On the Estimation of Panel Regression Models with Fixed Effects
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(9) Pd{lpxp Olnholkrrg hvwlpdwruv iru 4 lq +514, dv zhoo dv dq| uhdvrqdeoh JPP hvwlpdwru iru 4 ghshqg rq vhfrqg rughu vdpsoh prphqwv ri wkh gdwd1 Wkhvh hvwlpdwruv zloo eh frqvlvwhqw rqo| li wkhvh vdpsoh prphqwv frqyhujh lq suredelo0 lw| 7 8 1 Wkh vhfrqg rughu vdpsoh prphqwv fdq eh uhzulwwhq dv vxpv ri furvv0 vhfwlrqdo dyhudjhv ri vtxduhg ohyhov/ vtxduhg gl huhqfhv/ dqg2ru furvvsurgxfwv ri ohyhov dqg gl huhqfhv ri wkh gdwd1 Li wkh {hg h hfwv dqg lqlwldo frqglwlrqv vdwlvi| S 2 +s,olp <" ' c ' j 2 4/ wkhq ohppd 516 lpsolhv wkdw wkh furvv0vhfwlrqdo dyhudjhv ri wkh vtxduhg gl huhqfhv zloo frqyhujh lq suredelolw|/ zkhuhdv furvv0 vhfwlrqdo dyhudjhv ri wkh vtxduhg pd| zhoo glyhujh1 Wkh lpsolfdwlrq ri S ohyhov 2 ohppd 517 lv wkdw +s,olp <" ' j 2 4 lv lq idfw d qhfhvvdu| frqgl0 ' c wlrq iru wkh frqyhujhqfh lq suredelolw| ri wkh furvv0vhfwlrqdo dyhudjhv ri vtxduhg gl huhqfhv dqg wkhuhiruh iru wkh frqvlvwhqf| ri JPP ru PO hvwlpdwruv iru +514, zklfk ghshqg rq wkhvh dyhudjhv 9 1 Zh frqfoxgh wkdw d {hg h hfwv JPP ru PO hvwlpdwru iru +514, vkrxog rqo| ghshqg rq uvw gl huhqfhv ri wkh gdwd1 Pruhryhu/ d {hg h hfwv JPP ru PO hvwlpdwru iru +514, wkdw lv frqvlvwhqw iru dq| vhtxhqfhv i+c j dqg i> j grhv qrw h{lvwv1 Krzhyhu/ frqvlvwhqw uhvwulfwhg {hg h hfwv JPP dqg PO hvwlpdwruv iru prghov wkdw lqfoxgh dvvxpswlrq UIH4 grS h{lvw1 Zh qrwh wkdw UIH4/ ru +s,olp <" E{+c2 2 4/ lv d uhdvrqdeoh dv0 ' vxpswlrq wkdw lv phw lq prvw dssolfdwlrqv/ srvvleo| diwhu uhvfdolqj wkh gdwd1 5141 JPP hvwlpdwlrq xqghu {hg h hfwv1 Lq wkh dsshqgl{ zh suryh wkh iroorzlqj uhvxow= Wkhruhp 5181 Dvvxph wkdw VD dqg UIH krog/ j>c lv frqvwdqw/ A ' c dqg m4m 1SWkhq wkh vwdqgdug Duhoodqr0Erqg JPP hvwlpdwru e 4 lv frqvlvwhqw li 2 3232b > ' JE 1 ' Surri Vhh Dsshqgl{ D1 Qrwlfh wkdw 3232b. S. '. 7. >2 ' JE lv htxlydohqw wr V. n 2b1. Zh uhtxluh frqyhujhqfh lq suredelolw| ri wkhvh vdpsoh prphqwv lq rughu wr suryh xqlirup frqyhujhqfh ri wkh fulwhulrq ixqfwlrq +vhh h1j1 Qhzh| dqg PfIdgghq +4<<7,,1 8 Xqghu qrq0qrupdolw| ri wkh gdwd/ rqh pd| zdqw wr frqvlghu JPP hvwlpdwruv iru zklfk dovr h{sorlw lqirupdwlrq frqwdlqhg lq wklug dqg kljkhu rughu vdpsoh prphqwv1 Lq wkdw fdvh/ lvvxhv vlplodu wr wkrvh glvfxvvhg lq wklv sdshu zloo dulvh1 Lq sduwlfxodu/ vxfk d JPP hvwlpdwru lv d frqvlvwhqw {hg h hfwv hvwlpdwru rqo| li iyl>4 j vdwlv hv d jhqhudol}hg yhuvlrq ri frqglwlrq UIH41 Ehorz zh zloo frq qh rxu dwwhqwlrq wr hvwlpdwruv wkdw rqo| h{sorlw vhfrqg prphqwv ri wkh gdwd1 9 Dv fdq eh vhhq iurp +5143 , dqg +517,/ olnholkrrg ixqfwlrqv iru wkh frqglwlrqdo DU+4, prgho fdq dozd|v eh uhzulwwhq vxfk wkdw wkh| rqo| ghshqg rq furvv0vhfwlrqdo dyhudjhv ri vtxduhv dqg furvv0surgxfwv ri +u ,yl>4 > yl>4 > dqg wkh glvwxuedqfhv/ zkhuh u lv wkh yduldeoh lq wkh olnholkrrg ixqfwlrq wkdw fruuhvsrqgv wr wkh wuxh sdudphwhu 1. ;.
