Generalized Minimum Perpendicular Distance Square Method of Estimation

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Appled Mathematcs,, 3, 945-949 http://dx.do.org/.436/am..366 Publshed Ole December (http://.scrp.org/joural/am) Geeralzed Mmum Perpedcular Dstace Square Method of Estmato Rezaul Karm, Morshed Alam, M.

1 Appled Mathematcs,, 3, Publshed Ole December ( Geeralzed Mmum Perpedcular Dstace Square Method of Estmato Rezaul Karm, Morshed Alam, M. M. H. Chodhury, Forhad Hossa Departmet of Statstcs, Jahagragar Uversty, Savar, Bagladesh Emal: Receved September, ; revsed October, ; accepted October 3, ABSTRACT I case of heteroscedastcty, a Geeralzed Mmum Perpedcular Dstace Square (GMPDS) method has bee suggested stead of tradtoally used Geeralzed Least Square (GLS) method to ft a regresso le, th a am to get a better ftted regresso le, so that the estmated le ll be closest oe to the observed pots. Mathematcal form of the estmator for the parameters has bee preseted. A logcal argumet behd the relatoshp betee the slopes of the les ad has bee placed. Keyords: Heteroscedastcty; Ordary Least Square Method; Mmum Perpedcular Dstace Square Method; Geeralzed Least Square Method. Itroducto Lear regresso has a log hstory ts ay of developmet from the very beggg of eghteeth cetury tll today. A lot of lteratures are avalable ths area, these lteratures volves the estmato of regresso coeffcets ad costat by Ordary Least Square (OLS) method.e. by mmzg the sum of square of the vertcal dstaces betee the observed pots ad the assumed regresso le, ad estmate the regresso coeffcets tradtoally ko as OLS estmato procedure. M. F. Hossa ad G. Khalaf, (9) shoed that OLS method does ot mmze actual dstace from the observed pot to the ftted regresso le. They have suggested mmum perpedcular dstace square (MPDS) Method estmato for smple lear regresso case of homoscedastcty hch bols do the tradtoal OLS method. But regresso dsturbaces hose varaces are ot costat across observatos are heteroscedastc. Heteroscedastcty arses umerous applcatos, both cross-secto ad tme-seres data. For example, eve after accoutg for frm szes, e expect to observe greater varato the profts of large frms tha those of small oes. The varace of profts mght also deped o product dversfcato, research ad developmet expedture, ad dustry characterstcs ad therefore mght also vary across frms of smlar szes. Whe aalyzg famly spedg patters, e observe greater varato expedture o certa commodty groups amog hgh-come famles tha lo oes due to the greater dscreto alloed by hgher comes []. MPDS method s ot sutable for ths type of heteroscedastcty stuato because ths method as establshed oly for homoscedastcty cases. I ths paper e have cosdered mmum perpedcular dstace square method case of heteroscedastcty hch e called Geeralzed Mmum Perpedcular Dstace Square (GMPDS) method.. Problems of Ordary Least Square (OLS) ad Geeralzed Least Square (GLS) Method Suppose the smple lear regresso model s u here the respose varable s related to the explaatory varable through the regresso coeffcet, costat tercept ad radom dsturbace term u. We assume that the dsturbace terms u follo all assumptos of classcal lear regresso model. The estmato procedure of regresso coeffcet by Ordary Least Square (OLS) method ad Geeralzed Least Square (GLS) method s actually mmzg the sum of square of the vertcal dstaces u from the observed pots to the assumed regresso le. The OLS estmators are: SPxy SSx Copyrght ScRes.

