On Developing Ridge Regression Parameters: A Graphical investigation

Florda Ieraoal Uversy FIU Dgal Commos Dearme of Mahemacs ad Sascs College of Ars, Sceces & Educao 0 O Develog Rdge Regresso Parameers: A Grahcal vesgao Gsela Muz Dearme of Mahemacs ad Sascs, Florda Ieraoal Uversy, gmuz@fu.edu B.M. Goam Kbra Dearme of Mahemacs ad Sascs, Florda Ieraoal Uversy, brag@fu.edu Ghaz Shuur Jöög Uversy Follow hs ad addoal wors a: h://dgalcommos.fu.edu/mah_fac Par of he Physcal Sceces ad Mahemacs Commos Recommeded Cao Muz, Gsela; Kbra, B.M. Goam; ad Shuur, Ghaz, "O Develog Rdge Regresso Parameers: A Grahcal vesgao" (0). Dearme of Mahemacs ad Sascs.. h://dgalcommos.fu.edu/mah_fac/ Ths wor s brough o you for free ad oe access by he College of Ars, Sceces & Educao a FIU Dgal Commos. I has bee acceed for cluso Dearme of Mahemacs ad Sascs by a auhorzed admsraor of FIU Dgal Commos. For more formao, lease coac dcc@fu.edu.

O Develog Rdge Regresso Parameers: A Grahcal vesgao By Gsela Muz, B. M. Golam Kbra ad Ghaz Shuur 3 & Dearme of Mahemacs ad Sascs, Florda Ieraoal Uversy, Mam, Florda, USA 3 Dearme of Ecoomcs ad Sascs, Jöög Uversy, Swede ad Cere for Labour Mare Polcy (CAFO), Dearme of Ecoomcs ad Sascs, Växjö Uversy, Swede Absrac I hs aer we have revewed some exsg ad roosed some ew esmaors for esmag he rdge arameer. All all 9 dffere esmaors have bee suded. The vesgao has bee carred ou usg Moe Carlo smulaos. A large umber of dffere models were vesgaed where he varace of he radom error, he umber of varables cluded he model, he correlaos amog he exlaaory varables, he samle sze ad he uow coeffces vecors β have bee vared. For each model we have erformed 000 relcaos ad reseed he resuls boh erm of fgures ad ables. Based o he smulao sudy, we foud ha creasg he umber of correlaed varable, he varace of he radom error ad creasg he correlao bewee he deede varables have egave effec o he MSE. Whe he samle sze creases he MSE decreases eve whe he correlao bewee he deede varables ad he varace of he radom error are large. I all suaos, he roosed esmaors have smaller MSE ha he ordary leas squared ad some oher exsg esmaors. Key words: Lear Model; LSE; MSE; Moe Carlo smulaos; Mulcolleary; Rdge Regresso; AMS Subjec classfcao: Prmary 6J07, Secodary 6F. Iroduco I mos of he emrcal wors racoers ofe cocer abou he secfcao of he models uder cosderao, esecally wh regards o roblems assocaed wh he resduals, wh he am of assessg whe ose errors whch oher word mles ha he model s well secfed. Model mssecfcao ca be due o omsso of oe or several releva varables, cluso of u-ecessary varables, wrog fucoal form, mssecfed dyamcs, auocorrelao, heeroscedascy, ec. I he raccal wor, s recommeded ha racoers should coduc some dagosc ess order o assure he wheess of he model uder cosderao, oherwse he esmaed resuls ca be effce, based or cosse.

However, here are oher roblems ha also mgh flue he resuls wrog dreco, e.g. mulcolleary. Ths roblem haes suaos whe he exlaaory varables are hghly er-correlaed. The, becomes dffcul dseagle he searae effecs of each of he exlaaory varables o he exlaed varable. As a resul, esmaed arameers ca be wrogly -sgfca or have (uexecedly) wrog sgs. Noe ha mulcolleary s more a roblem wh he daa ha wh he model self, ad hece hs d of roblems ca o be defed by resdual aalyss. As a resul, a commo defcecy may aled sudes s he absece of ayg serous aeo o hs roblem. Ideed, alhough model mssecfcao s a mora area he sascal modellg, mulcolleary s a mora ssue oo. The hsory of mulcolleary daes bac a leas o he aer by Frsch (934) who roduced he coce o deoe a suao where he varables deal wh are subjec o wo or more relaos. Oe way o deal wh hs roblem s called he rdge regresso, frs roduced by Horel ad Keard (970 a,b). A hs sage, he ma eres les fdg a value of he rdge arameer, say K, such ha he reduco he varace erm of he sloe arameer s greaer ha he crease he squared bas of. The auhors roved ha here s a ozero value of such rdge arameer for whch he Mea Squared Errors (MSE) for he slo arameer usg he rdge regresso s smaller ha he varace of he Ordary Leas Square (OLS) esmaor of he resecve arameer. May auhors hereafer wored wh hs area of research ad develoed ad roosed dffere esmaes for he rdge regresso arameer. To meo few, McDoald ad Galareau (975), Lawless ad Wag (976), Saleh ad Kbra (996), Haq ad Kbra (996), Kbra (003), Khalaf ad Shuur (005) ad Alhams, Khalaf ad Shuur (006). I Kbra (003) ad Alhams, Khalaf ad Shuur (006), he auhors used smulao echques o sudy he roeres of some ew roosed esmaors ad comared her roeres wh some oular exsg esmaors. Uder cera codos, hey foud ha he MSEs of some of he ew roosed esmaors are smaller ha he corresodg MSE of he OLS esmaor ad oher ow exsg esmaors. Recely, Muz ad Kbra (009) develoed 5 ew rdge arameers based o Kbra (003) ad Khalaf ad Shuur (005) models wh wo exlaaory varables. They foud he ew arameers ouerform he revous oes erm of smaller MSEs.

I hs aer we am o exed he sudy by Muz ad Kbra (009) by develog 9 more ew rdge arameers ad o crease he dmeso of he models by cludg more exlaaory varables. We also sudy models wh 4 exlaaory varables ha are more realsc emrcal wor ha models wh oly varables. Processg hs maer, s ossble o vesgae he effec of he exra cluded varables o he MSEs. The aer s orgased as follows: I seco we rese he model we aalyse, ad gve he formal defo of he rdge regresso arameers used hs sudy. I Seco 3, he desg of our Moe Carlo exerme ogeher wh he facors ha ca affec he small samle roeres of hese roosed arameers are roduced. I Seco 4 we descrbe he resuls cocerg he varous arameers erm of MSE. The coclusos of he aer are reseed seco 5.. Mehodology I hs seco we rese he roosed rdge regresso esmaors. Ths cludes a bref bacgroud o he mehods suggesed by Hoerl ad Keard (970a), ad ha develoed by Khalaf ad Shuur (005), Alhams ad Shuur (008), Alhams, Khalaf ad Shuur (006) ad Muz ad Kbra (009). Moreover, he ew rdge arameer, (deoed by K AS ), ogeher wh he oher fve ew versos. Noaos ad some relmares The mulle lear regresso model ca be exressed as: y Xβ + e, (.) where y s a vecor of resoses, X s a observed marx of he regressors, β s a vecor of uow arameers, ad e s a vecor of errors. The ordary leas square esmaor (OLS) of he regresso coeffces β s defed as ( X ' X ) β X y, (.) Suose, here exss a orhogoal marx D such ha D CD Λ, where Λ dag λ, λ,..., λ ) are he egevalues of he marx C X X. The orhogoal (caocal ( form) verso of he mulle regresso model (.) s Υ Χ * α + e 3

where Χ * ΧD ad α D β. I case he marx X X s ll-codoed however ( he sese of here s a ear-lear deedecy amog he colums of he marx) he OLS of β has a large varace, ad mulcolleary s sad o be rese. Rdge regresso relaces X X wh X X + I, ( 0 ). The he geeralzed rdge regresso esmaors of α are gve as follows: ( X * X * + I ) X Y ( α ) * (.3) where dag,,..., ), 0 ad α Λ Χ * Υ ( s he ordary leas squares (OLS) esmaes of α. Accordg o Hoerl ad Keard (970) he value of whch mmzes he MSE ( ( α )) s, (.4) α where eleme of α. rereses he error varace of he mulle regresso model, ad α s he h. Proosed Esmaors I hs seco, we revew some already avalable esmaors ad roose some ew rdge arameers... Esmaors based o Hoerl ad Keard (970) Hocg, Seed ad Ly (976) showed ha for ow omal, he geeralzed rdge regresso esmaor s sueror o all oher esmaors wh he class of based esmaors hey cosdered. Neverheless, he omal value of fully deeds o he uow ad α, ad hey mus be esmaed from he observed daa. Hoerl ad Keard (970), suggesed o relace ad α by her corresodg ubased esmaors (.4). Tha s, (.5) α where s he resdual mea square esmae, whch s ubased esmaor of he h eleme of α, whch s a ubased esmaor of α. ad α s 4

