# On the Forecast Combination Puzzle

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## Abstract

**:**

## 1. Introduction

## 2. Additional Aspects of FCP

- A key factor missing in addressing the FCP is the true nature of the improvability of the candidate forecasts. While we all strive for better forecast performance than the candidates, that may not be feasible (at least for the methods considered). Thus, we have two scenarios (Yang 2004): (i) One of the candidates is pretty much the best we can hope for (within the considerations of course), and consequently, any attempt to beat it becomes futile. We refer to this scenario as “combining for adaptation” (CFA), because the proper goal of a forecast combination method under this scenario should be targeting the performance of the best individual candidate forecast, which is unknown. (ii) The other scenario is that a significant accuracy gain over all the individual candidates can be realized by combining the forecasts. We refer to this scenario as “combining for improvement” (CFI), because the proper goal of a forecast combination method under this scenario should be targeting the performance of the best combination of the candidate forecasts to overcome the defects of the candidates. In practical applications, both scenarios could be possible. Without factoring in this aspect, comparison of different combination methods may become somewhat misleading. In our view, bringing this lurking aspect into the analysis is beneficial to understand forecast combinations. With the above forecast combination scenarios spelled out, a natural question follows: can we design a combination method to bridge the two camps of methods proposed for the two scenarios? That is, in practical applications, without necessarily knowing the underlying forecast scenario, can we have a combination strategy adaptive to both scenarios?
- The methods being examined in the literature on FCP are mostly specific choices (e.g., least squares estimation). Can we do better with other methods (that may or may not have been invented yet) to mitigate relatively heavy estimation price? Furthermore, it is often assumed that the forecasts are unbiased and the forecast errors are stationary, which may not be proper for many applications. What happens when these assumptions do not hold?
- It has been stated in the literature that the simple methods (e.g., SA) are robust based on empirical studies. This may not be necessarily true in the usual statistical sense (rigorously or loosely). In many published empirical results, the candidate forecasts were carefully selected/built and thus well-behaved. Therefore, the finding in favor of the robustness of SA may be proper only for such situations in which the data analyst has extensive expertise on the forecasting problem and has done quite a bit of work on screening out poor/un-useful candidates; when allowing for the possibility of poor/redundant candidates for wider applications, the FCP may not be applicable anymore. It should be added that in various situations, the screening of forecasts are far from being an easy task, and the complexity may well be at the same level as model selection/averaging. Therefore, the view that we can do a good job in screening the candidate forecasts and then simply recruit SA can be overly optimistic. With the above, it is important to examine the robustness of SA in a broader context.

## 3. Problem Setup

#### AFTER Method

## 4. CFA versus CFI: A Hidden Source of FCP

**Case****1.**- Suppose ${y}_{t}$ ($t=1,\cdots ,T$) is generated by the linear model:$${y}_{t}={x}_{t}\beta +{\epsilon}_{t},$$$$\begin{array}{cc}\hfill \mathrm{Forecast}\phantom{\rule{4.pt}{0ex}}1:\phantom{\rule{4.pt}{0ex}}& {\widehat{y}}_{t,1}={x}_{t}{\widehat{\beta}}_{t};\hfill \\ \hfill \mathrm{Forecast}\phantom{\rule{4.pt}{0ex}}2:\phantom{\rule{4.pt}{0ex}}& {\widehat{y}}_{t,2}={\widehat{\alpha}}_{t},\hfill \end{array}$$

**Case****2.**- Suppose ${y}_{t}$ ($t=1,\cdots ,T$) is generated by the linear model:$${y}_{t}={x}_{t,1}{\beta}_{1}+{x}_{t,2}{\beta}_{2}+{\epsilon}_{t},$$$$\begin{array}{cc}\hfill \mathrm{Forecast}\phantom{\rule{4.pt}{0ex}}1:\phantom{\rule{4.pt}{0ex}}& {\widehat{y}}_{t,1}={x}_{t,1}{\widehat{\beta}}_{t,1};\hfill \\ \hfill \mathrm{Forecast}\phantom{\rule{4.pt}{0ex}}2:\phantom{\rule{4.pt}{0ex}}& {\widehat{y}}_{t,2}={x}_{t,2}{\widehat{\beta}}_{t,2},\hfill \end{array}$$

## 5. Multi-Level AFTER

Any combination of forecasts yields a single forecast. As a result, a particular combination of a given set of forecasts can itself be thought of as a forecasting method that could compete...

