# Is Climate Change Time-Reversible?

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## Abstract

**:**

## 1. Introduction

## 2. Time Reversibility

## 3. New Strategies to Detect Time Reversibility on Stationary Time Series

**Condition**

**1.**

#### 3.1. Strategy 1: For Detecting Time Reversibility

- We estimate a conventional autoregressive process (also called pseudo-causal model) by OLS, and the lag order p is selected using information criteria (for instance, AIC or BIC).
- We test the normality in the residuals of the AR(p). If the null hypothesis of Gaussianity is not rejected, we cannot identify a MAR(r,s) model, and for the reasons above, we have a time-reversible process. Moreover, if the null hypothesis of normality is rejected and the estimated p is an odd number, the condition $r=s$ can never be satisfied. According to Condition 1, this result would allow us to identify our process as time-irreversible. However, the selection of p might not be univocal and depend on the information criterion employed. As such, to have more robust results before proceeding to the next step, we increase p by one unit so that $r=s$ is still possible. In the alternative case that p is an even number, we directly proceed to the next step.
- We select a model among all MAR(r,s) specifications with $r+s=p$ if p is an even number; otherwise $r+s=p+1$. This step is performed using a maximum likelihood approach (see Giancaterini and Hecq 2022 and references therein). In the selection procedure, we also include the model given by the restricted likelihood that imposes commonalities in causal and noncausal parameters (the model with the same restrictions as in Condition 1). Note that when we compute the information criteria of the model with restricted likelihood, instead of estimating p parameters (or $p+1$ if p is an odd number), we estimate $p/2$ of them (or $(p+1)/2$), implying a smaller penalty term. Finally, we choose the model with the smallest information criteria.

#### 3.2. Strategy 2: For Detecting Time Reversibility

- 3.
- We select a model among all MAR(r,s) specifications with $r+s=p$ if p is an even number (otherwise $r+s=p+1$). Then, we choose the one with the largest likelihood (since we are considering models with the same number of parameters).
- 4.
- If the selected model is the one with $r=s$ (in our previous example, it was the MAR(1,1)), we compute a likelihood ratio test, taking into account the same restrictions as in Condition 1). If we do not reject the test’s null hypothesis, we have TR. On the other hand, if we reject the null hypothesis, we identify the process under investigation as time-irreversible.

#### 3.3. Simulation Study

- MAR(1,1): ${\varphi}_{0}=0.8$, ${\phi}_{0}=0.8$; time-reversible process;
- MAR(1,1): ${\varphi}_{0}=0.8$, ${\phi}_{0}=0.5$; time-irreversible process;
- MAR(1,1): ${\varphi}_{0}=0.8$, ${\phi}_{0}=0.1$; time-irreversible process;
- MAR(1,0): ${\varphi}_{0}=0.8$; time-irreversible process.

- MAR(1,1): ${\varphi}_{0}=0.95$, ${\phi}_{0}=0.95$; time-reversible process;
- MAR(1,1): ${\varphi}_{0}=0.95$, ${\phi}_{0}=0.5$; time-irreversible process;
- MAR(1,1): ${\varphi}_{0}=0.95$, ${\phi}_{0}=0.1$; time-irreversible process;
- MAR(1,0): ${\varphi}_{0}=0.95$; time-irreversible process.

## 4. Testing for Time Reversibility on Non-Stationary Processes

## 5. Is Climate Change Time-Reversible?

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Note

1 | Chen et al. (2000) state that to jointly test ${X}_{t,k}$ for a collection of k values, a portmanteau test is required. |

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**Figure 1.**Climate time series. (

