Data can be downloaded from: http://epp.eurostat.ec.europa.eu/portal/page/portal/esa95_supply_use_input_tables/data/workbookshttp://epp.eurostat.ec.europa.eu/portal/page/portal/esa95_supply_use_input_tables/data/workbooks

⁹Because, by definition, the weighting parameters range between 0 and 1, we opted to set the minimum number of points required to obtain a solution ($g=2$), to alleviate the computational burden in the experiment. In Golan et al. (1996, chapter 8), several simulation experiments are conducted to conclude that the outcomes of this type of estimator are not sensitive to the number of discrete points considered in the supporting vectors.

¹⁰We also follow the usual procedure of replacing the elements of L ($l_{ij}$) with $l_{ij}-{\delta }_{ij}$, where ${\delta }_{ij}$ is the Kronecker delta (${\delta }_{ij}=1$ if i = j; ${\delta }_{ij}=0$ otherwise).