# Detecting and Quantifying Structural Breaks in Climate

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Methodologies for Detecting Structural Breaks

#### 2.1. Impulse Indicator Saturation

**Example**

**1.**

- Estimate the model, including impulse indicator dummies for the first half of the sample, as represented by Figure 2a. That estimation is equivalent to estimating the model over the second half of the sample, ignoring the first half. Drop all statistically insignificant impulse indicator dummies and retain the statistically significant dummies (Figure 2b).

**Example**

**2.**

#### 2.2. Extensions

## 3. Background

## 4. Data

#### 4.1. Palmer Drought Severity Index

#### 4.2. Rainfall Data

#### 4.3. Remarks

## 5. Empirical Analysis

## 6. Remarks

Article 4.8 contains a specific commitment of financial assistance for “… [c]ountries with areas liable to drought and desertification …”. For such assistance, the decline in Mauritanian food production since 1970 can be reasonably attributed to the decline in rainfall. Put somewhat differently, a “date” is needed to establish the start of these impacts. A country may have a case for UNFCCC-based assistance if it can establish a reasonable benchmark for the decline in local food production. For Mauritania, the detected structural break in 1970 of the distribution of precipitation may serve as a plausible claim. That is, the adverse consequences of climate change and subsequent decline in food production could be “dated” to have started around 1970; and financial assistance could be based on this evidence. Additionally, at the 2015 UNFCCC Conference of the Parties (COP 21), 195 countries (Mauritania included) adopted the Paris Agreement on greenhouse gas emissions, mitigation, and financing from 2020. That agreement went into effect on November 4, 2016, shortly before the COP 22 in Marrakesh. More recently, the COP 27 in Sharm el-Sheikh approved a loss and damage fund to assist certain countries adversely affected by climate change.Cooperate in preparing for adaptation to the impacts of climate change; develop and elaborate appropriate and integrated plans for coastal zone management, water resources and agriculture, and for the protection and rehabilitation of areas, particularly in Africa, affected by drought and desertification, as well as floods …

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Actual and fitted values and the corresponding scaled residuals for the estimated model when the DGP does not have a break.

**Figure 2.**A characterization of bare-bones impulse indicator saturation with a target size of 5% when the DGP does not have a break.

**Figure 3.**Actual and fitted values and the corresponding scaled residuals for the estimated model when the DGP has a break and the model ignores that break.

**Figure 4.**A characterization of bare-bones impulse indicator saturation with a target size of 5% when the DGP has a break and the model ignores that break.

**Figure 5.**The Sahel Region in Africa (Source for map: Felix Koenig and T.L. Miles, WikiMedia Commons, commons.wikimedia.org/wiki/File:Sahel_Map-Africa_rough.png#, with added labels; accessed on 8 November 2022).

**Figure 7.**Eighteen Mauritanian raingauge stations (Source for map: the U.S. Central Intelligence Agency, available from the Perry-Castañeda Library Map Collection, University of Texas at Austin, www.lib.utexas.edu/maps/africa/mauritania_rel95.jpg, with added labels; accessed on 8 November 2022).

**Figure 14.**Actual and fitted values of rainfall R (as time series and as a cross-plot) for the baseline AR(2) model, the residual density and histogram, the scaled residuals, and the residual ACF and PACF.

**Figure 15.**Recursive least squares estimates of the first and second lags on the dependent variable R and on the intercept in the baseline AR(2) model, and their recursive t-ratios.

**Figure 16.**Recursive residual sums of squares, one-step residuals, innovation errors, and the one-step, breakpoint, and forecast Chow statistics for the baseline AR(2) model.

**Figure 17.**Actual, fitted, and forecast values of rainfall R (as time series and as a cross-plot) for the baseline AR(2) model, the residual density and histogram, the scaled residuals and forecast errors, the forecasts and ± twice their standard errors, and the residual ACF and PACF.

**Figure 18.**Actual and fitted values of rainfall R (as time series and as a cross-plot) for the modified (super-saturated) AR(2) model, the residual density and histogram, the scaled residuals, and the residual ACF and PACF.

Focus of Test Statistic | References |
---|---|

Covariance equality | Fisher (1922) |

Recursive estimates | Plackett (1950) |

Forecast errors | Chow (1960) |

Single unknown breakpoint in regression coefficients | Nyblom (1989); Andrews (1993), Hansen (1992) |

Multiple unknown breakpoints in the intercept | Bai and Perron (1998) |

Arbitrary unknown impulse breaks (impulse indicator saturation) | Hendry (1999), Hendry et al. (2008), Johansen and Nielsen (2009) |

Arbitrary unknown step breaks (step indicator saturation) | Castle et al. (2015) |

Step, trend, and coefficient breaks (super saturation, ultra saturation, and multi-saturation) | Ericsson (2011b) |

**Table 2.**Descriptive statistics on observed monthly rainfall as measured at 18 Mauritanian raingauge stations.

Station’s Location | Sample Period | Number of Observations | Average Monthly Rainfall (mm) | Maximum Monthly Rainfall (mm) | Standard Deviation of Monthly Rainfall (mm) |
---|---|---|---|---|---|