(10) Zkhq j>c lv qrw frqvwdqw/ V n2b lv qr orqjhu vx!flhqw iru wkh frqvlvwhqf| ri wkh vwdqgdug DE JPP hvwlpdwru1 Wkhruhp 5181 lpsolhv wkdw li wkh lqglylgxdo h hfwv duh wuhqglqj/ wkhq e 4 pd| eh d srru hvwlpdwru1 D sdqho ri up gdwd pd| w wkh frqglwlrqv ri wkh iroorzlqj h{dpsoh= H{dpsoh 5191 Ohw UIH krog zlwk .Ec ' f dqg .Ec2 : B : f ; 5 L/ dqg s S S _ > ' q Wkhq >2 ' fDq 2 E n ' E c dqg > ' > c $ ' ' S >2 .Ec2 1 Wkxv V ' Efc T @oE> / zkhuh T @oE> ' *4 <" 32 ' S _ + {0c $ Efc T @oE+ {0 / zkhuh dqg b ' f Pruhryhu/ + {0 ' ' c S T @oE+ {0 ' *4 <" 32 2>2 j2 1 Lw iroorzv wkdw V ' n 2bc dqg wkdw wkh ' vwdqgdug DE JPP hvwlpdwru/ e 4 / lv lqfrqvlvwhqw1 Wkh wkhruhp ehorz ghvfulehv krz wkh dv|pswrwlf glvwulexwlrq ri e 4 ghshqgv rq wkh ydoxhv ri V dqg b= Wkhruhp 51:1 Ohw dvvxpswlrqv VD dqg UIH krog/ ohw j >c eh frqvwdqw/ A ' c S .E> {0c 2 ' l c f l 4c dqg m4m Pruhryhu/ ohw *4 <" 33V ' S dqg ohw *4 <" 33V .E> 0c2 2 ' l 2 c f l 2 4 ' Li V ' b ' fc dqg i+c {0c j vdwlv hv wkh Olqghehuj frqglwlrq/ dqg li S .E+c {0c 2 4c wkhq dovr f l f ' *4 <" 3 ' _. fD Ee 4 4 $ Efc dE4 j2 n E4 j >c o32 l f Li V : fc dqg i> {0c j vdwlv hv wkh Olqghehuj frqglwlrq/ wkhq= li f. V. _. 4 4 $ Efc dE4 j2 n E4 j>c o32 l c / dqg b ' fc fDE3V Ee _. li f 2b V n 2bc fDE3Vn2b Ee 4 4 $ Efc dE4 j>c o32 l Ilqdoo|/ li V n 2bc dqg erwk i> {0c j dqg i> 0c2 j vdwlvi| wkh Olqghehuj frqglwlrq/ wkhq= _. li V ' dqg b ' fc Ee 4 4 $. Efcl E43 j2 nE43 j>c n Efcl 2 c _. li V ' n 2bc dqg b : fc Ee 4 4 $ _. li V : n 2bc Ee 4 4 $. Efcl Efcl 2 . Efcl E43 j>c n Efcl 2 c. . Surri Vhh Dsshqgl{ D1 Zkhq wkh lqglylgxdo h hfwv duh wuhqglqj/ udqgrp vhtxhqfhv olnh i> {0c j fdq vwloo vdwlvi| wkh Olqghehuj frqglwlrq= <.
(11) 2 2 H{dpsoh 51;1 > ' qk zlwk k : fD ; SOhw j ' j 2c ; 25 2L/ Sdqg 2ohw k 5 L Wkhq ' .E> {0c ' 2j q ' E 2kn Pruh jhqhudoo|/ li ' S f B j 2 { 4c ; 5 L/ wkhq zh vwloo kdyh .E> {0c 2 ' E 2kn ' Zh fdq vkrz wkdw i> {0c j vdwlv hv wkh Olqghehuj frqglwlrq e| dsso|lqj wkhruhp 5614; lq Gdylgvrq +4<<7,1 Wdnh f ' > {0c c dqg S ' > 1 Wkhq f2 *S2 ' E{0c 2 Vlqfh .Em0c| m2nB { 4c ; 5 Lc ; | 5 W / zh rewdlq wkdw {0c lv xqlirup u2nB erxqghg e| dsso|lqj wkh Plqnrzvnl lqhtxdolw|1 Lw iroorzv iurp wkhruhp 45143 S lq Gdylgvrq wkdw if2 *S2 j lv xqlirup lqwhjudeoh1 Ohw ' S ' > dqg 2 r ' .E> {0c 2 Vlqfh tT d 2 *r2 o ' tT E 32k q 2 2k ' 4c ' zh fdq frqfoxgh wkdw i> {0c j vdwlv hv wkh Olqghehuj frqglwlrq1. Zh kdyh wkh iroorzlqj dowhuqdwlyh wr wkhruhp 518/ zklfk lqyroyhv dvvxpswlrq UIH4 lqvwhdg ri dvvxpswlrq UIH4= Wkhruhp 51<1 Ohw dvvxpswlrqv VD/ UIH4 / dqg UIH5 krog/ A ' c dqg m4m Ohw S b eh wkh vpdoohvw srvlwlyh qxpehu +b f, vxfk wkdw 33b *4 <" .E> c ' S>c zlwk mS>c m 4 dqg S>c 9' f li b : f1 ' S 33V T @oE> dE4 c n 0c2 o ' l c f l 4c Pruhryhu/ ohw *4 <" ' iru vrph V fc dqg ohw i> dE4 c n 0c2 o .E> E4 c j vdwlvi| wkh Olqghehuj frqglwlrq1 Wkhq wkh vwdqgdug Duhoodqr0Erqg JPP hvwlpdwru/ e 4 / lv frqvlvwhqw li V n 2b Surri Vhh Dsshqgl{ D1 Wkh Duhoodqr0Erqg gl huhqfhv JPP hvwlpdwru/ e 4_s / xvhv odjjhg gl huhqfhv ri +c|3 dv lqvwuxphqwv lqvwhdg ri ohyhov/ wkdw lv/ lw h{sorlwv wkh iroorzlqj prphqw frqglwlrqv +518, .E{+cr {0c| ' fc 2 r | 2c | ' ec c A Wklv hvwlpdwru rqo| lqyroyhv gl huhqfhv ri wkh gdwd dqg zloo eh frqvlvwhqw iru dq| vhtxhqfh ri {hg h hfwv dv orqj dv dvvxpswlrq UIH +ru UIH , krogv1 Wkhruhp 51431 Dvvxph wkdw VD/ dqg UIH4 krog1 Wkhq wkh Duhoodqr0Erqg gl huhqfhv JPP hvwlpdwru/ e 4_s / lv frqvlvwhqw1 Surri Vhh Dsshqgl{ D1 Wkh dv|pswrwlf glvwulexwlrq ri e 4_s lv hdvlo| ghulyhg1 Zh pdnh wkh iroorzlqj dvvxpswlrqv= S Prphqw dvvxpswlrq 4 +PD4,= f je ' *4 <" je 4 ' 43.
