Abstract. Introduction

Transcription

1 THE IMPACT OF USING LOG-ERROR CERS OUTSIDE THE DATA RANGE AND PING FACTOR By Dr. Shu-Pg Hu Teclte Research Ic. 566 Hllster Ave. Ste. 30 Sata Barbara, CA 93 Abstract Ths paper dscusses the prs ad cs f usg the lg-errr mdel versus the Mmum- Ubased-Percetage Errr (MUPE mdel fr cst estmatg relatshp (CER develpmet. Further, we dscuss a very cmm ad crrect terpretat f the lg-errr CER result: t s cmmly assumed t be the mea f a lg-rmal dstrbut, whch s prve t be wrg. The theretcal dervats f tw avalable crrect factrs (Gldberg, PING t adjust fr the mea ut space fr lg-errr mdels are dscussed accrdgly. We als shw that the Gldberg equat utsde the data rage geerates cuter-tutve ad apprprately lw results by cmparg three cmmly mplemeted lg-lear equats: the Gldberger, the PING Factr, ad the ucrrected equats. Multplcatve errr terms are cmmly used the cst aalyss feld because experece tells us that the errr f a dvdual bservat (e.g., cst s geerally prprtal t the magtude f the bservat (t a cstat. The lg-errr mdel s a ppular way t geerate these types f CERs because rdary least squares (OLS ca be accmplshed lg space f the ftted equat s lg-lear. Hwever, the lg-lear CER wll be based lw whe trasfrmed back t ut space; t wll predct clser t the meda (t the mea f the CER rsk dstrbut ut space. Therefre, crrect factrs are requred t adjust the CER result t prduce the mea ut space. The MUPE regress mdel s a alteratve methd t hypthesze the multplcatve errr a CER. All f the Umaed Space Vehcle Cst Mdel, Eghth Edt (USCM8 CERs (Referece 5 have bee develped usg MUPE. The MUPE methd vlves a teratve, weghted least squares regress that prvdes ubased percetage errr regress results. N trasfrmat r adjustmet (t crrect the bas ut space s eeded fr fttg a MUPE equat. Itrduct Fr may CERs, the errr f a dvdual bservat (e.g., cst s apprxmately prprtal t the magtude f the bservat. I such cases, t s apprprate t hypthesze a multplcatve errr term fr the CER. Several ptmzat techques have bee used t mdel multplcatve errrs ver the years. Oe cmm practce was t wrk lg space by takg atural lgs f bth the depedet varable ad the equat frm. Whe the trasfrmed equat s lear lg space, OLS ca be appled t derve a Best Lear Ubased Estmatr (BLUE lg space, whch s als the Maxmum Lkelhd Estmatr (MLE lg space. If the equat frm s t lg-lear, the lg trasfrm ca stll be used t mdel a multplcatve errr, but the ptmzat wll be -lear SCEA Cferece - Jue 005 Page

2 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic. least squares. Althugh the mea ad meda are the same fr the lg-lear CERs lg space, whe trasfrmg the equat back t ut space the mea ad meda dffer. The ut space CER predcts clser t the meda stead f the mea. Therefre, the drect traslat f the equat back t ut space teds t uderestmate the mea value f the rgal ppulat. A multplcatve crrect factr (trduced as the PING Factr at Referece s used t adjust the CER result t mre clsely reflect the mea ut space. Lg Errr Mdel. The multplcatve errr mdel s geerally stated as fllws: Y = f ( x, ε fr =,, ( where: = sample sze Y = bserved cst f the th data pt, = t f (x, = the value f the hyptheszed equat at the th data pt = vectr f ceffcets t be estmated by the regress equat x = vectr f cst drver varables at the th data pt = errr term ε If the multplcatve errr term (ε s further assumed t fllw a lg-rmal dstrbut, the the errr ca be measured by the fllwg: e = l( ε = l( Y l( f (, ( x where l stads fr ature lgarthmc fuct. The bjectve s the t mmze the sum f squared e s (.e., (Σ(l(ε. If the trasfrmed fuct s lear lg space, the OLS ca be appled lg space t derve a slut fr. If t, we eed t apply the -lear regress techque t derve a slut. Althugh a least squares ptmzat lg space prduces a ubased estmatr lg space, the estmatr s lger ubased whe trasfrmed back t ut space (see Refereces,, ad 4. Hwever, the magtude f the bas ca be crrected wth a smple factr f the errrs are dstrbuted rmally lg space (see Refereces ad. Because f ths shrtcmg, the MUPE methd s recmmeded fr mdelg multplcatve errr drectly ut space t prduce ubased estmatrs. The MUPE Methd. The geeral specfcat fr a MUPE mdel s the same as gve abve (Equat, except that the errr term s assumed t have a mea f ad varace σ. Based up ths assumpt f a multplcatve mdel, a geeralzed errr term s defed by: e y f ( x, = (3 f ( x, where e w has mea f 0 ad varace σ. SCEA Cferece - Jue 005 Page

