Miscellanea Miscellanea

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1 Miscellanea Miscellanea Miscellanea Miscellanea Miscellanea CENRAL EUROPEAN REVIEW OF ECONOMICS & FINANCE Vol., No. (4) pp. -6 bigniew Śleszński USING BORDERED MARICES FOR DURBIN WASON D SAISIC EVALUAION Absrac In his paper he sage of bordered marices for Drbin-Wason d saisic evalaion in linear ime series model is presened. I is shown how o obain his saisic wiho esimaion of srcral parameers and vecor of residals. As an example he model of GDP growh in Poland, basing on empirical daa from 99 3 is shown. JEL Classificaion Code: C, C. Kewords: Aocorrelaion in he residals, Drbin Wason es, bordered marices. Inrodcion Le s consider he linear economeric model for empirical ime series daa. While esimaing model parameers sing leas sqares mehod we shold es if obained esimaors are minimm-variance nbiased esimaors (he bes in class of nbiased linear esimaors). wha means he are effecive (Gajda, 4). One of he main condiions for sing leas sqares mehod is no aocorrelaion in model residals. he mos poplar es, sed also for esing coinegraion among variables (Charemza, Deadman, 997), deecing he presence of firs order aocorrelaion, is Drbin-Wason es. I deermines if firs order aocorrelaion coefficien of residals is significanl differen from zero. While comping Drbin-Wason d saisic we have o know he vecor of residals, hs we have o esimae leas sqares parameers of he model o avoid sch procedre we will show how o esimae d saisic sing bordered marices, wiho model parameers and residals esimaion. Assisan Professor, Ph.D., K. Plaski Universi of echnolog and Hmaniies in Radom, Poland, Facl of Economics, Deparmen of Inernaional Bsiness & Finance.

2 Cenral Eropean Review of Economics & Finance Basic formlas An economeric model is considered: Y α + α + α α + ξ k k () assme we have empirical daa from n ime periods: [ z j ] and [ ] nx nx(k+ ) Denoing esimaed srcral parameers for model () sing leas sqared mehod as: we have: A [ a a a ] k (k+ ) () (3) ( ) A (4) Vecor of heoreical vales of dependen variable Y has he form: Vecor of residals is comped as follows: A () A (6) According o Drbin-Wason es, while esing for firs order aocorrelaion in model residals, we have o evalae d saisic: n n ( ) d (7) I is known ha (Kolpa i Śleszński, ): Le s denoe: n n 3 ( ) n (n) (8) (9)

3 . Śleszński, Using bordered marices for Drbin-Wason d... 3 Basing on (9) we have: ( n) () where vecors (-) and (-n) are obained from vecor b dropping respecivel he firs and he las componen. Denoing marices obained from marix given b () b dropping respecivel he firs and he las row b (-) and (-n) and vecors obained from vecor given b () b (-) and (-n) hen basing on (6) we have: Denoing: A ( ) A ( ) n n n n ( n) n () () ( n) (3) basing on (), sing (), () and (3) we have: A ( ) I s worh adding we have assmed ha here is inercep in he model (he firs colmn in marix conains onl ones and marix ( ) has zeros in firs colmn). Le s noice ha o evalae Drbin-Wason d saisic given wih eqaion (7) i is enogh o compe qoien of sqared componens of vecor ( ) given wih (4) b sm of sqares of residals given wih (8). Using bordered marices for and (-) evalaion In order o evalae (8) and (4) we will se bordered marices. For he prposes of his paper le s se slighl modified definiion of bordered marix. Given is a marix F [f ij ] wih p rows and q colmns, p, q, m < min{p-, q}. Marix F divided ino blocks according o scheme A B F C D () C D where inner marix A [a ij ] is a sqare non-singlar marix of order m, is called bordered marix. (4)

4 4 Cenral Eropean Review of Economics & Finance I is known ha (Kolpa, Śleszński, ) doing elemenar ransformaions on elemens of marix F given wih () sch ha in place of inner marix A we obain pper rianglar marix wih diagonal elemens eqal o one, in place of marix C and C zero marices, hen in place of D and D we obain D ~ and D ~ saisfing: D ~ D C A B D ~ (6) D C A B In order o deermine d saisic given wih (7) we will s bordered marix: F (7) Doing elemenar ransformaions on F, basing on (6), sing (8) and (4) we will obain: F ~ ( ) ( ) (k+ ) ( n) (k+ (8) Received in marix (8) nmber and vecor ( ) can be sed o compe Drbin Wason d saisic according o eqaion (7). In he nex par pracical example of described procedre will be shown. Example Le s consider a model: αz + αz + αz + α3z3 + α4z4 + ξ (9) Where: Y growh of GDP in Poland (in percens), variable idenicall eqal o, inflaion in Poland dring previos ear (in percens), growh of capial expendires in Poland dring previos ear (in percens), 3 binar variable eqal o dring , zero in oher cases, 4 binar variable eqal o dring 7, zero in oher cases. In he model empirical daa from saisical earbooks of Cenral Saisical Office from ears 99 3 will be sed, i means nmber of observaions n 3. Empirical daa is shown in able.

