Kayode Ayinde Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology P. M. B. 4000, Ogbomoso, Oyo State, Nigeria

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1 Journal of Mahemacs and Sascs 3 (4): 96-, 7 ISSN Scence Publcaons A Comparave Sudy of he Performances of he OLS and some GLS Esmaors when Sochasc egressors are boh Collnear and Correlaed wh Error Terms Kayode Aynde Deparmen of Pure and Appled Mahemacs, Ladoke Aknola Unversy of Technology P. M. B. 4, Ogbomoso, Oyo Sae, Ngera Absrac: The Classcal Lnear egresson Model assumes ha regressors are non sochasc, ndependen and uncorrelaed wh he error erms. These assumpons are no always enable especally where regressors are no ofen assumed fed n repeaed samplng. In hs paper, wh sochasc regressors, he performances of he Ordnary Leas Square (OLS) and some Generalzed Leas Square (GLS) esmaors are nvesgaed and compared under varous degree of non valdy of mulcollneary and correlaon beween regressor and error erms assumpons hrough Mone Carlo sudes a boh low and hgh replcaons. The mean squared error creron s used o eamne and compare he esmaors. esuls show ha he performances of he esmaors mproved wh ncreased replcaon. The ML and MLGD (GLS) esmaors compare favorably wh he OLS esmaor wh low replcaon. However wh ncreased replcaon, he OLS mehod s preferred among he esmaors n esmang all he parameers of he model n all level of correlaons. Key words: Sochasc egressors, Mulcollneary, Correlaon beween Sochasc egressor and Error Terms, OLS esmaor, Feasble GLS esmaors INTODUCTION egressors are assumed o be non sochasc (fed n repeaed samplng), ndependen and uncorrelaed wh he error erms n he Classcal Lnear egresson Model (CLM). These assumpons are no always sasfed especally n busness, economcs and socal scences. Auhors lke Neer and Wasserman [], Fomby e.al [], Maddala [3] have no only gven suaons and nsances where hese assumpons may be volaed bu have also dscussed her consequences on he Ordnary Leas Square (OLS) esmaor when used o esmae he model parameers.graybll [4], Sampson [5] [], Fomby e.al and many ohers emphaszed ha f regressors are sochasc and ndependen of he error erms; he OLS esmaor s unbased and has mnmum varance even hough s no Bes Lnear Unbased Esmaor (BLUE). They also poned ou ha he radonal hypohess esng s vald f he error erms are furher assumed normal bu modfcaon would be requred n he area of confdence nerval calculaed for each sample and he power of he es. When regressors are dependen (.e. here es mulcollneary), he OLS esmaes are sll unbased as long as mulcollneary s no perfec [6]. However when mulcollneary s hgh, only mprecse esmae may be avalable abou he ndvdual rue 96 regresson coeffcens whch are ofen sascally nsgnfcan because of s large sandard errors [7]. Neer and Wasserman [], Maddala [3] arbued a source of correlaon beween regressors and error erms o measuremen errors n he regressors. They noed ha f he OLS esmaor s appled o he CLM of hs form, he esmaes are no only bas bu lack propery of conssency. Maddala [3] emphaszed ha hs does no mply ha nferences abou he model parameers are no possble. Wh non sochasc regressors, OLS esmaor of gven as ( X X ) X Y () has been proved o be BLUE wh varance covarance mar of ( X X ) gven as V σ () [8]. When all he assumpons of he CLM hold ecep ha he error erms are no homoscedasc (.e. E( UU ) σ I ) bu are heeroscedasc n (.e. E( UU ) σ Ω ), he resulng model he Generalzed Leas Squares (GLS) Model. Aken [9]

2 J. Mah. & Sa., 3 (4): 96-, 7 has shown ha he GLS esmaor of gven as correlaon wh.e g( ( X Ω X ), ) (7) X Ω Y (3) (3) s effcen among he class of lnear unbased esmaors of wh varance covarance mar of gven as ( X Ω X ) V σ (4) where Ω s assumed o be known. However, Ω s no always known, s ofen esmaed by Ω o have wha s known as Feasble GLS esmaor. Many conssen esmaes of Ω can be obaned []. Wh frs order auocorrelaed error erms (A ()), among he Feasble GLS esmaors n leraures are he Cochrane and Orcu esmaor [], Hldreh and Lu esmaor [], Pras Wnsen esmaor [], Thornon esmaor [3], Durbn esmaor [4], Thel s esmaor [5], he Mamum Lkelhood esmaor and he Mamum Lkelhood Grd esmaor [6]. Some of hese esmaors have now been ncorporaed no Whe s SHAZAM program [7] and he new verson of he me seres processor (TSP) [8]. However, all of hese esmaors are known o be asympocally equvalen bu he queson on whch s o be preferred n small samples s he worry of researchers []. Assumng no auocorrelaon of he error erms, we eamne and compare he performances of some of hese Feasble GLS esmaors wh ha of he OLS esmaor when sochasc regressors are boh collnear and correlaed wh he error erms;and also denfy he esmaor whch s prefferred n esmang all he parameers of he CLM n all he levels of hese correlaons. MATEIALS AND MEHODS Consder he CLM wh sochasc regressors of he form y + e + + (5) where,,..., n ε ~ N (, σ ). correlaon wh e,.e f ( e, ) (6) 97 OLS esmaor dscussed earler can be used o oban esmaes of he model parameers. Also, consder he GLS model wh sochasc regressors and A () of he form Where u y + u + + u + ε (8) ( ),,..., n ε ~ N, σ s sad o have correlaon wh u,.e f ( ) u, (9) correlaon wh.e g( ), () Is parameer esmaon can be done usng he (feasble) GLS mehods. However for he purpose of comparson, model (8) s made o be equvalen wh model (5) by seng. Thus, he performances of he OLS esmaor and he followng feasble GLS esmaors were suded under model (5): Cochrane Orcu (COC), Hldreh - Lu (HILU), Mamum Lkelhood (ML) and he Mamum Lkelhood Grd (MLGD) esmaors. X ~ N µ, σ,. If Now, suppose ( ) hese varables are correlaed, hen X and X can be generaed wh he equaons X µ + σz () X µ + σ z + σ z where ~ N(,), and Z s he value of he correlaon beween he wo varables [9]. Mone Carlo epermens were performed for n, a small sample sze represenave of many me seres sudy [], wh four replcaon () levels (, 4, 8, ) and nne varous degree of mulcollneary and correlaon beween regressor and error erms (.e and -.99,-.75,-.5,.99 respecvely) ulzng equaon (5), (6) and (7). A a parcular choce of, and (a scenaro), each replcaon was frs obaned by generang e ~ N(,). Ne, ~ (,) was generaed N

3 J. Mah. & Sa., 3 (4): 96-, 7 usng equaon () havng correlaon e ~ N(,) as () ε + z ~ N(, wh Furhermore, ) was generaed usng equaon () havng correlaon ~ N(,) The values of as (3) + z3 wh y n equaon (5) were also calculaed by seng he rue regresson coeffcens as. Ths process connued unl all replcaons n hs scenaro were obaned. Anoher scenaro hen sared unl all he scenaros were compleed. Evaluaon and comparson of esmaors were eamned usng a creron whch conans boh bas and varance, he mean squared error (MSE) creron. Mahemacally, for any esmaor (5) MSE j (4) Var + B where Var, j B j j of of model and for,, and j,,,. For each of he esmaon mehods, a compuer program was wren usng TSP sofware o esmae he model parameers and o evaluae he creron. The four replcaon levels were furher grouped no low (, 4) and hgh ( 4, 8) and he effec of he correlaons on he performances of he esmaors (mehod) were eamned va he Analyss of Varance of he crera of each of he model parameers n he wo replcaon groups. Ths was also accomplshed by he LSD es of he esmaed margnal means of he hghes neracon effec wh _ mehod ha s sascally sgnfcan. A a parcular se of levels of correlaon, he esmaed margnal means of esmaors were preferred f hey are no sgnfcanly dfferen from he mos preferred one. An esmaor s mos preferred f s esmaed margnal mean s he smalles. SIMULATION ESULTS AND DISCUSSIONS The summary of our fndngs on he performances of he esmaors based on he crera for each of he model parameers n he wo replcaon groups s gven n able. In able he Analyss of Varance able s presened. Table : Summary of he ANOVA TABLE showng he sum of squares of he model parameers n he wo replcaon group G S d.f MSE ( ) MSE ( ) MSE ( ) 8.39* 33.58* 36.66* L * 3.843* M * 7.* * * O M 3.9* M * 9.57* M W EO TOTAL H 8.97* 3.39* * * * I M 4.3.3*.9* * 88.89* G M 3.*.843*.799* M * 6.3* M *.66* H EO TOTAL eplcaon Group, S Sources of varaon, d.f Degree of freedom * Compued F value s sgnfcan a.,, and M Mehod (Esmaor) From able, s observed ha he error sum of square and hence he mean square error (f esmaed) of all he esmaed parmeers reduce wh ncreased replcaons. Thus, he performances of he esmaors mprove wh ncreased replcaon. Also, he mulcollneary effec and any neracon effec wh mulcollneary are compleely nsgnfcan a he wo replcaon groups n esmang. 98