3 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic. Ths percetage errr dffers frm the tradtal percetage errr the dematr, where MUPE uses predcted cst stead f actual cst as the basele. The ptmzat bjectve s t fd the ceffcet vectr that mmzes the sum f squared e s: Mmze = y f ( x, = f ( x, = e (4 Teclte Research, Ic. has prpsed a MUPE regress techque t slve fr the fuct the umeratr separately frm the fuct the dematr (see Refereces ad 5. Ths s de thrugh a teratve prcess. Mmze y f ( x, k y f k ( x = = f ( x, k = f k ( x (5 where k s the terat umber ad the ther terms are as defed prevusly. The weghtg factr f each resdual the curret terat s equal t the recprcal f the predcted value frm the prevus terat. Sce the dematr Equat (5 s kept fxed thrughut the terat prcess, the MUPE techque turs ut t be a weghted least squares wth a addtve errr. The fal slut s derved whe the chage the estmated ceffcets ( vectr betwee the curret terat ad the last terat s wth the aalyst-specfed tlerace lmt. Ths ptmzat techque (Equat 5 s cmmly referred t as Iteratvely Reweghted Least Squares (IRLS; see Refereces 8 ad 9. PROS AND CONS OF MUPE AND LOG-ERROR MODELS Mdel develpers must decde the best way t mdel the errr f ther equat, chsg ether the MUPE mdel r the lg-errr mdel. Here are a few advatages f usg the MUPE methd: The MUPE CER has zer prprtal errr fr all pts the database ( sample bas. N trasfrmat r adjustmet (t crrect the bas ut space s eeded fr fttg a MUPE equat. Gdess-f-ft measures (r asympttc gdess-f-ft measures ca be appled t judge the qualty f the mdel uder the rmalty assumpt. The MUPE CER prduces csstet estmates f the parameters ad the mea f the equat. The estmated parameters usg the MUPE methd are als the maxmum lkelhd estmates (MLE f the parameters (by Matthew Gldberg, 00. A dsadvatage f usg the MUPE mdel s that t reles the -lear ptmzat techque t acheve a slut, whch ca be cumbersme. See the detaled descrpts f the MUPE methd Refereces ad 3. The fllwg ccers have bee rased abut usg the lg-errr mdel (Equat :. Errrs are t expressed meagful uts (.e., l(ε s are lg f dllars. SCEA Cferece - Jue 005 Page 3

4 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic.. Mmzg Σ(l(ε s t the same as mmzg Σ(ε. As a result, the stadard errr f estmate (SEE derved lg space fr lg-lear CERs cat be cmpared wth the SEE ut space fr -lear regress equats. 3. We must chse pwer frm CERs t use the lg-errr assumpt. 4. The resultat equat wll be based lw whe trasfrmg back t ut space. T bta a ubased estmate, we eed t multply the CER by a crrect factr t estmate the mea ut space. Althugh these ccers all appear t be legtmate, we thk ly the last e the lst s a vald ccer; the ther three are t. We wll address these ccers the rder gve abve. We uderstad that lg-errrs (Equat are t expressed dllars as uts. Hwever, the lgerrrs have a terestg terpretat: l(ε ca be vewed apprxmately as a percetage errr by Taylr seres expas. It actually matches the MUPE deft f percetage errr: l( ε ε = Y f ( x, = Y f ( x f ( x,, (6 Althugh mmzg Σ(l(ε s t the same as mmzg Σ(ε, ths s t a prblem because we hypthesze a multplcatve mdel (Equat, t a addtve e. As a geeral rule, the measure f SEE ft space (a ft measure cat be cmpared acrss dfferet mdels. We shuld t cmpare the ft measures betwee tw mdels f they are develped usg dfferet ft crtera. Fr example, cmparg the SEEs betwee a addtve mdel ad a multplcatve mdel s meagless because e mdel mmzes the sum f squared abslute errrs whle the ther mmzes the sum f squared percetage errrs. Nw let us address the thrd ccer the lst. Of curse, we have t chse lg-lear equats f we wat t apply OLS t ft equats lg space. Hwever, we are t frced t use OLS. T mdel the cst varat, we shuld ( hypthesze a CER frm based up gd lgc ad sld techcal gruds ad ( chse a apprprate errr term assumpt. Nte that the chce f the CER frm shuld t drve the errr term assumpt ad vce versa. If the regress equat cat be slved learly, we wll apply the -lear ptmzat techque t geerate a slut. Fr example, f a fxed cst term s guded by the egeerg judgmet ad errrs are assumed t be scaled wth the prject, we ca certaly hypthesze ths equat frm, a + bx c, wth a multplcatve errr term t expla the cst varat. I ths case, we eed t use the -lear regress techque t derve a slut. The last e the lst has bee a very cmm ccer fr years. Althugh a least squares slut s ubased lg space, the resultat CER s lger ubased whe trasfrmg back t ut space. Ths s the mst mprtat reas why the MUPE methd s suggested fr mdelg multplcatve errrs ut space (t avd the use f crrect factrs t adjust fr the mea. Despte the vald ccer that the resultat equat s based lw ut space, the lg-errr mdel s stll qute ppular whe mdelg multplcatve errrs. The reass are gve belw: SCEA Cferece - Jue 005 Page 4