2 946 R. KARIM ET AL. ad. The mportat assumpto for applyg OLS method s that the varace of each dsturbace term u, codtoal o the chose values of the explaatory varables, s some costat umber (s called homoscedastcty assumpto). If the data volet ths homoscedastcty assumpto that s the varace of each dsturbace term u codtoal o the chose values of the explaatory varables s radom (say ) the e ca ot apply OLS ad ths case e apply GLS estmato procedure for estmatg parameters []. The GLS estmators are: here, ad, ad The problem of OLS ad GLS estmato s that, actually they do t mmze real dstace from the observed pot to the ftted regresso le rather they mmze the vertcal dstace from the observe pot to the ftted regresso le. For ths reaso e have the ell ko theorem s. here s the estmated regresso coeffcet of o ad s the estmated regresso coeffcet of o. If OLS ad GLS mmze real dstace (error) the should be uty that s. But OLS ad GLS methods, t oly occurs f data are perfectly correlated, that s r. I real lfe problem ths type of perfect correlato occurs rare case. The Mmum Perpedcular Dstace Square Method suggested by Hossa ad Khalaf (9) produced the estmator hch gves for all cases ad t dcates that the errors are really mmzed ad gves more accurate result tha that of OLS [3]. Cocept of Mmum Perpedcular Dstace Square (MPDS) Estmato The real dstace of the assumed regresso le from the pots, ;,,, are ot the vertcal dstaces or heght of the pot mus heght of regresso le.e.. I fact the actual dstaces from the le to the pots, ;,,, are the perpedcular dstaces u s (as dcated Fgure ). These perpedcular dstaces ould also be postve ad egatve accordg to, s above the le u or belo the le u. Also assumg that u ~ N,. Hece estmatg ad by mmzg sum of the squares of these perpedcular dstaces ll produce the closest ftted regresso le from the pots, ;,,, hch may be used for more accurate predcto purposes. 3. The Method of Geeralzed Mmum Perpedcular Dstace Squares Method (GMPDSM) Let us cosder to-varable lear regresso fucto s u hch for ease of algebrac smplfcato e rte as u () here for each ad the respose varable s related to the explaatory varable through the regresso coeffcet, costat tercept ad radom dsturbace term u. We ko that oe of the mportat assumptos of the classcal lear regresso model s that the varace of each dsturbace term u, codtoal o the chose values of the explaatory varables s some costat umber equal to. Ths s the assumpto of homoscedastcty. Symbolcally, Var u E u ;,,, (x, y ) u u (x, y ) (x, y ) (x 4, y 4 ) Ŷ Fgure. Regresso les obtaed from OLS & MPDS method. Copyrght ScRes.

3 R. KARIM ET AL. 947 No f the codtoal varace of or u are ot same for each of the u..e., heteroscedastcty. Symbolcally, Var u E u ;,,, ad suppose the heteroscedastc varace are ko. The dvdg () by both sdes, e get u hch for ease of exposto e rte u () (3) here the trasformed varables are the orgal varables dvded by (the ko). We use the otato ad, the parameters of the trasformed model, to dstgush them from the usual MPDS parameters ad. No e see u Var u E u E E u sce s ko sce E u hch s a costat. That s, the varace of the trasformed dsturbace term u s o homoscedastc. Ths procedure of trasformg the orgal varables s doe such a ay that the trasformed varables satsfy the assumptos of the classcal model. No applyg MPDS method to ths trasformed model to estmate parameter e call Geeralzed Mmum Perpedcular Dstace Squares Method (GMPDSM). I short, GMPDS s MPDS o the trasformed varables that satsfy the classcal regresso assumptos. The estmators thus obtaed are kos as GMPDSM estmators. 3.. Perpedcular Dstace from the Pots to the Le Let us cosder to-varable lear regresso fucto or Dvdg both sdes by u e have u u For estmatg ad e eed to determe the perpedcular dstace from the observed pot, to the le. The perpedcular ds- (4) tace u from the pots, [4,5] s u to the ftted le 3.. Parameter Estmato Based o GMPDS Method To obta the GMPDS estmators, e mmze sum of square of perpedcular dstaces u from the pots, ;,,, to the ftted le follog steps are take. that s, here eghts u u (5) that s, the eghts are versely proportoal to the varace of u or codtoal o the gve,.e., var u var. Dfferetatg (5) th respect to, the puttg equal to zero ad settg for e get the ormal equato * Aga dfferetatg Equato (5) th respect to ad equatg zero th, e get d u d (7) Usg Equato (7) Equato (6) e get (6) Copyrght ScRes.

