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2 2 2 1 y = β 0 + β 1 x + u x u Cov(x, u) 0 β 0 β 1 x x u z Cov(z, u) = 0 Cov(z, x) 0 z x (1) z u (2)z x (3)z x Cov(z, u) = 0 Cov(z, x) 0 1 = 0 x = π 0 + π 1 z + v 1 Bowden and Turkington (1984) Baum (2006, Chapter8)

3 3 π 1 = Cov(z, x)/v ar(z) Cov(z, x) 0 π 1 0 π 1 0 ˆβ IV = Cov(z, y) Cov(z, x) = n (z i z)(y i ȳ) i=1 n (z i z) (x i x) i=1 ˆβ 0 = ȳ ˆβ IV x = r zy y y r zx x x z = x p lim( ˆβ IV ) = β 1 x u two stage least squared method: 2SLS k l l k x = π 0 + π 1 z i1 + π 2 z i2 + ω i X= Z(Z Z) 1 Z X P Z = Z(Z Z) 1 Z 2 β 2SLS = ( X X) 1 X y = {X Z(Z Z) 1 Z X} 1 {X Z(Z Z) 1 Z y} = (X P Z X) 1 X P Z y β 2SLS Z Z l l Z X k l = k 2SLS IV

4 4 1 û i = y i X β 2SLS X X β 2SLS 3 (l) (k) (1)l = k just identified (2)l > k over identified (3)l < k under identified (1)(2) (3) (2) (r) (l) k r = l k (1) (4) û (2) û (l) R 2 (3) û LM nr 2 χ 2 (l k) 2 2 STATA Sargan(1958) Basmann (1960) overid

5 5 4 4 t 4 Breusch and Pagan (1979) White (1980) Pagan and Hall (1983) Breusch and Pagan (1979) Pagan and Hall (1983) White (1982) 3 5 z x z u p lim ˆβ 1 = β 1 + Corr(z, u) Corr(z, x) σu σ x σ u σ x u x Corr(z, u) Corr(z, x) ˆβ 1 Weak Instrumental Variables Bound, Jaeger and Baker (1995) (R 2 ) (RSS Z2 RSS Z )/T SS RSS Z2 Z 2 RSS Z 3 STATA ivhettest Pagan and Hall (1983) White (1980) Breusch and Pargan (1979) Koenker (1981)

6 6 Shea(1997) i Rp 2 = (v i,i,ols )/(v i,i.iv ){(1 RIV 2 )/(1 R2 OLS )} v i,i 4 Andreson(1984) Hall, Rudebusch and Wilcox (1996) X Z canonical correlation Corr i i = 1, 2,..k Anderson l k + 1 Hall and Peixe (2000) (redundancy) Anderson 5 Hahn and Hausman (2002b) Staiger and Stock (1997) y 2 y 1 = β 0 + β 1 y 2 + β 2 z 1 + β 3 z 2 + u 1 z 1 z 2 z 3 z 4 4 Shea(1997) STATA ivreg2 first ffirst 5 STATA ivreg2 redundant ivreg2 Anderson and Rubin (1949) Cragg and Donald (1993) Stock and Wright (2000) 6 Nelson and Sartz (1990) Stock and Wright (2000) Stock, Wright and Yogo (2002) Hahn and Hausman (2002a, 2003) Andrews and Stock (2005) Chao and Swanson (2005) Stock and Yogo (2005) Hausman, Stock and Yogo (2005)

7 7 Hausman (1978) y 2 7 Durbin-Wu-Hausman (DWH) Hausman β e β c ( β c β e ) (var[ β c ] var[ β e ]) 1 ( β c β e ) χ(k 1 ) k 1 Wooldridge(2002, p.119) y 1 y 1 = β 0 + β 1 y 2 + β 2 z 1 + β 3 z 2 + u 1 y 2 y 2 = α 0 + α 1 z 1 + α 2 z 2 + α 3 z 3 + α 4 z 4 + v 2 z j u 1 v 2 u 1 y 2 u 1 1 = 0 y 2 u 1 u 1 = δ 1 v 2 + e 1 y 2 ˆv 2 y 1 y 1 = β 0 + β 1 y 2 + β 2 z 1 + β 3 z 2 + δ 1ˆv 2 + ε t 1 = 0 y 2 7 GMM Hayashi(2000) (GMM) 8 z 7 Durbin (1954) Wu(1973) Hausman(1978) Durbin-Wu-Hausman test Bowden and Turkington (1984, pp.50-52) Davidson and MacKinnon (2004, pp ) 8 Davidson and MacKinnon (2004 Chapter 9)