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(25) Zh qrwlfh wkdw PO dssolhg wr d prgho zlwk > ' deSEf o Z n +fi Pxqgodn +4<:;,/ dqg Fkdpehuodlq +4<;3/ 4<;7,,/ l1h1 zlwk % h ' eSEf / zloo ohdg wr d Iluvw e8 . Gl huhqfh IJOV w|sh hvwlpdwru iru qc zklfk lv dv|pswrwlfdoo| htxlydohqw wr q e , dv zhoo1 Wklv dssurdfk dovr |lhogv dq hvwlpdwru iru wc e +dqg q w ucZ / wkdw lv pruh -. h!flhqw wkdq wkh {hg h hfwv hvwlpdwru iru w1 Wklv fdq djdlq eh vhhq iurp wkh idf0 3 3 wrulvdwlrq ri s E dv }E( Ed Ej2D n T o3 Ej2D n T Lq wklv fdvh zh fdq uhsdudphwul}h wkh prgho vr wkdw d Ej2 n T 3 o3 2 Ej n T 3 ' d Ej2 n T 3 o3 Ej2 n T 3 E+ deSEf o Z h zlwk Z h ' d Ej 2 n T 3 o3 eSdq Ej2 n T 3 o n Z Vlqfh xvlqj wkh vshfl fdwlrq > ' Ej 2 n T 3 f 1 n D dprxqwv wr lpsrvlqj uhvwulfwlrqv rq Z zklfk lqyroyh wc l1h1 Z ' eSE1 Ej 2 n T Ew 3 c e w-. lv pruh h!flhqw wkdq e w ucZ 1 Zkhq Z lq > ' deSEf o Z n lv qrw uhvwulfwhg wr ghshqg rq w lq vrph zd|/ wkh lqirupdwlrq pdwul{ lv eorfn0gldjrqdo/ l1h1 wkh qrq0gldjrqdo hohphqwv lq wkh lqirupdwlrq pdwul{ fruuhvsrqglqj wr wkh vorsh sdudphwhuv q dqg Z rq wkh rqh kdqg dqg wkh fryduldqfh sdudphwhuv w rq wkh rwkhu kdqg duh }hur1 Lw iroorzv wkdw lw grhv qrw pdwwhu iru wkh h!flhqf| ri e w ucZ zkhwkhu wkh hohphqwv ri Z wkdw fruuhvsrqg wr wkrvh uhjuhvvruv lq f zklfk duh qrw fruuhodwhg zlwk > c duh uhvwulfwhg wr htxdo }hur ru qrw1 Krzhyhu/ lpsrvlqj }hur uhvwulfwlrqv rq Z +ru 1, grhv ohdg wr pruh h!flhqw PO hvwlpdwruv iru qc surylghg wkdw wkhvh uhvwulfwlrqv duh ydolg ri frxuvh1 7161 Jhqhudol}hg phwkrg ri prphqwv hvwlpdwruv iru fryduldqfh sdud0 phwhuv1 Wkhuh dovr h{lvw {hg dqg udqgrp h hfwv yhuvlrqv ri wkh Jhqhudol}hg Phwkrg riSPrphqwv iru hvwlpdwlqj w1 D Il{hg H hfwv JPP hvwlpdwru iru w plqlpl}hv S e e e8 . i eSd(E+ f q 8 . E+ f q 8 . ( (T ( oj` i eSd(E+ f q e8 . ( (T ( oj iru vrph zhljkwlqj pdwul{ ` D Udqgrp H hfwv JPP E+ f q S e8 . E+ f q e8 . j 2 hvwlpdwru iru w zrxog plqlpl}h i eSdE+ f q > S 2 e e T oj` i eSdE+ f q 8 . E+ f q 8 . j> T oj iru vrph zhljkwlqj pdwul{ ` Frqvlvwhqf| ri e wC c-. uhtxluhv wkdw wkh > r duh hvvhqwldoo| udqgrp h hfwv1 Wklv Udqgrp H hfwv hvwlpdwlrq surfhgxuh lv htxlydohqw wr d JPP hvwlpdwru e8 . E+ f q e8 . ( wkdw h{sorlwv wkh prphqw frqglwlrqv lq .ieSd(E+ f q e8 . E+ f q e8 . e (T e oj ' fc zkhuh e lv wkh (T ( oj ' fc dqg .id(E+ f q uvw froxpq ri wkh lghqwlw| pdwul{1 Vlqfh wkh UHJPP hvwlpdwru iru w h{sorlwv pruh prphqw frqglwlrqv wkdq wkh IHJPP hvwlpdwru/ zklfk duh qrw uhgxqgdqw/ wkh iruphu lv dv|pswrwlfdoo| pruh h!flhqw wkdq wkh odwwhu1 Wkh Rswlpdo IHJPP hvwlpdwru iru w lv dv|pswrwlfdoo| htxlydohqw wr wkh IHPOH iru w xqghu qrupdolw|1 Wkh Rswlpdo UHJPP hvwlpdwru iru w lq wxuq 57.
