Robins and Rotnitzky (2004) has established that U is an influence function for γ in the semiparametric model where assumptions (1)-(4) hold. This, in turn, implies that

E [∂U(ψ0, η, α, ω)

∂(η, α, ω) ∣(η,α,ω)=(η0,α0,ω0)] = 0.

Hence, under the null hypothesis (H0) i.e. that of no modelling error for the nuisance

models, ∑ i ̂ Ui≈ ∑ i {Ui+ E [ ∂U(ψ) ∂ψ ∣ψ=ψ0]E [S eff ψS effT ψ ] −1 Seff ψ,i} , where S eff

ψ is the efficient score

of ψ under our parametric model. By a property of influence functions,

E [∂U(ψ) ∂ψ ∣ψ=ψ0] = −E [US effT ψ ] . Thus, n−1/2∑ i ̂ U ≈ n−1/2∑ i {Ui− E [US effT ψ ] E [S eff ψ S effT ψ ] −1 Seff

ψ,i} and we have

E[{Ui− E [US effT ψ ] E [S eff ψS effT ψ ] −1 Seff ψ,i} 2 ] ≤ E {U2 i} (A.1)

since E [USeffT ψ ] E [S eff ψS effT ψ ] −1 Seff

ψ,i is the orthogonal projection of U onto the span of S

eff

ψ,i,

proving the first result. The second result follows from a standard application of Slutsky’s theorem and the central limit theorem.

Acknowledgments: The authors acknowledge research support from the National In- stitutes of Health (NIH). R. A. Matsouaka was supported by NIH grants 1R01MD006064, 1R21HD066312 (Theresa Osypuk, PI) and 1R21ES019712. E. Tchetgen Tchetgen was sup- ported by NIH grants 1R21ES019712, R01AI104459 and R01HL080644. We are also grateful to Nicole Schmidt for helpful comments and suggestions regarding the MTO data.

References

Abadie, A. (2003). Semiparametric instrumental variable estimation of treatment response models. Journal of Econometrics, 113(2):231–263.

Abadie, A., Angrist, J., and Imbens, G. (2002). Instrumental variables estimates of the effect of subsidized training on the quantiles of trainee earnings. Econometrica, 70(1):91–117.

Angrist, J. D. (2006). Instrumental variables methods in experimental criminological re- search: what, why and how. Journal of Experimental Criminology, 2(1):23–44.

Angrist, J. D., Imbens, G. W., and Rubin, D. B. (1996). Identification of causal effects using instrumental variables. Journal of the American statistical Association, 91(434):444–455.

Baiocchi, M., Cheng, J., and Small, D. S. (2014). Instrumental variable methods for causal inference. Statistics in medicine, 33(13):2297–2340.

Barnard, J., Frangakis, C. E., Hill, J. L., and Rubin, D. B. (2003). Principal stratification approach to broken randomized experiments: A case study of school choice vouchers in new york city. Journal of the American Statistical Association, 98(462):299–323.

Card, D. (1995). Using geographic variation in college proximity to estimate the return to schooling. In Christofides, L. N., Grant, E. K., and Swidinsky, R., editors, Aspects of Labor Market Behaviour: Essays in Honour of John Vanderkamp, pages 201–222. Toronto: University of Toronto Press.

Carneiro, P., Heckman, J. J., and Vytlacil, E. J. (2006). Estimating marginal and aver- age returns to education. Unpublished manuscript. University of Chicago, Department of Economics.

Clarke, P. S., Palmer, T. M., and Windmeijer, F. (2011). Estimating structural mean models with multiple instrumental variables using the generalised method of moments. Technical report, cemmap working paper.

Clarke, P. S. and Windmeijer, F. (2010). Identification of causal effects on binary outcomes using structural mean models. Biostatistics, 11(4):756–770.

Clarke, P. S. and Windmeijer, F. (2012). Instrumental variable estimators for binary out- comes. Journal of the American Statistical Association, 107(500):1638–1652.

