Prof Louis De Mesnard
email: louis.de-mesnard@u-bourgogne.fr

On boolean topological methods of structural analysis

Keywords: JEL classification. C67, D57, R15. KEYWORDS. Boolean, Structural Analysis, Qualitative, MFA

The properties of Boolean methods of structural analysis are used to analyze the intern structure of linear or non linear models. Here they are studied on the particular example of qualitative methods of input-output analysis. First, it is shown that these methods generate informational problems like biases when working in money terms instead of percentages, losses of information, increasing of computation time, and so on. Second, considering three ways to do structural analysis, analysis from the inverse matrix, from the direct matrix and from layers (intermediate flow matrices), these methods induce topological problems; the adjacency of the adjacency cannot be defined from the inverse matrix; the binary relation of influence may be non transitive from the direct matrix, with poorer results than with quantitative methods; for layers -based methods, the information carried out by layers is trivial.

I. Introduction

Structural analysis is useful to understand the functioning of a multivariable multiequation model \ One way to perform it is Boolean Topological Structural Analysis or Qualitative Structural Analysis 2 . For example to detect causalities between variables, one can replace the value of the coefficient that links one variable / with another variable y, i.e. the value of the oriented arc linking these two variables, by a boolean coefficient, 1 or 0, depending of the fact that the value exceeds or not a given value called a filter. This method seems interesting to have a global view of the structure because it allows to exhibit the "skeleton" of the structure.

However, there are some disadvantages to proceed like that.

In this paper, one will present the problems caused by boolean methods of structural analysis, confining on the particular and simple case of linear models for more clarity, specially the input-output (I-O) model, but the results will be transposable to other types of computable models. So the model that one will consider is the most simple in 1-0 analysis [START_REF] Leontief | Input-Output Economies[END_REF]): x = A x + y, where x is output vector, y is the final demand vector and A is the matrix of fixed technical coefficients calculated by A = Z x" 1 where Z is the matrix of flow given by the national accounting system, and the solution of the model is x = (I-A) _1 y, with oo (I-A)" 1 = X A*. This model will be largely sufficient to explain the problems, the most part of them being topological.

1 One a the first references seems to be [START_REF] Solow R | On the Structure of Linear Models[END_REF].

II. Remind: qualitative methods versus quantitative methods

There are two great types of structural analysis methods which can be divided each into two sub-types.

A

Working on one matrix only

One work on the direct matrix. It is possible to work in absolute terms, i.e. in monetary units, in dollars for example, on the flow matrix Z. Alternately, it is possible to work on relative terms, either on the technical coefficient matrix A or on the allocation coefficient matrix B.

Ax-

Technical coefficients are absolute elasticities: ay describes the variation of the output of sector / caused by a variation of 1 dollar of the output of sector j . The analysis can be also conducted on the matrix of allocation coefficients, by• = ^ or B = <x) _1 Z, in order to capture in percentage the effect of each sector over each sector: they are relative elasticities describing Ax' Ixi the variation ' of the sector / output in percentage caused by the variation of the sector j output; this approach eliminates completely the size effect (for the case of demand-driven models). Also, instead of A (or B), it is possible to work on normalized (or Markovian) matrices, of which column or row margins are unitary, to eliminate the effect of added value or the effect of final demand. Also, one can work on the inverse matrix (I -A)~l or even (I -B) _1 .

Qualitative methods build a binary relation of influence: the adjacency of a sector is defined as the sectors that are influenced by, or that are influencing, a sector. Denote H an input-output matrix (Z, or A, or B) and hy the terms of this matrix. In qualitative terms, a sector j is influencing a sector / if hy > (|), where § is the value of the filter: j Ri hy > ((). In

In 1-0 analysis, boolean methods of structural analysis are called improperly qualitative methods of input-output analysis, that is QIOA. They are found, with some variations that sometimes may lessen critics strongly, in most works like [START_REF] Mougeot | La structure productive française[END_REF], [START_REF] Kleine | Qualitative oder quantitative Input-Output-Analyse für die Konjunkturpolitik?[END_REF], [START_REF] Duru | Structural Analysis, Hierarchical Functions and Weak Structures[END_REF], Auray, Duru, [START_REF] Ancot | Analyse structurale des modeles input-output[END_REF], Holub and Schnabl (1985), [START_REF] Bon | Qualitative Input-Output Analysis[END_REF]), Schnabel (1992[START_REF] Schnabl | Ökonomische Strukturanalyse Mit Hilfe Von Verflechtungsschichten. Ein Neues Verfahren Der Qualitativen Input-Output Analyse[END_REF], 1995), [START_REF] Leicht | An Application of Qualitative Simulation in Economies[END_REF], [START_REF] Aroche-Reyes | Important Coefficients and Structural Change: A Multi-Layer Approach[END_REF], [START_REF] Cassetti | A New Method for the Identification of Patterns in Input-Output Matrices[END_REF], [START_REF] Gillen | Cassetti's New Method for the Identification of Patterns in Input-Output Matrices: An Alternative Formulation[END_REF], etc.

other terms, a boolean matrix W (1) (<|)) is deduced from the matrix H: w# = 1 <=> hy > <().

When a sector / is linked to another sector j by a "1" in W (1) (<|>), it is declared to be in the adjacency of this sector j 3 . In some works, <| > = 0, for example, [START_REF] Bon | Qualitative Input-Output Analysis[END_REF], in other works, (() > 0. This does not characterize qualitative methods because this process remains descriptive and, as said in [START_REF] Mesnard | A Note on Qualitative Input-Output Analysis[END_REF], it could be so interesting to use a visual representation of the structure. Searching the adjacency of the adjacency is characterizing qualitative methods: one search the vertices / that are influenced by a vertex / what is influenced itself by a vertex j or the sectors / that are influenced by j through /, or the sectors / that are influenced by /, i being influenced by/

To do this, the matrix W (2) (<()) is computed as a boolean product,

W (2) ((j)) = W {1) ((|)) * W (1) ((()), ( 2
)
where * is the following operation: w /y (0) = 1 if and only if there exists at least one sector / such that there is a direct path between / and /, i.e. w^\^) = 1 and a direct path between / and where one have w /y (<j >) = 1 if and only if there exists at least one sector / such that there is a direct link between j and /, i.e. w ; y = 1 and / is in the adjacency of the adjacency of the ( i t -i ) adjacency ... of the adjacency (k-1 times) of /, i.e. w h ((())= 1. This recursive formula can (arrows are oriented in the direction of influence)

J, i.e. Wy\$) = 1. Again, W (3) ((|)) is calculated from W (2) (( (
Then, a dependency matrix is computed: D = W w .