4 948 R. KARIM ET AL. here or, SPxy SSx SSy SPxy SSy SSx SPxy So the soluto of the above equato s: 4 SSx SSy SSx SSy SPxy SPxy Hece SSx SSy SSx SSy 4SPxy SPxy SSx SSy SSx SSy 4SPxy SPxy Usg ths result Equato (7) e ca estmate. Ad hece I ths method e get to regresso coeffcets, t could be proved that the + soluto.e. gves mmum of (5) ad hece e suggest the reader to use as the regresso coeffcet ad accordgly the regresso costat could be estmated by usg Equato (8) to ft the regresso le o.e.. (8) 3.3. Estmato of Regresso Coeffcet by Usg GMPDS for the Model β β To estmate regresso coeffcet ad regresso costat by mmzg sum of squares of the error term u s (assumed) the perpedcular dstaces from the ftted le to the pots,,,,, ; e do the smlar steps as e do Secto 3.7. That s, u u or u (9) Dfferetatg both sdes th respect to ad ad puttg equal to zero ad settg for ad, e get the follog solutos: SSy SSx SSy SSx 4SPxy SPxy Hece SSy SSx SSy SSx 4SPxy SPxy SSy SSx SSy SSx 4SPxy SPxy ad Here e also get to regresso coeffcets ad for the same rego as e have metoed Secto 3., e ll suggest the reader to use as regresso coeffcet ad accordgly the estmato of may be obtaed to ft the regresso le o Relatoshp betee Regresso Coeffcets If e cosder the GMPDS method to estmate regresso coeffcets ad as e have dcated Sectos 3. ad 3.3, by mmzg the error term u Copyrght ScRes.

5 R. KARIM ET AL. 949 ad u respectvely (the perpedcular dstaces from these les to the observed pots), e get S 4 Sx SSy SSx SSy SPxy SPxy for the le ad S 4 Sy SSx SSy SSx SPxy SPxy for the le e see that proportoal to.e. or, hch dcate that durg estmatg regresso coeffcet by usg GMPDS method case of heteroscedastcty, the error term s mmzed. Ths s a e agle to advocate the advatage our suggested method (GMPDSM) to estmate regresso coeffcets case of heteroscedastcty. 4. Cocludg Remarks The method of MPDS estmato actually mmze real dstaces from the observed pots to the ftted regres- s so le but OLS ad GLS method fal to do that by usg vertcal dstace from the observe pots to the ftted regresso le. But oe of the crucal assumptos of MPDS method ad also for tradtoal OLS method s that the varace of each dsturbace terms remas some costat umber. So e ca ot apply MPDS method he ths assumpto s volated. That s, presece of heteroscedastcty OLS ad MPDS s ot sutable. I ths paper our ma focus s o mmum perpedcular devatos case of heteroscedastcty, ad e have sho mathematcally that GMPDS method gves a estmator that the error term s really mmzed. Hece e propose GMPDS method case of heteroscedastcty. REFERENCES [] W. H. Greee, Ecoometrc Aalyss, 5th Edto, Pearso Educato, Sgapore, 3, [] D. Gujarat, Basc Ecoometrcs, 4th Edto, McGra- Hll, Ne ork, 3. [3] M. F. Hossa ad G. Khalaf, Mmum Perpedcular Dstace Square Method Estmato, Joural of Appled Statstcal Scece, Vol. 7, No., 9, pp [4] A. Mzrah ad M. Sullva, Calculus ad Aalytc Geometry, Wadsorth Publshg Compay, Beverly, 986. [5] M. R. Spegel ad Joh L, Mathematcal Hadbook of Formulas ad Tables, d Edto, Mcgra-Hll, Ne ork, 999. Copyrght ScRes.

Simple Linear Regression

Simple Linear Regression Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato