2004 1 1 1.1 Maddala(1993) Mátyás and Sevestre (1996) Hsiao(2003) Baltagi(2001) Lee(2002) Woolridge(2002a), Arellano(2003) Journal of Econometrics Econometrica Greene(2000) Maddala(2001) Johnston and Di- Nardo(1997) Woolridge(2000a) Handbook of Econometrics (North-Holland) Chamberlain (1984) Arellano and Honoré (2001) 1.2 1.2.1 Gauss, Carl Friedrich (Airy, George Biddell) (Poincaré, Henri)
2004 2 (p.2 ( ) 1 2 ( ) Fisher, Ronald A.1932, 1971, 1973a, 1973b) treatment group controll group 3 3 1 2 3 1 4 1 (error function) 2 (Legendre, Adrian Marie: 1752-1833) 1805) 1806 1809 1995 3 1992 1 4 2004 R.A. 1
2004 3 (erro components model) heterogeneous 1.1 1 y it = α + X itβ + u it i = 1,...N; t = 1,...T (1) i two-way error component 5 u it = µ i + λ t + ν it (2) µ i λ t ν it λ t = 0 (2) (one-way error component) u it = µ i + ν it (3) 5 n
2004 4 (1) OLS (pooling estimation) one-way fixed effect estimation (Least Squares Dummy Variable Model ; LSDV) (one-way random effect estimation) y it θy i X it θx i (Generalized Least Squares; GLS) θ 1.2.2 Baltagi(2001) Hsiao(2002)
2004 5 (self selectivity) (non-response) (attrition) Panel Study of Income Dynamics (PSID) 1968 4802 31000 5000 National Longitudinal Surveys of Labor Market Experience (NLS) 5 ( )1966 45-49 (5020 ) ( )1966 14-24 5225 ( )1967 30-44 5083 ( )1968 14-21 (5159 ) ( )1979 14-24 NLSY79 1986 1996 12-16 1997 NLSY97 12686 1993 (Statistics Canada) 15000 31000 The Canadian Survey of Labor Income Dynamics (SLID) 1984 5921 The German Social Economic Panel 1985 6471 The Belgian Socioeconomic Panel 1985 715 2092 The French Household Panel
2004 6 2059 The Hungarian Household Panel (1922-96) 1991 5000 The British Household Panel Survey (BHPS) The Dutch Socio-Economic Panel (ISEP) 1984-97 1992 The Russian Longitudinal Monitoring Survey (RLMS) 1999 5074 7799 The Swiss Household Panel (SHP) 1985 2012 6110 The Luxembourg Panel Socio-Economique Liewen zu L zebuerg (PSELL) 1994 2978 8232 EuroStat 1994 The European Community Household Panel (ECHP) ECHP 1.3 Bayes, Thomas 6 1 2 1950 (Savage, Leonard J) 6 Bayes theorem A H 1, H 2,... H k A H i P (H i A) P (A H i ) P (H i A) H 1, H 2,... H k H i H 2... H k = Ω P (H i A) = P (H i ) P (A H i ) sum{p (H j ) P (A H j )} P (H i ) H i prior probability P (H i A) posterior probability
2004 7 (Zellner, Arnold) 7 1.3.1 (Ordinary Least Squares; OLS) Y 1, Y 2... Y n µ Y var(y ) = σ 2 Y Y i = µ Y + ε i E(ε i ) = 0, var(ε i ) = σ 2 Y E(Y i ) = µ Y e i = Y i µ Y i = 1,..., n µ Y min n e 2 i = min n (y i µ Y ) 2 i=1 (sum of squared errors, SSE) µ Y SSE µ Y = d n (y i µ Y ) 2 i=1 i=1 = 2 n (Y i ˆµ Y ) = 0 dµ Y i=1 n Y i = nˆµ Y ˆµ Y = i=1 n Y i i=1 ˆµ Y (Best Linear Unbiased Estimator; BLUE) Gauss-Markov s theorem n 7 Koop(2003) Lancaster(2004)
