Unbalanced Panel Data Models

Transcription

1 Ubalacd Pal Data odls Chaptr 9 from Baltag: Ecoomtrc Aalyss of Pal Data 5 by Adrás alascs 4448

2 troducto balacd or complt pals: a pal data st whr data/obsrvatos ar avalabl for all crosssctoal uts th tr sampl prod ubalacd or complt pals: a pal data st whr som data/obsrvatos ar mssg for som cross-sctoal uts th sampl prod Radomly mssg obsrvatos Complt pal prso yar com ag sx complt pal prso yar com ag sx

3 h ubalacd o-way rror compot modl h modl: yt t ut whr u t,..., t,..., t t,, Vctor form: y u Z u whr x u Z xk Z,, Z dag h OLS o th ubalacd data s gv by: hs s BLUE wh. ˆ OLS Z Z f > th OLS s stll ubasd ad cosstt, but ts stadard rrors ar basd. Z y

4 Complt pal cas complt pal cas Z xk K x K x K x K x x Z, xk K x K x K x K x x Z, Z

5 Wth stmator: whr J J J E E dag y y y y x x x J x

6 complt pal Complt pal x x x E dag x x x J

7 Wth rsduals u for th ubalacd pals ar gv by: u y.. y y.... h Btw stmator ad th Btw rsduals ar obtad as follows: y t ˆ Btw Z PZ P dag J Z Py uˆ b y Zˆ Btw

8 h ubalacd two-way rror compot modl h fxd ffcts modl: Wasbk ad Kapty 989 cosdr th followg modl: y u t t,..., t t t u t t t,..., Furthrmor df th matrx that gvs th dummy-varabl structur for th complt data modl: D t s t x matrx obtad D D from by omttg th rows, corrspodg to D D dvduals ot obsrvd yar t D,..., D For complt pals: dag x D dag t t x

9 f μ ad λ t ar fxd, o has to ru th rgrsso wth th matrx of dumms o th prvous sld Howvr, t s fasbl for larg pals Wth trasformato dd complt cas: P P P ad Complt cas: P whr P ad dag dag t J

10 Extsos to hghr-ordr rror compot modl.g. 3-way: Davs,, 3 C B A P P 3 A B A C B A

11 h radom ffcts modl: Vctor form of th modl: whr Varac matrx: whr Usg th gral xprsso for th vrs of +, o obtas whr Davs shows that ths rsult ca b gralzd to a arbtrary umbr of radom rror compots,.g. 3-way modl:,,, uu E / / x x x u V P V V P V / / 3 3 uu E

12 Wasbk ad Kapty suggst a AOVA-typ quadratc ubasd stmator of th varac compots basd o th Wth rsduals Lt ad df By quatg to thr xpctd valus ad solvg ths thr quatos o gts UE of y P q P q q W W q q q,,,,

13 stg for dvdual ad tm ffcts usg ubalacd pal data Baltag ad L 99 drvd a corrspodg L tst for th ubalacd two-way rror compot modl Udr ormalty of dsturbacs th L statstc s gv by L A u r r t A r t / A u / u u / whch s asymptotcally dstrbutd as udr th ull hypothss H : f H : th L A / ad t s asymptotcally dstrbutd as f H : th L A / ad t s asymptotcally dstrbutd as

14 hs varac compots caot b gatv ad thrfor th altratv hypothss ar oulto ad Radolph 989: ad ad th o-sdd L statstcs ar gv by: H : : A E L H A / ad SL / var L L d u Du / u u D L A d E d var d L A Udr th ull hypothss thy hav asymptotc, dstrbuto Hoda s 985 o-sdd for th two-way modl wth ubalacd data s smply: HO L L / Baltag, Chag ad L 998: th locally ma most powrful osdd tst for ubalacd two-way rror compot modl s gv by Kg ad Wu 997: K L L Both tsts ar asymptotcally dstrbutd as, udr H

15 hs tsts ca b stadardzd ad th rsultg SL gv by: For Hoda s vrso: D For th Kg ad Wu vrso: D Sc L ad L ca b gatv for a spcfc applcato, spcally wh o or both varac compots ar small or clos to zro GH tst Gourroux, Holly ad ofort 98: m L L L L Rcommdato: f L>, L> f L>, L<= f L<=, L> f L<=, L<= ad h us of stadardzd vrso of ths tsts m 4 4

16 hak you for your attto!