Journal of Applied Input-Output Analysis, Vol. 11 & 12, 2006

A flexible approach to matrix balancing under partial information

By

Manfred Lenzen*, Blanca Gallegof and Richard Wood*
ISA, A28,The University of Sydney NSW 2006, Australia, http://www.isa.org.usyd.edu.au

Abstract
We present an extension of the standard RAS algorithm, referred to as 'constrained RAS',
or 'cRAS'. In contrast to existing RAS variants employing additional partial information ina particular aggregated form, our technique makes use of partial information of any kind.

This is achieved by imposing constraints on arbitrary-sized and -shaped sets of elements to
be balanced. The characteristics of the cRAS method are investigated in detail.

1 Introduction
A common problem in compiling and updating input-output tables is that of incom

plete data. Missing matrix elements may be due to a variety of reasons such as costly
and therefore incomplete industry surveys, or the suppression of confidential informa

tion. Unknowns almost always outnumber data points, resulting in the system being
underdetermined, that is exhibiting too many degrees of freedom to be solved analyti
cally, so that certain estimation, balancing or optimisation techniques have to be ap
plied.

Despite having received mixed responses in relation to the accuracy of its results1,
a widely applied remedy in this situation is the bi-proportional, or RAS method (Stone
and Brown 1962). Especially for the task of constructing or updating regional inputoutput tables from national data, the RAS method has been compared with other ap
proaches such as location quotients and commodity balances (Schaffer and Chu 1969;
Leontief 1953), gravity models (Leontief and Strout 1963), linear programming (Moses
1960; Matuszewski et al. 1964; Davis et al. 1977), or entropy and minimum informa
tion gain techniques (Uribe et al. 1966; Batten 1982; Golan et al. 1994). The originsof the method go back several decades (Deming and Stephan 1940); for a historicalbackground, see Bacharach 1970, Chap. 1-3, and Polenske 1997.

The RAS method - in its basic form - bi-proportionally scales a matrix Ao of unReceived March 2006,final version received January 2007.

* corresponding author; email m.lenzen@physics.usyd.edu.au, ph +61 / 2 / 9351-5985, fax +61 / 2 /9351-7725

t email blanca@physics.usyd.edu.au$ email rwood@physics.usyd.edu.au

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Journal of Applied Input-Output Analysis, Vol. 11 & 12, 2006

balanced preliminary estimates of an unknown real matrix A, using A's known row
and column sums. The balancing process is usually aborted when the discrepancy be
tween the row and column sums of Ao and A is less than a previously fixed threshold.
Bacharach 1970 has analysed the bi-proportional constrained matrix problem in great

detail, in particular in regard to the economic meaning of bi-proportional change2, theexistence and uniqueness of the iterative RAS solution, its properties of minimisationof distance metric3, and uncertainty associated with errors in row and column sum data
and with the assumption of bi-proportionality.
A special situation arises when some of the matrix elements of A are known in

addition to its row and column sums, for example from an industry survey. The 'modi
fied RAS' approach (Paelinck and Walbroeck 1963; Allen 1974; Lecomber 1975 deals
with this partial information as follows: the preliminary estimate Ao has to be "netted",
that is the known elements are subtracted, and Ao contains 0 at the corresponding positions. The net Ao is then subjected to the standard RAS procedure, and the known ele
ments are added back on after balancing. An operational example of this approach is
the GRIT (Generation of Regional Input-output Tables) method (Jensen et al. 1979;

West et al. 1984; Bayne and West 1989), which involves 4 steps: 1) preparation of thenational table, 2) preliminary estimate of the regional table using a non-survey ap

proach such as commodity balances or location quotients4, 3) insertion of superior,survey-based data, and 4) identification of important elements, and subsequent collec

tion and insertion of superior data. Within GRIT, aggregated superior data can only be
inserted after aggregating the preliminary table (Jensen et al. 1979, Sec. 6.4). Anothervariant of the modified RAS method takes into account the uncertainty of the prelimi
nary estimates, and contains the occurrence of perfectly known elements as a special

case5 (Allen and Lecomber 1975).
In practice, situations can arise where, in addition to certain elements of A, some1

See for example Uribe et al. 1966; Czamanski and Malizia 1969; Goldman 1969; Miernyk 1969;
Malizia and Bond 1974; Morrison and Smith 1974; Miernyk 1976; Hinojosa 1978; Parikh 1979;
Butterfleld and Mules 1980; Morrison and Thuman 1980. Jensen et al. 1979 summarise the dilemmaof regional input-output modeling as a trade-off between 'professional purity' and uncertain accu
racy.

2

When applied to the forecasting of monetary input-output matrices, bi-proportional changes havebeen 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 "substi
tutes 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 (seealso Giarratani 1975).

3 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 unbal
anced preliminary estimate. When applied to temporal forecasting, this property is explained as aconservative hypothesis of attributing inertia to inter-industrial relations (Bacharach 1970, p. 26).

4 For an overview of these techniques see Schaffer and Chu 1969, or Harrigan et al. 1980.
5 This is accomplished by introducing a matrix E of standard errors of the elements in Ao, which issubsequently balanced in order to take up the difference between the preliminary and true totals.

Where £/p0, the element in Ao remains unchanged. Allen and Lecomber 1975 also investigate theinfluence of errors in the "true" totals.

