Nonstationary Density Estimation and Kernel Autoregression

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS

AT YALE UNIVERSITY

Box 2125, Yale Station

New Haven, Connecticut 06520

COWLES FOUNDATION DISCUSSION PAPER NO. 1181

Note: Cowles Foundation Discussion Papers are preliminary materials

circulated to stimulate discussion and critical comment.

Requests for single copies of a Paper will be filled by the

Cowles Foundation within the limits of the supply. References

in publications to Discussion Papers (other than mere

acknowledgment by a writer that he has access to such

unpublished material) should be cleared with the author to

protect the tentative character of these papers.

NONSTATIONARY DENSITY ESTIMATION AND KERNEL

AUTOREGRESSION

Peter C. B. Phillips and Joon Y. Park

June 1998

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k *2_iO +w> v. d@,  O+w> v,j > l @ 4> ===> n l _ @k*2O+w> 3+v. d@,,  O+w> 3v,> l @ 4> ===> n l @ k%3*2*2_d*2_{iO+w> v} @ . %,  O+w> v@,j > l @ 4> ===> n l _ $k5+d,*2_X +O+w> v@,, > l @ 4> ===> n l _{_} @k5d*2_X +O+w> v,, > l @ 4> ===> n l > dv  $ 4> jlylqj wkh uhtxluhg uhvxow1

Iru sduw +g,/ zh xvh wkh iroorzlqj uhvxow iru vwdqgdug Eurzqldq prwlrq iurp Uhyx} dqg \ru +4<<4/ h{huflvh 5145/ s1 7;9,1

53*2qO+w> u ._v,  O+w> u,r$ T+O+w> u,> v,>_ zkhuh T lv d vwdqgdug Eurzqldq vkhhw1 Lw iroorzv wkdw

53_*2q_O +w> u ._v,  O+w> u, r _{_} @ 53_*2_O +w> 3+u . v@,,  O+w> 3u, _

$ T+O+w> u@,> v@,@ T+O+w> u@,> v,_ @ T+O_ +w> u,> v,>

dv uhtxluhg1 Wkh shqxowlpdwh glvwulexwlrqdo htxlydohqfh lq wkh deryh dujxphqw xvhv wkh idfw wkdw deT+w> v,@ T+d_ 2_{w> > e}2_{v, 0 h1j Uhyx} dqg \ru +4<<4/ h{huflvh 6144 s16:,0dqg vhwv d @ e @}*2₌

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dv uhtxluhg1 Uhvxow +43, lv suryhg lq d vlplodu zd|/ xvlqj wkh rffxsdwlrq irupxod +8, wr zulwh

3*e] "

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45

dv uhtxluhg iru +43,1

Wr suryh +44, zh surfhhg dv iroorzv1 Iluvw ohw T dqg T2 eh wzr lqghshqghqw vwdqgdug

Eurzqldq vkhhwv dqg ghqh T+w> v, @ T+w> v,4+v  3, . T2+w> v,4+v  3,= Qh{w/ iurp Ohppd

51<+g, zh ghgxfh wkdw dv  $ 4 53_*2q_O

+w> u ._v,  O+w> u,

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zkhuh T lv lqghshqghqw ri O1 Lw qrz iroorzv e| wkh frqwlqxrxv pdsslqj wkhruhp wkdw

*2] " 3"vN+v,O+w> u . v ,gv @ ] _" 3"vN+v, *2q_O +w> u ._v,  O+w> u, r gv _ $ 5] " 3"vN+v,T+O+w> u,> v,gv _ @ 5 iO+w> u,j*2 ] _" 3"vN+v,T+4> v,gv=

Qh{w ohw I +w, @ U_3"| vN+v,gv= Gluhfw fdofxodwlrq vkrzv wkdw U_3"" I +v,2_{gv @ *@7> dqg wkdw dv}

w $ 4> I +w,T+4> w, $ 3 d=v= 1 Wkhq/ lqwhjudwlrq e| sduwv |lhogv ] _" 3"vN+v,T+4> v,gv @  ] _" 3"I +v,gT+4> v, _ @ Q  3>] " 3"I +v, 2_gv_>

iurp zklfk wkh vwdwhg uhvxow iroorzv lpphgldwho|1

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H mP?+d> d . k> },  P?+d . nk> d . +n . 4,k> },m2o  F  k s q _o 4 . nk2_{orj q}o xqlirupo| lq } 5 U1

DS hLLu

Wkh uhvxow iroorzv iurp Dnrqrp +4<<6/ Ohppd 4,1 Qrwh wkdw

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46

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Q?+d> d . k,  Q?+d . nk> d . +n . 4,k, @ ?

