Bekker, P.A.

English

Tilburg University Repository

    Tilburg UniversityComment on: Identification in the linear errors in variables modelBekker, P.A.Publication date:1984Document VersionPublisher's PDF, also known as Version of recordLink to publication in Tilburg University Research PortalCitation for published version (APA):Bekker, P. A. (1984). Comment on: Identification in the linear errors in variables model. (Research MemorandumFEW). Faculteit der Economische Wetenschappen.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portalTake 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.Download date: 12. Nov. 20231!CBMR7626 ~TIJDSCHRIFTENBUREAUBestemming B1BLI~~`?'I-IL':~K Nr.~y 11984 ~" I I ~ ~ ~ ~159 ~aIsubfaculteit der econometrieRESEARCH MEMORANDUMi q N ~u iiNiuiui iu g nuNiqHi ip~u~ u~ i I~~ --. -TILBURG UNIVERSITYDEPARTMENT OF ECONOMICSPostbus 90153 - 5000 LE TilburgNetherlandsKt~.T:I:~`LIIrKEHOv;::S:.."r:~~OL1iLBU4i;FEW159COMMENT ON: IDENTIFICATION IN THF.LLNEAK F.RRORS IN VARIARLES MODELAy Paul A. Bekkerl)1) The author wishes to thank Arie Kapteyn and Tom Wansbeek for theirstimulating discussions on the subject. Financial support by the Nether-lands Organization for the Advancement of Pure Research (ZWO) is grate-fully acknowledged.1. IntroductionKapteyn and Wansbeek [1] considered the following multiple linear re-gression model with errors in variables:(1.1) Yj - ~~B f Ej. (j z 1,...,n)(1.2) xj - ~j f uj,where ~j, xj, uj and S are k-vectors, yj, Ej are scalars. The ~j areunobservable variables: instead the xj are observed. The measurementerrors u j are unobservable as well and ít is assumed that v j - N(O,S2)and e j- N(~,a2) for all j. The v j and e j are mutually independent andindependent of ~j. The ~j are considered as random drawings from some,as yet unspecified, multivariate distribution with zero mean.For the case k-1 Reiersbl [2] has shown that normality of ~j is the onlydistributional assumption which spoils identification. For the case k~1 and the components of ~j are mutually independent, Wilassen [3] hasshown that none of the components of ~j should be normally distributedto ~uarantee identifiabílity of S. Kapteyn and Wansbeek [1] did notassume independency of the components of ~j and they stated the follo-wing proposition: the parameter vector S is identified if and only ifthere does not exist a lineair combination of ~j which is normallydístributed. The necessity part in this proposition is incorrect, i.e.it may well be that a normally distributed linear combination of ~j doesnot spoil the identifiability of B. Here I present necessary and suf-ficient conditions for identification of s.22. Statement of the Result and ProofProposition: Under the assumptions above, the parameter vector 6 isidentified if and only if there does not exist a nonsin-gular matrix A-(a1,A2) such that l;~al is distributed no r-mally and independently of ~~A2.Proof: We first show that nonidentifiability of S implies the existenceof the matrix A. Let s be a scalar and t a k-vector. The characteristicfunction ~y ~x ( s,t) of y) and x~ isJ J(2.1) ~y ~x (s,t) - exp{-}(a2s2tt'S2t)} ~~(Bstt) ,J ~where m~(.) is the characteristic functíon of ~~. Assuming that B is notfully identified amoun[s [o saying that there exíst parameter2sets {B,aZ,f2} and {p~,a~ ,52~}, with S~ B~, generating the same distri-bution of y~, x~. Consequently ~ x(s,t) should be the same for bothsets of parameters: y~~ ~2(2.z) exp{-~(a2s2ft'f2t)} ~~(psft) 3 exp{-}(a~ sZft'S2~t)} m~(s~stt).