# Robust Estimation and Forecasting of Climate Change Using Score-Driven Ice-Age Models

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## Abstract

**:**

## 1. Introduction

## 2. Climate Econometrics

#### 2.1. Benchmark Ice-Age Model

#### 2.2. Score-Driven Ice-Age Models

#### 2.2.1. Score-Driven Homoskedastic Ice-Age Model

#### 2.2.2. Score-Driven Heteroskedastic Ice-Age Model

## 3. Empirical Results

#### 3.1. Data

#### 3.2. Estimation Results

#### 3.3. Forecasting Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Evolution of ${\mathrm{Ice}}_{t}$, ${\mathrm{CO}}_{2,t}$, and ${\mathrm{Temp}}_{t}$ from 798 thousand years ago to 1 thousand years ago.

**Figure 2.**Evolution of ${\mathrm{Ec}}_{t}$, ${\mathrm{Ob}}_{t}$, and ${\mathrm{Pr}}_{t}$ from 798 thousand years ago to 1 thousand years ago.

**Figure 3.**IRFs up to 20 lags for the benchmark ice-age model. Notes: The confidence interval is mean ± one standard deviation that is estimated for 2302 out of the 3 million simulations under the restrictions of Table 1. Ice uses the ${\delta}^{18}\mathrm{O}$ proxy; 1 ${\mathrm{CO}}_{2}$ is 780 gigatonnes; 1 Temp is 1 °C.

**Figure 4.**IRFs up to 20 lags for the score-driven ice-age model for the normal distribution. Notes: The confidence interval is mean ± one standard deviation that is estimated for 1771 out of the 3 million simulations under the restrictions of Table 1. Ice uses the ${\delta}^{18}\mathrm{O}$ proxy; 1 ${\mathrm{CO}}_{2}$ is 780 gigatonnes; 1 Temp is 1 °C.

**Figure 5.**IRFs up to 20 lags for the score-driven homoskedastic ice-age model for the t-distribution. Notes: The confidence interval is mean ± one standard deviation that is estimated for 2289 out of the 3 million simulations under the restrictions of Table 1. Ice uses the ${\delta}^{18}\mathrm{O}$ proxy; 1 ${\mathrm{CO}}_{2}$ is 780 gigatonnes; 1 Temp is 1 °C.

**Figure 6.**IRFs up to 20 lags for the score-driven heteroskedastic ice-age model for the t-distribution. Notes: The confidence interval is mean ± one standard deviation that is estimated for 1698 out of the 3 million simulations under the restrictions of Table 1. Ice uses the ${\delta}^{18}\mathrm{O}$ proxy; 1 ${\mathrm{CO}}_{2}$ is 780 gigatonnes; 1 Temp is 1 °C.

**Figure 7.**Robustness of the scaled score function to extreme values. Note: ${\u03f5}_{3,t}=0$ is assumed for this figure.

**Figure 8.**Multi-step ahead out-of-sample forecasts for the period of the last 100 thousand years. Notes: The confidence interval is mean ± one standard deviation. The true values are indicated by thick lines.

**Figure 9.**Multi-step ahead out-of-sample forecasts for the period of 223 to 124 thousand years ago. Notes: The confidence interval is mean ± one standard deviation. The true values are indicated by thick lines.

**Figure 10.**One-step ahead out-of-sample forecasts for the last 100 thousand years. Notes: The confidence interval is mean ± one standard deviation. The true values are indicated by thick lines.

${\mathbf{Ice}}_{\mathit{t}}$ Shock | ${\mathbf{CO}}_{2,\mathit{t}}$ Shock | ${\mathbf{Temp}}_{\mathit{t}}$ Shock | |
---|---|---|---|

${\mathrm{Ice}}_{t}$ | + | − | − |

${\mathrm{CO}}_{2,t}$ | − | + | + |

${\mathrm{Temp}}_{t}$ | − | + | + |

(a) Dependent Variables | ${\mathbf{Ice}}_{\mathit{t}}$ | ${\mathbf{CO}}_{2,\mathit{t}}$ | ${\mathbf{Temp}}_{\mathit{t}}$ |
---|---|---|---|

Variable | Ice volume | Atmospheric ${\mathrm{CO}}_{2}$ | Antarctic-based land surface temperature |

Start date | 798 thousand years ago | 798 thousand years ago | 798 thousand years ago |

End date | 1 thousand years ago | 1 thousand years ago | 1 thousand years ago |

Data frequency | 1 thousand years | 1 thousand years | 1 thousand years |

Measurement | Based on the ${\delta}^{18}\mathrm{O}$ proxy | 1 unit = 780 gigatonnes of ${\mathrm{CO}}_{2}$ | 1 unit = 1 Celsius degree |

Data source | Lisiecki and Raymo (2005) | Lüthi et al. (2008) | Jouzel et al. (2007) |