Hoerl ad Keard (970) suggesed o be where HK HK (.6) α α s he mum eleme of α. Now, whe gve smaller MSE ha he OLS. ad α are ow he HK wll Hoerl e al. (975), roosed a dffere esmaor of by ag he harmoc mea of. Tha s HK HKB (.7) α α α.. Esmaors based o Kbra (003) Kbra (003) roosed some ew esmaors based o geeralzed rdge regresso aroach. They are as follows: By usg he geomerc mea of, whch roduces he followg esmaor K GM α (.8) By usg he meda of, whch roduces he followg esmaor for 3 K MED Meda, α,,..., (.9)..3. Esmaors based o Khalaf ad Shuur (005) Khalaf ad Shuur (005) suggesed a ew mehod o esmae he rdge arameer, as a modfcao of HK as S KS (.) ( ) + α where s he mum egevalue of X X marx 5

6 Followg Kbra (003) ad Khalaf ad Shuur (005), Alhams Khalaf ad Shuur (006) roosed he followg esmaors for : ( ) + KS arh S α,,...,, (.) + 3 ) ( KS S α,,...,, (.) + 4 ) ( KS md S meda α. (.3)..4 Some roosed ew esmaors Followg Kbra (003), Khalaf ad Shuur (005), Alhams Khalaf ad Shuur (006) ad Alhams ad Shuur (008), we roosed he followg esmaors. Frs, followg Kbra (003) ad Khalaf ad Shuur (005), we roose he followg esmaor KS gm ) ( + α (.4) Now, usg equao (.4) ad square roo rasformaos (Alhams ad Shuur (008)), we roose he followg esmaors: α (.5) 3 α (.6) 4 α (.7) 5 α (.8)

6 meda α (.9) 7 meda α (.0) 8 ( ) + α (.) 9 ( ) + α (.) ( ) + α (.3) ( ) + α (.4) meda ( ) + α (.5) Noe ha he ew roosed esmaors: S (.), S (.), 3 (.6), 7.0), 8 (.), 9 (.), (.3), (.4) ad fally (.5) were o vesgaed Muz ad Kbra (009). ( 7

3. The Moe Carlo Desg The am of hs aer s o comare he erformace of our ew roosed esmaors he oher esmaors ogeher wh he OLS. Sce a heorecal comarso s o ossble, a smulao sudy has bee coduced hs seco. The desg of a good smulao sudy s deede o () wha facors are execed o affec he roeres of he esmaors uder vesgao ad () wha crera are beg used o judge he resuls. Sce rdge esmaors are suosed o have smaller MSE comared o OLS, he MSE wll be used as crera o measure he goodess of a esmaor, whle he frs queso wll be reaed shorly. All all abou 0 dffere esmaors have bee dscussed hs aer. From he relmary smulao sudy we have, however, seleced he followg bes 5 esmaors: HK, K, K, S, S,,, 4, 5, 6, 8, 9,,,. Amog hem he las 5 are ewly roosed. ad Sce he degree of colleary amog he exlaaory varable are of ceral morace, we followed Muz ad Kbra (009) geerag he exlaaory varable usg he followg devce, x j ( ) ( / ) zj + z,,,... j,,... (3.) where rereses he correlao bewee he exlaaory varables, ad z j are deede sadard seudo-radom umbers. The observaos for he deede varable are he deermed by: wher y β + β x + β x +... + β x + e e are..d. N (, ) of geeraly. 0,,,... (3.) 0 seudo-radom umbers, ad β 0 s ae o be zero whou loss Facors ha vary he Moe Carlo smulaos Sce our rmary eres les he erformace of our roosed esmaors accordg o he sregh of he mulcolleary, we used dffere degrees of correlao bewee he varables ad le ( ) 0.7, 0.8 ad 0.9. We also wa o see he effec of he samle szes o he erformace of he esmaors. Therefore, hs sudy, we cosdered, 0, 30, 40, 50 ad 0 whch wll cover models wh small, medum ad large samle szes. The umber of 8

he exlaaory varables s also of grea morace sce he bad mac of he colleary o he MSE mgh be sroger whe more varables he model are correlaed. We hece geeraed models wh ad ad 4 exlaaory varables. To see wheher he magude error varace have a sgfca effec of he erformaces of he roosed esmaors, we used dffere values of he error sadard devaos 0.0, 0.5,, 3, ad 5. For each se of exlaaory varables we cosdered he coeffce vecor ha corresoded o he larges ege value of X X marx subjec o he cosra ha β β. Newhouse ad Oma (97) saed ha f he mea squared error (MSE) s a fuco of β,, ad, ad f he exlaaory varables are fxed, he he MSE s mmzed whe we choose hs coeffce vecor. For gve values of,, β,, ad, he se of exlaaory varables are geeraed. The he exerme was reeaed 000 mes ad he average mea squared error was calculaed for all 5 esmaors. 4. Resuls Dscussos I hs seco we rese he resuls of our Moe Carlo exerme cocerg he MSEs of he dffere roosed esmaors comare o he OLS. A coveoal way o reor he resuls of a Moe Carlo exerme s o abulae he values of hese MSEs uder dffere codos. Whe deermg he maer of reseao, some accou has o be ae o he resuls obaed. Our orgal eo was o sar by reseg resuls for all he ma effecs erm of ables. However, sce he resuls are oo exesve, reseg he resuls erm of ables wll mae dffcul o follow he head le of he fdgs. We hece rese our mos mora fdgs form of fgures whch summares mos of he resuls hs wh resec o he dffere facures uder vesgaos. More exac resuls of he smulaed MSEs for he 5 esmaors are rovded he aedx (all resuls are o cluded he ables however bu are avalable u o reques from he auhors). Smulaed MSEs for fxed, ad ad dffere values of are reseed Table A., for fxed, ad ad dffere values of are reseed Table A., for fxed, ad ad dffere values of are reseed Table A.3. 9

The erformace of he esmaors wh resec o chages. I Table A. we have rovded he MSEs of he esmaors as a fuco of he varace of he errors ( ). Whe he value of creases, he MSE of he esmaors also creases. For all values of, he rdge regresso esmaors have smaller MSE ha he OLS. However, he erformace of he roosed esmaors 4, 5, 8,,, ad K, K s beer ha he erformace of he res of he aalyzed esmaors. Ths behavor was almos cosa for ay samle sze ad umber of varables cosdered. However, whe he sadard devao s large,.e. ( 5) roducg less MSE., he ew 8, ouerform all he oher esmaors erm of For gve 0. 70 ad, he erformace of esmaors as a fuco of he sadard devao of he errors for ad 4 are rovded Fgures ad resecvely. From hese fgures we observe ha as he sadard devao creases, he MSE also creases. The same s rue whe shfg from o 4 varables models esecally for he OLS, HK, S, (see fgure ). S, 40 MSE 35 30 5 0 5 5 0 0.0 0.5 5 Error's sadard devao OLS HK K K S3 S4 4 5 6 8 9 Fgure Performace of he esmaors as a fuco of whe

MSE 40 0 0 80 60 40 0 0 0.0 0.5 5 Error's sadard devao OLS HK K K S3 S4 4 5 6 8 9 Fgure Performace of he esmaors as a fuco of whe 4 Performace as a fuco of I Table A. we have rovded he MSEs of he esmaors as a fuco of he correlao bewee he exlaaory varables. For smaller sgma ( 0. 0) he chage he correlao bewee he exlaaory varables had almos o effec o he MSEs. I all suaos hey remaed almos he same for ay samle sze or umber of arameers, ad her MSEs are very small. Whe creases, he hgher correlao bewee he deede varables resuled a crease of he MSE of he -esmaors. I geeral, 4, 5, 8,, ad K, K erformed beer ha ohers. For gve ad 4 he erformace of esmaors as a fuco of he correlao bewee he exlaaory varables for 0 ad 50 are rovded Fgures 3 ad 4 resecvely. From hese fgures we observed ha as correlao creases, he MSE also creases. The MSE decreases however whe he umber of observaos creases from 0 o 50. All of he rdge esmaors have smaller MSE comared wh OLS ad hey are very close o each oher.

MSE.5.5 0.5 0 0.7 0.8 0.9 Correlao OLS HK K K S3 S4 4 5 6 8 9 Fgure 3 Performace of he esmaors as a fuco of whe 0 MSE 0.7 0.6 0.5 0.4 0.3 0. 0. 0 0.7 0.8 0.9 Correlao OLS HK K K S3 S4 4 5 6 8 9 Fgure 4 Performace of he esmaors as a fuco of whe 50 Performace as a fuco of I Table A.3 we have rovded he MSEs of he esmaors as a fuco of he samle sze. We observed ha, geeral, whe he samle sze creases he MSE decreases, or remaed he same. Eve for he large values of ad, f we crease he samle sze he MSE of esmaors decrease. Aga hs suao, as ad creased he erformace of 4, 5, 8,,, ad K, K s beer ha he res of he -esmaors.