5 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic. If the hyptheszed equat s lg-lear, e.g., y = ax b ε, the the regress ca be de lg space learly, whch des t vlve a teratve prcess. I ths case, we have the fllwg advatages: The tradtal gdess-f-ft measures ca be appled t judge the qualty f the ft lg space. The utlers ca be easly detfed fr further scruty. The predct tervals ca als be easly derved. Obvusly, the abve advatages are all restrcted t lg-lear mdels. They are lger vald fr -lear CERs, such as y = (ax b + cε. The stadard errr f estmate lg space (SEE L ca be regarded as a measure f a percetage errr at a certa gve x level ut space,.e., SEE L CV A at a gve x level. Nte that CV A detes the ceffcet f varat ut space expressed as a percetage. Ths apprxmat s derved by applyg Taylr seres expas t the rat f Equat 4 ver Equat. See Referece ( fr detals. The lg-errr ca be vewed apprxmately as the MUPE percetage errr (see Equat 6. Based up the abve dscusss, t appears that the MUPE methd s superr t the lg-errr mdel. I addt, ts errr term assumpt s mre geerc. Netheless, the chces betwee the MUPE ad lg-errr mdels shuld deped up the errr term assumpt: Chse MUPE f the errr term (ε s asscated wth a mea f e ad varace f σ. Chse lg-errr mdel f ε fllws a lg-rmal dstrbut wth a mea f zer ad varace σ lg space,.e., ε ~ LN(0, σ. The real purpse f usg a CER s fr predctg a future cst. The mpact f applyg crrect factrs t lg-errr CERs mght be mre severe tha we rgally thught whe the predct s utsde the database. We wll beg the dscuss by trducg the crrect factrs ad the related thery. DERIVATION OF CORRECTION FACTORS Theres fr Lg-Lear Mdels. Let us hypthesze a lg-lear equat wth a multplcatve errr term as gve Equat : where: Y 0 k ( = f k x, ε = e x x x ε fr =,..., (7 ε 's are depedetly, detcally dstrbuted (..d. radm varables asscated wth a lgrmal dstrbut wth mea f 0 ad varace σ lg space,.e., LN(0,σ,,,..., k, ad σ are ukw parameters, x, x,..., x k are the depedet varables fr the th data pt, ad k s the ttal umber f depedet varables the mdel. The abve mdel ca be equvaletly stated as k l( Y = 0 + j l( x j + l( ε fr =,..., (8 j= SCEA Cferece - Jue 005 Page 5

6 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic. The depedet varable lg space (l(y w fllws a rmal dstrbut because l(ε s dstrbuted as N(0,σ². Therefre, at a gve x value, say x = x = (x, x..., x k, the cdtal dstrbuts f Y bth lg ad ut space are gve by Equat 9 ad Equat 0, respectvely: k l( Y / x = x = 0 + j l( x j + l( ε = µ + l( ε ~ N( µ, σ (9 j= ( Y / x x = exp( µ + l( ε ~ LN( µ, σ (0 = Where µ = + Σ j l(x j It fllws frm Equat 9 that the cdtal mea ad meda value f Y ( lg space at ths gve value, x, are bth equal t µ : k E (l( Y / x = x = 0 + j l( x j = µ ( j= Hwever, t ca be easly shw by Equat 0 that the cdtal mea, meda, ad stadard devat f Y (at the gve x ut space are gve respectvely by the fllwg equats: 0 k E Y / x = x = exp( µ + σ / = e x x...x ( e = µ A ( ( k Meda Y / x = x = exp(µ = e x x...x = M A (3 ( 0 k k σ / 0 k Stdev Y / x = x = exp( µ + σ / e = e x x...x ( e e (4 ( (Nte: If a radm varable Y s dstrbuted as LN(µ, σ, the E(Y = exp(µ+σ /, Meda(Y = exp(µ, ad Var(Y = (E(Y (exp(σ -. Furthermre, the mde f Y at the gve x s ether the mea r the meda: σ k Mde Y / x = x = exp( µ σ = e x x...x e = MdeA (5 ( k 0 σ Therefre, the term e σ²/ ca be regarded as a factr explag the dfferece betwee µ A ad M A : µ A M A = exp( σ / (6 It s clear frm Equat 6 that the drect traslat f the slut frm lg space t ut space s t a gd estmatr f the ppulat mea, µ A, f the dfferece betwee mea (µ A ad meda (M A s t eglgble. Therefre, we have the fllwg cclus: k σ / σ The lg-lear equat wll be based lw f we d t apply crrect factrs. SCEA Cferece - Jue 005 Page 6

