Nonlinear and Nonparametric Regression and Instrumental Variables

Raymond J Carroll Raymond J. Carroll is Distinguished Professor, Department of Statistics and Faculties of Nutrition and Toxicology, Texas A&M University, College Station, TX 77843–3143 . David Ruppert is Andrew Schultz, Jr. Professor of Engineering, School of Operations Research & Industrial Engineering, Cornell University, Ithaca, NY 14853-3801 . Ciprian M. Crainiceanu is Assistant Professor, Department of Biostatistics, Johns Hopkins University, Baltimore, MD21205 . Tor D. Tosteson is Associate Professor, Community and Family Medicine, Dartmouth Medical School, Hanover, NH 03755 . Margaret R. Karagas is Associate Professor, Community and Family Medicine, Dartmouth Medical School, Hanover, NH 03755 . Carroll's research was supported by National Cancer Institute grant CA57030 and by the Texas A&M Center for Environmental and Rural Health via National Institute of Environmental Health Sciences grant P30-ES09106. Ruppert, Tosteson, and Karagas were supported by National Cancer Institute grant CA50597, and Tosteson and Karagas were supported by the National Cancer institute grants CA50597 and CA57494 and National Institute of Environmental Health Sciences grant ES07373. The authors thank Tailen Hsing for showing us Durrent's (1996) counterexample used in the Appendix.

David Ruppert Raymond J. Carroll is Distinguished Professor, Department of Statistics and Faculties of Nutrition and Toxicology, Texas A&M University, College Station, TX 77843–3143 . David Ruppert is Andrew Schultz, Jr. Professor of Engineering, School of Operations Research & Industrial Engineering, Cornell University, Ithaca, NY 14853-3801 . Ciprian M. Crainiceanu is Assistant Professor, Department of Biostatistics, Johns Hopkins University, Baltimore, MD21205 . Tor D. Tosteson is Associate Professor, Community and Family Medicine, Dartmouth Medical School, Hanover, NH 03755 . Margaret R. Karagas is Associate Professor, Community and Family Medicine, Dartmouth Medical School, Hanover, NH 03755 . Carroll's research was supported by National Cancer Institute grant CA57030 and by the Texas A&M Center for Environmental and Rural Health via National Institute of Environmental Health Sciences grant P30-ES09106. Ruppert, Tosteson, and Karagas were supported by National Cancer Institute grant CA50597, and Tosteson and Karagas were supported by the National Cancer institute grants CA50597 and CA57494 and National Institute of Environmental Health Sciences grant ES07373. The authors thank Tailen Hsing for showing us Durrent's (1996) counterexample used in the Appendix.

Ciprian M Crainiceanu Raymond J. Carroll is Distinguished Professor, Department of Statistics and Faculties of Nutrition and Toxicology, Texas A&M University, College Station, TX 77843–3143 . David Ruppert is Andrew Schultz, Jr. Professor of Engineering, School of Operations Research & Industrial Engineering, Cornell University, Ithaca, NY 14853-3801 . Ciprian M. Crainiceanu is Assistant Professor, Department of Biostatistics, Johns Hopkins University, Baltimore, MD21205 . Tor D. Tosteson is Associate Professor, Community and Family Medicine, Dartmouth Medical School, Hanover, NH 03755 . Margaret R. Karagas is Associate Professor, Community and Family Medicine, Dartmouth Medical School, Hanover, NH 03755 . Carroll's research was supported by National Cancer Institute grant CA57030 and by the Texas A&M Center for Environmental and Rural Health via National Institute of Environmental Health Sciences grant P30-ES09106. Ruppert, Tosteson, and Karagas were supported by National Cancer Institute grant CA50597, and Tosteson and Karagas were supported by the National Cancer institute grants CA50597 and CA57494 and National Institute of Environmental Health Sciences grant ES07373. The authors thank Tailen Hsing for showing us Durrent's (1996) counterexample used in the Appendix.

Tor D Tosteson Raymond J. Carroll is Distinguished Professor, Department of Statistics and Faculties of Nutrition and Toxicology, Texas A&M University, College Station, TX 77843–3143 . David Ruppert is Andrew Schultz, Jr. Professor of Engineering, School of Operations Research & Industrial Engineering, Cornell University, Ithaca, NY 14853-3801 . Ciprian M. Crainiceanu is Assistant Professor, Department of Biostatistics, Johns Hopkins University, Baltimore, MD21205 . Tor D. Tosteson is Associate Professor, Community and Family Medicine, Dartmouth Medical School, Hanover, NH 03755 . Margaret R. Karagas is Associate Professor, Community and Family Medicine, Dartmouth Medical School, Hanover, NH 03755 . Carroll's research was supported by National Cancer Institute grant CA57030 and by the Texas A&M Center for Environmental and Rural Health via National Institute of Environmental Health Sciences grant P30-ES09106. Ruppert, Tosteson, and Karagas were supported by National Cancer Institute grant CA50597, and Tosteson and Karagas were supported by the National Cancer institute grants CA50597 and CA57494 and National Institute of Environmental Health Sciences grant ES07373. The authors thank Tailen Hsing for showing us Durrent's (1996) counterexample used in the Appendix.

Margaret R Karagas Raymond J. Carroll is Distinguished Professor, Department of Statistics and Faculties of Nutrition and Toxicology, Texas A&M University, College Station, TX 77843–3143 . David Ruppert is Andrew Schultz, Jr. Professor of Engineering, School of Operations Research & Industrial Engineering, Cornell University, Ithaca, NY 14853-3801 . Ciprian M. Crainiceanu is Assistant Professor, Department of Biostatistics, Johns Hopkins University, Baltimore, MD21205 . Tor D. Tosteson is Associate Professor, Community and Family Medicine, Dartmouth Medical School, Hanover, NH 03755 . Margaret R. Karagas is Associate Professor, Community and Family Medicine, Dartmouth Medical School, Hanover, NH 03755 . Carroll's research was supported by National Cancer Institute grant CA57030 and by the Texas A&M Center for Environmental and Rural Health via National Institute of Environmental Health Sciences grant P30-ES09106. Ruppert, Tosteson, and Karagas were supported by National Cancer Institute grant CA50597, and Tosteson and Karagas were supported by the National Cancer institute grants CA50597 and CA57494 and National Institute of Environmental Health Sciences grant ES07373. The authors thank Tailen Hsing for showing us Durrent's (1996) counterexample used in the Appendix.

Abstract

We consider regression when the predictor is measured with error and an instrumental variable (IV) is available. The regression function can be modeled linearly, nonlinearly, or nonparametrically. Our major new result shows that the regression function and all parameters in the measurement error model are identified under relatively weak conditions, much weaker than previously known to imply identifiability. In addition, we exploit a characterization of the IV estimator as a classical “correction for attenuation” method based on a particular estimate of the variance of the measurement error. This estimate of the measurement error variance allows us to construct functional nonparametric regression estimators making no assumptions about the distribution of the unobserved predictor and structural estimators that use parametric assumptions about this distribution. The functional estimators uses simulation extrapolation or deconvolution kernels and the structural method uses Bayesian Markov chain Monte Carlo. The Bayesian estimator is found to significantly outperform the functional approach.

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Nonlinear and Nonparametric Regression and Instrumental Variables

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