**a**) Annual data for global land and ocean temperature anomalies. (

**b**) Annual data for global land temperature anomalies. (

**c**) Annual data for global ocean temperature anomalies. (

**d**) Annual data for solar activity. (

**e**) Annual data for greenhouses gas. (

**f**) Annual data for nitrous oxide. (

**g**) Monthly data for global component of climate at a glance. (

**h**) Monthly data for global surface temperature change. (

**i**) Monthly data for global mean sea level. (

**j**) Monthly data for Southern Oscillation index. (

**k**) Monthly data for North Atlantic Oscillation index. (

**l**) Monthly data for Pacific Decadal Oscillation index. (

**m**) Monthly data for Northern Hemisphere sea ice area. (

**n**) Monthly data for Southern Hemisphere sea ice area.

**Figure 2.**Cyclical components of the detrended time series. (

**a**) Annual data for cyclical component of global land and ocean temperature anomalies. (

**b**) Annual data for cyclical component of global land temperature anomalies. (

**c**) Annual data for cyclical component of global ocean temperature anomalies. (

**d**) Annual data for cyclical component of solar activity. (

**e**) Annual data for cyclical component of greenhouses gas. (

**f**) Annual data for cyclical component of nitrous oxide. (

**g**) Monthly data for cyclical component of global component of climate at a glance. (

**h**) Monthly data for cyclical component of global surface temperature change. (

**i**) Monthly data for cyclical component of global mean sea level.

**Table 1.**Frequencies with which time irreversibility is detected when the error term has a symmetric Student’s-t distribution ($\gamma $ = 1) and ${\nu}_{0}=3$. Finally, r and s are assumed as unknown and p as known.

MAR(1,1); ${\mathit{\varphi}}_{0}=0.8,\phantom{\rule{4pt}{0ex}}{\mathit{\phi}}_{0}=0.8,\phantom{\rule{4pt}{0ex}}{\mathit{\nu}}_{0}=3$, $\mathit{\gamma}=1$ | |||

Strategy 1 | Strategy 2 | $\mathit{RR}\left(\mathbf{1996}\right)$ | |

$T=100$ | 7.1% | 16.4% | 8.1% |

$T=200$ | 3.1% | 7.5% | 11.5% |

$T=500$ | 1.4% | 5.0% | 11% |

$T=1000$ | 0.8% | 4.5% | 13.6 % |

MAR(1,1); ${\mathit{\varphi}}_{0}=0.8,\phantom{\rule{4pt}{0ex}}{\mathit{\phi}}_{0}=0.5,\phantom{\rule{4pt}{0ex}}{\mathit{\nu}}_{0}=3$, $\mathit{\gamma}=1$ | |||

Strategy 1 | Strategy 2 | $\mathit{RR}\left(\mathbf{1996}\right)$ | |

$T=100$ | 51.4% | 63.7% | 20.7% |

$T=200$ | 77.9% | 84.8% | 29.4% |

$T=500$ | 99.0% | 99.5% | 40.7% |

$T=1000$ | 100% | 100% | 51.8 % |

MAR(1,1); ${\mathit{\varphi}}_{0}=0.8,\phantom{\rule{4pt}{0ex}}{\mathit{\phi}}_{0}=0.1,\phantom{\rule{4pt}{0ex}}{\mathit{\nu}}_{0}=3$, $\mathit{\gamma}=1$ | |||

Strategy 1 | Strategy 2 | $\mathit{RR}\left(\mathbf{1996}\right)$ | |

$T=100$ | 87.4% | 93.2% | 33.2% |

$T=200$ | 99.6% | 99.9% | 43.2% |

$T=500$ | 100% | 100% | 57.5% |

$T=1000$ | 100% | 100% | 68.4% |

MAR(1,0); ${\mathit{\varphi}}_{0}=0.8,\phantom{\rule{4pt}{0ex}}{\mathit{\nu}}_{0}=3$, $\mathit{\gamma}=1$ | |||

Strategy 1 | Strategy 2 | $\mathit{RR}\left(\mathbf{1996}\right)$ | |

$T=100$ | 91.5% | 93.2% | 34.2% |

$T=200$ | 99.6% | 99.9% | 43.8% |

$T=500$ | 100% | 100% | 58.9% |

$T=1000$ | 100% | 100% | 70.1% |

**Table 2.**Frequencies with which time irreversibility is detected when the error term has a symmetric Student’s-t distribution ($\gamma $ = 1) with ${\nu}_{0}=3$ and p, r, and s are assumed as unknown.

MAR(1,1); ${\mathit{\varphi}}_{0}=0.8,\phantom{\rule{4pt}{0ex}}{\mathit{\phi}}_{0}=0.8,\phantom{\rule{4pt}{0ex}}{\mathit{\nu}}_{0}=3$, $\mathit{\gamma}=1$ | |||

Strategy 1 | Strategy 2 | $\mathit{RR}\left(\mathbf{1996}\right)$ | |

$T=100$ | 20.9% | 21.6% | 8.1% |

$T=200$ | 9.5% | 12.6% | 11.5% |

$T=500$ | 3.5% | 7.1% | 11% |

$T=1000$ | 4.3% | 7.8% | 13.6 % |

MAR(1,1); ${\mathit{\varphi}}_{0}=0.8,\phantom{\rule{4pt}{0ex}}{\mathit{\phi}}_{0}=0.5,\phantom{\rule{4pt}{0ex}}{\mathit{\nu}}_{0}=3$, $\mathit{\gamma}=1$ | |||

