Ronner, Arjen E

Sterken, Elmer

Economics Letters

ARTS repository - University of Groningen

University of GroningenGeneralization of M(x)-Matrices; Application to Mathematical Economics and EconometricsRonner, Arjen E; Sterken, ElmerPublished in:Economics LettersDOI:10.1016/0165-1765(84)90183-6IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1984Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Ronner, A. E., & Sterken, E. (1984). Generalization of M(x)-Matrices; Application to MathematicalEconomics and Econometrics. Economics Letters, 16, 321-326. https://doi.org/10.1016/0165-1765(84)90183-6CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 24-12-2025Economics Letters 16 (1984) 321-326 North-Holland 321 GENERALIZATION OF M( x)-MATRICES Application to Mathematical Economics and Econometrics Arjen E. RONNER and Elmer STERKEN Unruersiiy of Groningen, Groningen, The Netherlunds Received 23 March 1984 Square matrices with properties such as non-negativity, diagonal dominance and equal row sums frequently occur in mathematical economics. In this paper Sierksma’s matrices of class M(x) are generalized to multisector models and are applied to the problem of pooling cross-section and time series data. 1. Introduction Matrix theory is widely used in analyzing economic systems. Economi- cal arguments often impose restrictions on the elements of matrices. Here we study a special class of matrices with some useful properties and applications. Input-output-modelling theory is one field of economical research with an increasing interest in matrix classes. The open Leontief model relates the final demand vector with the gross production vector (I,- T)y=c, where (1) I,, is the (n x n) unit matrix, T is the (n X n) input-coefficient matrix, y is the gross production vector, and c is the final demand vector. The impact of final demand on production in sectors depends on the sector-structure. Supposing the long-term sector-structure to be fixed, short-term changes in final demand will give rise to changes in gross 0165-1765/84/$3.00 0 1984, Elsevier Science Publishers B.V. (North-Holland) 322 A. E. Ronner, E. Sterken / Generdrution of M(x)- matrices production. An important topic in this analysis of changes in demand and production is the formulation of conditions, which are sufficient to prove that a certain sector will gain relatively more from a relative stronger rise of final demand in this sector. In other words: if the change of the final demand in sector i exceeds the change in sector j, will the gross production react in a similar way? Sierksma (1979) studied the following equivalence: Ay,:,Ay,oAc,<Ac,, i,j= 1 ,...,n, i#j. (2) He proved that, under regularity conditions, (2) is equivalent to the assertion that T belongs to a certain class of matrices, called M(x). Besides the open Leontief model this concept can be used in e.g., the Stolper-Samuelson framework Aw =p, where (3) A is the matrix of factor shares, p is the vector of product prices, and w is the vector of money rewards. Wegge and Kemp (1969) derived necessary and sufficient conditions regarding the factor shares matrix A for the problem Ap,>(), Ap,=O~~w,>o~ Aw,<o, j,j = 1 ,...,n, i #j. (4) However, they did not provide a class of matrices, satisfying these conditions. Matrices of class M(x) seem to be sufficient for (4), and necessary too, if statement (4) is put in a slightly more general framework like (2) [Sierksma, Steenge and Sterken (1983)]. In this paper we shall discuss a simultaneous vector approach in which elements of the matrix M are allowed to be ( p X p) matrices. Section 3 provides an application of class A( n, p, M,,, X) in mathematical eco- nomics and section 4 gives a useful application of symmetric matrices of class &(n, p, M,,, X) in econometrics. First some definitions and basic theorems are presented. Definition 1. A square (n x II) matrix M = {m,, } is said to belong to A. E. Ronner, E. Sterken / Generalization of M(x) matrices 323 classM(x)ifandonlyifm,,-m,,=x,x#O,i,j=l,..., n,i+j. This definition implies that if M E M(x), then mz2-x ... . . m,,-x “’ In this paper a simultaneous sector approach is introduced in which the elements of the matrix M are allowed to be ( p X p) matrices. Definition 2. A square (np X np) matrix M is said to belong to class &‘(n, p, M,,, X) if and only if M can be partitioned in (n X n) sub- matrices M,,(pXp)such that M,,-M,,=X, withX#O, i,j=l,..., n, i#j. The following generalization of Sierksma’s theorems 16 and 18 holds. Theorem 1. Let M EJl(n, p, M,,, X) and define S = IF=, M,, - (n - 1)X. Let S and X be non-singular, then (4 Mm’=(I,,cW’) ; [ X- M,, + S . . . X - Mm x -M,, ‘.. X -M,,,+S 1 x(zflc3x-‘), or equivulently, M-’ EJZ(n, p, S-‘( X- M,, + S)X-‘, X-l). (ii) M = IXI”~‘ISI. ProoJ: (i) Let A = k sne: 63 (M,, - X), and r=l B= ~s,,e;@(S-‘(X-M,,)X-I), 1=l (5) 324 A. E. Ronner, E. Sterken / Generaliratron of M(x) mafricrs with s, the n x 1 vector (1,. . . , 1)’ and e, the ith unit vector. Note that B= -(l,,@~s’)A(I,,@XX-‘)andAB=(I,@((S-X))B.Then M=A +(Z,, 8 X) and Mm’ =B+(I,, @ XP’), (6) since MM~‘=AB+A(Z,,Q?JX~‘)+(Z,,@X)B+Z,,,=(Z,,@((S-X))B - (Z,, @ S)B + (I,, 8 X)B + Z,?,, = I,,,,. The proof of (ii) is immediate. Q.E.D. In section 3 an application is given of the case M,, = M for each i=l , . . . ,n. Therefore the following corollary is useful: Corollary I. Zf the (np X np) matrix M ~d(n, p, M, X), then M=J,,@~+(Z,--J,)@A’ and M~‘=J,,~~S-‘+(Z,,-J,,)~~X-‘, where I,, = n ‘s,,s;. 2. On the application of class Jl(n, p, M;;, X) in mathematical economics Suppose that an np-sector input-output system can be divided into n subgroups of sectors, such that an M(x)-structure holds over the sub- groups. Notation I. We state that, if zl and z2 are (n X 1) vectors z, +C z2 = Vi = l,..., n, z; <z;, where zi is the ith element of z,. Definition 3. An (n x n) non-negative matrix T is called irreducible if Vi,j=l,..., n 3m = m(i, j)> 0 such that t,(,V’)> 0. Lemma I. Zf T is an (n x n) irreducible non-negative matrix and T bus a dominant eigenvalue p < q~ then (cpl - T) ’ >+= 0. Proof: See Debreu and Herstein (1953). Note that if TEA(n, 1, m,,, x) with rp > s > x > 0, (VI- T)-’ B 0. A. E. Ronner, E. Sterken / Generalization of M(x) mafrices 325 The open Leontief model is characterized by ‘p = 1. Now we are able to give a multisector alternative of Sierksma’s theorem 19. Theorem 2. If T ~.4’(n, p, M,,, X) with X a (p Xp) non-negative, irre- ducible matrix with dominant eigenvalue p < cp, then A-y, < A y, = AC, -=K AC J' i,j= 1 ,...,n, i#j (7) holds. Proof. AC, +C AC, e (cpZ,, - X)A y, -C (rpl,, - X)A y,. With the properties of X ( cpZP - X)) ’ B 0 holds according to Lemma 1. So A y, -=c A y,. Q.E.D. Note that the opposite is not true, otherwise the equivalence of the assertions in Sierksma’s theorem is violated. The same concept can be used in the Stolper-Samuelson framework. If A c&(n, p, M,,, X) both Aw, +z Aw, * Ap, -c Ap, and Ap, B 0, Ap, =O~Aw,~O,Aw,-=~Ohold,foreachi,j=l,..., n;i#j. 3. On the use of the class A(n, p, Mji, X) in pooling cross-section and time series data One of the most frequently used techniques in pooling and time series data is the analysis of variance. Suppose following model: y,,= i: x,,,P,, + u,/, i=l >...> n, t=l,..., T, s= 1 cross-section we have the (8) where y,,, x,(, and u,~ are (p x I)-vectors and p,, is a scalar. This model may describe the sales of p firms over n regions which are explained by the marketing instruments x, ,s. Given the covariance matrix of the disturbances the parameters are usually estimated by maximum likeli- hood methods, cf. Wansbeek and Kapteyn (1982). Therefore we need the inverse of the covariance matrix. All - at least, known to us - error components covariance matrices have a mathematical structure that corresponds to class A(n, p, M, X). Let us consider the two error 326 A. E. Ronner, E. Sterken / Generalizutron of M(x) mamc’es components model. We assume that the errors u,, (8) are decomposed as u,, = 77, + c,, 1 where nlr and F,, are independently distributed as n, - N,(O, r) and E,, - N,>(O, A ), with r positive semidefinite and A positive definite, both of order p. Define the ( p x n) matrices and ( pn x 1) vectors I/;=(u ,,,..., u,,,) and u,=vecV,, t=l,..., T. Then u ,,. . . ,uT are i.i.d. as N,,,,(O, Q,), with n, = r+A ... l’ 1 r . . . ;+A I. Note that L?, l &(n, p, r+ A, A), so according to Corollary 1 we can write S, = nr+ A and D, = _I,, @ S, + ( I,, - J,, ) 8 A, (9) ,n;‘=Jr,~Ss,‘+(I,-J,,)~Aa’. (10) The three error components model can be treated in a similar way. References Debreu, G. and I.N. Herstein, 1953, Nonnegative square matrices, Econometrica 21. 597-607. Sierksma, G.. 1979, Nonnegative matrices; the open Leontief model, Linear Algebra and its Applications 26, 175-210. Sierksma. G., A.E. Steenge and E. Sterken, 1983, A generalization of Stolper-Samuelson and Kemp-Wegge theorems, Working paper 83-15-WE (Econometrics Institute, Uni- versity of Groningen, Groningen). Wansbeek, T. and A. Kapteyn, 1982, A class of decompositions of the variance-covariance matrix of a generalized error components model, Econometrica 50. 713-724. Wegge, L.L.F. and M.C. Kemp, 1969. Generalizations of the Stolper-Samuelson and Samuelson-Rybczynski theorems in terms of conditional input&output coefficients. International Economic Review 10, 414-425. 