~ ~1.et R- S sft, then ~~(Ssft) -~~(((3-B )sflC) L~ ~ (s,R). Thus~~ca-s ),~(2.2) caries over into2(2.3) ~ ~ ( s,R) - exp{-}Is2(a~ -a2) fF'(s-s ),~t (R-B~s)'(R~-S2)(k-S~s)]} ~~(k).The characteristic function corresponding to the marginal distribution~of ~~(S-g ) is found by setting R- 0J3~2 ~~ 2 ~, ~ ~(2.4) ~ ~ ( s) - exp{-~s (a -a fs ( S2 -S2)S )} ,~' (s-a )which is the characteristic function of a normally distributed variable.In addition [o this result, which was obtained by Kapteyn andWansbeek [1], it wíll now be shown that nonidentifiability of S alsoimplies the existence of a matrix AZ such that ( a1,A2) is nonsingularand ~'al i s distributed i ndependently of ~~AZ. The characteristic func-Jtion corresponding to the marginal distributions of ~~ is found by set-ting s-0 in (2.3):(2.5) ~~(R) - exp{-~R.'(St~-St)k} ~~(R).Thus, we may rewrite (2.3) as(2.6) ~ ~ (s,R) - ~ ~ (s) ~~(k) exp{sB~'(St~-St)k}.~'(B-S ).~ ~'(B-B )~~ ~Let B be a(kx(k-1))-matrix of full column rank such that B (S2 -S2)B -0. F.quality (2.6) holds for all possible values of s and R. In particu-lar (2.6) holds if we let R vary such that R ~ Bm, where m is a(k-1)-vector:(2.7) ~ ~ (s,Bm) - ~ ~ (s) ~~(Bm).~'(6-B ),~ ~'(B-B )or equivalently,(2.8) ~ ~ (s.m) - ~ .~ (s) ~~,B(m).~'(S-B ,B) ~'(B-B )~rThus nonidentifiability of S implies the existence of a matrix (~-S ,B)~such that ~'.(B-B ) ís distributed normally and índependently ofJ ,~ ~r~'B. If rank (S-S ,B) - k then a matrix A is given by (s-S ,B). In the~ ~ ~trivial case where Rank (B-B ,B) - k-1, the variable ~~(B-S ) is distri-buted independently of itself and must therefore be equal to zero iden-tically (which is also considered as a normal distribution). In that~case any nonsingular matrix A whose first column equals ~-g wíll do.4To prove the necessity part of the Propositíon we assume thatthere exists a nonsingular matrix A-(a1,A2) such that ~~al is distri-buted normally and independently of ~~A2. If we substitute t- AR -a1R1 f A2R2 and B- AS - alsl } A2~2 (R1 and B1are scalars, R2 and 62are (k-1)-vectors) in (2.1), then the characteristic function of y~,x~takes the following form:(2.9) ~yJ~x,(s,AR) - exp{-}(o2s2tR.'A'S2AA.)} ~~(A(BsfR)).The characteristic function ~~(A(BstR)) can be rewritten as follows:(2.1(1) ~~(A(ssfA.)) - ~~,A(Bsi.k) - ~~,al(B1sfR1) ~~,A2(62sfX,2)- expl-~(BlsfRl)2Var(~'al)} ~~rA (62sfk2).2Using (2.1~), (2.9) carries over into(2.11) ~y ~x.(s1AR) - exp{-}(s,R')C(s,k')'} ~~,A (BZstk2) ,wherei t 2(2.12) C -a2 00 A'S2At Var(~'al)e B e e'The ~k(ktl) t 2 nonzero elements of C are functions of }k(k~-1) f 3 para-meters i n a2, St, 61 and Var(~'al), whereas the function ~~,A ( BZsi-RZ) is2not affected by these parameters. Clearly, different parameter values~enerate the same distribution of y~,x~. The existence of a nonsingulartransformation such that ~~al is distributed normally and independentlyof t;'A2 thus implies nonidentifiability of S. Q.E.D.753. DiscussionCompared to Kapteyn and Wansbeek's proposition, the sufficiency part ofthe proposition proved here is stronger. Nonidentifiability does notonly imply the existence of a normally distributed linear combina-tion ~~al , but also the existence of AZ such that ~~al and ~~A2 aremutually i ndependent. Consequently, the necessity part of their proposi-tion must be wrong, because they fail to invoke the existence of amatrix A2 such that ~~al and ~~A2 are mutually independent.As an example, consider the model with two regressors ~~1 andF~2, the first of which is normally distributed, ~~1 - N(O,a2), and thesecond i s a function of the first ~~z s~~1 - a2. According to Kapteynand Wansbeek this model would not be identified since ~~1 is normallydistributed. However, this point of view would be too pessimistic.Clearly there i s no nonsingular transformation ( al,a2) such that(~~l,~~Z)al is distributed independently of (~j1,~~2)a2 and so the modelis identified.References[1] Kapteyn, A. and T..J. Wansbeek: "Identification in the Linear Errorsin Variables Model", Econometrica 51 (1983), 1847-49.[2] Reíersbl, 0.: "Identifiability of a Linear Relation BetweenVariables Which are Subject to Error", Econometrica 18 (1950),375-389.