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zkhuh Q?+d> d.k, @S?_'4 ^d  V  d . k` lv wkh qxpehu ri ylvlwv ri Vwr wkh lqwhuydo ^d> d . k` =

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erxqg krogv xqlirupo| lq }=

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DH hLLu

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dqg qrwh wkdw D?+,  ^&$&?D?c&+,= Xvlqj wkh Pdunry lqhtxdolw| ri rughu 5u/ Ohppd 818/ dqg

wkh idfw wkdw D?+,  ^&$&?D?c&+,> zh rewdlq

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xqlirupo| lq }= Vlqfh n?  q> wkh uhvxow wkhq iroorzv iurp wkh Eruho0Fdqwhool ohppd e| vxlwdeoh

fkrlfhv ri u dqg d vhtxhqfh ? iru zklfk ?$ 3 dqg "

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lq +46, zh kdyh wkh uhsuhvhqwdwlrq

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vlqfh N +{, @ rm{m32odv m{m $ 41 Wkh vhfrqg whup ri +5:, lv qhjoljleoh zkhq 45u +4  , ? 3> zklfk zloo eh dvvxphg lq zkdw iroorzv1

Wkh qh{w sduw ri wkh surri iroorzv d olqh ri dujxphqw ghyhorshg lq Dnrqrp +4<<6/ Wkhruhp 5, wkdw dssolhv iru wkh fdvh ri d xqlirup nhuqho1 Wkh lghd lv wr vkrz wkdw wkh uvw sduw ri +5:, fdq eh uhsodfhg zlwk d fruuhvsrqglqj lqwhjudo h{suhvvlrq wkdw lqyroyhv wkh Eurzqldq prwlrq E= Wkh qdo sduw ri wkh dujxphqw wkhq pdnhv xvh ri wkh rffxsdwlrq irupxod1

Ohw n? eh dq lqwhjhu iru zklfk 4  n?@f?sq@orj+q,= Wkh uroh ri n? lv wr zlghq wkh edqgv

lq wkh irupxodh ehorz vr wkdw wkh hpehgglqj ri V3 lq wkh Eurzqldq prwlrq E fdq eh xvhg1 Zh

vhhn wr pdmrul}h wkh glhuhqfh f?   q4 ? [ ' N  f?  {  |_sf V3 q  4  {  |_s f q f4?  V_s3 q {  |sqf .f4?  ]  f N  f?  {  |_s f q  E+u,  4  {  |_s f q  4 f?  E+u,  {  |_s f q . 4 f?  gu  D+q, ._5n4 ?D2+q, . D+q,> xvlqj wkh whupv D+q, @ f?   q4 ? [ ' N  f?  {  |f_s V3 q  4  {  |_s f q f4?  V_s3 q {  |sqf .f4?   _5n4 ? 4 q ? [ ' N  f?  {  |_sf V3 q  4  {  |_s f q fn??  V_s3 q {  |sqf .fn??   >

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Wdnh hdfk ri wkhvh lq wxuq lq zkdw iroorzv1 Ohw *g+}, eh wkh fkdudfwhulvwlf ixqfwlrq ri wkh ghqvlw|

N +, = Ehfdxvh *g lv devroxwho| lqwhjudeoh/ zh kdyh

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Lq ylhz ri wkh vwurqj dssur{lpdwlrq +45, > V3@sq lv doprvw vxuho| ri glvwdqfh ohvv wkdq q*R@sq

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Solp lqi_?<"J?  @ 4= Khqfh S  olp vxs ?<"  D2+q,  f? ]  f N  f?  {  |_s f q  E+u,  4U ?3U?^E+u,` gu  @ 3= Xvlqj wkh rfffxsdwlrq irupxod +8, zh kdyh wkh htxlydohqfh

] _ f N  f?  {  |_s f q  E+u,  4U_?3U?^E+u,` gu @ 32 ] <sub>"</sub> 3"N  f?  {  |<sub>s</sub> f q  s  4U<sub>?3U?</sub>^s` O+4> s,gs @ 32] U ?3U? N  f?  {  |_s f q  s  O+4> s,gs> zkrvh prgxoxv lv erxqghg e| d frqvwdqw pxowlsoh ri