7 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic. Prcedure. I the fllwg paragraphs, we wll llustrate the prcedure f develpg crrect factrs. We wll als prve ( the lg-lear CER ut space (deted byyˆ fllws a lg-rmal dstrbut (see Equat 9 ad ( the varace f ths predctr depeds the lcat f the drver varables. T cmpute the mea ad stadard devat f Ŷ, let us frst trduce a varable r : r = l(x (X X - l(x t (7 where: l(x = (, l(x,..., l(x k, a rw vectr f gve drver values lg space ad s fr the tercept X = desg matrx lg space (The superscrpt t detes the traspse f a vectr r a matrx. Fr smplcty, the letter X s als used t dete the data matrx lg space. Nte that r s the s-called leverage value lg space f x s a vectr f depedet varables the data matrx. Bth the mea ad stadard devat f Yˆ (at the gve x are fucts f r. T be mre specfc, the stadard devat f a future predct lg space s equal t r σ (see Equat 8. I a e-depedet varable case, the value f r s gve by r = (, l( x ( X ' X = l( x + (l( x l( x (l( x l( x = + (l( x l( x SSx It s clear that the varable r captures the sample sze ad the dstace f the estmatg pt frm the ceter f the database terms f the sample stadard devat f the drver varable. If the classcal assumpts hld as explaed Equat 7, the OLS methd ca geerate a ubased estmatr fr the depedet varable lg space. Ths OLS predctr f Y lg space (at the gve x fllws a rmal dstrbut because t s a lear cmbat f the rmally dstrbuted l(y, l(y,..., ad l(y : ˆ ˆ l( Y / x = x = l( x ~ N(l(x, rσ = N( µ, rσ (8 where: = (,,..., k t, a clum vectr f the ukw ceffcets ˆ = (X X - X (l(y, the OLS slut fr l(y = (l(y, l(y,..., l(y t, a clum vectr f Y lg space l(x = (, l(x,..., l(x k, µ = l(x, ad r are all gve abve Nte that the mea ad varace f l( Yˆ / x = x are derved by matrx algebra (see Referece 9 ad clearly the varace f ths predctr s drve by the lcat f the data pt. By deft, therefre, the dstrbut f the drect traslat f Equat 8 t ut space fllws a lg-rmal dstrbut: SCEA Cferece - Jue 005 Page 7

8 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic. ˆ ˆ ˆ ˆ ˆ k Y / x = x = exp(l( x ˆ = e 0 x x... x ~ LN( µ, r σ (9 ( Based up Equats 9, the mea, meda, ad stadard devat f Yˆ, at a gve x vectr, are gve by Equats 0,, ad, respectvely: ˆ E ( Y / x = x = exp( µ + rσ / (0 Meda ( Yˆ / x = x = exp( µ ( ˆ Stdev Y / x = x = exp( µ + r σ / exp(r σ ( ( Ad the et crrect factr fr estmatg the mea value ut space at the gve x vectr s the rat betwee Equat ad Equat 0: ˆ σ Net CF fr Mea at x : E ( Y / x = x E( Y / x = x = exp(( r (3 Illustrats f Smple Cases. Let us use tw smple examples t llustrate Equat 3. I the uvarate case, where Y, Y,, Y are..d. radm varables asscated wth LN(µ,σ, the estmatr f the mea s equal t exp( l( Y. The dstrbuts f l(y ad exp( l( Y are gve by Equat 4 ad Equat 5, respectvely: l( Y = l( Y / ~ N( µ, σ (4 = exp(l( Y = Π Y ~ LN( µ, σ (5 It fllws frm Equat 5 that E( exp( l( Y = exp(µ + σ /. Thus, fr predctg the ppulat mea (.e., exp(µ + σ /, the et crrect factr fr Ŷ (.e., exp( l( Y ca be expressed as exp(σ / / exp(σ / = exp(( /σ / (6 The frst term Equat 6 dcates that the mea s uderestmated by exp(σ /, whch ca be regarded as a trasfrmat bas. The secd term Equat 5 dcates that the meda s verestmated by exp(σ /, whch ca be regarded as a samplg bas. Fr the e depedet varable mdel whe x = x, the predcted values f Y lg space ad ut space are gve by Equats 7 ad 8, respectvely: ˆ ˆ l( Y / x = x = (, l( x = ˆ + ˆ l( ~ ( + l(x, r σ ˆ x N (7 k SCEA Cferece - Jue 005 Page 8