English

Generalization of M(x)-Matrices; Application to Mathematical Economics and Econometrics

Arjen E. Ronner

Elmer Sterken

Crossref

Generalization of M(x)-matrices

   University of GroningenGeneralization of M(x)-Matrices; Application to Mathematical Economics and EconometricsRonner, Arjen E; Sterken, ElmerPublished in:Economics LettersDOI:10.1016/0165-1765(84)90183-6IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1984Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Ronner, A. E., & Sterken, E. (1984). Generalization of M(x)-Matrices; Application to MathematicalEconomics and Econometrics. Economics Letters, 16, 321-326. https://doi.org/10.1016/0165-1765(84)90183-6CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 21-06-2022Economics Letters 16 (1984) 321-326 North-Holland 321 GENERALIZATION OF M( x)-MATRICES Application to Mathematical Economics and Econometrics Arjen E. RONNER and Elmer STERKEN Unruersiiy of Groningen, Groningen, The Netherlunds Received 23 March 1984 Square matrices with properties such as non-negativity, diagonal dominance and equal row sums frequently occur in mathematical economics. In this paper Sierksma’s matrices of class M(x) are generalized to multisector models and are applied to the problem of pooling cross-section and time series data. 1. Introduction Matrix theory is widely used in analyzing economic systems. Economi- cal arguments often impose restrictions on the elements of matrices. Here we study a special class of matrices with some useful properties and applications. Input-output-modelling theory is one field of economical research with an increasing interest in matrix classes. The open Leontief model relates the final demand vector with the gross production vector (I,- T)y=c, where (1) I,, is the (n x n) unit matrix, T is the (n X n) input-coefficient matrix, y is the gross production vector, and c is the final demand vector. The impact of final demand on production in sectors depends on the sector-structure. Supposing the long-term sector-structure to be fixed, short-term changes in final demand will give rise to changes in gross 0165-1765/84/$3.00 0 1984, Elsevier Science Publishers B.V. (North-Holland) 322 A. E. Ronner, E. Sterken / Generdrution of M(x)- matrices production. An important topic in this analysis of changes in demand and production is the formulation of conditions, which are sufficient to prove that a certain sector will gain relatively more from a relative stronger rise of final demand in this sector. In other words: if the change of the final demand in sector i exceeds the change in sector j, will the gross production react in a similar way? Sierksma (1979) studied the following equivalence: Ay,:,Ay,oAc,<Ac,, i,j= 1 ,...,n, i#j. (2) He proved that, under regularity conditions, (2) is equivalent to the assertion that T belongs to a certain class of matrices, called M(x). Besides the open Leontief model this concept can be used in e.g., the Stolper-Samuelson framework Aw =p, where (3) A is the matrix of factor shares, p is the vector of product prices, and w is the vector of money rewards. Wegge and Kemp (1969) derived necessary and sufficient conditions regarding the factor shares matrix A for the problem Ap,>(), Ap,=O~~w,>o~ Aw,<o, j,j = 1 ,...,n, i #j. (4) However, they did not provide a class of matrices, satisfying these conditions. Matrices of class M(x) seem to be sufficient for (4), and necessary too, if statement (4) is put in a slightly more general framework like (2) [Sierksma, Steenge and Sterken (1983)]. In this paper we shall discuss a simultaneous vector approach in which elements of the matrix M are allowed to be ( p X p) matrices. Section 3 provides an application of class A( n, p, M,,, X) in mathematical eco- nomics and section 4 gives a useful application of symmetric matrices of class &(n, p, M,,, X) in econometrics. First some definitions and basic theorems are presented. Definition 1. A square (n x II) matrix M = {m,, } is said to belong to A. E. Ronner, E. Sterken / Generalization of M(x) matrices 323 classM(x)ifandonlyifm,,-m,,=x,x#O,i,j=l,..., n,i+j. This definition implies that if M E M(x), then mz2-x ... . . m,,-x “’ In this paper a simultaneous sector approach is introduced in which the elements of the matrix M are allowed to be ( p X p) matrices. Definition 2. A square (np X np) matrix M is said to belong to class &‘(n, p, M,,, X) if and only if M can be partitioned in (n X n) sub- matrices M,,(pXp)such that M,,-M,,=X, withX#O, i,j=l,..., n, i#j. The following generalization of Sierksma’s theorems 16 and 18 holds. Theorem 1. Let M EJl(n, p, M,,, X) and define S = IF=, M,, - (n - 1)X. Let S and X be non-singular, then (4 Mm’=(I,,cW’) ; [ X- M,, + S . . . X - Mm x -M,, ‘.. X -M,,,+S 1 x(zflc3x-‘), or equivulently, M-’ EJZ(n, p, S-‘( X- M,, + S)X-‘, X-l). (ii) M = IXI”~‘ISI. ProoJ: (i) Let A = k sne: 63 (M,, - X), and r=l B= ~s,,e;@(S-‘(X-M,,)X-I), 1=l (5) 324 A. E. Ronner, E. Sterken / Generaliratron of M(x) mafricrs with s, the n x 1 vector (1,. . . , 1)’ and e, the ith unit vector. Note that B= -(l,,@~s’)A(I,,@XX-‘)andAB=(I,@((S-X))B.Then M=A +(Z,, 8 X) and Mm’ =B+(I,, @ XP’), (6) since MM~‘=AB+A(Z,,Q?JX~‘)+(Z,,@X)B+Z,,,=(Z,,@((S-X))B - (Z,, @ S)B + (I,, 8 X)B + Z,?,, = I,,,,. The proof of (ii) is immediate. Q.E.D. In section 3 an application is given of the case M,, = M for each i=l , . . . ,n. Therefore the following corollary is useful: Corollary I. Zf the (np X np) matrix M ~d(n, p, M, X), then M=J,,@~+(Z,--J,)@A’ and M~‘=J,,~~S-‘+(Z,,-J,,)~~X-‘, where I,, = n ‘s,,s;. 2. On the application of class Jl(n, p, M;;, X) in mathematical economics Suppose that an np-sector input-output system can be divided into n subgroups of sectors, such that an M(x)-structure holds over the sub- groups. Notation I. We state that, if zl and z2 are (n X 1) vectors z, +C z2 = Vi = l,..., n, z; <z;, where zi is the ith element of z,. Definition 3. An (n x n) non-negative matrix T is called irreducible if Vi,j=l,..., n 3m = m(i, j)> 0 such that t,(,V’)> 0. Lemma I. Zf T is an (n x n) irreducible non-negative matrix and T bus a dominant eigenvalue p < q~ then (cpl - T) ’ >+= 0. Proof: See Debreu and Herstein (1953). Note that if TEA(n, 1, m,,, x) with rp > s > x > 0, (VI- T)-’ B 0. A. E. Ronner, E. Sterken / Generalization of M(x) mafrices 325 The open Leontief model is characterized by ‘p = 1. Now we are able to give a multisector alternative of Sierksma’s theorem 19. Theorem 2. If T ~.4’(n, p, M,,, X) with X a (p Xp) non-negative, irre- ducible matrix with dominant eigenvalue p < cp, then A-y, < A y, = AC, -=K AC J' i,j= 1 ,...,n, i#j (7) holds. Proof. AC, +C AC, e (cpZ,, - X)A y, -C (rpl,, - X)A y,. With the properties of X ( cpZP - X)) ’ B 0 holds according to Lemma 1. So A y, -=c A y,. Q.E.D. Note that the opposite is not true, otherwise the equivalence of the assertions in Sierksma’s theorem is violated. The same concept can be used in the Stolper-Samuelson framework. If A c&(n, p, M,,, X) both Aw, +z Aw, * Ap, -c Ap, and Ap, B 0, Ap, =O~Aw,~O,Aw,-=~Ohold,foreachi,j=l,..., n;i#j. 3. On the use of the class A(n, p, Mji, X) in pooling cross-section and time series data One of the most frequently used techniques in pooling and time series data is the analysis of variance. Suppose following model: y,,= i: x,,,P,, + u,/, i=l >...> n, t=l,..., T, s= 1 cross-section we have the (8) where y,,, x,(, and u,~ are (p x I)-vectors and p,, is a scalar. This model may describe the sales of p firms over n regions which are explained by the marketing instruments x, ,s. Given the covariance matrix of the disturbances the parameters are usually estimated by maximum likeli- hood methods, cf. Wansbeek and Kapteyn (1982). Therefore we need the inverse of the covariance matrix. All - at least, known to us - error components covariance matrices have a mathematical structure that corresponds to class A(n, p, M, X). Let us consider the two error 326 A. E. Ronner, E. Sterken / Generalizutron of M(x) mamc’es components model. We assume that the errors u,, (8) are decomposed as u,, = 77, + c,, 1 where nlr and F,, are independently distributed as n, - N,(O, r) and E,, - N,>(O, A ), with r positive semidefinite and A positive definite, both of order p. Define the ( p x n) matrices and ( pn x 1) vectors I/;=(u ,,,..., u,,,) and u,=vecV,, t=l,..., T. Then u ,,. . . ,uT are i.i.d. as N,,,,(O, Q,), with n, = r+A ... l’ 1 r . . . ;+A I. Note that L?, l &(n, p, r+ A, A), so according to Corollary 1 we can write S, = nr+ A and D, = _I,, @ S, + ( I,, - J,, ) 8 A, (9) ,n;‘=Jr,~Ss,‘+(I,-J,,)~Aa’. (10) The three error components model can be treated in a similar way. References Debreu, G. and I.N. Herstein, 1953, Nonnegative square matrices, Econometrica 21. 597-607. Sierksma, G.. 1979, Nonnegative matrices; the open Leontief model, Linear Algebra and its Applications 26, 175-210. Sierksma. G., A.E. Steenge and E. Sterken, 1983, A generalization of Stolper-Samuelson and Kemp-Wegge theorems, Working paper 83-15-WE (Econometrics Institute, Uni- versity of Groningen, Groningen). Wansbeek, T. and A. Kapteyn, 1982, A class of decompositions of the variance-covariance matrix of a generalized error components model, Econometrica 50. 713-724. Wegge, L.L.F. and M.C. Kemp, 1969. Generalizations of the Stolper-Samuelson and Samuelson-Rybczynski theorems in terms of conditional input&output coefficients. International Economic Review 10, 414-425. 