(26) lv dv|pswrwlfdoo| htxlydohqw wr e w ucZ / exw lq jhqhudo dv|pswrwlfdoo| ohvv h!flhqw e wkdq w uc-. / h{fhsw zkhq wkh uhjuhvvruv lq f duh qrw fruuhodwhg zlwk wkh > r. 81 Frqfoxvlrqv Lq wklv sdshu zh kdyh vwxglhg hvwlpdwlrq ri ydulrxv yhuvlrqv ri wkh DU+4, sdqho prgho zlwk lqlwldo frqglwlrqv1 Zh kdyh ghulyhg d qhfhvvdu| frqglwlrq iru frqvlv0 whqf| ri dq| hvwlpdwru iru wkh {hg h hfwv yhuvlrq ri wklv prgho1 Wklv frqglwlrq lqyroyhv wkh ghyldwlrqv ri wkh lqlwldo frqglwlrqv iurp wkh lqglylgxdo +phdq, h hfwv1 Wkh frqglwlrq doorzv iru furvv0vhfwlrqdo fruuhodwlrq dqg khwhurjhqhlw| ri wkh gdwd1 E| lpsrvlqj wklv frqglwlrq rq Fkdpehuodlq*v fruuhodwhg h hfwv irupxodwlrq ri wkh prgho/ zh rewdlqhg wkh Il{hg H hfwv +Txdvl, Pd{lpxp Olnholkrrg hvwlpdwru1 D uhodwhg exw shukdsv qrw yhu| vxusulvlqj uhvxow lv wkdw dq| frqvlvwhqw {hg h hfwv hvwlpdwru iru wkh DU+4, sdqho prgho zlwk lqlwldo frqglwlrqv lqyroyhv rqo| uvw gl huhqfhv ri wkh gdwd1 Zh kdyh vhhq wkdw li wkh lqglylgxdo h hfwv duh wuhqglqj/ d JPP hvwlpdwru zklfk ghshqgv rq ohyhov ri wkh gdwd/ h1j1 wkh Duhoodqr0Erqg JPP hvwlpdwru/ pd| kdyh srru surshuwlhv1 Zh kdyh hvwdeolvhg wkh frqqhfwlrq xqghu qrupdolw| dqg +furvv0vhfwlrqdo, kr0 prvnhgdvwlflw| ehwzhhq wkh IH+T,POH dqg wkh Rswlpdo Il{hg H hfwv Frqgl0 wlrqdo JPP hvwlpdwru/ zklfk h{sorlwv doo wkh vhfrqg rughu prphqw frqglwlrqv wkdw rqo| lqyroyh gl huhqfhv ri wkh gdwd1 Lw iroorzv wkdw wkh IH+T,POH lv dv0 |pswrwlfdoo| h!flhqw xqghu wkhvh frqglwlrqv1 Vlploduo|/ zh kdyh suryhq dv|ps0 wrwlf htxlydohqfh ri wkh UHPOH dqg wkh RUHFJPP hvwlpdwru gxh wr Dkq dqg Vfkplgw +4<<8,1 Wkh odwwhu lv vkrzq wr eh dv|pswrwlfdoo| pruh h!flhqw wkdq wkh RIHFJPP hvwlpdwru1 Wkh IHPOH zdv dovr frpsduhg zlwk dowhuqdwlyh olnholkrrg dssurdfkhv1 Wkh Ed|hvldq hvwlpdwru xvlqj Mh uh|v sulru/ ru htxlydohqwo|/ wkh Prgl hg POH/ lv dv|pswrwlfdoo| lqh!flhqw> pruhryhu/ wklv hvwlpdwru lv yhu| lpsuhflvh qhdu wkh xqlw urrw1 Wkh Frqglwlrqdo PO dssurdfk |lhogv dq lqfrqvlvwhqw hvwlpdwru1 Ilqdoo|/ lw lv vkrzq lq wklv sdshu wkdw wkh Il{hg H hfwv dssurdfk exw dovr Fkdpehuodlq*v Z0dssurdfk ohdg wr ohvv suhflvh hvwlpdwruv iru fryduldqfh sdudph0 whuv lq vwdwlf uhjuhvvlrq prghov wkdq wkh Fruuhodwhg Udqgrp H hfwv dssurdfk1. 58.
(27) D1 Surriv iru vhfwlrq 51 Surri ri Ohppd 515 Ohw { : f eh d jhqhulf frqvwdqw1 Uhfdoo wkdw c ' +c > Wkh prgho uhdgv dv E+c| > ' 4E+c|3 > n 0c| c ; | 5 W 1 S S E+c| > 2 ' 42 +s,olp <" E+c|3 > 2 n Wkhuhiruh/ solp <" ' ' S S dE+c|3 > 0c| on solp <" 02 / ; | 5 W 24 solp <" ' ' c| S Qrz/ solp <" E+c > 0c| ' fc ; | 5 W c ehfdxvh ri UIH4 +ru ' ehfdxvh ri VD dqg UIH4 zklfk lpso| .EE+c > 0c| ' f/ dqg S E+c2 > 0c| ' .EmE+c > 0c| mnB { ,1 Lw iroorzv wkdw solp <" ' S S 4 solp <" E+c > 0c| n solp <" 0 0 ' fc ; | 5 i c c A j ' ' c2 c| S E| lqgxfwlrq ryhu |/ zh wkhq rewdlq solp <" ' E+c|3 > 0c| ' fc dqg S solp <" E+c| > 2 4c ; | 5 W ' Surri ri Ohppd 516 Zh kdyh {+c| ' +c| > n > +c|3 ' E4 E+c|3 > n 0c| c ; | 5 W S S E{+c| 2 ' E4 2 +s,olp <" E+c|3 > 2 n Wkhuhiruh/ solp <" ' ' S S 2E4 solp <" E+c|3 > 0c| n solp <" 02 c ; | 5 W ' ' c| Xvlqj uhvxowv iurp wkh surri ri ohppd 515/ zh rewdlq e| lqgxfwlrq ryhu | wkdw S S solp <" E+ > 0 ' fc dqg solp E{+c| 2 4c ; | 5 W c| 3 c| <" ' ' Surri ri Ohppd 517 Zh kdyh {+c2 ' +c2 > n > +c ' E4 E+c > n 0c2 S S Wkhuhiruh/ E4 2 +s,olp <" E+c > 2 ' solp <" E{+c2 2 ' ' S S E+c > 0c2 solp <" 02 4 2E4 solp <" ' ' c2. 59.
