and stigma series are almost mirror images of each other. As stigma increases, use decreases. One might interpret these results as evidence that the reported measures are valid. That is, use decreases as perceptions of harm increase. Alternatively, these results are consistent with the idea that changes in stigma are associated with changes in invalid reporting. As stigma increases, false negative reports increase. The consensus view that the data reveal trends seems inconsistent with the view that the fraction of false reports varies with the stigmatization of drugs.

In the absence of an invariance assumption, the data do not identify the trends in illegal drug use. However, the data might reveal the direction of the trend. Consider the level estimates implied by assumption IR-1 and IR-2. Here, if P=0.30, the width of the bound on trends is 60 points—the observed trend plus and minus 0.30. In this case, the sign of the trend is identified if the observed prevalence rates change by more than 30 points over the period.

In practice, this bound implies that the data do not in general reveal the direction of the trend in prevalence rates. Since the observed annual (absolute) change in the prevalence rates never exceeds 5 points, the sign of the annual trend is not identified. Likewise, the observed increase in the fraction of adolescent users during the 1990s is at most estimated to be around 13 points. Thus, if one adopts the restriction that the invalid response rate is less than 30 percent, the observed data would not reveal whether use increased in the 1990s. If one were willing to impose a stronger restriction on misreporting (e.g., P=0.05), the observed data would be able to identify the sign of this trend.

Linking Stigma and Inaccurate Reporting

Imposing explicit restrictions on the behavior of invalid response over time can further narrow the bounds on the trends in prevalence rates. Suppose there is a positive relationship between false negative reporting and the stigmatization of illegal drugs. In particular, assume that

IR-3: the fraction of false positive reports, P[wt=1, yt=0], is fixed over time,

and that

IR-4: the fraction of users who claim to have abstained, P[wt=0|yt= 1], increases with an index, St, of the degree that illicit drugs are stigmatized. That is, if St+j=St, then P[wt+j=0|yt+j=1]=P[wt+j=0|yt= 1].

If stigma increases over time, assumptions IR 1–4 imply that

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