9 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic. ˆ ˆ ˆ ( Y / x = x = exp(ˆ + ˆ l( x = e x ~ LN( + l( x, r σ (8 where: (l( x l( x r = (, l( x ( X ' X = + l( x, (9 SSx ο ad are the ukw ceffcets; ˆ ad ˆ are ther estmated ceffcets, SSx s the sum f squared devat f the drver varable abut ts mea lg space, l(x s the average f the depedet varable lg space, ad x s a gve x value. The expected value f the predct whe x = x s the gve by E( Yˆ / x = x = exp( = e x + l( x + r σ / (l( x l( x exp ( + SSx σ (30 Whe Equat 30 s cmpared t Equat, the et crrect factr fryˆ at the gve x s exp ( r σ (l( x l( x = exp ( SSx σ Ubased Crrect Factr (Gldberger s Factr. I the fllwg sect, we wll demstrate that the et crrect factr, Gldberger s crrect factr (GF ca be expressed as GF = g (( r s exp (( r s (3 The fuct g s defed belw (see Equat 33. Let us use s t dete stadard errr f estmate lg space. Althugh the estmatr f the ppulat varace σ ( lg space s s, the statstc exp(s /, average, teds t verestmate the ukw factr exp(σ /: E( e s / ( E( s / ( σ / e = e (3 Nte that the equalty Equat 3 s based up Jese s equalty. Hece, t s ecessary t develp a fuct t geerate a ubased estmatr fr the et crrect factr gve Equat. Such a fuct, g, s gve belw: t ( p t ( p t g ( t = (33!!( p + 3!( p + ( p + 4 where p s the ttal umber f ceffcets t be estmated ad s the sample sze. 3 SCEA Cferece - Jue 005 Page 9

10 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic. The abve-defed fuct g has the fllwg prperty: aσ E ( g( a s = e fr ay real umber a (34 (See Referece fr math dervats f Equat 34. Hece, Gldberger suggests that we use Equats 35 ad 36, respectvely, as the ubased estmatrs fr the mea ad meda ut space: Mea Estmatr = e ˆ ˆ ˆ ˆ k k 0 x x x s g(( r / (35 Meda Estmatr = e ˆ ˆ ˆ ˆ k k 0 x x x s g( r / (36 The last term Equat 35, whch s t be multpled t the CER, s the s-called et crrect factr fr the mea r Gldberger s crrect factr (GF. Fr smplcty, ths factr ca be apprxmated by the fllwg term: (( r s exp (( r s GF = g (37 Ths theretcal crrect factr (Equat 37 suggested by Gldberger s a varable factr. It shuld be evaluated pt by pt ad multpled t the lg-errr CERs t bta the theretcal mea ut space (see Equat 35. Ths prcess s tedus ad ca get very cumbersme whe mre depedet varables are trduced t the CER. The PING Factr. The PING Factr (PF s gve by PF = g(( p s exp(( p s Sce r ( Equat 37 has t be evaluated at each dfferet x level, Equats 35 ad 36 are f lttle realstc use. We suggest usg p/ as a apprxmat f r fr ay gve x value, where p s the ttal umber f estmated ceffcets ad s the sample sze. Ths way, the crrect factr fr the mea (t be multpled t the CER ca be smplfed as gve belw: p s g(( (38 Equat 38 s cmmly referred t as the PING Factr. The term p/ the equat abve s the expected value f r. I ther wrds, we use the mea leverage value t apprxmate r as f x s sampled radmly frm the sample ppulat as x s the lg trasfrmed data matrx. The prf s gve by evaluatg the value f r the Hat matrx. Let X dete the desg matrx lg space as gve Equat 7. If x s a clum vectr the lg space data matrx, the r s smply the crrespdg dagal elemet f the by SCEA Cferece - Jue 005 Page 0