Sample size | 798 | 798 | 798 |

Minimum | $3.1000$ | $1.7269$ | $-10.2530$ |

Maximum | $5.0800$ | $2.9500$ | $3.7662$ |

Mean | $4.1707$ | $2.2382$ | $-5.2892$ |

Standard deviation | $0.4467$ | $0.2546$ | $2.9009$ |

(b) Explanatory Variables | ${\mathrm{Ec}}_{\mathbf{t}}$ | ${\mathrm{Ob}}_{\mathbf{t}}$ | ${\mathrm{Pr}}_{\mathbf{t}}$ |

Variable | Eccentricity of the Earth’s orbit | Obliquity | Precession of the equinox |

Start date | 798 thousand years ago | 798 thousand years ago | 798 thousand years ago |

End date | 1 thousand years ago | 1 thousand years ago | 1 thousand years ago |

Data frequency | 1 thousand years | 1 thousand years | 1 thousand years |

Measurement | Periodicity deriving from the | Periodicity deriving from the | Periodicity deriving from the |

changing non-circularity of the Earth’s orbit | changes in the tilt of the Earth’s rotational axis | precession of the equinox | |

(zero denotes circularity). | relative to the ecliptic (1 unit = 10 degrees). | (1 unit = 1 degree). | |

Data source | Paillard et al. (1996) | Paillard et al. (1996) | Paillard et al. (1996) |

Sample size | 798 | 798 | 798 |

Minimum | $0.0042$ | $2.2076$ | $0.0008$ |

Maximum | $0.0500$ | $2.4455$ | $0.3593$ |

Mean | $0.0271$ | $2.3342$ | $0.1802$ |

Standard deviation | $0.0119$ | $0.0591$ | $0.1039$ |

Benchmark | Score-Driven Homoskedastic | Score-Driven Homoskedastic | Score-Driven Heteroskedastic | ||||||
---|---|---|---|---|---|---|---|---|---|

Ice-Age Model | Gaussian Ice-Age Model | t Ice-Age Model | t Ice-Age Model | ||||||

${\gamma}_{0,1}$ | $1.3735$ *** $\left(0.3009\right)$ | ${\gamma}_{0,1}$ | $1.2697$ *** $\left(0.2884\right)$ | ${\gamma}_{0,1}$ | $1.2745$ *** $\left(0.2798\right)$ | ${\gamma}_{0,1}$ | $1.1817$ *** $\left(0.2812\right)$ | ${\omega}_{1}$ | $-1.1773$ *** $\left(0.2845\right)$ |

${\gamma}_{0,2}$ | $1.8718$ *** $\left(0.3023\right)$ | ${\gamma}_{0,2}$ | $1.6589$ *** $\left(0.3552\right)$ | ${\gamma}_{0,2}$ | $1.8197$ *** $\left(0.3579\right)$ | ${\gamma}_{0,2}$ | $1.4068$ *** $\left(0.3259\right)$ | ${\omega}_{2}$ | $-0.7409$ *** $\left(0.1416\right)$ |

${\gamma}_{0,3}$ | $-2.6657$ *** $\left(0.6905\right)$ | ${\gamma}_{0,3}$ | $-1.4256{\phantom{\rule{3.33333pt}{0ex}}}^{+}\left(0.9049\right)$ | ${\gamma}_{0,3}$ | $-1.4366{\phantom{\rule{3.33333pt}{0ex}}}^{+}\left(0.9252\right)$ | ${\gamma}_{0,3}$ | $-0.6955\left(0.6592\right)$ | ${\omega}_{3}$ | $-0.0455$ *** $\left(0.0139\right)$ |

${\mathsf{\Gamma}}_{1,1,1}$ | $0.8549$ *** $\left(0.0144\right)$ | ${\mathsf{\Gamma}}_{1,1,1}$ | $0.8733$ *** $\left(0.0155\right)$ | ${\mathsf{\Gamma}}_{1,1,1}$ | $0.8738$ *** $\left(0.0150\right)$ | ${\mathsf{\Gamma}}_{1,1,1}$ | $0.8824$ *** $\left(0.0162\right)$ | ${\beta}_{1}$ | $0.5266$ *** $\left(0.1144\right)$ |

${\mathsf{\Gamma}}_{1,1,3}$ | $-0.0208$ *** $\left(0.0020\right)$ | ${\mathsf{\Gamma}}_{1,1,3}$ | $-0.0175$ *** $\left(0.0024\right)$ | ${\mathsf{\Gamma}}_{1,1,3}$ | $-0.0175$ *** $\left(0.0024\right)$ | ${\mathsf{\Gamma}}_{1,1,3}$ | $-0.0172$ *** $\left(0.0025\right)$ | ${\beta}_{2}$ | $0.7628$ *** $\left(0.0446\right)$ |

${\mathsf{\Gamma}}_{1,2,2}$ | $0.8468$ *** $\left(0.0176\right)$ | ${\mathsf{\Gamma}}_{1,2,2}$ | $0.8410$ *** $\left(0.0287\right)$ | ${\mathsf{\Gamma}}_{1,2,2}$ | $0.8291$ *** $\left(0.0286\right)$ | ${\mathsf{\Gamma}}_{1,2,2}$ | $0.8528$ *** $\left(0.0231\right)$ | ${\beta}_{3}$ | $0.8796$ *** $\left(0.0286\right)$ |