Didelez, V., Meng, S., and Sheehan, N. A. (2010). Assumptions of iv methods for observa- tional epidemiology. Statistical Science, 25(1):22–40.

Ducharme, G. R. and LePage, Y. (1986). Testing collapsibility in contingency tables. Journal of the Royal Statistical Society. Series B (Methodological), pages 197–205.

Frangakis, C. E., Brookmeyer, R. S., Varadhan, R., Safaeian, M., Vlahov, D., and Strathdee, S. A. (2004). Methodology for evaluating a partially controlled longitudinal treatment using principal stratification, with application to a needle exchange program. Journal of the American Statistical Association, 99(465):239–249.

Glymour, M. M., Tchetgen, E. J. T., and Robins, J. M. (2012). Response to letters on “Credible mendelian randomization studies: approaches for evaluating the instrumental variable assumptions”. American journal of epidemiology, 176(5):458–459.

Goering, J., Carnevale, K., and Teodoro, M. (1996). Expanding housing choices for HUD- assisted families. US Dept. of Housing and Urban Development, Washington, DC.

Goering, J. M., Kraft, J., Feins, J., McInnis, D., Holin, M. J., and Elhassan, H. (1999). Moving to Opportunity for fair housing demonstration program: Current status and initial findings. US Department of Housing and Urban Development, Office of Policy Develop- ment and Research, Washington, DC.

Goldberger, A. S. (1972). Structural equation methods in the social sciences. Econometrica: Journal of the Econometric Society, pages 979–1001.

Greenland, S., Robins, J. M., and Pearl, J. (1999). Confounding and collapsibility in causal inference. Statistical Science, pages 29–46.

Guo, Z., Cheng, J., Lorch, S. A., and Small, D. S. (2014). Using an instrumental variable to test for unmeasured confounding. Statistics in medicine.

Heckman, J. (1997). Instrumental variables: A study of implicit behavioral assumptions used in making program evaluations. Journal of Human Resources, pages 441–462.

Hern´an, M. A. and Robins, J. M. (2006). Instruments for causal inference: an epidemiologist’s dream? Epidemiology, 17(4):360–372.

Hirano, K., Imbens, G. W., Rubin, D. B., and Zhou, X.-H. (2000). Assessing the effect of an influenza vaccine in an encouragement design. Biostatistics, 1(1):69–88.

Imbens, G. W. and Angrist, J. D. (1994). Identification and estimation of local average treatment effects. Econometrica, 62(2):467–475.

Joffe, M. M. and Brensinger, C. (2003). Weighting in instrumental variables and g-estimation. Statistics in medicine, 22(8):1285–1303.

Little, R. J. and Yau, L. H. (1998). Statistical techniques for analyzing data from prevention trials: Treatment of no-shows using rubin’s causal model. Psychological Methods, 3(2):147.

Martens, E. P., Pestman, W. R., de Boer, A., Belitser, S. V., and Klungel, O. H. (2006). Instrumental variables: application and limitations. Epidemiology, 17(3):260–267.

Meng, X.-L. (1994). Multiple-imputation inferences with uncongenial sources of input. Sta- tistical Science, pages 538–558.

Murray, M. P. (2006). Avoiding invalid instruments and coping with weak instruments. The Journal of Economic Perspectives, 20(4):111–132.

Neyman, J. (1923). Sur les applications de la th´eorie des probabilit´es aux experiences agri- coles: Essai des principes. Roczniki Nauk Rolniczych, 10:1–51.

Orr, L., Feins, J., Jacob, R., Beecroft, E., Sanbonmatsu, L., Katz, L. F., Liebman, J. B., and Kling, J. R. (2003). Moving to opportunity: Interim impacts evaluation. US Dept of Housing and Urban Development, Washington, DC.

Osypuk, T. L., Schmidt, N. M., Bates, L. M., Tchetgen-Tchetgen, E. J., Earls, F. J., and Glymour, M. M. (2012a). Gender and crime victimization modify neighborhood effects on adolescent mental health. Pediatrics, 130(3):472–481.

