Hsiao (2003, 6 ) Maddala, Li, Trost and Joutz (1997) Hsiao and Pesaran (2004) 4.2 y it = γy it 1 + x itβ + ε it i = 1, 2,..., N t = 1, 2,...T (
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1 Balestra and Nerlove (1966) (GMM) Arellano and Bond (1991) Arellano (2003) N T N T Smith and Fuerter (2004) 1 (the random coefficient model) Singer and Willett ( ) 2004)
2 Hsiao (2003, 6 ) Maddala, Li, Trost and Joutz (1997) Hsiao and Pesaran (2004) 4.2 y it = γy it 1 + x itβ + ε it i = 1, 2,..., N t = 1, 2,...T (1) γ x it K β K ε it ε it = µ i + u it (2) µ i iidn(0, σµ) 2, u it iidn(0, σu) 2 ε it N T 2 u it Maddala (2001) y it = x itβ + µ i + w it (3) w it = ρw it 1 + u it ρ < 1 (4) y it = γy it 1 + x itβ + µ i + u it (5) 2
3 Lillard and Willis (1978) AR(1) Lillard and Weiss (1979) Baltagi and Li (1991) Wansbeek (1992) (4) the Paris-Winsten (PW) transformation Maddala(2001) Nerlove(( ) (3) LSDV ŵ it 4) OLS ˆρ 3) y it = x itβ + µ i + u it (6) y it = y it ˆρy it 1 x it = x it ˆρx it 1 µ i = µ i (1 ˆρ) LSDV β µ i Bhargava, Franizini and Narendranathan (1982) Durbin-Watson Statistic w it = ρw it 1 + u it H 0 : ρ = 0 H 1 : ρ < 1 N T i=1 t=2 d = (ŵ wit ŵ wit 1 ) 2 N T i=1 t=1ŵ2 wit (7) û wit AR(1) AR(n) Baltagi and Li (1995) AR(1) AR(1) AR(1)
4 AR(1) MA(1) Balestra and Nerlove(1966) OLS Maddala (1971a, 1971b) Nickell (1981) (GMM) Maddala (2001) 4 Sevestre and Trognon (1996, pp ) Maddala (1971a) λ λ class β = 0 γ(λ) p lim ˆγ(0) < γ < p lim ˆγ(λ) < p lim ˆγ(1) < p within GLS poolingols lim between ˆγ( ) (8) GLS 3 Baltagi and Li (1995) Baltagi (2001,pp.90-95) (survival analysis)
5 Ridder and Wansbeek (1990, pp ) Trognon(1978) AR(1) x it = δx it 1 + w it w it N(0, σ 2 w) y i0 T γ Anderson and Hsiao (1981,1982) 4.3 Anderson and Hsiao (1981,1982) w it = γw it 1 + ρ z i + β x it + µ i + u it i = 1, 2,..N. t = 1, 2,...T (9) y it = w it + η i (10) µ i = (1 γ)η i E(η i ) = 0 V ar(η i ) = σ 2 η = σ 2 µ/(1 γ) 2 (11) z i 1 y i0 (µ i +ρ z i )/(1 γ) + β j=0 x it jγ j µ i y i0 y i0 µ i y i0 y i1 y i1... µ i 2 y i0 µ i u it y i0 = ȳ 0 + ε i ȳ 0 0 ε i iid ( 2a y i0 µ i ( 2b)y i0 µ i cov(y i0, µ i ) = ϕσy 2 0 y it [ϕε i /(1 γ)] = lim t E[y it ρ z i /(1 γ) β t 1 j=0 x it jγ j ε i ] 3 w i0 y it = w it + η i µ i = (1 γ)η i y it µ i y i0 η i + ρ z i /(1 γ) + β t 1 j=0 x it jγ j
6 w i0 3 w it 4 ( 4a)w i0 θ w σu/(1 2 γ 2 ) ( 4b)w i0 θ w σw 2 0 ( 4c)w i0 θ i0 σu/(1 2 γ 2 ) ( 4d)w i0 θ i0 σw0 2 Anderson and Hsiao (1981,1982) 8 5 L(γ, ρ, β, γ, η, σ 2 u, σ 2 w, σ 2 µ) = (2π) NT/2 v N/2 exp{ 1 2σ 2 (y it γy it 1 ρ z i β x it ) 2 } i t (12) v N T σ 2 µ = 0 Anderson and Hsiao (1981) (9) 10 z i µ i y it y it 1 = (x it x it 1 ) β + γ(y it 1 y it 2 ) + (u it u it 1 ) (13) u it 6 β γ 7 (y it 2 y it 3 ) 5 Anderson and Hsiao (1982) Hisao ( ) 6 y it 1 u it Hsiao (2003, pp.85-86)