5 . Śleszński, Using bordered marices for Drbin-Wason d... able. Vales of model (9) variables z z z z 3 z ,8 -, 99,6 7,3-4, 993 3,8 43,4 994, 3,3, , 8, ,8 7, 997 6,8 9,9 9, 998 4,8 4,9, 999 4,,8,3 4 7,3,9,,4,4, -9, 3 3,8,9-4 4,8,6, 3, 6, 6 6,, 7,7 7 6,6 6,8 8 4,8,,4 9,6 4,,7 3,9 3, -,8 4,3,6,,9 4,3,6 3,6 3,7 -,8 Sorce: saisical earbooks of Cenral Saisical Office. Basing on daa from able we esimae marices needed for bordered marix (7) consrcion. We have: , ,3 43,78 8, 4,9 8, 43,78 84, 47, 3, 8, 47, 7 4,9 3, 7 ()

6 6 Cenral Eropean Review of Economics & Finance [ ] 494, 8,4 8,8 83,8 769,7 83,8 (),3,4,4,3 3,,8,,6,,,4,4 3,,7,8,8,4, 9,6 3,4,6,4,7,9,,7 9,7,7 3,6, 9,,,,4,9,7,6,, 3,6,9 4,6 4,,8 9,4 4, 6,9 3, 3, 7,9 9 4,4,8 3,,9 7,7 4, 7,3 6,7 () We can now consrc bordered marix F given wih (7).

7 . Śleszński, Using bordered marices for Drbin-Wason d... 7,3 3,4,6,4,4,7,4,9,3,,7 3, 9,7,7,8 3,6,, 9,,,6,,4,,9,7,,6,,4, 3,6,4,9 4,6 3 4,,8, 9,4 4,,7 6,9 3, 3,8, 7,9 9 4,4,8,8 3,,4,9 7,7, 4, 7,3 9,6 6, 494, 8,4 8,8 83,8 769,7 83,8 8,4 7 3, 4,9 7 8,8 47, 8, 83,8 3, 47, 84, 43,78 8, 769,7 4,9 8, 43, , ,8 7 8, F (3)

8 8 Cenral Eropean Review of Economics & Finance On marix (3) we make elemenar ransformaions according o schema (8). As a resl we receive:,99 3,38,9 3,4,4,99,4,46,97,84,39,39 3,66,74,8,7,48,94,9,,7,34 33,784,4,6,33,,,,,8,,3 3,64,3,,7 38,87 F ~ (4) Basing on marix (4) we compe: , 3 ()

9 . Śleszński, Using bordered marices for Drbin-Wason d... 9 Finall, according o eqaion (7), we receive Drbin-Wason saisic: 3 d 3 ( ) 66,866, ,7838 (6) Even wiho checking in ables of Drbin-Wason es criical vales, according o rle of hmb, as d is close o, we can sa ha here is no firs order aocorrelaion in considered model (Górecki ). A he end, o confirm correcness of given compaions, le s have a look a a able from Grel program, where in pariclar d saisic is shown. able. Model : OLS, sing observaions 99-3 ( 3), Dependen variable: Y Coefficien Sd. Error -raio p-vale Mean dependen var S.D. dependen var.933 Sm sqared resid S.E. of regression R-sqared.8 Adjsed R-sqared.7848 F(4, 8).663 P-vale(F).7e-6 Log-likelihood Akaike crierion Schwarz crierion Hannan-Qinn 8.43 rho -.68 Drbin-Wason.9794 Sorce: op from Grel. Conclsions Presened procedre show ha i is possible o compe Drbin-Wason d saisic wiho esimaion of model parameers and vecor of residals. I is imporan as ver ofen deecing aocorrelaion in residals means necessi o specif model once again, hs resls of previos esimaions are seless. Using bordered marices makes i possible o compe sm of sqared residals and vecor of residals differences, and hs saisic d, wha makes compaional process shorer. I is worh noing ha elemenar ransformaions, even wih hge bordered marix, are eas o perform sing for example spreadshee.

10 6 Cenral Eropean Review of Economics & Finance Reference Charemza W. W., Deadman D. F Nowa ekonomeria. Warszawa: Polskie Wdawnicwo Ekonomiczne. Gajda J. B. 4. Ekonomeria Wkład i ławe obliczenia w programie komperowm. Warszawa: Wdawnicwo C.H. Beck. Górecki B. J.. Ekonomeria podsaw eorii i prakki. Warszawa: Wdawnicwo Ke ex. Kolpa M., Śleszński.. Algebra macierz brzegowch z zasosowaniami. Warszawa: Wdawnicwo C.H. Beck.