${\mathsf{\Gamma}}_{1,2,3}$ | $0.0136$ *** $\left(0.0016\right)$ | ${\mathsf{\Gamma}}_{1,2,3}$ | $0.0125$ *** $\left(0.0026\right)$ | ${\mathsf{\Gamma}}_{1,2,3}$ | $0.0137$ *** $\left(0.0026\right)$ | ${\mathsf{\Gamma}}_{1,2,3}$ | $0.0122$ *** $\left(0.0022\right)$ | ${\alpha}_{1}$ | $0.0500$ *** $\left(0.0192\right)$ |

${\mathsf{\Gamma}}_{1,3,2}$ | $0.8587$ *** $\left(0.2556\right)$ | ${\mathsf{\Gamma}}_{1,3,2}$ | $0.3714\left(0.3333\right)$ | ${\mathsf{\Gamma}}_{1,3,2}$ | $0.3771\left(0.3396\right)$ | ${\mathsf{\Gamma}}_{1,3,2}$ | $0.1382\left(0.2458\right)$ | ${\alpha}_{2}$ | $0.0930$ *** $\left(0.0154\right)$ |

${\mathsf{\Gamma}}_{1,3,3}$ | $0.8684$ *** $\left(0.0231\right)$ | ${\mathsf{\Gamma}}_{1,3,3}$ | $0.9008$ *** $\left(0.0297\right)$ | ${\mathsf{\Gamma}}_{1,3,3}$ | $0.8995$ *** $\left(0.0304\right)$ | ${\mathsf{\Gamma}}_{1,3,3}$ | $0.9377$ *** $\left(0.0228\right)$ | ${\alpha}_{3}$ | $0.1070$ *** $\left(0.0172\right)$ |

${\mathsf{\Gamma}}_{2,1,1}$ | $95.8353$ *** $\left(30.3012\right)$ | ${\mathsf{\Gamma}}_{2,1,1}$ | $88.7529$ *** $\left(29.2107\right)$ | ${\mathsf{\Gamma}}_{2,1,1}$ | $88.1550$ *** $\left(28.6783\right)$ | ${\mathsf{\Gamma}}_{2,1,1}$ | $84.6136$ *** $\left(28.7107\right)$ | ${\alpha}_{1}^{*}$ | $0.0513$ *** $\left(0.0163\right)$ |

${\mathsf{\Gamma}}_{2,1,4}$ | $-47.5937$ *** $\left(12.4804\right)$ | ${\mathsf{\Gamma}}_{2,1,4}$ | $-45.5349$ *** $\left(12.0258\right)$ | ${\mathsf{\Gamma}}_{2,1,4}$ | $-45.9952$ *** $\left(11.8462\right)$ | ${\mathsf{\Gamma}}_{2,1,4}$ | $-42.7119$ *** $\left(12.0538\right)$ | ${\alpha}_{2}^{*}$ | $-0.0199{\phantom{\rule{3.33333pt}{0ex}}}^{+}\left(0.0124\right)$ |

${\mathsf{\Gamma}}_{2,1,5}$ | $-5.2167$ *** $\left(1.0663\right)$ | ${\mathsf{\Gamma}}_{2,1,5}$ | $-5.4569$ *** $\left(1.0321\right)$ | ${\mathsf{\Gamma}}_{2,1,5}$ | $-5.6132$ *** $\left(1.0064\right)$ | ${\mathsf{\Gamma}}_{2,1,5}$ | $-4.8681$ *** $\left(0.9172\right)$ | ${\alpha}_{3}^{*}$ | $-0.0094\left(0.0123\right)$ |

${\mathsf{\Gamma}}_{2,2,1}$ | $13.8095$ *** $\left(3.8487\right)$ | ${\mathsf{\Gamma}}_{2,2,1}$ | $14.5128$ *** $\left(4.8910\right)$ | ${\mathsf{\Gamma}}_{2,2,1}$ | $14.8536$ *** $\left(4.8133\right)$ | ${\mathsf{\Gamma}}_{2,2,1}$ | $11.9581$ *** $\left(3.5543\right)$ | ||

${\mathsf{\Gamma}}_{2,2,7}$ | $0.2106$ *** $\left(0.0446\right)$ | ${\mathsf{\Gamma}}_{2,2,7}$ | $0.1688$ *** $\left(0.0518\right)$ | ${\mathsf{\Gamma}}_{2,2,7}$ | $0.1876$ *** $\left(0.0515\right)$ | ${\mathsf{\Gamma}}_{2,2,7}$ | $0.1270$ *** $\left(0.0477\right)$ | ||

${\mathsf{\Gamma}}_{2,3,1}$ | $-335.9696$ *** $\left(38.6135\right)$ | ${\mathsf{\Gamma}}_{2,3,1}$ | $-311.6342$ *** $\left(43.7423\right)$ | ${\mathsf{\Gamma}}_{2,3,1}$ | $-326.4885$ *** $\left(44.2197\right)$ | ${\mathsf{\Gamma}}_{2,3,1}$ | $-272.0190$ *** $\left(32.9400\right)$ | ||

${\mathsf{\Gamma}}_{2,3,4}$ | $254.2055$ *** $\left(28.4900\right)$ | ${\mathsf{\Gamma}}_{2,3,4}$ | $240.1468$ *** $\left(30.8981\right)$ | ${\mathsf{\Gamma}}_{2,3,4}$ | $258.2711$ *** $\left(30.4240\right)$ | ${\mathsf{\Gamma}}_{2,3,4}$ | $194.7069$ *** $\left(22.0173\right)$ | ||

${\mathsf{\Gamma}}_{2,3,5}$ | $26.6287$ *** $\left(8.3118\right)$ | ${\mathsf{\Gamma}}_{2,3,5}$ | $25.0557$ *** $\left(7.8012\right)$ | ${\mathsf{\Gamma}}_{2,3,5}$ | $28.2720$ *** $\left(7.6689\right)$ | ${\mathsf{\Gamma}}_{2,3,5}$ | $17.5525$ *** $\left(5.9466\right)$ | ||

${\mathsf{\Gamma}}_{3,1,1}$ | $-93.5393$ *** $\left(31.3704\right)$ | ${\mathsf{\Gamma}}_{3,1,1}$ | $-83.6310$ *** $\left(30.7764\right)$ | ${\mathsf{\Gamma}}_{3,1,1}$ | $-81.9865$ *** $\left(30.4458\right)$ | ${\mathsf{\Gamma}}_{3,1,1}$ | $-83.4944$ *** $\left(30.2589\right)$ | ||

${\mathsf{\Gamma}}_{3,1,2}$ | $-0.3706$ *** $\left(0.1233\right)$ | ${\mathsf{\Gamma}}_{3,1,2}$ | $-0.3519$ *** $\left(0.1164\right)$ | ${\mathsf{\Gamma}}_{3,1,2}$ | $-0.3544$ *** $\left(0.1131\right)$ | ${\mathsf{\Gamma}}_{3,1,2}$ | $-0.3307$ *** $\left(0.1135\right)$ | ||

${\mathsf{\Gamma}}_{3,1,4}$ | $46.7753$ *** $\left(12.9406\right)$ | ${\mathsf{\Gamma}}_{3,1,4}$ | $43.5364$ *** $\left(12.6666\right)$ | ${\mathsf{\Gamma}}_{3,1,4}$ | $43.5483$ *** $\left(12.5889\right)$ | ${\mathsf{\Gamma}}_{3,1,4}$ | $42.4409$ *** $\left(12.6434\right)$ | ||

${\mathsf{\Gamma}}_{3,2,1}$ | $-27.1270$ *** $\left(6.7356\right)$ | ${\mathsf{\Gamma}}_{3,2,1}$ | $-29.1279$ *** $\left(7.5563\right)$ | ${\mathsf{\Gamma}}_{3,2,1}$ | $-30.6263$ *** $\left(7.4143\right)$ | ${\mathsf{\Gamma}}_{3,2,1}$ | $-26.3809$ *** $\left(6.1789\right)$ | ||

${\mathsf{\Gamma}}_{3,2,2}$ | $-1.1138$ *** $\left(0.2191\right)$ | ${\mathsf{\Gamma}}_{3,2,2}$ | $-0.9235$ *** $\left(0.2529\right)$ | ${\mathsf{\Gamma}}_{3,2,2}$ | $-1.0217$ *** $\left(0.2528\right)$ | ${\mathsf{\Gamma}}_{3,2,2}$ | $-0.7289$ *** $\left(0.2339\right)$ | ||

${\mathsf{\Gamma}}_{3,2,4}$ | $5.6423$ ** $\left(2.2875\right)$ | ${\mathsf{\Gamma}}_{3,2,4}$ | $6.2517$ ** $\left(2.4809\right)$ | ${\mathsf{\Gamma}}_{3,2,4}$ | $6.7580$ *** $\left(2.4337\right)$ | ${\mathsf{\Gamma}}_{3,2,4}$ | $6.1482$ *** $\left(2.1916\right)$ | ||

${\mathsf{\Gamma}}_{3,3,4}$ | $-111.3537$ *** $\left(24.0465\right)$ | ${\mathsf{\Gamma}}_{3,3,4}$ | $-107.2597$ *** $\left(26.1398\right)$ | ${\mathsf{\Gamma}}_{3,3,4}$ | $-119.1342$ *** $\left(26.2451\right)$ | ${\mathsf{\Gamma}}_{3,3,4}$ | $-78.4289$ *** $\left(16.6672\right)$ | ||

${\mathsf{\Psi}}_{1,1,1}$ | NA | ${\mathsf{\Psi}}_{1,1,1}$ | $0.7982$ *** $\left(0.0343\right)$ | ${\mathsf{\Psi}}_{1,1,1}$ | $0.9020$ *** $\left(0.0438\right)$ | ${\mathsf{\Psi}}_{1,1,1}$ | $0.9651$ *** $\left(0.0551\right)$ | ||

${\mathsf{\Psi}}_{1,1,3}$ | NA | ${\mathsf{\Psi}}_{1,1,3}$ | $-0.0295$ *** $\left(0.0041\right)$ | ${\mathsf{\Psi}}_{1,1,3}$ | $-0.0325$ *** $\left(0.0050\right)$ | ${\mathsf{\Psi}}_{1,1,3}$ | $-0.0289$ *** $\left(0.0047\right)$ | ||

${\mathsf{\Psi}}_{1,2,2}$ | NA | ${\mathsf{\Psi}}_{1,2,2}$ | $1.1351$ *** $\left(0.0300\right)$ | ${\mathsf{\Psi}}_{1,2,2}$ | $1.3418$ *** $\left(0.0545\right)$ | ${\mathsf{\Psi}}_{1,2,2}$ | $1.3943$ *** $\left(0.0599\right)$ | ||

${\mathsf{\Psi}}_{1,2,3}$ | NA | ${\mathsf{\Psi}}_{1,2,3}$ | $0.0197$ *** $\left(0.0025\right)$ | ${\mathsf{\Psi}}_{1,2,3}$ | $0.0221$ *** $\left(0.0030\right)$ | ${\mathsf{\Psi}}_{1,2,3}$ | $0.0166$ *** $\left(0.0025\right)$ | ||

${\mathsf{\Psi}}_{1,3,2}$ | NA | ${\mathsf{\Psi}}_{1,3,2}$ | $4.1436$ *** $\left(0.5376\right)$ | ${\mathsf{\Psi}}_{1,3,2}$ | $4.9368$ *** $\left(0.7038\right)$ | ${\mathsf{\Psi}}_{1,3,2}$ | $4.7059$ *** $\left(0.6182\right)$ | ||

${\mathsf{\Psi}}_{1,3,3}$ | NA | ${\mathsf{\Psi}}_{1,3,3}$ | $0.8723$ *** $\left(0.0347\right)$ | ${\mathsf{\Psi}}_{1,3,3}$ | $1.0084$ *** $\left(0.0480\right)$ | ${\mathsf{\Psi}}_{1,3,3}$ | $0.9860$ *** $\left(0.0538\right)$ | ||

${\mathsf{\Omega}}_{1,1}$ | $0.0891$ *** $\left(0.0019\right)$ | ${\mathsf{\Omega}}_{1,1}$ | $0.0887$ *** $\left(0.0019\right)$ | ${\mathsf{\Omega}}_{1,1}$ | $0.0861$ *** $\left(0.0021\right)$ | ${\mathsf{\Omega}}_{1,1}$ | NA | ||

${\mathsf{\Omega}}_{2,1}$ | $-0.0102$ *** $\left(0.0018\right)$ | ${\mathsf{\Omega}}_{2,1}$ | $-0.0085$ *** $\left(0.0018\right)$ | ${\mathsf{\Omega}}_{2,1}$ | $-0.0096$ *** $\left(0.0018\right)$ | ${\mathsf{\Omega}}_{2,1}$ | $-0.1698$ *** $\left(0.0410\right)$ | ||

${\mathsf{\Omega}}_{2,2}$ | $0.0524$ *** $\left(0.0011\right)$ | ${\mathsf{\Omega}}_{2,2}$ | $0.0492$ *** $\left(0.0010\right)$ | ${\mathsf{\Omega}}_{2,2}$ | $0.0475$ *** $\left(0.0010\right)$ | ${\mathsf{\Omega}}_{2,2}$ | NA | ||

${\mathsf{\Omega}}_{3,1}$ | $-0.1755$ *** $\left(0.0267\right)$ | ${\mathsf{\Omega}}_{3,1}$ | $-0.1600$ *** $\left(0.0262\right)$ | ${\mathsf{\Omega}}_{3,1}$ | $-0.1672$ *** $\left(0.0274\right)$ | ${\mathsf{\Omega}}_{3,1}$ | $-0.2343$ *** $\left(0.0449\right)$ | ||

${\mathsf{\Omega}}_{3,2}$ | $0.3808$ *** $\left(0.0249\right)$ | ${\mathsf{\Omega}}_{3,2}$ | $0.3440$ *** $\left(0.0239\right)$ | ${\mathsf{\Omega}}_{3,2}$ | $0.3476$ *** $\left(0.0247\right)$ | ${\mathsf{\Omega}}_{3,2}$ | $0.5107$ *** $\left(0.0424\right)$ | ||

${\mathsf{\Omega}}_{3,3}$ | $0.6711$ *** $\left(0.0145\right)$ | ${\mathsf{\Omega}}_{3,3}$ | $0.6684$ *** $\left(0.0146\right)$ | ${\mathsf{\Omega}}_{3,3}$ | $0.6458$ *** $\left(0.0152\right)$ | ${\mathsf{\Omega}}_{3,3}$ | NA | ||

$\nu $ | NA | $\nu $ | NA | $\nu $ | $38.2860$ *** $\left(6.1464\right)$ | $\nu $ | $27.7970$ *** $\left(5.1586\right)$ |

^{+}indicate significance at the 1%, 5%, 10%, and 15% levels, respectively.

Benchmark | Score-Driven Homoskedastic | Score-Driven Homoskedastic | Score-Driven Heteroskedastic | |
---|---|---|---|---|

Ice-Age Model | Gaussian Ice-Age Model | t Ice-Age Model | t Ice-Age Model | |

LL | $1.5094$ | $1.5801$ | $1.6026$ | $1.7644$ |

AIC | $-2.9435$ | $-3.0699$ | $-3.1126$ | $-3.4134$ |

BIC | $-2.7675$ | $-2.8587$ | $-2.8955$ | $-3.1435$ |

HQC | $-2.8759$ | $-2.9887$ | $-3.0291$ | $-3.3097$ |

${C}_{\mu}$ | $0.9663$ | $0.9453$ | $0.9444$ | $0.9543$ |

${C}_{\lambda ,1}$ | NA | NA | NA | $0.5266$ |

${C}_{\lambda ,2}$ | NA | NA | NA | $0.7628$ |

${C}_{\lambda ,3}$ | NA | NA | NA | $0.8796$ |

LB ${v}_{1,t}$ (p-value) | $20.5200\phantom{\rule{3.33333pt}{0ex}}\left(0.8448\right)$ | $17.8671\phantom{\rule{3.33333pt}{0ex}}\left(0.9294\right)$ | $19.1502\phantom{\rule{3.33333pt}{0ex}}\left(0.8934\right)$ | $19.3022\phantom{\rule{3.33333pt}{0ex}}\left(0.8886\right)$ |

LB ${v}_{2,t}$ (p-value) | $118.9560$ *** $\phantom{\rule{3.33333pt}{0ex}}\left(0.0000\right)$ | $28.4522\phantom{\rule{3.33333pt}{0ex}}\left(0.4407\right)$ | $29.2620\phantom{\rule{3.33333pt}{0ex}}\left(0.3993\right)$ | $30.8129\phantom{\rule{3.33333pt}{0ex}}\left(0.3255\right)$ |

LB ${v}_{3,t}$ (p-value) | $41.8520$ ** $\left(0.0448\right)$ | $30.2398\phantom{\rule{3.33333pt}{0ex}}\left(0.3518\right)$ | $33.0351\phantom{\rule{3.33333pt}{0ex}}\left(0.2345\right)$ | $37.1410\phantom{\rule{3.33333pt}{0ex}}\left(0.1158\right)$ |

LB ${\u03f5}_{1,t}$ (p-value) | $20.5200\phantom{\rule{3.33333pt}{0ex}}\left(0.8448\right)$ | $17.867\phantom{\rule{3.33333pt}{0ex}}\left(0.9294\right)$ | $19.1502\phantom{\rule{3.33333pt}{0ex}}\left(0.8934\right)$ | $16.9958\phantom{\rule{3.33333pt}{0ex}}\left(0.9487\right)$ |

LB ${\u03f5}_{2,t}$ (p-value) | $98.1577$ *** $\left(0.0000\right)$ | $28.6594\phantom{\rule{3.33333pt}{0ex}}\left(0.4300\right)$ | $27.1411\phantom{\rule{3.33333pt}{0ex}}\left(0.5106\right)$ | $21.1773\phantom{\rule{3.33333pt}{0ex}}\left(0.8179\right)$ |

LB ${\u03f5}_{3,t}$ (p-value) | $47.0696$ ** $\left(0.0135\right)$ | $44.3814$ ** $\left(0.0255\right)$ | $45.6695$ ** $\left(0.0189\right)$ | $33.1683\phantom{\rule{3.33333pt}{0ex}}\left(0.2296\right)$ |

LB ${u}_{1,t}$ (p-value) | NA | NA | $19.9979\phantom{\rule{3.33333pt}{0ex}}\left(0.8645\right)$ | $21.7319\phantom{\rule{3.33333pt}{0ex}}\left(0.7935\right)$ |

LB ${u}_{2,t}$ (p-value) | NA | NA | $27.8256\phantom{\rule{3.33333pt}{0ex}}\left(0.4737\right)$ | $27.7605\phantom{\rule{3.33333pt}{0ex}}\left(0.4772\right)$ |

LB ${u}_{3,t}$ (p-value) | NA | NA | $33.5007\phantom{\rule{3.33333pt}{0ex}}\left(0.2178\right)$ | $35.0719\phantom{\rule{3.33333pt}{0ex}}\left(0.1678\right)$ |

LB ${e}_{1,t}$ (p-value) | NA | NA | NA | $34.1775\phantom{\rule{3.33333pt}{0ex}}\left(0.1951\right)$ |

LB ${e}_{2,t}$ (p-value) | NA | NA | NA | $25.8774\phantom{\rule{3.33333pt}{0ex}}\left(0.5798\right)$ |

LB ${e}_{3,t}$ (p-value) | NA | NA | NA | $22.7776\phantom{\rule{3.33333pt}{0ex}}\left(0.7441\right)$ |

Score-Driven | Score-Driven | |||||||
---|---|---|---|---|---|---|---|---|

Homoskedastic | Score-Driven | Score-Driven | Homoskedastic | Score-Driven | Score-Driven | |||

Benchmark | Gaussian | Homoskedastic | Heteroskedastic | Benchmark | Gaussian | Homoskedastic | Heteroskedastic | |

Ice-Age Model | Ice-Age Model | t Ice-Age Model | t Ice-Age Model | Ice-Age Model | Ice-Age Model | t Ice-Age Model | t Ice-Age Model | |

${\mathrm{Ice}}_{t}$ | MSE | MSE | MSE | MSE | MAE | MAE | MAE | MAE |

last 100,000 years | $\mathbf{0}.\mathbf{0917}$ | $0.0982$ | $0.0969$ | $0.1005$ | $0.2388$ | $0.2520$ | $0.2555$ | $\mathbf{0}.\mathbf{2380}$ |

last 90,000 years | $\mathbf{0}.\mathbf{1003}$ | $0.1075$ | $0.1058$ | $0.1098$ | $0.2535$ | $0.2680$ | $0.2713$ | $\mathbf{0}.\mathbf{2515}$ |

last 80,000 years | $\mathbf{0}.\mathbf{1081}$ | $0.1169$ | $0.1148$ | $0.1205$ | $\mathbf{0}.\mathbf{2661}$ | $0.2830$ | $0.2867$ | $0.2664$ |

last 70,000 years | $\mathbf{0}.\mathbf{1186}$ | $0.1284$ | $0.1258$ | $0.1352$ | $\mathbf{0}.\mathbf{2818}$ | $0.3008$ | $0.3044$ | $0.2881$ |

last 60,000 years | $\mathbf{0}.\mathbf{1259}$ | $0.1370$ | $0.1317$ | $0.1526$ | $\mathbf{0}.\mathbf{2848}$ | $0.3063$ | $0.3066$ | $0.3097$ |

last 50,000 years | $\mathbf{0}.\mathbf{1416}$ | $0.1549$ | $0.1472$ | $0.1765$ | $\mathbf{0}.\mathbf{3005}$ | $0.3265$ | $0.3238$ | $0.3373$ |

last 40,000 years | $\mathbf{0}.\mathbf{1712}$ | $0.1868$ | $0.1765$ | $0.2150$ | $\mathbf{0}.\mathbf{3444}$ | $0.3738$ | $0.3689$ | $0.3914$ |

last 30,000 years | $0.2148$ | $0.2292$ | $\mathbf{0}.\mathbf{2134}$ | $0.2691$ | $\mathbf{0}.\mathbf{3946}$ | $0.4187$ | $0.4080$ | $0.4471$ |

last 20,000 years | $0.3049$ | $0.3164$ | $\mathbf{0}.\mathbf{2911}$ | $0.3767$ | $0.5037$ | $0.5143$ | $\mathbf{0}.\mathbf{4948}$ | $0.5575$ |

last 10,000 years | $0.4889$ | $0.5027$ | $\mathbf{0}.\mathbf{4527}$ | $0.6162$ | $0.6935$ | $0.7032$ | $\mathbf{0}.\mathbf{6667}$ | $0.7810$ |

${\mathrm{CO}}_{2,t}$ | MSE | MSE | MSE | MSE | MAE | MAE | MAE | MAE |

last 100,000 years | $\mathbf{0}.\mathbf{0399}$ | $0.0432$ | $0.0424$ | $0.0466$ | $\mathbf{0}.\mathbf{1471}$ | $0.1503$ | $0.1506$ | $0.1556$ |

last 90,000 years | $\mathbf{0}.\mathbf{0440}$ | $0.0479$ | $0.0470$ | $0.0517$ | $\mathbf{0}.\mathbf{1580}$ | $0.1642$ | $0.1641$ | $0.1709$ |

last 80,000 years | $\mathbf{0}.\mathbf{0460}$ | $0.0509$ | $0.0494$ | $0.0550$ | $\mathbf{0}.\mathbf{1595}$ | $0.1675$ | $0.1664$ | $0.1755$ |

last 70,000 years | $\mathbf{0}.\mathbf{0513}$ | $0.0568$ | $0.0552$ | $0.0623$ | $\mathbf{0}.\mathbf{1719}$ | $0.1810$ | $0.1797$ | $0.1920$ |