[3] Willassen, Y.: "Extension of Some Results by Reiersbl to Multi-variate Models", Scandinavian Journal of Statistics 6(1979),89-91.iIN 1983 REF,DS VERSCHENF.N126 H.H. TigelaarIdentification of noisy linear systems with multíple arma inputs.127 J.P.C. KleijnenStatistical Analysis of Steady-State Simulations: Survey of RecentProgress.128 A.J. de ZeeuwTwo notes on Nash and Information.129 H.L. Theuns en A.M.L. Passier-GrootjansToeristische ontwikkeling - voorwaarden en systematiek; een selec-tief literatuuroverzicht.130 J. Plasmans en V. SomersA Maximum Likelihood Estimation Method of a Three Market Disequili-brium Model.131 R. van Montfort, R. Schippers, R. HeutsJohnson SU transformations for parameter estimation in arma-modelswhen data are non-gaussian.132 J. Glombowski en M. Kruger~n the R81e of Distribution in Different Theories of CyclicalGrowth.133 J.W.A. Vingerhoets en li.J.A. CoppensInternationale Grondstoffenovereenkomsten.Effecten, kosten en oligopolisten.134 W.J. OomensThe economic interpretation of the advertising effect of LydiaPinkham.135 J.P.C. KleijnenRegression analysis: assiunptions, alternatives, applications.136 J.P.C. Kleijnen~n the interpretation of variables.137 G. van der Laan en A.J.J. TalmanSimplicial approximation of solutions to the nonlinear complemen-tarity problem with lower and upper bounds.11IN 1984 REEDS VERSCHENEN138 C.J. Cuypers, J.P.C. Kleijnen en J.W.M. van RooyenTesting the Mean of an Asymetric Population:Four Procedures Evaluated139 T. Wansbeek en A. KapteynEstimation in a linear model with serially correlated errors whenobservations are missing140 A. Kapteyn, S. van de Geer, H. van de Stadt, T. WansbeekInterdependen[ preferences: an econometric analysis141 W..J.H. van GroenendaalDiscrete and continuous univariate modelling142 J.P.C. Kleijnen, P. Cremers, F. van BelleThe power of weighted and ordinary least squares with estimatedunequal variances in experimental design143 J.P.C. KleijnenSuperefficient estimation of power functions in simulationexperiments144 P.A. Bekker, D.S.G. PollockIdentification of linear stochastic models with covariancerestrictions.145 Max D. Merbis, Aart J. de 7.eeuwFrom structural form to state-space form146 T.M. Doup and A.J.J. TalmanA new variable dimension simplicial algorithm to find equilibria onthe product space of unit simplices.147 G. van der Laan, A.J.J. Talman and L. Van der HeydenVariable dimensíon algorithms for unproper labellings.148 G.J.C.'rh. van SchijndelDynamic firm behaviour and financial leverage clienteles149 M. Plattel, J. PeilThe ethicr political and theoretical reconstruction of contemporaryeconomic doctrínes150 F.J.A.M. Hoes, C.W. VroomJapanese Business Policy: The Cash Flow Trianglean exercise in sociological demystification151 T.M. Doup, G. van der Laan and A.J.J. TalmanThe (2TT}1-2)-ray algorithm: a new simplicial algorithm to computeeconomic equílibriaiiiIN 1984 RF.EDS VERSCHENEN (vervolg)l52 A.L. Hempenius, P.G.H. MulderTotal Mortality Analysis of the Rotterdam Sample of the Kaunas-Rotterdam Intervention Study (KRIS)153 A. Kapteyn, P. KooremanA disaggregated analysis of the allocation of time within thehousehold.154 T. Wansbeek, A. KapteynStatistically and Computationally Efficien[ Estimation of theGravity Model.155 P.F.P.M. NederstigtOver de kosten per zíekenhuisopname en levensduurmodellen156 R.R. MeijboomAn input-output like corporate model including multipletechnologies and make-or-buy decisions157 P. Kooreman, A. KapteynF.stimation of Ra[ioned and Unrationed Household Labor SupplyFunctions Using Flexible Functional Forms158 R. Heuts, J. van LieshoutAn implementation of an inventory model with stochastic lead time~ i ~ ~~ui~~~ ~iáiu~~~u i~ 7 000 O~ 059784 8