4 J. Mah. & Sa., 3 (4): 96-, 7 A he low replcaon group, he neracon effec of * M s sgnfcan n esmang whle ha of * M s sgnfcan n esmang and. Hence, he performances of he esmaors are affeced by correlaon beween regressor and error erms n esmang and by mulcollneary n esmang and. Ther esmaed margnal means can be found n he sudy done by Aynde []. From he sudy, was observed ha he esmaed margnal means of decrease as ncreases whle ha of and ncrease as ncreases. Also, he performances of he OLS, ML and he MLGD esmaors are no sgnfcanly dfferen from one anoher a all he levels of correlaon. Furhermore, from able a he hgh replcaon group, he neracon effec of * M s sgnfcan n esmang whle ha of * * M s sgnfcan n esmang and. Hence, he performances of he esmaors are affeced by correlaon beween regressor and error erms n esmang and by he jon effec of mulcollneary and correlaon beween regressor and error erms n esmang and. From her esmaed margnal means [], s observed ha n esmang he OLS mehod s mos effcen n all he levels of correlaon. Is esmaed margnal means decrease as ncreases. However, he GLS mehods compee wh he OLS when. In esmang and, he esmaed margnal means ofen ncrease n all he levels of as ncreases. Ecep when and. and when 75, and he OLS esmaor may ofen be mos effcen even hough s performances are no sgnfcanly dfferen from oher GLS esmaors. When and, he GLS esmaors compee favorably wh he OLS esmaor; and. and when 75 he OLS esmaor s no only mos effcen bu s performances s sgnfcanly dfferen from ohers. Therefore, can 99 be nferred ha he OLS esmaor s conssenly preferred n esmang all he model parameers a all he levels of correlaons. CONCLUSION Among he four GLS esmaors eamned, he ML and MLGD esmaors can only compee wh he OLS esmaor when replcaon s low. However, wh ncreased replcaon, he OLS esmaor s mos preferred among he esmaors n esmang all he model parameers a all he levels of correlaon even hough he performances of he GLS mehods a mes may no be bad. EFEENCES. Neer,J. and Wasserman, W.,974. Appled Lnear Model. chard D. Irwn, Inc.. Fomby, T.B., Hll,.C. and Johnson, S.., 984. Advanced Economerc Mehods. Sprnger Verlag, New York Berln Hedelberg London Pars Tokyo. 3. Maddala G.S.,. Inroducon o Economercs. John Wley and sons L, England. 3 rd Edon 4. Graybll, F.A., 96. An nroducon o Lnear Sascal Models. New York. McGraw Hll. 5. Sampson, A.P., 974. A ale of wo regressons. Journal of he Amercan Sascal Assocaon 69, Johnson, J., 984. Economerc Mehods, Thrd Edon, New York, McGraw Hll. 7. Chaerjee, S., Had, A.S. and Prce, B.,. egresson Analyss by Eample. Thrd Edon. A Wley Inerscence Pubcaon. John Wley and Sons. 8. Markov,A.A.,9.Wahrschenlchkesrechnug. Lepzg: Tuebner 9. Aken, A.C., 935. On Leas Squares and Lnear combnaons of observaons. Proceedngs of he oyal Sascal Socey. Ednburgh, 55, Cochrane, D. and Orcu, G.H Applcaon of Leas Square o elaonshp Conanng Auocorrelaed Error Terms. Journal of he Amercan Sascal Assocaon 44, Hldreh, C. and Lu, J.Y., 96. Demand elaonshps wh Auocorrelaed Dsurbances. Mchgan Sae Unversy. Agrculural Epermen Sascal Bullen 76, Eas Lansng, Mchgan

5 J. Mah. & Sa., 3 (4): 96-, 7. Pars, S.J. and Wnsen, C.B., 954. Trend Esmaors and Seral Correlaon.Unpublshed Cowles Commsson, Dscusson Paper, Chcago. 3. Thornon, D.L., 98. The approprae auocorrelaon ransformaon when he auocorrelaon process has a fne pas. Federal eserve Bank of S. Lous Paper, 8 4. Durbn, J., 96. Esmaon of Parameers n Tme Seres egresson Models. Journal of he oyal Sascal Socey, B., Thel, H., 97. Prncple of Economercs. New York, John Wley and sons. 6. Beach, C.M. and Macknnon, J.S., 978. A Mamum Lkelhood Procedure for egresson wh auocorrelaed errors. Economerca 46, No., Whe, K.J., 978. A General Compuer Program for Economerc Mehods SHAZAM. Economerca, 46, TSP.,5. Users Gude and eference Manual. Tme Seres Processor. New York. 9. Aynde, K. and Oyejola, B.A., 7. A Comparave sudy of he Peformances of he OLS and Some GLS esmaors when regressors are correlaed wh error erms. esearch Journal of Appled Scences, (3): 5.. Park,.E. and Mchell, B.M., 98. Esmang he Auocorrelaed Error Model wh Trended Daa. Journal of Economercs, 3, 85.. Aynde, K. 6. A Sudy of obusness of Some mehods of Parameer Esmaon n egresson Model o Correlaons. Unpublshed Ph.D Thess. Unversy of Ilorn, Ngera.