2 Tobin (1958) 2 limited dependent variables: LDV 2 corner solution 2 truncated censored x top coding censor from above censor from below 2 Heck
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1 % % 2 Heckit 6 1
2 2 Tobin (1958) 2 limited dependent variables: LDV 2 corner solution 2 truncated censored x top coding censor from above censor from below 2 Heckit
3 3 3 Tobin (1958) Goldberger (1964) Amemiya(1985, Chapter 10) Maddala(1983,Chapter 6) 1 y i = x iβ + u i u i N(0, σ 2 u) y i = yi if yi > 0 y i = 0 if yi 0 y i = yi > 0 N 1 y i = 0 N 0 N 1 y i F i = F (x iβ, σ 2 ) = f i = f(x iβ, σ 2 ) = x i β 1 /2σ 2 dt 2πσ 2 e t2 1 2πσ 2 e (1/2σ2 )(x i β)2 y = 0 Φ φ rescaliong y > 0 f(y i y i > 0) = 1 ( 0 x ) [ σ φ β / 1 Φ( 0 ] x β ) σ σ (0 < y < ) σ y (inverse Mills ratio) 1 Amemiya(1985,Chapter 10) Maddala(1983,Chapter 6) Winkelmann and Boes (2006,pp , pp )
4 4 λ(δ) = φ(δ)/φ(δ) < δ < λ (δ) = λ(δ) [λ(δ) + δ] < 0 λ δ c δ δ = c 2 λ c 1 y = 0 y E(y i y i > 0) = µ + σ φ( x β/σ) 1 Φ( x β/σ) = µ + σ φ(x β/σ) Φ(x β/σ) = µ + σλ(x β/σ) µ y y i = 0 P P (y i = 0) = P (u i < x iβ) = (1 F i ) y i > 0 P P (y i > 0) f(y i y i > 0) = 1 2πσ 2 e (1/2σ2 )(y i x i β)2 L = N 0 (1 F i ) N 1 1 2πσ 2 e (1/2σ2 )(y i x i β)2 β β = β OLS σ(x 1x 1 ) 1 x 0λ 0 (x β/σ) β OLS N 1 y(> 0) λ 0 y = 0 2 E( u u > c) = φ(c)/(1 Φ(c)) = λ( c) = λ(δ) δ = c
5 5 P (y = 0 x) x l = φ(x β/σ)β l /σ P (y > 0 x) x l = φ(x β/σ)β l /σ (1) E(y y > 0, x) x l = β l {1 λ(x β/σ)[x β/σ + λ(x β/σ)]} (2) l β l x β/σ E(y x) = P (y > 0 x)e(y y > 0, x) E(y x) P (y > 0 x) E(y y > 0, x) = E(y y > 0, x) + P (y > 0 x) (3) x l x l x l (1)(2) (3) E(y x) x l = β l Φ(x β/σ) β l Φ(x β/σ) OLS 3 Amemiya (1985) Amemiya 1985) Heckman(1974) Gronau(1973) ( ) 3 OLS OLS Φ(x β/σ) ) 2005, 5
6 6 y i = x iβ + σλ(x iα) + ε i y i > 0 α = β/σ E(ε i ) = 0 λ V ar(ε i ) = σ 2 σ 2 x iαλ(x iα) σ 2 λ(x iα) 2 ˆα ˆα ( ) y i > 0 ˆβ σ White robust ) x 1i x 2i x 1i x 2i % Mroz(1987) 1975 The Panel Study of Income Dynamics(PSID) Wooldridge (2003a,b) inlf 1975 hours lwage 7 5 STATA heckman heckprob
7 7 (faminc) nwifeinc=famincwage*hours)/1000) educ exper expersq age 6 kidslt kidsge nwifeinc inlf=1 inlf= nwifeinc OLS OLS Winkelmann and Boes (2006, p.248, Exercise 7.13) Wooldridge (2003a, Table 17.1, p.562, Table 17.2, p.570) OLS
8 inlf 2 2 nwifeinc age kidslt6 kidsge6 2 OLS 2 8 Wald OLS kidsge6 8 Wooldridge (2003a, Table 17.5, p.590)
9 9 (2005, 5 ) 7 STATA Mroz(1987) Cameron and Trivedi (2005) Winkelmann and Boes (2006) Wooldridge(2003b) MROZ.DTA 1 Cameron and Trivedi (2005, p.540, Figure 16.2 /***Generate Inverse Mills Ratio ***/ set obs 100 gen c = 4*(50- n)/100 gen PHIc = norm(c) gen phic = normden(c) gen lamdac = phic/(1-phic) summarize /*Graph of Mills ratio and cdf and density 1*/ graph twoway (scatter lamdac c, c(l) msize(vtiny) clstyle(p1) clwidth(medthick)) /* */ (scatter PHIc c, c(l) msize(vtiny) clstyle(p3) clwidth(medthick)) /* */ (scatter phic c, c(l) msize(vtiny) clstyle(p2) clwidth(medthick)), /* */ scale (1.2) plotregion(style(none)) /* title( Inverse Mills Ratio as Cutoff Varies ) */ xtitle( Cutoff point c, size(medlarge)) xscale(titlegap(*5)) /*