For gve 0. 90 ad, he erformace of he esmaors as a fuco of he samle sze for 0. 5 ad 5 are rovded Fgures 5 ad 6 resecvely. From hese fgures, we observed ha as he samle sze creases, he MSE decreases. Exce for a few suaos, hs aer was cosa for all of he esmaors. Noe he huge crease he MSE whe shfg from 0. 5 o 5. MSE 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.000 0.00 0.0000 0 30 40 50 0 Samle sze OLS HK K K S3 S4 4 5 6 8 9 Fgure 5 Performace of he esmaors as a fuco of whe 0. 5 MSE 70.0000 60.0000 50.0000 40.0000 30.0000 0.0000.0000 0.0000 0 30 40 50 0 Samle sze OLS HK K K S3 S4 4 5 6 8 9 Fgure 6 Performace of he esmaors as a fuco of whe 5 3

5. Cocludg Remars I hs aer we have revewed ad roosed some ew esmaors for esmag he rdge arameer. The ew roosed esmaors are defed based o he wor of Kbra (003), Khalaf ad Shuur (005) ad Alhams Khalaf ad Shuur (006). The erformace of he esmaors deeds o he varace of he radom error ( ), he correlaos amog he exlaaory varables ( ), he samle sze ( ) ad he uow coeffces vecors β. Based o he smulao sudy, some coclusos mgh be draw. However, hese coclusos mgh be resrced o he se of exermeal codos whch are vesgaed. We used he MSE crera o measure he goodess of he esmaors. The crease of umber of correlaed varable, ad he crease of he correlao bewee he deede varables have a egave effec he MSE, he sese ha also creases. Whe he samle sze creases he MSE decreases, eve whe he correlao bewee he deede varables ad are large. I all suaos, he roosed esmaors have smaller MSE ha he ordary leas squared esmaors. Fve of hem, 4, 5, 8,, K, K, ad he erformed beer ha he res he sese of smaller MSE. Fally, aears ha he roosed esmaors 4, 5, 8,, are useful ad may be recommeded o he racoers. The 8, esmaors are arcularly also recommeded whe worg wh model wh large resdual varaces sce hey ouerform all he ohers such cases. Refereces Alhams, M., Khalaf, G., Shuur, G. (006). Some modfcaos for choosg rdge arameers. Commucaos Sascs- Theory ad Mehods, 35: 005-00. Alhams, M., ad Shuur, G. (008). Develog rdge arameers for SUR model. Commucaos Sascs- Theory ad Mehods 37(4): 544-564. Demser, A.P., Schazoff, M., Wermuh, N. (977) A smulao sudy of aleraves o ordary leas squares. Joural of he Amerca Sascal Assocao 7:77-9. Frsch, R. (934). Sascal Cofluece Aalyss by Meas of Comlee Regresso Sysems, Publcao 5 (Oslo: Uversy Isue of Ecoomcs, 934). Galo, Sr Fracs. (885). Regresso owards medocry heredy saure. Joural of Ahroologcal Isue 5: 46-63. Gbbos, D.G. (98). A smulao sudy of some rdge esmaors. Joural of he Amerca 4

Sascal Assocao 76:3-39. Hocg, R.R., Seed, F.M., Ly, M.J. (976). A class of based esmaors lear regresso. Techomercs 8: 45-438 Hoerl, A.E., Keard R.W. (970). Rdge regresso: based esmao for o-orhogoal roblems. Techomercs : 55-67. Hoerl, A.E., Keard, R.W., Baldw, K.F. (975) Rdge regresso: some smulao. Commucaos Sascs 4: 5-3 Khalaf, G., Shuur, G. (005) Choosg rdge arameers for regresso roblems. Commucaos Sascs- Theory ad Mehods, 34:77-8. Kbra B.M.G. (003). Performace of some ew rdge regresso esmaors Commucaos Sascs- Theory ad Mehods, 3:49-435. Lawless, J.F., Wag, P.(976). A smulao sudy of rdge ad oher regresso esmaors. Commucaos Sascs A, 5: 307-33. McDoald, G.C., Galareau D.I. (975) A Moe Carlo evaluao of some rdge-ye esmaors. Joural of he Amerca Sascal Assocao 70:407-46. Mogomery, D.C., Pec, E.A. ad Vg, G.G. (00) Iroduco o lear regresso aalyss, Thrd Edo, Joh Wley, New Yor. Muz, G., ad Kbra, B. M. G. (009). O some rdge regresso esmaors: A Emrcal Comarsos. Commucaos Sascs-Smulao ad Comuao, 38:3, 6-630. Myers, R.H. (990). Classcal ad Moder Regresso wh Alcaos, secod edo. Duxbury. Belmo,CA. Newhouse, J.P., Oma S.D. (97). A evaluao of rdge esmaors. Rad Cororao, P- 76-PR. Sgh, S., Tracy, D.S. (999). Rdge-regresso usg scrambled resoses. Mera 47-57. 5

APPENDIX A Table A. Smulaed MSE for fxed,, ad ad dffere values of,, 0.7 \ OLS HK K K S3 S4 4 5 6 8 9 0.0 0.47 0.46 0.46 0.46 0.46 0.46 0.389 0.5 0.9 0.7 0.5 0.5 0.389 0.9 0.7 0.5 0.5 0.765 0.56 0.336 0.69 0.57 0.57 0.304 0.7 0.7 0.89 0.7 0.36 0.304 0.49 0. 0.36.799.05 0.438 0.97.04.04 0.479 0.70 0.3 0.5 0.70 0.43 0.479 0.63 0.9 0.43 5 36.39 8.75 3.898.70 8.36 8.36 6.05.733.446.04.733 0.37 6.05 0.547.373 0.37,, 0.8 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.073 0.073 0.073 0.073 0.073 0.073 0.057 0. 0.094 0.05 0. 0. 0.057 0.095 0.05 0.0 0.5 0.438 0.44 0.7 0.084 0.43 0.43 0.087 0.065 0.053 0.047 0.065 0.084 0.087 0.063 0.048 0.084.608 0.866 0.54 0.33 0.840 0.840 0.7 0. 0.084 0.08 0. 0.089 0.7 0.075 0.5 0.089 5 38.59 0.7 4.069.577 0.9 0.9 6.65.7.384.38.7 0.7 6.65 0.477 3.35 0.7,, 0.9 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.073 0.073 0.073 0.073 0.073 0.073 0.05 0.0 0.094 0.00 0.0 0.05 0.094 0.00 0. 0.05. 0.5 0.68 0.39 0.5 0.076 0.33 0.33 0.087 0.063 0.05 0.043 0.063 0.085 0.087 0.063 0.04 0.085.445.67 0.96 0.37.9.9 0.30 0.33 0.95 0.08 0.33 0.094 0.30 0.074 0.9 0.090 5 6.65 3.8 4.630.54 30.49 30.49 7.3.890.4..890 0.56 7.3 0.438.03 0.56 0,, 0.7 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.035 0.03 0.03 0.03 0.03 0.03 0.030 0.03 0.03 0.00 0.03 0.03 0.030 0.03 0.00 0.03 0.5 0.090 0.06 0.045 0.04 0.07 0.07 0.043 0.09 0.05 0.05 0.09 0.038 0.043 0.09 0.08 0.038 0.64 0.5 0.080 0.064 0.09 0.09 0.9 0.059 0.047 0.043 0.059 0.040 0.9 0.035 0.068 0.040 5 5.59.83.0 0.645 4.99 4.99.393.98 0.853 0.54 0.98 0.6.393 0.46.93 0.6 0,, 0.8 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.89 0.89 0.89 0.89 0.89 0.89 0.74 0.055 0.057 0.3 0.055 0.055 0.74 0.057 0.3 0.057 0.5 0.334 0.5 0.49 0.0 0.55 0.55 0.56 0.077 0.077 0.086 0.077 0.05 0.56 0.065 0.7 0.059 0.776 0.444 0.8 0.7 0.535 0.535 0.65 0.7 0.099 0.098 0.7 0.06 0.65 0.07 0.46 0.06 5 3.77 6.66.3 0.69 8.96 8.96 3.500.65 0.703 0.5.65 0.3 3.500 0.7.80 0.3 0,, 0.9 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.03 0.03 0.03 0.03 0.03 0.03 0.05 0.03 0.03 0.0 0.03 0.03 0.05 0.03 0.0 0.03 0.5 0.90 0.4 0.049 0.035 0. 0. 0.047 0.08 0.03 0.00 0.08 0.040 0.047 0.09 0.0 0.040 0.666 0.340 0.094 0.056 0.49 0.49 0.55 0.050 0.037 0.037 0.050 0.04 0.55 0.034 0.058 0.04 5 5.76 7.69.44 0.599 9.60 9.60 3.440.303 0.635 0.457.303 0.093 3.440 0.76.4 0.093 50,, 0.7 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.03 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.03 0.008 0.0 0.0 0.0 0.0 0.008 0.0 0.5 0.07 0.00 0.06 0.05 0.05 0.05 0.08 0.0 0.0 0.0 0.0 0.06 0.08 0.0 0.06 0.06 0.076 0.046 0.07 0.03 0.07 0.07 0.049 0.03 0.00 0.07 0.03 0.06 0.049 0.04 0.030 0.06 5.56 0.783 0.33 0.3.444.444 0.994 0.434 0.39 0.04 0.434 0.09 0.994 0.075 0.578 0.09 50,, 0.8 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.073 0.07 0.07 0.07 0.07 0.07 0.068 0.00 0.0 0.054 0.00 0.00 0.068 0.0 0.054 0.00 0.5 0.096 0.070 0.053 0.047 0.089 0.089 0.067 0.033 0.03 0.035 0.033 0.0 0.067 0.05 0.048 0.0 0.7 0.7 0.06 0.049 0.56 0.56 0.6 0.050 0.043 0.039 0.050 0.0 0.6 0.06 0.067 0.0 5.38.84 0.388 0.50.099.099.46 0.467 0.3 0.7 0.467 0.035.46 0.08 0.66 0.035 6

