Distortions in the elicitation of economic variables arise frequently. A common problem in household surveys is that reported values exhibit a significant degree of rounding. We interpret rounding as a filter that allows limited information about the relationship of interest to pass. We argue that rounding is an active decision of the survey respondent, and propose a general structural model that helps to explain some of the typical distortions that arise out of this active decision. Specifically, we assume that there is insufficient ability of individuals to acquire, process and recall information, and that rational individuals aim at using the scarce resources they devote to a survey in an optimal fashion. This model implies selection and places some structure on the selection equation. We use the formal model to correct for some of the distorting effects of rounding on the relationship of interest, using all the data available. Finally, we show how the concepts developed in this paper can be applied in consumer demand analysis by exploiting a controlled survey experiment, and obtain plausible results.