A flexible approach to matrix balancing under partial information

3

aggregates of elements of A are known. For example, a published table AG of national
aggregates may constitute partial information when constructing a multi-regional input-

output system, or a more disaggregated national table. Accordingly, Oosterhaven et al.

1986 add a "national cell constraint" to the standard row and column sum constraints.

Similarly, Jackson and Comer 1993 use partition coefficients for groups of cells of adisaggregated base year matrix to disaggregate cells in an updated but aggregated ma

trix. Batten and Martellato 1985, p. 52-55) discuss further constraints structures, involving intermediate and final demand data. Gilchrist and St Louis 1999 propose athree-stage RAS for the case when aggregation rules exist under which the partial ag

gregated information AG can be constructed from its disaggregated form A. These rules
take the form of matrix operators P and Q, with AG = PAQ. The essence of their
TRAS method is that balancing is carried out alternately at the disaggregated and at

the aggregated level, with the aggregated row and column scaling operators RG and SG
being disaggregated via multiplication with the transposed operators Pl and Q\ Subject
ing an input-output matrix to random censoring, Gilchrist and St Louis 1999 demon
strate that the inclusion of partial aggregated information into the RAS procedure leads
to superior outcomes than applying the standard RAS method.

In the above approaches the partial information has to assume a particular aggre

gated form. A number of authors6 have expressed the table estimation as an economet
ric or a linear or quadratic programming problem, which allows the balancing of negative elements and the inclusion of partial information of any form. Lahr and de Mesnard generalised the method 1) by formulating it as a Lagrangian multiplier calculus,
as an econometric estimation, or as an optimisation, 2) by enabling balancing of negative elements, and 3) by allowing for partial information of any form, by imposing
constraints on arbitrary-sized and -shaped sets of elements of Ao. Lahr and de Mesnard
2004 provide a recent overview.

The aim of this work is to test the performance of approaches that feature arbi
trary constraints. In particular we investigate whether

1) it is better to constrain a few large elements, or many small elements,

2) it is better to have a few constraints spanning many elements, or many con
straints on single elements,

3) the shape of the constraints (adjacent elements or disjunct elements) influences
the performance.

We enumerate performance using a range of common distance measures, and apply
them to the flow matrix, the Leontief inverse, and some multipliers. We restrict ourselves to testing a RAS-type approach (referred to as 'constrained RAS', or 'cRAS'),and ignore Lagrangian and optimisation approaches.

This article is organised as follows: We describe our proposed cRAS method, thenpresent an application to a generalised regional input-output system for Australia, andconclude.

6 See for example Byron 1978, Morrison and Thuman 1980, van der Ploeg 1982 and Cole 1992 on the
Lagrangean method, Gerking 1976; 1979a and Toh 1998 on econometric estimation, and Tarancon
and Del Rio 2005 on optimisation. See Junius and Oosterhaven 2003 and Jackson and Murray 2004

on the problem of negative entries. See Morrison and Thuman 1980, Cole 1992, Thissen and Lofgren 1998 and Tarancon and Del Rio 2005 on generalized constraints.

4

Journal of Applied Input-Output Analysis, Vol. 11 & 12, 2006

2 A representative cRAS method
In the standard RAS method, the preliminary estimate Ao is alternately row- andcolumn-scaled using diagonal matrices f and s, so that after one round of balancing,.

Ai = fAo§. Ai is then subjected to the next scaling operation. One may start with eitherrows, Ai = (fA0)s, or columns, Ai = f(A0§). In the first round, fa = a, I 2,Ao,y and §M =cij I 2/Ao.y, and so on, where the ays are row/column totals of A.
Let C be the number of additional available data points to be used as constraints

on the Ay. Let Ec be the number of elements of A participating in constraint c. Let (ic,e,Ecjce) be the coordinates of element e in constraint c. Let Bc = 2 Aw^be the value pree= 1

scribed for the sum of elements participating in constraint c. A typical cRAS algorithm

can then be described as a sequence:- Generate preliminary estimate Ao;- scale rows of Ao by pre-multiplication with f;- scale constraint 1 by multiplying all E\ participating elements with
/Ec

e=l

Ec

I Ec

e=l

/ e=l

- scale constraint 2 by multiplying all E2 participating elements with ;
I Ec

Ec

I Ec

Bi/1^ Aoti2ej2e = 2-j Ai2 j2e I Zj Ao,i2ej2e;
/ e=l

e=l

I e=l

- and so on, until constraint C;- scale columns by post-multiplication with s;- result of the above operations is Ai;- repeat scaling process for Ai until all scaling factors deviate from 1 by less thana specified threshold.

Any permutation of this sequence is also a valid cRAS algorithm. Constraints can include any number of elements, which may be fully, partly or non-adjacent7. Constraints
may also exclude some of the row and column totals (compare Thissen and Lofgren
1998, p. 1994).

3 Performance evaluation
In the following, we will examine the performance of the cRAS method for the example of estimating the use matrix U of the 1996-97 Australian input-output tables fromits true row and column sums, as well as varying numbers of subsets of true elements,
7

Constraints with Ec = 1 need not be part of the scaling procedure, but could be "netted out" using
the "modified RAS" method.