Comment on: Identification in the linear errors in variables model

    Tilburg UniversityComment on: Identification in the linear errors in variables modelBekker, P.A.Publication date:1984Document VersionPublisher's PDF, also known as Version of recordLink to publication in Tilburg University Research PortalCitation for published version (APA):Bekker, P. A. (1984). Comment on: Identification in the linear errors in variables model. (Research MemorandumFEW). Faculteit der Economische Wetenschappen.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portalTake 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.Download date: 12. May. 20211!CBMR7626 ~TIJDSCHRIFTENBUREAUBestemming B1BLI~~`?'I-IL':~K Nr.~y 11984 ~" I I ~ ~ ~ ~159 ~aIsubfaculteit der econometrieRESEARCH MEMORANDUMi q N ~u iiNiuiui iu g nuNiqHi ip~u~ u~ i I~~ --. -TILBURG UNIVERSITYDEPARTMENT OF ECONOMICSPostbus 90153 - 5000 LE TilburgNetherlandsKt~.T:I:~`LIIrKEHOv;::S:.."r:~~OL1iLBU4i;FEW159COMMENT ON: IDENTIFICATION IN THF.LLNEAK F.RRORS IN VARIARLES MODELAy Paul A. Bekkerl)1) The author wishes to thank Arie Kapteyn and Tom Wansbeek for theirstimulating discussions on the subject. Financial support by the Nether-lands Organization for the Advancement of Pure Research (ZWO) is grate-fully acknowledged.1. IntroductionKapteyn and Wansbeek [1] considered the following multiple linear re-gression model with errors in variables:(1.1) Yj - ~~B f Ej. (j z 1,...,n)(1.2) xj - ~j f uj,where ~j, xj, uj and S are k-vectors, yj, Ej are scalars. The ~j areunobservable variables: instead the xj are observed. The measurementerrors u j are unobservable as well and ít is assumed that v j - N(O,S2)and e j- N(~,a2) for all j. The v j and e j are mutually independent andindependent of ~j. The ~j are considered as random drawings from some,as yet unspecified, multivariate distribution with zero mean.For the case k-1 Reiersbl [2] has shown that normality of ~j is the onlydistributional assumption which spoils identification. For the case k~1 and the components of ~j are mutually independent, Wilassen [3] hasshown that none of the components of ~j should be normally distributedto ~uarantee identifiabílity of S. Kapteyn and Wansbeek [1] did notassume independency of the components of ~j and they stated the follo-wing proposition: the parameter vector S is identified if and only ifthere does not exist a lineair combination of ~j which is normallydístributed. The necessity part in this proposition is incorrect, i.e.it may well be that a normally distributed linear combination of ~j doesnot spoil the identifiability of B. Here I present necessary and suf-ficient conditions for identification of s.22. Statement of the Result and ProofProposition: Under the assumptions above, the parameter vector 6 isidentified if and only if there does not exist a nonsin-gular matrix A-(a1,A2) such that l;~al is distributed no r-mally and independently of ~~A2.Proof: We first show that nonidentifiability of S implies the existenceof the matrix A. Let s be a scalar and t a k-vector. The characteristicfunction ~y ~x ( s,t) of y) and x~ isJ J(2.1) ~y ~x (s,t) - exp{-}(a2s2tt'S2t)} ~~(Bstt) ,J ~where m~(.) is the characteristic functíon of ~~. Assuming that B is notfully identified amoun[s [o saying that there exíst parameter2sets {B,aZ,f2} and {p~,a~ ,52~}, with S~ B~, generating the same distri-bution of y~, x~. Consequently ~ x(s,t) should be the same for bothsets of parameters: y~~ ~2(2.z) exp{-~(a2s2ft'f2t)} ~~(psft) 3 exp{-}(a~ sZft'S2~t)} m~(s~stt).