] U ?3U? O+4> s,gs @ R@r # q*R s q $ = Lw iroorzv wkdw D2+q, @ R@rf?q*R@sq> dqg 4 n?D2+q, @ R@r # f?q*R n?sq $ = +5<,

Wr qg wkh rughu ri pdjqlwxgh ri D+q,> zh fdq xvh dowhuqdwh yhuvlrqv ri Ohppdv 818 dqg 81:

4:

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odujh hqrxjk wkdw wkh whup r@r+f32oE3? , lv qhjoljleoh lq uhodwlrq wr wkh rwkhu wzr whupv/ zklfk

fdq eh duudqjhg e| prgli|lqj Dvvxpswlrq 517 dv qhhghg1 E| Dvvxpswlrq 519 q3B_k2 ?$ 4/ dqg k?@qE3B*2$ 3 iru vrph  A 3> vr wkdw r@r # q0_kn*2_? q*e_q*e $ . R@r  q0 q_k32_?  @ r@r # q0_qE3BEn*2*2 q*e_q*e $ . r@r # q0_qE32E3B*2 q $ @ r@r # q0_qEn*2*2 q*e_q*e $ . r@r # q0_qE32*2 q $ @ r@r  4 qen2eD3230  . r@r  q0 q23*2  @ r@r  4 qen2eD3S30  . r@r  q0 q23*2  @ r@r  4 qenS30 

zklfk jlyhv wkh uhtxluhg uhvxow +48,1 Wr suryh +49, zh qhhg wr vkrz wkdw f? ] _ f N  f?  {  |_s f q  E+u,  4  {  |_s f q  4 f?  E+u,  {  |_s f q . 4 f?  gu@r$ O@cV > +63, iru wkhq s q ei?+{, @ s4 qk2_? ? [ |' N  {  ||3 k? 

4; @ f? ] _ f N  f?  {  |_s f q  E+u,  4  {  |_s f q f4?  E+u,  {  |_s f q .f4?  .r@r  4 q*en*S30  @r_{$ O}@cV  = +64,

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dv lq +5:, deryh1 Krzhyhu/ dffruglqj wr +47, {  |_s f q @r_{$ d  E} f+, +66, dqg/ vlqfh f? @ sq@k? $ 4 dv q $ 4> O+4> s, lv frqwlqxrxv lq s> dqg U_3"" N+v,gv @ 4> zh kdyh f? ] _" 3"N  f?  {  |_s f q  s  O+4> s,gs $@rO+4> d  Ef+,, @ O@cV >

dv uhtxluhg iru +63,/ dqg +64, wkhq iroorzv1

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4 qk? ? [ |' N  {  ||3 k?  ||3 @ s_qk4 ? ? [ |' N  {  ||3 k?  |_s|3 q @ f? ] _ f N  f?  {  |_s f q  E+u,   E+u, . s|f q  gu .r@r  4 q*en*S30  +67, @ f? ] _" 3"  s . s|f q   N  f?  {  |_s f q  s  O+4 > s,  gs .r@r  4 q*en*S30  = Vlqfh _I+f ? $@rEf+,> dqg lq ylhz ri +66, dqg wkh frqwlqxlw| ri sO+4 > s,/ zh rewdlq 4 qk? ? [ |' N  {  ||3 k?  ||3$@rdO+4> d  Ef+,, @ dO@cV > dv uhtxluhg iru +4;,1

4<

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udwkhu wkdq 4@qk? lq +67, dqg ohw  5 +4@5> 4,1 Wkhq/ iroorzlqj wkh vdph dssurdfk/ zh kdyh

4 s qk2 ? ? [ |' N  {  ||3 k?  ||3 @ f? ] _ f N  f?  {  |_s f q  E+u,  s qE+u, . |fgu .r@r # q*R q*en*S30 $ = +68,

