Identification and estimation of counterfactual outcomes

As counterfactual outcomes are important objects of inference, one may be inter­ested in the identification and estimation of counterfactual outcomes. The pos­sible identification of counterfactual outcomes follows from model structures and observed decisions and outcomes (Heckman, 1990; Lee, 1995). Observed outcomes and choice probabilities provide sample information. Latent variable models provide prior structural restrictions.

Professor C. Manski in a series of articles put aside the latent-variable model perspective to go back to probabilistic basics. His results are summarized in Manski (1994). The main findings provide informative bounds on some counter­factual outcomes. Without latent variable modeling, if one is not satisfied with just bounds, the identification and evaluation of a counterfactual outcome would require extra prior restrictions on some other counterfactual outcomes. Statisti­cians approach the selected sample as a mixture problem. A widely-used method of evaluation in statistics is the method of matching (Rubin, 1987). Heckman, Ichimura, and Todd (1998) show that the fundamental identification condition (or assumption) for the matching method is a condition imposed on a specific counterfactual outcome. Corresponding to the two-sector model in (18.2)-(18.3), this identification condition requires that o2e = 0 (Heckman et al., 1998, pp. 268-9). Professor J. Heckman and his associates in a series of forthcoming papers contrast the econometrics and statistical approaches on program evaluation. Some preliminary review can be found in M. J. Lee (1997).

Note

* The author acknowledges research support from the Research Grants Council of Hong Kong under grant HKUST595/96H for his research.

References

Ahn, H., and J. L. Powell (1993). Semiparametric estimation of censored selection models with a nonparametric selection mechanism. Journal of Econometrics 58, 3-29.

Ai, C. (1997). A semiparametric maximum likelihood estimator. Econometrica 65, 933-63.

Amemiya, T. (1973). Regression analysis when the dependent variable is truncated normal. Econometrica 41, 997-1016.

Amemiya, T. (1974). Multivariate regression and simultaneous equation models when the dependent variables are truncated normal. Econometrica 42, 999-1012.

Amemiya, T. (1979). The estimation of a simultaneous equation tobit model. International Economic Review 20, 169-81.

Amemiya, T. (1983). A comparison of the Amemiya GLS and the Lee-Maddala-Trost G2SLS in a simultaneous equations tobit model. Journal of Econometrics 23, 295-300.

Amemiya, T. (1984). Tobit models: a survey. Journal of Econometrics 24, 3-61.

Andrews, D. W.K. (1991). Asymptotic normality of series estimators for nonparametric and semiparametric regression models. Econometrica 59, 307-45.

Andrews, D. W.K., and M. M.A. Schafgans (1998). Semiparametric estimation of the inter­cept of a sample selection model. Review of Economic Studies 65, 497-517.

Bjorklund, A., and R. Moffitt (1987). Estimation of wage gains and welfare gains in self­selection models. Review of Economics and Statistics 69, 42-9.

Chamberlain, G. (1986). Asymptotic efficiency in semiparametric models with censoring. Journal of Econometrics 32, 189-218.

Chamberlain, G. (1992). Efficiency bounds for semiparametric regression. Econometrica 60, 567-96.

Chen, S. (1997). Semiparametric estimation of the Type-3 tobit model. Journal of Econometrics 80, 1-34.

Chen, S., and L. F. Lee (1998). Efficient semiparametric scoring estimation of sample selec­tion models. Econometric Theory 14, 423-62.

Cosslett, S. R. (1991). Semiparametric estimation of regression model with sample selectiv­ity. In W. A. Barnett, J. Powell, and G. Tauchen (eds.) Nonparametric and Semiparametric Methods in Econometrics and Statistics. pp. 175-97. Cambridge: Cambridge University Press.

Dubin, J., and D. McFadden (1984). An econometric analysis of residential electric appli­ance holdings and consumption. Econometrica 52, 345-62.

Gallant, A. R., and D. W. Nychka (1987). Semiparametric maximum likelihood estimation.

Econometrica 55, 363-93.

Goldberger, A. S. (1983). Abnormal selection bias. In S. Karlin, T. Amemiya, and L. A. Goodman (eds.) Studies in Econometrics, Time Series and Multivariate Statistics. New York: Wiley.

Griliches, Z., B. H. Hall, and J. A. Hausman (1978). Missing data and self-selection in large panels. Annals de l’INSEE 30 -31, 137-76.