A flexible approach to matrix balancing under partial information

5

starting with different preliminary estimates. We will also appraise estimates for the Le-

ontief inverse L = (I-Ux"1)"1, and output and CO2 multipliers £ = 1L and X = cL, re
spectively, where I is a unity matrix, x is the diagonalised gross output vector, 1 is aunity row vector, and c is a row vector of sectoral CO2 intensities.
While a performance evaluation based on U emphasises the importance of the ta

ble elements themselves, evaluations based on L, £ and X recognise that these derived
quantities are ultimately of interest in applied models. With each subsequent step offurther processing U (through inversion to L, summing to £ and weighting to X), thederived quantities embody more widely inter-linked information. Evaluations of thesequantities (rather than table elements) are the subject of a more integrated and applied
measure of overall, 'holistic', or operational accuracy (Jensen 1980). While table accuracy represents the conventional understanding of a measure of the deviation of all esti
mated matrix elements from their "true" values, holistic accuracy is concerned with therepresentativeness of a table of the synergistic characteristics of an economy. In thisperspective, the accuracy of single elements may be unimportant, as long as the resultsof modeling exercises yield a realistic picture for the purpose of the analyst or decision-maker. Table accuracy implies holistic accuracy, but not vice versa; nevertheless, theformer may not be achievable, while the latter may be more realistic in pragmatic
terms. However, a drawback is that general criteria for holistic accuracy are hard to define, and hence are a characteristic of the model output, such as the derived quantities
above.

3.1 Programming features
A number of stochastic experiments of up to 250 runs were carried out, in which thepreliminary estimate, the number of constraints, the numbers Ec and coordinates (iw,jCif)of constrained subset elements, and other parameters were varied.
3.1.1 Balancing threshold

It is in principle possible to terminate the balancing process based on absolute or relative criteria. For example, a threshold can be formulated as max, ( |w, - un,] / w, ), whereu contains the row and column sums of U, and un those of the nth iteration of Uo.
However, industry sectors would receive equal weights independent of their size.

Therefore, we adopted the criterion |w, - uni\<Ua, for the row and column totals, andalso for all additional constraints, where £/= Af~2 2*y £/*/ is the average absolute valueof the true flow matrix, and a is a scaling factor. This way, the balancing result will
exhibit an overall absolute proximity to constraints, which is better by a certain factor
than the average magnitude of the matrix itself.
3.1.2 Preliminary estimate

We allowed three different types of preliminary estimates Uo: 1) the 1994-95 Australianflow matrix U95, 2) a weighting U[ = (w, uj) I S™ Umn according to industry size, and 3)
random numbers U* with mean U. Option 1 represents a classical updating task. Option 2 resembles an estimation task (often of regional tables) in the absence of any data
except prescribed row and column sums; the formulation is the most simple prediction
for inter-industry flows. It is based on the assumption of independence, that is it im-

6

Journal of Applied Input-Output Analysis, Vol. 11 & 12, 2006

Fig. 1: Constraint structures and coordinate locations in the flow matrix U, for theexample of distributing 50 constraints with 20 elements each. Set elements

can be either disjunct (left) or adjacent (right). Their coordinates can assume either random positions (top) or be skewed towards top-ranking ele
ments (bottom). Note that in the numerical experiments the number of set
elements does not have to be constant.
1 -"r"
•* ""

- *
• Z

*r

!

.* :» i» -::"rV :•: -V -II 1 B
s i g

"* "Z

"ar - *0a •-->•to - ="u

JB " " *1h-_J_J
1

—ri=—

e

H—

m

1

1

plies that flows from industry i to industry j are large if i produces much and j uses
much. Since the production recipe, transport distances or other parameters do not play
any role, this estimate is generally quite unrealistic (Uribe et al. 1966). Note that in theabsence of additional subset constraints, option 2 already satisfies row and column con

straints, since £ (w, uj) I E™ Umn = uh and therefore cannot precede simple RAS balancing.

3.1.3 Constraint size and structure
In principle, the elements belonging to any particular constraint can be an arbitrary-

shaped and -sized subset of the elements of U. This was simulated by randomly selecting the number and size of constraints as well as the coordinates of elements. The con
straint value is the sum over values of the target matrix elements that belong to the respective constraint. In this work, all constraints are consistent by definition, because

their values are determined from the 1996-97 U matrix.
The elements belonging to any constraint can be anything from completely dis
junct (referring to completely different pairs of industries) to all adjacent (having in
dustries in common). One common example for completely disjunct constraints are socalled "rectangular" constraints which arise out of constraining all elements U[f con
necting industries i and j and regions r and s of a multi-regional input-output table, by
the national estimate Uij. More random disjunct constraints can occur if a table expressed in a certain industry sector classification is constrained using data expressed inA flexible approach to matrix balancing under partial information

Fig. 2: Number of constrained elements (A, left y-axis) and overlap / duplication

ratio (O, right y-axis) as a function of the number of set elements (all con
straint structures, equal selection of constraint element coordinates).
12000

O - - Overlap / duplication ratrcr

1

100

1000

10000

100000

Number of set elements

another industry sector classification. In Australia, for example, superior data often ex

ists classified as ANZSIC8, while input-output tables are classified in IOPC9. The mostcommon versions of these, ANZSIC 4-digit and IOPC 4-digit, overlap in a sense thatfor some parts of the economy, ANZSIC is more detailed, for others IOPC. Moreover,while IOPC is structured to minimise the output of secondary products for all sectors,
ANZSIC could include entire establishments with multiple output commodities in one

sector10. The result of constraining an IOPC system with ANZSIC data are constraints
that are scattered over non-adjoining rows as well as columns (Fig. 1 left).
It is equally likely to encounter constraints that incorporate adjacent elements,

which mostly result from aggregated superior information, for example the intermediate

input of a whole group of commodities, or the output of a whole group of industries.