B. Working on a sequence of matrices

There are two great manners for working on sequence of matrices.

1.

Working on the sequence of layers {Z 0 , Zi, Z 2 , Z*,...}

Starting from Z = A <x>, where Z is the flow matrix, as x = (I -A)" 1 y = (jL^ A* j y, one can write: Z = A(y) + A(A y> + A<A 2 y> + ...+A<A* y> + ... What Schnabl (1992[START_REF] Schnabl | Ökonomische Strukturanalyse Mit Hilfe Von Verflechtungsschichten. Ein Neues Verfahren Der Qualitativen Input-Output Analyse[END_REF], 1995) calls layers, i.e. the matrices Z 0 = A(y>, Zi = A (A y), Z2=A(A 2 y), ZA=A(A^y>, can be interpreted as the successive flow matrices generated by an initial demand vector y at steps 1, 2, 3, £, respectively: so, layers based methods work in absolute terms.

Bon, which is a supporter of qualitative methods uses the term (Bon, 1989, p. 223).

At this point, Schnabl's method (1992Schnabl's method ( , 1994Schnabl's method ( , 1995)), MFA ox Minimal Flow Analysis, consists into constructing a boolean matrix W* from each layer Z* to indicate if there is a link between vertices in Z*. Then matrices W* are combined by the recursive formula:

with W (0) = I. This formula is set by analogy with the recursive formula (1) used by traditional qualitative input-output analysis 5 , but the first term, W (1) , in the right side of equation ( 1) is fixed and independent to k, when the corresponding term, W*_i, in equation ( 2) is variable and dependent to k.

The result of formula ( 2) is the following at the first step: W (1) = W 0 * W (0) = W 0 , i.e.

= 1 and only if there exists at least one sector / such that there is a direct path with a length equal to 1 between j and /, i.e. (w 0 ) /y = l. At the second step, W (2) = Wi * W (1) = Wi * Wo, i.e. wff = 1 if and only if there exists at least one sector / such that there is a direct link between j and / (w 0 ) /y = 1 in the first layer Zo, i.e. in the matrix of flows after an impulsion of final demand, and / is in direct relation with / i.e. {w \ ) H = 1 in the second layer Zi.

(2)

/ j j Graph for W (1) = Wi* W <0) = W, Graph for W (2) = W i *W (1)
(arrows are oriented in the direction of influence) 6

See Lantner (1972aLantner ( , 1972b[START_REF] Lantner | Recherche sur l'interprétation du déterminant d'une matrice input-output[END_REF].

7

At this point, qualitative methods could become boolean in the sense that the same procedure than described above could be applied, but without topological considerations (the adjacency of the adjacency will not be calculated). Boolean methods are working on simple oriented graphs. In this paper, they will often be compared to quantitative methods for two categories of problems, informational problems and topological problems.

III. Topological problems of one-matrix boolean methods

Topological problems are perhaps the most important because they affect the intern coherence of boolean methods, when preceding problems either can be corrected (by working in relative terms) either affect only the cost and the interest of the method. Here, it is recalled why boolean methods are topologically wrong [START_REF] Mesnard | A Note on Qualitative Input-Output Analysis[END_REF]. There are two main ways to proceed when making use of topological methods: pretopological methods by working on the inverse matrix (I -A)" 1 or by working on the direct matrix A (or even (I -B) -1 and B) and layer-based methods (by working on intermediate matrices in absolute terms).

There are two great types of pretopological methods, those that work on the inverse matrix and those that work on the direct matrix.

A

Boolean analysis on the inverse matrix

Denote m,y as the terms of the inverse matrix (I -A) -1 , A being the direct matrix of technical coefficients. In quantitative terms, it is indicating how the final demand of commodity j influences the output of any commodity /. In qualitative terms, the above matrix H becomes the inverse matrix and if m,y > 0, then = 1.

The problem is that the adjacency of the adjacency has no meaning. As the inverse matrix is yet the limit of an infinite sum of power matrices, (I-A) -1 = £^0 A', the multiplier captures all the effects between a commodity j and another sector /. Conversely, if one says that the adjacency of the adjacency has a meaning, one says that there is something after infinite...

In other words, assume that > (J) <=> w)^ = 1 and mji >(()<=» wff = 1; for qualitative 1-0 analysis, wff -1 but this has no meaning, even if transitivity is assumed, because the corresponding attitude in quantitative 1-0 analysis would be to calculate nty mji or nff = 2y m ij m ji which is not allowed because the term mu yet indicates all the effects of the demand of commodity / on sector i. So, in the inverse matrix, one is able to consider paths with one and only one arc, i.e. / ->y andy -» /, but not paths with more than one arc like / ->y -> /: the influence from / over / via y does not exist because one have an influence of / over j initiated by a exogenous impulsion of demand, i.e. Axj = mji A v/ (or Axj -/w// A yi to the total) but the influence of / does not continue over / because the influence of j over / comes from another exogenous impulsion of demand, i.e. Ax,-= my Ayj (or Ax/ = X, Ayj to the total). Therefore the interest of boolean analysis on the inverse matrix is very low, or at least not greater than the simple study of the inverse matrix.

Boolean analysis on the direct matrix

In the direct matrix, things are quite different because one will be able to calculate the adjacency of the adjacency. The above matrix H becomes the direct matrix (here, the technical coefficient matrix A is used, but the flow matrix Z or the allocation coefficient matrix B can also be used to develop the same arguments): an > <)) <=> w]P = 1.

The relation W used by qualitative 1-0 analysis is transitive if:

w]^ = 1 for all /, / and for at least oney.

In many cases, the relation is not transitive, as in the following example:

aji ><()<=> WjP = 1 an <(()<=> wlP = 0

It is even possible that there is no direct relation between / and / : au<ty <=> = 0 , but only a reverse relation -so, a circuit -between / and i: au>ty <=> = 1.