~ ~1.et R- S sft, then ~~(Ssft) -~~(((3-B )sflC) L~ ~ (s,R). Thus~~ca-s ),~(2.2) caries over into2(2.3) ~ ~ ( s,R) - exp{-}Is2(a~ -a2) fF'(s-s ),~t (R-B~s)'(R~-S2)(k-S~s)]} ~~(k).The characteristic function corresponding to the marginal distribution~of ~~(S-g ) is found by setting R- 0J3~2 ~~ 2 ~, ~ ~(2.4) ~ ~ ( s) - exp{-~s (a -a fs ( S2 -S2)S )} ,~' (s-a )which is the characteristic function of a normally distributed variable.In addition [o this result, which was obtained by Kapteyn andWansbeek [1], it wíll now be shown that nonidentifiability of S alsoimplies the existence of a matrix AZ such that ( a1,A2) is nonsingularand ~'al i s distributed i ndependently of ~~AZ. The characteristic func-Jtion corresponding to the marginal distributions of ~~ is found by set-ting s-0 in (2.3):(2.5) ~~(R) - exp{-~R.'(St~-St)k} ~~(R).Thus, we may rewrite (2.3) as(2.6) ~ ~ (s,R) - ~ ~ (s) ~~(k) exp{sB~'(St~-St)k}.~'(B-S ).~ ~'(B-B )~~ ~Let B be a(kx(k-1))-matrix of full column rank such that B (S2 -S2)B -0. F.quality (2.6) holds for all possible values of s and R. In particu-lar (2.6) holds if we let R vary such that R ~ Bm, where m is a(k-1)-vector:(2.7) ~ ~ (s,Bm) - ~ ~ (s) ~~(Bm).~'(6-B ),~ ~'(B-B )or equivalently,(2.8) ~ ~ (s.m) - ~ .~ (s) ~~,B(m).~'(S-B ,B) ~'(B-B )~rThus nonidentifiability of S implies the existence of a matrix (~-S ,B)~such that ~'.(B-B ) ís distributed normally and índependently ofJ ,~ ~r~'B. If rank (S-S ,B) - k then a matrix A is given by (s-S ,B). In the~ ~ ~trivial case where Rank (B-B ,B) - k-1, the variable ~~(B-S ) is distri-buted independently of itself and must therefore be equal to zero iden-tically (which is also considered as a normal distribution). In that~case any nonsingular matrix A whose first column equals ~-g wíll do.4To prove the necessity part of the Propositíon we assume thatthere exists a nonsingular matrix A-(a1,A2) such that ~~al is distri-buted normally and independently of ~~A2. If we substitute t- AR -a1R1 f A2R2 and B- AS - alsl } A2~2 (R1 and B1are scalars, R2 and 62are (k-1)-vectors) in (2.1), then the characteristic function of y~,x~takes the following form:(2.9) ~yJ~x,(s,AR) - exp{-}(o2s2tR.'A'S2AA.)} ~~(A(BsfR)).The characteristic function ~~(A(BstR)) can be rewritten as follows:(2.1(1) ~~(A(ssfA.)) - ~~,A(Bsi.k) - ~~,al(B1sfR1) ~~,A2(62sfX,2)- expl-~(BlsfRl)2Var(~'al)} ~~rA (62sfk2).2Using (2.1~), (2.9) carries over into(2.11) ~y ~x.(s1AR) - exp{-}(s,R')C(s,k')'} ~~,A (BZstk2) ,wherei t 2(2.12) C -a2 00 A'S2At Var(~'al)e B e e'The ~k(ktl) t 2 nonzero elements of C are functions of }k(k~-1) f 3 para-meters i n a2, St, 61 and Var(~'al), whereas the function ~~,A ( BZsi-RZ) is2not affected by these parameters. Clearly, different parameter values~enerate the same distribution of y~,x~. The existence of a nonsingulartransformation such that ~~al is distributed normally and independentlyof t;'A2 thus implies nonidentifiability of S. Q.E.D.753. DiscussionCompared to Kapteyn and Wansbeek's proposition, the sufficiency part ofthe proposition proved here is stronger. Nonidentifiability does notonly imply the existence of a normally distributed linear combina-tion ~~al , but also the existence of AZ such that ~~al and ~~A2 aremutually i ndependent. Consequently, the necessity part of their proposi-tion must be wrong, because they fail to invoke the existence of amatrix A2 such that ~~al and ~~A2 are mutually independent.As an example, consider the model with two regressors ~~1 andF~2, the first of which is normally distributed, ~~1 - N(O,a2), and thesecond i s a function of the first ~~z s~~1 - a2. According to Kapteynand Wansbeek this model would not be identified since ~~1 is normallydistributed. However, this point of view would be too pessimistic.Clearly there i s no nonsingular transformation ( al,a2) such that(~~l,~~Z)al is distributed independently of (~j1,~~2)a2 and so the modelis identified.References[1] Kapteyn, A. and T..J. Wansbeek: "Identification in the Linear Errorsin Variables Model", Econometrica 51 (1983), 1847-49.[2] Reíersbl, 0.: "Identifiability of a Linear Relation BetweenVariables Which are Subject to Error", Econometrica 18 (1950),375-389.