9 1 (1) (2) (3) (4) (5) (1)-(5) (i) (i + 1) 4 (1) (2) (3) (4) (5) (1)-(2) (1)-(5) (5) 1
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1 9 1 (1) (2) (3) (4) (5) (1)-(5) (i) (i + 1) 4 (1) (2) (3) (4) (5) (1)-(2) (1)-(5) (5) 1
2 2 2 y i = 1, 2, 3,...J (1 < 2 < 3 <... < J) y i yi 1 y i = x iβ + u i i = 1, 2,..., n x u y i 2 y i = j κ j 1 < yi < κ j j = 1, 2,...J (threshold mechanism) J J κ 0 < κ 1 < κ 2 <... < κ J y i = 1 κ 0 < yi < κ 1 κ 0 x iβ < u i < κ 1 x iβ y i = 2 κ 1 < yi < κ 2 κ 1 x iβ < u i < κ 2 x iβ.. y i = J κ J 1 < yi < κ J κ J 1 x iβ < u i < κ J x iβ κ 0 =, κ J =. 1 Winkelmann and Bose (2006, pp )
3 3 1 J = 3 κ 1 κ 2 2 y i = 1 y i = 2 κ 1 y i = 2 y i = 3 κ 2 1 u i f(u i x i ) 2 y i π ij = P (y i = j x i ) = F (κ j x iβ) F (κ j 1 x iβ) j = 1, 2, 3, F ( ) = 0, F ( ) = 1 Φ(u) Λ(u) π ij = Φ( κ j x i β σ ) Φ( κ j 1 x i β ) j = 1, 2,..J σ κ β σ = 1 π ij = P (y i = j x i ) = P (y i j x i ) P (y i j 1 x i ) = Λ(κ j x iβ) Λ(κ j 1 x iβ) j = 1, 2,..J (odds ratio) P (y i j x i ) P (y i > j x i ) = exp(κ j x iβ) = exp(κ j ) exp( x iβ) ln( P (y i j x i ) P (y i > j x i ) ) = κ j x iβ j proportional odds model x i κ j κ m P (y i j x i )/P (y i > j x i ) P (y i m x i )/P (y i > m x i ) = exp(κ j) exp(κ m ) 2 F (u) f(u) = df (u)/du
4 4 f(y i x i ; β, κ 1, κ 2,..κ J 1 ) = (π i1 ) d i1 (π i2 ) d i2 (π ij ) d ij = J (π ij ) d ij { 1 y i = j) d ij = 0 n log L(β, κ 1, κ 2,...κ J 1 ; y, x) = n i=1j=1 J d ij log π ij x i x il κ j x il β x il β MP E j=1 MP E ijl = π ij x il = [f(κ j x iβ) f(κ j 1 x iβ)] β l β J 1 κ j κ j β J 1 2 j x i MP E ijl MP E ijm = β l β m β j MP E
5 5 3 β j κ j j β 3 κ ij x i κ ij = κ j + x iγ j j = 1, 2,...J κ j π ij = F ( κ j + x iγ j x iβ) F ( κ j 1 + x iγ j 1 x iβ) = F ( κ j x iβ j ) F ( κ j 1 x iβ j 1 ) β j = β γ j β γ j j β j κ j 4 β j j Long (1997, pp ) 2 (Score Test) β 1 = β 2 =... = β J 1 = β K(J 2) 5 K Brant(1990) β j j j j Wald Test) β j x il β jl 3 Williams(2006) Long(1997) Winkelmann and Bose (2005, pp ) 4 Williams (2006) Stata program gologit2 5 2
6 6 MP E ijl = f( κ j x i β j)β jl f( κ j 1 x i β j 1)β j 1l MP E ijm f( κ j x i β j)β jm f( κ j 1 x i β j 1)β j 1m β j κ j 4 Papke (1998) (k) % Mehra andprescott (1985) Weil(1989) Kocherlakota (1996) (1) (2) 6 Wooldridge (2002, pp ) 7 Papke(2004) Poterba et. al.(2005,2006)
7 7 (1) Mostly bonds (2) (Mixed) (3) (Mostly stocks) 3 (1)(2)(3) 1992 National Longitudinal Survey (NLS) of Mature Women % (pctstock) 3 (1) (2) (3) female 0.60 marrried age ( 60 ) educ choice 1 ( 0.62) years in pension plan profit-sharing plan family inc 6 net wealth in ( 198,000 ) black stock in ( 0.32) IRA in IRA(Individual Retirement Account)
8 8 Papke (1998) pctstock (1)(2)(3) (OLS ) 9 1 OLS OLS Papke(1998) 1 iii 1 1.6% 3 choice 13% profit-sharing plan 14% 10 28% stock in % Papke(1998) IRA in % 1 OLS 10 1 OLS β i 9 Papke (1998, p.213) OLS 10 11
9 9 β j 1 κ j 2 5 sequential Winkelmann and Bose (2006, pp ) 6 STATA Leslie E Papke (1998) How Are Participants Investing Their Accounts in Participant Directed Individual Account Pension Plans American Economic Review, 88(2), pp Wooldridge / ec/faculty/wooldridge/book2.htm. PENSION.DTA use PENSION.DTA, clear