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zklfk jlyhv wkh uhtxluhg olplw +4:, lq wklv fdvh1 Wkh vhfrqg whup lq +69, lv qhjoljleoh/ dv zh qrz vkrz1 Iluvw/U_3"" vN+v,gvO4> q3*2+{  |f,@ 3 e| yluwxh ri wkh idfw wkdw U_3"" vN+v, @ 3 e|

wkh v|pphwu| frqglwlrq lq Dvvxpswlrq 5171 Zh pd| wkhuhiruh zulwh  k? ] _" 3"vN+v,O+4> q 3*2_{+{  |} f,  v@f?,gv @ k? ] _" 3"vN+v, q O  4> q3*2_{+{  |} f,  v@f?   O  4> q3*2_{+{  |} f, r gv  frqvw= k?  k? s q *23#] " 3"mvm *23#_{mN+v,mgv> d=v= +6;,}

iru dq| A 3/ gxh wr wkh K‚roghu frqwlqxlw| ri O+4> ,1 Wkh uljkw vlgh +6;, whqgv wr }hur e|

yluwxh ri Dvvxpswlrq 517+d, surylghg q3*2_kn2*E32#_{? $ 3/ zklfk krogv zkhqhyhu k}

? @ fq&

dqg n ? 4@9  > vlqfh  fdq eh duelwudulo| vpdoo1 Wkxv/ iurp +69, > +6:, > dqg +6;, zh jhw wkh frqyhujhqfh _] " 3"i{  k?vj N+v,O  4>{  |s f q fv?  gv@r$ {fOfcf_

iru wkh vdph udqjh ri edqgzlgwkv k?= Wklv suryhv wkh vwdwhg uhvxow +4:, iru { @ {f {hg dqg

53

D

Ohw  5 +4@5> 4, dqg iroorzlqj wkh vdph olqh ri dujxphqw dv lq wkh surri ri Wkhruhp 614/ zh qg sf? k?sqk2? ? [ |' N  {  ||3 k?  +||3 {, @ f*2?_q [? |' N  {  ||3 k?   ||3 { k?  @ f*2 ? _q4 ? [ ' N  f?  {  |_sf V3 q   f?  {  |_sf V3 q  4{  |sf V_q 3  _f4 ?  .f*2? _q4 ? [ ' N  f?  {  |_sf V3 q   f?  {  |_sf V3 q  4{  |sf V3 q   A _f4 ?  @ f*2 ? _q4 ? [ ' N  f?  {  |_sf V3 q   f?  {  |_sf V3 q  4{  |sf V_q 3  _f4 ?  .r@r+f*2nE332oE3? ,>

dqg wkh huuru lv duelwudulo| vpdoo dv q $ 4 surylghg u A ₂ . _eE3 > zklfk fdq eh duudqjhg e| vxlwdeoh uhvwulfwlrq ri wkh nhuqho ixqfwlrq e| prgli|lqj Dvvxpswlrq 517 dqg wklv lv dvvxphg lq zkdw iroorzv1

Surfhhglqj dv lq uvw sduw ri wkh surri ri Wkhruhp 614/ zh kdyh sf? k?sqk2? ? [ |' N  {  ||3 k?  +||3 {, @ f*2 ? ] _ f N  f?  {  |_s f q  E+u,   f?  {  |_s f q  E+u,  4  {  |_s f q  4 f?  E+u,  {  |_s f q . 4 f?  gu . r@r # f*2? q f3? f*2? q*en0 $ . r@r+4, @ f*2 ? ] _ f N  f?  {  |_s f q  E+u,   f?  {  |_s f q  E+u,  gu .r@r 3 Cf D 232 ? qe q 0 4 D _+6<, Revhuyh wkdw f_?D232 qe q 0_@ q D e3en0 qek D 232 ? @ q  23en0 kD232 ?

dqg ohw  @ 4   iru  A 3 vpdoo1 Wkhq q23en0 kD232 ? @ q  eBn0 qekn  2B ?

Vlqfh % lv duelwudulo| vpdoo dqg  fdq eh fkrvhq duelwudulo| vpdoo/ zh ghgxfh wkdw qeBn0 qekn  2B ? @ r+4,