Moreover, there is a difficulty of definition. Clearly, 1) and W (2) are two distinct relations. So transitivity must be defined in a modified relation, the relation W, as 'Vy = 1 if and only if / is in the adjacency of y, directly or indirectly". However even in this case, the relation can be not transitive. Confusion must not be done. For example, in the following expression in W (1) ,

ay > 0 <=> = 1 (2) t w / 7 =1 tfy/ ><()«=> Wji =1 but W (
Qij >(()<=> wf = 1 aji > 0 « WjP = 1 > (i) a a > <|) <=> w u = 1
the two first items, that are implying wj z 2) = 1 in W (2) , are not incompatible with the third, i.e.

= 1, because W (1) and W (2) are distinct, even if wj, 0 = 1 and wj, 0 = 1 are able to coexist.

The true no-transitivity which is in question here is concerning a contradiction between qualitative and quantitative 1-0 analysis: in relative terms, one can have ajj ] < <j ) even if aij > 0 and aji > ((), where a y

(2) is a term of the power matrix A 2 . Consider again the example where in qualitative terms,

Qij > <(><=> = 1 aji ><(><=> WjP -1 (2) ì •w u =1

IV. Informational problems

A. Lost of information, choice of filters and volume of computation

In qualitative methods, some information is obviously lost because all values under the filter are putted to zero, and all values equal or above the filter are putted to 1. This is not a serious problem if it is only like a rounding that gives the correct result at the end, but here it is not the case. In theory, it is possible to change the value of the filter to explore all the possible set but there is and infinite number of possible values for the filter, so it is not really possible. So, in real applications, a finite number of values of the filter are chosen. Often l0 , equidistant values are chosen, but perhaps a different set of values can be chosen (a well-known problem 9

One assumes that the value of the filter does not decrease with the parameter k. Like in MFA [START_REF] Schnabl | Ökonomische Strukturanalyse Mit Hilfe Von Verflechtungsschichten. Ein Neues Verfahren Der Qualitativen Input-Output Analyse[END_REF][START_REF] Schnabl | Ökonomische Strukturanalyse Mit Hilfe Von Verflechtungsschichten. Ein Neues Verfahren Der Qualitativen Input-Output Analyse[END_REF][START_REF] Schnabl | Ökonomische Strukturanalyse Mit Hilfe Von Verflechtungsschichten. Ein Neues Verfahren Der Qualitativen Input-Output Analyse[END_REF].

but in quantitative terms one have a^+ l) < a]f generally, so < an generally, where af^ = 21. ciij Qji is the indirect coefficient from / to / and is a term of the matrix A 2 , and often af? < <() which implies that the corresponding term in the boolean matrix is zero: w-, 2) = 0, by defining the relation \V (K) as = 1 <=> a^ > 0 for all / andy, where a^ is a term of A*.

This often holds because for a productive economy, in the open 1-0 model, A* -> 0; as an illustration [START_REF] Mesnard | A Note on Qualitative Input-Output Analysis[END_REF] one can assume that all coefficients are equal, i.e. a tJ = ^-j-y for all / and y, with n productive sector and one exogenous sector, then analysis may find a relation between / and /, where quantitative 1-0 analysis may not find; this is called an artefact 9 . [START_REF] Mougeot | La structure productive française[END_REF] practice both methods.

in econometrics and data analysis, when a quantitative variable must be converted into a qualitative variable), for example by determining dividing the interval between the higher term and the lower term of the matrix into N shares, either in billion of dollars when matrix H is expressed in absolute terms, either in percentage when H is expressed in relative terms.

Another method consist into dividing the set of vectors into N of sensibly equal shares, say 10 A consequence of all this is that the volume of computation remains larger with boolean methods than with quantitative methods: this is a real paradox, because qualitative methods are normally used when the amount of computation is so large that it can discourage anybody: by replacing real numbers by boolean numbers, all things would normally become more easy and faster to compute.

Boolean mathematics are more easy than ordinary mathematics to compute by hand for small matrices, so, before the computing era, this approach was justified but now, as everybody can compute easily with its personal computer, it is not acceptable to lost information. In the other hand and paradoxically, for large matrices, the volume of calculation is important.

S. Concealing of successive boolean matrices

Remembering that W (/r) is reflecting a set of couples of vertices, denoted S (k \ one have

S ( k) c S ( k -\) c £ 5 (i) genera ii y u so D = ^ W (« = W (i) = Wo So? boolean methods lead
generally to the same result regarding to the dependency matrices i.e. the first boolean matrix conceals the boolean matrices that follow it.

C. Bias caused by working on absolute coefficients instead on relative coefficients

Often, traditional qualitative 1-0 analysis, and even more sophisticated methods like Schnabl's MFA, are working on absolute coefficients, i.e. in 1-0 analysis on the matrix of flows Z expressed in dollars. When the filter is expressed in money units and when Z is used, small sectors 13 will be removed from the analysis because all their flows are naturally small, and it will remain only the larger sectors. That is why it is preferable to work on A, or B, with a filter expressed in relative terms, rather than on Z, with a filter expressed in absolute terms, allowing more correct comparisons by eliminating the size effect: even a small buying sector can have large technical coefficients or a small selling sector can have large allocation coefficients l4 . This point is very important regarding to the following one. This is demonstrated in (Bon, 1989, p. 224).

Like "Glass" or "Leather and Shoes" sector in the French national accounting system.

An alternative method to eliminate the size effect when working on Z would be to use

D. Non discerning 'layers 9 '

The layers based method MFA is probably better than many other boolean methods. It is apparently more discriminating than traditional boolean methods, because that there are much more zeros in W* than in W*-i: as coefficients become smaller from A*" 1 to A*, the process can be stopped in practice for k > 5, unlike for traditional boolean methods. As Wj indicates the adjacency for indirect influences, analyzing this matrix is topologically more satisfactory than analyzing matrices of adjacency of adjacency.

However, one will demonstrate that it is more interesting to work on the sequence I, A, A 2 , A* rather than on layers. First, remember that it is preferable to work in relative terms, when layers based methods work in absolute terms. Second, note that the same output \k can be generated either by Z*_i s (s is the sum vector) or by A* y because one have: Finally, in relative terms, the information brought by layers is always the same with both manners to do relative terms, A or B. Taking into account the fact that working in absolute terms is less preferable than working in relative terms, layers are not discerning.

Example.

