References
Conclusions cont.
In practice, the data often dictate the method. If one has access to
experimental data, one worries less about selection (though IV is often used to correct for selection of treatment status contrary to assignment). Given observational data, if one can find a discontinuity in expected treatment with respect to an observable assignment variable, one uses RD; if one can conceive of plausible excluded instruments, one uses IV. In the absence of these features of the data, repeated measures may used to control for invariant unobservables, or observations may be matched on observables.
Checking that your model is not badly misspecified, and conducting various kinds of sensitivity tests, is perhaps the most valuable way to minimize bias in published estimates. Nichols (2007, 2008) offers a kind of “checklist” of things to look at in these models (in Stata) and there will be a monograph with more user-friendly text and examples later this year (forthcoming from Stata Press).
Overview Matching and Reweighting Panel Methods Instrumental Variables (IV) Regression Discontinuity (RD) More
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References
Abadie, Alberto and Guido W. Imbens. 2006. “On the Failure of the Bootstrap for Matching Estimators.”NBER technical working paper 325.
Abadie, Alberto, David Drukker, Jane Leber Herr, and Guido W. Imbens, 2004. “Implementing matching estimators for average treatment effects in Stata,” Stata Journal 4(3): 290-311
Abadie, Alberto, Joshua D. Angrist, and G. Imbens, (2002), ”Instrumental Variables Estimation of Quantile Treatment Effects,” Econometrica 70(1): 91-117.
Abadie, Alberto, and Guido W. Imbens. 2002. “Simple and Bias-Corrected Matching Estimators for Average Treatment Effects,”NBER technical working paper 283.
Abowd, J., Creecy, R. and Kramarz, F. 2002. “Computing person and firm effects using linked longitudinal employer-employee data.” Technical Paper 2002-06, U.S. Census Bureau.
Anderson, T. W. and H. Rubin (1949). “Estimators of the Parameters of a Single Equation in a Complete Set of Stochastic Equations.” Annals of Mathematical Statistics, 21: 570-582.
Andrews, Donald W. K.; Marcelo J. Moreira; and James H. Stock. 2007. “Performance of Conditional Wald Tests in IV Regression with Weak Instruments.” Journal of Econometrics, 139(1): 116-132.Working paper version onlinewith supplementsat Stock’s website. Andrews, Donald W. K.; Marcelo J. Moreira; and James H. Stock. 2006. “Optimal Two-Sided Invariant Similar Tests for Instrumental Variables Regression.” Econometrica 74: 715-752. Earlier version published asNBER Technical Working Paper No. 299with supplements at Stock’s website.
Andrews, Martyn, Thorsten Schank, and Richard Upward. “Practical fixed effects estimation methods for the three-way error components model.”University of Nottingham Working Paper.
Angrist, Joshua D. and Alan B. Krueger. 2000. “Empirical Strategies in Labor Economics,” in A. Ashenfelter and D. Card eds. Handbook of Labor Economics, vol. 3. New York: Elsevier Science.
Angrist, Joshua D., Guido W. Imbens and D.B. Rubin. 1996. “Identification of Causal Effects Using Instrumental Variables.” Journal of the American Statistical Association 91, 444-472.
Angrist, Joshua D., Guido W. Imbens, and Alan B.Krueger. 1999. “Jackknife Instrumental Variables Estimation.” Journal of Applied Econometrics 14(1): 57-67.
Angrist, Joshua D. and J¨orn-Steffen Pischke. 2009. Mostly Harmless Econometrics Princeton: Princeton University Press. Austin Nichols Causal inference with observational data
Overview Matching and Reweighting Panel Methods Instrumental Variables (IV) Regression Discontinuity (RD) More
Sensitivity Testing
Connections across method types Conclusions
References
Arellano, Manuel. 1987. “Computing Robust Standard Errors for Within-Groups Estimators.” Oxford Bulletin of Economics and Statistics, 49: 431-34.
Arellano, M., and S. Bond. 1991. “Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations.” Review of Economic Studies 58: 277-297.
Athey, Susan and Guido W. Imbens. 2006. “Identification and Inference in Nonlinear Difference-in-Differences Models.” Econometrica 74 (2): 431-497.
Autor, David H., Lawrence F. Katz, Melissa S. Kearney. 2005. “Rising Wage Inequality: The Role of Composition and Prices.”NBER Working Paper 11628.
