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) Hahn,Hausman and Kuerteiner (2002) Arellano and Bond (1991) Ahn and Schmidt (1995) GMM 2 (19) µ i y it y it 1 =(x it x it 1 ) 0 β + δ(y it 1 y it 2 )+(ν it ν it 1 ) (3) v it 3 1 4.3 2 Arellano and Bover (1995), Blundell and Bond (1998) 3 y it 1 ν it 1 19 1
c 2000 2 (y it 2 y it 3 ) y it 2 y it 3 (y it 1 y it 2 ) 4 Arellano and Bond (1991) Ahn and Schmidt (1995) 2 y 5 E[y is, (ν it ν it 1 )] = 0, s =0, 1,...t 2, t =2,...T (4) GMM) 1 P n i=1 n y is[(y it y it 1 ) (x it x it 1 ) 0 β δ(y it 1 y it 2 )] = 0 (5) s =0,..., t 2, t =2,..., T Ahn and Schmidt(1995) y y (ν it ν it 1 ) ( ) E[y is (ν is+1 ν is ) y is+1 (ν is+2 ν is+1 )] = 0 (6) E[(y it x it 0 β)y it (y it 1 x it 1 0 β)y it 1 ]=0 (7) t =2,...T Binder, Hsiao, and Pesaran (2000) Hsiao, Pesaran and Tahmiscioglu (2002) Hsiao (2002) GMM GMM T=5 N=50 2500 1 GMM 15 20 GMM Hsiao, Pesaran and Tahmiscioglu (2002) Fujiki, Hsiao and Shen (2002) (Minimum Distance Estimation: MDE) 6 4 Arellano(1989) y it 2 y it 3 5 (orthogonality conditions) Holtz-Eakin(1988) Holtz-Eakin, Newey and Rosen(1988) 6 MDE Lee(2002,Chap3) 2
c 2000 3 2 (β, δ) P min[ N 4νi Ω 1 4νi ] (8) i=1 Ω 4νi 4νi = [4y i1 β4x i1 δ4y i0, 4y i2 β4x i2 δ4y i1,...] N MDE GMM MDE GMM Hahn, Hausman and Kuersteiner (2002) y n y n 3 (long differences;ld) (MDE) Arellano and Bond (1991) Kiviet (1995) Ziliak 1997 Blundell and Bond (1998) Alonso-Borrego and Arellano (1999) GMM GMM N Blundell, Bond and Windmeijer(2000) GMM GMM 6.2 spurious Levin-Lin (LL) test(1992,1993) Im-Pesaran-Shin (IPS) test(1997) Maddala-Wu (MW) test (1999) 3
c 2000 4 y it = ρy it 1 + e it i =1, 2,...N (9) t H 0 : ρ 1 =1 vs H 1 : ρ 1 < 1 (10) Levin-Lin (LL) test H 0 : ρ 1 = ρ 2 =... = ρ N = ρ =1 vs H 1 : ρ 1 = ρ 2 =... = ρ < 1 (11) O Connell(1998) Levin-Lin test Im-Pesaran-Shin (IPS) test Levin-Lin test H 0 : ρ i =1for all i vs H 1 : ρ 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 P N i=1 log e P i 2N χ N P λ test Maddala and Wu (1999) Fisher Choi(1999a) Fisher Fisher 4
c 2000 5 6.3 7 y it i t p it E(y it )=1 p it +0 (1 p it )=p it (x it ) p it =Pr[y it =1]=E(y it x it )=F (x 0 itβ) (12) F (x 0 itβ) F (x 0 itβ) =Φ(x 0 itβ) = xz 0 it β 0 ex it β 1 2π e u2 /2 du (13) F (x 0 itβ) = (14) 1+e x0 it β y it = x0 it β + u it. y it =1 if yit > 0 (15) y it =0 if yit 6 0 Pr[y it =1]=Pr[y it > 0]=Pr[u it > x 0 itβ] =F (x 0 itβ) (16) 7 Maddala(1983 1987) Gourieroux (2000) Lee (2002) Maltinominal logit, ordered probit, sequential Tobit Count data 5
c 2000 6 6.3.1 µ i yit = x 0 itβ + µ i + ν it (17) Pr[y it =1]=Pr[yit > 0] = Pr[ν it > x 0 itβ µ i ]=F(x 0 itβ + µ i ) (18) T µ i N µ i T 8 µ i β 9 Chamberlain (1980) µ i P T t=1 y it β L c = N Q i=1 Pr(y i1,...y it P T t=1 y it) (19) β µ i Chamberlin Hausman χ 2 Chamberlin 10 Liang and Zeger (1986) Generalized Estimating Equations (GEE) Population-averaged Model 8 Neyman and Scott (1948) (the classical incidental parameter problem) Lancaster (2000) 9 Hsiao(2002) β µ i β 10 N T Heckman (1981b) 6
c 2000 7 6.3.2 11 yit = x 0 itβ + µ i + ν it ν it IIN(0, σν) 2 (20) y it = y it if y it > 0 (21) y it =0 otherwise d it =1 if yit > 0, d it =0 otherwise LogL = P i,t (1 d it )LogΦ( x itβ µ i )+ P d it { 1 σ i,t 2 log σ2 1 2σ (y 2 it x 0 it β µ i) 2 } (22) β σ µ i µ i T β σ Heckman and MaCurdy (1980) (iterative methods) 12 β σ µ i β σ β σ β σ Honoré(1992) 62 b = β u ist (b) =max{y is, (x is x it )b} max{0, (x is x it )b} (23) u ist (β) =max{y is, (x is x it )β} max{0, (x is x it )β} (24) =max{µ i + ν it, x is β, x it β} max{ x is β, x it β} 11 Hausman and Wise (1979) 12 µ i 1980, P.59) 7
