beliefs downward. note that the marginal investor in the futures market likely changes over time, and this change can be an additional source of fluctuations in the gap between market - implied rates and mean projections. unfortunately, we have no direct measure of the beliefs of the marginal investor. but we can obtain evidence on whether the dispersion in beliefs fell from december to march by looking at the dispersion of beliefs among survey respondents. one measure of such dispersion in beliefs is the extent to which the klic varies across respondents. a high ( low ) klic means that the respondent ’ s forecast distribution requires substantial ( little ) tilting to deliver the market - implied rate. slide 14 shows, in the bars outlined in light blue, the klic distribution brodsky et al. ( 2016b ). note that this distribution is constructed by tilting the average of respondents ’ probability distributions. alternatively, we could have tilted the probability distribution for each respondent and then averaged across these tilted distributions. while the two approaches do not, in general, deliver the same answer, in this case they almost do. bis central bankers ’ speeches across respondents for the 2018 projections in december, and in the dark blue bars, the distribution for the 2018 projections in january. it is clear that dispersion in beliefs fell dramatically between the two surveys. the right panel shows that dispersion increased slightly in march but remains below the december levels. belief heterogeneity slide 14 already illustrates that forecasts are very different across survey respondents. i ’ ll now provide further evidence of how pervasive and important heterogeneity in beliefs is. recall that in both the recent march and january surveys, respondents were asked about their subjective probability of “ returning to the zlb at some point in 2016 – 2018. ” slide 15 plots the distribution across respondents of this probability. it shows that respondents belonged to three groups, roughly speaking : a few who believed this probability to be low ( less than 15 percent of respondents ), a majority who thought the probability was about one in four ( between 20 and 35 percent of respondents ), and a sizable minority who assessed this probability as fairly large ( more than 40 percent of respondents ). this heterogeneity in the probability assigned to a return to the zlb translates of course into heterogeneity in the probability distribution for the federal funds rate. the left panel of slide 16 shows the smoothed density forecasts for the year - end 2017

up [ down ] on the average, during the next 5 to 10 years? ” only respondents who give a response over 5 percent are then asked the clarifying follow - up question “ would that be [ x ] percent per year, or is that the total for prices over the next 5 to 10 years? ” respondents who answer “ total ” are then asked for a “ per year ” amount. further, the clarifying question used in the michigan survey that asks whether respondents meant their response to reflect price changes per year or over the entire time period induced significant revisions. the follow - up question is administered only to respondents who give expectations over 5 percent, thus failing to correct misinterpretations among those who gave lower responses. in our experimental surveys, we instead administered the follow - up to everyone. we found that if we administered the follow - up question only to those giving responses over 5 percent, then the median long - term expectation would be 4. 3 percent instead of 3. 7 percent in our test sample. thus, we have reasons to believe that the 5 percent follow - up question in the michigan survey leads to a systematic overestimation of actual average responses. in contrast, for the survey of primary dealers discussed later, we added a question in 2007 that directly elicited uncertainty over cpi inflation from 5 to 10 years ahead. see potter ( 2011, 2012 ). bis central bankers ’ speeches as food or gasoline. 20 in related research, we have shown that respondents tend to express more extreme inflation expectations when they are prompted to think of specific prices in answering the question rather than overall inflation. 21 in the sce, we therefore ask directly for expectations about the rate of inflation or deflation. 22 probabilistic questions on the sce as i noted earlier, a key feature of our survey is its use of probabilistic questions. following the current practice in the literature for the elicitation of expectations about continuous variables, we tested questions that elicited density forecasts by asking respondents to assign the percent chance that the value of interest would fall within different pre - specified ranges or “ bins. ” for instance, for inflation, we ask for the subjective probability distribution over a range of possible future inflation outcomes. this is shown in slide 3. 23 for each respondent, we fit a flexible parametric probability density function ( pdf ) to their bin probabilities. we can then compute several moments of interest from the fitted density forecast