Baker, Michael, Dwayne Benjamin, and Shuchita Stanger. 1999. “The Highs and Lows of the Minimum Wage Effect: A Time-Series Cross-Section Study of the Canadian Law.”Journal of Labor Economics, 17(2): 318-350.
Basmann, R.L. 1960. “On Finite Sample Distributions of Generalized Classical Linear Identifiability Test Statistics.” Journal of the American Statistical Association 55(292): 650-59.
Baum, Christopher F. 2006. “Time-series filtering techniques in Stata.”Presented at NASUG5.
Baum, Christopher F., Mark E. Schaffer, Steven Stillman, and Vince Wiggins. 2006. “overid: Stata module to calculate tests of overidentifying restrictions after ivreg, ivreg2, ivprobit, ivtobit, reg3.”RePEcor findit overid.
Baum, Christopher F., Mark E. Schaffer, and Steven Stillman. 2007. “Enhanced routines for instrumental variables/GMM estimation and testing.” Unpublished working paper.
Baum, Christopher F., Mark E. Schaffer, and Steven Stillman. 2003. “Instrumental variables and GMM: Estimation and testing.” Stata Journal 3(1), 1-31. AlsoBoston College Department of Economics Working Paper No 545
Becker, Sascha O. and Andrea Ichino. 2002. “Estimation of average treatment effects based on propensity scores”, The Stata Journal 2(4): 358-377. Also findit pscore for updates (e.g. Stata Journal 5(3): 470).
Becker, Sascha O. and Marco Caliendo, 2007. “mhbounds - Sensitivity Analysis for Average Treatment Effects.”IZA Discussion Paper 2542.
Overview Matching and Reweighting Panel Methods Instrumental Variables (IV) Regression Discontinuity (RD) More
Sensitivity Testing
Connections across method types Conclusions
References
Blinder, Alan S. 1973. “Wage Discrimination: Reduced Form and Structural Estimates.”The Journal of Human Resources 8(4): 436-455. Bound, John, David A. Jaeger, and Regina Baker. 1993. “The Cure Can Be Worse than the Disease: A Cautionary Tale Regarding Instrumental Variables.”NBER Technical Working Paper No. 137.
Bound, John, David A. Jaeger, and Regina Baker. 1995. “Problems with Instrumental Variables Estimation when the Correlation Between the Instruments and the Endogenous Explanatory Variables is Weak.”Journal of the American Statistical Association, 90(430), 443-450. Cameron, A. Colin, Jonah B. Gelbach, and Douglas L. Miller. 2006. “Robust Inference with Multi-Way Clustering.”NBER Technical Working Paper T0327.
Cameron, A. Colin, Jonah B. Gelbach, and Douglas L. Miller. 2007. “Bootstrap-Based Improvements for Inference with Clustered Errors” FSU College of Law, Law and Economics Paper 07/002.
Chao, John C. and Norman R. Swanson. (2005). “Consistent Estimation with a Large Number of Weak Instruments.” Econometrica, 73(5), 1673-1692.Working paper version available online.
Cochran, William G. 1965. “The Planning of Observational Studies of Human Populations” and discussion.Journal of the Royal Statistical Society A128(2): 234-266.
Cochran, William G., and Donald B. Rubin. 1973. ”Controlling Bias in Observational Studies: A Review.” Sankhya 35: 417-46. Cook, Thomas D. 2008. “Waiting for Life to Arrive: A History of the Regression-Discontinuity Design in Psychology, Statistics and Economics.” Journal of Econometrics, 142(2).
Cragg, J.G. and S.G. Donald. (1993). “Testing Identifiability and Specification in Instrumental Variable Models,” Econometric Theory, 9, 222-240.
Dahl, Gordon and Lance Lochner. 2005. “The Impact of Family Income on Child Achievement.”NBER Working Paper 11279. Davidson, J. and MacKinnon. 2006. “The Case against JIVE.” Journal of Applied Econometrics 21: 827–833.
Devereux, Paul J. 2007. “Improved Errors-in-Variables Estimators for Grouped Data.” Journal of Business and Economic Statistics, 25(3): 278-287.
Overview Matching and Reweighting Panel Methods Instrumental Variables (IV) Regression Discontinuity (RD) More
Sensitivity Testing
Connections across method types Conclusions
References
DiNardo, John. 2002. “Propensity Score Reweighting and Changes in Wage Distributions”University of Michigan Working Paper. DiNardo, John and David Lee. 2002. “The Impact of Unionization on Establishment Closure: A Regression Discontinuity Analysis of Representation Elections.”NBER Working Paper 8993.