Strategy 1 | Strategy 2 | $\mathit{RR}\left(\mathbf{1996}\right)$ | |

$T=100$ | 61.6% | 67.1% | 20.7% |

$T=200$ | 79.0% | 85.2% | 29.4% |

$T=500$ | 99.0% | 99.5% | 40.7% |

$T=1000$ | 100% | 100% | 51.8 % |

MAR(1,1); ${\mathit{\varphi}}_{0}=0.8,\phantom{\rule{4pt}{0ex}}{\mathit{\phi}}_{0}=0.1,\phantom{\rule{4pt}{0ex}}{\mathit{\nu}}_{0}=3$, $\mathit{\gamma}=1$ | |||

Strategy 1 | Strategy 2 | $\mathit{RR}\left(\mathbf{1996}\right)$ | |

$T=100$ | 92.2% | 93.8% | 33.2% |

$T=200$ | 99.8% | 99.9% | 43.2% |

$T=500$ | 100% | 100% | 57.5% |

$T=1000$ | 100% | 100% | 68.4% |

MAR(1,1); ${\mathit{\varphi}}_{0}=0.8,\phantom{\rule{4pt}{0ex}}{\mathit{\nu}}_{0}=3$, $\mathit{\gamma}=1$ | |||

Strategy 1 | Strategy 2 | $\mathit{RR}\left(\mathbf{1996}\right)$ | |

$T=100$ | 95% | 95.7% | 34.2% |

$T=200$ | 99.9% | 99.9% | 43.8% |

$T=500$ | 100% | 100% | 58.9% |

$T=1000$ | 100% | 100% | 70.1% |

**Table 3.**Frequencies with which time irreversibility is detected on non-stationary time series; r and s are assumed as unknown and p as known. Data are considered to have quarterly frequency ($\lambda =1600$).

${\mathit{cc}}^{\mathit{Y}}:\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathbf{MAR}(1,1);\phantom{\rule{4pt}{0ex}}{\mathit{\varphi}}_{0}=0.8,\phantom{\rule{4pt}{0ex}}{\mathit{\phi}}_{0}=0.8;\phantom{\rule{4pt}{0ex}}{\mathit{\nu}}_{0}=3$, $\mathit{\gamma}=1$ | ||

Strategy 1 | Strategy 2 | |

$T=100$ | 15.1% | 33.3% |

$T=200$ | 6.7% | 8.1% |

$T=500$ | 1.4% | 5.0% |

$T=1000$ | 0.8% | 4.5% |

${\mathit{cc}}^{\mathit{Y}}:\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathbf{MAR}(1,1);\phantom{\rule{4pt}{0ex}}{\mathit{\varphi}}_{0}=0.8,\phantom{\rule{4pt}{0ex}}{\mathit{\phi}}_{0}=0.5;\phantom{\rule{4pt}{0ex}}{\mathit{\nu}}_{0}=3$, $\mathit{\gamma}=1$ | ||

Strategy 1 | Strategy 2 | |

$T=100$ | 40.6% | 59.5% |

$T=200$ | 58.0% | 69.7% |

$T=500$ | 86.7% | 94.8% |

$T=1000$ | 99.2% | 100% |

${\mathit{cc}}^{\mathit{Y}}:\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathbf{MAR}(1,1);\phantom{\rule{4pt}{0ex}}{\mathit{\varphi}}_{0}=0.8,\phantom{\rule{4pt}{0ex}}{\mathit{\phi}}_{0}=0.1;\phantom{\rule{4pt}{0ex}}{\mathit{\nu}}_{0}=3$, $\mathit{\gamma}=1$ | ||

Strategy 1 | Strategy 2 | |

$T=100$ | 71.9% | 89.2% |

$T=200$ | 92.0% | 97.3% |

$T=500$ | 99.6% | 100% |

$T=1000$ | 100% | 100% |

${\mathit{cc}}^{\mathit{Y}}:\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathbf{MAR}(1,0);\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathit{\varphi}}_{0}=0.8;\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathit{\nu}}_{0}=3$, $\mathit{\gamma}=1$ | ||

Strategy 1 | Strategy 2 | |

$T=100$ | 75.1% | 91.1% |

$T=200$ | 93.4% | 98.8% |

$T=500$ | 100% | 100% |

$T=1000$ | 100% | 100% |

**Table 4.**Estimated coefficients of the time series identified as non-Gaussian. The figures in parentheses are the standard errors computed by using the Hessian matrix.

