MATRIX BALANCING UNDER CONFLICTING INFORMATION

Abstract

We have developed a generalised iterative scaling method (KRAS) that is able to balance and reconcile input–output tables and SAMs under conflicting external information and inconsistent constraints. Like earlier RAS variants, KRAS can: (a) handle constraints on arbitrarily sized and shaped subsets of matrix elements; (b) include reliability of the initial estimate and the external constraints; and (c) deal with negative values, and preserve the sign of matrix elements. Applying KRAS in four case studies, we find that, as with constrained optimisation, KRAS is able to find a compromise solution between inconsistent constraints. This feature does not exist in conventional RAS variants such as GRAS. KRAS can constitute a major advance for the practice of balancing input–output tables and Social Accounting Matrices, in that it removes the necessity of manually tracing inconsistencies in external information. This quality does not come at the expense of substantial programming and computational requirements (of conventional constrained optimisation techniques).

Keywords:

• Matrix balancing
• Inconsistent constraints
• Constrained optimisation
• RAS

Notes

¹ Lahr and de Mesnard (2004) provide a recent overview of extensions to the classic RAS technique. Huang et al. (2008) explore different objective functions for input-output matrix updating.

² Recently, Canning and Wang (2005) note that “another important advantage of the mathematical programming model over scaling methods is in its flexibility. Additional constraints can be easily imposed, such as […] incorporating an associate term in the objective function to penalize solution deviations from the initial row and column total estimates when they are not known with certainty.”

³ Möhr et al. (1987) make this problem feasible by adding an “augmentation” which ensures that matrix elements are non-zero where constraints require it. While this is a solution to these authors’ problem, it does not address constraint conflicts in general.

⁴ Barker et al. (1984, p. 475) write: “… we observed that the income, expenditure, production and financial estimates of data are typically inconsistent. The presence of such accounting inconsistencies emphasises the unreliable nature of economic data.” See also Smith et al. (1998).

⁵ Barker et al. (1984, p. 475) remark that “… trading off the relative degrees of uncertainty of the various data items in the system in order to adjust the prior data to fit the accounting identities […] is essentially what national income accountants do during the last stages of compiling the accounts when faced with major discrepancies between data from different sources”. Dalgaard and Gysting (2004, p. 170) from Statistik Denmark report that many analysts responsible for compiling input–output tables favour manual adjustment, because “based on the experience that many errors in primary statistics are spotted in the course of a balancing process that is predominantly manual, compilers are typically convinced that a (mainly) manual balancing process yields results of higher quality than those emanating from a purely automatic balancing of the accounts. From that point of view, the resources involved in manual balancing are justified as a very efficient consistency check on the accounts.”

⁶ When applied to the forecasting of monetary input–output matrices, bi-proportional changes have been interpreted as productivity, substitution or fabrication effects (Leontief, 1941; Stone and Brown, 1962) affecting industries over time. Miernyk's (1976) view however is that the RAS method “substitutes computational tractability for economic logic”, and that the production interpretation loses its meaning when the entire input–output table is balanced, and not only inter-industry transactions (see also Giarratani, 1975).

⁷ The RAS, Linear Programming and minimum information gain algorithms yield a balanced matrix estimate that is – in terms of some measure of multidimensional “distance” – closest to the unbalanced preliminary estimate. When applied to temporal forecasting, this property is explained as a conservative hypothesis of attributing inertia to inter-industrial relations (Bacharach, 1970, p. 26). While the classic RAS method is aimed at maintaining the value structure of the balanced matrix, the closely related cross-entropy methods (Robinson et al., 2001) are aimed at maintaining the coefficient structure.

⁸ See Kruithof (1937) as cited in Lamond and Stewart (1981); Deming and Stephan (1940); attributed to 1930s Leningrad architect Sheleikhovskii by Bregman (1967a).

⁹ Dalgaard and Gysting (2004) do describe balancing matrices with “unequal net row and column sum” and “macro differences between supply and use”. However, rather than inconsistencies in external information, this means correct differences in the sum over supply by industry and use by product, which naturally occur in asymmetric commodity-by-industry supply and use tables.

