We consider two distinct possibilities. In the first, which we refer to as Model A, reticent respondents are always candid in responding to the first part of the CQ, i.e. the question about the visit. However, they may behave reticently when responding to the sensitive second part of the CQ, i.e. the question about the bribe. The second possibility, which we refer to as Model B, allows reticent respondents to avoid answering the sensitive second part of the CQ by responding “No” to the first part of the CQ even if a visit did occur. Although the first part of the CQ is not _per se_ sensitive, a “No” answer can put the respondent in the position of avoiding an inquiry on a sensitive act.<sup>12</sup> Specifically, we assume that reticent respondents who received a visit but not a request for a bribe respond candidly to both parts of the CQ. In contrast, reticent respondents receiving a visit and a bribe request and choosing to behave reticently deny that the visit occurred and thereby avoid the sensitive second question.<sup>13</sup> For notational convenience, we now define a random variable that fully summarizes the possible responses to the two-part CQ. Specifically, if no visit is acknowledged, if a visit is DD acknowledged but no bribe is reported, and if a visit and a bribe are both acknowledged. Note DD= 1 DD= 2 that is defined in such a way that it is non-missing for all respondents that answered either part of the DD= 3 CQ. This is in contrast with the bribe question itself, which is non-missing only for those respondents DD who acknowledge a visit in the first part of the CQ. The use of a structural model integrating responses on both parts of the CQ therefore will allow us to use the data on all respondents in estimating rates of corruption. This is in contrast to standard estimates that focus only on the subset of the sample corresponding to respondents who acknowledge a visit.

Table 3 spells out our assumptions on the likelihood of observing these three possible responses to the CQ. The panels of the table reflect combinations of the models (A and B) and the types of respondents (candid and reticent), with only three panels needed since candid respondents behave the same in both models. Within panels, the rows reflect the three possibilities for the respondents' actual experiences on visits and bribery, i.e. (i) a visit actually did not occur, (ii) a visit did occur without a bribe request, or (iii) a visit did occur and a bribe was requested. Of course, these actual experiences are unobserved for the researcher, who only sees survey responses. The columns of the table correspond to the three possible responses to the CQ, which are observed by the researcher. The cell entries spell out the probability of the (observed) response corresponding to the column, given the model, the type of respondent, and the (unobserved) actual experience indicated in the row of the table. Of course, even though the probability of two different values of D can both be positive, the value of D is unique in any given observation in the survey.

> 12 On the WBES questionnaires, this tax question appears in the middle (or in Nigeria, at the end) of a series of twopart questions each of which is identical in structure to the question on taxes, but referring to other government agencies.  Thus by the time the respondents reach the tax question they should know that acknowledgement of a visit will be followed by a question about a bribe request.

> 13 We also considered a third model, similar to model B except that all reticence respondents who chose to behave reticently did so on the visit question whether or not a visit had resulted in a bribe request.  When evaluating the performance of each model—see Section 6—this third model was the least preferred for all countries and therefore we have not reported any results for this model.

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