Testing Forecast Rationality for Measures of Central Tendency

Abstract

Rational respondents to economic surveys may report as a point forecast any measure of the central tendency of their (possibly latent) predictive distribution, for example the mean, median, mode, or any convex combination thereof. We propose tests of forecast rationality when the measure of central tendency used by the respondent is unknown. These tests require us to overcome an identification problem when the measures of central tendency are equal or in a local neighborhood of each other, as is the case for (exactly or nearly) symmetric and unimodal distributions. As a building block, we also present novel tests for the rationality of mode forecasts. We apply our tests to survey forecasts of individual income, Greenbook forecasts of U.S. GDP, and random walk forecasts for exchange rates. We find that the Greenbook and random walk forecasts are best rationalized as mean, or near-mean forecasts, while the income survey forecasts are best rationalized as mode forecasts.

Is the mode elicitable relative to unimodal distributions?

“…For a more general definition of the mode for probability distributions not admitting a Lebesgue or counting density, we refer to Dearborn and Frongillo (2020). Together with the mean and the median, the mode is one the three main measures of central tendency in statistics (Dimitriadis et al, 2019), which is reflected in many introductory textbooks to statistics as well as the Bank of England's (2019) quarterly Inflation Report, which features all three point forecasts. The accuracy of point forecasts for the mean, median, mode, or any other functional T on some class of probability densities F on R can be measured by loss or scoring functions.…”

“…Neither of these cases is covered by the original proof given in Heinrich (2014). In the meantime, the literature has made some progress regarding the elicitability question of the mode by showing that it is asymptotically elicitable Dimitriadis et al (2019) under some restrictions, but generally even fails to be indirectly elicitable Dearborn and Frongillo (2020). The question as to whether the mode is elicitable relative to the relevant class of distributions with a continuous and unimodal Lebesgue density has been explicitly stated as an open problem (ibidem).…”