DiNardo, John, Nicole M. Fortin, and Thomas Lemieux. 1996. “Labor Market Institutions and the Distribution of Wages, 1973-1992: A Semiparametric Approach.”Econometrica, 64(5): 1001-1044.
DiNardo, John and Justin L. Tobias. 2001. “Nonparametric Density and Regression Estimation.”The Journal of Economic Perspectives, 15(4): 11-28.
DiPrete, Thomas A. and Markus Gangl. 2004. “Assessing Bias in the Estimation of Causal Effects: Rosenbaum Bounds on Matching Estimators and Instrumental Variables Estimation with Imperfect Instruments.” Sociological Methodology, 34: 271-310.Stata code to estimate Rosenbaum bounds
Dufour, Jean-Marie. 2003. “Identification, Weak Instruments, and Statistical Inference in Econometrics.” Canadian Journal of Economics, 36, 767-808.
Dufour, Jean-Marie and Mohamed Taamouti. 1999; revised 2003. “Projection-Based Statistical Inference in Linear Structural Models with Possibly Weak Instruments.”Manuscript, Department of Economics, University of Montreal.
Dufour, Jean-Marie and Mohamed Taamouti. 2007. “Further results on projection-based inference in IV regressions with weak, collinear or missing instruments.” Journal of Econometrics, 139(1): 133-153.
Eliason, Scott R. 2007. “Calculating Rosenbaum Bounds in Stata: Average Causal Effects of College v. HS Degrees on Wages Example.” University of Minnesota paper. See alsothis website.
Fisher, Ronald A. 1918. “The causes of human variability.” Eugenics Review 10: 213-220. Fisher, Ronald A. 1925. Statistical Methods for Research Workers. Edinburgh: Oliver and Boyd.
Fisher, Ronald A. 1926. “The arrangement of field experiments.” Journal of the Ministry of Agriculture of Great Britain, 33:503–513. Fr¨olich, Markus. 2007a. “Nonparametric IV Estimation of Local Average Treatment Effects with Covariates.” Journal of Econometrics 139 (1), 35-75. AlsoIZA Discussion Paper 588.
Overview Matching and Reweighting Panel Methods Instrumental Variables (IV) Regression Discontinuity (RD) More
Sensitivity Testing
Connections across method types Conclusions
References
Fr¨olich, Markus. 2007b. “Propensity score matching without conditional independence assumptionwith an application to the gender wage gap in the United Kingdom.” The Econometrics Journal 10(2), 359407.
Fr¨olich, Markus. 2004. “What is the Value of Knowing the Propensity Score for Estimating Average Treatment Effects?” Econometric Reviews 23(2): 167-174. AlsoIZA Discussion Paper 548.
Goldberger, Arthur S. 1972. “Selection bias in evaluating treatment effects: Some formal illustrations.” Discussion paper 123-72, Institute for Research on Poverty, University of Wisconsin, Madison.
Goldberger, Arthur S., and 0tis D. Duncan. 1973. Structural Equation Models in the Social Sciences. New York: Seminar Press. Gomulka, Joanna, and Nicholas Stern. 1990. “The Employment of Married Women in the United Kingdom 1970-83.” Economica 57: 171-199.
Hahn, Jinyong, Petra Todd, and Wilbert Van der Klaauw. 2001. “Identification and Estimation of Treatment Effects with a Regression-Discontinuity Design.” Econometrica 69(1): 201-209.
Hardin, James W., Henrik Schmiediche, and Raymond J. Carroll. 2003. “Instrumental variables, bootstrapping, and generalized linear models.” Stata Journal 3(4): 351-360. See alsohttp://www.stata.com/merror/.
Heckman, James J. and Edward Vytlacil. 1999. “Local Instrumental Variables and Latent Variable Models for Identifying and Bounding Treatment Effects.” Proceedings of the National Academy of Sciences of the United States of America 96:4730-34.
Heckman, James J. and Edward Vytlacil. 2000. “The Relationship between Treatment Parameters within a Latent Variable Framework.” Economics Letters 66:33-39.
Heckman, James J. and Edward Vytlacil. 2004. “Structural Equations, Treatment Effects and Econometric Policy Evaluation.” Econometrica 73(3): 669738.