54

surylghg k? lv qrw wrr vpdoo/ lq sduwlfxodu iru

k?A 4

qe3>

iru vrph dueluwudulo| vpdoo  A 3= Lw iroorzv wkdw +6<, krogv zlwk d qhjoljleoh huuru dv q $ 4= Iurp wkh rffxsdwlrq irupxod zh ghgxfh wkdw f*2 ? ] _ f N  f?  {  |_s f q  E+u,   f?  {  |_s f q  E+u,  gu @ f*2? ] _" 3"N  f?  {  |_s f q  s   f?  {  |_s f q  s  O+4> s,gs>

dqg wudqviruplqj s :$ v @ f?^+{  |f, @sq  s` > wklv h{suhvvlrq ehfrphv

f*2 ? ] _" 3"vN +v, O  4>  {  |_s f q  v f?  gv= +73, Vlqfh U_3"" vN +v, gv @ 3> +73, htxdov ] " 3"vN+v,f *2 ?  O  4>  {  |_s f q  v f?   O  4>{  |s f q  gv _ $  3**2XO+4> d  Ef+,,> +74,

lq ylhz ri +44, ri Ohppd 5143 dqg wkh idfw wkdw q3*2_{+{  |f,}@r_{$ d  Ef+,1 Lq wkh olplw +74, > *}

lv ghqhg e| * @ 5U_3"" U_3"" md  emdeN+d,N+e,gdge/ dqg X+, lv d vwdqgdug Eurzqldq prwlrq lqghshqghqw ri E= Zh ghgxfh wkdw sf? k?sqk2? ? [ |' N  {  ||3 k?  +|_|3{,$  *_ *2_X _O@cV  @ *_ *23X O@cV > +75, dv uhtxluhg iru +4<,1

D2

Iroorzlqj wkh olqh ri dujxphqw xvhg lq wkh uvw sduw ri wkh surri ri Wkhruhp 614/ zh qg 4 es_qk2_? ? [ |' N  {  ||3 k?  x| @ f*2? s4_q ? [ ' N  f?  {  |_sf V3 q  4{  |sf V3 q    _f4 ?  x .f*2 ? s4_q ? [ ' N  f?  {  |_sf V3 q  4{  |sf V3 q   A _f4 ?  x @ f*2? s4_q ? [ ' N  f?  {  |_sf V3 q  4{  |sf V3 q    _f4 ?  x .RR  f*2 ? f32oE3? 

55 @ f*2? s4_q ? [ ' N  f?  {  |_sf V3 q  4{  |sf V3 q    _f4 ?  x .RR  f*2? f32oE3?  = +76, Qrz f232oE3

? @ r+4, dv q $ 4> iru u A 6@7+4  ,> zklfk lv khuhe| dvvxphg/ dqg wkhq wkh odvw

whup ri +76, fdq eh qhjohfwhg1 Wkh uvw whup lv d pduwlqjdoh zlwk txdgudwlf yduldwlrq 2_f ?_q4 ? [ ' N  f?  {  |_sf V3 q ₂ 4{  |sf V3 q    _f4 ?  =

Dv lq wkh surri ri Wkhruhp 614 zh fdq ghwhuplqh wkh dv|pswrwlf ehkdylrxu ri wklv frqglwlrqdo yduldqfh ixqfwlrq xvlqj wkh vwurqj dssur{lpdwlrq1 Lq sduwlfxodu/ zh kdyh

2_f ?4_q ? [ ' N  f?  {  |_sf V3 q 2 4{  |sf V3 q    _f4 ?  @ 2f? ] _ f N  f?  {  |_s f q  E+u, 2 4{  |s f q  E+u,    _f4 ?  gu .r@r  4 q*en*S30  @ f? ] _" 3"N  f?  {  |_s f q  s, ₂ 4{  |s f q  s    _f4 ?  O+4> s, gs . r@r+4, @ ] " 3"N +v, 2₄_{mvm  f}3 ? O  4>{  |s_qf  _fv ?  gv . r@r+4, $ @r ] _" 3"N +v, 2_gv_O +4> d  Ef+,, @ n2O@cV_ +4,> +77,

zkhuh O@cV_ +u, @ O_+u> d  E_f+,, dqg n2 @U_3"" N +v,2gv=

Lw qrz iroorzv iurp wkh jhqhudol}hg pduwlqjdoh fhqwudo olplw wkhruhp +h1j Kdoo dqg Kh|gh/ 4<;3/ s18;, wkdw wkh uvw whup ri +76, kdv d pl{hg qrupdo olplw glvwulexwlrq zlwk frqglwlrqdo yduldqfh +77,1 Khqfh/ 4 e s qk2 ? ? [ |' N  {  ||3 k?  x|$_PQ3> n2O@cV +4,> ru/ htxlydohqwo|/ 4 e s qk2_? ? [ |' N  {  ||3 k?  x|$_Y n2O@cV_ +4,>

zkhuh Y lv d Eurzqldq prwlrq wkdw lv lqghshqghqw ri E1 Wklv jlyhv wkh vwdwhg uhvxow1

