drugs.¹⁰ Thus, despite considerable resources devoted to reducing misreporting in the national drug use surveys, inaccurate response remains an inherent concern. Surely some respondents fail to provide valid information about whether they consume illegal drugs.

Inaccurate reporting in drug use surveys is conceptually different from the nonresponse problem examined above. While the fraction of nonrespondents is known, the data do not reveal the fraction of respondents who give invalid responses to the questionnaire. It might be that all positive reports are invalid, in which case the usage rate may be zero. Alternatively, it might be that all negative reports are invalid, in which case the entire population may have consumed illegal drugs. Thus, to draw inferences about the fraction of users in the United States, one must impose assumptions about self-reporting errors.

To evaluate the impact of invalid response on the ability to infer levels of use, I introduce notation that distinguishes between self-reports and the truth. Let w_t be the self-reported measure in period t, where w_t= 1 if the respondent reported use and 0 otherwise. Let y_t be the truth, where y_t=1 if the respondent consumed drugs and 0 otherwise. We are interested in learning probability of use, P[y_t=1]. Formally, we can relate this unobserved prevalence rate to the self-reported usage rates as follows: