Uncertainty, information, and disagreement of economic forecasters

ABSTRACT

An information framework is proposed for studying uncertainty and disagreement of economic forecasters. This framework builds upon the mixture model of combining density forecasts through a systematic application of the information theory. The framework encompasses the measures used in the literature and leads to their generalizations. The focal measure is the Jensen–Shannon divergence of the mixture which admits Kullback–Leibler and mutual information representations. Illustrations include exploring the dynamics of the individual and aggregate uncertainty about the US inflation rate using the survey of professional forecasters (SPF). We show that the normalized entropy index corrects some of the distortions caused by changes of the design of the SPF over time. Bayesian hierarchical models are used to examine the association of the inflation uncertainty with the anticipated inflation and the dispersion of point forecasts. Implementation of the information framework based on the variance and Dirichlet model for capturing uncertainty about the probability distribution of the economic variable are briefly discussed.

KEYWORDS:

• Bayesian hierarchical model
• entropy
• Kullback-Leibler
• mixture density
• mutual information
• survey of professional forecasters

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• C11
• C18
• C41

Appendix

Disagreement based on Rényi information divergence.

The expression for Rényi information divergence between two normal distributions $f_i=N({\mu }_i,{\sigma }_i^2),i=1,2$ can be written in the following form:

${\displaystyle \begin{array}{c}\multicolumn{1}{c}{K_r(f_1:f_2)=\frac{r{({\mu }_1-{\mu }_2)}^2}{2{\sigma }_r^2}+\frac{1}{2(r-1)}log\frac{{\sigma }_2^2}{{\sigma }_r^2}-\frac{1}{2}log\frac{{\sigma }_1^2}{{\sigma }_2^2},}\end{array}}$

which is defined when ${\sigma }_r^2=(1-r){\sigma }_1^2+r{\sigma }_2^2>0$. In Example 2, $f_1=f_{y|x}$ and $f_2=f_y$. Noting that ${\sigma }_{y|x}^2=(1-{\rho }^2){\sigma }_y^2$, we have ${\sigma }_r^2=[1+(r-1\left){\rho }^2\right]{\sigma }_y^2>0$ which holds for all r>0. Then

${\displaystyle \begin{array}{c}\multicolumn{1}{c}{K_r(f_{y|X}:f_y)=\frac{r{(\alpha +\beta X-{\mu }_y)}^2}{2[1+(r-1\left){\rho }^2\right]{\sigma }_y^2}+\frac{1}{2(r-1)}log\frac{1}{[1+(r-1){\rho }^2}-\frac{1}{2}log(1-{\rho }^2).}\end{array}}$

The last term is the mutual information (17). Letting $\alpha ={\mu }_y-\beta {\mu }_x,\beta =\rho \frac{{\sigma }_y}{{\sigma }_x}$, and taking the expectation $E\left[K_r\right(f_{y|X}:f_y\left)\right]$, we obtain (18).

Proof of Proposition 1. The ME model subject to the moment constraints (27) is given by:

(31) ${\displaystyle \begin{array}{c}\multicolumn{1}{c}{f_c^{\ast }\left(y\right)=Cexp\left(-\sum _{j=1}^J{\lambda }_{c,j}{y}^j\right),}\end{array}}$(31)

where ${\lambda }_{c,1},\dots ,{\lambda }_{c,J}$ are the Lagrange multipliers for constraints (26) and $C=C({\lambda }_{c_1},\dots ,{\lambda }_{c,J})$ is the normalizing factor. (The ME model can be a probability vector or a continuous density). Breaking the log-ratio in $K(f_{y|x}^{\ast }:f_a^{\ast })$, we have:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{E_g\left[K\right(f_{y|X}^{\ast }:f_c^{\ast }\left)\right]}& =\hfill & E_g\left[-H\left(f_{y|X}^{\ast }\right)-E_{y|x}[log{f}_c^{\ast }(Y\left)\right]\right]\hfill \\ =\hfill & -E_g\left[H\left(f_{y|X}^{\ast }\right)\right]-logC+E_g\left[\sum _{j=1}^J{\lambda }_{c,j}{E}_{y|x}\left(Y^j\right)\right]\hfill \\ =\hfill & -E_g\left[H\left(f_{y|X}^{\ast }\right)\right]-logC+\sum _{j=1}^r{\lambda }_{c,j}{E}_g\left[E_c\left(Y^j\right)\right]\hfill \\ =\hfill & -E_g\left[H\left(f_{y|X}^{\ast }\right)\right]-logC+\sum _{j=1}^r{\lambda }_{c,j}{\mu }_{c,j}.\hfill \end{array}}$

The second equality is obtained using (31) and the last equality is from (27). The last result is found by noting that

${\displaystyle \begin{array}{c}\multicolumn{1}{c}{H\left(f_c^{\ast }\right)=-logC+\sum _{j=1}^J{\lambda }_{c,j}{\mu }_{c,j}.}\end{array}}$

Acknowledgments

We would like to thank, but not implicate, two anonymous reviewers, Kajal Lahiri, Paul Nystrom, Jeffrey Racine, Minchul Shin, and Kenneth Wallis for their valuable feedbacks and comments that led to improving the exposition of this paper. Shoja’s research was partially supported by an Info-Metrics Institute’s Summer Fellowship. Soofi’s research was supported by a Roger L. Fitzsimonds Distinguished Scholar Award.