Rounding is the familiar practice of reporting one value whenever a real number lies in an interval. Uncertainty about the extent of rounding is common when researchers analyze survey responses to numerical questions. The prevalent practice has been to take numerical responses at face value, even though many may in fact be rounded. This paper studies the rounding of responses to survey questions that ask persons to state the percent-chance that some future event will occur. We analyze data from the Health and Retirement Study and find strong evidence of rounding, the extent of rounding differing across respondents. We propose use of a person's response pattern across different questions to infer his rounding practice, the result being interpretation of reported numerical values as interval data. We then bring to bear recent developments on statistical analysis of interval data to characterize the potential consequences of rounding for empirical research. Finally, we propose enrichment of surveys by probing to learn the extent and reasons for rounding.

%B Journal of Business Economics Statistics %I CCCB CCCBC %V 28 %P 219-231 %G eng %N 2 %2 PMC2847785 %4 Interval data/Partial identification/Probabilistic expectations/Rounding/Survey data %$ 22820 %R 10.1198/jbes.2009.08098 %0 Journal Article %J Annals of Applied Statistics %D 2008 %T SKIP SEQUENCING: A DECISION PROBLEM IN QUESTIONNAIRE DESIGN. %A Charles F Manski %A Molinari, Francesca %K questionnaire design %XThis paper studies questionnaire design as a formal decision problem, focusing on one element of the design process: skip sequencing. We propose that a survey planner use an explicit loss function to quantify the trade-off between cost and informativeness of the survey and aim to make a design choice that minimizes loss. We pose a choice between three options: ask all respondents about an item of interest, use skip sequencing, thereby asking the item only of respondents who give a certain answer to an opening question, or do not ask the item at all. The first option is most informative but also most costly. The use of skip sequencing reduces respondent burden and the cost of interviewing, but may spread data quality problems across survey items, thereby reducing informativeness. The last option has no cost but is completely uninformative about the item of interest. We show how the planner may choose among these three options in the presence of two inferential problems, item nonresponse and response error.

%B Annals of Applied Statistics %I 2 %V 2 %P 264-285 %8 2008 Mar 01 %G eng %N 1 %2 PMC2858349 %4 Survey Design/Nonresponse/Response Error %$ 25300 %R 10.1214/07-aoas134 %0 Journal Article %J Econometrica %D 2002 %T Inference on Regressions with Interval Data on a Regressor or Outcome %A Charles F Manski %A Tamer, Elie %K Identification %K interval data %K Regression Analysis %X This paper examines inference on regressions when interval data are available on one variable, the other variables being measured precisely. Let a population be characterized by a distribution P(y, x, v, v0, v1), where y ε R1, x ε Rk, and the real variables (v, v0, v1) satisfy v0 ≤ v ≤ v1. Let a random sample be drawn from P and the realizations of (y, x, v0, v1) be observed, but not those of v. The problem of interest may be to infer E(y|x, v) or E(v|x). This analysis maintains Interval (I), Monotonicity (M), and Mean Independence (MI) assumptions: (I) P(v0 ≤ v ≤ v1) = 1; (M)E(y|x, v) is monotone in v; (MI) E(y|x, v, v0, v1) = E(y|x, v). No restrictions are imposed on the distribution of the unobserved values of v within the observed intervals [v0, v1]. It is found that the IMMI Assumptions alone imply simple nonparametric bounds on E(y|x, v) and E(v|x). These assumptions invoked when y is binary and combined with a semiparametric binary regression model yield an identification region for the parameters that may be estimated consistently by a modified maximum score (MMS) method. The IMMI assumptions combined with a parametric model for E(y|x, v) or E(v|x) yield an identification region that may be estimated consistently by a modified minimum-distance (MMD) method. Monte Carlo methods are used to characterize the finite-sample performance of these estimators. Empirical case studies are performed using interval wealth data in the Health and Retirement Study and interval income data in the Current Population Survey. %B Econometrica %I 70 %V 70 %P 519-46 %G eng %N 2 %L pubs_2002_Manski_CEcmetrica.pdf %4 Econometric Methods: Single Equation Models: General/Regression %$ 1068 %R https://doi.org/10.1111/1468-0262.00294 %0 Report %D 2001 %T Social Security Expectations and Retirement Savings Decisions %A Dominitz, Jeff %A Charles F Manski %A Heinz, Jordan %K Consumption and Savings %K Social Security %X Retirement savings decisions should depend on expectations of Social Security retirement income. Persons may be uncertain of their future Social Security benefits for several reasons, including uncertainty about their future labor earnings, the formula now determining social security benefits, and the future structure of the Social Security system. To learn how Americans perceive their benefits, we have elicited Social Security expectations from respondents to the Survey of Economic Expectations. We have also performed a more intensive face-to-face survey on a small sample of respondents. This paper presents the empirical findings. It also illustrates how data on expectations may help predict how Social Security policy affects retirement savings. %B NBER Working Paper %I The National Bureau of Economic Research %C Cambridge, MA %G eng %4 Social Security expectations/Retirement Saving %$ 10862 %R https://www.nber.org/papers/w8718 %0 Newspaper Article %B The New York Times %D 2000 %T Why Polls Are Fickle %A Charles F Manski %K Methodology %B The New York Times %I The New York Times Co. %C New York, NY %8 Monday, Oct. 16, 2000 %G eng %1 4 %4 Survey Methods %$ 9802 %! Why Polls Are Fickle %& Op-Ed %0 Book Section %B Wealth, Work and Health: Innovations in Measurement in the Social Sciences %D 1999 %T The Several Cultures of Research on Subjective Expectations %A Dominitz, Jeff %A Charles F Manski %E James P Smith %E Robert J. Willis %K Expectations %K Methodology %B Wealth, Work and Health: Innovations in Measurement in the Social Sciences %I University of Michigan Press %C Ann Arbor, MI %P 15-33 %@ 0472110268 %G eng %U https://books.google.com/books?id=lKvp4D1HuH8C&pg=PA209&lpg=PA209&dq=The+Size+Distribution+of+Wealth+in+the+United+States:+A+Comparison+Among+Recent+Household+Surveys&source=bl&ots=hFIAdSeWob&sig=ACfU3U2nIQ6QSOJ4wEBUDcbZOo-x7n8b7g&hl=en&sa=X&ved=2ahUKEwjQ %4 Methodology/Subjective Expectations %$ 8188 %! The Several Cultures of Research on Subjective Expectations