Surri ri Wkhruhp 619 Zulwh

p?+{, @ S_? |'N _%3+ |3 ?  ||3 S_? |'N _%3+ |3 ?  . S_? |'N _%3+ |3 ?  x| S_? |'N _%3+ |3 ?  = +78,

56

Iurp Wkhruhp 614 dqg Ohppd 616/ zh ghgxfh wkdw

 s ?2 ? S_? |'N _%3+ |3 ?  ||3  s ?2 ? S_? |'N  %3+|3 ?  @r$ {OfcV OfcV @ { iru { @ il{hg> +79, dqg  ?? S_? |'N _%3+ |3 ?  ||3  s ?2 ? S_? |'N _%3+ |3 ?  @r$ dO @cV  O@cV @ d iru { @ s qd=

Pruhryhu/ iurp Ohppd 617 lw iroorzv wkdw 4 s qk2 ? ? [ |' N  {  ||3 k?  x| $ 3R iru { @  {hg s qd Zh ghgxfh wkdw p?+{,$ {R iru { @ il{hg> dqg 4 s qp?+{, R $ d iru { @sqd> dv uhtxluhg1

Surri ri Wkhruhp 61: Vwduw zlwk wkh fdvh zkhuh { lv {hg1 Iurp +78, zh kdyh

e s qk2 ?ip?+{,  {j @  e s ?2 ? S_? |'N _%3+ |3 ?  +||3 {,  s ?2 ? S_? |'N _%3+ |3 ?  .  e s ?2 ? S_? |'N _%3+ |3 ?  x|  s ?2 ? S_? |'N _%3+ |3 ?  = +7:,

Lq ylhz ri Ohppd 617/ wkh uvw whup rq wkh uljkw vlgh ri +7:, lv

e s qk2_?   s ?2 ? S_? |'N _%3+ |3 ?  +||3 {,   s ?2 ? S_? |'N _%3+ |3 ?  @ RR # k?se qk2? sf? $ @ RRk2?$ 3 =

zkhq k? $ 3= Rq wkh rwkhu kdqg/ wkh vhfrqg whup ri +7:, kdv wkh iroorzlqj olplw e| yluwxh ri

Wkhruhp 614 dqg Ohppd 618  e s ?2 ? S_? |'N _%3+ |3 ?  x|  s ?2_? S_? |'N  %3+|3 ?  $_ q n22OfcV r_*2 Y +4, OfV _ @qn22@OfcV r_*2 Y +4,= Wkxv/ zkhq k?$ 3 zh kdyh e s qk2 ?ip?+{,  {j$_ q n22@OfcV r*2 Y +4,= zklfk suryhv +53,1

57

Zkhq k? $ 4 wkh uvw whup lq +7:, grplqdwhv dqg zh kdyh xvlqj +4:, ri Ohppd 616 dqg

+4<, ri Ohppd 617 sf? k? ip?+{,  {j @ I_S ? ?s?2? S_? |'N _%3+ |3 ?  +||3 {,  s ?2 ? S_? |'N  %3+|3 ?  . I_S ? ?s?2? S_? |'N _%3+ |3 ?  x|  s ?2 ? S_? |'N  %3+|3 ?  +7;, @ IS? ?s?2? S_? |'N _%3+ |3 ?  +||3 {,  s ?2 ? S_? |'N  %3+|3 ?  . RR  4 k2 ?  +7<, _ $ q *32_OfcV  r*2 X+4, OfcV @q*32_@OfcV  r*2 X+4,> zklfk frpsohwhv wkh surri ri +56,1

Zkhq k?@ k @ frqvwdqw/ erwk whupv ri +7:, dhfw wkh olplw glvwulexwlrq dqg zh jhw

s_f ?ip?+{,  {j @ I_S ? s ?2_? S_? |'N _%3+ |3 ?  +||3 {,  s ?2 ? S_? |'N _%3+ |3 ?  . I_S ? s ?2_? S_? |'N _%3+ |3 ?  x|  s ?2 ? S_? |'N _%3+ |3 ?  _ $ k q *32_OfcV  r*2 X+4, OfcV . k 3q_n22_OfcV  r*2 Y +4, OfV @ kq*32@OfcV r*2 X+4, . k3qn22@OfcV r*2 Y +4,> ohdglqj wr +55,1

