Panel Data Analysis Introduction

Transcription

1 Panl Data Analysis Introduction Modl Rprsntation N-first or T-first rprsntation Poold Modl Fixd Effcts Modl Random Effcts Modl Asymptotic Thory N, or T N, T Panl-Robust Infrnc

2 Panl Data Analysis Introduction Th Modl y x yit xit ui vt it u v y x u it it it it i t it On-Way (Individual) Effcts: Unobsrvd Htrognity Cross Sction and Tim Sris Corrlation Cov( u, u ) 0, Cov(, ) 0, i j i j it jt Cov(, ) 0, t it i it it i it

3 Panl Data Analysis Introduction N-first Rprsntation y x β u it it i it i 1,,..., N; t 1,,..., T y Xβ u i i i i T i y Xβ ( I i ) u Dummy Variabls Rprsntation N T T-first Rprsntation y x β u ti ti i ti t 1,,..., T; i 1,,..., N y Xβ u t t t y Xβ ( i I ) u T y Xβ Du D I i or D i I N N T T N

4 Panl Data Analysis Introduction Notations yi 1 x x 1 1, i1 x, i1 xk, i1 i i1 1 y x i i 1, i x, i x K, i i i, x y Xi, i, β y x it x it 1, it x, it xk, it it K y t y y y t1 t tn, X t x x t1 1, t1 x, t1 xk, t1 t1 u1 x t 1, t x, t x K, t t u x, t, u x x tn 1, tn x, tn xk, tn tn un

5 Exampl: Invstmnt Dmand Grunfld and Grilichs [1960] I F C it i it it it i = 10 firms: GM, CH, GE, WE, US, AF, DM, GY, UN, IBM; t = 0 yars: I it = Gross invstmnt F it = Markt valu C it = Valu of th stock of plant and quipmnt

6 Poold (Constant Effcts) Modl y x β u ( i 1,,..., N; t 1,,..., T ) it it i it assuming u u i y x β u, or y it it it β x 1 u y Xβ it it it E( X) 0, Var( X) I i

7 Fixd Effcts Modl y x β u ( i 1,,..., N; t 1,,..., T) it it i it u i is fixd, indpndnt of it, and may b corrlatd with x it. Cov( u, ) 0, Cov( u, x ) 0 i it i it yi Xi uiit i, i 1,,..., N yt Xt u t, t 1,,..., T

8 Fixd Effcts Modl Fixd Effcts Modl Classical Assumptions Strict Exognity: Homoschdasticity: No cross sction and tim sris corrlation: Var Extnsions: ( u, X) I NT E ( ux, ) 0 Panl Robust Varianc-Covarianc Matrix it Var( ux, ) Var( u, X) it

9 Random Effcts Modl Error Componnts y x β it it it u ( i 1,,..., N; t 1,,..., T) it i it u i is random, indpndnt of it and x it. Cov( u, ) 0, Cov( u, x ) 0, Cov(, x ) 0 i it i it it it Dfin th rror componnts as it = u i + it yi Xi ( uiit i ), i 1,,..., N yt Xt ( u t ), t 1,,..., T

10 Random Effcts Modl Random Effcts Modl Classical Assumptions Strict Exognity E( X) 0, E( u X) 0 E( X) 0 it i it X includs a constant trm, othrwis E(u i X)=u. Homoschdasticity Var Var u Cov u ( it X), ( i X) u, ( i, it ) 0 Var( X) it u Constant Auto-covarianc (within panls) Var( ε X) I i i i T u T T

11 Random Effcts Modl Random Effcts Modl Classical Assumptions (Continud) Cross Sction Indpndnc Var( ε X) I i i Extnsions: i T u T T Var( ε X) Ω I N Panl Robust Varianc-Covarianc Matrix

12 Fixd Effcts Modl Estimation Within Modl Rprsntation y x β u it it i it y x β u i i i i y y it i ( xit xi ) β ( it i ) y x β it it it y Xβ i i i or Qy QXβQ i i i 1 whr Q IT itit, ( QiT 0, Q Q Q) T

