# Bayesian Calibration of Generalized Pools of Predictive Distributions

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## Abstract

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## 1. Introduction

## 2. Combination and Calibration

#### 2.1. A General Combination Model

- Linear opinion pool ($m=1$), i.e., $\phi \left(x\right)=x$:$$\begin{array}{c}\hfill {H}_{1t}\left(y\right|\omega )=\sum _{k=1}^{K}{\omega}_{k}{F}_{kt}\left(y\right)\end{array}$$
- Harmonic opinion pool ($m=2$), i.e., $\phi \left(x\right)=1/x$:$$\begin{array}{c}\hfill {H}_{2t}\left(y\right|\omega )={\left(\sum _{k=1}^{K}{\omega}_{k}{F}_{kt}{\left(y\right)}^{-1}\right)}^{-1}\end{array}$$
- Logarithmic opinion pool ($m=3$), i.e., $\phi \left(x\right)=log\left(x\right)$:$$\begin{array}{c}\hfill {H}_{3t}\left(y\right|\omega )=\prod _{k=1}^{K}{F}_{kt}{\left(y\right)}^{{\omega}_{k}}\end{array}$$

- Linear opinion pool ($m=1$):$$\begin{array}{c}\hfill {h}_{1t}\left(y\right|\omega )=\sum _{k=1}^{K}{\omega}_{k}{f}_{kt}\left(y\right)\end{array}$$
- Harmonic opinion pool ($m=2$):$$\begin{array}{c}\hfill {h}_{2t}\left(y\right|\omega )={H}_{2t}{\left(y\right|\omega )}^{2}\sum _{k=1}^{K}{\omega}_{k}{F}_{kt}{\left(y\right)}^{-2}{f}_{kt}\left(y\right)\end{array}$$
- Logarithmic opinion pool ($m=3$):$$\begin{array}{c}\hfill {h}_{3t}\left(y\right|\omega )={H}_{3t}\left(y\right|\omega )\prod _{k=1}^{K}{\omega}_{k}{F}_{kt}{\left(y\right)}^{-1}{f}_{kt}\left(y\right)\end{array}$$

#### 2.2. A Calibration Model

- The combination formula is flexibly dispersive if for the class $\mathcal{F}$ of fixed, non-random cdfs, for all ${F}_{0t}\in \mathcal{F}$ and ${F}_{1t},\dots ,{F}_{Kt}\in \mathcal{F}$, $L\left(y\right)={F}_{0}$, then $H\left(({F}_{1t}\left(y\right),\cdots ,{F}_{Kt}\left(y\right))\right|\theta )$ is a neutrally-dispersed forecast (i.e., $\mathbb{V}ar\left(L\left(Y\right|{F}_{1t},\dots ,{F}_{Kt})\right)=1/2$).
- The combination formula is exchangeably flexible dispersive if for the class $\mathcal{F}$ of fixed, non-random cdfs, for all ${F}_{0t}\in \mathcal{F}$ and ${F}_{1t},\dots ,{F}_{Kt}\in \mathcal{F}$, $L={F}_{0t}$, then H is anonymous, i.e., $H\left(({F}_{\pi \left(1\right)t},\cdots ,{F}_{\pi \left(K\right)t})\right|\theta )=H\left(({F}_{1t},\cdots ,{F}_{Kt})\right|\theta )$, and a neutrally-dispersed forecast.

#### 2.3. A Beta Mixture Calibration and Combination Model

## 3. Bayesian Inference

**θ**given the observations is $\pi \left(\theta \right|{\mathbf{y}}_{1:T})\propto g(\mu ,\nu ,\omega )L\left({\mathbf{y}}_{1:T}\right|\theta )$, where $g(\mu ,\nu ,\omega )$ corresponds to the prior density of the parameters.