University of Groningen Research Database

  
 University of Groningen
Generalization of M(x)-Matrices; Application to Mathematical Economics and Econometrics
Ronner, Arjen E; Sterken, Elmer
Published in:
Economics Letters
DOI:
10.1016/0165-1765(84)90183-6
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
it. Please check the document version below.
Document Version
Publisher's PDF, also known as Version of record
Publication date:
1984
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Ronner, A. E., & Sterken, E. (1984). Generalization of M(x)-Matrices; Application to Mathematical
Economics and Econometrics. Economics Letters, 16, 321-326. https://doi.org/10.1016/0165-
1765(84)90183-6
Copyright
Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the
author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).
Take-down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the
number of authors shown on this cover page is limited to 10 maximum.
Download date: 22-05-2019
Economics Letters 16 (1984) 321-326 
North-Holland 
321 
GENERALIZATION OF M( x)-MATRICES 
Application to Mathematical Economics and Econometrics 
Arjen E. RONNER and Elmer STERKEN 
Unruersiiy of Groningen, Groningen, The Netherlunds 
Received 23 March 1984 
Square matrices with properties such as non-negativity, diagonal dominance and equal row 
sums frequently occur in mathematical economics. In this paper Sierksma’s matrices of 
class M(x) are generalized to multisector models and are applied to the problem of pooling 
cross-section and time series data. 
1. Introduction 
Matrix theory is widely used in analyzing economic systems. Economi- 
cal arguments often impose restrictions on the elements of matrices. Here 
we study a special class of matrices with some useful properties and 
applications. Input-output-modelling theory is one field of economical 
research with an increasing interest in matrix classes. The open Leontief 
model relates the final demand vector with the gross production vector 
(I,- T)y=c, where (1) 
I,, is the (n x n) unit matrix, 
T is the (n X n) input-coefficient matrix, 
y is the gross production vector, and 
c is the final demand vector. 
The impact of final demand on production in sectors depends on the 
sector-structure. Supposing the long-term sector-structure to be fixed, 
short-term changes in final demand will give rise to changes in gross 
0165-1765/84/$3.00 0 1984, Elsevier Science Publishers B.V. (North-Holland) 
322 A. E. Ronner, E. Sterken / Generdrution of M(x)- matrices 
production. An important topic in this analysis of changes in demand 
and production is the formulation of conditions, which are sufficient to 
prove that a certain sector will gain relatively more from a relative 
stronger rise of final demand in this sector. In other words: if the change 
of the final demand in sector i exceeds the change in sector j, will the 
gross production react in a similar way? 
Sierksma (1979) studied the following equivalence: 
Ay,:,Ay,oAc,<Ac,, i,j= 1 ,...,n, i#j. (2) 
He proved that, under regularity conditions, (2) is equivalent to the 
assertion that T belongs to a certain class of matrices, called M(x). 
Besides the open Leontief model this concept can be used in e.g., the 
Stolper-Samuelson framework 
Aw =p, where (3) 
A is the matrix of factor shares, 
p is the vector of product prices, and 
w is the vector of money rewards. 
Wegge and Kemp (1969) derived necessary and sufficient conditions 
regarding the factor shares matrix A for the problem 
Ap,>(), Ap,=O~~w,>o~ Aw,<o, j,j = 1 ,...,n, i #j. (4) 
However, they did not provide a class of matrices, satisfying these 
conditions. Matrices of class M(x) seem to be sufficient for (4), and 
necessary too, if statement (4) is put in a slightly more general framework 
like (2) [Sierksma, Steenge and Sterken (1983)]. 
In this paper we shall discuss a simultaneous vector approach in which 
elements of the matrix M are allowed to be ( p X p) matrices. Section 3 
provides an application of class A( n, p, M,,, X) in mathematical eco- 
nomics and section 4 gives a useful application of symmetric matrices of 
class &(n, p, M,,, X) in econometrics. First some definitions and basic 
theorems are presented. 
Definition 1. A square (n x II) matrix M = {m,, } is said to belong to 
A. E. Ronner, E. Sterken / Generalization of M(x) matrices 323 
classM(x)ifandonlyifm,,-m,,=x,x#O,i,j=l,..., n,i+j. 
This definition implies that if M E M(x), then 
mz2-x ... 
. . 
m,,-x “’ 
In this paper a simultaneous sector approach is introduced in which the 
elements of the matrix M are allowed to be ( p X p) matrices. 
Definition 2. A square (np X np) matrix M is said to belong to class 
&‘(n, p, M,,, X) if and only if M can be partitioned in (n X n) sub- 
matrices M,,(pXp)such that M,,-M,,=X, withX#O, i,j=l,..., n, 
i#j. 
The following generalization of Sierksma’s theorems 16 and 18 holds. 