10 10 ytitle( Inverse Mills, pdf and cdf, size(medlarge)) yscale(titlegap(*5)) */ legend(pos(11) ring(0) col(1)) legend(size(small)) /* */ legend( label(1 Inverse Mills ratio ) label(2 N[0,1] Cumulative df ) label(3 N[0,1] Density )) graph save mills.gph, replace Mroz use MROZ.DTA, clear set more off /*** limited depemdemt vatiable ***/ reg inlf nwifeinc educ exper expersq age kidslt6 kidsge6 ovtest hettest logit inlf nwifeinc educ exper expersq age kidslt6 kidsge6 probit inlf nwifeinc educ exper expersq age kidslt6 kidsge6 dprobit inlf nwifeinc educ exper expersq age kidslt6 kidsge6 /*** Tobit ***/ tobit hours nwifeinc educ exper expersq age kidslt6 kidsge6, ll reg hours nwifeinc educ exper expersq age kidslt6 kidsge6 ovtest hettest /*** Heckman s selection bias model ***/ reg lwage educ exper expersq heckman lwage educ exper expersq, select(nwifeinc age kidslt6 kidsge6) mills(mymills) heckman lwage educ exper expersq, twostep select(nwifeinc age kidslt6 kidsge6) rhosigma heckman lwage educ exper expersq, twostep select(inlf = nwifeinc age kidslt6 kidsge6) rhosigma heckman lwage educ exper expersq, twostep select(hours = nwifeinc age kidslt6 kidsge6) rhosigma [1] 2005)
11 11 [2] 1997 Probit, Logit, Tobit II 4 pp [3] 2001 [4] Amemiya, T.(1985) Advanced Econometrics, Harvard University Press. [5] Cameron, A.C. and Trivedi, P.K.(2005) Microeconometrics: Methods and Applications, Cambridge University Press. [6] Goldberger, A.S.(1964) Econometric Theory, Wiley. [7] Gronau, R.(1973) The Effects of Children on the Housewife s Value of Time, Journal of Political Economy, 81(2-2), S.168-S.199. [8] Gronau, R.(1974) Wage Comparisons-A Selectivity Bias, Journal of Political Economy, 82(6), pp [9] Heckman, J.J. 1974) Shadow Prices, Market Wages, and Labor Supply, Econometrica, 42(4), pp [10] Heckman, J.J.(1979) Sample Selection as a Specification Erro, Econometrica, 47(1). pp [11] Maddala, G.S.(1983) Limited-Dependent and Qualitative Variables in Economics, Cambridge University Press. [12] Mroz, T.A.(1987) The Sensitivity of an Empirical Model of married Women s Hours of Work to Economic and Statistical Assumptions, Econometrica, 55(4), pp [13] Tobin, J.(1958) Estimation of Relationships for Limited Dependent Variables, Econometrica, 26. pp [14] Winkelmann, Rainer and Boes, Stefan.(2006) Analysis of Microdata, Springer. [15] Wooldridge, jeffrey. M.(2003a) Introductory Econometrics, Thomson. [16] Wooldridge, Jeffrey. M.(2003b) Econometric Analysis of Cross Section and Panel Data, The MIT Press