7 ( γ iv β iv ) [ ( N T (y i,t 1 y i,t 2 )(y it 2 y it 3 ) (y it 2 y it 3 )(x it x it 1 ) ) = (x it x it 1 )(y it 2 y it 3 ) (x it x it 1 )(x it x it 1 ) i=1t=3 [ ( N T i=1t=3 (14) ) ] y it 2 y it 3 (y it y it 1 ) x it x it 1 )] 1 y it 2 ( γ iv β iv ) [ ( N T y it 2 (y i,t 1 y i,t 2 ) y it 2 (x it x it 1 ) ) = (x it x it 1 )y it 2 (x it x it 1 )(x it x it 1 ) i=1t=2 [ ( ) ] N T y it 2 (y it y it 1 ) i=1t=2 x it x it 1 )] 1 (15) (y i,t 1 y i,t 2 ) (y it 2 y it 3 ) y it ˆβ ˆγ 9 ρ ȳ it ˆγȳ it 1 ˆβ x it = ρ z i + µ i + ū it i = 1,..N (16) ȳ i = T t=1 y it/t, x i = T t=1 x it/t, ū i = T t=1 u it/t 3 σ 2 u σ2 µ N T σu 2 i=1 t=2 = it y it 1 ) ˆγ(y it 1 y it 2 ) ˆβ (x it x it 1 )] 2 2N(T 1) (17) N σµ 2 i=1 = i ˆγȳ i, 1 ˆρ z i ˆβ x i ) 1 N T ˆσ2 u (18) N T γ β σ 2 u ρ σµ 2 N N T Anderson and Hsiao (1982) N T T N 8 8 Arellano (1989) y it 2 y it 3 (y it 2 y it 3 )
8 γ 3 T N ρ N T 3 T N ρ N T γ ρ γ N T ( 3) T N T N Hsiao, Pesaran and Tahmiscioglu (2002) Fujiki, Hsiao and Shen (2002) Chamberlain (1982,1984) (Minimum Distance Estimation: MDE) 9 2 (β, γ) min[ N u i=1 i Ω 1 u i ] (19) Ω u i u i = [ y i1 β x i1 γ y i0, y i2 β x i2 γ y i1,...] N 4.4 Arellano and Bond (1991) Ahn and Schmidt (1995) 2 y 10 E[y is, (u it u i,t 1 )] = 0, s = 0, 1,...t 2, t = 2,...T (20) Arellano and Bond (1991) GMM) 1 n i=1 n y is[(y it y i,t 1 ) (y i,t 1 y i,t 2 ) γ (x it x i,t 1 ) β] = 0 (21) s = 0,..., t 2, t = 2,..., T 9 MDE Chamberlain (1982,1984) Lee(2002, 3 ) 10 Holtz-Eakin(1988) Holtz-Eakin, Newey and Rosen(1988)
9 (y i1, y i2, y i3,...y it 2 ) 11 [y i1 ] W i = 0 [y i1, y i2 ] (22) [y i1,...y it 2 ] 20) E(W i u i ) = 0 (23) 1 (y it y it 1 ) = (y it 1 y it 2 ) γ + (x it x it 1 ) β + (u it u it 1 ) (24) y it = y it 1γ+ x itβ+ u it i = 1, 2...N W i Arellano and Bond (1991) (GMM) (24) W y it = W y it 1γ + W x itβ + W u it (25) γ β ˆγ GMM = [( y it 1 ) W ˆβ GMM = [( x it 1 ) W 1 ˆV N W ( y it 1 )] 1 [( y it 1 ) 1 W ˆV N W ( y it )] (26) 1 ˆV N W ( x it 1 )] 1 [( x it 1 ) 1 W ˆV N W ( y it )] (27) V N = N i=1 W i ( u i)( u i ) W i 12 x it E(x it u is ) = 0, t, s = 1, 2,..., T x it µ i 24) x it x it (predetermined) E(x it u is ) 0 for s < t E(x it u is ) = 0 for s t (x i1, x i2,..., x is 1 ) W i 11 Baltagi (2001, p Arellano and Bond (1991, p.279) 25) one-step GMM two-step iid
10 Arellano and Bond (1991) GMM GMM Arellano and Bond (1991) j 13 1 r j = T 3 j T t=4+j r tj (28) r tj = E( u it u it j ) H 0 : r j = 0 m j = ˆr j SE(ˆr j ) (29) ˆr j û it ˆr tj = N 1 N i=1 û it û it j Arellano and Bond (1991) Sargan (1958) s = û W [ N i=1 W i ( û i )( û i ) W i ] 1 W ( û) χ 2 p k 1 (30) p W û 25) Arellano and Bond (1991) Ahn and Schmidt(1995) y y (u it u it 1 ) () E(y is u it ) = 0 t = 2,...T, s = 0, 1,...t 2 (31) E(u it u it ) = 0 t = 2,...T 1 (32) T (T 1)/2 + (T 2) (32) γ 1 σµ/σ 2 u 2 Ahn and Schmidt(1995) (31)(32) 1 i t cov(u it, y i0 ) cov(u it, y i0 ) = 0 13 Arellano (2003, pp )
11 i t cov(u it, µ i ) cov(u it, µ i ) = 0 3 i t s cov(u it, u is ) cov(u it, u is ) = GMM Chamberlain (1982,1984) (Minimum Distance Estomator) Blundell and Bond (1998) GMM GMM Anderson and Hsiao (1981,1982) GMM GMM Arellano and Bond GMM γ 1 µ i Blundell and Bond (1998) T=3 E(y i1 u i3 ) = 0 γ GMM y i2 = πy i1 + µ i + u i2 i = 1, 2,...N (33) γ 1 µ i π 0 y i1 y i2 E(y i1 µ i ) > 0 σ 2 µ = var(µ i ) σ 2 u = var(u it ) π k p lim ˆπ = (γ 1) (σµ/σ 2 u) 2 + k k = (1 γ) (1 + γ) (34) Blundell and Bond (1998) GMM Nelson amd Startz (1990) Staiger and Stock (1997) Ahn and Schumidt (1995) T-3 E(u it y it 1 ) = 0 t = 4, 5,...T (35) y i2 E(u i3 y i2 ) = 0 (36) y i y i0 y i0 t