2004 8 1.3.2 89 y = β 0 + β 1 x + u (4) x u Cov(x, u) 0 (5) β 0 β 1 x x u z Cov(z, u) = 0 (6) Cov(z, x) 0 (7) x Cov(z, u) = 0 Cov(z, x) 0 1 = 0 x = π 0 + π 1 z + v (8) π 1 = Cov(z, x)/v ar(z) (7) π 1 0 ( ) ˆβ 1 = n (z i z)(y i ȳ) i=1 n (z i z) (x i x) i=1 (9) 8 Bowden and Turkington (1984) Wooldridge (2003) 15 Wooldridge(2003) 9 Philip G. Wright(1928) B Stock and Trebbi (2003) Philip Wright Sewall Wright Philip Wright
2004 9 ˆβ 0 = ȳ ˆβ 1 x (10) z = x (6) (7) p lim( ˆβ 1 ) = β 1 (6) (7) x u 10 z x z u p lim ˆβ 1 = β 1 + Corr(z, u) Corr(z, x) σu σ x (11) σ u σ x u x Corr(z, u) Corr(z, x) ˆβ 1 11 y 2 y 1 = β 0 + β 1 y 2 + β 2 z 1 + β 3 z 2 + u 1 (12) z 1 z 2 z 3 z 4 Hausman (1978) y 2 12 y 2 y 2 = α 0 + α 1 z 1 + α 2 z 2 + α 3 z 3 + α 4 z 4 + v 2 (13) z j u 1 v 2 u 1 y 2 u 1 1 = 0 y 2 u 1 10 6 7 4 9 10 11 Weak Instrumental Variables Staiger and Stock (1997) Nelson and Sartz (1990) Gary Chamberlain Jerry Hausman Christopher Sims 12 Durbin-Wu-Hausman test Wu-Hausman test Bowden and Turkington (1984, pp.50-52) Davidson and MacKinnon (2004, pp.338-340)
2004 10 u 1 = δ 1 v 2 + e 1 (14) 13 ˆv 12 y 1 = β 0 + β 1 y 2 + β 2 z 1 + β 3 z 2 + δ 1ˆv 2 + ε (15) t 1 = 0 y 2 (l) k l k (4) 6 (1) (4) û (2) û (l) R 2 (3) û nr 2 χ 2 (l k) (16) 1.3.3 (Method of Moments) (Pearson, Karl) f(x θ) θ
2004 11 E(x) = xf(x θ)dx = g(θ) (17) E(x) θ θ θ = g 1 (E(x)) (18) E(x) x ˆθ = g 1 ( x) (19) ˆθ θ 13 (Generalized Method of Moments; GMM) 14 (overidentified) 15 p q 2 Q(θ) arg min θ Q(θ) (20) Q(θ) = f(θ) Af(θ) A (weighting matrix) Q(θ) 0 f(θ) = 0 Q(θ) = 0 E(f(θ)) = 0 q f(θ) 13 14 Hansen(1982) Hansen and Singleton(1982) 15 (Hansen(1982))
2004 12 1.3.4 (Edgeworth, Francis Ysidro) 16 5 y = (y 1, y n ) θ = (θ 1, θ 2, θ p ) likelihood L(θ) 17 L(θ) θ θ maximum likelihood estimator θ L(θ) logl(θ) θ log(lθ) θ = 0 (21) y i = βx i + ε i i = 1,, n (22) ε i N(0, σ 2 ) 16 Pratt (1976) 1997 pp.108-116 Silvey (1970) Cox and Hinkley (1974) (1995) 17 1992
2004 13 { log L(β) = log [(2πσ 2 ) n2 exp (y }] βx) (y βx) 2σ 2 = n 2 log(2πσ2 ) 1 2σ 2 (y βx) (y βx) β (23) β = ˆβ = x y/x x (24) log L(β) 2 log(lβ) β 2 = x x/σ 2 (25) β 2 log L(β) β I(θ) y y f θ (y) { 2 } log L(θ) I(θ) = E θ 2 { 2 } log f θ (y) = E θ 2 (26) I(θ) V {t(y)} 1 I(θ) (27) θ Z H 0 : θ = θ 0 n( θ θ 0 ) N(0, 1/I 1 (θ 0 )) θ 1 > θ 0 ni1 (θ 0 )( θ θ 0 ) > Z α (28) I 1 (θ) = I(θ)/n Z α α Z
