# _3.2 Modeling Responses to the Random-Response Questions_

For both models, the interview process for RRQs is modeled as in Kraay and Murrell (2015). The key assumption here is that reticent behavior is equally prevalent in responses to the RRQ as it is in the CQ. Recall that on each of the questions in the RRQ, the respondent is instructed to answer the sensitive question if the coin comes up tails, and to answer “Yes” if the coin comes up heads. Our definition of a reticent respondent is one who gives knowingly false answers with a nonzero probability when honest answers to a question could generate the inference that the respondent might have committed a sensitive act. Given the assumptions above and this definition, there is therefore a probability that a reticent respondent will respond “No” even when the coin comes up heads. Similarly, when the coin comes up tails, there is a probability that the reticent respondent will answer qq “No” even if the respondent has done the sensitive act in question. The data in Table 2 provide qq reassurance on this assumption. If the outcome of a head on the coin toss actually encourages candor, then all entries in that table would lie above 50%. However, as discussed in the previous section, for the truly sensitive questions the responses are greater than 50% only in two out of 49 cases (seven sensitive questions in seven countries).<sup>14</sup> As discussed in more detail in Kraay and Murrell (2015), these assumptions imply that the probability of a “Yes” response on any single RRQ is . To see this, note that respondents are supposed to answer “Yes” either if they are guilty (with probability ) or if they are not 0.5(1 + gg)(1 −qq) guilty and the coin comes up heads (with probability ). These two probabilities sum to gg but must be scaled down by , the probability that a reticent respondent provides an 0.5(1 −gg) honest “Yes” response. For candid respondents, the probability of a “Yes” response on a given RRQ is 0.5(1 + gg) (1 −qq) . Candid respondents can have a lower guilt probability than reticent respondents (i.e., ), but always answer honestly (i.e. is not relevant to them). 0.5(1 + kkgg) Define the random variable as the number of “Yes” responses on the 7 sensitive RRQs that are kkgg≤gg qq given in bold in Table 2. Since both the coin toss and the decision to behave reticently on a given XX question are independent across questions in the battery of RRQs, is binomially distributed, with different success probabilities for reticent and candid respondents as noted above, i.e. a success XX probability of for candid respondents and for reticent respondents. It is also useful to briefly note why we use several RRQs and not just a single RRQ. As discussed 0.5(1 + kkgg) 0.5(1 + gg)(1 −qq) in Section 2, in modeling the data generating process it is important to allow for the possibility that the reticent do not always behave reticently, that is to include in the model a parameter But then if , which is a possibility that we do not want to rule out _a priori_ , the effects of and on data qq< 1. gathered on any single CQ or single RRQ are identical.<sup>15</sup> Hence, these parameters would not be kk= 1 qq rr

> 14 Since the assumption that rates of reticence on the CQ and the RRQ are the same is so critical to our procedures, and is non-standard in the context of the existing literature on RRQs, we elaborate on this point in Appendix B. Importantly, in that appendix we report the results of some simulations showing that if our assumption is not correct in the sense that reticent behaviour is less prevalent in the RRQ, then we _underestimate_ the degree of downward bias in standard estimates of corruption.

> 15 This point is trivial to show using equations (1) and (2) and the information appearing in the paragraphs immediately above .

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