The calculus of dissent: Bias and diversity in FOMC projections

Abstract

Economic projections by the Federal Open Market Committee (FOMC) were very inaccurate in the years during and after the Great Recession. Relying on a model of collective prediction that weighs the “wisdom of crowds” against shared biases, we examine GDP forecast errors in a panel dataset of FOMC projections from 1992 through 2016. Consistent with the model, we find that diversity of projections reduces collective error, while shared bias magnifies collective error. Collective error is associated strongly with errors by the Federal Reserve Board staff. The benefits of diversity often are statistically significant, especially for projections with terms longer than 1 year.

Appendix A. Applying the diversity prediction theorem to FOMC projections

Assume agents want to predict the true value of some variable $X$. Let $x_{im}$ be independent predictions of $X$ by individual $i$ at meeting $m$. Let $s_m$ be the mean prediction of $i$ individuals at meeting $m$.

$$\begin{aligned} s_m = \frac{1}{k}\sum _{i=1}^k x_{im} \end{aligned}$$

(16)

Let $c$ be the mean of $s_m$ over $m$ meetings.

$$\begin{aligned} c = \frac{1}{n}\sum _{m=1}^n s_m \end{aligned}$$

(17)

Let collective error be the squared difference between the collective error and the target value $(c-X)^2$. We use the diversity prediction theorem (DPT) to solve for collective error in terms of meeting diversity, the squared difference between individual and collective predictions at each FOMC meeting $(s_m-c)^2$, and meeting error, the squared difference between the mean meeting prediction and the actual value $(s_M-X)^2$. We begin by expanding the polynomial of collective error, inserting an additional square of collective error $(c^2-c^2)$, and then rearranging the terms.

$$\begin{aligned} (c-X)^2&= c^2-2cX+X^2 \end{aligned}$$

(18)

$$\begin{aligned}&= c^2-2cX+X^2 +(c^2-c^2) \end{aligned}$$

(19)

$$\begin{aligned}&= 2c^2-c^2-2cX+X^2 \end{aligned}$$

(20)

Since $c$ is the mean of $s_m$ as shown in equation 17, we can replace two of the $c$ variables with the mean of $s_m$. We then insert a square of the mean of $s_m$ minus itself. We then rearrange the terms into groups of $X$ and $c$.

$$\begin{aligned} (c-X)^2&= 2s_m c-c^2-2s_m X+X^2 \end{aligned}$$

(21)

$$\begin{aligned}&= 2s_m c-c^2-2s_m X+X^2+\bigg [\frac{1}{n}\sum _{m=1}^n(s_m^2-s_m^2)\bigg ] \end{aligned}$$

(22)

$$\begin{aligned}&= \bigg [\frac{1}{n}\sum _{m=1}^n(s_m^2)\bigg ]-2s_m X+X^2-\bigg [\frac{1}{n}\sum _{m=1}^n(s_m^2)\bigg ]+2s_m c-c^2 \end{aligned}$$

(23)

Since $X$ and $c$ are constant in $m$, these terms can be moved inside a single summation.

$$\begin{aligned} (c - X)^2 =\frac{1}{n}\sum _{m=1}^n\bigg [(s_m^2-2s_m X+X^2)-(s_m^2-2s_m c+c^2)\bigg ] \end{aligned}$$

(24)

Factoring the polynomials gives us the DPT used in Eq. 1.

$$\begin{aligned} (c - X)^2 =\frac{1}{n}\mathop {\sum }_{m=1}^n \bigg [-(s_m-c)^2+(s_m-X)^2\bigg ] \end{aligned}$$

(25)

Since $s_m$ is the mean of individual predictions per meeting, we can follow the same process of the DPT to expand into components of meeting diversity, the squared difference between individual predictions and the mean meeting prediction $(x_{im}-s_m)^2$, and the individual error, the squared difference between individual predictions and actual values $(x_{im}-X)^2$.

$$\begin{aligned} (s_m-X)^2&= s_m^2-2s_m X+X^2 \end{aligned}$$

(26)

$$\begin{aligned}&= s_m^2-2s_m X+X^2 +(s_m^2-s_m^2) \end{aligned}$$

(27)

$$\begin{aligned}&= 2s_m^2-s_m^2-2s_m X+X^2 \end{aligned}$$

(28)

$$\begin{aligned}&= 2x_{im} s_m-s_m^2-2x_{im} X+X^2 \end{aligned}$$

(29)

$$\begin{aligned}&= 2x_{im} s_m-s_m^2-2s_{im} X+X^2+\bigg [\frac{1}{k}\sum _{i=1}^k(x_{im}^2-x_{im}^2)\bigg ] \end{aligned}$$

(30)

$$\begin{aligned}&= \bigg [\frac{1}{k}\sum _{i=1}^k(x_{im}^2)\bigg ]-2x_{im} X+X^2-\bigg [\frac{1}{k}\sum _{i=1}^k(x_{im}^2)\bigg ]+2x_{im} s_m-s_m^2 \end{aligned}$$

(31)

$$\begin{aligned}&= \frac{1}{k}\sum _{i=1}^k\bigg [(x_{im}^2-2x_{im} X+X^2)-(x_{im}^2-2x_{im} s_m+s_m^2)\bigg ]\end{aligned}$$

(32)

$$\begin{aligned}&= \frac{1}{k}\mathop {\sum }_{i=1}^k \bigg [-(x_{im}-s_m)^2+(x_{im}-X)^2\bigg ] \end{aligned}$$

(33)

Inserting Eqs. 33 into 25 gives us Eq. 34. Collective error $(c-X)^2$ equals the mean of individual errors $(x_{im}-X)^2$ less group diversity $(s_m-c)^2$ and mean meeting diversity $(x_{im}-s_m)^2$.

$$\begin{aligned} (c - X)^2 = \frac{1}{n}\mathop {\sum }_{m=1}^n \bigg [-(s_m-c)^2+\frac{1}{k}\sum _{i=1}^{k}\bigg (-(x_{im}-s_m)^2+(x_{im}-X)^2\bigg ) \bigg ] \end{aligned}$$

(34)