2004 14 H 0 : θ = θ 0 H 1 : θ θ 0 H 0 θ 0 χ 2 (1) 2 log L( θ) L(θ 0 ) > χ2 α(1)(= Z 2 α/2 ) (29) χ 2 α(1) = Z 2 α/2 (Wald Tests) r r(θ) = 0 r(θ) V ar(r(ˆθ)) R(θ 0 )V ar(ˆθ)r (θ 0 ) (30) R(θ) r i (θ)/ θ i r k V ar(ˆθ) W = r (ˆθ)(R(ˆθ) V ar(ˆθ)r (ˆθ)) 1 r(ˆθ) (31) H 0 : θ = θ 0 χ 2 (r) (Lagrange Multiplier Tests) l(θ) r(θ) = 0 θ l(θ) r (θ)λ (32) g( θ) R ( θ) λ = 0 (33) r( θ) = 0 λ LM = λ R( θ)ĩ 1 R ( θ) λ (34) H 0 : θ = θ 0 χ 2 (r) 18 18 Davidson and MacKinnon (2004, 10
2004 15 1.4 19 1.4.1 3 µ 1, µ 2,... µ a (a 3) analysis of variance: ANOVA (factor) (level (treatment 20 A A 1,..., A a r 1,..., r a A i j y ij y ij = µ i + ε ij i = 1, 2,..., a; j = 1,..., r i (35) µ i i ε ij N(0, σ 2 ) n = Σr i r i µ i µ = r i µ i n grand mean 19 3 1992 1969 20 completely randomized design (randomized block design) 1969
2004 16 (effect) α i = µ i µ r i α i = 0 (30) y ij = µ + α i + ε ij i = 1, 2,... ; j = 1,..., r i (36) ( µ)+ i α i + ε ij ) one-way layout A H 0 : µ 1 = µ 2 =... µ a H 0 : α 1 = α 2 =... α a = 0 S e = (y ij ȳ i ) 2 i j = yij 2 yi 2 r i (37) i j i S e σ 2 ν e = n a H 0 : µ 1 = µ 2 =... µ a y ij = µ + ε ij µ ȳ S T = i H 0 (y ij ȳ i ) 2 = yij 2 ȳi 2 n (38) j i j S A = S T S e = yi 2 r i y 2 n i = r i (ȳ i ȳ) 2 (39) i S A S e ν A = a 1 F = S A ν A S e ν e (40) ν A ν e F (ν A, ν e ) (ANOVA Test) (35) (S A ) (S A ν A ) (S e ) (S e ν e )
2004 17 1.4.2 21 V i = (y ij ȳ i ) 2 (r i 1) i = 1,..., a i V e = (r i 1)V i (n a) = c i (y ij ȳ i ) 2 (n a) j B = (n a) log V e (r i 1) log V i (41) i H 0 : σ1 2 = σ2 2 =... = σa 2 a 1 B = 1 + 1 3(a 1) B { i 1 r 1 i 1 n a } (42) F 2 ( main effect () (one factor at a time experiment) ( (factorial experiment) ) 22 1.4.3 A B 21 1992,pp.122-1228) 22
2004 18 A i B j k y ijk AB a, b r y ijk = µ ij + e ijk i = 1, 2,... a; j = 1,..., b; k = 1, 2,..., r (43) µ ij A i B j e ijk N(0, σ 2 ) n = abr A i A i B j B j µ i = µ ij b, µ j = µ ij a, µ j i i µ ij ab (44) µ A i B j µ main effect α i = µ i µ, i = 1,... a; β j = µ j µ, j = 1,... b (45) µ ij A, B (interaction) (αβ) ij = µ ij (µ + α i + β j ) = µ ij µ i µ j + µ i = 1, 2,... a; j = 1, 2..., b (46) (38) (41) y ijk = µ + α i + β j + (αβ) ij + ε ijk (47) ()+( A )+( B )+( AB )+( ) Σα i = 0, Σβ j = 0, (αβ) ij = 0, j = 1,... b; (αβ) ij = 0, i = i j 1,... a S A, S B, S AB S e j v T = n 1, v A = a 1, v B = b 1, v AB = (a 1)(b 1), v e = ab(r 1) (48) V A = S A v A, V B = S B v B, V AB = S AB v AB, V e = S e v e (49) F H 0 : (αβ) ij 0 F AB = V AB V e H 0 : α i 0 F A = V A V e H 0 : β j 0 F B = V B V e
2004 19 y ij = µ + α i + β j + (αβ) ij + ε ij (50) S e
2004 20 1.1