Theorem 1. Let M EJl(n, p, M,,, X) and define S = IF=, M,, - (n - 1)X. 
Let S and X be non-singular, then 
(4 Mm’=(I,,cW’) ; 
[ 
X- M,, + S . . . X - Mm 
x -M,, ‘.. X -M,,,+S 1 
x(zflc3x-‘), 
or equivulently, M-’ EJZ(n, p, S-‘( X- M,, + S)X-‘, X-l). 
(ii) M = IXI”~‘ISI. 
ProoJ: (i) Let 
A = k sne: 63 (M,, - X), and 
r=l 
B= ~s,,e;@(S-‘(X-M,,)X-I), 
1=l 
(5) 
324 A. E. Ronner, E. Sterken / Generaliratron of M(x) mafricrs 
with s, the n x 1 vector (1,. . . , 1)’ and e, the ith unit vector. Note that 
B= -(l,,@~s’)A(I,,@XX-‘)andAB=(I,@((S-X))B.Then 
M=A +(Z,, 8 X) and Mm’ =B+(I,, @ XP’), (6) 
since MM~‘=AB+A(Z,,Q?JX~‘)+(Z,,@X)B+Z,,,=(Z,,@((S-X))B 
- (Z,, @ S)B + (I,, 8 X)B + Z,?,, = I,,,,. The proof of (ii) is immediate. 
Q.E.D. 
In section 3 an application is given of the case M,, = M for each 
i=l , . . . ,n. Therefore the following corollary is useful: 
Corollary I. Zf the (np X np) matrix M ~d(n, p, M, X), then 
M=J,,@~+(Z,--J,)@A’ and M~‘=J,,~~S-‘+(Z,,-J,,)~~X-‘, 
where 
I,, = n ‘s,,s;. 
2. On the application of class Jl(n, p, M;;, X) in mathematical economics 
Suppose that an np-sector input-output system can be divided into n 
subgroups of sectors, such that an M(x)-structure holds over the sub- 
groups. 
Notation I. We state that, if zl and z2 are (n X 1) vectors z, +C z2 = Vi 
= l,..., n, z; <z;, where zi is the ith element of z,. 
Definition 3. An (n x n) non-negative matrix T is called irreducible if 
Vi,j=l,..., n 3m = m(i, j)> 0 such that t,(,V’)> 0. 
Lemma I. Zf T is an (n x n) irreducible non-negative matrix and T bus a 
dominant eigenvalue p < q~ then (cpl - T) ’ >+= 0. 
Proof: See Debreu and Herstein (1953). 
Note that if TEA(n, 1, m,,, x) with rp > s > x > 0, (VI- T)-’ B 0. 
A. E. Ronner, E. Sterken / Generalization of M(x) mafrices 325 
The open Leontief model is characterized by ‘p = 1. Now we are able to 
give a multisector alternative of Sierksma’s theorem 19. 
Theorem 2. If T ~.4’(n, p, M,,, X) with X a (p Xp) non-negative, irre- 
ducible matrix with dominant eigenvalue p < cp, then 
A-y, < A y, = AC, -=K AC 
J' 
i,j= 1 ,...,n, i#j (7) 
holds. 
Proof. AC, +C AC, e (cpZ,, - X)A y, -C (rpl,, - X)A y,. With the properties 
of X ( cpZP - X)) ’ B 0 holds according to Lemma 1. So A y, -=c A y,. 
Q.E.D. 
Note that the opposite is not true, otherwise the equivalence of the 
assertions in Sierksma’s theorem is violated. 
The same concept can be used in the Stolper-Samuelson framework. 
If A c&(n, p, M,,, X) both Aw, +z Aw, * Ap, -c Ap, and Ap, B 0, Ap, 
=O~Aw,~O,Aw,-=~Ohold,foreachi,j=l,..., n;i#j. 
3. On the use of the class A(n, p, Mji, X) in pooling cross-section and 
time series data 
One of the most frequently used techniques in pooling 
and time series data is the analysis of variance. Suppose 
following model: 
y,,= i: x,,,P,, + u,/, i=l >...> n, t=l,..., T, 
s= 1 
cross-section 
we have the 
(8) 
where y,,, x,(, and u,~ are (p x I)-vectors and p,, is a scalar. This model 
may describe the sales of p firms over n regions which are explained by 
the marketing instruments x, ,s. Given the covariance matrix of the 
disturbances the parameters are usually estimated by maximum likeli- 
hood methods, cf. Wansbeek and Kapteyn (1982). Therefore we need the 
inverse of the covariance matrix. All - at least, known to us - error 
components covariance matrices have a mathematical structure that 
corresponds to class A(n, p, M, X). Let us consider the two error 
326 A. E. Ronner, E. Sterken / Generalizutron of M(x) mamc’es 
components model. We assume that the errors u,, (8) are decomposed as 
u,, = 77, + c,, 1 
where nlr and F,, are independently distributed as 
n, - N,(O, r) and E,, - N,>(O, A ), 
with r positive semidefinite and A positive definite, both of order p. 
Define the ( p x n) matrices and ( pn x 1) vectors 
I/;=(u ,,,..., u,,,) and u,=vecV,, t=l,..., T. 
Then u ,,. . . ,uT are i.i.d. as N,,,,(O, Q,), with 
n, = 
r+A ... l’ 
1 
r . . . ;+A I. 
Note that L?, l &(n, p, r+ A, A), so according to Corollary 1 we can 
write S, = nr+ A and 
D, = _I,, @ S, + ( I,, - J,, ) 8 A, (9) 
,n;‘=Jr,~Ss,‘+(I,-J,,)~Aa’. (10) 
The three error components model can be treated in a similar way. 
References 
Debreu, G. and I.N. Herstein, 1953, Nonnegative square matrices, Econometrica 21. 
597-607. 
Sierksma, G.. 1979, Nonnegative matrices; the open Leontief model, Linear Algebra and its 
Applications 26, 175-210. 
Sierksma. G., A.E. Steenge and E. Sterken, 1983, A generalization of Stolper-Samuelson 
and Kemp-Wegge theorems, Working paper 83-15-WE (Econometrics Institute, Uni- 
versity of Groningen, Groningen). 
Wansbeek, T. and A. Kapteyn, 1982, A class of decompositions of the variance-covariance 
matrix of a generalized error components model, Econometrica 50. 713-724. 
Wegge, L.L.F. and M.C. Kemp, 1969. Generalizations of the Stolper-Samuelson and 
Samuelson-Rybczynski theorems in terms of conditional input&output coefficients. 
International Economic Review 10, 414-425. 