8 Australian and New Zealand Standard Industry Classification.
9 Input-Output Product Classification.
10 An example is broadacre farming, where combined beef-wheat farms are classified in one ANZSIC
sector, but split into two IOPC sectors.

8

Journal of Applied Input-Output Analysis, Vol. 11 & 12, 2006

Fig. 3: Probabilities for selecting elements from a ranked list, obtained from 2000
runs. The curve parameter is the number r of randomisations, r = 1 selects
all ranks equally.
100.%

c

o

1

L%

O

0.1%

Rank bracket

These will appear as either vertical (inputs) or horizontal (outputs) bands of the table
to be estimated (Fig. 1 right).

In general, subsets of randomly selected elements will be mostly disjunct. In orderto generate adjacent-cell subsets, a seed element was first selected randomly, and then

only adjacent cells were selected for the remaining elements up to the respective con

straint size. In this work, only row-and column-segment-shaped constraints (withrandom orientation) were considered as adjacent-cell subsets.

With increasing number and/or size of constraints, overlaps and/or duplications
become more likely. In the following we refer to the (gross) number of elements in all
constraints as 'set elements', while a smaller number of elements (net of overlap) are
actually 'constrained'. If set elements are few, they are likely not overlapping (Fig. 2,
set elements = constrained elements, triangles and solid line coincide; overlap ratio =1). As the number of set elements increases, overlap/duplication increases: the numberof actually constrained elements falls below the number of set elements contained in all
constraints (Fig. 2, set elements > constrained elements, triangles and solid line devi
ate; overlap ratio >1).

3.1.4 Coordinate location

In principle, constraints can incorporate arbitrary elements, or focus on important ele
ments. In order to cover these alternatives in our simulations, set elements (Fig. 1 left)or seed elements (right) were either selected equally (top) or preferentially (bottom)

A flexible approach to matrix balancing under partial information

9

Fig. 4: Distribution of element numbers and values in the 1994/95 Australian flow
matrix across size brackets. While there are large numbers of elements

near zero and between A$lm and A$100m9 most of the total intermediatetransaction value is concentrated in a few elements of A$lb and larger.
3000 T

T 45%

2000

■ 30%

73

*o

B

(4-1

0
S-4

0)

1000

■ 15%

3

6

9

Flow value bracket log( Uij $)from a ranking of important elements. The latter were defined as those elements thathave the largest value in the preliminary estimates U95 or U1.
Preferential selection (Fig. 1 bottom) can be realised by generating random rank

numbers (in this case from 1 to N2 = 1062) with a larger probability for lower numbers(upper ranks) in order to ensure that these are always part of a constraint subset. Sucha probability distribution can be constructed by repeatedly applying a random number

Rand e[0,l[ as the upper bound of a subsequent random number. An algorithm that car
ries out this task is quite simple:

x = 1. : For k = 1 To r : x = x x Rand ([0,l[) :Next k : rank = x x N2
The mean of random values x generated in such a way decreases with the number ofrandomisations r. Based on once-randomised (r = 1) values in [0,l[, elements of rank
rank are selected with equal probability (curve 1). Since low numbers x€ [O,jc[ occur
with increasing frequency after repeated randomisations (r > 1), top-ranking elements

are selected with increasing probability (Fig. 3).

A streamlined constraint selection is much in the spirit of 'GRIT' and similar
techniques: Jensen and West 1980 report that a surprisingly large number of smaller
elements in a flow matrix can even be removed before output multipliers show a sig
nificant change, because the value of these elements is often negligible compared to the

combined value of a few large flow values (see Fig. 4). Hewings and Jensen (1989, p.

314) note that not necessarily the largest elements are the most important in terms of10

Journal of Applied Input-Output Analysis, Vol. 11 & 12, 2006

their effect on multipliers, but that both coefficient size and location matter. Jensen andWest 1980, West 1981 and Hewings 1984 propose generating a ranking of matrix ele
ments in terms of their effect on multipliers, in order to identify elements for whichmore accurate information should be collected. These considerations could determine
selection procedures similar to the simple one chosen here.
3.1.5 Distance measures

The results of the balancing process are compared with the published 1996-97 matrices. A number of authors11 examine concepts of relative distance between two matrices
in order to characterise the comparative performance of matrix balancing methods. Ac

cording to Butterfield and Mules (1980, p. 293), "there exists no single statistical test

for assessing the accuracy with which one matrix corresponds to another. Analysts
working in this area have tended to use a number of [complementary] tests." Accord

ingly, we chose seven measures (compare Harrigan et al. 1980, Gunluk-Senesen andBates 1988, and Lahr 2001):l]\UiJ -Unij\- the relative arithmetic mean of absolute differences AMAD =

^ JT
2

;

U

/y 1/7

jj ..12the relative geometric mean of absolute differences12 GMAD = —- the Isard/Romanoff Similarity Index13 SIM = 1 —- the X2 distribution of absolute differences CHI = 2