${\mathit{cc}}^{\mathit{GHG}}$ | ${\mathit{cc}}^{\mathit{N}2\mathit{O}}$ | ${\mathit{cc}}^{\mathit{GCAG}}$ | ${\mathit{cc}}^{\mathit{GISTEMP}}$ | ${\mathit{cc}}^{\mathit{GMSL}}$ | SOI | NAO | PDO | NH | |
---|---|---|---|---|---|---|---|---|---|

${\widehat{\varphi}}_{1}$ | 0.9620 | 0.9818 | 0.4417 | 0.4003 | 1.1233 | −0.0933 | −0.0966 | / | 0.1657 |

(0.0841) | (0.0722) | (0.0225) | (0.0239) | (0.0231) | (0.0327) | (0.00342) | (0.0300) | ||

${\widehat{\varphi}}_{2}$ | −0.3230 | −0.2413 | 0.1443 | 0.1245 | −0.1169 | −0.1315 | / | / | −0.0071 |

(0.0822) | (0.0980) | (0.0225) | (0.0239) | (0.0347) | (0.0326) | (0.0306) | |||

${\widehat{\varphi}}_{3}$ | / | −0.0016 | / | / | −0.6729 | / | / | / | −0.0824 |

(0.1030) | (0.0336) | (0.0304) | |||||||

${\widehat{\varphi}}_{4}$ | / | −0.2028 | / | / | 0.3971 | / | / | / | 0.0025 |

(0.0753) | (0.0335) | (0.0303) | |||||||

${\widehat{\varphi}}_{5}$ | / | / | / | / | 0.0898 | / | / | / | −0.0025 |

(0.0346) | (0.0303) | ||||||||

${\widehat{\varphi}}_{6}$ | / | / | / | / | −0.1798 | / | / | / | −0.0390 |

(0.0229) | (0.0303) | ||||||||

${\widehat{\varphi}}_{7}$ | / | / | / | / | / | / | / | / | 0.0058 |

(0.0303) | |||||||||

${\widehat{\varphi}}_{8}$ | / | / | / | / | / | / | / | / | −0.0136 |

(0.0303) | |||||||||

${\widehat{\varphi}}_{9}$ | / | / | / | / | / | / | / | / | −0.0926 |

(0.0303) | |||||||||

${\widehat{\varphi}}_{10}$ | / | / | / | / | / | / | / | / | 0.0496 |

(0.0304) | |||||||||

${\widehat{\varphi}}_{11}$ | / | / | / | / | / | / | / | / | 0.1039 |

(0.0305) | |||||||||

${\widehat{\varphi}}_{12}$ | / | / | / | / | / | / | / | / | 0.7015 |

(0.0300) | |||||||||

${\widehat{\phi}}_{1}$ | / | / | / | / | 0.0880 | 0.4951 | 0.2925 | 0.9183 | 0.7655 |

(0.0225) | (0.0313) | (0.0328) | (0.0215) | (0.0391) | |||||

${\widehat{\phi}}_{2}$ | / | / | / | / | 0.2709 | 0.3169 | / | −0.1365 | −0.0512 |

(0.0225) | (0.0313) | (0.0291) | (0.0391) | ||||||

${\widehat{\phi}}_{3}$ | / | / | / | / | / | / | / | 0.0063 | / |

(0.0291) | |||||||||

${\widehat{\phi}}_{4}$ | / | / | / | / | / | / | / | 0.0664 | / |

(0.0214) | |||||||||

$\widehat{\nu}$ | 19.8 | 13.0 | 5.3 | 9.9 | 6.8 | 8.0 | 96.2 | 9.1 | 8.2 |

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**MDPI and ACS Style**

Giancaterini, F.; Hecq, A.; Morana, C.
Is Climate Change Time-Reversible? *Econometrics* **2022**, *10*, 36.
https://doi.org/10.3390/econometrics10040036

**AMA Style**

Giancaterini F, Hecq A, Morana C.
Is Climate Change Time-Reversible? *Econometrics*. 2022; 10(4):36.
https://doi.org/10.3390/econometrics10040036

**Chicago/Turabian Style**

Giancaterini, Francesco, Alain Hecq, and Claudio Morana.
2022. "Is Climate Change Time-Reversible?" *Econometrics* 10, no. 4: 36.
https://doi.org/10.3390/econometrics10040036