(28) Surri ri Wkhruhp 518 Uhfdoo wkdw c ' +c > Z1o1r1j1 zh kdyh dvvxphg wkdw A ' Wkhq Duhoodqr dqg Erqg*v JPP hvwlpdwru h{sorlwv rqo| rqh prphqw frqglwlrq/ .d+c {0c o ' fc zkhuh {0c ' {+c 4{+c2 G S S S S e 4 ' d +c {+c2 o3 d +c {+c o ' 4 n d +c {+c2 o3 d +c {0c o S S S +c {0c ' > {0c n c {0c c Zh fdq zulwh wkh qxphudwru dv S S dqg wkh ghqrplqdwru dv +c {+c2 ' +c dE4 E+c > n 0c2 o ' S S 2 dE4 c n c 0c2 o n > dE4 c n 0c2 o Ohw { eh d jhqhulf srvlwlyh frqvwdqw1 E| dvvxpswlrq VD zh kdyh .E> {0c ' f dqg .E02c| { 4c ; 5 L1 S S S Khqfh T @oE 33b > {0c ' 3232b .dE> {0c 2 o 3232b >2 { S S Wkhuhiruh/ *4 <" T @oE 33b > {0c ' f li 3232b >2 ' JE 1 S S ^6 Lw iroorzv wkdw 33b > {0c $ f li 3232b >2 ' JE 1 S S ^6 Qrz/ 33b > {0c $ f lpsolhv wkdw s*4 <" 33b > {0c ' f S S Wkhuhiruh/ s*4 <" 33b > {0c ' f li 3232b >2 ' JE 1 S S Olnhzlvh/ s*4 <" 33b > 0c2 ' f li 3232b >2 ' JE S Vlqfh .Ec 0c| ' fc zh rewdlq wkdw s*4 <" dE4 c2 n c 0c2 o ' S E4 j 2 Lq dgglwlrq/ zh qg wkdw s*4 <" c {0c ' f S 33b E4 > c ' E4 j >c 1 Ilqdoo|/ wkh dvvxpswlrqv lpso| wkdw s*4 <" Wkxv wkh suredelolw| olplw ri wkh ghqrplqdwru ri e 4 zloo eh gl huhqw iurp }hur1 . 5:.
(29) Surri ri Wkhruhp 51: Uhfdoo wkdw j 2 ' *4 <". . . S. c2 . '. S 33V. 4. Lq dgglwlrq/ *4 <" .E> {0c 2 ' l c zlwk f l 4c ' S .E> 0c2 2 ' l 2 c zlwk f l 2 4c dqg *4 <" 33V ' S *4 <" 33b > c ' j >c / f b fDV ' Wkhq wkh iroorzlqj uhvxowv fdq hdvlo| eh yhul hg= S Li V : fc s*4 <" 3fDEnV c {0c ' f S Li V : c s*4 <" 3V dE4 c2 n c 0c2 o ' f S Li f 2b V n 2bc s*4 <" 33b > dE4 c n 0c2 o ' E4 j>c S Li f l f ' *4 <" 3 .E+c {0c 2 4c dqg i+c {0c j vdwlv hv ' S _ wkh Olqghehuj frqglwlrq/ wkhq 3fD + {0c $ Efc l f ' c S + {+c2 ' Li f V dqg b ' fc wkhq s*4 <" 3 ' c S S s*4 <" 3 dE4 c2 n c 0c2 ons*4 <" 3 > dE4 c n 0c2 o ' E4 j 2 n E4 j>c S + {+c2 ' E4 j>c Li f 2b V n 2bc wkhq s*4 <" 33b ' c S 3 Vr li V ' b ' fc f l f ' *4 <" .E+c {0c 2 4c dqg i+c {0c j ' vdwlv hv wkh Olqghehuj frqglwlrq/ wkhq _. 4 4 $ Efc dE4 j2 n E4 j >c o32 l f fD Ee Li V : f dqg i> {0c j vdwlv hv wkh Olqghehuj frqglwlrq/ wkhq S S S _ 3fDEnV +c {0c ' 3fDEnV c {0c n 3fDEnV > {0c $ Efc l ( li f. V. _. dqg b ' fc fDE3V Ee 4 4 $ Efc dE4 j2 n E4 j>c o32 l ( _. li f 2b V n 2bc fDE3Vn2b Ee 4 4 $ Efc dE4 j>c o32 l Ilqdoo|/ li V n 2b dqg i> 0c2 j vdwlv hv wkh Olqghehuj frqglwlrq/ wkhq S _ li V ' dqg b ' fc 3fDEnV +c {+c2 $ E4 j2 n E4 j>c n Efc l 2 ( S _ li V ' n 2b dqg b : f/ 3fDEnV +c {+c2 $ E4 j >c n Efc l 2 ( S _ li V : n 2bc 3fDEnV +c {+c2 $ Efc l 2 Vr li i> {0c j dqg i> 0c2 j vdwlvi| wkh Olqghehuj frqglwlrq/ wkhq _. li V ' dqg b ' fc Ee 4 4 $. Efcl E43 j2 nE43 j>c n Efcl 2 ( _. li V ' n 2b dqg b : f/ Ee 4 4 $ _. li V : n 2bc Ee 4 4 $. Efcl Efcl 2 . . Efcl E43 j>c n Efcl 2 (. 5;.