[3] Willassen, Y.: "Extension of Some Results by Reiersbl to Multi-variate Models", Scandinavian Journal of Statistics 6(1979),89-91.iIN 1983 REF,DS VERSCHENF.N126 H.H. TigelaarIdentification of noisy linear systems with multíple arma inputs.127 J.P.C. KleijnenStatistical Analysis of Steady-State Simulations: Survey of RecentProgress.128 A.J. de ZeeuwTwo notes on Nash and Information.129 H.L. Theuns en A.M.L. Passier-GrootjansToeristische ontwikkeling - voorwaarden en systematiek; een selec-tief literatuuroverzicht.130 J. Plasmans en V. SomersA Maximum Likelihood Estimation Method of a Three Market Disequili-brium Model.131 R. van Montfort, R. Schippers, R. HeutsJohnson SU transformations for parameter estimation in arma-modelswhen data are non-gaussian.132 J. Glombowski en M. Kruger~n the R81e of Distribution in Different Theories of CyclicalGrowth.133 J.W.A. Vingerhoets en li.J.A. CoppensInternationale Grondstoffenovereenkomsten.Effecten, kosten en oligopolisten.134 W.J. OomensThe economic interpretation of the advertising effect of LydiaPinkham.135 J.P.C. KleijnenRegression analysis: assiunptions, alternatives, applications.136 J.P.C. Kleijnen~n the interpretation of variables.137 G. van der Laan en A.J.J. TalmanSimplicial approximation of solutions to the nonlinear complemen-tarity problem with lower and upper bounds.11IN 1984 REEDS VERSCHENEN138 C.J. Cuypers, J.P.C. Kleijnen en J.W.M. van RooyenTesting the Mean of an Asymetric Population:Four Procedures Evaluated139 T. Wansbeek en A. KapteynEstimation in a linear model with serially correlated errors whenobservations are missing140 A. Kapteyn, S. van de Geer, H. van de Stadt, T. WansbeekInterdependen[ preferences: an econometric analysis141 W..J.H. van GroenendaalDiscrete and continuous univariate modelling142 J.P.C. Kleijnen, P. Cremers, F. van BelleThe power of weighted and ordinary least squares with estimatedunequal variances in experimental design143 J.P.C. KleijnenSuperefficient estimation of power functions in simulationexperiments144 P.A. Bekker, D.S.G. PollockIdentification of linear stochastic models with covariancerestrictions.145 Max D. Merbis, Aart J. de 7.eeuwFrom structural form to state-space form146 T.M. Doup and A.J.J. TalmanA new variable dimension simplicial algorithm to find equilibria onthe product space of unit simplices.147 G. van der Laan, A.J.J. Talman and L. Van der HeydenVariable dimensíon algorithms for unproper labellings.148 G.J.C.'rh. van SchijndelDynamic firm behaviour and financial leverage clienteles149 M. Plattel, J. PeilThe ethicr political and theoretical reconstruction of contemporaryeconomic doctrínes150 F.J.A.M. Hoes, C.W. VroomJapanese Business Policy: The Cash Flow Trianglean exercise in sociological demystification151 T.M. Doup, G. van der Laan and A.J.J. TalmanThe (2TT}1-2)-ray algorithm: a new simplicial algorithm to computeeconomic equílibriaiiiIN 1984 RF.EDS VERSCHENEN (vervolg)l52 A.L. Hempenius, P.G.H. MulderTotal Mortality Analysis of the Rotterdam Sample of the Kaunas-Rotterdam Intervention Study (KRIS)153 A. Kapteyn, P. KooremanA disaggregated analysis of the allocation of time within thehousehold.154 T. Wansbeek, A. KapteynStatistically and Computationally Efficien[ Estimation of theGravity Model.155 P.F.P.M. NederstigtOver de kosten per zíekenhuisopname en levensduurmodellen156 R.R. MeijboomAn input-output like corporate model including multipletechnologies and make-or-buy decisions157 P. Kooreman, A. KapteynF.stimation of Ra[ioned and Unrationed Household Labor SupplyFunctions Using Flexible Functional Forms158 R. Heuts, J. van LieshoutAn implementation of an inventory model with stochastic lead time~ i ~ ~~ui~~~ ~iáiu~~~u i~ 7 000 O~ 059784 8

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