^2

*v

;

(JJ • — JJ •• )2lJ TT niJ—"

- the arithmetic mean of relative differences AMRD = -J

JJ"

N2

- the information index INFO = 2 U,, ij logs-yr^ ; andij- the correlation coefficient CORR =Uij/Var (Uijwhere Uw is the n*1 iteration of Uo, and N = 106 the dimension of U and the tL
3.1.6 Stochastic experiments

In the following, each experiment will be characterised by a selection of parameters,such as11 Czamanski and Malizia 1969; McMenamin and Haring 1974; Morrison and Smith 1974; Lecomber
1975; Harrigan et al. 1980; Jackson and Comer 1993

12 The square of this measure has also been referred to as 'Theil's inequality index' (Lahiri 1984).
13 1 - SIM has been used as a "Dissimilarity Index" by Thissen and Lofgren 1998.

A flexible approach to matrix balancing under partial information11

the preliminary estimate: 1994/95 flow matrix U95, independent flows U1, ran
dom numbers UR;
the constraint structure: disjunct, adjacent;the selection of constrained elements, r: 1: equally, 2 and 3: increasing degreeof preference for top-ranking flows;
constraint size: number of elements contained in constraints;

total $ value of set elements: percentage of total $ value of flow matrix U;
number of constraints;

accuracy type (table -► holistic): distance test performed on flow matrix U, Le-ontief inverse L, output multipliers £, CO2 multipliers X.

4 Results
The following results are a selection from thousands of balancing runs carried out on acontemporary desktop computer, programmed in simple Visual Basic. While in mostruns the threshold was reached after 30 seconds or so, only a minority of the runs took
up to 5 minutes computing time.

4.1 Influence of the balancing thresholdIn order to illustrate the effect of varying the balancing threshold14, we have chosen an
example with a potentially long balancing procedure, where an initially unrealistic esti

mate (U1) is substantially altered during balancing, based on a large number (2500) ofadditional data points. All distance measures reach a stationary state already when thebalance criterion an =\ui -uni\IU falls below 10 (Fig. 5, left graph, A). This may
seem surprisingly large, however considering the size distribution of elements (Fig. 4),

10 U is in the order of 107A$, which is well below the value range that contains thebulk of the total transaction value.

As expected the correlation and similarity indices increase, while absolute differ

ences and the information gap decrease (Fig. 5). Interestingly, the subsequent row andcolumn balance improves the balance criterion (grey parts of A, left graph), but counteracts the distance measures by unbalancing the constraint subsets (black parts). This

alternating process of constraint and row/column balancing causes all performancemeasures to perform a damped oscillation.

The absolute mean of relative differences (AMRD) is much higher then the absolute mean of absolute differences (AMAD), since the former is influenced by a large
number of small elements that are not well approximated. Moreover, the AMRD as

well as the x2 and information measures do not seem to recover from the effect of thesecond row/column balancing just after step 2500. We could not understand and ex
plain this feature in detail, but re-examining the formulae in Sec. 1.1.5 shows that these
14 Polenske (1997, p. 79) already pointed out that "analysts should consider [... the issue of the] mini
mum level of discrepancy allowed between control totals and estimated totals", and that "information
concerning these issues is so limited".

12

Journal of Applied Input-Output Analysis, Vol. 11 & 12, 2006

Fig. 5: Convergence of a typical cRAS run for the example of "initial estimate UI,2500 disjunct constraints of size 1 and total $ value 50%U, selected with r=
2 preference, distance tes^on U". The curves show a record of the balance

criterion an =\m<-uni\/U, and all distance measures. The intermittent

steps (some visible in grey) show the log of row and column adjustments(counted as 1 step for all rows, and 1 step for all columns), while the remaining black parts represent the adjustments (1 step each) of the addi
tional constraints.
1000

1.2

2.4

INFO —hx AMRD
-

a—

1.2

GMAD—ho CHI

-~-^,

10 -■

%

~o

:

0.1 --X•
0.001—4 alpha — + AMA=2-1

CORR—x SIM

ioo

step ioooo

0
1000000

0.0

0

100

10000

1000000

Step

measures are the only ones that feature a single term [//, in the denominator, and not asum. It appears that because of this characteristic, these three measures are quite sus

ceptible to suboptimal balancing outcomes of small (and potentially unimportant) ele
ments, which was confirmed in further simulations. The GMAD behaves very similarly
to the AMAD, but varies less. Similarly, CORR and SIM vary less over a single andbetween runs. The above results were confirmed when the constraint structure, size,
numbers and value were changed, and as a consequence, the AMAD was chosen as thepreferred performance measure in most of the analyses below.