12 y i1 = µ i 1 γ + u i1 (37) t = 2 y it (36) E[(µ i + u i3 )(u i2 + (γ 1)u i1 )] = 0 (38) E(u i1 µ i ) = E(u i1 u i3 ) = 0 i = 1, 2,...N (39) y i0 u i1 µ i /(1 γ) Blundell and Bond (1998) γ 1 σ µ/σ 2 u 2 (35)(36) GMM GMM Z + i GMM Z i y i Z + i = 0 0 y i (40) y it 1 Z i (T-2) m GMM y i Z i = 0 y i1 y i (41) y i1... y it 2 GMM σµ/σ 2 u 2 = 1, T = 4 GMM GMM γ = γ = γ = γ GMM GMM γ 1 σµ/σ 2 u 2
13 (GMM) GMM (OLS) GLS) (IV) (MDE) GMM GMM GMM Blundell and Bond (1998) GMM empirical likelihood empirical likelihood Owen (2001) Mittelhammer, Judge and Miller (2000) 16 STATA Johnston and DiNardo (
14 GMM Arellano and Bond (1991) N=100 T= GMM 2 GMM (IV) 3 two-step GMM Ziliak (1997) Ziliak (1997) N=532 T=8 iid GMM Keane and Runkle (1992) forward filter 2SLS(FF) 18 Ahn and Schmidt (1999) Crepon, Kramarz and Trognon (1997) Alonso-Borrego and Arellano (1999) N=100 T=4, GMM Alvarez and Arellano (2003) one-step GMM (LIML) N T 17 seed STATA seed Hayashi and Sims (1983) forward filtering
15 T/N 1/T 1/N 1/(2N-T) T GMM LIML T LIML GMM Blundell and Bond (1998) GMM N=100,200,500 T=4 GMM Binder, Hsiao, and Pesaran (2000) Hsiao, Pesaran and Tahmiscioglu (2002) Hsiao (2003) GMM GMM T=5 N=50, % GMM 15-20% GMM MDE GMM MDE GMM Hahn, Hausman and Kuersteiner (2002) 3 y n y n 3 (long differences;ld) Wansbeek and Bekker (1996) GMM Ahn and Schmidt (1995) Alvarez and Arellano (2003) T N T/N N/T N T
16 Frankel and Rose (1996) Pedroni(2001) Sala-i-Martin (1996) Nerlove (2000) Quah (1996) 2 Nagahata, Saita, Sekine and Tachibana (2004) spurious Levin-Lin (LL) test(1992,1993) Im-Pesaran-Shin (IPS) test(2003) Maddala-Wu (MW) test (1999) y it = γy it 1 + u it i = 1, 2,...N (42) t H 0 : γ 1 = 1 vs H 1 : γ 1 < 1 (43) Levin-Lin (LL) test H 0 : γ 1 = γ 2 =... = γ N = γ = 1 vs H 1 : γ 1 = γ 2 =... = γ N = γ < 1 (44)
17 O Connell(1998) Levin-Lin test Im-Pesaran-Shin (IPS) test Levin-Lin test H 0 : γ i = 1 for all i vs H 1 : γ i < 1 at least one i Maddala (2001, p.554) N Levin-Lin test Augmented Dickey-Fuller test N t M σ 2 t t M σ 2 /N Maddala-Wu test N Ronald A. Fisher (1973a) P i i p λ = 2 N i=1 log e P i 2N χ N P λ test Maddala and Wu (1999) Fisher Choi(1999) Fisher Fisher 4.7 STATA Wooldridge (2003) ( msu.edu/ ec/wooldridge/book2.htm) WAGEPAN.DTA Vella and Verbeek (1998) Wooldridge (2003) ln wage it = α + γ ln wage it 1 + β exp er it + δ exp er 2 it + ζhours it + ηhours it 1 + θunion i + κeduc i + λmarried i + νpoorhlth i + µ i + ν t + u it