11 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic. symmetrcal Hat matrx H (= X(X X - X. Sce H s a square, demptet matrx (.e., H*H = H, the trace f H s equal t ts rak, whch s the umber f estmated ceffcets. (The trace f a square matrx, by deft, s the sum f ts dagal elemets. Therefre, the mea value f r s the mea value f dagal elemets f the Hat matrx H,.e., the umber f estmated ceffcets dvded by the sample sze. It s clear that the PING Factr s a geeral crrect fr the level f the fuct; t s evaluated wth the rage f the database. Cmpared t Gldberger s Factr, the PING Factr s a hady, cstat crrect factr fr the etre equat. I mst stuats (whe gets suffcetly large ad s s mderately small, say < 0.8, the PING Factr ca be further apprxmated by PF e ( p s = exp(( p s (39 Ths smplfed PING Factr (Equat 39 s a gd apprxmat f the theretcal e (Equat 38 fr mst cases (see Referece fr detals. Hwever, the dfferece betwee Equats 38 ad 39 ca be e percet r mre f the sample sze s small ad/r stadard errr f estmate s farly large. Fr example, f = 6, p =, ad s = 0.95, the the smplfed PING Factr s.35 whle the theretcal PING Factr s.33. I mst applcats wth mderate stadard errr f estmates, we recmmed usg the smplfed PING Factr. Just as Gldberger s Factr, the frst term the PING Factr s used t adjust the dwward bas betwee the mea ad the meda, whch ca be regarded as a trasfrmat bas. As fr the secd term Equat 39, t s used t adjust the upward bas fr estmatg the meda. Ths bas ca be regarded as a samplg bas because t vashes as the sample sze appraches fty. Sme aalysts wder why the PING Factr gets bgger whe the sample sze creases whle the values f s ad p rema fxed (see Equat 39. Ths s because the samplg bas (fr estmatg the meda gets smaller fr a larger sample. We ly eed t adjust the trasfrmat bas betwee the mea ad meda whe the sample sze appraches fty r gets suffcetly large. SENSITIVITY ANALYSIS Nw let us exame the value f r Equat 37. As defed Equat 7, t s gve by r = l(x (X X - l(x t Ths value shuld be greater tha r equal t zer because r s part f the varace f the predctr at a gve x lg space (see Equat 8. It s als buded abve by e f x s a vectr the data matrx. Hwever, ths restrct (r lger exsts f x s t wth the data rage. I ther wrds, the value f r ca be larger tha e f t s evaluated utsde the data rage. I ths case, Gldberger s crrect factr (.e., g((- r s²/ r exp(( r s²/ s less tha e. Whe ths happes, Gldberger s equat prduces a estmate that s actually smaller tha the ucrrected equat! As shw by Equat 3, the further the drver varable mves away frm the ceter f the database, the smaller ( magtude the crrect factr becmes due t creased leverage value. I learg curve aalyss, the ubased frst ut cst SCEA Cferece - Jue 005 Page

12 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic. ˆ ( r s / (.e., e e may be eve less tha % f the ucrrected frst ut cst (.e., due t the huge dwward crrect f the meda fr the frst ut cst (.e., T. Ths kd f ubased T, as well as may ther ubased estmates fr predcts utsde the data rage, are suspcusly lw ad hece shuld t be csdered useful. I mathematcal terms, the rat betwee the theretcal ubased Gldberger s Factr (Equat 37 ad the PING Factr (Equat 38 s gve by s p s p s g (( r (( exp g ( r (40 As explaed abve, ths rat ca be substatally dfferet frm e f r s evaluated utsde the data rage. Let us exame whether the PING Factr s a gd substtute fr the Gldberger Factr fr the pts wth the data rage. A e-depedet-varable mdel wll be used as a llustratve example. Gldberger s Factr reaches ts maxmum at the ceter f the database lg space, ad t decreases whe mvg away frm that pt. As gve Equat 3, whe the dstace betwee the depedet varable ad the ceter s wth e sample stadard devat (evaluated lg space, Gldberger s equat s hgher tha the equat multpled by the Pg Factr (.e., the PING Factr equat. Gldberger s equat les belw the Pg Factr equat whe the depedet varable mves away frm ths rage. Fr the purpse f llustrat, we chse a pwer equat wth eght data pts ad a farly large stadard errr f estmate (0.5. We tced the average dfferece betwee the Gldberger ad PING Factr equats s abut % fr the pts wth the data rage, creasg t a cuple f percet twards the budares f the data pts. Hwever, whe the depedet varable devates abut 30% beyd the budares, there s a crss-ver betwee Gldberger s equat ad the ucrrected equat. If the stadard errr f estmate s 0.5 r larger, Gldberger s equat decreases very rapdly whe the depedet varable mves further ad further away frm the budares, whch makes the prject qute ucerta. We llustrate three equats the graph Fgure : the ubased Gldberger s equat, the PING Factr equat, ad the ucrrected equat (wth crrect factrs appled. Fgure llustrates these equats whe 0.8 s chse as the stadard errr f estmate ad the drver varable has a smaller expet tha that f Fgure. e ˆ SCEA Cferece - Jue 005 Page