   University of GroningenGeneralization of M(x)-Matrices; Application to Mathematical Economics and EconometricsRonner, Arjen E; Sterken, ElmerPublished in:Economics LettersDOI:10.1016/0165-1765(84)90183-6IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1984Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Ronner, A. E., & Sterken, E. (1984). Generalization of M(x)-Matrices; Application to MathematicalEconomics and Econometrics. Economics Letters, 16, 321-326. https://doi.org/10.1016/0165-1765(84)90183-6CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 17-07-2023Economics Letters 16 (1984) 321-326 North-Holland 321 GENERALIZATION OF M( x)-MATRICES Application to Mathematical Economics and Econometrics Arjen E. RONNER and Elmer STERKEN Unruersiiy of Groningen, Groningen, The Netherlunds Received 23 March 1984 Square matrices with properties such as non-negativity, diagonal dominance and equal row sums frequently occur in mathematical economics. In this paper Sierksma’s matrices of class M(x) are generalized to multisector models and are applied to the problem of pooling cross-section and time series data. 1. Introduction Matrix theory is widely used in analyzing economic systems. Economi- cal arguments often impose restrictions on the elements of matrices. Here we study a special class of matrices with some useful properties and applications. Input-output-modelling theory is one field of economical research with an increasing interest in matrix classes. The open Leontief model relates the final demand vector with the gross production vector (I,- T)y=c, where (1) I,, is the (n x n) unit matrix, T is the (n X n) input-coefficient matrix, y is the gross production vector, and c is the final demand vector. The impact of final demand on production in sectors depends on the sector-structure. Supposing the long-term sector-structure to be fixed, short-term changes in final demand will give rise to changes in gross 0165-1765/84/$3.00 0 1984, Elsevier Science Publishers B.V. (North-Holland) 322 A. E. Ronner, E. Sterken / Generdrution of M(x)- matrices production. An important topic in this analysis of changes in demand and production is the formulation of conditions, which are sufficient to prove that a certain sector will gain relatively more from a relative stronger rise of final demand in this sector. In other words: if the change of the final demand in sector i exceeds the change in sector j, will the gross production react in a similar way? Sierksma (1979) studied the following equivalence: Ay,:,Ay,oAc,<Ac,, i,j= 1 ,...,n, i#j. (2) He proved that, under regularity conditions, (2) is equivalent to the assertion that T belongs to a certain class of matrices, called M(x). Besides the open Leontief model this concept can be used in e.g., the Stolper-Samuelson framework Aw =p, where (3) A is the matrix of factor shares, p is the vector of product prices, and w is the vector of money rewards. Wegge and Kemp (1969) derived necessary and sufficient conditions regarding the factor shares matrix A for the problem Ap,>(), Ap,=O~~w,>o~ Aw,<o, j,j = 1 ,...,n, i #j. (4) However, they did not provide a class of matrices, satisfying these conditions. Matrices of class M(x) seem to be sufficient for (4), and necessary too, if statement (4) is put in a slightly more general framework like (2) [Sierksma, Steenge and Sterken (1983)]. In this paper we shall discuss a simultaneous vector approach in which elements of the matrix M are allowed to be ( p X p) matrices. Section 3 provides an application of class A( n, p, M,,, X) in mathematical eco- nomics and section 4 gives a useful application of symmetric matrices of class &(n, p, M,,, X) in econometrics. First some definitions and basic theorems are presented. Definition 1. A square (n x II) matrix M = {m,, } is said to belong to A. E. Ronner, E. Sterken / Generalization of M(x) matrices 323 classM(x)ifandonlyifm,,-m,,=x,x#O,i,j=l,..., n,i+j. This definition implies that if M E M(x), then mz2-x ... . . m,,-x “’ In this paper a simultaneous sector approach is introduced in which the elements of the matrix M are allowed to be ( p X p) matrices. Definition 2. A square (np X np) matrix M is said to belong to class &‘(n, p, M,,, X) if and only if M can be partitioned in (n X n) sub- matrices M,,(pXp)such that M,,-M,,=X, withX#O, i,j=l,..., n, i#j. The following generalization of Sierksma’s theorems 16 and 18 holds. Theorem 1. Let M EJl(n, p, M,,, X) and define S = IF=, M,, - (n - 1)X. Let S and X be non-singular, then (4 Mm’=(I,,cW’) ; [ X- M,, + S . . . X - Mm x -M,, ‘.. X -M,,,+S 1 x(zflc3x-‘), or equivulently, M-’ EJZ(n, p, S-‘( X- M,, + S)X-‘, X-l). (ii) M = IXI”~‘ISI. ProoJ: (i) Let A = k sne: 63 (M,, - X), and r=l B= ~s,,e;@(S-‘(X-M,,)X-I), 1=l (5) 324 A. E. Ronner, E. Sterken / Generaliratron of M(x) mafricrs with s, the n x 1 vector (1,. . . , 1)’ and e, the ith unit vector. Note that B= -(l,,@~s’)A(I,,@XX-‘)andAB=(I,@((S-X))B.Then M=A +(Z,, 8 X) and Mm’ =B+(I,, @ XP’), (6) since MM~‘=AB+A(Z,,Q?JX~‘)+(Z,,@X)B+Z,,,=(Z,,@((S-X))B - (Z,, @ S)B + (I,, 8 X)B + Z,?,, = I,,,,. The proof of (ii) is immediate. Q.E.D. In section 3 an application is given of the case M,, = M for each i=l , . . . ,n. Therefore the following corollary is useful: Corollary I. Zf the (np X np) matrix M ~d(n, p, M, X), then M=J,,@~+(Z,--J,)@A’ and M~‘=J,,~~S-‘+(Z,,-J,,)~~X-‘, where I,, = n ‘s,,s;. 2. On the application of class Jl(n, p, M;;, X) in mathematical economics Suppose that an np-sector input-output system can be divided into n subgroups of sectors, such that an M(x)-structure holds over the sub- groups. Notation I. We state that, if zl and z2 are (n X 1) vectors z, +C z2 = Vi = l,..., n, z; <z;, where zi is the ith element of z,. Definition 3. An (n x n) non-negative matrix T is called irreducible if Vi,j=l,..., n 3m = m(i, j)> 0 such that t,(,V’)> 0. Lemma I. Zf T is an (n x n) irreducible non-negative matrix and T bus a dominant eigenvalue p < q~ then (cpl - T) ’ >+= 0. Proof: See Debreu and Herstein (1953). Note that if TEA(n, 1, m,,, x) with rp > s > x > 0, (VI- T)-’ B 0. A. E. Ronner, E. Sterken / Generalization of M(x) mafrices 325 The open Leontief model is characterized by ‘p = 1. Now we are able to give a multisector alternative of Sierksma’s theorem 19. Theorem 2. If T ~.4’(n, p, M,,, X) with X a (p Xp) non-negative, irre- ducible matrix with dominant eigenvalue p < cp, then A-y, < A y, = AC, -=K AC J' i,j= 1 ,...,n, i#j (7) holds. Proof. AC, +C AC, e (cpZ,, - X)A y, -C (rpl,, - X)A y,. With the properties of X ( cpZP - X)) ’ B 0 holds according to Lemma 1. So A y, -=c A y,. Q.E.D. Note that the opposite is not true, otherwise the equivalence of the assertions in Sierksma’s theorem is violated. The same concept can be used in the Stolper-Samuelson framework. If A c&(n, p, M,,, X) both Aw, +z Aw, * Ap, -c Ap, and Ap, B 0, Ap, =O~Aw,~O,Aw,-=~Ohold,foreachi,j=l,..., n;i#j. 3. On the use of the class A(n, p, Mji, X) in pooling cross-section and time series data One of the most frequently used techniques in pooling and time series data is the analysis of variance. Suppose following model: y,,= i: x,,,P,, + u,/, i=l >...> n, t=l,..., T, s= 1 cross-section we have the (8) where y,,, x,(, and u,~ are (p x I)-vectors and p,, is a scalar. This model may describe the sales of p firms over n regions which are explained by the marketing instruments x, ,s. Given the covariance matrix of the disturbances the parameters are usually estimated by maximum likeli- hood methods, cf. Wansbeek and Kapteyn (1982). Therefore we need the inverse of the covariance matrix. All - at least, known to us - error components covariance matrices have a mathematical structure that corresponds to class A(n, p, M, X). Let us consider the two error 326 A. E. Ronner, E. Sterken / Generalizutron of M(x) mamc’es components model. We assume that the errors u,, (8) are decomposed as u,, = 77, + c,, 1 where nlr and F,, are independently distributed as n, - N,(O, r) and E,, - N,>(O, A ), with r positive semidefinite and A positive definite, both of order p. Define the ( p x n) matrices and ( pn x 1) vectors I/;=(u ,,,..., u,,,) and u,=vecV,, t=l,..., T. Then u ,,. . . ,uT are i.i.d. as N,,,,(O, Q,), with n, = r+A ... l’ 1 r . . . ;+A I. Note that L?, l &(n, p, r+ A, A), so according to Corollary 1 we can write S, = nr+ A and D, = _I,, @ S, + ( I,, - J,, ) 8 A, (9) ,n;‘=Jr,~Ss,‘+(I,-J,,)~Aa’. (10) The three error components model can be treated in a similar way. References Debreu, G. and I.N. Herstein, 1953, Nonnegative square matrices, Econometrica 21. 597-607. Sierksma, G.. 1979, Nonnegative matrices; the open Leontief model, Linear Algebra and its Applications 26, 175-210. Sierksma. G., A.E. Steenge and E. Sterken, 1983, A generalization of Stolper-Samuelson and Kemp-Wegge theorems, Working paper 83-15-WE (Econometrics Institute, Uni- versity of Groningen, Groningen). Wansbeek, T. and A. Kapteyn, 1982, A class of decompositions of the variance-covariance matrix of a generalized error components model, Econometrica 50. 713-724. Wegge, L.L.F. and M.C. Kemp, 1969. Generalizations of the Stolper-Samuelson and Samuelson-Rybczynski theorems in terms of conditional input&output coefficients. International Economic Review 10, 414-425. 