Gronau, R. (1974). Wage comparisons: a selectivity bias. Journal of Political Economy 82, 119-43.

Hay, J., and R. J. Olsen (1984). Let them eat cake: a note on comparing alternative models of the demand for medical care. Journal of Business and Economic Statistics 2, 279-82.

Heckman, J. J. (1974). Shadow prices, market wages, and labor supply. Econometrica 42, 679-94.

Heckman, J. J. (1979). Sample selection bias as specification error. Econometrica 47, 153-61.

Heckman, J. J. (1990). Varieties of selection bias. American Economic Association Papers and Proceedings 313-18.

Heckman, J. J., and R. Robb (1985). Alternative methods for evaluating the impact of inter­ventions. In J. Heckman and B. Singer (eds.) Longitudinal Analysis of Labor Market Data. Cambridge: Cambridge University Press.

Heckman, J. J., and B. E. Honore (1990). The empirical content of the Roy model. Econometrica 58, 1121-49.

Heckman, J. J., H. Ichimura, and P. Todd (1998). Matching as an econometric evaluation estimator. Review of Economic Studies 65, 261-94.

Honore, B. E., E. Kyriazidou, and C. Udry (1997). Estimation of type 3 tobit models using symmetric trimming and pairwise comparisons. Journal of Econometrics 76, 107-28.

Ichimura, H. (1993). Semiparametric least squares estimation of single index models. Journal of Econometrics 58, 71-120.

Ichimura, H., and L. F. Lee (1991). Semiparametric estimation of multiple index models: single equation estimation. In W. A. Barnett, J. Powell, and G. Tauchen (eds.) Nonparametric and Semiparametric Methods in Econometrics and Statistics, ch. 1. New York: Cambridge University Press.

Lee, L. F. (1978). Unionism and wage rates: a simultaneous equation model with qual­itative and limited dependent variables. International Economic Review 19, 415-33.

Lee, L. F. (1981). Simultaneous equations models with discrete endogenous variables. In C. F. Manski and D. McFadden (eds.) Structural Analysis of Discrete Data and Econometric Applications, ch. 9. Cambridge, MA: MIT Press.

Lee, L. F. (1982). Some approaches to the correction of selectivity bias. Review of Economic Studies 49, 355-72.

Lee, L. F. (1983). Generalized econometrics models with selectivity. Econometrica 51, 507-12.

Lee, L. F. (1984). Tests for the bivariate normal distribution in econometric models with selectivity. Econometrica 52, 843-63.

Lee, L. F. (1992a). Amemiya’s generalized least squares and tests of overidentification in simultaneous equation models with qualitative or limited dependent variables. Eco­nometric Reviews 11, 319-28.

Lee, L. F. (1992b). On efficiency of methods of simulated moments and maximum simu­lated likelihood estimation of discrete response models. Econometric Theory 8, 518-52.

Lee, L. F. (1994a). Semiparametric two-stage estimation of sample selection models subject to tobit-type selection rules. Journal of Econometrics 61, 305-44.

Lee, L. F. (1994b). Semiparametric instrumental variable estimation of simultaneous equa­tion sample selection models. Journal of Econometrics 63, 341-88.

Lee, L. F. (1995). The computational of opportunity costs in polychotomous choice models with selectivity. Review of Economics and Statistics 77, 423-35.

Lee, L. F. (1996). Simulation estimation of sample selection models. Working Paper no. 96/ 97-4, Department of Economics, Hong Kong University of Science and Technology.

Lee, L. F. (1998). Semiparametric estimation of simultaneous-equation microeconometric models with index restrictions. Japanese Economic Review 49, 343-80.

Lee, L. F., and A. Chesher (1986). Specification testing when some test statistics are identi­cally zero. Journal of Econometrics 31, 121-49.

Lee, L. F., G. S. Maddala, and R. P. Trost (1980). Asymptotic covariance matrices of two – stage probit and two-stage tobit methods for simultaneous equations models with selectivity. Econometrica 48, 491-503.

Lee, M.-J. (1997). Econometric methods for sample selection and treatment effect models. Manuscript, Institute of Policy and Planning Science, University of Tsukuba, Japan.

Leung, S. F., and S. Yu (1996). On the choice between sample selection and two-part models. Journal of Econometrics 72, 197-229.

Manning, W. G., N. Duan, and W. H. Rogers (1987). Monte Carlo evidence on the choice between sample selection and two-part models. Journal of Econometrics 35, 59-82.

Maddala, G. S. (1983). Limited Dependent and Qualitative Variables in Econometrics. Cam­bridge: Cambridge University Press.

Maddala, G. S. (1985). A survey of the literature on selectivity bias as it pertains to health care markets. Advances in Health Economics and Health Services Research 6, 3-18.

Manski, C. (1975). Maximum score estimation of the stochastic utility model of choice. Journal of Econometrics 3, 205-28.

Manski, C. (1994). The selection problem. In C. Sims (ed.) Advances in Econometrics, ch. 4, pp. 143-70. Cambridge: Cambridge University Press.

Melino, A. (1982). Testing for selection bias. Review of Economic Studies 49, 151-3.

McFadden, D. (1973). Conditional logit analysis of qualitative choice behavior. In P. Zarembka (ed.) Frontiers in Econometrics. New York: Academic Press.

McFadden, D. (1978). Modeling the choice of residential location. In A. Karlquist et al. (eds.) Spatial Interaction Theory and Residential Location. Amsterdam: North-Holland.

McFadden, D. (1989). A method of simulated moments for estimation of discrete response models without numerical integration, Econometrica 57, 995-1026.

Mroz, T. A. (1987). The sensitivity of empirical models with sample-selection biases. Econometrica 55, 765-99.

Nawata, K. (1993). A note on the estimation with sample-selection bias. Economics Letters 42, 15-24.

Nawata, K., and N. Nagase (1996). Estimation of sample selection bias models. Econometric Reviews 15, 387-400.

Nelson, F. D. (1977). Censored regression models with unobserved, stochastic censoring thresholds. Journal of Econometrics 6, 309-27.

Nelson, F. (1984). Efficiency of the two step estimator for models with endogenous sample selection. Journal of Econometrics 24, 181-96.

Newey, W. K. (1987). Efficient estimation of limited dependent variable models with endogenous explanatory variables. Journal of Econometrics 36, 231-50.

Newey, W. K. (1988). Two step estimation of sample selection models. Manuscript, Depart­ment of Economics, Princeton University.

Newey, W. K. (1990). Semiparametric efficiency bounds. Journal of Applied Econometrics 5, 99-135.

Newey, W. K., J. L. Powell, and J. R. Walker (1990). Semiparametric estimation of selection models: some empirical results. AEA Papers and Proceedings 80, 324-8.

Olsen, R. (1980). A least squares correction for selectivity bias. Econometrica 48, 1815-20.

Olsen, R. (1982). Distributional tests for selectivity bias and a more robust likelihood estimator. International Economic Review 23, 223-40.

Pagan, A., and F. Vella (1989). Diagnostic tests for models based on individual data: a survey. Journal of Applied Econometrics 4, S29-S59.

Pakes, A., and D. Pollard (1989). Simulation and the asymptotic of optimization esti­mators. Econometrica 57, 1027-57.

Powell, J. L. (1986). Symmetrically trimmed least squares estimation for tobit models. Econometrica 54, 1435-60.

Powell, J. L. (1987). Semiparametric estimation of bivariate latent variable models. Dis­cussion paper no. 8704, Social Systems Research Institute, University of Wisconsin, Madison, Wl.

Powell, J. L. (1994). Estimation of semiparametric models. In R. F. Engle and D. L. McFadden (eds.) Handbook of Econometrics, Volume 4, ch. 14. Amsterdam: North-Holland.

Robinson, P. M. (1988). Root-n-consistent semiparametric regression. Econometrica 56, 931-54.

Roy, A. (1951). Some thoughts on the distribution of earnings. Oxford Economic Papers 3, 135-46.

Rubin, D. (1987). Multiple Imputation for Nonresponse in Surveys. New York: John Wiley.

Sattinger, M. (1978). Comparative advantage in individuals. Review of Economics and Statis­tics 60, 259-67.

Schmertmann, C. P. (1994). Selectivity bias correction methods in polychotomous sample selection models. Journal of Econometrics 60, 101-32.

Stein, C. (1956). Efficient nonparametric testing and estimation. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 1 pp. 187-95. Berkeley: University of California Press.

Vella, F. (1998). Estimating models with sample selection bias: a survey. Journal of Human Resources 33, 127-69.

Wales, T. J., and A. D. Woodland (1980). Sample selectivity and the estimation of labour supply functions. International Economic Review 21, 437-68.

Willis, R. J., and S. Rosen (1979). Education and self-selection. Journal of Political Economy 87, S7-S36.

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