More Efficient Kernel Estimation in Nonparametric Regression with Autocorrelated Errors

More Efficient Kernel Estimation in

Nonparametric Regression with Autocorrelated Errors By

Zhijie Xiao, Oliver Linton, Raymond J. Carroll and E. Mammen June 2002

COWLES FOUNDATION DISCUSSION PAPER NO. 1375

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY

Box 208281

New Haven, Connecticut 06520-8281 http://cowles.econ.yale.edu/

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e 'JREE*L}AA3*23_*2f n^fc dqg _'REA3*2*2 Lq dgglwlrq/ 4@ r m6eEfr6Efrm'RE ^ f nA3*23_*2f E*L}A*2c wkxv m'A em'JREA3*23_*2 UHIHUHQFHV

Dndlnh/ K1/ 4<:6/ Lqirupdwlrq Wkhru| dqg dq H{whqvlrq ri wkh Pd{lpxp Olnholkrrg Sulqflsoh/ lq Shwury dqg Fvdnl +hgv,5qg Lqwhuqdwlrqdo V|psrvlxp rq Lqirupdwlrq Wkhru|/ Exgdshvw1

Dqghuvhq/ W1Z1 +4<:4,1 Wkh Vwdwlvwlfdo Dqdo|vlv ri Wlph Vhulhv1 Qhz \run= Mrkq Zloh| dqg Vrqv1

Dxhvwdg/ E1 dqg Wmûvwkhlp/ G1/ +4<<4,1 Ixqfwlrqdo lghqwlfdwlrq lq qrqolqhdu wlph vhulhv1 Lq

Qrqsdudphwulf Ixqfwlrqdo Hvwlpdwlrq dqg Uhodwhg Wrslfv/ hg1 J1 Urxvvdv/ Noxzhu Dfdghplf= Dpvwhugdp1 ss 7<683:1

Fkhq/ [1/ dqg R1 Olqwrq +5334,1 Dq Dowhqdwlyh zd| ri frpsxwlqj h!flhqw vhplsdudphwulf lqvwuxphqwdo yduldeohv hvwlpdwruv1

Frqoh|/ W1J1/ O1S1 Kdqvhq/ H1J1M1 Oxwwphu/ dqg M1D1 Vfkhlqnpdq +4<<:,1 Vkruw0Whup Lqwhuhvw Udwhv dv Vxeruglqdwhg Glxvlrqv/ Wkh Uhylhz ri Ilqdqfldo Vwxglhv 43/ 85808::1

Idq/ M/ H1 Pdpphq/ dqg Z1 Kçugoh +4<<;,1 Gluhfw hvwlpdwlrq ri orz glphqvlrqdo frpsrqhqwv lq dgglwlyh prghov1 Dqq1 Vwdwlvw1/ 59/ <760<:41

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

Kdqqdq/ H1M1/ dqg P1 Ghlvwohu +4<;;,1 Wkh Vwdwlvwlfdo Wkhru| ri Olqhdu V|vwhpv1 Mrkq Zloh|> Qhz \run1

Kduw/ M1G1 +4<<4,1 Nhuqho Uhjuhvvlrq Hvwlpdwlrq zlwk Wlph Vhulhv Huuuruv1 Mrxuqdo ri wkh Ur|do Vwdwlvwlfdo Vrflhw|1 86/ 4:604;:1

Kdvwlh/ W1 dqg U1 Wlevkludql+4<<4,1 Jhqhudol}hg Dgglwlyh Prghov1 Fkdspdq dqg Kdoo/ Orqgrq1

Nulvwhqvhq/ G1 dqg R1 Olqwrq +5334,= Dq Dowhuqdwlyh JOV0olnh Wudqvirupdwlrq lq Uhjuhvvlrq Prghov zlwk DU+4, huuruv1 Sureohp Fruqhu/ Hfrqrphwulf Wkhru|1