NARCIS 

University of Groningen

   University of GroningenGeneralization of M(x)-Matrices; Application to Mathematical Economics and EconometricsRonner, Arjen E; Sterken, ElmerPublished in:Economics LettersDOI:10.1016/0165-1765(84)90183-6IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1984Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Ronner, A. E., & Sterken, E. (1984). Generalization of M(x)-Matrices; Application to MathematicalEconomics and Econometrics. Economics Letters, 16, 321-326. https://doi.org/10.1016/0165-1765(84)90183-6CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 27-04-2023Economics Letters 16 (1984) 321-326 North-Holland 321 GENERALIZATION OF M( x)-MATRICES Application to Mathematical Economics and Econometrics Arjen E. RONNER and Elmer STERKEN Unruersiiy of Groningen, Groningen, The Netherlunds Received 23 March 1984 Square matrices with properties such as non-negativity, diagonal dominance and equal row sums frequently occur in mathematical economics. In this paper Sierksma’s matrices of class M(x) are generalized to multisector models and are applied to the problem of pooling cross-section and time series data. 1. Introduction Matrix theory is widely used in analyzing economic systems. Economi- cal arguments often impose restrictions on the elements of matrices. Here we study a special class of matrices with some useful properties and applications. Input-output-modelling theory is one field of economical research with an increasing interest in matrix classes. The open Leontief model relates the final demand vector with the gross production vector (I,- T)y=c, where (1) I,, is the (n x n) unit matrix, T is the (n X n) input-coefficient matrix, y is the gross production vector, and c is the final demand vector. The impact of final demand on production in sectors depends on the sector-structure. Supposing the long-term sector-structure to be fixed, short-term changes in final demand will give rise to changes in gross 0165-1765/84/$3.00 0 1984, Elsevier Science Publishers B.V. (North-Holland) 322 A. E. Ronner, E. Sterken / Generdrution of M(x)- matrices production. An important topic in this analysis of changes in demand and production is the formulation of conditions, which are sufficient to prove that a certain sector will gain relatively more from a relative stronger rise of final demand in this sector. In other words: if the change of the final demand in sector i exceeds the change in sector j, will the gross production react in a similar way? Sierksma (1979) studied the following equivalence: Ay,:,Ay,oAc,<Ac,, i,j= 1 ,...,n, i#j. (2) He proved that, under regularity conditions, (2) is equivalent to the assertion that T belongs to a certain class of matrices, called M(x). Besides the open Leontief model this concept can be used in e.g., the Stolper-Samuelson framework Aw =p, where (3) A is the matrix of factor shares, p is the vector of product prices, and w is the vector of money rewards. Wegge and Kemp (1969) derived necessary and sufficient conditions regarding the factor shares matrix A for the problem Ap,>(), Ap,=O~~w,>o~ Aw,<o, j,j = 1 ,...,n, i #j. (4) However, they did not provide a class of matrices, satisfying these conditions. Matrices of class M(x) seem to be sufficient for (4), and necessary too, if statement (4) is put in a slightly more general framework like (2) [Sierksma, Steenge and Sterken (1983)]. In this paper we shall discuss a simultaneous vector approach in which elements of the matrix M are allowed to be ( p X p) matrices. Section 3 provides an application of class A( n, p, M,,, X) in mathematical eco- nomics and section 4 gives a useful application of symmetric matrices of class &(n, p, M,,, X) in econometrics. First some definitions and basic theorems are presented. Definition 1. A square (n x II) matrix M = {m,, } is said to belong to A. E. Ronner, E. Sterken / Generalization of M(x) matrices 323 classM(x)ifandonlyifm,,-m,,=x,x#O,i,j=l,..., n,i+j. This definition implies that if M E M(x), then mz2-x ... . . m,,-x “’ In this paper a simultaneous sector approach is introduced in which the elements of the matrix M are allowed to be ( p X p) matrices. Definition 2. A square (np X np) matrix M is said to belong to class &‘(n, p, M,,, X) if and only if M can be partitioned in (n X n) sub- matrices M,,(pXp)such that M,,-M,,=X, withX#O, i,j=l,..., n, i#j. The following generalization of Sierksma’s theorems 16 and 18 holds. Theorem 1. Let M EJl(n, p, M,,, X) and define S = IF=, M,, - (n - 1)X. Let S and X be non-singular, then (4 Mm’=(I,,cW’) ; [ X- M,, + S . . . X - Mm x -M,, ‘.. X -M,,,+S 1 x(zflc3x-‘), or equivulently, M-’ EJZ(n, p, S-‘( X- M,, + S)X-‘, X-l). (ii) M = IXI”~‘ISI. ProoJ: (i) Let A = k sne: 63 (M,, - X), and r=l B= ~s,,e;@(S-‘(X-M,,)X-I), 1=l (5) 324 A. E. Ronner, E. Sterken / Generaliratron of M(x) mafricrs with s, the n x 1 vector (1,. . . , 1)’ and e, the ith unit vector. Note that B= -(l,,@~s’)A(I,,@XX-‘)andAB=(I,@((S-X))B.Then M=A +(Z,, 8 X) and Mm’ =B+(I,, @ XP’), (6) since MM~‘=AB+A(Z,,Q?JX~‘)+(Z,,@X)B+Z,,,=(Z,,@((S-X))B - (Z,, @ S)B + (I,, 8 X)B + Z,?,, = I,,,,. The proof of (ii) is immediate. Q.E.D. In section 3 an application is given of the case M,, = M for each i=l , . . . ,n. Therefore the following corollary is useful: Corollary I. Zf the (np X np) matrix M ~d(n, p, M, X), then M=J,,@~+(Z,--J,)@A’ and M~‘=J,,~~S-‘+(Z,,-J,,)~~X-‘, where I,, = n ‘s,,s;. 2. On the application of class Jl(n, p, M;;, X) in mathematical economics Suppose that an np-sector input-output system can be divided into n subgroups of sectors, such that an M(x)-structure holds over the sub- groups. Notation I. We state that, if zl and z2 are (n X 1) vectors z, +C z2 = Vi = l,..., n, z; <z;, where zi is the ith element of z,. Definition 3. An (n x n) non-negative matrix T is called irreducible if Vi,j=l,..., n 3m = m(i, j)> 0 such that t,(,V’)> 0. Lemma I. Zf T is an (n x n) irreducible non-negative matrix and T bus a dominant eigenvalue p < q~ then (cpl - T) ’ >+= 0. Proof: See Debreu and Herstein (1953). Note that if TEA(n, 1, m,,, x) with rp > s > x > 0, (VI- T)-’ B 0. A. E. Ronner, E. Sterken / Generalization of M(x) mafrices 325 The open Leontief model is characterized by ‘p = 1. Now we are able to give a multisector alternative of Sierksma’s theorem 19. Theorem 2. If T ~.4’(n, p, M,,, X) with X a (p Xp) non-negative, irre- ducible matrix with dominant eigenvalue p < cp, then A-y, < A y, = AC, -=K AC J' i,j= 1 ,...,n, i#j (7) holds. Proof. AC, +C AC, e (cpZ,, - X)A y, -C (rpl,, - X)A y,. With the properties of X ( cpZP - X)) ’ B 0 holds according to Lemma 1. So A y, -=c A y,. Q.E.D. Note that the opposite is not true, otherwise the equivalence of the assertions in Sierksma’s theorem is violated. The same concept can be used in the Stolper-Samuelson framework. If A c&(n, p, M,,, X) both Aw, +z Aw, * Ap, -c Ap, and Ap, B 0, Ap, =O~Aw,~O,Aw,-=~Ohold,foreachi,j=l,..., n;i#j. 3. On the use of the class A(n, p, Mji, X) in pooling cross-section and time series data One of the most frequently used techniques in pooling and time series data is the analysis of variance. Suppose following model: y,,= i: x,,,P,, + u,/, i=l >...> n, t=l,..., T, s= 1 cross-section we have the (8) where y,,, x,(, and u,~ are (p x I)-vectors and p,, is a scalar. This model may describe the sales of p firms over n regions which are explained by the marketing instruments x, ,s. Given the covariance matrix of the disturbances the parameters are usually estimated by maximum likeli- hood methods, cf. Wansbeek and Kapteyn (1982). Therefore we need the inverse of the covariance matrix. All - at least, known to us - error components covariance matrices have a mathematical structure that corresponds to class A(n, p, M, X). Let us consider the two error 326 A. E. Ronner, E. Sterken / Generalizutron of M(x) mamc’es components model. We assume that the errors u,, (8) are decomposed as u,, = 77, + c,, 1 where nlr and F,, are independently distributed as n, - N,(O, r) and E,, - N,>(O, A ), with r positive semidefinite and A positive definite, both of order p. Define the ( p x n) matrices and ( pn x 1) vectors I/;=(u ,,,..., u,,,) and u,=vecV,, t=l,..., T. Then u ,,. . . ,uT are i.i.d. as N,,,,(O, Q,), with n, = r+A ... l’ 1 r . . . ;+A I. Note that L?, l &(n, p, r+ A, A), so according to Corollary 1 we can write S, = nr+ A and D, = _I,, @ S, + ( I,, - J,, ) 8 A, (9) ,n;‘=Jr,~Ss,‘+(I,-J,,)~Aa’. (10) The three error components model can be treated in a similar way. References Debreu, G. and I.N. Herstein, 1953, Nonnegative square matrices, Econometrica 21. 597-607. Sierksma, G.. 1979, Nonnegative matrices; the open Leontief model, Linear Algebra and its Applications 26, 175-210. Sierksma. G., A.E. Steenge and E. Sterken, 1983, A generalization of Stolper-Samuelson and Kemp-Wegge theorems, Working paper 83-15-WE (Econometrics Institute, Uni- versity of Groningen, Groningen). Wansbeek, T. and A. Kapteyn, 1982, A class of decompositions of the variance-covariance matrix of a generalized error components model, Econometrica 50. 713-724. Wegge, L.L.F. and M.C. Kemp, 1969. Generalizations of the Stolper-Samuelson and Samuelson-Rybczynski theorems in terms of conditional input&output coefficients. International Economic Review 10, 414-425. 