10 10 set more off /*data analysis*/ tabulate pctstck choice, chi2 tabulate pctstck age, chi2 /**OLS **/ reg pctstck female married age educ choice pyears prftshr finc25 finc35 finc50 finc75 finc100 finc101 wealth89 black stckin89 irain89 /**ordered logit analysis **/ ologit pctstck female married age educ choice pyears prftshr finc25 finc35 finc50 finc75 finc100 finc101 wealth89 black stckin89 irain89 estimates store A quietly ologit pctstck female married educ choice pyears prftshr finc25 finc35 finc50 finc75 finc100 finc101 wealth89 black stckin89 irain89 estimates store B lrtest B A ologit pctstck female married age educ choice pyears prftshr finc25 finc35 finc50 finc75 finc100 finc101 wealth89 black stckin89 irain89 estimates store C quietly ologit pctstck female married age educ pyears prftshr finc25 finc35 finc50 finc75 finc100 finc101 wealth89 black stckin89 irain89 estimates store D lrtest D C quietly ologit pctstck female married age educ choice pyears prftshr finc25 finc35 finc50 finc75 finc100 finc101 wealth89 black stckin89 irain89 predict zero fifty hundred sum zero fifty hundred list zero fifty hundred in 1/10 /* */ graph twoway qfitci zero age, clpattern(solid) clwidth(thick) qfitci fifty age, clpattern(tight dot) clwidth(thick) qfitci hundred age, clpattern(dash) clwidth(thick) ytitle(probability) xtitle(age) graph save pension choice.gph, replace ologit pctstck female married age educ choice pyears prftshr finc25 finc35 finc50 finc75 finc100 finc101 wealth89 black stckin89 irain89, robust /**ordered probit analysis **/ oprobit pctstck female married age educ choice pyears prftshr finc25 finc35 finc50 finc75 finc100 finc101 wealth89 black stckin89 irain89
11 11 oprobit pctstck female married age educ choice pyears prftshr finc25 finc35 finc50 finc75 finc100 finc101 wealth89 black stckin89 irain89, robust gologit2 Richard Williams Stata findit gologit2 /*autofit */ gologit2 pctstck female married age educ choice pyears prftshr finc25 finc35 finc50 finc75 finc100 finc101 wealth89 black stckin89 irain89, autofit /*proportional line*/ gologit2 pctstck female married age educ choice pyears prftshr finc25 finc35 finc50 finc75 finc100 finc101 wealth89 black stckin89 irain89, pl lrforce store(constrained) /*non-proportional line*/ gologit2 pctstck female married age educ choice pyears prftshr finc25 finc35 finc50 finc75 finc100 finc101 wealth89 black stckin89 irain89, npl lrforce store(unconstrained) /*gamma*/ gologit2 pctstck female married age educ choice pyears prftshr finc25 finc35 finc50 finc75 finc100 finc101 wealth89 black stckin89 irain89, auto gamma lrf [1] 2005) [2] Brant, R.(1990) Assessing Proportionality in The Proportional Odds Model for Ordinal Logstic Regression, Biometrics, 46(4), pp [3] Cameron, A.C. and Trivedi, P.K.(2005) Microeconometrics: Methods and Applications, Cambridge University Press. [4] Hensher, David A., Rose, John M. and Greene, William H.(2005) Applied Choice Analysis, Cambridge University Press. [5] Kocherlakota, Narayana R.(1996) The Equity Premium: It s Still a Puzzle, Journal of Economic Literature, 34, pp [6] Long, J.Scott.(1997) Regression Models for Categorical and Limited Dependent Varaibles, SAGE Publications.