•

V. Conclusion

Four great type of structural analysis methods have been studied: two types of methods based on the analysis of one single matrix, the direct matrix or the inverse matrix, and two types of methods based on the analysis of a sequence of matrices, either the layers (the successive intermediate flow matrices), either the successive power matrices. The first three are qualitative, i.e. boolean based methods, the last one is quantitative.

Boolean topological and pretopological methods of structural analysis are wrong when they work on the inverse matrix, and fall into the problem of non transitivity when the work on the direct matrix.

One have proved that boolean methods lead to lost information and to a larger volume of computation than quantitative methods, it is more adequate to work on relative terms (in percentage) than in absolute terms (in dollars) to avoid to remove small sectors from the analysis. For layers based methods, when layers are converted into relative terms instead of absolute terms, one layer provide the same information than the following, so these methods are not informationally discerning.

The above developments prove why quantitative analysis is preferable to boolean or qualitative analysis, with tools like eigenvalues, determinant, multipliers, etc.. If it is necessary to have an idea of the intern dynamic of the model 16 , in quantitative analysis, one will work on the sequence of power matrices I, A, A 2 , A*, that provide paths of length i6

See for example [START_REF] Mesnard | A Note on Qualitative Input-Output Analysis[END_REF].

equal to 0, 1, 2, £, and on its algebraic summation £ti A 1 that provides the influence flowing by paths of a length equal or lower than k.

The transposition of these developments to other fields than input-output is easy, and even for non linear models, because the remarks on non transitivity are very general, and because the remarks about the inverse matrix can be applied to the equilibrium of a non linear model, even if the corresponding matrix cannot be computed simply. The remarks on layers are concerning models where the intern dynamic is important.

8

  According to Ranko Bon idea.In other words, W w =W*-i *W*_ 2 *... *Wj. For Schnabl, "W w is reflecting the connections of sectors of the length of k steps".Quantitative methods work on the sequence power of matrices. By quantitative methods, one must understand the methods that are staying on valuated oriented graphs and that analyze the structure by calculating the eigenvalues of A, the determinant of I -A 6 , the multipliers, etc.for the global level. At a more disaggregated level, the sequence of power matrices{I, A, A2 , A*, ...}will be analyzed: paths can be ordered from those with the stronger influence to those with the lower influence (remembering that the stronger paths are often the shorter), one can search the stronger paths of a given length k (measured in number of arcs, by examining the matrix A*) 7 , or the influence flowing by paths of a length equal or lower than k by examining Xt 2 A ; , etc. Even if a boolean analysis is required (for example, for very large matrices 8 ), one can build a binary relation of influence on the successive power matrices I, A, A 2 , A*,... by doing a® >(()<=> = 1 , where is a term of the power matrix A*; in this case, what one may want is to detect paths of any length between two sectors j and /, so, it could be sufficient to analyze the boolean summation of the sequence of matrices W (A ' } .

  10

15

  corresponds to the same iteration than Z*-i, but A* <y> * Z*_i: analyzing the power matrix A* is not the same thing than analyzing the layer Z*_i. What is preferable? All layers, evaluated in absolute terms, carry out the same information in relative terms. To prove it, one must define first column coefficients: column coefficients are not exactly technical coefficients, but that are close to them because they are calculated in column without taking into account the added value of each sector:A* is such that a* = S iJ ^k where {tij) k is a term of Z*, or in matrix terms, one value of the filter for each sector, i.e. a vectorial form instead of a scalar form, but it is really more complicated and perhaps less defensible than working on A or B.So, the information carried out by matrices W* in MF A is trivial: the sequence of matrices Z k carries out the same information in terms of column coefficients. This is caused by the A fact that all matrices A* of column coefficients, deduced from layers Z* at each step k y are A A identical: A* = Ao for all k\ so, all layers carry again and again the same information because all boolean matrices W* found from the A* are the same (in other terms, a MFA with relative coefficients instead of absolute coefficients gives the same result for all layers). It is easy to prove this property. One have: A* = Z k W Z k y l = A (A* y) <s' A (A* y»" 1 A _ j ç=> A K = A (A* y) [(s ; A) (A* y)] (because if b and c are two vectors, (W (c)> = (b) (c» <=> A* = A (A* y) (A* y)" 1 W A)" 1 <=> A K = A (s ; A)~l so Â* is a constant. A Similarly, all matrices of allocation coefficients B*, deduced from layers Z k , are identical: (Z k s)" 1 Z k = «y B*> B s)" 1 <y B*> B « B, = [<y B*> <B s)]" 1 <y B*> B = <B s)" 1 <y B*>~1 (y B*> B (because if b and c are two vectors, ((c) b) = (c> (b) ^>B K = (B s) 1 B Remark. When one consider the Leontief model, only the matrix A plays a role from one iteration to the following. One can write: Z*+i = A (Z k s), because x*+i = Z k s = A* +1 y, and so the only change between Z*+i and Z k is brought by A, but it is also the case for A* +1 and Similar reasoning comes from matrix B. A*, A* +1 = AA* 15 . So, the lack of information between two iterations when layers are considered in relative terms does not lies on this fact.

  carried out by each layer in relative terms is:

  parts, and then fix the values of the filter to obtain such parts, so, even if the values of filter are any number, there are between to successive values of the filter a share equal to of the whole set of sectors 11 . Both methods are obviously arbitrary, but it could be acceptable in practice if the number of levels for the filter are sufficiently numerous and if the model is sufficiently "smooth", what is often the case with linear model but what cannot be guaranteed with non linear models. However, the question that raise is: what is the sensibility of the

results to the choice of these values? Sometimes, a small variation of the value of the filter may generate a large variation on the boolean matrix: some arcs appear or disappear, what can change strongly the results.


Prof Author Louis De Mesnard
email: louis.de-mesnard@u-bourgogne.fr

Keywords: TEL classification. C63, C67, D57. KEYWORDS. Input-Output, Change, Causative, RAS, Biproportion

The causative-matrix method to analyze temporal change assumes that a matrix transforms one Markovian transition matrix into another by a left multiplication of the first matrix; the method is demand-driven when applied to input-output economics. An extension is presented without assuming the demand-driven or supply-driven hypothesis. Starting from two flow matrices X and Y, two diagonal matrices are searched, one premultiplying and the second postmultiplying X, to obtain a result the closer as possible to Y by least squares. The paper proves that the method is deceptive because the diagonal matrices are unidentified and the interpretation of results is unclear.