   University of GroningenGeneralization of M(x)-Matrices; Application to Mathematical Economics and EconometricsRonner, Arjen E; Sterken, ElmerPublished in:Economics LettersDOI:10.1016/0165-1765(84)90183-6IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1984Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Ronner, A. E., & Sterken, E. (1984). Generalization of M(x)-Matrices; Application to MathematicalEconomics and Econometrics. Economics Letters, 16, 321-326. https://doi.org/10.1016/0165-1765(84)90183-6CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019Economics Letters 16 (1984) 321-326 North-Holland 321 GENERALIZATION OF M( x)-MATRICES Application to Mathematical Economics and Econometrics Arjen E. RONNER and Elmer STERKEN Unruersiiy of Groningen, Groningen, The Netherlunds Received 23 March 1984 Square matrices with properties such as non-negativity, diagonal dominance and equal row sums frequently occur in mathematical economics. In this paper Sierksma’s matrices of class M(x) are generalized to multisector models and are applied to the problem of pooling cross-section and time series data. 1. Introduction Matrix theory is widely used in analyzing economic systems. Economi- cal arguments often impose restrictions on the elements of matrices. Here we study a special class of matrices with some useful properties and applications. Input-output-modelling theory is one field of economical research with an increasing interest in matrix classes. The open Leontief model relates the final demand vector with the gross production vector (I,- T)y=c, where (1) I,, is the (n x n) unit matrix, T is the (n X n) input-coefficient matrix, y is the gross production vector, and c is the final demand vector. The impact of final demand on production in sectors depends on the sector-structure. Supposing the long-term sector-structure to be fixed, short-term changes in final demand will give rise to changes in gross 0165-1765/84/$3.00 0 1984, Elsevier Science Publishers B.V. (North-Holland) 322 A. E. Ronner, E. Sterken / Generdrution of M(x)- matrices production. An important topic in this analysis of changes in demand and production is the formulation of conditions, which are sufficient to prove that a certain sector will gain relatively more from a relative stronger rise of final demand in this sector. In other words: if the change of the final demand in sector i exceeds the change in sector j, will the gross production react in a similar way? Sierksma (1979) studied the following equivalence: Ay,:,Ay,oAc,<Ac,, i,j= 1 ,...,n, i#j. (2) He proved that, under regularity conditions, (2) is equivalent to the assertion that T belongs to a certain class of matrices, called M(x). Besides the open Leontief model this concept can be used in e.g., the Stolper-Samuelson framework Aw =p, where (3) A is the matrix of factor shares, p is the vector of product prices, and w is the vector of money rewards. Wegge and Kemp (1969) derived necessary and sufficient conditions regarding the factor shares matrix A for the problem Ap,>(), Ap,=O~~w,>o~ Aw,<o, j,j = 1 ,...,n, i #j. (4) However, they did not provide a class of matrices, satisfying these conditions. Matrices of class M(x) seem to be sufficient for (4), and necessary too, if statement (4) is put in a slightly more general framework like (2) [Sierksma, Steenge and Sterken (1983)]. In this paper we shall discuss a simultaneous vector approach in which elements of the matrix M are allowed to be ( p X p) matrices. Section 3 provides an application of class A( n, p, M,,, X) in mathematical eco- nomics and section 4 gives a useful application of symmetric matrices of class &(n, p, M,,, X) in econometrics. First some definitions and basic theorems are presented. Definition 1. A square (n x II) matrix M = {m,, } is said to belong to A. E. Ronner, E. Sterken / Generalization of M(x) matrices 323 classM(x)ifandonlyifm,,-m,,=x,x#O,i,j=l,..., n,i+j. This definition implies that if M E M(x), then mz2-x ... . . m,,-x “’ In this paper a simultaneous sector approach is introduced in which the elements of the matrix M are allowed to be ( p X p) matrices. Definition 2. A square (np X np) matrix M is said to belong to class &‘(n, p, M,,, X) if and only if M can be partitioned in (n X n) sub- matrices M,,(pXp)such that M,,-M,,=X, withX#O, i,j=l,..., n, i#j. The following generalization of Sierksma’s theorems 16 and 18 holds. Theorem 1. Let M EJl(n, p, M,,, X) and define S = IF=, M,, - (n - 1)X. Let S and X be non-singular, then (4 Mm’=(I,,cW’) ; [ X- M,, + S . . . X - Mm x -M,, ‘.. X -M,,,+S 1 x(zflc3x-‘), or equivulently, M-’ EJZ(n, p, S-‘( X- M,, + S)X-‘, X-l). (ii) M = IXI”~‘ISI. ProoJ: (i) Let A = k sne: 63 (M,, - X), and r=l B= ~s,,e;@(S-‘(X-M,,)X-I), 1=l (5) 324 A. E. Ronner, E. Sterken / Generaliratron of M(x) mafricrs with s, the n x 1 vector (1,. . . , 1)’ and e, the ith unit vector. Note that B= -(l,,@~s’)A(I,,@XX-‘)andAB=(I,@((S-X))B.Then M=A +(Z,, 8 X) and Mm’ =B+(I,, @ XP’), (6) since MM~‘=AB+A(Z,,Q?JX~‘)+(Z,,@X)B+Z,,,=(Z,,@((S-X))B - (Z,, @ S)B + (I,, 8 X)B + Z,?,, = I,,,,. The proof of (ii) is immediate. Q.E.D. In section 3 an application is given of the case M,, = M for each i=l , . . . ,n. Therefore the following corollary is useful: Corollary I. Zf the (np X np) matrix M ~d(n, p, M, X), then M=J,,@~+(Z,--J,)@A’ and M~‘=J,,~~S-‘+(Z,,-J,,)~~X-‘, where I,, = n ‘s,,s;. 2. On the application of class Jl(n, p, M;;, X) in mathematical economics Suppose that an np-sector input-output system can be divided into n subgroups of sectors, such that an M(x)-structure holds over the sub- groups. Notation I. We state that, if zl and z2 are (n X 1) vectors z, +C z2 = Vi = l,..., n, z; <z;, where zi is the ith element of z,. Definition 3. An (n x n) non-negative matrix T is called irreducible if Vi,j=l,..., n 3m = m(i, j)> 0 such that t,(,V’)> 0. Lemma I. Zf T is an (n x n) irreducible non-negative matrix and T bus a dominant eigenvalue p < q~ then (cpl - T) ’ >+= 0. Proof: See Debreu and Herstein (1953). Note that if TEA(n, 1, m,,, x) with rp > s > x > 0, (VI- T)-’ B 0. A. E. Ronner, E. Sterken / Generalization of M(x) mafrices 325 The open Leontief model is characterized by ‘p = 1. Now we are able to give a multisector alternative of Sierksma’s theorem 19. Theorem 2. If T ~.4’(n, p, M,,, X) with X a (p Xp) non-negative, irre- ducible matrix with dominant eigenvalue p < cp, then A-y, < A y, = AC, -=K AC J' i,j= 1 ,...,n, i#j (7) holds. Proof. AC, +C AC, e (cpZ,, - X)A y, -C (rpl,, - X)A y,. With the properties of X ( cpZP - X)) ’ B 0 holds according to Lemma 1. So A y, -=c A y,. Q.E.D. Note that the opposite is not true, otherwise the equivalence of the assertions in Sierksma’s theorem is violated. The same concept can be used in the Stolper-Samuelson framework. If A c&(n, p, M,,, X) both Aw, +z Aw, * Ap, -c Ap, and Ap, B 0, Ap, =O~Aw,~O,Aw,-=~Ohold,foreachi,j=l,..., n;i#j. 3. On the use of the class A(n, p, Mji, X) in pooling cross-section and time series data One of the most frequently used techniques in pooling and time series data is the analysis of variance. Suppose following model: y,,= i: x,,,P,, + u,/, i=l >...> n, t=l,..., T, s= 1 cross-section we have the (8) where y,,, x,(, and u,~ are (p x I)-vectors and p,, is a scalar. This model may describe the sales of p firms over n regions which are explained by the marketing instruments x, ,s. Given the covariance matrix of the disturbances the parameters are usually estimated by maximum likeli- hood methods, cf. Wansbeek and Kapteyn (1982). Therefore we need the inverse of the covariance matrix. All - at least, known to us - error components covariance matrices have a mathematical structure that corresponds to class A(n, p, M, X). Let us consider the two error 326 A. E. Ronner, E. Sterken / Generalizutron of M(x) mamc’es components model. We assume that the errors u,, (8) are decomposed as u,, = 77, + c,, 1 where nlr and F,, are independently distributed as n, - N,(O, r) and E,, - N,>(O, A ), with r positive semidefinite and A positive definite, both of order p. Define the ( p x n) matrices and ( pn x 1) vectors I/;=(u ,,,..., u,,,) and u,=vecV,, t=l,..., T. Then u ,,. . . ,uT are i.i.d. as N,,,,(O, Q,), with n, = r+A ... l’ 1 r . . . ;+A I. Note that L?, l &(n, p, r+ A, A), so according to Corollary 1 we can write S, = nr+ A and D, = _I,, @ S, + ( I,, - J,, ) 8 A, (9) ,n;‘=Jr,~Ss,‘+(I,-J,,)~Aa’. (10) The three error components model can be treated in a similar way. References Debreu, G. and I.N. Herstein, 1953, Nonnegative square matrices, Econometrica 21. 597-607. Sierksma, G.. 1979, Nonnegative matrices; the open Leontief model, Linear Algebra and its Applications 26, 175-210. Sierksma. G., A.E. Steenge and E. Sterken, 1983, A generalization of Stolper-Samuelson and Kemp-Wegge theorems, Working paper 83-15-WE (Econometrics Institute, Uni- versity of Groningen, Groningen). Wansbeek, T. and A. Kapteyn, 1982, A class of decompositions of the variance-covariance matrix of a generalized error components model, Econometrica 50. 713-724. Wegge, L.L.F. and M.C. Kemp, 1969. Generalizations of the Stolper-Samuelson and Samuelson-Rybczynski theorems in terms of conditional input&output coefficients. International Economic Review 10, 414-425. 

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