Heckman, James J., Hidehiko Ichimura, and Petra E. Todd. 1997. “Matching as an Econometric Evaluation Estimator: Evidence from Evaluating a Job Training Programme.”Review of Economic Studies 64(4): 605-654.
Hirano, Keisuke, Guido W. Imbens, and Geert Ridder, 2003. “Efficient Estimation of Average Treatment Effects Using the Estimated Propensity Score.” Econometrica 71(4): 1161-1189. (See alsoNBER WP T0251.
Overview Matching and Reweighting Panel Methods Instrumental Variables (IV) Regression Discontinuity (RD) More
Sensitivity Testing
Connections across method types Conclusions
References
Holland, Paul W. 1986. “Statistics and causal inference.”Journal of the American Statistical Association 8(396): 945-960. Imbens, Guido and Karthik Kalyanaraman. 2009. “Optimal Bandwidth Choice for the Regression Discontinuity Estimator,NBER working paper 14726.
Imbens, Guido and Jeffrey Wooldridge. “What’s New in Econometrics.”NBER Summer Institute Course notes.
Imbens, Guido and Thomas Lemieux. 2008. “Regression Discontinuity Designs: A Guide to Practice.” Journal of Econometrics, 142(2). See alsoNBER Working Paper 13039.
Imbens, Guido W. 2004. “Nonparametric Estimation of Average Treatment Effects Under Exogeneity: A Review.” Review of Economics and Statistics 86(1): 4-29, 06. Earlier version available asNBER Technical Working Paper 0294.
Imbens, Guido W. 2006. “Matching methods for estimating treatment effects using Stata.”Presented at NASUG6.
Imbens, Guido W. and Joshua D. Angrist. 1994. “Identification and Estimation of Local Average Treatment Effects.” Econometrica 62(2): 467-75
Jann, Ben. 2007. “Univariate kernel density estimation.”Working paperandStata package.
Juhn, Chinhui, Kevin M. Murphy, Brooks Pierce. 1993. “Wage Inequality and the Rise in Returns to Skill.” Journal of Political Economy 101(3): 410-442.
Juhn, Chinhui, Kevin M. Murphy, Brooks Pierce. 1991. “Accounting for the Slowdown in Black-White Wage Convergence.” in Workers and Their Wages, ed. Marvin Kosters, Washington, DC: AEI Press.
Kaushal, Neeraj. 2007. “Do Food Stamps Cause Obesity? Evidence from Immigrant Experience.”NBER Working Paper No. 12849 K´ezdi, G´abor. 2004. “Robust Standard Error Estimation in Fixed-Effects Panel Models.”Hungarian Statistical Review Special(9): 96-116. Kinal, Terrence W. 1980. “The Existence of Moments of k-Class Estimators.”Econometrica, 48(1), 241-250.
Kleibergen, Frank. 2002. “Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression.”Econometrica, 70(5), 1781-1803.
Overview Matching and Reweighting Panel Methods Instrumental Variables (IV) Regression Discontinuity (RD) More
Sensitivity Testing
Connections across method types Conclusions
References
Kleibergen, Frank. 2007. “Generalizing weak instrument robust IV statistics towards multiple parameters, unrestricted covariance matrices and identification statistics.” Journal of Econometrics 139(1): 181-216.
Kleibergen, Frank and Richard Paap. 2006. “Generalized reduced rank tests using the singular value decomposition.” Journal of Econometrics 133(1): 97-126.Preprint.
Klein, Roger W. and Francis Vella. 2009. “A Semiparametric Model for Binary Response and Continuous Outcomes under Index Heteroscedasticity” Journal of Applied Econometrics, 24(5): 735–762.
Klein, Roger W. and Richard H. Spady. 1993. “An efficient semiparametric estimator for discrete choice models” Econometrica, 61: 387–421.http://www.jstor.org/stable/pdfplus/2951556.pdf
Lee, David S., Enrico Moretti, and Matthew J. Butler. 2004. “Do Voters Affect Or Elect Policies? Evidence From The U. S. House.” Quarterly Journal of Economics 119(3): 807-859.