13 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic Gldberger's Eq y = a x^b * PF y = a x^b Cst Data Rage X Fgure : Cmparg a Lg-Lear CER wth the Gldberger ad PING Factr Equats Usg 0.5 as the Stadard Errr Lg Space 50 Gldberger's Eq y = a x^b * PF y = a x^b 00 Cst 50 Data Rage X Fgure : Cmparg a Lg-Lear CER wth the Gldberger ad PING Factr Equats Usg 0.8 as the Stadard Errr Lg Space SCEA Cferece - Jue 005 Page 3

14 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic. Based up the abve graph, f the depedet varable represets weght puds, the the cst f a 5000-pud bx s cheaper tha the cst f a 3000-pud bx usg the Gldberger equat. Ths s certaly cuter-tutve ad very dubtful. CAUTIONARY NOTES IN PREDICTION Sme refereces csder exp( 0 t be represetatve f the level f the (cdtal meda fr the etre fuct. Ths culd be very msleadg f the tercept term (whe the drver varables are set at e s far away frm the mass f the data pts. I ths crcumstace, the varace the estmate f the tercept ca be very large, larger fact tha the ppulat varace f the regress le. Usg the meda value f Y at the tercept (e.g., the frst ut cst learg curves t crrect the upward bas f the meda fr the etre equat ut space ca cause the crrected fuct t le utsde the rage f the data set. Ths practce shuld be avded because Gldberger s Factr shuld be evaluated pt by pt. The prmary use f a CER s t make future predcts based future drver values, whch may r may t be the CER data rage. Hwever, the use f a CER fr extraplat s always rsky, especally fr lg-errr mdels. Gve the wde avalablty f cmputers, aalysts may use Gldberger s equat fr predct eve f the future x value les utsde the data rage. But f the future predct s far away frm the ceter f the database, Gldberger s equat wll t prduce a lgcal r tutvely crrect result as llustrated abve, especally whe the stadard errr f estmate s mderately large. Ths ptfall s a bg ccer whe usg lg-lear CERs. Geerally, the PING Factr shuld be appled t all equats ft lg space by re-specfyg each equat's cstat term as the prduct f ts rgal cstat ad the crrect factr. Hwever, exercse caut f dummy varables are used t stratfy bservats wth dfferet attrbutes (e.g., arbre versus grud-based ateas. A CER's predctve capablty may t be mprved by applyg e adjustmet factr t tw r mre dfferet ppulats f the dvdual sample varaces asscated wth these dfferet categrcal data are t equal. I thery, these dfferet ppulats shuld have smlar varaces (ad smlar slpe parameters rder t be aalyzed e equat wth dummy varables just t dfferetate the tercept term. Hwever, fr small samples, t s fte hard t fd suffcet evdece t reject the ull hypthess that the varaces are equal. A mre detaled dscuss f the PING Factr ad the dervat f Equat 34 ca be fud Referece. CONCLUSIONS The lg-errr mdel ad MUPE mdel are tw ppular techques used t hypthesze the multplcatve errr term the CERs. If the multplcatve errr term fllws a lg-rmal dstrbut ut space, the the use f lg-errr mdel s apprprate. If the multplcatve errr term s symmetrcal arud e wth a mea f e ad varace f σ, the chse the MUPE methd. Therefre, the chces betwee the MUPE ad lg-errr mdels shuld be based up the errr term assumpt. There are prs ad cs asscated wth dfferet fttg techques. The advatages f usg the lg-errr mdel are gve belw: If the hyptheszed equat s lg-lear, e.g., y = ax b ε, the the regress ca be de lg space learly uder the lgarthmc trasfrmat. As a result, ths prcess s a OLS SCEA Cferece - Jue 005 Page 4