**Proposition**

**1.**

## 6. Is SA Really Robust?

**Case****3.**- Suppose a new information variable ${x}_{t,3}$ has the same distribution as ${x}_{t,1}$ and is independent of ${\mathbf{z}}_{t-1}$, ${y}_{t}$, and $({x}_{t,1},{x}_{t,2})$. A new candidate forecast ${\widehat{y}}_{t,3}={x}_{t,3}{\widehat{\beta}}_{t,3}$ joins the candidate pool in Case 2, where ${\widehat{\beta}}_{t,3}$ is obtained from OLS estimation with historical data.
**Case****4.**- A new candidate forecast ${\widehat{y}}_{t,3}={x}_{t,2}{\widehat{\beta}}_{t,2}$ identical to Forecast 2 joins the candidate pool in Case 2.
**Case****5.**- A new candidate forecast ${\widehat{y}}_{t,3}={\tilde{x}}_{t,2}{\tilde{\beta}}_{t,2}$ is generated using a transformed information variable ${\tilde{x}}_{t,2}=exp\left({x}_{t,2}\right)$, where ${\tilde{\beta}}_{t,2}$ is obtained from OLS estimation with historical data. This new forecast joins the candidate pool in Case 2.

## 7. Improper Weighting Formulas: A Source of the FCP Revisited

## 8. Linking Forecast Model Screening to FCP

## 9. Real Data Evaluation

## 10. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Assumptions of Proposition 1

**Assumption**

**A1.**

**Assumption**

**A2.**

#### Appendix A.2. Propositions and Proofs

**Proposition**

**A1.**

**Proposition**

**A2.**

**Proof**

**of**

**Proposition**

**A2.**

#### Appendix A.3. Additional Numerical Results

**Figure A1.**(Case 1) Comparing the average forecast risk of different forecast combination methods with AR(1) information variables (the dashed line represents the SA baseline; the x-axis is in logarithmic scale).

**Figure A2.**(Case 2) Comparing the average forecast risk of different forecast combination methods with AR(1) information variables (the dashed line represents the SA baseline; the x-axis is in logarithmic scale).

**Figure A3.**(Case 1) Performance of mAFTER under the adaptation scenario with AR(1) information variables (the dashed line represents the SA baseline; the x-axis is in logarithmic scale).

**Figure A4.**(Case 2) Performance of mAFTER under improvement scenario with AR(1) information variables (the dashed line represents the SA baseline; the x-axis is in logarithmic scale).

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**Figure 1.**(Case 1) Comparing the average forecast risk of different forecast combination methods (the dashed line represents the simple average (SA) baseline; the x-axis is in logarithmic scale). BG, Bates and Granger; LinReg, linear regression.

**Figure 2.**(Case 2) Comparing the average forecast risk of different forecast combination methods (the dashed line represents the SA baseline; the x-axis is in the logarithmic scale).

**Figure 3.**(Case 1) Performance of mAFTER under the adaptation scenario (the dashed line represents the SA baseline; the x-axis is in the logarithmic scale).

**Figure 4.**(Case 2) Performance of mAFTER under the improvement scenario (the dashed line represents the SA baseline; the x-axis is in logarithmic scale).

**Figure 6.**Comparing normalized MSFEs of different forecast combination methods with REG-Imputed SPF datasets (pre-1990 period). Left panel: PGDP variable. Right panel: RGDP variable. For each method, the bars from left to right represent 1-, 2-, 3-, and 4-quarter ahead forecasting results, respectively. The dashed line represents the SA baseline.

**Table 1.**Comparing the normalized average forecast risk of different combination methods under structural breaks.

SA | LinReg | BG | AFTER | |
---|---|---|---|---|

standard | 1.000 | 1.026 (0.011) | 1.005 (0.003) | 1.047 (0.010) |

$rw=40$ | 1.000 | 1.060 (0.033) | 0.992 (0.002) | 0.991 (0.009) |

$rw=20$ | 1.000 | 1.64 (0.42) | 0.980 (0.003) | 0.952 (0.007) |

**Table 2.**Comparing the normalized average forecast risk of different forecast combination methods after the procedure of screening and selecting the best X% models for subsequent forecast combining.