50,, 0.9 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.03 0.0 0.0 0.0 0.0 0.0 0.009 0.00 0.05 0.004 0.0 0.0 0.009 0.05 0.004 0.0 0.5 0.07 0.039 0.07 0.0 0.054 0.054 0.03 0.0 0.008 0.007 0.0 0.06 0.03 0.0 0.009 0.06 0.48 0.3 0.033 0.09 0.89 0.89 0.079 0.00 0.03 0.03 0.00 0.06 0.079 0.0 0.05 0.06 5 6.3.978 0.464 0.99 4.57 4.57.858 0.45 0.5 0.69 0.45 0.06.858 0.05 0.539 0.06 0,, 0.7 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.0 0.006 0.006 0.006 0.006 0.006 0.005 0.009 0.008 0.004 0.009 0.009 0.005 0.008 0.004 0.009 0.5 0.04 0.0 0.008 0.007 0.03 0.03 0.0 0.005 0.005 0.005 0.005 0.008 0.0 0.006 0.006 0.008 0.039 0.03 0.03 0.0 0.038 0.038 0.08 0.0 0.009 0.008 0.0 0.008 0.08 0.007 0.07 0.008 5 0.8 0.4 0.63 0.3 0.779 0.779 0.57 0. 0.65 0. 0. 0.0 0.57 0.03 0.36 0.0 0,, 0.8 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.0 0.006 0.006 0.006 0.006 0.006 0.005 0.009 0.007 0.003 0.009 0.009 0.005 0.007 0.003 0.009 0.5 0.06 0.0 0.008 0.007 0.05 0.05 0.0 0.005 0.004 0.004 0.005 0.008 0.0 0.006 0.006 0.008 0.048 0.07 0.03 0.0 0.045 0.045 0.03 0.0 0.008 0.008 0.0 0.008 0.03 0.006 0.06 0.008 5.06 0.58 0.78 0.4 0.968 0.968 0.653 0. 0.50 0.096 0. 0.0 0.653 0.08 0.033 0.0 0,, 0.9 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.036 0.034 0.034 0.034 0.034 0.034 0.03 0.0 0.0 0.03 0.0 0.0 0.03 0.0 0.03 0.0 0.5 0.058 0.039 0.06 0.0 0.053 0.053 0.036 0.04 0.04 0.06 0.04 0.0 0.036 0.0 0.03 0.0 0.8 0.067 0.09 0.0 0.6 0.6 0.064 0.0 0.08 0.07 0.0 0.0 0.064 0.0 0.03 0.0 5.5.005 0. 0.8.86.86 0.960 0.6 0.3 0.099 0.6 0.03 0.960 0.09 0.03 0.03, 4, 0.7 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.84 0.83 0.83 0.83 0.83 0.83 0.07 0.59 0.3 0.063 0.59 0.59 0.33 0.3 0.063 0.59 0.5.48 0.847 0.90 0.7 0.664 0.664 0.33 0.40 0.4 0.5 0.40 0.4 0.34 0.3 0. 0.4 5.564 3.088 0.39 0.359.78.78 0.39 0.37 0.85 0.88 0.37 0.39 0.70 0.5 0.33 0.39 5 9.0 70. 6.667 6.9 50.63 50.63 6.5 6.86.659.66 6.86 0.568 4.7.587 3.9 0.568, 4, 0.8 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.70 0.69 0.69 0.69 0.69 0.69 0.9 0.69 0.90 0.93 0.69 0.69 0.93 0.90 0.93 0.69 0.5.04 6.365 0.34 0.89 3.308 3.308 0.64 0.7 0.3 0.3 0.7 0.83 0.409 0.05 0.34 0.83 47.6 4.49 0.66 0.495.43.43 0.485 0.384 0.54 0.83 0.306 0.384 0.96 0.47 0.359 0.96 5 90. 63.5.40 6.984 9.0 9.0 7.455 5.690.463.868 5.690 0.58.58.507 4.37 0.58, 4, 0.9 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.88 0.77 0.79 0.79 0.79 0.79 0.09 0.64 0.60 0.84 0.64 0.64 0.43 0.59 0.84 0.64 0.5 3.7.50 0.405 0.3 5.5 5.5 0.70 0.90 0.8 0.97 0.90 0.64 0.480 0.73 0.5 0.64 94.7 48.44 0.865 0.67.6.6 0.57 0.406 0.4 0.75 0.406 0.73.78 0.05 0.336 0.73 5 33 74 3.87 9.66 553.3 553.3 8.05 6.60.0.583 6.60 0.45 6.49.87 3.965 0.45 0, 4, 0.7 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.063 0.063 0.063 0.063 0.063 0.063 0.060 0.060 0.047 0.054 0.060 0.060 0.06 0.047 0.054 0.060 0.5 0.359 0.3 0.08 0.079 0. 0. 0.08 0.067 0.059 0.060 0.067 0.05 0.7 0.053 0.068 0.05.85 0.646 0. 0.4 0.673 0.673 0.60 0.45 0.097 0.093 0.45 0.057 0.88 0.075 0. 0.057 5 8.5 4.95.64.495 5.5 5.5.85.599.390.08.599 0.09 5.958 0.785.9 0.09 0, 4, 0.8 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.063 0.064 0.064 0.063 0.063 0.063 0.063 0.063 0.5 0.367 0.4 0. 0. 0.80 0.80 0. 0.087 0.080 0.08 0.087 0.067 0.54 0.076 0.088 0.067.95 0.756 0.8 0.96 0.936 0.936 0.98 0.54 0.9 0. 0.54 0.069 0.49 0.090 0.39 0.069 5 30.56 6.70.5 3.0.69.69 3.336.67.47.04.67 0.69 8.985 0.588.850 0.69 7

0, 4, 0.9 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.0630 0.060 0.06 0.06 0.06 0.06 0.049 0.06 0.057 0.0 0.06 0.06 0.054 0.057 0.05 0.06 0.5 0.579 0.344 0. 0.3 0.49 0.49 0.085 0.070 0.063 0.064 0.070 0.059 0.7 0.060 0.070 0.059.099.70 0. 0.48.500.500 0.79 0.36 0.083 0.087 0.36 0.06 0.530 0.070 0. 0.06 5 5.09 7.7 3. 35.6 35.6 3.55.74 8.54 0.835 0.80.74 0.9.745 0.46.478 0.9 50, 4, 0.7 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.03 0.0 0.0 0.0 0.0 0.0 0.08 0.0 0.06 0.06 0.0 0.0 0.0 0.06 0.06 0.0 0.5 0.07 0.049 0.06 0.07 0.066 0.066 0.030 0.03 0.00 0.00 0.03 0.09 0.047 0.08 0.04 0.09 0. 0.4 0.040 0.043 0.9 0.9 0.069 0.047 0.034 0.03 0.047 0.09 0.6 0.04 0.046 0.09 5 4.868.686 0.489 0.55 4.43 4.43.386 0.969 0.57 0.408 0.969 0.04.87 0.5 0.8 0.04 50, 4, 0.8 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.0 0.00 0.00 0.00 0.00 0.00 0.00 0.0 0.0 0.0 0.0 0.0 0.00 0.0 0.0 0.00 0.5 0. 0.073 0.035 0.035 0.098 0.098 0.037 0.09 0.08 0.08 0.09 0.03 0.06 0.06 0.03 0.03 0.374 0.4 0.056 0.059 0.39 0.39 0.078 0.053 0.037 0.037 0.053 0.03 0.76 0.030 0.050 0.03 5 9. 4.978 0.698 0.9 7.735 7.735.48 0.94 0.394 0.356 0.94 0.040 4.065 0.76 0.730 0.040 50, 4, 0.9 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.03 0.0 0.0 0.0 0.0 0.0 0.05 0.0 0.07 0.05 0.0 0.0 0.09 0.07 0.05 0.0 0.5 0.66 0.099 0.03 0.033 0.38 0.38 0.030 0.0 0.09 0.09 0.0 0.00 0.07 0.08 0.0 0.00 0.600 0.335 0.063 0.07 0.494 0.494 0.074 0.045 0.07 0.08 0.045 0.00 0.35 0.0 0.040 0.00 5 4.84 8.068 0.999.504.04.04.56.03 0.34 0.30.03 0.035 5.570 0.39 0.635 0.035 0, 4, 0.7 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.008 0.0 0.008 0.008 0.0 0.0 0.0 0.008 0.008 0.0 0.5 0.030 0.0 0.0 0.0 0.09 0.09 0.05 0.0 0.0 0.0 0.0 0.009 0.03 0.009 0.0 0.009 0.089 0.053 0.09 0.00 0.085 0.085 0.037 0.04 0.08 0.06 0.04 0.0 0.063 0.0 0.04 0.0 5.966.067 0.3 0.45.879.879 0.77 0.440 0.53 0.99 0.440 0.06.365 0.3 0.43 0.06 0, 4, 0.8 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.009 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.040 0.07 0.05 0.05 0.038 0.038 0.08 0.03 0.0 0.0 0.03 0.0 0.08 0.0 0.04 0.0 0.3 0.077 0.03 0.05 0.3 0.3 0.04 0.06 0.08 0.07 0.06 0.0 0.083 0.03 0.05 0.0 5 3.065.679 0.78 0.330.869.869 0.86 0.458 0.9 0.80 0.458 0.05.889 0.084 0.409 0.05 0, 4, 0.9 OLS HK K K S3 S4 4 5 6 8 9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.069 0.043 0.08 0.08 0.063 0.063 0.09 0.03 0.0 0.0 0.03 0.0 0.040 0.0 0.04 0.0 0.36 0.3 0.09 0.035 0.3 0.3 0.045 0.06 0.06 0.06 0.06 0.0 0.5 0.03 0.04 0.0 5 5.757 3.39 0.40 0.599 5.95 5.95 0.93 0.43 0.65 0.58 0.43 0.05.967 0.068 0.369 0.05 8