## 4. Empirical Results

#### 4.1. Simulation Study

- the equally-weighted model (EW):$$\begin{array}{cc}\hfill {H}_{1t}(y,\omega )& =\omega F\left(y\right|-1,1)+(1-\omega \left)F\right(y|0.5,3),\hfill \\ \hfill {H}_{2t}(y,\omega )& =\omega {\left(F\left(y\right|-1,1)\right)}^{-1}+(1-\omega ){\left(F\left(y\right|0.5,3)\right)}^{-1},\hfill \\ \hfill {H}_{3t}(y,\omega )& =exp\left(\omega log\left(F\right(y|-1,1))+(1-\omega )log(F\left(y\right|0.5,3\left)\right)\right),\hfill \end{array}$$
- the beta calibration model (BC1):$$\begin{array}{c}\hfill {G}_{mt}\left(y\right|\theta )={B}_{{\alpha}_{1},{\beta}_{1}}\left({H}_{mt}\left(y\right|{\omega}_{1})\right)\end{array}$$
- the two-component beta mixture calibration model (BC2):$$\begin{array}{c}\hfill {G}_{mt}\left(y\right|\theta )=\rho {B}_{{\alpha}_{1},{\beta}_{1}}\left({H}_{mt}\left(y\right|{\omega}_{1})\right)+(1-\rho ){B}_{{\alpha}_{2},{\beta}_{2}}\left({H}_{mt}\left(y\right|{\omega}_{2})\right),\end{array}$$

**θ**) are reported in Table 1 for the linear combination models, in Table 2 for the harmonic combination models and in Table 3 for the logarithmic combination models, according to ${p}_{i}$. In the tables, ${\alpha}_{1}$ and ${\beta}_{1}$ stand for the parameters of the beta distribution and ${\omega}_{1}$ for the combination weight in the BC1 model and in the first component of the BC2 model, while the parameters of the second component of BC2 are referred to as ${\alpha}_{2}$, ${\beta}_{2}$ and ${\omega}_{2}$.

#### 4.2. Financial Application: Standard&Poors500 Index

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix. Computational Details

**θ**.

## References

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**Figure 1.**Density function for linear (red line), harmonic (blue line) and logarithmic (green line) combination models. The combination weight is ${\omega}_{1}=0.5$.

**Figure 2.**Combination densities for the three schemes (different rows) for ${\omega}_{1}=0.9$ (solid lines) and ${\omega}_{1}=0.1$ (dashed lines).

**Figure 3.**Empirical cdfs of the probability integral transforms (PITs) generated by ${F}_{1}$ (red line), ${F}_{2}$ (green line) and the true model (black line), for the simulated realizations of the variable of interest Y.

**Figure 4.**Beta calibration of ${F}_{1}$ (first row) and ${F}_{2}$ (second row) by using a beta calibration function (i.e., $J=1$) and the Bayesian estimates of the calibration parameters, that is ${\alpha}_{1}=0.773$ and ${\beta}_{1}=1.352$ for ${F}_{1}$ and ${\alpha}_{1}=7.485$ and ${\beta}_{1}=7.477$ for ${F}_{2}$.

**Figure 5.**(Left column): PITs’ cdf for the linear pool at different values of $\mathbf{p}$; (Right column): contribution of calibration components for BC1 (green) and BC2 (blue), where BC2${}_{1}$ (solid) is the fist component of the beta mixture in BC2 and BC2${}_{2}$ (dashed) the second component.

**Figure 6.**(Left column): PITs’ cdf for the harmonic pool at different values of $\mathbf{p}$; (Right column): contribution of calibration components for BC1 (green) and BC2 (blue), where BC2${}_{1}$ (solid) is the fist component of the beta mixture in BC2 and BC2${}_{2}$ (dashed) the second component.

**Figure 7.**(Left column): PITs cdf for the harmonic pool at different values of $\mathbf{p}$; (Right column): contribution of calibration components for BC1 (green) and BC2 (blue), where BC2${}_{1}$ (solid) is the fist component of the beta mixture in BC2 and BC2${}_{2}$ (dashed) the second component.

**Figure 8.**Different behavior of the equally-weighted (EW) model for the three pool schemes applied to the S&P500 daily percent log return.

**Figure 9.**PITs’ cdf of the ideal model C (black line), EW (magenta), BC1 (red) and BC2 (green) for linear (top right), harmonic (bottom left) and logarithmic (bottom right) pools, and the PITs of the EW models (top left) for linear (red), harmonic (blue) and logarithmic (green), in the first data subsample: pre-crisis period.

**Figure 10.**PITs cdf of the ideal model C (black line), EW (magenta), BC1 (red) and BC2 (green) for linear (top right), harmonic (bottom left) and logarithmic (bottom right) pools, and the PITs of the EW models (top left) for linear (red), harmonic (blue) and logarithmic (green), in the second data subsample: in-crisis period.