8 8 u Cov(z, u) = 0 GMM E[zu] = 0 l l g i (β) = Z iu i = Z i(y i x i β) g i l g(β) = 1 N i=1 N g i(β) = 1 N i=1 N Z i(y i x i β) = 1 N Z u N l = k g( β GMM ) = 0 β GMM β IV l > k min J( β GMM ) = Ng( β GMM ) W g( β GMM ) W iid g( β GMM ) l l k J( β)/ β = 0 β GMM = (X ZWZ X) 1 X ZWZ y W I N Hansen(1982) W = S 1 S = E[Z u u Z] = E[Z ΩZ] S l l) β EGMM = (X ZS 1 Z X) 1 X ZS 1 Z y β EGMM S 2 2SLS Ω Ŝ 2 GMM feasible efficient teo-step GMM estimator: FEGMM

9 9 β F EGMM = (X ZŜ 1 Z X) 1 X ZŜ 1 Z y Ω = σ 2 I N β EGMM = β IV Ŝ 2SLS û i i Z i Ŝ = 1 N N i=1û2 i Z iz i GMM Hansen(1982) J GMM J( β EGMM ) = Ng( β EGMM ) Ŝ 1 g( β EGMM ) χ 2 (l k) 8 Griliches(1976) Blackburn and Neumark(1992) the National Longitudinal Survey of Young Men (NLS) 1980 Wooldridge ec/faculty/wooldridge/book2.htm Blackburn and Newmark(1992) IQ 1968 IQ 9 the knowledge of the world of work (KW W ) % Blackburn and Neumark (1992) IQ 3 1 iq

10 10 1 IQ 2 Griliches, Hall and Hausman(1978) IQ Blackburn abd Neumark (1992) ln w = X β + D γ + γ A A + ε w X D A A IQ KW W IQ = A + ϵ I, KW W = γ K A + ϵ K A Z A = Z γ Z + ϵ Z ln w = X β + D γ + γ A IQ + ε IQ = Z γ Z + ϵ Z Cov(Z, ε) = 0, Cov(ε, ϵ Z ) = 0 Z Griliches(1976, 1977) Blackburn and Neumark(1992) Griliches IQ Griliches(1976) VI Griliches IQ Griliches(1976, S80) IQ

11 11 IQ IQ IQ IQ Griliches(1976) 4 IQ KW W IQ KW W Sargan Basmann 7 2 GMM 5 6 Anderson Canonical Correlation Cragg and Donald Andreson and Rubin Stock and Wright Hall and Peixe IQ IQ Shea

12 12 Pagan and Hall White/Koenker Breusch and Pagan 7 Hausman Wu-Hausman Durbin- Wu-Hasuman Griliches(1978,1979) Baum(2006 Chapter8) IQ IQ 9 10 STATA use WAGE2.DTA, clear /**Regression**/ reg lwage educ exper tenure married black south urban /* 3*/

13 13 reg lwage educ exper tenure married black south urban, robust/* 3*/ reg educ IQ KWW sibs brthord meduc feduc hettest ovtest reg educ IQ KWW meduc feduc/* 4*/ hettest ovtest reg educ IQ KWW meduc feduc, robust/* 4*/ /**Endogeneity Issues: IVreg 5**/ ivreg lwage exper tenure married black south urban (educ = IQ KWW meduc feduc) overid ivreg lwage exper tenure married black south urban (educ = IQ KWW meduc feduc), robust /**download ivreg2 from STATA corporation**/(stata findit ivreg2 ivreg2 ) ivreg2 lwage exper tenure married black south urban (educ=iq KWW meduc feduc), gmm2s orthog(iq) /* 6*/ ivhettest, all ivreg2 lwage exper tenure married black south urban (educ=iq KWW meduc feduc), gmm2s orthog(iq KWW)/* 6*/ ivhettest, all ivreg2 lwage exper tenure married black south urban (educ= IQ KWW meduc feduc), ffirst redundant(iq) ivreg2 lwage exper tenure married black south urban (educ= IQ KWW meduc feduc), ffirst redundant(kww) ivreg2 lwage exper tenure married black south urban (educ= IQ KWW meduc feduc), ffirst redundant(meduc) ivreg2 lwage exper tenure married black south urban (educ= IQ KWW meduc feduc), ffirst redundant(feduc) ivreg2 lwage exper tenure married black south urban (educ= IQ KWW meduc feduc), ffirst redundant(iq KWW) ivreg2 lwage exper tenure married black south urban (educ= IQ KWW meduc feduc), ffirst redundant(meduc feduc) /**Durbin-Wu-Hausamn tests for endogeneity in IV estimation 7 **/ quietly ivreg2 lwage exper tenure married black south urban (educ=iq KWW meduc feduc), small

14 14 estimates store iv quietly regress lwage exper tenure married black south urban educ hausman iv., constant sigmamore quietly ivreg2 lwage exper tenure married black south urban (educ=iq KWW meduc feduc), orthog(iq)small ivendog [1] (2005) [2] Anderson, T.W. (1984) Introduction to Multivariate Statistical Analysis, Wiley. [3] Anderson, T.W. and Rubin, H.(1949) Estimators of the Paramters of a Single Equation in a Complete Set of Stochastic Equations, Annals of Mathematical Statistics, 21, pp [4] Andrew, Donald W.K. and Stock, James H.(2005) Inference with Weal Instruments, NBER Technical Working Paper 313. [5] Angrist, J.D.and Krueger, A.B.(1991) Does Compulsory School Attendance Affect Schooling and Earnings?, Quarterly Journal of Economics, 106, pp [6] Angrist, Joshua D. and Krueger, Alan B.(2001) Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experiments, Journal of Economic Perspectives, 15(4), pp [7] Basmann, R.L. (1960) On Finite Sample Distributions of Generalized Classical Linear Identifiability Test Statistics, Journal of the American Statistical Association, 55(292), pp [8] Baum, Christopher. (2006) An Introduction to Modern Econometrics Using Stata, Stata Press. [9] Blackburn, McKinley and Neumark, David.(1992) Unobserved Ability, Efficiency Wages, and Interindustry Wage Differentials, Quaterly Journal of Economics, 107(4), pp [10] Bound, John., Jaeger, David.A. and Baker, Regina. M.(1995) Problems with Instrumental Variables Estimation when the Correlation

15 15 between the Instruments and the Endogenous Explanatory Variable is Weak, Journal of the American Statistical Association, 90(430), pp [11] Bowden, R.J. and Turkington, D.A.(1984) Instrumental Variables, Cambridge University Press. [12] Breusch, Trevor., Qian, Hailong., Schmidt, Peter., and Wyhowski, Donald.(1999) Redundancy of Moment Conditions, Journal of Econometrics, 91, pp [13] Cameron, A.C.and Trivedi, P.K.(1998) Regression Analysis of Count Data, Cambridge University Press. [14] Chao, John.C. and Swanson, Norman R.(2005) Consistent Estimation with a Large Number of Weak Instruments, Econometrica, 73(5), PP [15] Cameron, A.C. and Trivedi, P.K.(2005) Microeconometrics: Methods and Applications, Cambridge University Press. [16] Cragg, John G.and Donald, Stephen G.(1993) Testing Identifiability and Specification in Instrumental Varaible Models, Econometric Theory, 9, pp [17] Davidson, Russell and MacKinnon, James G.(2004) Econometric Theory and Methods, Oxford University Press. [18] Durbin, J.(1954) Errors in variables, Review of the Internatinal Statistical Institute, 22, pp [19] Griliches, Zvi.(1976) Wages of Very Young Men, Journal of Political Economy, 84(4. Part 2), pp. S69-S85. [20] Griliches, Zvi.(1977) Estimating the Returns to Schooling: Some Econometric Problems, Econometrica, 45(1), pp [21] Griliches, Zvi., Hall, Bronwyn., and Hausoman, Jerry.(1978) Missing Data and self-selection in Large Panels, Annales de L INSEE, XXX- XXXI, pp [22] Hahn, Jinyoung and Hausman, Jerry. (2002a) A New Specification Test for the Validity of Instrumental variables, Econometrica, 70(1), pp