(30) Surri ri Wkhruhp 51< Wkh surri lv vlplodu wr wkh surri ri wkhruhp 5181 zlwk rqh prgl fdwlrq1 Lw lv dvvxphg Swkdw b lv wkh vpdoohvw srvlwlyh qxpehu/ b f/ vxfk wkdw *4 <" 33b .E> c ' S>c zlwk mS>c m 4 dqg S>c 9' f li b : f1 ' S 33V T @oE> dE4 c n 0c2 o ' l c zlwk f l 4c Pruhryhu/ *4 <" ' iru vrph V fc dqg i> dE4 c n0c2 o.E> E4 c j vdwlv hv wkh Olqghehuj frqglwlrq1 S >2 T @oE0c| 41 Lw iroorzv iurp wkhvh dvvxpswlrqv wkdw *4 <" 33V ' S 33b Wkhuhiruh/ li V n 2b / zh rewdlq > dE4 c o ' S wkdw s*4 <" 33b > {0c ' f E4 S>c / dqg s*4 <" Surri ri Wkhruhp 5143 Wulyldo1 Qrwlfh wkdw wkh uvw0gl huhqfhv ri wkh gdwd {+c| / | ' 2c c A / rqo| ghshqg rq +c dqg > wkurxjk c 1 Surri ri Wkhruhp 5146 Zh zloo suryh wkh wkhruhp e| vkrzlqj iru duelwudu| A wkdw wkhuh h{lvwv d prphqw frqglwlrq wkdw lv l, h{sorlwhg e| wkh RUHFJPP hvwlpdwru exw qrw e| wkh RIHFJPP hvwlpdwru/ dqg ll, qrw uhgxqgdqw uhodwlyh wr wkh prphqw frqglwlrqv wkdw duh rswlpdoo| h{sorlwhg e| wkh IHFJPP hvwlpdwru1 Zh zloo vkrz wkdw .E+c {0c ' f lv vxfk d prphqw frqglwlrq1 Iluvw qrwh wkdw wkh IHFJPP hvwlpdwru grhv qrw h{sorlw .E+c {0c ' fc ehfdxvh wklv prphqw frqglwlrq lqyroyhv gdwd lq ohyhov1 Rq wkh rwkhu kdqg/ wkh prphqw frqglwlrqv h{sorlwhg e| wkh RUHFJPP hvwlpdwru fdq eh irupxodwhg lq vxfk d zd| wkdw .E+c {0c ' f lv rqh ri wkhp +vhh +515,,/ wkdw lv/ wkh RUHFJPP hvwlpdwru h{sorlwv .E+c {0c ' fc hlwkhu h{solflwo| ru lpsolflwo|1 Qrz/ wkh prphqw frqglwlrqv wkh zhljkwlqjS pdwul{ ri wkh RIHFJPP hvwl0 S dqg S 3 2 3 2 3 pdwru rqo| ghshqg rq + wkurxjk ' E+c > 2 ' c ' c ' +l1h1 j 2 ,1 Krzhyhu/ wkh rswlpdo zhljkwlqj pdwul{ iru wkh RUHFJPP hvwlpdwru S 2 zklfk h{sorlwv .E+c {0c ' f dovr ghshqgv rq 3 + vhsdudwho|= ' c S S 2 2 2 3 + {0 o ' 2j + 1 .dE 3 c ' c ' c Zh frqfoxgh wkdw .E+c {0c ' f lv qrw uhgxqgdqw/ dqg wkdw T @oEe 4-. C T @oEe 48 . C . 5<.
(31) Surri ri Wkhruhp 5147 Wkh frqglwlrqdo DU[+4, sdqho prgho zlwk {hg h hfwv +dqg g h{rjhqrxv uhjuhv0 vruv, fdq eh vwdwhg dv + ' 4+c3 n E 4 > n f q n 0 c 4 zlwk 0 mf c +c c > Efc j2 U c ; 5 L1 Lq dgglwlrq/ zh zloo dvvxph Ohw xv gh qh 3 f E E E 4 ' E E E C 4A 3. +D14,. wkdw dvvxpswlrq VD krogv/ dqg j2 : f f f f 4 f 4 4. 3 4 f f E 4 F f f F E E 42 F f F E c ) ' E 11 F E 1 F E A3 D f C 4 f 4A 32. 4 F F F F Fc F F D. +D15,. wkhq E+c > % q*E 4 n +c3 > % q*E 4 ' ) 'f q n 0 ~ n 0 c. +D16,. zkhuh ~ ' ) E+c > % q*E 4 n 'f q Ohw c ' +c > % q*E 4 Zh zloo dgg wkh iroorzlqj dvvxpswlrqv 'f duh l1k1g1 zlwk .Em{%&cc| m2nB { 4c ; & 5 ic c gj/ ; 5 L/ ; | 5 W / [ E{%cr E{% c| ' Prc| c ;rc | 5 W c dqg s *4 <" '. +D17,. dqg s *4 <". [ 2 ' j2 ' c. dqg s *4 <". 4c s *4 <". [ c 0 ' fc '. [ c {%&ccr ' P&c| zlwk mP&c| m '. ; & 5 ic c gj/ ; | 5 W 1. Dvvxpswlrq +D18, jhqhudol}hv dvvxpswlrq UIH41 63. 4c. +D18,.
(32) Wkh prgl hg orj0olnholkrrg ixqfwlrq iru wkh frqglwlrqdo DU[+4, sdqho prgho +glylghg e| , lv jlyhq e| A 2 +D19, *L} RhEoc r2 c K ' 1Eo *L} r2 2 [ 2 E+ o+c3 f K 'E+ o+c3 f K 2r ' A 32. zkhuh 1Eo '. [A | | o A |' |. dqg wkh prgl hg orj0olnholkrrg htxdwlrqv duh jlyhq e| wkh vxpv +ryhu , ri E+ o+c3 f K '+c3 ' fc +D1:, r2 A 2 [j2 c Eoc r2 c K ' n 2r2 E+ o+c3 f K 'E+ o+c3 f K ' fc 2re [qc Eoc r2 c K ' 2 E+ o+c3 f K 'f ' f r [4c Eoc r2 c K ' 1 Eo n. Zh fdq uhzulwh wkhvh ixqfwlrqv xvlqj + o+c3 f K ' E4 o +c3 n f Eq K n 0. +D1;,. Lw iroorzv iurp ' ' fc +D16, dqg +D1;, wkdw 'E+ o+c3 f K ' 'EE4 o E+c3 > % q*E 4 n f Eq K n 0 ' 'EE4 o E~ n 0 n f Eq K n 0 . +D1<,. Zh zloo suryh wkdw rqh ri wkh vroxwlrqv ri wkh prgl hg orj0olnholkrrg htxdwlrqv lv d frqvlvwhqw hvwlpdwru iru E4c j 2 c q e| yhuli|lqj wkh frqglwlrqv ri wkhruhp 714151 ri Dphpl|d +4<;9,1 E| xvlqj +D1<,/ lw fdq hdvlo| eh vkrzq xqghu wkh Sdvvxpswlrqv2 ri wkh prgho *L} Rh Eoc r c K frqyhujhv +lqfoxglqj +D17, dqg +D18,, wkdw *L} RhEoc r2 c K ' ' xqlirupo| lq suredelolw| wr A 2 j2 2 *L} Rh Eoc r c K( 4c j c q ' 1Eo *L} r 2 |oE' +D143, 2 2r 2 dEo 4 2 dP5^5 n j2 |oE ' o 2Eo 4 j 2 |oE' E4 2r nEK q P%^% EK q n 2Eo 4 P5^% EK q oc 2. 2. 64.