last 60,000 years | $\mathbf{0}.\mathbf{0590}$ | $0.0653$ | $0.0634$ | $0.0702$ | $\mathbf{0}.\mathbf{1900}$ | $0.1995$ | $0.1986$ | $0.2064$ |

last 50,000 years | $\mathbf{0}.\mathbf{0692}$ | $0.0769$ | $0.0746$ | $0.0831$ | $\mathbf{0}.\mathbf{2128}$ | $0.2254$ | $0.2240$ | $0.2352$ |

last 40,000 years | $\mathbf{0}.\mathbf{0842}$ | $0.0935$ | $0.0902$ | $0.1015$ | $\mathbf{0}.\mathbf{2462}$ | $0.2622$ | $0.2598$ | $0.2754$ |

last 30,000 years | $\mathbf{0}.\mathbf{1104}$ | $0.1214$ | $0.1165$ | $0.1313$ | $\mathbf{0}.\mathbf{3089}$ | $0.3263$ | $0.3212$ | $0.3404$ |

last 20,000 years | $0.1269$ | $0.1338$ | $\mathbf{0}.\mathbf{1224}$ | $0.1467$ | $0.3261$ | $0.3353$ | $\mathbf{0}.\mathbf{3212}$ | $0.3528$ |

last 10,000 years | $0.1891$ | $0.1978$ | $\mathbf{0}.\mathbf{1733}$ | $0.2221$ | $0.4280$ | $0.4379$ | $\mathbf{0}.\mathbf{4088}$ | $0.4664$ |

${\mathrm{Temp}}_{t}$ | MSE | MSE | MSE | MSE | MAE | MAE | MAE | MAE |

last 100,000 years | $\mathbf{4}.\mathbf{1809}$ | $4.5136$ | $4.5168$ | $4.8663$ | $\mathbf{1}.\mathbf{6976}$ | $1.7704$ | $1.7906$ | $1.8453$ |

last 90,000 years | $\mathbf{4}.\mathbf{4536}$ | $4.8600$ | $4.8533$ | $5.2546$ | $\mathbf{1}.\mathbf{7714}$ | $1.8574$ | $1.8754$ | $1.9406$ |

last 80,000 years | $\mathbf{4}.\mathbf{5747}$ | $5.1051$ | $5.0628$ | $5.4916$ | $\mathbf{1}.\mathbf{7939}$ | $1.9107$ | $1.9207$ | $1.9848$ |

last 70,000 years | $\mathbf{5}.\mathbf{0599}$ | $5.6824$ | $5.6177$ | $6.0398$ | $\mathbf{1}.\mathbf{9174}$ | $2.0581$ | $2.0621$ | $2.1106$ |

last 60,000 years | $\mathbf{5}.\mathbf{7960}$ | $6.5246$ | $6.4482$ | $6.8760$ | $\mathbf{2}.\mathbf{1167}$ | $2.2824$ | $2.2886$ | $2.3143$ |

last 50,000 years | $\mathbf{6}.\mathbf{4533}$ | $7.3840$ | $7.2948$ | $7.8500$ | $\mathbf{2}.\mathbf{2670}$ | $2.4787$ | $2.4871$ | $2.5283$ |

last 40,000 years | $\mathbf{7}.\mathbf{3939}$ | $8.3779$ | $8.1927$ | $9.0115$ | $\mathbf{2}.\mathbf{4913}$ | $2.7003$ | $2.6894$ | $2.7802$ |

last 30,000 years | $\mathbf{8}.\mathbf{8750}$ | $9.7268$ | $9.3292$ | $10.5446$ | $\mathbf{2}.\mathbf{8029}$ | $2.9516$ | $2.8995$ | $3.0520$ |

last 20,000 years | $10.0692$ | $10.5303$ | $\mathbf{9}.\mathbf{5685}$ | $11.9260$ | $2.9566$ | $3.0268$ | $\mathbf{2}.\mathbf{8845}$ | $3.2107$ |

last 10,000 years | $14.0302$ | $14.6377$ | $\mathbf{12}.\mathbf{8303}$ | $16.8965$ | $3.7069$ | $3.7892$ | $\mathbf{3}.\mathbf{5384}$ | $4.0858$ |

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## Share and Cite

**MDPI and ACS Style**

Blazsek, S.; Escribano, A.
Robust Estimation and Forecasting of Climate Change Using Score-Driven Ice-Age Models. *Econometrics* **2022**, *10*, 9.
https://doi.org/10.3390/econometrics10010009

**AMA Style**

Blazsek S, Escribano A.
Robust Estimation and Forecasting of Climate Change Using Score-Driven Ice-Age Models. *Econometrics*. 2022; 10(1):9.
https://doi.org/10.3390/econometrics10010009

**Chicago/Turabian Style**

Blazsek, Szabolcs, and Alvaro Escribano.
2022. "Robust Estimation and Forecasting of Climate Change Using Score-Driven Ice-Age Models" *Econometrics* 10, no. 1: 9.
https://doi.org/10.3390/econometrics10010009