16 16 [23] Hahn, Jinyoung and Hausman, Jerry. (2002b) Notes on Bias in Estimators for Simultaneous Equation Models, Economics Letters, 75. pp [24] Hahn, Jinyoung and Hausman, Jerry. (2003) Weak Instruments: Diagnosis and Cures in Empirical Econometrics, American Economic Review, 93(2), pp [25] Hall, Alastair R., Rudebusch, Glenn D. and Wilcox, David W.(1996) JUdging Instrument Relevance in Instrumental Variables Estimation, International Economic Review, 37(2), pp [26] Hall, Alastair R. and Peixe, Fernanda P.M.(2000) A Consistent Method for the Selection of Relevant Instruments, A paper presented at Econometric Society World Congress [27] Hansen, Lars.P (1982) Large Sample Properties of Generalized Method of Moments Estimators, Econometrica, 50(4), pp [28] Hausman, Jerry. (1978) Specification tests in econometrics, Econometrica, 46, pp [29] Hausman, Jerry., Stock, James H. and Yogo, Motohiro.(2005) Asymptotic Properties of the Hahn-Hausman Test for Weak-Istruments, Economics Letters, 89, pp [30] Hayashi, Fumio.(2000) Econometrics, Princeton University Press. [31] Koenker, Roger.(1981) A Note on Studentizing a test for Heteroscedasticity, Journal of Econometrics, 17., pp [32] Nelson, Charles R. and Startz, Richard.(1990a) The Distribution of the Instrumental Variables Estimator and Its t-ratio When the Instrument is a Poor One, Journal of Business, 63(1, Part.2), pp. S125-S140. [33] Nelson, Charles R.and Startz, Richard.(1990) Some Further Results on the Exact Small Sample Properties of the Instrumental Variable Estimator, Economerica, 58(4), pp [34] Pagan, A.R. and Hall, D. (1983) Diagnostic Tests as Residual Analysis, Econometric Reviews, 2(2), pp [35] Ruud, P.A. (2000) An Introduction to Classical Econometric Theory, Oxford University Press.

17 17 [36] Sargan, J.D. (1958) The Estimation of Economic Relationships Using Instrumental Variables, Econometrica, 26(3), pp [37] Shea, John.(1997) Instrument Relevance in Multivariate Linear Models: A Simple Measure, Review of Economics and Statistics, 79(2), pp [38] Staiger, Douglas. and Stock, James.H. (1997) Instrumental Variables Regression with Weak Instrumetns, Econometrica, 65(3), pp [39] Stock, James H. and Wright Jonathan H. (2000) GMM with Weak Identification, Econometrica, 68(5), pp [40] Stock, James H., Wright, Jonathan H. and Yogo, Motohiro. (2002) A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments, Journal of Business and Economic Statistics, 20(4), pp [41] Stock, James H. and Yogo, Motohiro. (2005) Testing for Weak Instruments in Linear IV Regression, in Andrews, D.W.K. and Stock, J.H.(eds) Identification and Inference for Econometric Models: Essays in Honor of Thomas Rothenberg, Cambridge University Press. pp [42] Winklemann, Rainer and Boes, Stefan. (2005) Analysis of Microdata, Springer. [43] White, Halbert. (1980) A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity, Econometrica, 48(4), pp [44] White, Halbert. (1982) Instrumental Variables Regression with Independent Observations, Econometrica, 50(2), pp [45] Wooldridge, Jeffrey. M. (2002) Econometric Analysis of Cross Section and Panel Data, The MIT Press [46] Wu, D-M. (1973) Alternative tests of independence between stochastic regressors and disturbances, Econometrica, 41, pp