**Figure 11.**PITs cdf of the ideal model C (black line), EW (magenta), BC1 (red) and BC2 (green) for linear (top right), harmonic (bottom left) and logarithmic (bottom right) pools, and the PITs of the EW models (top left) for linear (red), harmonic (blue) and logarithmic (green), in the third data subsample: post-crisis period.

**Figure A1.**BC1 model ($J=1$): 100,000 MCMC samples (left column) and MCMC progressive averages (right column) for the parameters ${\omega}_{1}$, ${\alpha}_{1}$ and ${\beta}_{1}$ (different rows).

**Figure A2.**BC2 model ($J=2$): 100,000 MCMC samples (left column) and MCMC progressive averages (right column) for the parameters ${\omega}_{1}$, ${\omega}_{2}$, ${\alpha}_{1}$ and ${\alpha}_{2}$ (different rows).

**Figure A3.**BC2 model ($J=2$): 100,000 MCMC samples (left column) and MCMC progressive averages (right column) for the parameters ${\beta}_{1}$, ${\beta}_{2}$ and ρ (different rows).

**Table 1.**Parameter estimates in the linear combination model for different choices of the mixture probabilities $\mathbf{p}$ of the data-generating process.

p | (1/5, 1/5, 3/5) | (1/7, 1/7, 5/7) | (3/5, 1/5, 1/5) | (5/7, 1/7, 1/7) | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{\theta}$ | BC1 | BC2 | BC1 | BC2 | BC1 | BC2 | BC1 | BC2 |

${\alpha}_{1}$ | 0.755 | 3.293 | 0.921 | 6.970 | 0.461 | 0.452 | 0.496 | 0.650 |

${\beta}_{1}$ | 0.642 | 0.953 | 0.639 | 0.937 | 0.816 | 3.744 | 0.812 | 0.876 |

${\omega}_{1}$ | 0.015 | 0.191 | 0.000 | 0.500 | 0.256 | 0.925 | 0.342 | 0.230 |

${\alpha}_{2}$ | 0.692 | 0.665 | 0.550 | 0.707 | ||||

${\beta}_{2}$ | 3.093 | 0.713 | 0.827 | 13.033 | ||||

${\omega}_{2}$ | 0.150 | 0.233 | 0.063 | 0.315 | ||||

ρ | 0.697 | 0.512 | 0.215 | 0.806 |

**Table 2.**Parameter estimates in the harmonic combination model for different choices of the mixture probabilities $\mathbf{p}$ of the data-generating process.

p | (1/5, 1/5, 3/5) | (1/7, 1/7, 5/7) | (3/5, 1/5, 1/5) | (5/7, 1/7, 1/7) | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{\theta}$ | BC1 | BC2 | BC1 | BC2 | BC1 | BC2 | BC1 | BC2 |

${\alpha}_{1}$ | 0.744 | 7.026 | 0.906 | 7.775 | 0.416 | 0.383 | 0.457 | 0.457 |

${\beta}_{1}$ | 0.634 | 0.878 | 0.632 | 1.013 | 0.755 | 0.827 | 0.747 | 0.778 |

${\omega}_{1}$ | 0.042 | 0.529 | 0.024 | 0.456 | 0.363 | 0.734 | 0.507 | 0.511 |

${\alpha}_{2}$ | 0.615 | 0.665 | 3.720 | 0.462 | ||||

${\beta}_{2}$ | 0.929 | 0.651 | 1.133 | 0.734 | ||||

${\omega}_{2}$ | 0.380 | 0.302 | 0.093 | 0.474 | ||||

ρ | 0.453 | 0.415 | 0.824 | 0.456 |

**Table 3.**Parameter estimates in the logarithmic combination model for different choices of the mixture probabilities $\mathbf{p}$ of the data-generating process.

p | (1/5, 1/5, 3/5) | (1/7, 1/7, 5/7) | (3/5, 1/5, 1/5) | (5/7, 1/7, 1/7) | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{\theta}$ | BC1 | BC2 | BC1 | BC2 | BC1 | BC2 | BC1 | BC2 |

${\alpha}_{1}$ | 0.751 | 7.062 | 0.917 | 6.514 | 0.441 | 2.587 | 0.469 | 2.180 |

${\beta}_{1}$ | 0.639 | 0.950 | 0.640 | 0.966 | 0.764 | 1.109 | 0.753 | 0.869 |

${\omega}_{1}$ | 0.018 | 0.517 | 0.000 | 0.431 | 0.370 | 0.031 | 0.465 | 0.411 |

${\alpha}_{2}$ | 0.578 | 0.645 | 0.367 | 0.515 | ||||

${\beta}_{2}$ | 0.823 | 0.680 | 0.875 | 2.770 | ||||

${\omega}_{2}$ | 0.426 | 0.379 | 0.843 | 0.423 | ||||

ρ | 0.484 | 0.510 | 0.274 | 0.389 |

**Table 4.**Parameter estimates in the different combination models for the pre-crisis data subsample: 1 January 2007–5 October 2007.