Lee, David S. 2001. “The Electoral Advantage to Incumbency and Voters’ Valuation of Politicians’ Experience: A Regression Discontinuity Analysis of Elections to the U.S. House.”NBER Working Paper 8441. New version “Randomized Experiments from Non-random Selection in U.S. House Elections” forthcoming in Journal of Econometrics with aSupplemental Mathematical Appendix. Lee, David S. 2005. “Training, Wages, and Sample Selection: Estimating Sharp Bounds on Treatment Effects.”NBER Working Paper 11721witherrata. Previous version: Trimming for Bounds on Treatment Effects with Missing Outcomes, NBER Technical Working Paper 277.
Lee, David S. and David Card. 2008. “Regression Discontinuity Inference with Specification Error.” Journal of Econometrics, 142(2). See alsoNBER Technical Working Paper 322and previous version: Center for Labor Economics Working Paper 74.
Lee, Lung-Fei. 1992. “Amemiya’s Generalized Least Squares and Tests of Overidentification in Simultaneous Equation Models with Qualitative or Limited Dependent Variables.” Econometric Reviews 11(3): 319-328.
Leibbrandt, Murray, James Levinsohn, and Justin McCrary. 2005. “Incomes in South Africa Since the Fall of Apartheid.”NBER Working Paper 11384.
Leuven, Edwin and Barbara Sianesi. 2003. “psmatch2: Stata module to perform full Mahalanobis and propensity score matching, common support graphing, and covariate imbalance testing.”RePEcor findit psmatch2.
Machado, Jos and Jos Mata. 2005. “Counterfactual Decompositions of Changes in Wage Distributions Using Quantile Regression.” Journal of Applied Econometrics 20(4): 445-65.
Manski, Charles F. 1995. Identification Problems in the Social Sciences. Cambridge, MA: Harvard University Press. Austin Nichols Causal inference with observational data
Overview Matching and Reweighting Panel Methods Instrumental Variables (IV) Regression Discontinuity (RD) More
Sensitivity Testing
Connections across method types Conclusions
References
McCrary, Justin. 2007. “Manipulation of the Running Variable in the Regression Discontinuity Design: A Density Test.”NBER Technical Working Paper 334.
Mikusheva, Anna. 2005. “Robust Confidence Sets in the Presence of Weak Instruments.” Unpublished manuscript, Harvard University. Mikusheva, Anna and Brian P. Poi. 2006. “Tests and confidence sets with correct size in the simultaneous equations model with potentially weak instruments.”The Stata Journal 6(3): 335-347. Working paper.
Moreira, Marcelo J. 2001. “Tests With Correct Size When Instruments Can Be Arbitrarily Weak.”Working paper available online. Moreira, Marcelo J. 2003. “A Conditional Likelihood Ratio Test for Structural Models.” Econometrica, 71 (4), 1027-1048.Working paper version available on Moreira’s website.
Morgan, Stephen L. and David J. Harding. 2006. “Matching Estimators of Causal Effects: Prospects and Pitfalls in Theory and Practice.” Sociological Methods and Research 35(1)3-60.
Nannicini, Tommaso. 2006. “A simulation-based sensitivity analysis for matching estimators.”presented at NASUG5,working paper online.
Newey, W.K. 1987. “Efficient Estimation of Limited Dependent Variable Models with Endogeneous Explanatory Variables”. Journal of Econometrics 36: 231-250.
Neyman, Jerzy. 1923. “On the Application of Probability Theory to Agricultural Experiments: Essay on Principles, Section 9,” translated with an introduction by D. M. Dabrowska and T. P. Speed. 1990.Statistical Science 5(4): 465-472.
Neyman, Jerzy, K. Iwaskiewicz, and St. Kolodziejczyk. 1935. “Statistical problems in agricultural experimentation” (with discussion). Supplement to the Journal of the Royal Statistical Society 2(2): 107-180.
Nichols, Austin. 2006. “Weak Instruments: An Overview and New Techniques.”presented at NASUG5. Nichols, Austin. 2007. “Causal inference with observational data.”Stata Journal 7(4): 507-541. Nichols, Austin. 2008. “Erratum and discussion of propensity score reweighting.”Stata Journal 8(4): 532-539.
Overview Matching and Reweighting Panel Methods Instrumental Variables (IV) Regression Discontinuity (RD) More
Sensitivity Testing
Connections across method types Conclusions
References
Orr, Larry L., Howard S. Bloom, Stephen H. Bell, Fred Doolittle, Winston Lin, and George Cave. 1996. Does training for the disadvantaged work? Evidence from the national JTPA Study. Washington, DC: The Urban Institute Press.