Wkh fdvh zkhuh { @sqd lv suryhg lq wkh vdph zd| dqg wkh olplwv lqyroyh O@cV_ udwkhu wkdq OfcV =

S +iuihi?Uit

D‚lw Vdkdold/ \1 +4<<9, _Qrqsdudphwulf sulflqj ri lqwhuhvw udwh ghulydwlyh vhfxulwlhv%1 Hfrqrphw0

ulfd/ 97/ 85:08931

D‚lw Vdkdold/ \1 +4<<:, _Pd{lpxp olnholkrrg hvwlpdwlrq ri glvfuhwho| vdpsohg glxvlrqv= d forvhg irup dssurdfk%1 Xqlyhuvlw| ri Fklfdjr/ plphrjudskhg1

Dnrqrp/ M1 +4<<6,1 _Frpsruwhphqw dv|pswrwltxh gx whpsv g*rffxsdwlrq gx surfhvvxv ghv vrpphv sduwlhoohv%1 Dqqdov ri wkh Lqvwlwxwh ri Khqul Srlqfduh/ 5</ 8:0;41

Elhuhqv/ K1 +4<:;,1 _Nhuqho Hvwlpdwruv ri Uhjuhvvlrq Ixqfwlrqv/% lq W1 Ehzoh| +hg1,/ Dgydqfhv

lq Hfrqrphwulfv/ Iliwk Zruog Frqjuhvv/ Yro1 41 Fdpeulgjh= Fdpeulgjh Xqlyhuvlw| Suhvv/

ss1 <<~4771

Froorpe/ J1 +4<;8d,1 _Qrqsdudphwulf wlph vhulhv dqdo|vlv dqg suhglfwlrq= xqlirup doprvw vxuh frqyhujhqfh ri wkh zlqgrz dqg n0QQ dxwruhjuhvvlrq hvwlpdwhv1% Vwdwlvwlfv/ 49/ 5<:063:1

58

Froorpe/ J1 +4<;8e,1 _Qrqsdudphwulf uhjuhvvlrq= dq xs0wr0gdwh eleolrjudsk|1% Vwdwlvwlfv/ 49/ 63<06571

Froorpe/ J1 dqg Z1 Kdugoh +4<;9,1 _Vwurqj xqlirup frqyhujhqfh udwhv lq urexvw qrqsdudphwulf wlph vhulhv dqdo|vlv dqg suhglfwlrq= Nhuqho uhjuhvvlrq hvwlpdwlrq iurp ghshqghqw revhuyd0 wlrqv/% Vwrfkdvwlf Surfhvvhv dqg wkhlu Dssolfdwlrqv/ 56/ ::~;<1

Fv‚rujr/ P1 dqg O1 Kruydwk +4<<6,1 Zhljkwhg Dssur{lpdwlrqv lq Suredelolw| dqg Vwdwlvwlfv1 Zloh|= Qhz \run1

Kdugoh/ Z1 +4<<3,1 Dssolhg Qrqsdudphwulf Uhjuhvvlrq1 Fdpeulgjh= Fdpeulgjh Xqlyhuvlw| Suhvv1

Kdugoh/ Z1 dqg R1 E1 Olqwrq +4<<8,1 _Dssolhg qrqsdudphwulf phwkrgv/% lq G1I1 PfIdgghq dqg U1I1 Hqjoh +hgv1, Wkh Kdqgerrn ri Hfrqrphwulfv Yro LY/ Dpvwhugdp= Qruwk Kroodqg1 Kdugoh/ Z1 dqg S1 Ylhx +4<<5,1 _Nhuqho uhjuhvvlrq vprrwklqj ri wlph vhulhv/% Mrxuqdo ri Wlph

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Mldqj J1 M1 dqg M1 O1 Nqljkw +4<<:, _D qrqsdudphwulf dssurdfk wr wkh hvwlpdwlrq ri glxvlrq surfhvvhv/ zlwk dq dssolfdwlrq wr d vkruw0whup lqwhuhvw udwh prgho%1 Hfrqrphwulf Wkhru|/ 46/ 94809781

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