(33) S S zkhuh P5^5 ' R *4 <" ~ '~ 4c P%^% ' R *4 <" f 'f ' ' S ~ 'f c dqg Eo r2 K 5 X ' Ec Efc 4 ?g 1 4/ P5^% ' R *4 <" ' Qrwlfh wkdw dvvxpswlrq +D18, lv d qhfhvvdu| frqglwlrq iru xqlirup frqyhujhqfh +lq suredelolw|, ri *L} RhEoc r2 c K 1 Qh{w/ zh kdyh wr vkrz wkdw wkh suredelolw| olplw ri wkh prgl hg orj0olnholkrrg ixqfwlrq lv pd{lpl}hg dw wkh wuxh ydoxhv1 Zh zloo uvw vkrz wkdw s *4 <" s *4 <". [ [4c E4c j2 c q ' f '. +D144,. [ [j2 c E4c j2 c q ' f '. s *4 <". [ [qc E4c j2 c q ' f '. Hydoxdwlqj wkh whupv ri wkh prgl hg orj0olnholkrrg htxdwlrqv dw wkh wuxh ydoxhv ri wkh sdudphwhuv/ zh rewdlq 0 '+c3 j2 A 2 0 '0 [j2 c E4c j 2 c q ' n 2j2 2je 0 'f [qc E4c j 2 c q ' 2 j [4c E4c j 2 c q ' 1 E4 n. +D145,. Wkhq s *4 <". [ [4c E4c j2 c q ' 1 E4 n |oE' E4 '. +D146,. ' 1 E4 1 E4 ' f. s *4 <". [ A 2 j2 2 2 [j c E4c j c q ' n e |oE' ' f ' 2j2 2j. s *4 <". [ [ 2 [qc E4c j c q ' *4 <" .E0 'f ' f 2 ' j '. zkhuh zh kdyh xvhg wkdw |oE' E4 ' |oE E4 EA 3 E4 ' 1 E4 . 65.
(34) Ilqdoo|/ zh kdyh wr yhuli| wkdw wkh Khvvldq ri wkh suredelolw| olplw ri wkh prgl hg orj0olnholkrrg ixqfwlrq lv d qhjdwlyh gh qlwh v|pphwulf pdwul{ dw wkh wuxh ydoxhv ri wkh sdudphwhuv1 Wklv Khvvldq lv jlyhq e| 3 4 2 1 E4 P5^5 nj j|o2 E ' 1 jE24 Pj5^% 2 E F 1 E4 A 32 M ' C f D j2 2je P5^% P%^% j2 f j2 . . . . Zh fdq uhzulwh M dv 4 3 P5^5 3 4 j2 f Pj5^% 1 E4 |oE E4 ' E4 1 jE24 f 2 1 E4 C f f D +D147, A2j32e f D n C f j2 P5^% P%^% f f f j2 f j2 # $ 1 E4 |oE E4 ' E4 1 jE24 Lw fdq eh vkrzq wkdw lv qhjdwlyh gh qlwh 1 E4 A2j32e j2 v|pphwulf iru 4 1 Zkhq 4 ' / M lv vlqjxodu= ljqrulqj wkh srzhuv ri j wkh gldjrqdo hohphqwv ri M htxdo 2 EA 2 dqg wkh r 0gldjrqdo hohphqwv ri M htxdo 2 EA 2 Vr zkhq 4 ' c wkh sdudphwhuv 4 dqg j 2 duh qrw xqltxho| lghqwl hg1 Wkh 2 2 pdwul{ rewdlqhg diwhu holplqdwlqj wkh plggoh urz dqg froxpq iurp wkh vhfrqg pdwul{ lq +D147, lv dovr qhjdwlyh gh qlwh v|pphwulf dqg khqfh M lv qhjdwlyh gh qlwh v|pphwulf zkhq 4 1 Wkh dv|pswrwlf glvwulexwlrq ri wkh Prgl hg POH iru wkh sdudphwhuv lq wkh DU[+4, sdqho prgho +D14, lv jlyhq e| 3 4 e 4 u 4 s e2 u j2 D $ fc EM 3 U EM 3 C j +D148, e qu q . . . . . zkhuh wkh Prgl hg Lqirupdwlrq Pdwul{ +U , lv jlyhq e| 3 4 2 |oE' ' n P5^5 nj j|o2 E ' 1 jE24 Pj5^% 2 E F A 32 U ' C 1 jE24 f D 2j e P5^% P%^% f j2 j2 . . . +D149,. . Wr ghulyh U zh kdyh xvhg wkh jhqhudo uhvxow .EE0 0 E0 2 0 ' j e E|oE |oE2 n |oE 2 n 2 Dq h{solflw h{suhvvlrq iru |oE' ' fdq eh irxqg xvlqj Pdsoh1 Lw fdq hdvlo| eh fkhfnhg wkdw |oE' ' 9' 1 E4 dqg wkhuhiruh M 9' U 1 66.