I.

Introduction

Jackson, [START_REF] Jackson Randall | A Causative Matrix Approach to Interpreting Structural Change[END_REF] present an extension to input-output of the causative matrix approach to evaluate the change between two matrices. Denoting A the technical coefficient matrix and B the inverse Leontief matrix, B = (I-A)~1, there are two possibilities, working on the direct matrix A or the inverse matrix B. They choose the second possibility and they compute Markovian matrices: B'^B' (m'J 1 andB^=B /+1 (M* 1 )" 1 where M is the diagonal matrix whose diagonal element m z/ is the sum of column / of matrix B.

Then the transition matrix B^1 is assumed to be linked to the transition matrix B^ by the formula:

(1)

BS'-CBJ,

The matrix C is the causative matrix, that explain change between the Markovian matrices Bf and B#. One have b%\ = X* c ik b' MtJ .

Matrix C is found by inverting B^:

(2) C = B^1(B^)" 1

Matrix C should be compared to the identity matrix: all diagonal elements should be compared to 1, and all off-diagonal elements will be compared to 0. This matrix is called the left causative matrix by Jackson et alii (1990, p. 261-262). It is linked to the idea of backward linkages.

This should not be confused with the idea of right causative matrix exposed by Jackson et alii

This approach that can be qualified as "multiplicative", is one way to compare two matrices of column coefficients. An alternative approach to do the same thing consists into calculating the difference matrix D = B^1 -B^ (or B' +1 -B'; and similarly with the direct matrix); this matrix should be compared to the null matrix: all elements should be compared to zero. The model can be qualified as "additive" because the model is BjJ 1 =D +B^. Matrix D is also a type of causative matrix, but additive. Note that the multiplicative model could have be chosen instead of the additive model for the biproportional filter: z' +1 =ck(tj, Z' +1 j instead of Z' +1 = D + A^Z', Z' +1 j , where K denotes the biproportional projector (Mesnard, 1990a andb, 1997). However, to follow the term chosen by [START_REF] Rogerson | Modeling temporal change in flow matrices[END_REF], [START_REF] David | Dynamic Flow Modeling with Interregional Dependency Effects[END_REF], Jackson, [START_REF] Jackson Randall | A Causative Matrix Approach to Interpreting Structural Change[END_REF], the term causative will be reserved to the multiplicative model applied to Markovian matrices found from technical coefficient matrices (direct or inverse).

Interpretation of matrix C is not easy because it is completely filled, nor sparse nor diagonal: it contains n 2 terms. So, to understand the change from one matrix with n 2 elements to another matrix with n 2 elements, this requires to analyze n 2 elements, the same number of elements than for a simple difference-based method, like the computation of B^ -B^1 or like the biproportional filter on two flow matrices Z' and Z /+1 (Mesnard, 1990a andb, 1997). It is a good idea to try to reduce the complexity of the analysis. In the biproportional filtering method, the number of (3) (1990).» Remark. A reverse comparison is possible, t +1 on t instead of t on t + 1: = C B^1, where C is the causative matrix for the reverse analysis. So:

parameters is decreased by computing systematically only the variability of complete rows or columns, i.e., the variability of supplying or demanding sectors (technically, the Frobenius norm of row or column vectors of the difference matrix Z ,+1 -k(Z', Z' +1 ) ), that is 2 n elements. In the causative method, to reduce complexity, Jackson et alii (1990) focus on only two set of elements:

the set of the diagonal elements of C and the set of the off-diagonal elements of C, that is 2 n elements also.

Remark. Matrix C may contain negative terms: a positive (negative) c,* implies that the probability of transition between state i and state / is increased (reduced). "Remark. Note that a right causative matrix can also be defined for sales coefficients and forward linkages:

Remark. In the causative method, each Markovian matrix should be square, what implies additional hypotheses.

Jackson et alii (1990, p. 268) have proposed a promising idea, that they called the double causative model:

However, as they said themselves,

The use of a double causative matrix would require both additional data and additional consideration of appropriate estimation techniques (4) This is obviously true but it can be the source of some complementary ideas. First, estimation problems are effectively great, because one cannot find directly C matrices by computing the inverse of as in equation ( 2), and the number of parameters to be found is equal to the number of known coefficients: 2 n 2 , so it is too large.

Second, the interpretation of the two (completely filled) causative matrices CL and CR is more complicated than with the simple form of equation (1) because:

(

5) b^ = ±ic Lll b' Mlt c Rkj h=\ hi
Third, this approach is restricted by the fact that is limits itself to Markovian chains, with matrices calculated in column, what involves a clear choice for the demand-driven model, excluding the supply-driven model, i.e. what predetermines the sense of causality. Remember that two incompatible hypotheses are possible: Leontief (technical coefficients, calculated in columns, are constant) or Ghosh (allocation coefficients, calculated in rows, are constant) and the analysis on technical coefficient cannot be compared to the analysis on allocation coefficients; this problem disappear with the biproportional approach (Mesnard, 1990a(Mesnard, , 1990b[START_REF] De | A biproportional filter to compare technical and allocation coefficient variations[END_REF]. There is a large discussion about it [START_REF] Bon | Comparative Stability Analysis of Demand-Side and Supply-side Input-Output Models[END_REF], [START_REF] Oosterhaven | The Supply-Driven Input-Output Model: A New Interpretation but Still Implausible[END_REF][START_REF] Oosterhaven | The Supply-Driven Input-Output Model: A New Interpretation but Still Implausible[END_REF][START_REF] Oosterhaven | The Supply-Driven Input-Output Model: A New Interpretation but Still Implausible[END_REF], [START_REF] Miller | Stability of Supply Coefficients and Consistency of Supply-Driven and Demand-Driven Input-Output Models: a Comment[END_REF], [START_REF] Gruver | A Comment on the Plausibility of Supply-Driven Input-Output Models[END_REF], [START_REF] Rose | On the Plausibility of the Supply-Driven Input-Output Model: Empirical Evidence on Joint Stability[END_REF], etc., and there is a certain contradiction between the fact to introduce two causative matrices -the sense of causality seems to be undetermined -and to stay with matrices proportional by rows -the sense of causality seems to be determined -. So, it could be interesting to keep the idea of causative matrices, without predetermining the sense of causality, i.e., without predetermining the role of rows and the role of columns, by an extension of the causative-matrices method outside of the Markovian chains view. This is why one can think to propose a modification of the double causative model, in which two diagonal matrices L and R will replace matrices CL and CR: their interpretation will be more easy because the number of