Olq/ [1 dqg Fduuroo/ U1 M1 +5333,/ Qrqsdudphwulf Ixqfwlrq Hvwlpdwlrq iru Foxvwhuhg Gdwd Zkhq wkh Suhglfwru lv Phdvxuhg Zlwkrxw2Zlwk Huuru/ Mrxuqdo ri wkh Dphulfdq Vwdwlvwlfdo Dvvrfldwlrq/ <8/ 85308671

Olqwrq/ R1E1 +4<<:,1 H!flhqw hvwlpdwlrq ri dgglwlyh qrqsdudphwulf uhjuhvvlrq prghov1 Elr0 phwulnd/;7/ 79<07:71

Olqwrq/ R1E1 dqg H1 Pdpphq1 +5335,1 Hvwlpdwlqj dq DUFK+4, prgho e| nhuqho uhjuhvvlrq phwkrgv1 Pdqxvfulsw/ OVH1

Olqwrq/ R1E1 dqg M1S1 Qlhovhq1 +4<<8,1 Hvwlpdwlqj vwuxfwxuhg qrqsdudphwulf uhjuhvvlrq e| wkh nhuqho phwkrg1 Elrphwulnd ;5/ <604341

Pdpphq/ H/ R1E1 Olqwrq/ dqg M1S1 Qlhovhq1 +4<<<,1 Wkh h{lvwhqfh dqg dv|pswrwlf surshuwlhv ri d edfnwwlqj surmhfwlrq dojrulwkp xqghu zhdn frqglwlrqv1 Wkh Dqqdov ri Vwdwlvwlfv 5:/ 47760 47<31

Pdvu|/ H1 +4<<9d,1 Pxowlyduldwh uhjuhvvlrq hvwlpdwlrq= Orfdo sro|qrpldo wwlqj iru wlph vhulhv1

Vwrfkdvwlf Surfhvvhv dqg wkhlu Dssolfdwlrqv1 98/ ;404341

Pdvu|/ H1 +4<<9e,1 Pxowlyduldwh orfdo sro|qrpldo uhjuhvvlrq iru wlph vhulhv= Xqlirup vwurqj frq0 vlvwhqf| dqg udwhv1 M1 Wlph Vhu1 Dqdo1 4:/ 8:408<<1

Rsvrphu/ M1/ \1 Zdqj/ dqg \1 \dqj+5334,1 Qrqsdudphwulf Uhjuhvvlrq zlwk Fruuhodwhg Huuruv1 Pdqxvfulsw1

Skloolsv/ S1F1E1 dqg Y1 Vror/+4<<5,/ Dv|pswrwlfv iru Olqhdu Surfhvvhv/Dqqdov ri Vwdwlvwlfv/

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Urelqvrq/ S1P1 +4<;6,1 Qrqsdudphwulf Hvwlpdwruv iru Wlph Vhulhv1 Mrxuqdo ri Wlph Vhulhv Dqdo|vlv 7/ 4;8053:1

Urvhqeodww/ P1+4<89,1 D fhqwudo olplw wkhruhp dqg vwurqj pl{lqj frqglwlrqv/Surf1 Qdw1 Dfdg1 Vfl1 7/ 7607:1

Uxfnvwxko/ D1/ Zhovk/ D1 K1 dqg Fduuroo/ U1 M1 +5333,/ Qrqsdudphwulf Ixqfwlrq Hvwlpd0 wlrq ri wkh Uhodwlrqvkls Ehwzhhq Wzr Uhshdwhgo| Phdvxuhg Yduldeohv/ Vwdwlvwlfd Vlqlfd/ 43/ 84:41

Uxsshuw/ G1/ dqg P1 Zdqg +4<<7,1 Pxowlyduldwh orfdoo| zhljkwhg ohdvw vtxduhv uhjuhvvlrq1 Dqq1 Vwdwlvw1 55/ 4679046:31