12 表 1 女性労働供給に関するロジット プロビット トービット OLS 推定 Explanatory Variables Logit Dependent Variable: inlf Probit Dependent Variable: hours dprobit Tobit OLS Coefficient z-value Coefficient z-value df/dx z-value Coefficient z-value Coefficient z-value nwifeinc educ exper expersq age kidslt kidsge _cons Number of obs LR chi2(7) Pseude R2 Adj R
13 表 2 女性労働供給に関する OLS ヘックマン 2 段階推定 Dependent Variable OLS Heckit Explanatory Variables Coefficient t-value Coefficient z-value lwage educ exper expersq _cons selection: inlf nwifeinc age kidslt kidsge _cons mills 比 lambda rho sigma Number of obs Adj R Wald chi2(3) Prob>chi
14 図 1 逆ミルズ比と切断点 C との関係 Inverse Mills ratio N[0,1] Cumulative df N[0,1] Density Cutoff point c 出典 :Cameron and Trivedi (2005) Figure p.540
2 1,2, , 2 ( ) (1) (2) (3) (4) Cameron and Trivedi(1998) , (1987) (1982) Agresti(2003)
3 1 1 1 2 1 2 1,2,3 1 0 50 3000, 2 ( ) 1 3 1 0 4 3 (1) (2) (3) (4) 1 1 1 2 3 Cameron and Trivedi(1998) 4 1974, (1987) (1982) Agresti(2003) 3 (1)-(4) AAA, AA+,A (1) (2) (3) (4) (5) (1)-(5) 1 2 5 3 5 (DI)
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1 y x y = α + x β+ε (1) x y (2) x y (1) (2) (1) y (2) x y (1) (2) y x y ε x 12 x y 3 3 β x β x 1 1 β 3 1
1 y x y = α + x β+ε (1) x y (2) x y (1) (2) (1) y (2) x y (1) (2) y x y ε x 12 x y 3 3 β x β x 1 1 β 3 1 2 2 N y(n 1 ) x(n K ) y = E(y x) + u E(y x) y u(n 1 ) y = x β + u β Ordinary Least Squares:OLS (min
80 X 1, X 2,, X n ( λ ) λ P(X = x) = f (x; λ) = λx e λ, x = 0, 1, 2, x! l(λ) = n f (x i ; λ) = i=1 i=1 n λ x i e λ i=1 x i! = λ n i=1 x i e nλ n i=1 x
80 X 1, X 2,, X n ( λ ) λ P(X = x) = f (x; λ) = λx e λ, x = 0, 1, 2, x! l(λ) = n f (x i ; λ) = n λ x i e λ x i! = λ n x i e nλ n x i! n n log l(λ) = log(λ) x i nλ log( x i!) log l(λ) λ = 1 λ n x i n =
AR(1) y t = φy t 1 + ɛ t, ɛ t N(0, σ 2 ) 1. Mean of y t given y t 1, y t 2, E(y t y t 1, y t 2, ) = φy t 1 2. Variance of y t given y t 1, y t
87 6.1 AR(1) y t = φy t 1 + ɛ t, ɛ t N(0, σ 2 ) 1. Mean of y t given y t 1, y t 2, E(y t y t 1, y t 2, ) = φy t 1 2. Variance of y t given y t 1, y t 2, V(y t y t 1, y t 2, ) = σ 2 3. Thus, y t y t 1,
2 2 2 (Poisson Distribution) P (y = j) = e λ λ j λ > 0, j = 0, 1, 2... j! j! j E(y) = V ar(y) = λ λ y x λ = λ(x iβ) f(y i x iβ) = exp( exp(x i β)) exp
11 1 1 Cameron and Trivedi (1998,2005) Winkelmann(1997) 1 Bortkiewicz(1898) 1 2 2 2 (Poisson Distribution) P (y = j) = e λ λ j λ > 0, j = 0, 1, 2... j! j! j E(y) = V ar(y) = λ λ y x λ = λ(x iβ) f(y i x
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雇用と年金の接続 在職老齢年金の就業抑制効果と老齢厚生年金受給資格者の基礎年金繰上げ受給要因に関する分析 The Labour Market Behaviour of Older People: Analysing the Impact of the Reformed "Earning Test"
Powered by TCPDF (www.tcpdf.org) Title 雇用と年金の接続 : 在職老齢年金の就業抑制効果と老齢厚生年金受給資格者の基礎年金繰上げ受給要因に関する分析 Sub Title The labour market behaviour of older people: analysing the impact of the reformed "Earning test"
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