Best X% | 10% | 20% | 40% | 60% | 80% |
---|---|---|---|---|---|

$\sigma =2$, $\rho =0$ | |||||

AFTER | 0.998 | 0.989 | 0.966 | 0.951 | 0.945 |

BG | 1.000 | 0.999 | 0.997 | 0.997 | 0.996 |

LinReg | 1.017 | 1.024 | 1.056 | 1.098 | 1.151 |

$\sigma =2$, $\rho =0.5$ | |||||

AFTER | 0.996 | 0.990 | 0.968 | 0.956 | 0.951 |

BG | 1.000 | 0.998 | 0.997 | 0.997 | 0.996 |

LinReg | 1.013 | 1.024 | 1.043 | 1.095 | 1.159 |

$\sigma =4$, $\rho =0$ | |||||

AFTER | 0.994 | 0.987 | 0.984 | 0.981 | 0.974 |

BG | 0.999 | 0.998 | 0.998 | 0.998 | 0.997 |

LinReg | 1.002 | 1.012 | 1.056 | 1.101 | 1.163 |

$\sigma =4$, $\rho =0.5$ | |||||

AFTER | 0.995 | 0.990 | 0.976 | 0.969 | 0.961 |

BG | 1.000 | 0.999 | 0.998 | 0.997 | 0.997 |

LinReg | 1.004 | 1.010 | 1.030 | 1.086 | 1.136 |

**Table 3.**Comparing the performance of forecast combination methods with the Society of Professional Forecasters (SPF) datasets (pre-1990 period). Values shown are normalized MSFEs averaged over 1-, 2-, 3-, and 4-quarter-ahead forecasting. mAFTER, multi-level AFTER; RGDP, growth rate of real GDP; UNEMP, quarterly average of the monthly unemployment rate; REG, regression.

Target Variable | SA | LinReg | BG | AFTER | mAFTER |
---|---|---|---|---|---|

REG-imputed | |||||

PGDP | 1.00 | 1.88 | 0.95 | 0.90 | 0.90 |

RGDP | 1.00 | 1.64 | 1.00 | 1.11 | 1.01 |

UNEMP | 1.00 | 1.79 | 0.99 | 0.98 | 0.98 |

SA-imputed | |||||

PGDP | 1.00 | 2.17 | 0.98 | 0.95 | 0.95 |

RGDP | 1.00 | 1.83 | 1.00 | 1.13 | 1.03 |

UNEMP | 1.00 | 1.69 | 0.99 | 0.97 | 0.98 |

**Table 4.**Comparing the performance of the forecast combination methods with SPF datasets (post-2000 period). Values shown are normalized MSFEs averaged over 1-, 2-, 3-, and 4-quarter ahead forecasting.

Target Variable | SA | LinReg | BG | AFTER | mAFTER |
---|---|---|---|---|---|

REG-imputed | |||||

PGDP | 1.00 | 5.70 | 1.00 | 1.15 | 1.02 |

RGDP | 1.00 | 6.55 | 1.00 | 1.03 | 1.02 |

UNEMP | 1.00 | 1.03 | 0.95 | 0.90 | 0.91 |

SA-imputed | |||||

PGDP | 1.00 | 8.05 | 1.00 | 1.15 | 1.02 |

RGDP | 1.00 | 3.03 | 1.02 | 1.03 | 1.02 |

UNEMP | 1.00 | 1.01 | 0.96 | 0.92 | 0.93 |

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**MDPI and ACS Style**

Qian, W.; Rolling, C.A.; Cheng, G.; Yang, Y. On the Forecast Combination Puzzle. *Econometrics* **2019**, *7*, 39.
https://doi.org/10.3390/econometrics7030039

**AMA Style**

Qian W, Rolling CA, Cheng G, Yang Y. On the Forecast Combination Puzzle. *Econometrics*. 2019; 7(3):39.
https://doi.org/10.3390/econometrics7030039

**Chicago/Turabian Style**

Qian, Wei, Craig A. Rolling, Gang Cheng, and Yuhong Yang. 2019. "On the Forecast Combination Puzzle" *Econometrics* 7, no. 3: 39.
https://doi.org/10.3390/econometrics7030039