Vhyhulql/ W1 D1 dqg Vwdqlvzdolv/ M1 J1 +4<<7,/ Txdvlolnholkrrg Hvwlpdwlrq lq Vhplsdudphwulf Prghov/ Mrxuqdo ri wkh Dphulfdq Vwdwlvwlfdo Dvvrfldwlrq/ ;</ 8348441

Wmûvwkhlp/ G1/ dqg E1 Dxhvwdg +4<<7,1 Qrqsdudphwulf lghqwlfdwlrq ri qrqolqhdu wlph vhulhv= surmhfwlrqv1 M1 Dp1 Vwdw1 Dvvrf1 ;</ 46<;0473<1

Ylodu0Ihuqdqgh}/ M1P1 dqg P1 Iudqflvfr0Ihuqdqgh} +5334,1 Orfdo sro|qrpldo uhjuhvvlrq vprrwkhuv zlwk DU0huuru vwuxfwxuh1 Iruwkfrplqj lq WHVW1

Zlog/ F1 M1 dqg \hh/ W1 Z1 +4<<9,/ Dgglwlyh H{whqvlrqv wr Jhqhudol}hg Hvwlpdwlqj Htxdwlrq Phwkrgv/ Mrxuqdo ri wkh Ur|do Vwdwlvwlfdo Vrflhw|/ Vhulhv E/ 8;/ :440:581

Zx/ F1 R1/ Fkldqj/ F1 W1 dqg Krryhu/ G1 U1 +4<<;,/ Dv|pswrwlf Frqghqfh Uhjlrqv iru Nhuqho Vprrwklqj ri d Ydu|lqj Frh!flhqw Prgho zlwk Orqjlwxglqdo Gdwd/ Mrxuqdo ri wkh Dphulfdq Vwdwlvwlfdo Dvvrfldwlrq/ <6/ 46;;47351

\rvklkdud/ N1+4<:9,1 Olplwlqj ehkdylru ri X0vwdwlvwlfv iru vwdwlrqdu|/ devroxwh uhjxodu surfhvvhv1

]1 Zdkuvfkhlqolfknhlwvwkhrulh yhuz1 Jhelhwh/ 68/ 56:05851

]hjhu/ V1 O1 dqg Gljjoh/ S1 M1 +4<<7,/ Vhplsdudphwulf Prghov iru Orqjlwxglqdo Gdwd Zlwk Dssolfdwlrq wr FG7 Fhoo Qxpehuv lq KLY Vhurfrqyhuwhuv/ Elrphwulfv/ 83/ 9;<09<<1

E Wdeohv dqg Iljxuhv

WDEOH 4= Uhodwlyh H!flhqf| +UH,/? ' ff

+DU+5, Suhzklwhqlqj/ Hvwlpdwlqj6E%dw %' f, DUPD Sdudphwhuv Uhodwlyh H!flhqf|

k k2  2 UH4 UH5 UH3 3 3 3 3 413:3 4137; 413333 314 41379 41345 31<<34 315 41355 31<;9 31<948 318 31<;4 31<95 31;333 31: 31<95 31<75 319:44 31< 31<96 31<66 318857 314 41375 4136< 31<<33 315 41357 31<<7 31<933 318 31<;3 31<:4 31:833 31: 31<94 31<;5 318433 31< 31<;3 31<<3 314<33 314 314 41359 31<<< 31<945 314 315 4133; 31<;9 31<499 315 315 31<<4 31<;6 31;8:4 318 315 31<8< 31<:5 31937; 31: 315 31<83 31<95 316;97 31< 315 31<:3 31<;9 31468: 314 318 31<:4 31<87 31:666 315 318 31<93 31<83 319954 318 318 31<75 31<74 3175;9 31: 318 31<73 31<76 315948 31< 318 31<:9 31<;3 313;;7 318 315 31<:6 31<:9 31::85 315 315 41349 31<<: 31<58< 315 31: 31<;9 31<;: 319869 315 315 31<<3 31<;; 31<333 318 315 31<97 31<:< 318;83 31: 315 31<;3 31<<3 315583