Aioun el Atrouss | 1946–2011 | 673 | 20.4 | 245 | 37.9 |

Akjoujt | 1931–2011 | 749 | 7.0 | 131 | 17.2 |

Aleg | 1921–1997 | 918 | 20.6 | 290 | 40.8 |

Atar | 1921–2009 | 907 | 7.0 | 121 | 15.3 |

Bir Moghrein | 1942–2003 | 622 | 3.3 | 74 | 10.0 |

Boghe | 1919–1997 | 941 | 23.7 | 310 | 44.0 |

Boutilimit | 1921–2009 | 944 | 13.6 | 200 | 28.5 |

Chinguetti | 1931–1997 | 774 | 4.4 | 99 | 10.4 |

F’Derik | 1938–1997 | 685 | 4.2 | 117 | 11.9 |

Kaedi | 1905–1997 | 956 | 31.2 | 364 | 56.4 |

Kiffa | 1922–2011 | 944 | 25.5 | 325 | 47.6 |

Moudjeria | 1911–1997 | 955 | 16.7 | 229 | 34.2 |

Nema | 1922–2011 | 963 | 21.1 | 205 | 35.8 |

Nouadhibou | 1906–2011 | 1137 | 1.8 | 83 | 5.8 |

Nouakchott | 1930–2011 | 845 | 9.7 | 191 | 23.7 |

Rosso | 1934–2011 | 754 | 21.9 | 498 | 45.9 |

Tichitt | 1921–1997 | 826 | 6.4 | 102 | 14.2 |

Tidjikja | 1907–2011 | 980 | 10.8 | 173 | 22.7 |

**Table 3.**Regression coefficients, estimated standard errors (in parentheses), and diagnostic statistics for several models of rainfall ${R}_{t}$.

Model Description | ||||
---|---|---|---|---|

Regressor or Diagnostic Statistic | Baseline AR(2) | Baseline AR(2), with Forecasts for 1971–1997 | AR(2) with IIS (1%) | AR(2) with Super Satur-ation (1%) |

Estimation Sample | 1921–1997 | 1921–1970 | 1921–1997 | 1921–1997 |

Intercept | 81.1 (20.6) | 201.5 (40.1) | 78.7 (18.9) | 191.8 (25.6) |

${R}_{t-1}$ | 0.37 (0.12) | −0.08 (0.15) | 0.44 (0.11) | 0.11 (0.10) |

${R}_{t-2}$ | 0.12 (0.11) | −0.02 (0.15) | 0.05 (0.10) | −0.16 (0.10) |

${I}_{1927t}$ | — | — | 129.1 (42.4) | 104.8 (37.1) |

${I}_{1969t}$ | — | — | 115.4 (43.0) | — |

${S}_{1970t}$ | — | — | — | −72.6 (12.7) |

${S}_{1994t}$ | — | — | — | 68.9 (20.7) |

$\widehat{\sigma}$ | $45.70$ | $41.40$ | $41.92$ | $36.45$ |

AR(2) statistic | 1.59 [0.2118] F(2,72) | 2.30 [0.1121] F(2,45) | 0.96 [0.3884] F(2,70) | 0.40 [0.6735] F(2,69) |

ARCH(1) statistic | 0.01 [0.9437] F(1,75) | 0.02 [0.9003] F(1,48) | 0.50 [0.4812] F(1,75) | 0.15 [0.6988] F(1,75) |

Normality statistic | 1.30 [0.5218] ${\chi}^{2}$(2) | 0.75 [0.6860] ${\chi}^{2}$(2) | 0.87 [0.6481] ${\chi}^{2}$(2) | 0.27 [0.8738] ${\chi}^{2}$(2) |

Heteroscedasticity statistic #1 | 2.48 [0.0511] F(4,72) | 1.63 [0.1843] F(4,45) | 1.29 [0.2824] F(4,70) | 1.74 [0.1254] F(6,69) |

Heteroscedasticity statistic #2 | 2.70 [0.0273] F(5,71) | 1.86 [0.1214] F(5,44) | 1.21 [0.3126] F(5,69) | 1.52 [0.1758] F(7,68) |

RESET statistic | 0.78 [0.4633] F(2,72) | 0.29 [0.7520] F(2,45) | 0.97 [0.3858] F(2,70) | 0.84 [0.4361] F(2,69) |

Number of Breaks | Schwarz | LWZ | F(m) | F(m ∣ m − 1) |
---|---|---|---|---|

0 | 7.77 | 7.90 | — | — |

1 | <7.59> | <7.84> | 9.96 | 9.96 |

2 | 7.65 | 8.03 | 6.61 | 2.60 |

3 | 7.73 | 8.24 | 5.34 | 2.14 |

4 | 7.77 | 8.42 | 5.00 | 2.72 |

5 | 7.82 | 8.60 | 4.78 | 2.46 |

6 | 7.86 | 8.78 | 4.73 | 2.58 |

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Ericsson, N.R.; Dore, M.H.I.; Butt, H.
Detecting and Quantifying Structural Breaks in Climate. *Econometrics* **2022**, *10*, 33.
https://doi.org/10.3390/econometrics10040033

**AMA Style**

Ericsson NR, Dore MHI, Butt H.
Detecting and Quantifying Structural Breaks in Climate. *Econometrics*. 2022; 10(4):33.
https://doi.org/10.3390/econometrics10040033

**Chicago/Turabian Style**

Ericsson, Neil R., Mohammed H. I. Dore, and Hassan Butt.
2022. "Detecting and Quantifying Structural Breaks in Climate" *Econometrics* 10, no. 4: 33.
https://doi.org/10.3390/econometrics10040033