18 ln wage = exp er = hours = union = 1 educ = married = 1 poorhlth = 1 ν t = STATA /**Dynamic Panel **/ /*Pooled OLS*/ reg lwage lwage 1 d81 d82 d83 d84 d85 d86 d87 exper expersq hours hours 1 union educ married poorhlth /*LSDV*/ xtreg lwage lwage 1 d81 d82 d83 d84 d85 d86 d87 exper expersq hours hours 1 union educ married poorhlth, fe est store fixed xtreg lwage lwage 1 d81 d82 d83 d84 d85 d86 d87 exper expersq hours hours 1 union educ married poorhlth, re xttest0 est store random hausman fixed random /*Anderson-Hsiao IV Estimation*/ xtivreg lwage lwage 1 d81 d82 d83 d84 d85 d86 exper expersq ( hours hours 1 = union educ married poorhlth ), re ec2sls /*Anderson-Hsiao Maximum Likelihood Estimation*/ xtreg lwage lwage 1 d81 d82 d83 d84 d85 d86 d87 exper expersq hours hours 1 union educ married poorhlth, mle /*Arellano-Bond GMM Estimation*/ xtabond lwage hours hours 1 d81 d82 d83 d84 d85 d86 d87, lags(1) inst(exper expersq union educ married poorhlth) artests(2) xtabond lwage hours hours 1 d81 d82 d83 d84 d85 d86 d87, lags(1) inst(exper expersq union educ married poorhlth) artests(2) robust
19 OLS γ F E(0.092) < MLE(0.17) < RE(0.49) = OLS(0.49) (IV) one-step GMM γ GMM IV GMM(0.31) < IV (0.57)
20
21 Dependent Variable: lwage Estimated Coefficient Pool OLS t-statistics Estimated Coefficient t-statistics Estimated Coefficient z-statistics Estimated Coefficient z-statistics lwage_ d d d d d d d (dropped) exper expersq hours hours_ union educ (dropped) married poorhlth _cons Fixed Random MLE Diagnostic Test Number of observation Number of groups (ari) R-sq: within between overall Log Likelihood F test that all u_i=0: sigma_u sigma_e rho Breusch and Pagan Lagrangian multiplier test for random effects: Hausman specification test Likelihood-ratio test of sigma_u = 0 for MLE F(544, 3757) = 6.19 Prob>F = chi2(1) = Prob > chi2 = chi2(13) = chibar2(01) = Prob>chibar2 =
22 Dependent Variable: lwage Estimated Coefficient z-statistics hours hours_ lwage_ d d d d d d exper expersq _cons Diagnostic Test Number of observation Number of groups R-sq: Wald test sigma_u sigma_e rho within between overall IV chi2(11) = Prob>chi2 = hours hours_1 lwage_1 d81 d82 d83 d84 d85 d86 exper Baltagi(2001) the error component two-stage least square (EC2SLS)
23 Dependent Variable: lwage one-step results Estimated Coefficient Robust z-statistics lwage_ hours hours_ d d d d d _cons Diagnostic Test Number of observation Number of groups Sargan test Wald test Arellano-Bond test for residual AR(1) Arellano-Bond test for residual AR(2) GMM chi2(26) = Prob>chi2 = chi2(8) = z = Prob>z = z = 2.20 Prob>z =
2004 2 µ i ν it IN(0, σ 2 ) 1 i ȳ i = β x i + µ i + ν i (2) 12 y it ȳ i = β(x it x i ) + (ν it ν i ) (3) 3 β 1 µ i µ i = ȳ i β x i (4) (least square d
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9 AR (i) (ii) MA (iii) (iv) (v) 9.1 2 1 AR 1 9.1.1 S S y j = (α i + β i j) D ij + η j, η j = ρ S η j S + ε j (j =1,,T) (1) i=1 {ε j } i.i.d(,σ 2 ) η j (j ) D ij j i S 1 S =1 D ij =1 S>1 S =4 (1) y j =
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c 2000 1 6.1 y it = δy it 1 + x 0 itβ + u it i =1, 2,..., N t =1, 2,...T (1) δ x 0 it K β K u it u it = µ i + ν it (2) µ i IID(0, σµ) 2, ν it IID(0, σν) 2 u it N T 1 v it Anderson and Hsiao (1981) Arellano(1989)
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