Pearl, Judea. 2000. Causality: Models, Reasoning, and Inference. Cambridge: Cambridge University Press. Poi, Brian P. 2006. “Jackknife instrumental variables estimation in Stata.” Stata Journal 6(3): 364-376.
Rabe-Hesketh, Sophia, Anders Skrondal, and Andrew Pickles. 2002. “Reliable estimation of generalised linear mixed models using adaptive quadrature.” Stata Journal 2: 1-21.See also [http://gllamm.org].
Roodman, David M. 2006. “How to Do xtabond2: An Introduction to Difference and System GMM in Stata.”CGDev WP 103and presentation at NASUG5.
Rosenbaum, Paul R. 2002. Observational Studies. New York: Springer.
Rosenbaum, Paul R. and Donald B. Rubin. 1983. “The Central Role of the Propensity Score in Observational Studies for Causal Effects.” Biometrika 70(1): 41-55.
Rubin, Donald B. 1974. “Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies.” Journal of Educational Psychology 66: 688-701.
Sargan, J.D. 1958. “The Estimation of Economic Relationships Using Instrumental Variables.” Econometrica 26: 393-415. Schaffer, Mark E., and Stillman, Steven. 2006. “xtoverid: Stata module to calculate tests of overidentifying restrictions after xtivreg, xtivreg2, xthtaylor.” http://ideas.repec.org/c/boc/bocode/s456779.html
Schaffer, Mark E. 2007. “xtivreg2: Stata module to perform extended IV/2SLS, GMM and AC/HAC, LIML and k-class regression for panel data models.” http://ideas.repec.org/c/boc/bocode/s456501.html
Schechtman, Edna and Shlomo Yitzhaki. 2001. “The Gini Instrumental Variable, or ’The Double IV Estimator’.”at SSRN. Smith, Jeffrey A., and Petra E. Todd. 2005. “Does Matching Overcome Lalonde’s Critique of Nonexperimental Estimators?” Journal of Econometrics 125: 305-353.
Overview Matching and Reweighting Panel Methods Instrumental Variables (IV) Regression Discontinuity (RD) More
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References
Smith, Jeffrey A., and Petra E. Todd. 2001. “Reconciling Conflicting Evidence on the Performance of Propensity-Score Matching Methods.”The American Economic Review 91(2): 112-118.
Song, Kyungchul. 2009. “Efficient Estimation of Average Treatment Effects under Treatment-Based Sampling.”PIER Working Paper 09-011.
Staiger, Douglas and James H. Stock (1997). “Instrumental Variables Regression with Weak Instruments.”Econometrica, 65, 557-586. Stock, James H. and Motohiro Yogo (2005), “Testing for Weak Instruments in Linear IV Regression.” Ch. 5 in J.H. Stock and D.W.K. Andrews (eds), Identification and Inference for Econometric Models: Essays in Honor of Thomas J. Rothenberg, Cambridge University Press. Originally published 2001 asNBER Technical Working Paper No. 284; newer version (2004)available at Stock’s website. Stock, James H.; Jonathan H. Wright; and Motohiro Yogo. (2002). “A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments.” Journal of Business and Economic Statistics, 20, 518-529.Available from Yogo’s website. Stock, James H. and Mark W. Watson. 2006. “Heteroskedasticity-Robust Standard Errors for Fixed Effects Panel Data Regression.” NBER Technical Working Paper 323.
Thistlewaite, D. L., and Campbell, Donald T. (1960). “Regression-Discontinuity Analysis: An Alternative to the Ex-Post Facto Experiment.” Journal of Educational Psychology 51: 309-317.
Tukey, J. W. 1954. “Causation, regression and path analysis,” in Statistics and Mathematics in Biology. Ames: Iowa State College Press. Wold, Herman. 1956. “Causal inference from observational data: A review of ends and means.”Journal of the Royal Statistical Society. A119(1): 28-61.
Wooldridge, J.M. 2002. Econometric Analysis of Cross Section and Panel Data. Cambridge, MA: MIT Press.Available from Stata.com. Yun, Myeong-Su. 2004. “Decomposing Differences in the First Moment.” Economics Letters 82(2): 275-280. See alsoIZA Discussion Paper 877.
Yun, Myeong-Su. 2005a. “A Simple Solution to the Identification Problem in Detailed Wage Decompositions.” Economic Inquiry 43(4): 766-772. See alsoIZA Discussion Paper 836.