Jul 21, 2001 - the shape of conditional distributions are expected to vary with .... ing upon X and with scale and shape parameters independent of X. We use ...
PARAMETER APPROXIMATIONS FOR QUANTILE REGRESSIONS WITH MEASUREMENT ERROR Andrew Chesher
THE INSTITUTE FOR FISCAL STUDIES DEPARTMENT OF ECONOMICS, UCL
cemmap working paper CWP02/01
Sdudphwhu Dssur{lpdwlrqv iru Txdqwloh Uhjuhvvlrqv zlwk Phdvxuhphqw Huuru Aaij jtji jA|ij uNi WiNaB|B j|Nat BAa iBW|Wj YAt|||j uNi 8tWB, 7|ajt BAa PAjit|+ N,,j}j wNAaNA Mxo| 54vw 5334 Ot|iBW| Ai 4T@U| Lu UL@h@|i 4i@thi4i?| ihhLh L? ^@?|*i hi}hittL? u?U|L?t t ?it|}@|i_ t?} @ t4@** @h@?Ui @TThL 4@|L? Ai @TThL 4@|L? tLt L |i ihhLh UL?|@4?@|i_ @?_ ihhLh uhii ^@?|*i hi}hittL? u?U|L?t @hi hi*@|i_c @ !i) u@U|Lh Mi?} |i _t|hM|L? Lu |i ihhLh uhii UL@h@|i , @U| U@*U*@ |L?t ThLMi |i @UUh@U) Lu |i @TThL 4@|L? Ai Lh_ih Lu |i @TThL 4@|L? ihhLh t ?U@?}i_ u |i ihhLh uhii UL@h@|i _i?t|) t hiT*@Ui_ M) |i ihhLh UL?|@4?@|i_ _i?t|) W| t |i? TLttM*i |L ti |i @TThL 4@|L? |L ?it|}@|i |i ti?t||) Lu it|4@|it |L @h)?} @4L?|t Lu 4i@thi4i?| ihhLh lj+NiatG 4i@thi4i?| ihhLhc T@h@4i|ih @TThL 4@|L?tc ^@?|*i hi}hittL?
YA|iNaW|NA
Txdqwloh uhjuhvvlrq ixqfwlrqv duh ri sduwlfxodu lqwhuhvw lq sureohpv zkhuh wkh glvshuvlrq ru wkh vkdsh ri frqglwlrqdo glvwulexwlrqv duh h{shfwhg wr ydu| zlwk fryduldwh ydoxhv1 Vlqfh wkh vhplqdo sdshu ri Nrhqnhu dqg Edvvhww +4 lq dqdo|vlv ri wkh g|qdplfv dqg glvwulexwlrq ri zdjhv dqg hduqlqjv/ Exvklqvn| +5334/ 4 {, d ghshqghqfh zh pdnh h{solflw zkhuh rwkhuzlvh wkhuh pljkw eh frqixvlrq1 Frqvlghulqj yduldwlrqv lq {/ dqg T vxemhfw wr +D414, wkhuh lv [ I \ +Tm{,gT . I l +Tm{,g{l @ g +D415, l
zkhuh \
I +Tm{, @ I l +Tm{, @
C \ I +|m{, C| |@T C \ I +|m{, = C{l |@T
Vkruwo| vhfrqg sduwldo ghulydwlyhv dsshdu/ I \ \ / I \ l dqg I lm / ghqhg vlploduo|1 Htxdwlrq +D415, ohdgv gluhfwo| wr wkh iroorzlqj h{suhvvlrqv iru wkh uvw sduwldo ghulydwlyhv ri wkh frqglwlrqdo txdqwloh ixqfwlrq1 4 I \ +Tm{, I l +Tm{, Tl + > {, @ \ I +Tm{,
T + > {, @
+D416, +D417,
Wkh vhfrqg rughu sduwldo ghulydwlyhv ri wkh txdqwloh ixqfwlrq iroorz rq glhuhqwldwlqj +D416, dqg +D417,1 T + > {, @
I \ \ +Tm{, I \ \ +Tm{, T + > {, @ \ \ 5 I +Tm{, I +Tm{,6
\l 4 I +Tm{, . I \ \ +Tm{,Tl + > {, 5 +Tm{, I \ l +Tm{, I \ \ +Tm{,I l +Tm{, @ \ . 5 I +Tm{, I \ +Tm{,6
T l + > {, @
I\
\l 4 I +Tm{,Tm + > {, . I lm +Tm{, +Tm{, I l +Tm{, \ \ . \ I +Tm{,Tm + > {, . I \ m +Tm{, 5 I +Tm{, I lm +Tm{, I \ l +Tm{,I m +Tm{, I \ m +Tm{,I l +Tm{, @ \ . . I +Tm{, I \ +Tm{,5 I \ +Tm{,5 I \ \ +Tm{,I l +Tm{,I m +Tm{, I \ +Tm{,6
Tlm + > {, @
+D418,
+D419,
I\
+D41:,
BiB6j|ji VViN!6B|NAt uNi #BA|,j -j}ijttNAt | jBtij6jA| .iiNi 54
Lq wkh pdlq wh{w zh qrwhg wkdw m +}, 4 I\lmm[ +T] m}, I\l m[ +T] m},j[ CT] @ C lm @3 5 I\\ m[ +T] m}, I\\m[ +T] m},
+D41;,
zklfk zh qrz zlvk wr h{suhvv lq whupv ri wkh frqglwlrqdo TUI dqg lwv ghulydwlyhv1 Wkh ohdglqj whup lv jlyhq gluhfwo| e| +D417, zlwk vxlwdeoh h{sdqvlrq ri qrwdwlrq1 Qrz qrwh wkdw/ iurp +D419,/ I \ l +Tm{,I m +Tm{, T l + > {,Tm + > {, T + > {,Tl + > {,Tm + > {, @ = \ 5 I +Tm{, T + > {, T + > {,5 dqg iurp +D41:,/ h{sorlwlqj +D416, dqg +D417, T l + > {,Tm + > {, T m + > {,Tl + > {, T + > {,Tl + > {,Tm + > {, I lm +Tm{, @ Tlm + > {,. . \ I +Tm{, T + > {, T + > {, T + > {,5 Vxevwlwxwlqj wklv qdo h{suhvvlrq lq +D41;, jlyhv htxdwlrq +5, lq wkh pdlq wh{w1
VVjAa! 4H Cj juujW| NA |j BVViN!6B|NA Nu tA} |j ,N} ajAt|+ Nu ] iB|ji |jA [1 Fkhvkhu +4w
vw
vw i[ +}, . r+ , i[ +},
dqg wkhlu ghulydwlyhv vdwlvi| m +}, j]
@
m j[ +},
.