4.2 Influence of the preliminary estimate
The importance of the preliminary (pre-RAS) estimate is well known from experiencesin the estimation of regional input-output tables, and our results confirm this. If the

1994-95 flow matrix U95 is used as a start matrix in an updating exercise, the resulting
balanced estimate is much closer to the 1996-97 flow table U97 (Fig. 6, left diagram)than in the case of the independence estimate U1: Similarity and AMAD are about 0.2
points up, and 0.6 points down, respectively, and the correlation is close to unity. At
the other extreme, a random matrix with mean U (Fig. 6, right diagram) is quite a bad
approximation, but this cRAS outcome performs similarly to that resulting from the independence estimate (Fig. 5), even though the industry-size-weighted form makes moreintuitive sense (compare Hewings 1977, p. 940). Varying the additional constraints

yielded table accuracy AMADs of around 95%, 85% and 20%, for UR, U1, and U95 respectively. This finding confirms Polenske's (1997) review15, and also Miernyk's (1969;

A flexible approach to matrix balancing under partial information

13

Fig. 6: Convergence of cRAS for the example in Fig. 5, except initial estimates U95(left diagram) and UR (right diagram). The curves show a record of thebalance criterion an =\ui -uni\/U, and the distance measures AMAD,
CORR and SIM. The intermittent steps (some visible in grey) show the logof row and column adjustments (counted as 1 step), while the remaining
black parts represent the adjustments (1 step each) of the additional con
straints.
1.2

1000

1000 a

1.2

-6

-0.8

10

10 =

A

alrjha+ASs/LAD

0.8

O CQRRx Slfyl

."X
0.1

!

zz=A alpha—o COItt —

0.4

-0.4

0.1 -•
.X

0.001
100

10000

1000000

0

0.001

Step

100

Step

10000

1000000

1976) view that - at least in terms of table accuracy - regional input-output estimates
can be an inadequate substitute for real data. In all cases the main impetus towards improving table accuracy are the additional constraints, and not the row and column con

straints. Additional constraints carry much more information about U97 than both U1 andUR, making them appear almost equal in comparison.

4.3 Influence of the constraint selection and size
In practice, the analyst is most likely to be confronted with finding data for a certain
number of constraints, while limited in time and resources. In looking for superior information, the analyst can choose between constraints of different size and $ value,
subject of course to data availability. A crucial question is then which constraints

would lead to good results in terms of approximating the target matrix: a few large

constraints spanning many elements, or many small constraints? Is it efficient to focus
on potentially large elements? In the following we investigate these issues by varying
constraint selection and size. We test for disjunct structures only, since results for adja
cent structures are quite similar (see also Sec. 1.4).
Generally speaking, adding more constraints improves table accuracy, provided the

15 Polenske (1997, p. 81) finds that analysts often only report on comparative differences in balancing
outcomes, but rarely state the absolute values of table inaccuracy, which are around 30%. Exception
ally "low errors are attributable to special situations, such as the a priori introduction of large
amounts of actual data, and to favourable economic conditions, such as a small degree of structuralchange".

14

Journal of Applied Input-Output Analysis, Vol. 11 & 12, 2006

Fig. 7: Performance of cRAS runs for the example of 'Initial estimate U1, N dis
junct constraints of varying size and undefined total $ value, selected with
r=l,2,3 preference, distance test on U". Increasing the number of con
straints has a larger impact on accuracy of U than increasing constraint
size or selectivity.
110%

A N=100,r=l100%

X N=100,r=2
90%
o N=100,r=3
Q

<

80%

<

O N=1000,r=l70%i
X N=1000,r=2

60%
a N=1000,r=3
50%

10

100

Average constraint size

data are not conflicting. In our example, an increase in the number of constraints from100 to 1000 led to a 30-40% decrease in the AMAD (Fig. 7, compare curves AxO

for 100 constraints with curves <>*□ for 1000 constraints). In the case of 100 con
straints, altering the constraint size (1-100 elements) or selection (equally r = 1, increasing focus on large transactions r = 2, 3, see Fig. 3) did not change the AMAD

significantly. The first finding is remarkable since as it says that for the estimation ac
curacy to be expected from 100 constraints, it does not matter much whether thesecontain 100 or 10,000 set elements.

With increasing number of constraints, the way of selection and their size become
important (compare Hewings and Janson 1980, p. 848; Lynch 1986, pp. 280-282).

First, focusing on large transactions is always benefitial, although r = 3 produces only
slightly better results than r = 2. Constraint size increases are also benefitial, however
exceeding 10 elements improves accuracy only marginally, at least for preferential selection. The declining impact of the preferential selection is due to the fact that alreadyat r = 2, almost all top elements have been selected. The declining impact of constraint

size increase can be understood in the sense that the information contained in larger
constraints is less specific to potentially important single elements within the con
straint, and hence table accuracy is diminished.

A flexible approach to matrix balancing under partial information

15

Fig. 8: Performance of cRAS runs for the example in Fig. 7, but with varying total constraint $ value, and undefined size. Compared to covering more $value with constraints, selectivity makes a minor difference to accuracy ofU.
110%
A N=100, r=l100%

X N=100, r=2

90%

"-

o N=100,r=3
80%

O N=1000, r=l70%

X N=1000, r=260%□ N=1000,r=3
50%
1%

10%

100%

1000%

10000%

Constraint value

Rather than constraint size and hence number of set elements, the total transaction
value covered by the constraints appears to be a more important determinant (Fig. 8).

100 constraints can potentially cover 100% of the transaction value (and more for pref
erential selection, due to overlap/duplication), but only if the constraint size is near 100(less for preferential selection). At these constraint sizes, the non-specificity prevents
the constraints from having an impact on table accuracy. Selecting 1000 constraints cancover more than 100% transaction value at sizes of less than 10 elements, and hence

improve the AMAD significantly16. Above coverages of 1000% no more changes occur
because of the high degree of duplication in the constraint information. Running morethan 3000 low-sized constraints, the AMAD could be reduced to less than 30%.

