2004 1 4 4.1 Balestra and Nerlove (1966) 1960 1980 (GMM) Arellano and Bond (1991) Arellano (2003) N T N T Smith and Fuerter (2004) 1 (the random coefficient model) 1 1995 2001 Singer and Willett (2003 9-11 2001-2) 2004)
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
2004 3 4.2.1 Lillard and Willis (1978) AR(1) Lillard and Weiss (1979) 1960-70 Baltagi and Li (1991) Wansbeek (1992) (4) the Paris-Winsten (PW) transformation Maddala(2001) Nerlove((2002 7 ) (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)
2004 4 AR(1) MA(1) 3 4.2.2 Balestra and Nerlove(1966) OLS Maddala (1971a, 1971b) Nickell (1981) (GMM) Maddala (2001) 4 Sevestre and Trognon (1996, pp.130-33) 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) 4 19 1970 1970 1970 1970 1970 (survival analysis)
2004 5 Ridder and Wansbeek (1990, pp.566-571) 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
2004 6 4 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 (2003 4 ) 6 y it 1 u it 1 1 7 Hsiao (2003, pp.85-86)
2004 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) 2 3 3 (y i,t 1 y i,t 2 ) (y it 2 y it 3 ) y it 2 8 2 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 )
2004 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)
2004 9 (y i1, y i2, y i3,...y it 2 ) 11 [y i1 ] 0... 0 W i = 0 [y i1, y i2 ]... 0 0 0... 0 (22) 0 0... [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.132 12 Arellano and Bond (1991, p.279) 25) one-step GMM two-step iid
2004 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.121-23)
2004 11 2 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 ) = 0 1-3 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 i1 14 14 y i0 y i0 t
2004 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 0 0... 0 0 y i2 0... 0 Z + i = 0 0 y i3... 0 (40)...... 0 0 0 0... y it 1 Z i (T-2) m GMM y i1 0 0... 0... 0 Z i = 0 y i1 y i2... 0... 0........... (41) 0 0 0... y i1... y it 2 GMM σµ/σ 2 u 2 = 1, T = 4 GMM GMM γ = 0 1.75 γ = 0.5 3.26 γ = 0.9 55.40 γ GMM GMM γ 1 σµ/σ 2 u 2
2004 13 4.5 4.3 4.4 (GMM) GMM (OLS) GLS) (IV) (MDE) 15 4.1 4.1 GMM GMM 16 15 1 GMM Blundell and Bond (1998) GMM empirical likelihood empirical likelihood Owen (2001) Mittelhammer, Judge and Miller (2000) 16 STATA Johnston and DiNardo (1997 11
2004 14 17 GMM Arellano and Bond (1991) N=100 T=7 1000 1 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=4,7 1000 GMM Alvarez and Arellano (2003) one-step GMM (LIML) N T 17 seed STATA seed 123456789 18 Hayashi and Sims (1983) forward filtering
2004 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=50,500 2500 1% 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
2004 16 4.6 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)
2004 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) (http://www. 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
2004 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
2004 19 4.2-4.4 4.2 OLS γ F E(0.092) < MLE(0.17) < RE(0.49) = OLS(0.49) (IV) one-step GMM 4.3-4.4 γ GMM IV GMM(0.31) < IV (0.57)
Dependent Variable: lwage Estimated Coefficient Pool OLS t-statistics Estimated Coefficient t-statistics Estimated Coefficient z-statistics Estimated Coefficient z-statistics lwage_1 0.4900 44.31 0.0924 8.04 0.4900 44.31 0.1696 14 d81 0.3583 14.91 0.7329 4.14 0.3583 14.91 0.1518 7.04 d82 0.3121 12.18 0.0148 0.88 0.3121 12.18 0.1182 4.33 d83 0.3176 11.30-0.0135-0.81 0.3176 11.30 0.1188 3.41 d84 0.3453 11.37-0.0143-0.88 0.3453 11.37 0.1458 3.41 d85 0.3547 10.87-0.0201-1.25 0.3547 10.87 0.1659 3.26 d86 0.3813 10.95-0.0115-0.72 0.3813 10.95 0.2005 3.39 d87 0.3959 10.72 (dropped) 0.3959 10.72 0.2341 3.47 exper 0.0335 3.30 0.1406 17.73 0.0335 3.30 0.0949 7.5 expersq -0.0016-2.64-0.0056-9.88-0.0016-2.64-0.0047-8.44 hours -0.0001-11.72-0.0001-13.68-0.0001-11.72-0.0001-13.44 hours_1 0.0001 10.09 0.0001 6.18 0.0001 10.09 0.0001 7.62 union 0.0893 7.03 0.0614 4.06 0.0893 7.03 0.0764 5.39 educ 0.0489 12.45 (dropped) 0.0489 12.45 0.0765 8.68 married 0.0663 5.70 0.0320 2.22 0.0663 5.70 0.0493 3.69 poorhlth -0.0480-1.16-0.0082-0.22-0.0480-1.16-0.0148-0.41 _cons -0.1671-2.72 1.0394 24.87-0.1671-2.72 0.0701 0.56 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 4316 4316 4316 545 545 545 --- 0.2764 0.1735 --- 0.0672 0.8268 0.4623 0.1484 0.4623 --- --- --- F(544, 3757) = 6.19 Prob>F = 0.0000 0.3633 0 0.2740 0.2740 0.6374 0 chi2(1) = 580.65 Prob > chi2 = 0.0000 chi2(13) = -6324.07 chibar2(01) = 936.32 Prob>chibar2 = 0.000 4316 544 --- --- --- -1149.4376 0.273 0.2753 0.4958
Dependent Variable: lwage Estimated Coefficient z-statistics hours 0.0000 0.25 hours_1 0.0006 6.66 lwage_1 0.5692 34.36 d81 0.4165 8.83 d82 0.2822 7.38 d83 0.2314 6.03 d84 0.1848 5.56 d85 0.1491 4.90 d86 0.1509 5.06 exper 0.0702 4.43 expersq -0.0033-3.48 _cons -1.1540-7.55 Diagnostic Test Number of observation Number of groups R-sq: Wald test sigma_u sigma_e rho within between overall IV 4316 545 0.0532 0.3976 0.1900 chi2(11) = 1610.58 Prob>chi2 = 0.0000 0.0000 0.4483 0.0000 hours hours_1 lwage_1 d81 d82 d83 d84 d85 d86 exper Baltagi(2001) the error component two-stage least square (EC2SLS)
Dependent Variable: lwage one-step results Estimated Coefficient Robust z-statistics lwage_1 0.3111 7.81 hours -0.0002-8.95 hours_1 0.0001 6.33 d82-0.0284-1.76 d83-0.0187-1.42 d84-0.0021-0.17 d85-0.0100-0.89 d86-0.0055-0.37 _cons 0.0382 9.59 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 3189 545 chi2(26) = 84.13 Prob>chi2 = chi2(8) = 138.34 z = -8.63 Prob>z = 0.0000 z = 2.20 Prob>z = 0.0280