Osypuk, T. L., Tchetgen, E. J. T., Acevedo-Garcia, D., Earls, F. J., Lincoln, A., Schmidt, N. M., and Glymour, M. M. (2012b). Differential mental health effects of neighborhood relocation among youth in vulnerable familiesresults from a randomized trialmental health effects of neighborhood relocation. Archives of general psychiatry, 69(12):1284–1294.

Richardson, T. S., Evans, R. J., and Robins, J. M. (2011). Transparent parameterizations of models for potential outcomes. Bayesian Statistics, 9:569–610.

Richardson, T. S. and Robins, J. M. (2010). Analysis of the binary instrumental variable model. Heuristics, Probability and Causality: A Tribute to Judea Pearl, pages 415–444.

exposure periodapplication to control of the healthy worker survivor effect. Mathematical Modelling, 7(9):1393–1512.

Robins, J. and Rotnitzky, A. (2004). Estimation of treatment effects in randomised trials with non-compliance and a dichotomous outcome using structural mean models. Biometrika, 91(4):763–783.

Robins, J. M. (1989). The analysis of randomized and non-randomized aids treatment trials using a new approach to causal inference in longitudinal studies. Health service research methodology: a focus on AIDS, 113:159.

Robins, J. M. (1994). Correcting for non-compliance in randomized trials using structural nested mean models. Communications in Statistics-Theory and methods, 23(8):2379–2412.

Robins, J. M., Rotnitzky, A., and Scharfstein, D. O. (2000). Sensitivity analysis for selection bias and unmeasured confounding in missing data and causal inference models. In Statis- tical models in epidemiology, the environment, and clinical trials, pages 1–94. Springer.

Rubin, D. B. (1978). Bayesian inference for causal effects: The role of randomization. The Annals of Statistics, pages 34–58.

Rubin, D. B. (1980). Randomization analysis of experimental data: The fisher randomization test comment. Journal of the American Statistical Association, 75(371):591–593.

Rubin, D. B. (1986). Statistics and causal inference: Comment: Which ifs have causal answers. Journal of the American Statistical Association, 81(396):961–962.

Sofer, T., Cornelis, M. C., Kraft, P., and Tchetgen, E. J. T. (2014). Control function assisted ipw estimation with a secondary outcome in case-control studies. arXiv preprint arXiv:1407.4413.

Tan, Z. (2006b). Regression and weighting methods for causal inference using instrumental variables. Journal of the American Statistical Association, 101(476):1607–1618.

Tan, Z. (2010). Marginal and nested structural models using instrumental variables. Journal of the American Statistical Association, 105(489):157–169.

Tchetgen Tchetgen, E. J. (2014). A general regression framework for a secondary outcome in case–control studies. Biostatistics, 15(1):117–128.

Tchetgen Tchetgen, E. J., Robins, J. M., and Rotnitzky, A. (2010). On doubly robust estimation in a semiparametric odds ratio model. Biometrika, 97(1):171–180.

Tchetgen Tchetgen, E. J. and Vansteelandt, S. (2013). Alternative identification and infer- ence of the effect of treatment on the treated with an instrumental variable. (submitted).

Tsiatis, A. A. (2006). Semiparametric theory and missing data. Springer.

Turner, M. A. (1998). Moving out of poverty: Expanding mobility and choice through tenant-based housing assistance. Housing policy debate, 9(2):373–394.

Vansteelandt, S., Bowden, J., Babanezhad, M., and Goetghebeur, E. (2011). On instrumental variables estimation of causal odds ratios. Statistical Science, 26(3):403–422.

Vansteelandt, S. and Goetghebeur, E. (2003). Causal inference with generalized structural mean models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 65(4):817–835.

Vansteelandt, S. and Goetghebeur, E. (2005). Sense and sensitivity when correcting for observed exposures in randomized clinical trials. Statistics in medicine, 24(2):191–210.

Wright, S. (1928). Appendix to The Tariff on Animal and Vegetable Oils, by P.G. Wright. New York: MacMillan.

Wright, S. (1934). “The method of path coefficients”. Annals of Mathematical Statistics, 5:161–215.

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