Bounding parameter estimates with nonclassical measurement error

Dan A. Black, Mark C. Berger, Frank A. Scott

Research output: Contribution to journalArticlepeer-review

107 Scopus citations


The bias introduced by errors in the measurement of independent variables has increasingly been a topic of interest among researchers estimating economic parameters. However, studies typically use the assumption of classical measurement error; that is, the variable of interest and its measurement error are uncorrelated, and the expected value of the mismeasured variable is equal to the expected value of the true measure. These assumptions often arise from convenience rather than conviction. When a variable is bounded, it is likely that the measurement error and the true value of the variable are negatively correlated. We consider the case of a noisily measured variable with a negative covariance between the measurement error and the true value of the variable. We show that, asymptotically, the parameter in a univariate regression is bounded between the ordinary least squares (OLS) estimator and an instrumental variables (IV) estimator. Further, we demonstrate that the OLS bound can be improved in the case where there are two noisy reports on the variable of interest. In the case of continuous variables, this lower-bound estimate is a consistent estimate of the parameter. In the case of binary or discrete noisily measured variables, we also identify point estimates using a method-of-moments framework. We then extend our bounding results to simple multivariate models with measurement error. We provide empirical applications of our analytical results using employer and employee reports on health insurance coverage and wage growth, and reports of identical twins on the level of schooling and wages. Using OLS, health insurance coverage is associated with a reduction in wage growth of 6.5–7.4%, whereas IV estimates suggest a 11.2–11.8% reduction associated with health insurance coverage. We are able to improve the lower bound estimate to 8.2% using our bounding strategy and obtain a point estimate of 8.8% using the method-of-moments framework. The estimates using the data for identical twins, though not correcting for problems such as endogenous determination of the level of schooling, do illustrate the potential usefulness of correcting for measurement error as a complement to other approaches. Using the multiple reports on the level of schooling and the our proposed estimators, we are able to tighten the spread between the upper- and lower-bound estimates of the returns to schooling from 7–10 percentage points to approximately 4 percentage points.

Original languageEnglish
Pages (from-to)739-748
Number of pages10
JournalJournal of the American Statistical Association
Issue number451
StatePublished - Sep 1 2000

Bibliographical note

Funding Information:
Dan A. Black is Professor of Economics and Senior Research Associate of the Center for Policy Research at Syracuse University and Senior Fellow, Carnegie Mellon Census Research Data Center, Center for Policy Research, Syracuse University, Syracuse, NY 13244 (E-mail: dablac01 @maxwell.syredu). Mark C. Berger is the William B. Sturgill Professor of Economics and Director of Center for Business and Economic Research and Frank A. Scott is Gatton Professor of Economics, Department of Economics, Gatton College of Business and Economics, The University of Kentucky, Lexington, KY 40506 (E-mail: mberger@pop.ukyedu; This research was supported by U.S. Agency for Health Care Policy and Research grant 1-RO1-HS081 88-01 and National Institute for Child Health and Human Development grant I-R01-HD36073-01A2. The authors thank Alan Krueger for his data; Mukhtar Ah, Chris Bollinger, Steve Klepper, Seth Sanders, Jim Spletzer, and Zheng Wang for useful conversations and comments; and the editor, associate editor, and two referees for detailed, insightful comments that substantially improved the article.


  • Errors in variables
  • Instrumental Variables
  • Replicates

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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