4.4 Influence of the constraint structure
We now turn to variations of constraint structure, as illustrated in Fig. 1. Our first ob
servation is that for single-element constraints (size = 1 in Fig. 9) the curves for dis16 A number of runs with fixed total constraint $ values of 10%U, 30%U and 50%U were carried out,
showing that many small constraints performed better than a few large constraints.

16

Journal of Applied Input-Output Analysis, Vol. 11 & 12, 2006

Fig. 9: Comparison of balancing adjacent (A) and disjunct (D) structures, underequal (1) and preferential (2) selection (U1 initial estimate, total $-value undefined, 500 constraints, test on U). The diagram is organised as in Fig. 7.
105%

Al

■ Dl

A2

* D2

100%
A
A

95%

a

90%

A?

■ r •.-■*■

-

A

A o n

AA A

H

85%

■

-S--

75%
20

1

—r~

40

60

80

100

Constraint size

junct and adjacent structures coincide for equal (DA) and preferential (xH) selection.
This is to be expected, since an adjacent structure of size 1 is essentially a disjunctstructure. For constraint sizes of 30 and larger however, the adjacent constraint structures perform worse than the disjunct ones, and above size 90, their curves (AS) join.

This deterioration is due to the fact that the constraint length approaches the dimensionof the flow matrix, so that long constraints more or less coincide with already existing
row and column sum constraints, thus effectively reducing the numbers of constraints
through increasing duplication.

Increasing the focus on large elements (A+O in Fig. 10) improves the perform
ance of adjacent structures at low constraint sizes, but not at large sizes. For r = 3, a U-shaped form becomes apparent, which is due to two effects for declining performance:
1) small numbers of set elements at low sizes, and 2) overlap with row/column sum
constraints at large sizes. In order to concentrate constraints even more on large flows,
we selected seed elements from the ranking list, starting with the top element for the
first constraint, and then proceeding down the list, with all other set elements attachedin random directions, as before (labeled Aoo).
Once again, performance improves at
low constraint sizes, but not at large sizes. In practice, however, it is unlikely that information can be found exclusively for top-ranking flows, so that in the following wewill restrict ourselves to the preferential selections as described in Sec. 1.1.4.

A flexible approach to matrix balancing under partial information

17

Fig. 10: Performance of balancing adjacent (A) structures under equal (1) andpreferential (2, 3, °°) selection (U1 initial estimate, total $-value undefined,
500 constraints, test on U). The diagram is organised as in Figs. 7 and 9.
105%

100%

-'

95% -■

90% -•

85% -bi O/O70%.... vy

'V

-r

0

20

40

60

80

100

Constraint size

4.5 Influence of the accuracy type
We will now compare the balancing effects on the flow matrix, the Leontief inverse,
and multipliers. It is well known that the uncertainty of a sum of many numbers can beconsiderable lower than the uncertainties of any of the numbers. This effect can be ob
served when the flow matrix U is converted into a Leontief inverse L= (I-Ux"1)"1, and- further increasing the degree of summation - into output and CO2 multipliers <? = 1L

and x = cL (Fig. II)17. The table accuracy of L in terms of the AMAD is considerably
lower at around 30% than that of U, which is due to the stochastic error cancellations
during the numerous additions and multiplications during the matrix inversion. How
ever, multipliers (and not L) are likely to be the objects of practical applications. Thus
aggregating even further, the (holistic) accuracy of output multipliers £ is extremely

low at around 2%. CO2 multipliers X are less accurate at around 20%, because in thecase of Australia CO2 intensities c are exceptionally high for only a few industries suchas beef cattle grazing, beef products, aluminium or electricity generation, so that theoverall uncertainty is largely determined by these few contributions, leading to a lower
degree of error cancellation.
Probably first observed by Hewings (1977, p. 934).

18

Journal of Applied Input-Output Analysis, Vol. 11 & 12, 2006

Fig. 11: Performance of cRAS runs for the initial estimate U1, N disjunct con
straints of varying size and selectivity, and undefined $ value, tested on the

Leontief inverse L (top), output multipliers f (middle), and CO2 multipliers
X (bottom). The diagrams are organised as in Figs. 7, 9 and 10.
37%

A N=100,
r=l35%

N=100,
33%

I

r=3

ft o ° o

4 31%

N=1000,
r=l

29%

D

UD

□

N=1000,
r=3

27%

-1

1

1

I I I 1 |-100

A N=100,
2.6%

r=l

N=100,

o

o

o

r=3

,2.2%

N=1000,
1.8%

r=l

'd-d;

D N=1000,
r=3
-i—i

1.4%

i

i i i i i

10

25%

23%

21%

N=1000,

19%

r=l
17%

N=1000,
r=3

15%

1

10

Average constraint size100

A flexible approach to matrix balancing under partial information

19

Tab. 1: AMAD of cRAS runs on 500 adjacent constraint structures of varying sizeand value, but with preferentially selected seed elements (r = 2), including
its standard deviation over constraint sizes 1 to 100 (curve shapes as inFig. 9), for different preliminary estimates and accuracy types.
Accuracy type

U

L

UR

97.1±5.3

33.5±1.1

2.0±0.3

23.7±1.9U185.9±9.530.8±3.3

1.8±0.3

22.4±2.9

u95

20.3±1.0

7.2±0.3

0.4±0.04

3.0±0.6

Preliminary estimateX

We repeated the above numerical experiment for adjacent structures, and undervariation of the initial estimate (Tab. 1). In summary, holistic accuracy is always higher

than table accuracy (improving from U via L to f and X), and updating (U95) yields

much better results than estimating (U1 and UR), with the assumption of independence
performing only slightly better than a random preliminary estimate (compare Sec. 1.2).