c 2000 8 u ist (β) s t ν it i.i.d. u ist (β) u its (β) i.i.d. E[(ξ(ψ(u its (β) ψ(u ist (β)))) x it,µ i ]=0 (25) β N β 6.4 attrition (weighted least square=wls) (self-selection resones) 13 ANOVA( ) 14 (ML) 15 ANOVA 13 (New Jersey) Gary Hausman and Wise (1979) 14 ANOVA (Searle (1971) Townsend and Searle (1971) Wallace and Hussain (1969) Swamy and Arora (1972) Fuller and Battese (1974) Henderson (1953) 15 Jennrich and Sampson (1976) Harville (1977) Das (1979) Corbeil and Searle (1976a,b) Hocking (1985) 8
c 2000 9 Townsend and Searle (1971) Baltagi and Chang (1994) ANOVA ANOVA (variance component ratio) 16 [1] Ahn, S.C. and Schmidt, P. (1995) Efficient Estimation of Models for Dynamic Panel Data, Journal of Econometrics, 68, pp.5-28. [2] Alonso-Borrego,C. and Arellano,M.(1999) Symmetrically Normalized Instrumental-variable Estiantion Using Panel Data, Journal of Business and Economic Statistics, 17, pp.36-49. [3] Anderson, T.W. and Hsiao, C. (1981) Estimation of Dynamic Models with Error Components, Journal of the American Statistical Association, 76, pp.598-606. [4] Arellano, M. (1989) A Note on the Anderson-Hsiao Estimator for Panel Data, Economics Letters, 31, pp.337-341. [5] Arellano, M. and Bond, S. (1991) Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations, Review of Economic Studies, 58, pp.277-297. 16 9
c 2000 10 [6] Arellano, M. and Bover, O. (1995) Another Look at the Instrumental Variable Estimation of Error-components Models, Journal of Econometrics, 68, pp.29-52. [7] Baltagi, B.H. and Y.J. Chang (1994) Incomplete Panels: A Comparative Study of Alternative Estimators for the Unbalanced One-Way Error Components Regression Model, Journal of Econometrics, 62, pp.67-89. [8] Binder, M., C. Hsiao and M.H. Pesaran (2000) Estimation and Inference in Short Panel Vector Autoregression with Unit Roots and Cointegration, mimeo. [9] Blundell, R. and Bond, S. (1998) Initial Conditions and Moment Restrictions in Dynamic Panel Data Models, Journal of Econometrics, 87, pp.115-143. [10] Blundell, R., S. Bond and F. Windmeijer (2000) Estimation in Dynamic Panel Data Models: Improving on the Performance of the Standard GMM Estimator, Advances in Econometrics 15, pp.53-91. [11] Chamberlain, G. (1980) Analysis of Covariance with Qualitative Data, Review of Economic Studies, 47, pp.225-238. [12] Choi, I. (1999a) Unit Root Tests for Panel Data, Working Paper, Department of Economics, Kookmin University, Korea. [13] Corbeil, R.R. and S.R. Searle (1976a) A Comparison of Variance Component Estimators, Biometrics, 32, pp.779-791. [14] Corbeil, R.R. and S.R. Searle (1976b) Restricted Maximum Likelihood (REML), Estimation of Variance Components in the Mixed Model, Technometrics 18, pp.31-38.das, K. (1979) Asymptotic Optimality of Restricted Maximum Likelihood Estimates for the Mixed Model, Calcutta Statistical Association Bulletin 28, pp.125-142. [15] Das, K. (1979) Asymptotic Optimality of Restricted Maximum Likelihood Estimates for the Mixed Model, Calcutta Statistical Association Bulletin 28, pp.125-142. [16] Fisher, R.A.(1973a) Statistical Methods for Research Workers, 14th ed, New York: Hafner Publishing. [17] Fujiki, H., C. Hsiao, and Y. Shen.(2002) Is There a Stable Money Demand Function under the Low Interest Rate Policy? A Panel Data Analysis, Monetary and Economic Studies, 20(2), pp.1-23. 10
c 2000 11 [18] Fuller, W.A. and G.E. Battese (1974) Estimation of Linear Models with Cross-Error Structure, Journal of Econometrics, 2, pp.67-78. [19] Gourieroux, C. (2000) Econometrics of Qualitative Dependent Variables, Cambridge: Cambridge University Press. [20] Hahn,J., Hausman, J. and Kuersteinerm G.