Table A. Smulaed MSE for fxed,, ad ad dffere values of,, 0.0 OLS HK K K S3 S4 4 5 6 8 9 0.70 0.47 0.46 0.46 0.46 0.46 0.46 0.389 0.5 0.9 0.7 0.5 0.5 0.389 0.9 0.7 0.5 0.80 0.073 0.073 0.073 0.073 0.073 0.073 0.057 0. 0.094 0.05 0. 0. 0.057 0.095 0.05 0.0 0.90 0.073 0.073 0.073 0.073 0.073 0.073 0.05 0.0 0.094 0.00 0.0 0.05 0.094 0.00 0. 0.05.,, OLS HK K K S3 S4 4 5 6 8 9 0.70.799.05 0.438 0.97.04.04 0.479 0.70 0.3 0.5 0.70 0.43 0.479 0.63 0.9 0.43 0.80.608 0.866 0.54 0.33 0.840 0.840 0.7 0. 0.084 0.08 0. 0.089 0.7 0.075 0.5 0.089 0.90.445.67 0.96 0.37.9.9 0.30 0.33 0.95 0.08 0.33 0.094 0.30 0.074 0.9 0.090,, 5 OLS HK K K S3 S4 4 5 6 8 9 0.70 36.39 8.75 3.898.70 8.36 8.36 6.05.733.446.04.733 0.37 6.05 0.547.373 0.37 0.80 38.59 0.7 4.069.577 0.9 0.9 6.65.7.384.38.7 0.7 6.65 0.477 3.35 0.7 0.90 6.65 3.8 4.630.54 30.49 30.49 7.3.890.4..890 0.56 7.3 0.438.03 0.56, 4, 0.0 OLS HK K K S3 S4 4 5 6 8 9 0.70 0.84 0.83 0.83 0.83 0.83 0.83 0.07 0.59 0.3 0.063 0.59 0.59 0.33 0.3 0.063 0.59 0.80 0.70 0.69 0.69 0.69 0.69 0.69 0.9 0.69 0.90 0.93 0.69 0.69 0.93 0.90 0.93 0.69 0.90 0.88 0.77 0.79 0.79 0.79 0.79 0.09 0.64 0.60 0.84 0.64 0.64 0.43 0.59 0.84 0.64, 4, OLS HK K K S3 S4 4 5 6 8 9 0.70 5.564 3.088 0.39 0.359.78.78 0.39 0.37 0.85 0.88 0.37 0.39 0.70 0.5 0.33 0.39 0.80 47.6 4.49 0.66 0.495.43.43 0.485 0.384 0.54 0.83 0.306 0.384 0.96 0.47 0.359 0.96 0.90 94.7 48.44 0.865 0.67.6.6 0.57 0.406 0.4 0.75 0.406 0.73.78 0.05 0.336 0.73, 4, 5 OLS HK K K S3 S4 4 5 6 8 9 0.70 9 70. 6.667 6.9 50.63 50.63 6.5 6.86.659.66 6.86 0.568 4.7.587 3.9 0.568 0.80 90 63.5.40 6.984 9 9 7.455 5.690.463.868 5.690 0.58.58.507 4.37 0.58 0.90 33. 74 3.87 9.66 553.3 553.3 8.05 6.60.0.583 6.60 0.45 6.49.87 3.965 0.45 0,, 0.0 OLS HK K K S3 S4 4 5 6 8 9 0.70 0.035 0.03 0.03 0.03 0.03 0.03 0.030 0.03 0.03 0.00 0.03 0.03 0.030 0.03 0.00 0.03 0.80 0.89 0.89 0.89 0.89 0.89 0.89 0.74 0.055 0.057 0.3 0.055 0.055 0.74 0.057 0.3 0.057 0.90 0.03 0.03 0.03 0.03 0.03 0.03 0.05 0.03 0.03 0.0 0.03 0.03 0.05 0.03 0.0 0.03 0,, OLS HK K K S3 S4 4 5 6 8 9 0.70 0.64 0.5 0.080 0.064 0.09 0.09 0.9 0.059 0.047 0.043 0.059 0.040 0.9 0.035 0.068 0.040 0.80 0.776 0.444 0.8 0.7 0.535 0.535 0.65 0.7 0.099 0.098 0.7 0.06 0.65 0.07 0.46 0.06 0.90 0.666 0.340 0.094 0.056 0.49 0.49 0.55 0.050 0.037 0.037 0.050 0.04 0.55 0.034 0.058 0.04 0,, 5 OLS HK K K S3 S4 4 5 6 8 9 0.70 5.59.83.0 0.645 4.99 4.99.393.98 0.853 0.54 0.98 0.6.393 0.46.93 0.6 0.80 3.77 6.66.3 0.69 8.96 8.96 3.500.65 0.703 0.5.65 0.3 3.500 0.7.80 0.3 0.90 5.76 7.69.44 0.599 9.60 9.60 3.440.303 0.635 0.457.303 0.093 3.440 0.76.4 0.093 9

0, 4, 0.0 OLS HK K K S3 S4 4 5 6 8 9 0.70 0.063 0.063 0.063 0.063 0.063 0.063 0.060 0.060 0.047 0.054 0.060 0.060 0.06 0.047 0.054 0.060 0.80 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.063 0.064 0.064 0.063 0.063 0.063 0.063 0.063 0.90 0.063 0.06 0.06 0.06 0.06 0.06 0.049 0.06 0.057 0.0 0.06 0.06 0.054 0.057 0.05 0.06 0, 4, OLS HK K K S3 S4 4 5 6 8 9 0.70.85 0.646 0. 0.4 0.673 0.673 0.60 0.45 0.097 0.093 0.45 0.057 0.88 0.075 0. 0.057 0.80.95 0.756 0.8 0.96 0.936 0.936 0.98 0.54 0.9 0. 0.54 0.069 0.49 0.090 0.39 0.069 0.90.099.70 0. 0.48.500.500 0.79 0.36 0.083 0.087 0.36 0.06 0.530 0.070 0. 0.06 0, 4, 5 OLS HK K K S3 S4 4 5 6 8 9 0.70 8.5 4.95.64.495 5.5 5.5.85.599.390.08.599 0.09 5.958 0.785.9 0.09 0.80 30.56 6.70.5 3.0.69.69 3.336.67.47.04.67 0.69 8.985 0.588.850 0.69 0.90 5.09 7.7 3. 35.6 35.6 3.55.74 8.54 0.835 0.80.74 0.9.745 0.46.478 0.9 50,, 0.0 OLS HK K K S3 S4 4 5 6 8 9 0.70 0.03 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.03 0.008 0.0 0.0 0.0 0.0 0.008 0.0 0.80 0.073 0.07 0.07 0.07 0.07 0.07 0.068 0.00 0.0 0.054 0.00 0.00 0.068 0.0 0.054 0.00 0.90 0.03 0.0 0.0 0.0 0.0 0.0 0.009 0.00 0.05 0.004 0.0 0.0 0.009 0.05 0.004 0.0 50,, OLS HK K K S3 S4 4 5 6 8 9 0.70 0.076 0.046 0.07 0.03 0.07 0.07 0.049 0.03 0.00 0.07 0.03 0.06 0.049 0.04 0.030 0.06 0.80 0.7 0.7 0.06 0.049 0.56 0.56 0.6 0.050 0.043 0.039 0.050 0.0 0.6 0.06 0.067 0.0 0.90 0.48 0.3 0.033 0.09 0.89 0.89 0.079 0.00 0.03 0.03 0.00 0.06 0.079 0.0 0.05 0.06 50,, 5 OLS HK K K S3 S4 4 5 6 8 9 0.70.56 0.783 0.33 0.3.444.444 0.994 0.434 0.39 0.04 0.434 0.09 0.994 0.075 0.578 0.09 0.80.38.84 0.388 0.50.099.099.46 0.467 0.3 0.7 0.467 0.035.46 0.08 0.66 0.035 0.90 6.3.978 0.464 0.99 4.57 4.57.858 0.45 0.5 0.69 0.45 0.06.858 0.05 0.539 0.06 50, 4, 0.0 OLS HK K K S3 S4 4 5 6 8 9 0.70 0.03 0.0 0.0 0.0 0.0 0.0 0.08 0.0 0.06 0.06 0.0 0.0 0.0 0.06 0.06 0.0 0.80 0.0 0.00 0.00 0.00 0.00 0.00 0.00 0.0 0.0 0.0 0.0 0.0 0.00 0.0 0.0 0.00 0.90 0.03 0.0 0.0 0.0 0.0 0.0 0.05 0.0 0.07 0.05 0.0 0.0 0.09 0.07 0.05 0.0 50, 4, OLS HK K K S3 S4 4 5 6 8 9 0.70 0. 0.4 0.040 0.043 0.9 0.9 0.069 0.047 0.034 0.03 0.047 0.09 0.6 0.04 0.046 0.09 0.80 0.374 0.4 0.056 0.059 0.39 0.39 0.078 0.053 0.037 0.037 0.053 0.03 0.76 0.030 0.050 0.03 0.90 0.600 0.335 0.063 0.07 0.494 0.494 0.074 0.045 0.07 0.08 0.045 0.00 0.35 0.0 0.040 0.00 50, 4, 5 OLS HK K K S3 S4 4 5 6 8 9 0.70 4.868.686 0.489 0.55 4.43 4.43.386 0.969 0.57 0.408 0.969 0.04.87 0.5 0.8 0.04 0.80 9. 4.978 0.698 0.9 7.735 7.735.48 0.94 0.394 0.356 0.94 0.040 4.065 0.76 0.730 0.040 0.90 4.84 8.068 0.999.504.04.04.56.03 0.34 0.30.03 0.035 5.570 0.39 0.635 0.035 0