18 表 1 変数定義 wage hours IQ KWW educ exper tenure age married black south urban sibs brthord meduc feduc lwage 月給週間平均労働時間 IQスコア労働意識 (knowledge of world work) スコア教育年数労働経験年数現職の在職年数年齢結婚ダミー黒人ダミー南部ダミー都市部ダミー兄弟姉妹数兄弟姉妹における自分の順位母親の教育年数父親の教育年数月給の対数表示

19 表 2 基本統計量観察値平均標準偏差最小最大 wage ,078 hours IQ KWW educ exper tenure age married black south urban sibs brthord meduc feduc lwage

20 表 3 賃金関数の OLS 推定被説明変数 :Iwage Coefficient Robust t- ratio 説明変数 educ exper tenure married black south urban _cons 観察値 935 R-squared Root MSE Breusch- Pagan/Cook- Weisberg test for heteroskedasticity Ramsey RESET test for omitted variables Chi2(1)=3.69 Prob>Chi2=0.055 F(3,924)=0.69 Prob>F=0.5556

21 表 4 教育年数の OLS 推定被説明変数 :educ Coefficient Robust t- ratio 説明変数 IQ KWW meduc feduc _cons 観察値 772 F(4, 717) Prob>F R-squared Root MSE Breusch- Pagan/Cook-Weisberg test for heteroskedasticity Ramsey RESET test for omitted variables Chi2(1)=3.37 Prob>Chi2=0.066 F(3,714)=14.26 Prob>F=0.000

22 表 5 賃金関数の操作変数方 (IV) 推定被説明変数 :Iwage Coefficient Robust t- ratio 説明変数 educ exper tenure married black south urban _cons 観察値 772 R-squared Root MSE Tests of overidentifying restrictions Sargan N*R-sq test Basmann test Chi-squ(3) P-value= Chi-squ(3) P-value=0.920 Instrumented: edu Instruments: exper, tenure, married, black, south, urban, IQ, KWW, medc, feduc

23 表 6 賃金関数の 2 段階 GMM 推定被説明変数 :Iwage Coefficient z-ratio 説明変数 educ exper tenure married black south urban _cons 観察値 722 Centered R2 Uncentered R2 Root MSE Identification Tests Underidentification test (Anderson Canonical Correlation LM statistic) Chi-sq(4)= P-value=0.000 Weak identification test (Cragg-Donald Wald F statistic) Weak instrument-robust inference Anderson-Rubin Wald F test Anderson-Rubin Wald Chi-sq test Stock-Wright LM S stat Sargan statistic (overidentification test of all instruments) IV redundancy test (LM test for IQ) Shea Partial R F(4,711)=21.81 P-value=0.000 Chi-sq(4)=88.58 P-value=0.000 Chi-sq(4)=78.90 P-value=0.000 Chi-sq(3)=0.503 P-value=0.918 Chi-sq(1)= P-value=0.000 R2= P-value=0.000 IV heteroskedasticity tests Pagan-Hall general test statistic: Pagan-Hall test w/assumed normality: White/Koenker nr2 test statistic: Breusch-Pagan/Godfrey/Cook-Weisberg: Chi-squ(10) P-value=0.020 Chi-squ(10) P-value=0.000 Chi-squ(10) P-value=0.019 Chi-squ(10) P-value=0.000

24 表 7 内生性に関する Durbin-Wu-Hausman 検定被説明変数 :Iwage (b) iv (B) Coefficient (b-b) Difference sqrt(diag(v_b- V_B)) S.E. 説明変数 educ exper tenure married black south urban _cons Tests of endogeneity of: educ Wu-Hausman F test: Durbin-Wu-Hasuman chi2 test: chi2(8)=25.52 Prob>Chi2= F(1, 713) P-value= Chi-sq(1) P-value=0.000

DVIOUT-swp00

DVIOUT-swp00 第 9 講マッチング手法先に述べたように政策を実施した処理群のデータはあるが対照群のデータは不在であり外部情報から対処群をみつけてマッチングさせる必要が出てくる場合には次のような手法を用いる実際に外部データが十分にあり処理群に含まれる個別サンプルの全ての変数属性にぴったり一致するような対照群サンプルを選ぶことが出来ればこれを完全一致マッチング (exact matching) と呼ぶが