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. We overcome an identification problem that arises 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 distributions. As a building block, we also present novel tests for the rationality of mode forecasts. We apply our tests to income forecasts from the Federal Reserve Bank of New York's Survey of Consumer Expectations. We find these forecasts are rationalizable as mode forecasts, but not as mean or median forecasts. We also find heterogeneity in the measure of centrality used by respondents when stratifying the sample by past income, age, job stability, and survey experience.

Figures (25)

• Figure 1: Power of the mode rationality test. This figure plots the empirical rejection frequencies against the degrees of misspecification $\kappa$ for different sample sizes in the vertical panels and for the two DGPs in the horizontal panels. The misspecification follows the bias design described in the main text and we use the instrument vector $(1,X_t)$ and a nominal significance level of $5\%$.
• Figure 2: Confidence set coverage rates under the alternative. This figure shows the percentage values in how many of the simulation runs the respective confidence set includes a given point in the triangle. The six subfigures contain different true forecasts (whose weights are highlighted in red color) satisfying the null hypothesis in Assumption \ref{['assu:GMMWeakIDRegCond']} (D) for some $\theta_0 \in \Theta$, respectively. We use the instruments $\mathbf{h}_{t} = (1,X_t)$, the iid DGP, $n=5000$ and $\gamma = 0.5$ throughout this figure.
• Figure 3: Confidence sets for income survey forecasts. This figure shows the measures of centrality that "rationalize" the New York Federal Reserve income survey forecasts. The circles that comprise each triangle correspond to specific convex combinations of the vertices, which are the mean, median and mode functionals. Black dots indicate that the measure is inside the Stock-Wright 90% confidence set, grey dots indicate that the measure is inside the 95% confidence set, and white dots indicate that rationality for that measure of centrality can be rejected at the 5% level. The left panel uses only a constant as the instrument; the right panel uses a constant and the forecast.
• Figure 4: Confidence sets for income survey forecasts, stratified by income. This figure shows the measures of centrality that "rationalize" the New York Federal Reserve income survey forecasts, for low-, middle- and high-income respondents. Groups are formed using terciles of lagged reported income. Black dots indicate that the measure is inside the Stock-Wright 90% confidence set, grey dots indicate that the measure is inside the 95% confidence set, and white dots indicate that rationality for that measure of centrality can be rejected at the 5% level. All panels use a constant and the forecast as test instruments.
• Figure 5: Confidence sets for income survey forecasts, stratified by age and income. See the notes to Figure \ref{['fig:ResultsSCEIncomeTercile']} for additional details.

Theorems & Definitions (41)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Theorem 2.6
• Theorem 2.7
• Corollary 2.8
• Theorem 2.9
• Remark 3.1
• Remark 3.2