0,, 0.0 OLS HK K K S3 S4 4 5 6 8 9 0.70 0.0 0.006 0.006 0.006 0.006 0.006 0.005 0.009 0.008 0.004 0.009 0.009 0.005 0.008 0.004 0.009 0.80 0.0 0.006 0.006 0.006 0.006 0.006 0.005 0.009 0.007 0.003 0.009 0.009 0.005 0.007 0.003 0.009 0.90 0.036 0.034 0.034 0.034 0.034 0.034 0.03 0.0 0.0 0.03 0.0 0.0 0.03 0.0 0.03 0.0 0,, OLS HK K K S3 S4 4 5 6 8 9 0.70 0.039 0.03 0.03 0.0 0.038 0.038 0.08 0.0 0.009 0.008 0.0 0.008 0.08 0.007 0.07 0.008 0.80 0.048 0.07 0.03 0.0 0.045 0.045 0.03 0.0 0.008 0.008 0.0 0.008 0.03 0.006 0.06 0.008 0.90 0.8 0.067 0.09 0.0 0.6 0.6 0.064 0.0 0.08 0.07 0.0 0.0 0.064 0.0 0.03 0.0 0,, 5 OLS HK K K S3 S4 4 5 6 8 9 0.70 0.8 0.4 0.63 0.3 0.779 0.779 0.57 0. 0.65 0. 0. 0.0 0.57 0.03 0.36 0.0 0.80.06 0.58 0.78 0.4 0.968 0.968 0.653 0. 0.50 0.096 0. 0.0 0.653 0.08 0.033 0.0 0.90.5.005 0. 0.8.86.86 0.960 0.6 0.3 0.099 0.6 0.03 0.960 0.09 0.03 0.03 0, 4, 0.0 OLS HK K K S3 S4 4 5 6 8 9 0.70 0.0 0.0 0.0 0.0 0.0 0.0 0.008 0.0 0.008 0.008 0.0 0.0 0.0 0.008 0.008 0.0 0.80 0.0 0.0 0.0 0.0 0.0 0.0 0.009 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.90 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0, 4, OLS HK K K S3 S4 4 5 6 8 9 0.70 0.089 0.053 0.09 0.00 0.085 0.085 0.037 0.04 0.08 0.06 0.04 0.0 0.063 0.0 0.04 0.0 0.80 0.3 0.077 0.03 0.05 0.3 0.3 0.04 0.06 0.08 0.07 0.06 0.0 0.083 0.03 0.05 0.0 0.90 0.36 0.3 0.09 0.035 0.3 0.3 0.045 0.06 0.06 0.06 0.06 0.0 0.5 0.03 0.04 0.0 0, 4, 5 OLS HK K K S3 S4 4 5 6 8 9 0.70.966.067 0.3 0.45.879.879 0.77 0.440 0.53 0.99 0.440 0.06.365 0.3 0.43 0.06 0.80 3.065.679 0.78 0.330.869.869 0.86 0.458 0.9 0.80 0.458 0.05.889 0.084 0.409 0.05 0.90 5.757 3.39 0.40 0.599 5.95 5.95 0.93 0.43 0.65 0.58 0.43 0.05.967 0.068 0.369 0.05

Table A.3 Smulaed MSE for fxed, ad ad dffere values of, 0.7, 0.0 OLS HK K K S3 S4 4 5 6 8 9 0.47 0.46 0.46 0.46 0.46 0.46 0.389 0.5 0.9 0.7 0.5 0.5 0.389 0.9 0.7 0.5 0 0.035 0.03 0.03 0.03 0.03 0.03 0.030 0.03 0.03 0.00 0.03 0.03 0.030 0.03 0.00 0.03 30 0. 0. 0. 0. 0. 0. 0.7 0.035 0.037 0.093 0.035 0.035 0.7 0.037 0.093 0.035 40 0.089 0.089 0.089 0.089 0.089 0.089 0.086 0.06 0.08 0.07 0.06 0.06 0.086 0.08 0.07 0.06 50 0.03 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.03 0.008 0.0 0.0 0.0 0.0 0.008 0.0 0 0.0 0.006 0.006 0.006 0.006 0.006 0.005 0.009 0.008 0.004 0.009 0.009 0.005 0.008 0.004 0.009, 0.7, 0.5 OLS HK K K S3 S4 4 5 6 8 9 0.765 0.56 0.336 0.69 0.57 0.57 0.304 0.7 0.7 0.89 0.7 0.36 0.304 0.49 0. 0.36 0 0.090 0.06 0.045 0.04 0.07 0.07 0.043 0.09 0.05 0.05 0.09 0.038 0.043 0.09 0.08 0.038 30 0.6 0.8 0.089 0.078 0.45 0.45 0.7 0.056 0.055 0.060 0.056 0.038 0.7 0.044 0.079 0.038 40 0.4 0.086 0.067 0.059 0.7 0.7 0.083 0.043 0.04 0.046 0.043 0.08 0.083 0.03 0.063 0.08 50 0.07 0.00 0.06 0.05 0.05 0.05 0.08 0.0 0.0 0.0 0.0 0.06 0.08 0.0 0.06 0.06 0 0.04 0.0 0.008 0.007 0.03 0.03 0.0 0.005 0.005 0.005 0.005 0.008 0.0 0.006 0.006 0.008, 0.7, OLS HK K K S3 S4 4 5 6 8 9.799.05 0.438 0.97.04.04 0.479 0.70 0.3 0.5 0.70 0.43 0.479 0.63 0.9 0.43 0 0.64 0.5 0.080 0.064 0.09 0.09 0.9 0.059 0.047 0.043 0.059 0.040 0.9 0.035 0.068 0.040 30 0.79 0.7 0.098 0.080 0.40 0.40 0.58 0.083 0.073 0.066 0.083 0.039 0.58 0.047 0.4 0.039 40 0.94 0.4 0.073 0.059 0.77 0.77 0.6 0.066 0.058 0.05 0.066 0.09 0.6 0.035 0.084 0.09 50 0.076 0.046 0.07 0.03 0.07 0.07 0.049 0.03 0.00 0.07 0.03 0.06 0.049 0.04 0.030 0.06 0 0.039 0.03 0.03 0.0 0.038 0.038 0.08 0.0 0.009 0.008 0.0 0.008 0.08 0.007 0.07 0.008, 0.7, 5 OLS HK K K S3 S4 4 5 6 8 9 36.39 8.75 3.898.70 8.36 8.36 6.05.733.446.04.733 0.37 6.05 0.547.373 0.37 0 5.59.83.0 0.645 4.99 4.99.393.98 0.853 0.54 0.98 0.6.393 0.46.93 0.6 30 4.47.77 0.757 0.465 3.654 3.654.04 0.795 0.553 0.40 0.795 0.07.04 0.60.0 0.07 40.685.378 0.534 0.358.373.373.470 0.587 0.46 0.95 0.587 0.049.470 0.4 0.786 0.049 50.56 0.783 0.33 0.3.444.444 0.994 0.434 0.39 0.04 0.434 0.09 0.994 0.075 0.578 0.09 0 0.8 0.4 0.63 0.3 0.779 0.779 0.57 0. 0.65 0. 0. 0.0 0.57 0.03 0.36 0.0, 0.8, 0.0 OLS HK K K S3 S4 4 5 6 8 9 0.073 0.073 0.073 0.073 0.073 0.073 0.057 0. 0.094 0.05 0. 0. 0.057 0.095 0.05 0.0 0 0.035 0.03 0.03 0.03 0.03 0.03 0.030 0.03 0.03 0.00 0.03 0.03 0.030 0.03 0.00 0.03 30 0.0 0.00 0.00 0.00 0.00 0.00 0.08 0.034 0.07 0.009 0.034 0.034 0.034 0.07 0.009 0.034 40 0.089 0.089 0.089 0.089 0.089 0.089 0.086 0.06 0.08 0.07 0.06 0.06 0.086 0.08 0.07 0.06 50 0.073 0.07 0.07 0.07 0.07 0.07 0.068 0.00 0.0 0.054 0.00 0.00 0.068 0.0 0.054 0.00 0 0.0 0.006 0.006 0.006 0.006 0.006 0.005 0.009 0.007 0.003 0.009 0.009 0.005 0.007 0.003 0.009, 0.8, 0.5 OLS HK K K S3 S4 4 5 6 8 9 0.438 0.44 0.7 0.084 0.43 0.43 0.087 0.065 0.053 0.047 0.065 0.084 0.087 0.063 0.048 0.084 0 0.334 0.5 0.49 0.0 0.55 0.55 0.56 0.077 0.077 0.086 0.077 0.05 0.56 0.065 0.7 0.059 30 0.084 0.05 0.03 0.05 0.065 0.065 0.03 0.07 0.04 0.03 0.07 0.06 0.03 0.09 0.06 0.06 40 0.045 0.09 0.00 0.08 0.039 0.039 0.03 0.03 0.0 0.0 0.03 0.09 0.03 0.04 0.03 0.09 50 0.096 0.070 0.053 0.047 0.089 0.089 0.067 0.033 0.03 0.035 0.033 0.0 0.067 0.05 0.048 0.0 0 0.06 0.0 0.008 0.007 0.05 0.05 0.0 0.005 0.004 0.004 0.005 0.008 0.0 0.006 0.006 0.008, 0.8, OLS HK K K S3 S4 4 5 6 8 9.608 0.866 0.54 0.33 0.840 0.840 0.7 0. 0.084 0.08 0. 0.089 0.7 0.075 0.5 0.089 0 0.776 0.444 0.8 0.7 0.535 0.535 0.65 0.7 0.099 0.098 0.7 0.06 0.65 0.07 0.46 0.06 30 0.73 0.43 0.053 0.037 0.07 0.07 0.099 0.038 0.07 0.05 0.038 0.07 0.099 0.043 0.07 0.099 40 0.40 0.076 0.036 0.07 0.0 0.0 0.068 0.08 0.0 0.09 0.08 0.00 0.068 0.06 0.034 0.00 50 0.7 0.7 0.06 0.049 0.56 0.56 0.6 0.050 0.043 0.039 0.050 0.0 0.6 0.06 0.067 0.0 0 0.048 0.07 0.03 0.0 0.045 0.045 0.03 0.0 0.008 0.008 0.0 0.008 0.03 0.006 0.06 0.008