15 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic. lg space ad all the gdess-f-ft measures ca be evaluated that space. Ths advatage des t exst f the CER has a -lear fuctal frm ut space but t cat be trasfrmed t a lear equat lg space. The stadard errr f estmate lg space (SEE L ca be regarded as a measure f a percetage errr at a certa gve x level ut space,.e., SEE L CV A at a gve x level. Nte that CV A detes the ceffcet f varat the ut space expressed as a percetage. See Referece ( fr detals. Lg-errrs (Equat ca be vewed apprxmately as the MUPE percetage errr. The dsadvatages f usg the lg-errr mdel are summarzed belw: It vlves a tw-step prcess. Frst, we eed t trasfrm bth the depedet ad depedet varables t perfrm OLS lg space. After develpg the CER lg space, we eed t trasfrm the results back t ut space. We eed t derve a crrect factr (by ether Gldberger s methd r the PING Factr t adjust the ut space CER result t bta a ubased estmate, sce the CER result s clser t the meda tha the mea ut space. Ths s yet ather extra step. We must be extremely cautus whe the future predct les utsde the data rage. Gldberger s Factr s t recmmeded whe the drver varables are utsde the rage f the data used t create the CER. The PING Factr s easer t use, just as accurate as the Gldberg Factr wth the data rage, ad far mre sutable utsde the data rage (althugh care must be take. Here are sme salet pts whe usg Gldberger s Factr r the PING Factr t acheve the ubased estmate ut space fr lg-lear CERs: Gldberger s Factr (Equat 36 ad the PING Factr geerally match each ther very clsely wth the data rage. There are tw terms vlved bth factrs: e s fr adjustg the dwward bas betwee the mea ad the meda (a trasfrmat bas; the ther s used t adjust the upward bas fr estmatg the meda (a samplg bas. Gldberger s Factr s a varable factr. It shuld be evaluated pt by pt ad multpled t the lg-errr CERs fr the etre fuct rder t bta the theretcal mea ut space. Ths prcess s tedus, as shw Equats 35 ad 36; t ca becme very cumbersme whe mre depedet varables are trduced t the equat. The PING Factr s a hady, cstat factr, whch s used t adjust the level f the etre fuct. A cmm msuse f Gldberger s Factr s t derve a adjustmet at sme extreme pt, such as T, ad the multply t t the etre equat. Ths practce emplys the crrected equat well utsde the majrty f the data pts ad shuld be avded. The PING Factr ad MUPE equats match each ther clsely mst cases (Referece 3. The PING Factr shuld be used wth caut f dummy varables are specfed the equat. Ths s because a cstat crrect factr (.e., the PING Factr may t be adequate t crrect the dwward bas fr tw r mre ppulats wth pssble uequal varaces. Whe makg a predct utsde the data rage, the theretcal ubased Gldberger Factr shuld be used wth caut because ths factr may be csderably less tha e whe the predct les utsde the data rage. SCEA Cferece - Jue 005 Page 5

16 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic. Bgraphy Dr. Shu-Pg Hu: Educated at Natal Tawa Uversty (B.S., Mathematcs ad Uversty f Calfra, Sata Barbara (M.S., Mathematcs ad Ph.D., Statstcs. Dr. Hu s a techcal expert at Teclte Research, Ic. She jed Teclte 984 ad has served as a cmpay expert all statstcal matters. She s expereced depedet cst estmat, cst research, ad rsk aalyss. She advcated a teratve regress techque (MUPE t mdel a multplcatve errr term wthut bas ad develped crrect factrs (the PING Factr t adjust the dwward bas lg-errr mdels. She has ver 0 years f experece USCM CER develpmet ad the related database. She als has 7 years f experece the desg, develpmet, mdfcat, ad tegrat f statstcal sftware packages fr fttg varus types f regress equats, learg curves, cst rsk aalyss, ad ther PC-based mdels. Tel: ( Fax: ( E-mal: shu@teclte.cm SCEA Cferece - Jue 005 Page 6

17 The Impact f Usg Lg-CERs Outsde the Data Rage ad PING Factr Teclte Research, Ic. REFERENCES. Hu, S. ad Sjvld, A. R., Errr Crrects fr Ubased Lg-Lear Least Square Estmates, TR-006/, March Hu, S. ad Sjvld, A. R., Multplcatve Errr Regress Techques, 6 d MORS Sympsum, Clrad Sprgs, Clrad, 7-9 Jue Hu, S., The Mmum-Ubased-Percetage-Errr (MUPE Methd CER Develpmet, 3 rd Jt Aual ISPA/SCEA Iteratal Cferece, Vea, VA, -5 Jue Gldberger, A. S., The Iterpretat ad Estmat f Cbb-Duglas Fucts, Ecmetrca, Vl. 35, July-Oct 968, pp Nguye, P., N. Lzz, et al., Umaed Space Vehcle Cst Mdel, Eghth Edt, U. S. Ar Frce Space ad Mssle Systems Ceter (SMC/FMC, Octber Hllebradt, P., Kllgswrth, P., et al., Umaed Space Vehcle Cst Mdel, Sxth Edt, U. S. Ar Frce Space Dvs (AFSC, Nvember Dua, N., Smearg Estmate: A Nparametrc Retrasfrmat Methd, Jural f the Amerca Statstcal Asscat, Vl. 78, Sep 983, N. 383, pp Seber, G. A. F., ad C. J. Wld, Nlear Regress, New Yrk: Jh Wley & Ss, 989, pages 37, 46, Wesberg, S., Appled Lear Regress, d Edt, New Yrk: Jh Wley & Ss, 985, pages SCEA Cferece - Jue 005 Page 7