WDEOH 5= Uhodwlyh H!flhqf| +UH,/? ' 2ff

+DU+5, Suhzklwhqlqj,

DUPD Sdudphwhuv Uhodwlyh H!flhqf|

k k2  2 UH4 UH5 3 3 3 3 4138; 41379 314 41357 41354 315 4133: 31<<9 318 31<:8 31<93 31: 31<99 31<6; 31< 31<97 31<5< 314 41345 41359 315 41334 31<<9 318 31<6; 31<86 31: 31<3; 31<4< 31< 31<66 31<73 314 314 41337 41334 314 315 31<;; 31<;9 315 315 31<:3 31<:5 318 315 31<4< 31<4< 31: 315 31;<9 31;<7 31< 315 31<5< 31<5< 314 318 31<94 31<78 315 318 31<7: 31<5< 318 318 31<3: 31;;: 31: 318 31;;< 31;:7 31< 318 31<59 31<55 318 315 31<79 31<85 315 315 31<:3 31<<3 315 31: 31<89 31<95 315 314 31<:< 31<;: 315 315 31<8; 31<:4 318 315 31<66 31<7; 31: 315 31<69 31<84

WDEOH 6= Uhodwlyh H!flhqf| +UH,/? ' Dff +DU+5, Suhzklwhqlqj, k k2  2 UH4 UH5 3 3 3 3 41365 41359 314 41336 41334 315 31<;5 31<<5 318 31<84 31<83 31: 31<73 31<59 31< 31<6; 31<48 314 31<<< 31<<: 315 31<:3 31<<3 318 31<38 31<63 31: 31;9: 31;;4 31< 31;94 31;99 314 314 31<:; 31<<5 314 315 31<94 31<:< 315 315 31<75 31<94 318 315 31;;: 31;<9 31: 315 31;8: 31;8; 31< 315 31;8: 31;88 314 318 31<68 31<65 315 318 31<53 31<47 318 318 31;:9 31;98 31: 318 31;83 31;74 31< 318 31;87 31;7; 318 315 31<4; 31<6: 315 315 31<74 31<:; 315 31: 31;<: 31<54 315 314 31<7< 31<:9 315 315 31<59 31<88 318 315 31;9: 31;;4 31: 315 31;94 31;99

WDEOH 7 Uhodwlyh H!flhqf| +UH,/? ' fff k k2  2 UH4 UH5 3 3 3 3 41349 41345 314 31<<7 31<<9 315 31<:4 31<;; 318 31<7< 31<7; 31: 31<76 31<5; 31< 31<73 31<54 314 31<;; 31<<: 315 31<9: 31<;: 318 31<45 31<64 31: 31;;3 31;<3 31< 31;:4 31;;9 314 314 31<:4 31<;< 314 315 31<8; 31<:9 315 315 31<75 31<8< 318 315 31;<< 31<36 31: 315 31;:8 31;:7 31< 315 31;:9 31;:8 314 318 31<6: 31<66 315 318 31<58 31<4; 318 318 31;<3 31;;4 31: 318 31;:4 31;95 31< 318 31;:8 31;:4 318 315 31<57 31<6: 315 315 31<74 31<:8 315 31: 31<33 31<5: 315 315 31<5< 31<86 318 315 31;;6 31;<4 31: 315 31;:; 31;;3

WDEOH 8= Uhodwlyh H!flhqf| +UH,/? ' ff +DU+4, Suhzklwhqlqj, k k2  2 UH4 UH5 3 3 3 3 41376 4135< 314 41357 41349 315 41339 41338 318 31<99 31<<4 31: 31<86 31<:< 31< 31<7; 31<:7 314 41357 4146; 315 4133: 4133< 318 31<99 31<;< 31: 31<85 31<:9 31< 31<:4 31<<3 315 315 31<:9 31<<9