(2002) Bias Corrected Instrumental Variables Estimation for Dynamic Panel Models with Fixed Effects, MIT, mimeo. [21] Harville, D.A. (1977) Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems, Journal of the American Statistical Association 72, pp.320-340. [22] Hausman, J.A. and D. Wise (1979) Attrition Bias in Experimental and Panel Data: the Gary Income Maintenance Experiment, Econometrica, 47, pp.455-473. [23] Heckman, J.J. (1981b) The Incidental Parameters Problem and the Problem of Initial Conditions in Estimating a Discrete time-discrete Data Stochastic Process, in C.F. Manski and D. McFadden (eds.), Structural Analysis of Discrete Data with Econometric Applications, MIT Press, Cambridge. [24] Heckman, J.J. and T.E. MaCurdy (1980) A Life-Cycle Model of Female Labor Supply, Review of Economic Studies, 52, pp.681-690. [25] Henderson, C.R., Jr. (1953) Estiation of Variance Components, Biometrics, 9, pp.226-252. [26] Hocking, R.R. (1985) The Analysis of Linear Models, Monterey: Brooks/Cole Company. [27] Holtz-Eakin,D.(1988) Testing for Individual Effects in Autoregressive Models, Journal of Econometrics, 39, pp.297-307. [28] Holtz-Eakin, D., Newey, W. and Rosen, H.S. (1988) Estimating Vector Autoregressions with Panel Data, Econometrica, 56, pp.1371-1395. [29] Honoré, B.E. (1992) Trimmed LAD and Least Squares Estimation of Truncated and Censored Regression Models with Fixed Effects, Econometrica, 60, pp.533-565. [30] Hsiao, C. (2002) Analysis of Panel Data 2nd ed., Cambridge: Cambridge University Press. 11
c 2000 12 [31] Hsiao, C., M.H. Pesaran and A.K. Tahmiscioglu (2002) Maximum Likelihood Estimation of Fixed Effects Dynamic Panel Data Models Covering Short Time Periods, Jounral of Econometrics, 109, pp.107-150. [32] Im, K., M.H. Pesaran and Y. Shin (1997) Testing for Unit Roots in Heterogeneous Panels, Econometrica, forthcoming. [33] Jenrich, R.I. and P.F. Sampson (1976) Newton-Raphson and Related Algorithms for Maximum Likelihood Variance Component Estimation, Technometrics 18, pp.11-17. [34] Kiviet, H.H. (1995) On Bias Inconsistency and Efficiency in Various Estimators of Dynamic Panel Data Models, Journal of Econometrics, 68, pp.53-78. [35] Lancaster, T. (2000) The Incidental Parameter Problem Since 1948, Journal of Econometrics 95, pp.391-413. [36] Lee, M.J.(2002) Panel Data Econometrics: Methods-of-Moments and Limited Dependent Variables, San Diego: Academic Press. [37] Levin, A. and C.F. Lin (1992) Unit Root Test in Panel Data: Asymptotic and Finite Sample Properties, Discussion Paper #92-93 (University of California at San Diego). [38] Levin, A. and C.F. Lin (1993) Unit Root Test in Panel Data: New Results, Discussion Paper #93-56 (University of California at San Diego). [39] Liang, K.Y. and S.L. Zeger (1986) Longitudinal Data Analysis Using Generalized Linear Models, Biometrika 73, pp.13-22. [40] Maddala, G.S. (1983) Limited-Dependent and Qualitative Variables in Econometrics, Cambridge: Cambridge University Press. [41] Maddala, G.S. (1987) Limited Dependent Variable Models Using Panel Data, The Journal of Human Resources, 22, pp.307-338. [42] Maddala, G.S. and Wu, S.(1999) A Comparative Study of Panel Data Unit Root Tests and a New Simple Test, Oxford Bulletin of Economics and Statistics, 61, pp.631-652. [43] Neyman, J. and E.L. Schott (1948) Consistent Estimates Based on Partially Consistent Observations, Econometrica, 16, pp.1-32. 12
c 2000 13 [44] Searle, S.R. (1971) Linear Models, Wiley, New York. [45] Swamy, P.A.V.B. and S.S. Arora (1972) The Exact Finite Sample Properties of the Estimators of Coefficients in the Error Components Regression Models, Econometrica, 40, pp.261-275. [46] Townsend, E.C. and S.R. Searle (1971) Best Quadratic Unbiased Estimation of Variance Components from Unbalanced Data in the One- Way Classification, Biometrics 27, pp.643-657. [47] Wallace, T.D. and A. Hussain (1969) The Use of Error Components Models in Combining Cross-Section and time-series Data, Econometrica, 37, pp.55-72. [48] Ziliak,J.P. 1997 Efficient Estimation with Panel Data When Instruments are Predetermined: An Empirical Comparison of Moment- Condition Estimators, Journal of Business and Economic Statistics, 15, pp.419-431. 13