Flexible Approximation of Subjective Expectations Using Probability Questions

Abstract

We use spline interpolation to approximate the subjective cumulative distribution function of an economic agent over the future realization of a continuous (possibly censored) random variable. The method proposed exploits information collected using a small number of probability questions on expectations and requires a weak prior knowledge of the shape of the underlying distribution. We find that eliciting 4 or 5 points on the cumulative distribution function of an agent is sufficient to accurately approximate a wide variety of underlying distributions. We show that estimated moments of general functions of the random variable can be computed analytically and/or using standard simulation techniques. We illustrate the usefulness of the method by estimating a simple model to asses the impact of expectations on investment decisions in a commonly used trust game. JEL Classification:C81, C10, D39

Heterogeneity in consumers’ income and pension expectations

“…Following Bellemare et al (2012) and De Bresser and van Soest (2015), these six points are used to nonparametrically estimate the complete subjective distribution of the future RIRR for each observation. The distribution function is obtained by linking the six points corresponding to the reported probabilities using splines, imposing monotonicity.…”

What can we learn from the fifties?

“…Hence, if these Q 1 do not indicate a lack of knowledge of the stock market as "don't know" and answers identified as uninformative, they reflect at least a clearly imprecise knowledge of the stock market. The analysis will 13 Instead of assuming a normal distribution to fit each respondent-specific distribution, it would have been possible to use a more flexible method proposed recently by Bellemare, Bissonnette, and Kröger (2012) which maintains much weaker assumptions on the shape of each underlying distribution. This approach is, however, beyond the scope of this paper.…”

Retirement Expectations and Satisfaction with Retirement Provisions

“…Our second approach, adapted from Bellemare et al . (), uses spline interpolation to fit a subjective distribution that passes through the points corresponding to the probabilities reported by the respondents. This procedure is non‐parametric, in the sense that it does not assume any parametric form of the underlying distribution .…”

“…The first, proposed in Dominitz and Manski (1997), fits an assumed underlying (log-normal) distribution for each observation by minimizing the squared difference between the probabilities implied by the assumed distribution and those reported in the data. Our second approach, adapted from Bellemare et al (2012), uses spline interpolation to fit a subjective distribution that passes through the points corresponding to the probabilities reported by the respondents. This procedure is non-parametric, in the sense that it does not assume any parametric form of the underlying distribution.…”