M I S C E L L A N E A Mchał Kolupa, bgnew Śleszyńsk SOME EMAKS ON COINCIDENCE OF AN ECONOMETIC MODEL Absrac In hs paper concep of concdence of varable and mehods for checkng concdence of model and varables are presened. Parcularly Hellwg s hypohess and mehods for consrucng model wh dfference compensaors are descrbed. I makes possble keepng non concdenonal varables n model. Keywords: marx, varables, vecor, coeffcen, economerc model, concdence JEL: C Problem of concdence of an economerc model s very mporan. Usually lack of varable concdence means lack of possbly for correc nerpreaon of srucural parameers esmaon. Many economercans were workng wh hs problem. Ths arcle descrbes he mos mporan achevemens concernng concdence of varable and model. In he paper an economerc model s consdered Y e k k () Is varables are sandardzed. Gven s a marx of observaons and all varables of he model () Q y () where marx s of order k n (s rank equals k) and y s an n dmensonal vercal vecor. Because varables of he model () are sandardzed hence: n T T k y k (3) n The marx (k) and he vecor (k) are marx and vecor of correlaon coeffcens beween varables n par respecvely, and, Y, =, k. In oher words coeffcens r r, componens of he vecor (k). are elemens of he marx (k) and r ry, Henceforh we shall alk abou a par of correlaon k k (see (Hellwg, 976)) are,. I s a regular par f 39
r k Mchał Kolupa, bgnew Śleszyńsk r r (4) The par of correlaon exss f and only f, 987) r where (Hauke & Pomanowska, T r k k k k (5) Ths coeffcen s measurng qualy of model () or he correlaon par k k,. Esmaon of a of parameers of he model () obaned by LSM are componens of he vecor A(k) sasfyng he sysem: k A k k (6) The problem of concdence of he model () (or of he correlaon par k k been gven by. Hellwg n hs paper (Hellwg, 976)., ) has Defnon The explanaory varable k of he model () has he propery of concdence f sgn a sgn (7) r Where a and r are componens of vecors respecvely A(k) and (k). Defnon If he relaon (7) s sasfed for all I =, k hen he model () (or he correlaon par k, k ) has he propery of concdence. Of course, f he par k k sgn, s regular hen (7) leads o a (8) In oher words he model () (or he correlaon par k k, ) has he propery of concdence f and only f all componens of he vecor A(k) are posve. M. Kolupa has proved ha he explanaory varable k of he model () has he propery of concdence f and only f r (9) Where s he h row of marx (k) whou s h elemen, marx s obaned by droppng he h row and h column of marx (k) and he vecor s creaed by droppng he h componen of he vecor (k) (Kolupa, 98). In order o ulze he nequaly (8) we use a bordered marx U () r * The marx U s ransformed no marx U by elemenary ransformaons as follows: 4
Some emarks on Concdence of an Economerc Model - he marx s ransformed o an upper rangular one wh s dagonal elemens equal - he vecor s ransformed o a zero vecor Hence U where U d r d () If d > (d < ) hen explanaory varable has he propery of concdence (has no hs propery). The heory of bordered marces and her applcaons s gven n monograph (Kolupa & Szczepańska-Gruźlewska, 99). The propery of concdence of he model () allows o gve a proper economcal nerpreaon of esmaors of he model s parameers. I explans he neres aken for he concdence propery.. Hellwg has defned an unversal marx n paper (Hellwg, 976) as follows. Defnon 3 G k where g Marx k k for g (3) r r for s called an unversal one. The properes of hs marx are gven n (Kolupa, 98). In 976. Hellwg has gven he followng hypohess: If Ik k Gk (4) where I(k), (k), G(k) are marces of order k k respecvely un, correlaon and unversal, hen an economerc model descrbed by a regular correlaon par k, k has he propery of concdence. Many neresng resuls concerned wh Hellwg s hypohess have been obaned and hey are presened n (Kolupa, 98). Hellwg s hypohess was proved by M. Kolupa (see (Kolupa, 996)). Because he propery of concdence s very mporan for pracce s he reason for reang. Ths propery s reached by elmnaon from he model () of non concdenonal varables. I s no good procedure as can be seen from he followng example. Le us consder a model gven par of correlaons: () 4
, 3, Mchał Kolupa, bgnew Śleszyńsk 3,,,3 (5) Ths model s non concdenal, because: a sgn a sgn a (6) sgn 3 The explanaory varable s elmnaed from he model (5). We obaned a model descrbed by a new par:,,3 (7) whch has propery of concdence. The same resul can be obaned by elmnaon from he model (5) of varable 3 whch has he propery of concdence n he model gven by he par (4).. Hellwg has nroduced a new concep how o guaranee he propery of concdence of he model () whou elmnaon of explanaory varables whch have no he propery of concdence (Hellwg, 987). Whou loss of generaly we suppose he non concdenal varables n he model () are, f and he concdenal ones are f + k. Among varables f + k we pck ou one for example f k and pu: v v for, f (8) for, f, (9) We oban a new model descrbed by he par k k,. Is explanaory varables are ones gven n (8) and (9). The varables gven by (8) are called Hellwg s dfference compensaors. For hs reason he model descrbed by he par k, k s called a model wh compensaors (Kolupa & Śleszyńsk, 989). Is qualy s gven by he r k where T k k k k coeffcen r () Le b I =, k denoe componens of he vecor B(k) sasfyng he sysem k Bk k () Hence (Kolupa & Marcnkowska-Lewandowska & adzo, 99), (Kolupa & adzo, 99) (adzo, 99) b d a, f () b b where a, f, k (3) a a a f (4) 4
d,, Some emarks on Concdence of an Economerc Model r (5) From () (3) and (4) can be seen ha he model descrbed by he par k, k has he propery of concdence f and only f a a a f a (6) r k k, k and k k k r k Coeffcens r (k) and calculaed for he correlaon pars respecvely, are equal: r (7) A and, we shall gve some remarks abou leraure concerned wh he problem of concdence of he model (). The frs paper was Hellwg s work (Hellwg, 976) n 976. I caused a grea neres among many Polsh economercans. Today here exss over 5 works concerned wh he problem of concdence of he model (). We have quoed only some of hem whch were drecly nvolved n our problem. eferences Hauke J. & Pomanowska J. (987). wązk korelacyne w śwele kryerum neuemne określonośc macerzy blokowe. Przegląd Saysyczny, 3, 9 4. Hellwg. (987). Model z kompensaorem różncowym. Przegląd Saysyczny,, 3 7. Hellwg. (976). Przechodność skorelowana zmennych losowych płynące sąd wnosk ekonomeryczne. Przegląd Saysyczny,, 3. Kolupa M. (996). Dowód hpoezy. Hellwga. Przegląd Saysyczny, 3, 7-34. Kolupa M. (98). O pewnych własnoścach macerzy unwersalne. Przegląd Saysyczny, 3, 76-84. Kolupa M. & Śleszyńsk. (989). O koncydennośc kompensaora różncowego. Przegląd Saysyczny,, 3-7. Kolupa M. & Szczepańska-Gruźlewska E. (99). Macerze brzegowe, eora, algorymy, zasosowana ogólne ekonomeryczne. Warszawa: PWE. Kolupa M. & Marcnkowska-Lewandowska W. & adzo A. (99). Koncydenca model ekonomerycznych eora zasosowana. Insyu Cyberneyk arządzana. Warszawa: SGPS. adzo A. (99). Kompensaory różncowe sudum eoreyczne, Monografe opracowana. Warszawa: SGPS. Kolupa M. & adzo A. (99). O własnoścach kompensaora różncowego. Przegląd Saysyczny,, 37 4. 43