, 0.8, 5 OLS HK K K S3 S4 4 5 6 8 9 38.59 0.7 4.069.577 0.9 0.9 6.65.7.384.38.7 0.7 6.65 0.477 3.35 0.7 0 3.77 6.66.3 0.69 8.96 8.96 3.500.65 0.703 0.5.65 0.3 3.500 0.7.80 0.3 30 6.36 3.05 0.788 0.430 4.653 4.653.68 0.76 0.469 0.39 0.76 0.058.68 0.4 0.898 0.058 40 3.33.549 0.480 0.94.660.660.490 0.559 0.37 0.5 0.559 0.038.490 0.08 0.7 0.038 50.38.84 0.388 0.50.099.099.46 0.467 0.3 0.7 0.467 0.035.46 0.08 0.66 0.035 0.06 0.58 0.78 0.4 0.968 0.968 0.653 0. 0.50 0.096 0. 0.0 0.653 0.08 0.033 0.0, 0.9, 0.0 OLS HK K K S3 S4 4 5 6 8 9 0.073 0.073 0.073 0.073 0.073 0.073 0.05 0.0 0.094 0.00 0.0 0.05 0.094 0.00 0. 0.05. 0 0.03 0.03 0.03 0.03 0.03 0.03 0.05 0.03 0.03 0.0 0.03 0.03 0.05 0.03 0.0 0.03 30 0.0 0.00 0.00 0.00 0.00 0.00 0.06 0.034 0.07 0.007 0.034 0.034 0.06 0.07 0.007 0.034 40 0.090 0.089 0.089 0.089 0.089 0.089 0.079 0.06 0.07 0.053 0.06 0.06 0.079 0.07 0.054 0.06 50 0.03 0.0 0.0 0.0 0.0 0.0 0.009 0.00 0.05 0.004 0.0 0.0 0.009 0.05 0.004 0.0 0 0.036 0.034 0.034 0.034 0.034 0.034 0.03 0.0 0.0 0.03 0.0 0.0 0.03 0.0 0.03 0.0, 0.9, 0.5 OLS HK K K S3 S4 4 5 6 8 9 0.68 0.39 0.5 0.076 0.33 0.33 0.087 0.063 0.05 0.043 0.063 0.085 0.087 0.063 0.04 0.085 0 0.90 0.4 0.049 0.035 0. 0. 0.047 0.08 0.03 0.00 0.08 0.040 0.047 0.09 0.0 0.040 30 0.3 0.066 0.030 0.0 0.087 0.087 0.035 0.07 0.04 0.03 0.07 0.07 0.035 0.09 0.05 0.07 40 0.95 0.5 0.070 0.053 0.5 0.5 0.084 0.035 0.035 0.039 0.035 0.07 0.084 0.09 0.050 0.07 50 0.07 0.039 0.07 0.0 0.054 0.054 0.03 0.0 0.008 0.007 0.0 0.06 0.03 0.0 0.009 0.06 0 0.058 0.039 0.06 0.0 0.053 0.053 0.036 0.04 0.04 0.06 0.04 0.0 0.036 0.0 0.03 0.0, 0.9, OLS HK K K S3 S4 4 5 6 8 9.445.67 0.96 0.37.9.9 0.30 0.33 0.95 0.08 0.33 0.094 0.30 0.074 0.9 0.090 0 0.666 0.340 0.094 0.056 0.49 0.49 0.55 0.050 0.037 0.037 0.050 0.04 0.55 0.034 0.058 0.04 30 0.449 0.30 0.06 0.034 0.304 0.304 0.6 0.034 0.03 0.03 0.034 0.07 0.6 0.0 0.039 0.07 40 0.499 0.75 0.093 0.057 0.358 0.358 0.60 0.05 0.044 0.045 0.05 0.07 0.60 0.03 0.07 0.07 50 0.48 0.3 0.033 0.09 0.89 0.89 0.079 0.00 0.03 0.03 0.00 0.06 0.079 0.0 0.05 0.06 0 0.8 0.067 0.09 0.0 0.6 0.6 0.064 0.0 0.08 0.07 0.0 0.0 0.064 0.0 0.03 0.0, 0.9, 5 OLS HK K K S3 S4 4 5 6 8 9 6.65 3.8 4.630.54 30.49 30.49 7.3.890.4..890 0.56 7.3 0.438.03 0.56 0 5.76 7.69.44 0.599 9.60 9.60 3.440.303 0.635 0.457.303 0.093 3.440 0.76.4 0.093 30.65 5.77 0.686 0.38 7.35 7.35.78 0.754 0.385 0.30 0.754 0.05.78 0. 0.86 0.05 40 9.84 4.804 0.76 0.30 6.9 6.9.565 0.538 0.88 0.56 0.538 0.04.565 0.085 0.698 0.04 50 6.3.978 0.464 0.99 4.57 4.57.858 0.45 0.5 0.69 0.45 0.06.858 0.05 0.539 0.06 0.5.005 0. 0.8.86.86 0.960 0.6 0.3 0.099 0.6 0.03 0.960 0.09 0.03 0.03 4, 0.7, 0.0 OLS HK K K S3 S4 4 5 6 8 9 0.84 0.83 0.83 0.83 0.83 0.83 0.07 0.59 0.3 0.063 0.59 0.59 0.33 0.3 0.063 0.59 0 0.063 0.063 0.063 0.063 0.063 0.063 0.060 0.060 0.047 0.054 0.060 0.060 0.06 0.047 0.054 0.060 30 0.379 0.378 0.378 0.378 0.378 0.378 0.03 0.037 0.033 0.030 0.037 0.037 0.036 0.033 0.030 0.037 40 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.06 0.0 0.05 0.05 0.0 0.06 0.0 0.05 50 0.03 0.0 0.0 0.0 0.0 0.0 0.08 0.0 0.06 0.06 0.0 0.0 0.0 0.06 0.06 0.0 0 0.0 0.0 0.0 0.0 0.0 0.0 0.008 0.0 0.008 0.008 0.0 0.0 0.0 0.008 0.008 0.0 4, 0.7, 0.5 OLS HK K K S3 S4 4 5 6 8 9.48 0.847 0.90 0.7 0.664 0.664 0.33 0.40 0.4 0.5 0.40 0.4 0.34 0.3 0. 0.4 0 0.359 0.3 0.08 0.079 0. 0. 0.08 0.067 0.059 0.060 0.067 0.05 0.7 0.053 0.068 0.05 30 0.43 0.097 0.05 0.05 0.3 0.3 0.05 0.043 0.038 0.039 0.043 0.035 0.084 0.036 0.043 0.035 40 0.089 0.057 0.07 0.07 0.078 0.078 0.08 0.0 0.08 0.08 0.0 0.0 0.049 0.06 0.0 0.0 50 0.07 0.049 0.06 0.07 0.066 0.066 0.030 0.03 0.00 0.00 0.03 0.09 0.047 0.08 0.04 0.09 0 0.030 0.0 0.0 0.0 0.09 0.09 0.05 0.0 0.0 0.0 0.0 0.009 0.03 0.009 0.0 0.009 3