II. Computation of bicausative matrices

The flow matrices are used or computed if necessary: Then, the change between these flow matrices is assumed to be of the form: L' Z / R'. This form justifies the denomination bicausative by analogy with the concept of biproportion, even if the form is not biproportional, despite its likeness: even if L' and R' are diagonal matrices, this is not a biproportion as the expression of the terms V and R' will shown it. These terms can be obtained by minimizing the sum of squares of the differences between z* 1 and / • z\j rj:

(6) Z' = A' <x'> (
Remark. Matrices 7J and Z' +1 do not need to be square: they could be as well as rectangular because the above formulas do not imply that n = m. This is a real advantage on the causative method because transforming a naturally rectangular matrix into a square matrix requires additional hypothesis; this is required for the traditional Leontief model, but not for the Stone (8) parameters becomes equal to 2 n instead of n 2 or2n 2 . This model will be called the bicausative model. The aim of this paper is to examine the pros and the cons of this idea.

model (Make / Use matrices); the biproportional filter does not require square matrices also [START_REF] De | A biproportional filter to compare technical and allocation coefficient variations[END_REF]. Note that there is not matter of a "left model" and a "right model" as for causative matrix model. •

The result is the following:

(9) l' = S\ , ,"forall/ Zfo) (/!)• -,

for ally

This is denoted:

(10) LS(L', Z' +1 ) = L'Z'R' If Z' and Z' +1 are of dimensions («, m), L' is of dimensions (n, n) and R' is of dimensions (m, m).
Change can be also evaluated in reverse form from Z' +1 to Z':

(11) LS(Z' +1 , Z') = L' +1 Z' +1 R' +1 i.e., (12) 7
' +, = >!
for all i /=i for all j

Unlike with causative matrices, results will be not the same than for the direct form and the reverse form with bicausative matrices, but as equation ( 3) has suggested it, to compare direct and reverse results with causative matrices, it is sufficient to inverse matrices h M and R /+1 .

Matrix LS^Z', Z' +1 j =L' Z' R' is on the space of year when matrix LS^Z' +1 , Z'j = L' +1 Z' +1 R' +1 is on the space of year t. So, to compare direct and reverse method, that is to say to go from year t to year one should compute

Ls(z', Z< +1

)=L'Z'R< and(L' +1 ) 'isjz" 1 , Z'j (R' + ') 1 =Z' +1 that is on the space of year /+1: matrix R' should be compared to matrix ^ matrix (L' +1 ^J should be compared to matrix V.

III. Properties of bicausative matrices

A.

Iterative resolution of the algorithm

For more generality, the problem is presented in a form that is independent to the choice made on the nature of matrices, Markovian or not, and direct or inverse. One consider two matrices X(nxm) and Y(rtxw), that can be either Z / or Z /+1 . The problem is to find two diagonal matrices U(«XA) and V(mxw), knowing the two matrices X and Y, with a symmetrical role to rows and columns and without specifying if n is greater than m or m is greater than n. The form of the model is Y = U X V and one computes matrices U and V by the least squares:

n m (13) min SS;SS=ZX(y iJ -u i XijV J y •=ij=i
By deriving SS with respect of the ut and v 7 , one obtain: ( 14)

ILyyXijVj u t =^ , for all i n ,=1
, for ally

V j = n E L 2
x y uj

1=1

We denote it: LS(X, Y) = U X V.

This system cannot be solved analytically: it must be solved recursively. As for the algorithms of biproportion, one calculates it recursively, beginning by a set of coefficients vj 0) = 1, for all j, for example, then repeating the following operations for the step k:

(15)

\1

The solution exists and is unique because it is found from the minimization of a quadratic

n m A 2 SS function: SS = J^J^(yij-yij)
, by denoting pij^UiXyVj. This function, < K"x9? m ->9t, of the n x m terms y t j is continuous, derivable, on its interval of variation, what is itself compact. The function -SS is convex, so it has a unique maximum on its interval of variation. Finally, SS has a unique minimum.

However, the difficulty lies in the fact that the solution of this minimization program cannot be found analytically, but only iteratively. So some problems may occur for the initialization of the iterative process and for its convergence.

S.

Non identification of bicausative matrices

Independence by respect to initialization

Property 1. All terms of matrices U and V are non negative if all terms of matrices X and Y and non negative.

Proof. Consider the expression of u and v terms:

m XytfXijVj(k) (17) for all / ILyyXyUtik+l) Vy(*+1) =
, for all / As the equilibrium is independent to the initialization, initial coefficients can be chosen arbitrarily.

One can choose them all non negative. If all v,(0) are non negative, all w 7 (l) are non negative, and then all vy(l) are non negative. By recursion, all Uj(k+1) are non negative and all v/(fc+ 1) are non negative, so all u* and all vj are non negative. •

The solutions of the algorithm, u*, for all i and vj 9 for all j 9 are independent to the initialization.

Property 2. The algorithm may be initialized by any set of identical values, v y (0) = A,, for ally, instead of v,(0) = 1 , for all j 9 without change anything.

Proof. The initialization by v/(0) = X 9 for allj 9 gives: The above developments show that the estimators U and V are specified at a coefficient of proportionality. For example, if one multiplies the coefficients of U by X 9 the symmetrical coefficients of V are divided by X. Therefore, the true values are specified at a coefficient of proportionality. It looks like a problem of identification, to the sense of econometrics, but as the products u* v*, for all /, j, remain fixed, these products are identified. This is the major inconvenient of the method: two vectors are searched, but they are not identified}.

Remark. When projecting a matrix X on the margins of a matrix Y, to compare X to Y, one computes AT(X, Y) = D X E, where K is the operator of biproportion (what is a generalization ofRAS), with:

dt = -, for all i 2 %ij ej (20) * ej = "
for ally

ILxijdi

/=i

So, even if with biproportion the same problem of identification (Mesnard, 1994) occurs, it is not a difficulty for the biproportional filtering method (Mesnard, 1990a(Mesnard, , 1990b[START_REF] De | A biproportional filter to compare technical and allocation coefficient variations[END_REF] " 2 1 4 " " 2 1 3 "

x =
3 1 2 andY = 3 9 1 4 5 2 _ _ 4 5 7

When initializing by y,(0) = 1 for all j, one obtain:

LS(X, Y) = 0.