(35) Surri ri Wkhruhp 5148 Wkh frqglwlrqdo DU+4, sdqho prgho fdq eh uhzulwwhq dv + +c ' ) E4 d+c > o n E4 0 / ' c c c. +D14:,. zkhuh ) E4 dqg E4 kdyh ehhq gh qhg lq wkh pdlq wh{w1 Uhfdoo wkdw T E4 ' j2 E4 E4 Wkh Frqglwlrqdo PO hvwlpdwruv iru 4 dqg j 2 lv edvhg rq wkh iroorzlqj vx!flhqw vwdwlvwlfv iru wkh +c > = +c > ' E) T 3 ) 3 ) T 3 E+ +c '. E4 3 E+ +c A . +D14;,. Wkh Frqglwlrqdo PO hvwlpdwruv iru 4 dqg j2 lq wkh frqglwlrqdo DU+4, sdqho prgho duh gh qhg dv wkh pd{lpl}huv ri wkh frqglwlrqdo olnholkrrg ixqfwlrq T uS E+ +c3 m+c > c +c ' Wkh iroorzlqj uhvxowv duh hdvlo| yhul hg= E+ +c3 .E+ +c3 m+c > c +c ' E4 '0 c T @oE+c > m+c ' A 3 j2 c JE+ +c3 c +c > m+c '. . A 3. j 2 E4 c. T @oE+ +c3 m+c > c +c ' j2 dT E4 A 3 E4 E4 o ' j2 E4 ' E4 c dqg 0 ' E4 d E4 ' E4 o3 E4 '0 ' 0 '0 Lw iroorzv wkdw wkh frqglwlrqdo olnholkrrg ixqfwlrq lv htxdo wr wkh mrlqw qrupdo ghqvlw| ri wkh E4 '0 c ' c c T Wkh Frqglwlrqdo PO hvwlpdwru iru 4 edvhg rq uS E+ +c3 m+c > c +c ' lv lqfrqvlvwhqw/ ehfdxvh wkh suredelolw| olplw ri wkh frqglwlrqdo olnholkrrg lv qrw pd{lpl}hg dw wkh wuxh ydoxh 41 Wklv fdq eh vhhq e| frpsdulqj wkh suredelolw| olplwv ri wkh frqglwlrqdo olnholkrrg htxdwlrq iru 4 dqg wkh prgl hg olnholkrrg htxdwlrq iru 4 zkhq erwk duh hydoxdwhg dw wkh wuxh ydoxhv ri wkh sdudphwhuv1Wkh _ *L} m E4 ' E4 mn ' s E4 9' fc zkhuh mfmn gl huhqfh ehwzhhq wkhp lv 1 E4 n 2 _4 lv wkh surgxfw ri qrq0}hur hljhqydoxhv ri wkh pdwul{ f Vlqfh wkh Prgl hg PO hvwlpdwruv iru 4 dqg j 2 duh frqvlvwhqw/ lw iroorzv wkdw wkh Frqglwlrqdo PO hvwlpdwru iru 4 pxvw eh lqfrqvlvwhqw1 Qrwlfh wkdw wkh Frqglwlrqdo POH iru 4 lv gl huhqw iurp wkh Zlwklq POH2 OVGY hvwlpdwru iru 4/ zklfk krzhyhu lv dovr lqfrqvlvwhqw1. 67.
(36) E1 Uhvxowv iru vhfwlrq 61 Frqvlvwhqf| ri UHPOH iru wkh frqglwlrqdo DU+4, sdqho prgho= Wklv PO hvwlpdwru lv gh qhg dv wkh pd{lpxp ri d surgxfw ri surshu ghqvlw| ixqfwlrqv1 Wklv olnholkrrg ixqfwlrq grhv qrw lqyroyh dq| lqflghqwdo sdudphwhuv1 Frqvlvwhqf| ri wkh UHPOH wkhuhiruh iroorzv e| vwdqgdug dujxphqwv1 Qhyhuwkh0 ohvv/ lw lv lqwhuhvwlqj wr vhh wkdw wkh dv|pswrwlf uvw rughu frqglwlrqv duh vdwlv hg dw wkh wuxh sdudphwhu ydoxhv= Uhfdoo wkdw n 0 / zkhuh erwk dqg 0 duh l1l1g1/ zlwk .E+c ' fc ' .E ' j2 U n j 2 dqg .E+c 0 ' fc ^dqg .E 0 ' f`/ ; 5 W Ohw P 3 ' fc vlqfh .E+ P 3 ' .d|oEP 3 + o ' Wkhq .E+c 3 P c3 c3 5 3 64 5 646 5 3 f 4 42 2 2 f 2 n 4 E 9 f f 4 :F 9 f 2 2 n 4 2 :F: 9 E 9 E :F 2 :F: E 3 9 f f f :F 9 E 3 9 . 9|o EP 9 f n 4 :F: ' |o EP P 9 :F 2 1 1 1 :F 9 E 7 8D8 7 C f f f f 7 C 8D 11 11 11 111 1 1 1 11 11 11 11 111 1 1 1 1 ' f 3 ' fc .d|oE P Y P 2 o ' fc dqg Lw lv dovr hdvlo| yhul hg wkdw .E+c P Yj YP .d|oE P 2 o ' f 3. 3. . Yj . Surri ri Wkhruhp 614 Wkh IH +Txdvl, POH iru i4c j2 j lq wkh frqglwlrqdo DU+4, sdqho prgho lv edvhg rq wkh +txdvl, olnholkrrg ixqfwlrq fruuhvsrqglqj wr wkh iroorzlqj +dx{loldu|, prgho + +c ' 4E+c3 +c n c. +E14,. zkhuh ' E 4 c n 0 Efc E 4 2 j 2 n j2 U Uhfdoo wkdw c ' E+c > Zh dvvxph wkdw 4 c j2 : fc dqg wkdw dvvxpswlrqv VD dqg UIH4 krog1 Qrwlfh wkdw li 4 ' c wkh whup ' E4 c zrxog ydqlvk dqg wkh sdudphwhu S 2 j ' s*4 <" 2 zrxog dovr ydqlvk1 ' c Lw lv frqyhqlhqw wr uhsdudphwul}h wkh dx{loldu| prgho/ wkdw lv/ wr uhsodfh h E 4 2 j 2 e| j h2 Ohw .E ' j h2 n j2 U eh ghqrwhg e| P1 Wr suryh frqvlvwhqf| ri wkh IH+T,POH/ zh zloo yhuli| wkh frqglwlrqv ri wkh0 ruhp 714141 lq Dphpl|d +4<;9,1 Wkh pdlq gl huhqfh zlwk d surri iru vwdqgdug PO hvwlpdwruv lv wkdw wkh c duh qrw l1l1g1 exw vdwlvi| UIH41. 68.
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