WDEOH 9= Uhodwlyh H!flhqf| +UH,/? ' 2ff

+DU+4, Suhzklwhqlqj, k k2  2 UH4 UH5 314 41333 41334 315 31<;3 31<<6 318 31<68 31<93 31: 31<53 31<77 31< 31<47 31<6: 314 31<<< 31<<< 315 31<:< 31<<6 318 31<56 31<7; 31: 31;<: 31<49 31< 31<5; 31<6< 315 315 31<76 31<9;

WDEOH := Uhodwlyh H!flhqf| +UH,/? ' Dff +DU+4, Suhzklwhqlqj, k k2  2 UH4 UH5 314 31<<8 41333 315 31<9: 31<<5 318 31<4: 31<84 31: 31<34 31<64 31< 31;<7 31<55 314 31<<3 31<<< 315 31<99 31<<5 318 31<33 31<63 31: 31;97 31;;4 31< 31;94 31;99 315 315 31<58 31<8<

WDEOH ;= Uhodwlyh H!flhqf| +UH,/ ?' fff

+DU+4, Suhzklwhqlqj, k k2  2 UH4 UH5 314 31<;6 31<<9 315 31<97 31<;; 318 31<58 31<83 31: 31<45 31<67 31< 31<3; 31<5: 314 31<;6 31<<9 315 31<93 31<;: 318 31<3< 31<63 31: 31;;3 31;<3 31< 31;:: 31;;4 315 315 31<63 31<83

WDEOH <= Uhodwlyh H!flhqf| +UH,/? ' ff

+DU+5, Suhzklwhqlqj/6E% '%/ hvwlpdwh doo srlqwv ,

DUPD Sdudphwhuv Lqwhjudwhg Vtxduhg Huuruv Uhodwlyh H!flhqf|

k  LVH3 LVH4 LVH5 UH4 UH5 UH3 3 3 313974 31398; 313994 4135: 41364 413333 314 313::3 313::9 313::; 4133: 41343 31<<34 315 313<45 313<38 313<34 31<<5 31<;; 31<948 318 314746 314696 314664 31<97 31<75 31;333 31: 314;3< 314:66 314993 31<8: 31<49 319:44 31< 315588 31548: 315377 31<89 31<39 318857 314 313:;7 313:;< 313:<4 41339 4133< 31<<33 315 313<;6 313<9; 313<:3 31<;7 31<;9 31<933 318 3156<6 315549 31555: 31<59 31<64 31:833 31: 3193:6 31873< 3186:8 31;<4 31;;8 318433 31< 61958: 61488< 6145<: 31;:3 31;96 314<33 315 315 314735 314676 314674 31<8: 31<89 31;8:4

WDEOH 43= Uhodwlyh H!flhqf| +UH,/? ' 2ff

+DU+5, Suhzklwhqlqj/ 6E% '%/ dwdoo srlqwv ,

DUPD Sdudphwhuv Lqwhjudwhg Vtxduhg Huuruv Uhodwlyh H!flhqf|

k  LVH3 LVH4 LVH5 UH4 UH5 UH3 3 3 31367;; 3136858 3136864 413443 413453 413333 314 3137538 313749; 31374:4 31<<43 31<<4; 31<<34 315 3137<<4 3137;:4 3137<3: 31<:94 31<;67 31<948 318 313::89 313:6:< 313:5;: 31<847 31<6<9 31;333 31: 313<<74 313<6<: 313<44; 31<786 31<4:6 319:44 31< 3145733 31449<3 3144593 31<766 31<3:9 318857 314 31375;: 3137574 3137585 31<;<3 31<<4; 31<<33 315 31386<; 313855; 3138635 31<9;7 31<;55 31<933 318 31467;3 31455;3 3145743 31<43: 31<4<: 31:833 31: 3168;83 3164763 3164663 31;:9; 31;:69 318433 31< 51855<3 5147943 5145;43 31;839 31;768 314<33 315 315 313::65 313:5<7 313:67< 31<767 31<838 31;8:4