2.

Ineffectiveness of a separable modification

We call separable modification of a matrix, a modification of X that can be reduced to the left product by a diagonal matrix *F of size n, and to the right product by another diagonal matrix Q ofsize/w:X = ¥XQ.

Property 3. A separable modification of X, X = *P X Q, is ineffective: LS^X, Y j = LS(X, Y).

Proof. One denote LSI (x, Y] = U X V and LS(X, Y) = U X V; then:

m / \2' X* 2 v,(*) /=1 V /
by denoting Ui{k+ 1) = \|/, w,(£+ 1) and v 7 (£) = co, vj(k).

C. Convergence of the solution of the algorithm

Some problems of convergence may occur: there can be some problems of local equilibrium.

Example 2. It is again the following of example 1. When initializing by vi(0) = l, V2(0) = 0, v 3 (0) = 0, one obtain another solution but with a lower value of SS: To avoid this, one must be careful, always looking at the value of SS, trying a large (an even infinite) set of initializing values and doing repeatedly the computation.

LS(X, Y) = 0.
Remark. Similar problems of convergence may occur in very simple models known as chaotic.

For example, in the "May equation", JC,+I = ax t (1 -*,), the solution is given by solving

x* = a x* (1 -x*), that is x* = ^T^> ^ut w hen a exceeds 3, the solution becomes chaotic. See figures 5 and 6. •

IV. Problems of interpretation

A. Interpretation of bicausative matrices

The two diagonal matrices R' and L' are summarizing the information about the change from matrices Z T to Z t +i.

For example, consider a 3x3 case:

V V R' - l\ 0 fi 0 ft ¿11 ¿12 ¿13 J t t ¿21 ¿22 ¿23 ¿31 ¿32 ¿33 ri l\ z U r \ l \ z \2 r 2 h z \3 r 3 h z 2l r \ h z 22 r 2 h z 23 r 3 h ¿31 r \ h z 32 r 2 h ¿33 r 3
• The causal diagonal matrix R / is interpreted as a column bicausative matrix: it affects equally all terms of a column, i.e. it affects equally the intermediary purchases of a sector: it is formally similar to an input (i.e., technical) effect.

• The causal diagonal matrix L' is interpreted as a row bicausative matrix: it affects equally all terms of a row, i.e. it affects equally all the intermediary sales of a sector. It corresponds to an allocation or output effect and it is formally similar to a price effect. It is not exactly a price because it is possible to remove the effect of price change from 7J to Z' +1 by using the same price base, for example, the price base of the date t.

However, the above interpretation is false because the terms /, and r, are not identified.

Remark. A consequence of this problem of identification is that the method -application of least squares on two matrices Z / and Z /+1 to obtain two vectors V and r'--cannot be applied to the prevision of another matrix Z' +2 at a later date /+2 by doing Z' +1+A = 1' Z' +2 r'.

Remark. The same interpretation could be used for the biproportional projection of a matrix to the margins of another matrix. When calculating the matrix A^Z', Z' +1 j = C Z' D', that is the biproportional projection of Z' on the margins of Z' +1 , where C and D' are diagonal matrices, one can interpret the terms dj as a technical effect and the terms c, as a "price" effect. However, C and D are not identified and the similarity stops at this point because Z' is projected on the margins of Z' +1 and not on Z M itself as it is done with the bicausative method.

Note that [START_REF] Paelinck | Etude empirique sur l'évolution de coefficients «input-output». Essai d'application de la procédure «RAS» de Cambridge au tableau interindustriel belge[END_REF] give a rather different interpretation, because they work on technical coefficient matrices instead of flow matrices. Considering two technical coefficient matrices, A' and A M , the price effect from the date t to the date /4-1 is given by expressing the matrix A' to the prices of the date f +1: A /7 ' +1 = p A' p" 1 , where p is the price vector of the date t+1 with the date t as price base. Then the transformation of A ///+1 such a this matrix has the same margins than A /+1 is given by: £(A', A' +1 ) = C A' //+1 D'. In this case the d/s are interpreted as a substitution effect and the C/'s as a fabrication or transformation effect. However, it is partially incorrect to do a biproportional projection on a technical coefficient matrix, because the biproportional projection is yet a generalization for both column and rows of the computation of column coefficients [START_REF] De | A biproportional filter to compare technical and allocation coefficient variations[END_REF]. Moreover, as the terms C and D are not identified, they cannot be interpreted for themselves as a substitution effect and a fabrication effect. •

fi. Interpretation of the bias

The causative method is generalized by the bicausative method to eliminate the effect of the demand-driven-model hypothesis. The bicausative method allows to detect change in the structure by the exhibition of a left diagonal matrix and of a right diagonal matrix, but some problems may arise in comparison with other methods as the biproportional filtering method (Mesnard, 1990a(Mesnard, andb, 1997)).

Remark. Remember that in the biproportional filter method, one calculates Z /+1 -K\ 7J, Z' + or Z' -K[Z T+1 , Z' J, that is in general terms Y -£(X, Y) : as £(X, Y) have the same row and column margins than Y, one is able to compare both matrices without the differential growth effect of sectors, i.e., with only the structural effect. This generalizes the simple comparison of two technical coefficient matrices, A' +1 -A' (same column margins) or two allocation coefficient matrices B' +1 -B' (same row margins), without implying a choice between demand-driven and supply-driven models, i.e. this generalizes the "shift-and-share method".

However, as U and V are diagonal in LS(X, Y) = U X V, because of the properties of biproportion [START_REF] De | A biproportional filter to compare technical and allocation coefficient variations[END_REF],

The bias becomes:

(27)

There can be a bias between Y and LS (X, Y): the expression Y -LS (X, Y) can be more or less different to 0. The quality of the analysis is depending on the size of this bias.

Remark. Such a problem does not arise with the biproportional filtering method, because it is the difference Y -K(X, Y) that is itself analyzed: this difference is not a bias, it is the subject of the analysis."