4, 0.7, OLS HK K K S3 S4 4 5 6 8 9 5.564 3.088 0.39 0.359.78.78 0.39 0.37 0.85 0.88 0.37 0.39 0.70 0.5 0.33 0.39 0.85 0.646 0. 0.4 0.673 0.673 0.60 0.45 0.097 0.093 0.45 0.057 0.88 0.075 0. 0.057 30 0.455 0.66 0.083 0.087 0.383 0.383 0.9 0.09 0.063 0.060 0.09 0.037 0.5 0.048 0.083 0.037 40 0.9 0.67 0.047 0.050 0.53 0.53 0.077 0.055 0.036 0.033 0.055 0.0 0.50 0.05 0.050 0.0 50 0. 0.4 0.040 0.043 0.9 0.9 0.069 0.047 0.034 0.03 0.047 0.09 0.6 0.04 0.046 0.09 0 0.089 0.053 0.09 0.00 0.085 0.085 0.037 0.04 0.08 0.06 0.04 0.0 0.063 0.0 0.04 0.0 4, 0.7, 5 OLS HK K K S3 S4 4 5 6 8 9 9.0 70. 6.667 6.9 50.63 50.63 6.5 6.86.659.66 6.86 0.568 4.7.587 3.9 0.568 0 8.5 4.95.64.495 5.5 5.5.85.599.390.08.599 0.09 5.958 0.785.9 0.09 30.56 5.809.060.78 8.75 8.75.4.668 0.849 0.709.668 0.096 4.8 0.4.304 0.096 40 6.643 3.564 0.639 0.73 5.760 5.760.634.54 0.69 0.58.54 0.056 3.380 0.9 0.978 0.056 50 4.868.686 0.489 0.55 4.43 4.43.386 0.969 0.57 0.408 0.969 0.04.87 0.5 0.8 0.04 0.966.067 0.3 0.45.879.879 0.77 0.440 0.53 0.99 0.440 0.06.365 0.3 0.43 0.06 4, 0.8, 0.0 OLS HK K K S3 S4 4 5 6 8 9 0.70 0.69 0.69 0.69 0.69 0.69 0.9 0.69 0.90 0.93 0.69 0.69 0.93 0.90 0.93 0.69 0 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.064 0.063 0.064 0.064 0.063 0.063 0.063 0.063 0.063 30 0.040 0.039 0.039 0.039 0.039 0.039 0.036 0.038 0.037 0.038 0.038 0.038 0.038 0.038 0.037 0.038 40 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.07 0.03 0.06 0.06 0.03 0.07 0.03 0.06 50 0.0 0.00 0.00 0.00 0.00 0.00 0.00 0.0 0.0 0.0 0.0 0.0 0.00 0.0 0.0 0.00 0 0.0 0.0 0.0 0.0 0.0 0.0 0.009 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4, 0.8, 0.5 OLS HK K K S3 S4 4 5 6 8 9.04 6.365 0.34 0.89 3.308 3.308 0.64 0.7 0.3 0.3 0.7 0.83 0.409 0.05 0.34 0.83 0 0.367 0.4 0. 0. 0.80 0.80 0. 0.087 0.080 0.08 0.087 0.067 0.54 0.076 0.088 0.067 30 0.84 0. 0.059 0.059 0.54 0.54 0.059 0.047 0.044 0.044 0.047 0.038 0.094 0.04 0.049 0.038 40 0.3 0.07 0.09 0.09 0.097 0.097 0.030 0.03 0.08 0.09 0.03 0.04 0.058 0.07 0.0 0.0 50 0. 0.073 0.035 0.035 0.098 0.098 0.037 0.09 0.08 0.08 0.09 0.03 0.06 0.06 0.03 0.03 0 0.040 0.07 0.05 0.05 0.038 0.038 0.08 0.03 0.0 0.0 0.03 0.0 0.08 0.0 0.04 0.0 4, 0.8, OLS HK K K S3 S4 4 5 6 8 9 47.6 4.49 0.66 0.495.43.43 0.485 0.384 0.54 0.83 0.306 0.384 0.96 0.47 0.359 0.96 0.95 0.756 0.8 0.96 0.936 0.936 0.98 0.54 0.9 0. 0.54 0.069 0.49 0.090 0.39 0.069 30 0.653 0.385 0. 0.9 0.53 0.53 0.9 0.093 0.064 0.063 0.093 0.040 0.8 0.050 0.085 0.040 40 0.378 0.5 0.055 0.059 0.3 0.3 0.083 0.057 0.035 0.033 0.057 0.03 0.8 0.04 0.050 0.03 50 0.374 0.4 0.056 0.059 0.39 0.39 0.078 0.053 0.037 0.037 0.053 0.03 0.76 0.030 0.050 0.03 0 0.3 0.077 0.03 0.05 0.3 0.3 0.04 0.06 0.08 0.07 0.06 0.0 0.083 0.03 0.05 0.0 4, 0.8, 5 OLS HK K K S3 S4 4 5 6 8 9 90 63.40 6.984 9.0 9.0 7.455 5.690.463.868 5.690 0.58.58.507 4.37 0.58 0 30.56 6.70.5 3.0.69.69 3.336.67.47.04.67 0.69 8.985 0.588.850 0.69 30 4.9 7.74.60.633.44.44.90.643 0.703 0.606.643 0.083 5.743 0.38.59 0.083 40 8.66 4.74 0.760 0.890 7.35 7.35.70.83 0.579 0.473.83 0.056 4.0 0.57 0.933 0.056 50 9. 4.978 0.698 0.9 7.735 7.735.48 0.94 0.394 0.356 0.94 0.040 4.065 0.76 0.730 0.040 0 3.065.679 0.78 0.330.869.869 0.86 0.458 0.9 0.80 0.458 0.05.889 0.084 0.409 0.05 4, 0.9, 0.0 OLS HK K K S3 S4 4 5 6 8 9 0.038 0.038 0.038 0.038 0.038 0.038 0.03 0.037 0.033 0.03 0.037 0.037 0.035 0.033 0.03 0.037 0 0.063 0.06 0.06 0.06 0.06 0.06 0.049 0.06 0.057 0.0 0.06 0.06 0.054 0.057 0.05 0.06 30 0.038 0.038 0.038 0.038 0.038 0.038 0.03 0.037 0.033 0.03 0.037 0.037 0.035 0.033 0.03 0.037 40 0.08 0.08 0.08 0.08 0.08 0.08 0.04 0.07 0.04 0.03 0.07 0.07 0.06 0.04 0.04 0.07 50 0.03 0.0 0.0 0.0 0.0 0.0 0.05 0.0 0.07 0.05 0.0 0.0 0.09 0.07 0.05 0.0 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4

4, 0.9, 0.5 OLS HK K K S3 S4 4 5 6 8 9 0.308 0.84 0.065 0.066 0.47 0.47 0.053 0.040 0.036 0.037 0.040 0.035 0.8 0.035 0.04 0.035 0 0.579 0.344 0. 0.3 0.49 0.49 0.085 0.070 0.063 0.064 0.070 0.059 0.7 0.060 0.070 0.059 30 0.308 0.84 0.065 0.066 0.47 0.47 0.053 0.040 0.036 0.037 0.040 0.035 0.8 0.035 0.04 0.035 40 0.7 0.5 0.043 0.044 0.44 0.44 0.04 0.030 0.07 0.08 0.030 0.06 0.080 0.05 0.03 0.06 50 0.66 0.099 0.03 0.033 0.38 0.38 0.030 0.0 0.09 0.09 0.0 0.00 0.07 0.08 0.0 0.00 0 0.069 0.043 0.08 0.08 0.063 0.063 0.09 0.03 0.0 0.0 0.03 0.0 0.040 0.0 0.04 0.0 4, 0.9, OLS HK K K S3 S4 4 5 6 8 9.49 0.65 0.3 0.47 0.900 0.900 0. 0.09 0.05 0.05 0.09 0.036 0.383 0.04 0.070 0.036 0.099.70 0. 0.48.500.500 0.79 0.36 0.083 0.087 0.36 0.06 0.530 0.070 0. 0.06 30.49 0.65 0.3 0.47 0.900 0.900 0. 0.09 0.05 0.05 0.09 0.036 0.383 0.04 0.070 0.036 40 0.600 0.336 0.073 0.084 0.490 0.490 0.09 0.067 0.039 0.039 0.067 0.06 0.4 0.030 0.055 0.06 50 0.600 0.335 0.063 0.07 0.494 0.494 0.074 0.045 0.07 0.08 0.045 0.00 0.35 0.0 0.040 0.00 0 0.36 0.3 0.09 0.035 0.3 0.3 0.045 0.06 0.06 0.06 0.06 0.0 0.5 0.03 0.04 0.0 4, 0.9, 5 OLS HK K K S3 S4 4 5 6 8 9 7.93 5.4.876.760.46.46.69.690 0.509 0.507.690 0.068 8.79 0.46 0.987 0.068 0 5.09 7.7 3. 35.6 35.6 3.55.74 8.54 0.835 0.80.74 0.9.74 0.46.478 0.9 30 7.93 5.4.876.760.46.46.69.690 0.509 0.507.690 0.068 8.79 0.46 0.987 0.068 40 9.84 4.804 0.76 0.30 6.9 6.9.565 0.538 0.88 0.56 0.538 0.04.565 0.085 0.698 0.04 50 4.84 8.068 0.999.504.04.04.56.03 0.34 0.30.03 0.035 5.570 0.39 0.635 0.035 0 5.757 3.39 0.40 0.599 5.95 5.95 0.93 0.43 0.65 0.58 0.43 0.05.967 0.068 0.369 0.05 5