12 12 [7] Long, J.Scott. and Freese, Jeremy. (2006) Regression Models for Categorical Dependent Variables Using Stata, 2nd ed.,stata Press. [8] Mehra, Rajnish and Prescott, Edward C.(1985) The Equity Premium: A Puzzle, Journal of Monetary Economics, 15(2), pp [9] Papke, Leslie, E.(1998) How Are Participants Investing Their Accounts in Participants Directed Individual Account Pension Plans, American Economic Review, 88(2), pp [10] Papke, Leslie, E.(2004) Choice and Other Determinants of Employee Contributions to Defined Contribution Plans, Social Security Bulletin, 65(2), pp [11] Poterba, James., Rauh, Joshua., Venti, Steven., and Wise, David. (2005) Lifecycle Asset Allocation Strategies and The Distribution of 401(k) Retirement Wealth, The MIT, mimeo. [12] Poterba, James., Rauh, Joshua., Venti, Steven., and Wise, David. (2006) Defined Contribution Plans, Defined Benefit Plans and The Accumulation of Retirement Wealth, The MIT, mimeo. [13] Singer, Judith D. and Willett, John B.(2003) Applied Longitudinal Data Analysis, Oxford University Press. [14] Weil, Philippe. (1989) The Equity Premium Puzzle and The Risk-Free Rate Puzzle, Journal of Monetary Economics, 24, pp [15] Williams, Ricahrd. (2006) Generalized Ordered Logit/Partial Proportional Odds Models for Ordinal Dependent Variables, The Stata Journal, 6(1), pp A pre-publication version is avalable at rwilliam/gologit2.pdf [16] Winkelmann, Rainer and Boes, Stefan.(2006) Analysis of Microdata, Springer. [17] Wooldridge, Jeffrey. M.(2003) Econometric Analysis of Cross Section and Panel Data, The MIT Press
13 表 1 OLS 順序プロビット 順序ロジット推定 Dependent variable: pctstock OLS Ordered Probit Ordered Logit 係数 t 値係数 z 値係数 z 値 female married age educ choice years in pension plan profit-sharing plan (=1) <family inc<= <family inc<= <family inc<= <family inc<= <family inc<=100, ,000<family inc net wealth in black stock in 1989 (=1) IRA in 1989 (=1) _cons cut1 (κ 1 ) cut2 (κ 2 ) Number of Obs Adj R-squared Root MSE Wald chi2(17) Log-likelihood value LR chi2(17)
14 表 2 一般化順序ロジット推定 Dependent variable: pctstock 係数 z 値係数 z 値係数 z 値係数 z 値 female married age educ choice years in pension plan profit-sharing plan (=1) <family inc<= <family inc<= <family inc<= <family inc<= <family inc<=100, ,000<family inc net wealth in black stock in 1989 (=1) IRA in 1989 (=1) _cons Number of Obs Pseudo R2 Root MSE Wald chi2(17) Log-likelihood value LR chi2(34) Wald test of parallel-lines assumptions chi2(12) Prb>chi2 Constrained Mostly bonds Generalized Ordered Logit Unconstrained Mixed Mostly bonds Mixed
15 図 1 順序選択モデルの閾値
16 図 2 一般化順序選択モデルにおける確率変化
17 図 3 年金投資選択確率と年齢の関係 Probability Age 95% CI Fitted values 95% CI Fitted values 95% CI Fitted values
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
10 2 1 2007 4 6 25-44 57% 2017 71% 2 Heckit 6 1 2 Tobin (1958) 2 limited dependent variables: LDV 2 corner solution 2 truncated 50 50 censored x top coding censor from above censor from below 2 Heckit
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)
遺産相続、学歴及び退職金の決定要因に関する実証分析 『家族関係、就労、退職金及び教育・資産の世代間移転に関する世帯アンケート調査』
2-1. (2-1 ) (2-2 ) (2-3 ) (Hayashi [1986]Dekle [1989]Barthold and Ito [1992] [1996]Campbell [1997] [1998]Shimono and Ishikawa [2002]Shimono and Otsuki [2006] [2008]Horioka [2009]) 1 2-1-1 2-1-1-1 8 (1.
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 =
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
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,
わが国企業による資金調達方法の選択問題
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