However, this manner to calculate the bias in the bicausative method could appear as mixing up two phenomenons, the bias caused by the differences in the sector size and the bias caused by the real structural effect because both matrices X and Y have not the same margins. So it could seem correct to give to both matrices the same margins by a biproportion to eliminate the size effect of the differential growth of margins and to found what can be called "the structural bias of the bicausative method"; it is logical to choose the last matrix margins of each couple of matrices, then:

what becomes similar to the result of the biproportional filtering method (Mesnard, 1990a andb, 1997). Then, if one tries to evaluate the structural bias of the bicausative method, one is conducted to evaluate the structural change as found by the biproportional filtering method.

Remark. An additional problem is that this hold for any method, denoted M 9 used instead of least squares LS, so long as M has a form like M(X, Y) = *FXO, where *F and O are diagonal:

However, the bicausative method does not measure change in margins of matrix X when one tries to go to another matrix Y because if a matrix X is transformed so that it has the margins of a matrix Y, to obtain the matrix K(X, Y), the bicausative method applied from X to Y gives the same results than the bicausative method applied to K(X, Y), Y :

Here, K (X, Y) = P X Q have the same margins than Y, but the associated bicausative matrices are equivalent to those simply obtained with X.

Finally, what is the interpretation of the bicausative method?

Remark. One have even:

(28) (29) LS [K(X, Y), Y] = LS [P X Q, Y] = LS (X, Y) (30) LS [M (X, Y), Y] = LS OF X O, Y) = LS (X, Y)
where M is the same as above.

V. "Application" for France 1980-1996

The bicausative-matrices method will be applied to the period 1980-1996. This is not a true application, because the method is deceptive. Price effects are removed as possible by using the price base of 1980. To reduce computer time, sectors are regrouped in "10 sectors" tables (in fact, 9x10 sectors) from the French "40 sectors" tables (in fact, 34x36 sectors), following the rule indicated by table 3.

Remember that sectors T25 (Trade) and T38 (Non Marketable Services) do not appear in row, but only in column, in the French national accounting system. So, sector Trade in the "10 sectors"

VI. Conclusion

The bicausative-matrices method could appear as a method suitable to analyze the change from one flow matrix X to another flow matrix Y, without predetermining the orientation of the data, that is to say considering columns as variables and rows as observations, but giving to rows and columns the same role. In this sense, it allows to perform the same type of analysis than the biproportional filtering method.

However, the bicausative-matrices method is deceptive. It is based on the calculation of two diagonal matrices U and V such that the product U X V is a the nearest as possible of Y in the sense of least squares. The searched diagonal matrices U and V, that are calculated iteratively, are not identified; the product U X V is identified but this is not useful for the interpretation of the method, because the comparison of U X V and Y indicates only the bias of estimation.

When analyzing this bias, Y -U X V, if one try to give to U X V the same rows and column than Y by a biproportion, K(U XV, Y), in order to remove the effect of the differences in sector sizes, the resulting bias Y -K(XJ X V, Y) becomes identical to the result of the biproportional filtering method applied from X to Y, Y -£(X, Y).

Moreover, the method does not analyze the differential growth of sectors between X and Y because if the margins of Y are given to X, i.e., if X is projected on Y by a biproportion, K(X, Y), and if the bicausative-matrices method is applied from the resulting matrix to Y, the result, LS(.K(X, Y), Y), is equal to the result of the bicausative method applied from X to Y, LS(X, Y).

  7) Z' +1 =A' +1 (x' +1 > Remark. One can work on direct or on inverse matrices, the choice depending on what one wants to study. Direct technical coefficient matrices are close to flow matrices, i.e., A = Z (x)" 1 , when inverse technical coefficient matrices indicate the link between final demand y and output x at equilibrium, i.e., x = (I -A)" 1 y. •

  vy(l), for ally and the product u* Xy v* remains unchanged: (19) u* Xy V* = u* Xij v* 9 for all / and j, w,-(l), v/(l), w* and v y * denoting the values obtained after an initialization by v/(0) = 1, for all7.By induction, the dual result is proved: things are unchanged if one initializes by w,(0) = A, instead ofn,-(0)=l for all/." Remark. When the computation is initialized by a set of not identical values, i.e. by 3 j\,J2 I Y/i(0) * v/ 2 (0), for allj 9 then result is not predictable.

  because matrices D and E are not searched for themselves; what is searched is the product matrix D XE to be compared to Y: Y-D XE; but as D XE is identified and fixed, Y-D XE is itself identified and fixed. • A very simple example will help to understand. Example 1.

  one initialize by v 7 (0) = v/(0), for all j 9 then w,(l) = w,(l), for all /, and v) = vj , for all j, and by induction, ut(k) = w,(&), for all i and Vj(k) = v y (&), for all y. Therefore, u* = w*, for all i and v* = v*, for all j, and w* x f y v* = u* \|/, x,y Coy v* = w* x# v* = w* x,y v* and soUXV = UXV." V However, this property does not hold with a separable modification of Y: Y = *F Y Q. and

  Figure1 here

Figure 2 .

 2 Figure 2. u and v for Vl (0) = 1, v 2 (0) = 0, v 3 (0) = 0.436357, LS(X, Y)

•

  All these "curious" properties are caused by the fact that both methods are not identified in the

	econometric sense.
	Example 3. It is the following of preceding examples.
	0.888	0	2 1 4	0.629
	*(X, Y) =	2.410	3 1 2	1.511
	0	1.326	4 5 2	0.998
	1.116 1.341 3.543
	4.548 3.642 4.810
	3.336 10.017 2.647
		1.154	0	1.116 1.341 3.543	0.594	0
	LS[K(X, Y), Y] =	1.437		4.548 3.642 4.810	1.519
		0	0.387	3.336 10.017 2.647	0	0.402
		1.116 1.341 3.543
		4.548 3.642 4.810
		3.336 10.017 2.647

Table 1 here Table 2 here Table 3 here After

 1

300 iterations (but convergence was obtained before), the matrices Z5

(Z 1980 , Z 1996

) and I5

(Z 1996 , Z 1980

) are indicated by tables 4 and 5.1 and r vectors are not indicated because they are not identified.

Table 4 hereTable 5 here

 45 Moreover, the analyze of the bias is deceptive, as shown by table 6.

Table 6 here

 6 table contains sector T25 in column but not in row; sector Non Marketable Services do not appear in row in the "10 sectors" table.

Table 1 .

 1 Sectors of the "40 sectors table" Groups of sectors

	Regrouped sectors