4.6

Sequence^in)variance

As already shown for ordinary RAS by Bacharach 1970, in order to fulfil the mini

mum information gain property, the cRAS balance outcome has to be unique, and inparticular independent of the order of the constraint sequence. A large number ofcRAS runs of the adjacent, slightly selective type (r=2) were carried out, with con
straint numbers between 50 and 500, and constraint sizes between 5 and 100, andtested on U. In all instances, the balancing result was unique.

4.7 Other issues
As with the Linear Programming and minimum information gain methods, (c) RAS
preserves any preliminary zero values. Zero-value data in a preliminary matrix to be

updated (for example U95) may sometimes reflect technological impossibilities but - as
Bacharach (1970, p. 26) pointed out - sometimes actually be an approximation of avery small positive value18, in which case the zero should not be preserved. On theother hand, setting preliminary zero-value data to an arbitrarily small positive value

may allow for positive updated values, but may also lead to non-zero transactions thatin reality are technically impossible. We compared two runs of the adjacent, preferen
tially selected (r = 2) type, with one run starting with the original 1995 flow matrix

U95, and the other having all zero elements in U95 set to a small value. We obtained no
discernible differences in the AMAD of the cRAS estimates. This is probably due tothe fact that zero elements in U95 are either zero in U97, or positive but small, and thus
their contribution to the AMAD is negligible in any case.
In real-world applications, data for additional constraints may not actually supe18 Or, in the case of temporal forecasting, technical innovation may turn a truly zero input into a positive input.

20

Journal of Applied Input-Output Analysis, Vol. 11 & 12, 2006

rior, but even conflicting. Due to the nature of industry surveys proceeding as separate
sales and purchasing (supply and demand) recordings, even corresponding row and column totals of gross output and input may be different. Even a small discrepancy be
tween nominally equivalent constraints can prevent convergence of the (c)RAS method,most obviously if the given balancing threshold is smaller than the discrepancy. Recon

ciliation is therefore necessary, possibly by using a RAS technique on the conflicting
constraints themselves, or by using Lagrangian or optimisation technique with uncer
tainty or reliability terms embedded in the function to be minimised19.

5

Conclusions

We have described the performance of a RAS matrix balancing variant, called 'con
strained RAS' or cRAS. In contrast to previous approaches for the estimation of inputoutput tables, cRAS allows incorporating constraints on arbitrary-sized and -shaped
sets of elements into the preliminary estimate. These elements may be disjunct or adjacent, and exhibit a widely varying degree of overlap. We have tested cRAS's perform

ance for the example of the 1996-97 Australian input-output flow matrix, using a num
ber of distance measures.

First, the preliminary estimate is of crucial importance for the accuracy of the bal
ancing outcome. Our experiments yielded table accuracies of balanced matrices ofaround 90% for estimated preliminaries, and around 20% for the previous-year preliminary. They showed that a preliminary based on some distribution assumption and row
and column sums may even not be much better than a random matrix.

Second, including more constraints into the balancing process always improves thebalance outcome. This is especially the case if constraints are small, numerous, disjunct, and concentrated on important matrix cells. This result agrees with previous findings by Jensen and West 1980, in the sense that for many analytical purposes, it ismore important to estimate a few strategic table elements as accurately as possible,
than to estimate as many elements as possible. In this context, we also found it important to "pin-point" such strategic elements in disjunct, small constraints, rather thanhave the value of say one important element "buried" in an adjacent, large constraint.
While the size of a constraint directly diminishes the ability to uncover one particular
strategic element from its summed value, an adjacent structure means that this uncer

tainty affects the industry sector to which the strategic element belongs.
Third, the accuracy of derived quantities (such as inverse matrices and aggregates
thereof) is always higher than the accuracy of the flow matrix itself. This is due to cancellation of stochastic error components during numerous additions and multiplications
that lead to the derived quantities. Holistic accuracy - as this effect has been termed 19 A number of authors discuss balancing techniques that include uncertainty and reliability, using
either RAS (Jensen and McGaurr 1976; 1977; Lahiri 1984), variance minimisation (Gerking 1976;
1979a; b; Miernyk 1979), or Lagrangian minimization (van der Ploeg 1982). All methods have incommon the problem that probabilities for conflicting data are mostly unavailable, so that subjective
judgment has to be consulted.

A flexible approach to matrix balancing under partial information

21

is therefore certainly a concept to be kept in mind when compiling and using interindustry tables. Finally, as ordinary RAS, cRAS produces a unique balancing outcome.
Given the scarcity of superior data, especially when estimating regional or physi
cal input-output tables, the main value of cRAS is that the analyst can make use of any
kind of information they can find, irrespective of the degree of aggregation.

Acknowledgements
This research was supported financially by the Australian Research Council (ARC)
through a Linkage Project, and by Sydney Water Corporation. The authors also thankan anonymous referee for valuable comments.

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