[ v>w
# vw
vwm m vw +}, i[ +},i[ +}, i[ i[ +}, i[ +},5
$ . r+ ,=
Lw iroorzv lpphgldwho| wkdw [ [ m m lm Tl[ + > },j[ +}, lm Tl[ + > },j] +}, @ r+ , l>m
l>m
dqg wkhq gluhfwo| wkdw wkh rughu ri wkh dssur{lpdwlrq huuru lq +5, lv qrw lqfuhdvhg rq m m vxevwlwxwlqj j] +}, iru j[ +},1
BiB6j|ji VViN!6B|NAt uNi #BA|,j -j}ijttNAt | jBtij6jA| .iiNi 55
VVjAa! H .!VNAjA|B, Nji $t|iO|NAtH `BA|,jt BAa iBAaN6 A6Oji }jAjiB|NA Ohw V kdyh dq h{srqhqwldo srzhu glvwulexwlrq zlwk phdq dqg yduldqfh 5 dqg vkdsh sdudphwhu 5 +4> 4,1 Wkh suredelolw| ghqvlw| ixqfwlrq ri V lv # 5 $ v 4. iV +v, @ D h{s E zkhuh D dqg E duh ghqhg dv iroorzv1 4@5 + 65 +4.,, 4 D@ 6@5 +4.,+ 45 +4.,,
E@
+ 65 +4.,,
4 4.
+ 45 +4.,,
Ohw J kdyh d Jdppd glvwulexwlrq zlwk phdq dqg yduldqfh 1 Wkh ghqvlw| ixqfwlrq ri J 5 ^3> 4` lv iJ +j, @ +,4 j4 h{s+j,= Txdqwlohv1 Pdq| vwdwlvwlfdo sdfndjhv kdyh idvw urxwlqhv iru fdofxodwlqj Jdppd txdq0 wlohv1 Wkhvh fdq eh xvhg wr fdofxodwh HS txdqwlohv/ dv iroorzv1 Ohw TJ + > , eh wkh 0txdqwloh ri J1 Ohw TV + > > > , eh wkh 0txdqwloh ri V1 Txdqwlohv ri V duh uhodwhg wr txdqwlohv ri J dv iroorzv1 3
4 4. 5 4@5 5 plq+ > 4 , 4 4 C D TV + > > > , @ . vljq+ 3=8, E TJ +4 6 > +4 . ,, 4 +4 . , 5 45 +4 . , 7 5 Svhxgr0udqgrp qxpehu jhqhudwlrq1 Wkh HS txdqwloh irupxod ohdgv gluhfwo| wr idvw svhxgr0udqgrp qxpehu jhqhudwlrq ehfdxvh/ li N kdv d xqlirup glvwulexwlrq rq ^3> 4`/ wkhq TV +N> > > , kdv dq HS glvwulexwlrq zlwk phdq / yduldqfh 5 dqg vkdsh sdudphwhu 1
W"AN,ja}6jA|t L dp judwhixo wr wkh HVUF iru vxssruw ri sduw ri wklv uhvhdufk xqghu judqw qxpehu U335:;69/ dqg wr wkh Ohyhukxoph Wuxvw iru vxssruw dw wkh Fhqwuh iru Plfurgdwd Phwkrgv dqg Sudfwlfh1 -jujijAWjt OBaje e A}it|e ce BAa D Y6OjAt +5334,= _Lqvwuxphqwdo yduldeohv hvwlpdwhv ri wkh hhfw ri vxevlgl}hg wudlqlqj rq wkh txdqwlohv ri wudlqhh hduqlqjv%/ Hfrqrphwulfd/ iruwkfrplqj1 OiBB+Be ce +5334,= _Wkh hhfwv ri ghprjudsklfv dqg pdwhuqdo ehkdylru rq wkh glvwulexwlrq ri eluwk rxwfrphv%/ Hpslulfdo Hfrqrplfv/ 4