13 Fixd Effcts Modl Estimation Modl Assumptions E ( x ) 0 it it Var( x ) (1 1/ T ) it it Cov T t s Var ( it, is xit, xis ) ( 1/ ) 0, 1 T ( i Xi ) Q ( IT itit ) Var( X) Ω I N

14 Fixd Effcts Modl Estimation: OLS Within Estimator: OLS y Xβ y Xβ 1 1 N N OLS i1 i i i1 i i βˆ ( X X) X y X X X y Var ˆ ( βˆ ) ( X X) X ΩX( X X) ˆ i i i OLS ˆ N N N X 1 ix i i i 1 iq X X i X i1 ixi 1 N ˆ XX i1 i i ˆ ˆ / ( NT N K), ˆ y Xβ ˆ

15 Fixd Effcts Modl Estimation: ML Normality Assumption y x β u ( t 1,,..., T ) it it i it y Xβ u i ( i 1,,..., N) i i i T i 0 I i ~ normal iid(, T ) i y Xβ with y Qy, X QX, Q, i i i i i i i i i 1 Q IT itit T ~ normal(0, ), whr QQ Q i

16 Fixd Effcts Modl Estimation: ML Log-Liklihood Function T 1 1 ll ( β, y, X ) ln ln T T 1 1 ln ln( ) ln Q Q 1 i i i i i 1 i i Sinc Q is singular and Q =0, w maximiz T T 1 ll ( β, y, X ) ln ln( ) i i i i i

17 Fixd Effcts Modl Estimation: ML ML Estimator ˆ N ( β, ) ML arg max ll (,, ) i 1 i β yi Xi N ˆ ˆ i1 i i 1 ˆ 1 ˆ, ˆ ˆ i yi Xiβ NT T ˆ ˆ T ˆ ˆ T 1 N( T 1)

18 Fixd Effcts Modl Hypothsis Tsting Pool or Not Pool F-Tst basd on dummy variabl modl: constant or zro cofficints for D w.r.t F(N-1,NT-N-K) F-tst basd on fixd ffcts (unrstrictd) modl vs. poold (rstrictd) modl UR ˆ ˆ, ˆ ˆ UR FEFE RSSR POPO y x β u it it i it vs. ( u u, i) i y x β u it it it ( RSSR RSSUR ) / N 1 F ~ F ( N 1, NT N K ) RSS / ( NT N K) RSS

19 Random Effcts Modl Estimation: GLS Th Modl y Xβ ε, ε u i i i i i i T i E( ε X ) 0 i i Var( ε X ) I i i i i T u T T Tu Q IT Q 1 1 whr Q IT itit, IT Q itit T T

20 Random Effcts Modl Estimation: GLS GLS N 1 N 1 GLS i 1 i i i1 i i βˆ ( XΩ X) XΩ y X X X y Var( βˆ ) ( XΩ X) X X 1 1 N 1 GLS i1 i i u whr IT i TiT Q IT Q T u T u 1 1 and Q IT Q T u 1

21 Random Effcts Modl Estimation: GLS Fasibl GLS Basd on stimatd rsiduals of fixd ffcts modl ˆ ˆ ˆ / NT ( 1) 1 ˆ ˆ ˆ ˆ ˆ /, ˆ ˆ T ˆ ˆ ˆ βgls ( XΩ X) XΩ y Var( βˆ ) ( XΩˆ X) GLS T 1 T u T N whr i t 1 it 1 1 ˆ 1 1 whr Q I Q, ˆ ˆ T ˆ 1 T 1 u ˆ ˆ 1

22 Random Effcts Modl Estimation: GLS Fasibl GLS Within Modl Rprsntation 1 1 y y 1 T u it yi ( xit xi ) ( it i ) x it it it (1 ) u ( ) it it i i it i E( ) 0, Var( ) it it Cov(, ) Cov(, ) 0 it i it jt