¹⁰ Using the trivial case of starting with an initial estimate A that already satisfies all prescribed row and column totals, Lenzen et al. (2007) construct a case where Junius and Oosterhaven's (2003) GRAS balancing algorithm leads to a solution X that is inferior to the initial estimate in terms of their target function. They show that the factor e has to be taken out of the Junius and Oosterhaven formulation in order to correct the problem.

¹¹ Tarancon and Del Rio (2005) present an interesting variant of the bounded optimisation problem, by deriving lower and upper bounds from criteria for the stable structural evolution of input–output coefficients, and introducing supplementary variables to take up the slack between the bounds and the matrix entries. If the model turns out to be inconsistent because some constraints cannot be met within those bounds, then the analyst manually chooses certain constraints to be relaxed, until no variable exceeds the bounds.

¹² Single-element constraints need not be part of the scaling procedure, but could be “netted out” using the “modified RAS” method.

¹³ Here, the modulus function a mod b refers to the remainder of the division of a by b.

¹⁴ Non-unity constraints can appear when there is knowledge of relative as opposed to absolute values for some matrix elements. For example, in the construction of multi-regional input–output systems we may use information regarding the fraction of value added or gross output allocated to a given region.

¹⁵ See Bregman (1967b), Elfving (1980) and Lamond and Stewart (1981). Elfving (1980) distinguishes “Gauss-Seidel type schemes”, where only one Lagrange multiplier is adjusted in every step, and “Jacobi type methods” where the non-linear system in Equation 12 is solved simultaneously for all Lagrange multipliers. The latter methods require the Jacobian of Equation 12 to have full rank, i.e. all constraint equations have to be linearly independent (compare Eriksson, 1980; Erlander, 1981).

¹⁶ The problem can be restated so that all exponents are positive, by multiplying Equation 14 with .

¹⁷ Using Newton's method, a root r of a function f(x), in the vicinity of x ₀, is approached by first truncating the Taylor expansion around x ₀: f(x) = f(x ₀)+f'(x ₀)(x – x ₀)+½ f'(x ₀) (x – x ₀)²+…, to 0 = f(r) ≈ f(x ₀)+f'(x ₀) (r – x ₀), and then iteratively solving x _n  = x _n–1 – f(x _n–1)/f'(x _n–1).

¹⁸ If external data are not normally distributed, the adjustment of constraints ασ _i in Equation 23 could be adapted to the characteristics of the alternative distribution. For example, the shape parameter of a Weibull distribution could take the place of the standard deviation σ in Equation 23. Further, if conflicting values were distributed uniformly, adjustments could be made in proportion to the uniform ranges. In the case of subjective reliability scores, one could in a similar fashion, that is, by taking normalised scores as step-wise adjustments.

¹⁹ Cole (1992) does not give any information on reliability. All standard deviations were set by the authors.

²⁰ The small δ value has been incorporated to handle the case of σ _i  = 0.

²¹ The run time for this example was in the order of several seconds for complete covergence, using the intel fortran90 compiler v.7.1 on a RHEL-WS24 linux Kernel with a Xenon 32 bit and 2×3.1 GHz CPU and 4Gb RAM.

²² New South Wales, Victoria, Queensland, South Australia, Western Australia, Tasmania, Northern Territory, Australian Capital Territory.

²³ See http://www.isa.org.usyd.edu.au/research/ISA_TBL_Indicators.pdf.

²⁴ It has been assumed that these {σi} represent the standard deviations of normally distributed random errors in the survey data. Possible systematic errors in the data sources have not been captured in this model.

²⁵ A small δ value was applied in cases of σi = 0.

²⁶ The initial RAS iterations took up to 5 min of run time each, due to the time spent on the Newton algorithm looking for a solution of Equation 16 when far from the initial estimate. However, this time shortened as the run advanced and the average run time for a KRAS iteration was about 5 sec.