On Combining Statistical and Fuzzy Techniques: Detection of Business Cycles From Uncertain Data

Detecting the beginning and the end of the business cycle is an important and di cult economic problem. One of the reasons why this problem is di cult is that for each year, we have only expert estimates (subjective probabilities) indicating to what extent the economy was in growth or recession. In our previous papers, we used fuzzy techniques to process this uncertain information; namely, we used the operation min(a; b) to combine the subjective probabilities (expert estimates) of two events into a probability that both events happen. This function corresponds to the most optimistic estimate of the joint probability. In this paper, we use another operation which corresponds to the most cautious (pessimistic) estimate for joint probability. It turns out, unexpectedly, that as we get better extrapolations for subjective probabilities, the resulting change times become fuzzier and fuzzier until, for the best (least sensitive) extrapolation, we get the largest fuzziness. We explain this phenomenon by showing that in the presence of noise, an arbitrary continuous (e.g., fuzzy) system can be well described by its discrete analog, but as the description gets more accurate, the continuous description becomes necessary.