23 Random Effcts Modl Estimation: ML Log-Liklihood Function y x β ( u ) x β ( t 1,,..., T ) it it i it it it y Xβ ε ( i 1,,..., N) i i i ε ~ normal iid( 0, ) i ll T 1 1 ( β,, y, X ) ln ln ε ε 1 i u i i i i

24 Random Effcts Modl Estimation: ML whr T u IT u itit Q ( I ) T Q u ( ) IT i TiT Q I T Q T u T u ( ) ( ) T 1 T u T u IT i TiT

25 Random Effcts Modl Estimation: ML ML Estimator ˆ ˆ N ( β,, ˆ u ) ML arg max ll (,,, ) i 1 i β u yi Xi whr T 1 1 ll ( β,, y, X ) ln ln ε ε T 1 T 1 i u i i i i u ln ln 1 T u T ( y ) ( ) t 1 it it y t1 it it x β T x β u

26 Random Effcts Modl Hypothsis Tsting Pool or Not Pool Tst for Var(u i ) = 0, that is Cov( ) Cov( u u ) Cov( ) it, is i it, i is it, is For balancd panl data, th Lagrang-multiplir tst statistic (Brusch-Pagan, 1980) is:

27 Random Effcts Modl Hypothsis Tsting Pool or Not Pool (Cont.) LM NT ˆ ( J ) ˆ T I N T 1 ˆ ˆ N T NT ˆ i1 t1 it T 1 ˆ ˆ β whr ˆ it y it xit 1 uˆ 1 ~ (1) N T i1 t1 it Poold 1

28 Random Effcts Modl Hypothsis Tsting Fixd Effcts vs. Random Effcts H Cov u random ffcts 0 : ( i, xit ) 0 ( ) H Cov u fixd ffcts 1 : ( i, xit ) 0 ( ) Estimator GLS or RE-OLS (Random Effcts) LSDV or FE-OLS (Fixd Effcts) Random Effcts E(u i X i ) = 0 Consistnt and Efficint Consistnt Infficint Fixd Effcts E(u i X i ) =/= 0 Inconsistnt Consistnt Possibly Efficint

29 Random Effcts Modl Hypothsis Tsting Fixd ffcts stimator is consistnt undr H 0 and H 1 ; Random ffcts stimator is fficint undr H 0, but it is inconsistnt undr H 1. Hausman Tst Statistic βˆ ˆ ( ˆ ) ( ˆ ) 1 ˆ ˆ RE β FE β RE βfe βre βfe H Var Var ˆ ˆ ˆ ~ (# βfe ), providd # βfe # βre ( no intrcpt)

30 Random Effcts Modl Hypothsis Tsting Altrnativ Hausman Tst Estimat th random ffcts modl y x β x γ ( u ) it it i i it F Tst that = 0 H0 γ H0 Cov u i x it : 0 : (, ) 0

31 Extnsions Random Cofficints Modl y x β it it i it βi β ui Mixd Effcts Modl y Two-Way Effcts x β ( x u ) it it it i it Nstd Random Effcts y x β ( z γ ) it it it i it y x β u v it it i t it y x β u v ijt ijt i ij ijt

32 Exampl: U. S. Productivity Munnll [1988] Productivity Data 48 Continntal U.S. Stats, 17 Yars: STATE = Stat nam, ST ABB=Stat abbrviation, YR =Yar, 1970,...,1986, PCAP =Public capital, HWY =Highway capital, WATER =Watr utility capital, UTIL =Utility capital, PC =Privat capital, GSP =Gross stat product, EMP =Employmnt,

33 Rfrncs B. H. Baltagi, Economtric Analysis of Panl Data, 4th d., John Wily, Nw York, 008. W. H. Grn, Economtric Analysis, 7th d., Chaptr 11: Modls for Panl Data, Prntic Hall, 011. C. Hsiao, Analysis of Panl Data, nd d., Cambridg Univrsity Prss, 003. J. M. Wooldridg, Economtric Analysis of Cross Sction and Panl Data, Th MIT Prss, 00.