P | Linear | Harmonic | Logarithmic | |||
---|---|---|---|---|---|---|

$\mathit{\theta}$ | BC1 | BC2 | BC1 | BC2 | BC1 | BC2 |

${\alpha}_{1}$ | 5.840 | 0.000 | 0.084 | 17.573 | 2.468 | 34.692 |

${\beta}_{1}$ | 5.807 | 0.000 | 0.371 | 15.114 | 2.867 | 34.462 |

${\omega}_{1}$ | 1.000 | 0.000 | 1.000 | 0.863 | 1.000 | 0.706 |

${\alpha}_{2}$ | 5.812 | 0.020 | 1.781 | |||

${\beta}_{2}$ | 5.651 | 0.466 | 2.166 | |||

${\omega}_{2}$ | 1.000 | 0.199 | 0.93 | |||

ρ | 0.000 | 0.7926 | 0.269 |

**Table 5.**Parameter estimates in the different combination models for the in-crisis data subsample: 20 June 2008–26 March 2009.

P | Linear | Harmonic | Logarithmic | |||
---|---|---|---|---|---|---|

$\mathit{\theta}$ | BC1 | BC2 | BC1 | BC2 | BC1 | BC2 |

${\alpha}_{1}$ | 7.025 | 278.600 | 0.977 | 0.944 | 0.974 | 1.010 |

${\beta}_{1}$ | 6.646 | 803.260 | 0.865 | 1.014 | 1.292 | 1.018 |

${\omega}_{1}$ | 1.000 | 1.000 | 0.740 | 0.263 | 0.821 | 0.031 |

${\alpha}_{2}$ | 6.760 | 0.975 | 1.131 | |||

${\beta}_{2}$ | 6.334 | 1.010 | 0.972 | |||

${\omega}_{2}$ | 1.000 | 0.247 | 0.298 | |||

ρ | 0.000 | 0.000 | 0.000 |

**Table 6.**Parameter estimates in the different combination models for the pre-crisis data subsample: 27 March 2009–31 December 2009.

P | Linear | Harmonic | Logarithmic | |||
---|---|---|---|---|---|---|

$\mathit{\theta}$ | BC1 | BC2 | BC1 | BC2 | BC1 | BC2 |

${\alpha}_{1}$ | 6.542 | 47110.000 | 1.031 | 0.972 | 1.127 | 1.007 |

${\beta}_{1}$ | 6.071 | 0.000 | 0.419 | 0.942 | 2.275 | 1.066 |

${\omega}_{1}$ | 1.000 | 1.000 | 0.823 | 0.967 | 0.186 | 0.406 |

${\alpha}_{2}$ | 6.710 | 1.039 | 0.891 | |||

${\beta}_{2}$ | 6.307 | 0.938 | 1.015 | |||

${\omega}_{2}$ | 1.000 | 0.920 | 0.921 | |||

ρ | 0.000 | 0.000 | 0.000 |

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**MDPI and ACS Style**

Casarin, R.; Mantoan, G.; Ravazzolo, F. Bayesian Calibration of Generalized Pools of Predictive Distributions. *Econometrics* **2016**, *4*, 17.
https://doi.org/10.3390/econometrics4010017

**AMA Style**

Casarin R, Mantoan G, Ravazzolo F. Bayesian Calibration of Generalized Pools of Predictive Distributions. *Econometrics*. 2016; 4(1):17.
https://doi.org/10.3390/econometrics4010017

**Chicago/Turabian Style**

Casarin, Roberto, Giulia Mantoan, and Francesco Ravazzolo. 2016. "Bayesian Calibration of Generalized Pools of Predictive Distributions" *Econometrics* 4, no. 1: 17.
https://doi.org/10.3390/econometrics4010017