# Temperature Anomalies, Long Memory, and Aggregation

## Abstract

**:**

## 1. Introduction

## 2. Temperature Anomalies

## 3. Methods

#### 3.1. Long Memory

**Definition**

**1.**

- (i).
- In the spectral sense if ${f}_{x}\left(\lambda \right)\sim {C}_{f}{\lambda}^{-2d}$ as $\lambda \to 0$ with ${C}_{f}$ a constant;
- (ii).
- In the self-similar sense if ${m}^{1-2d}\mathrm{Cov}\left(\right)open="("\; close=")">{x}_{t}^{\left(m\right)},{x}_{t+k}^{\left(m\right)}$ as $k,m\to \infty $ where ${x}_{t}^{\left(m\right)}=({x}_{tm-m+1}+\cdots +{x}_{tm})/m$, with $m\in \mathbb{N}$, $m/k\to 0$, and ${C}_{m}$ is a constant;
- (iii).
- In the covariance sense if ${\gamma}_{x}\left(k\right)\sim {C}_{x}{k}^{2d-1}$ as $k\to \infty $ with ${C}_{x}$ a constant.

#### 3.2. Cross-Sectional Aggregation

#### 3.3. Semiparametric Estimators of Long Memory

## 4. Results

#### 4.1. Monthly Data with 1200 km Smoothing Radius and Optimal Bandwidth

#### 4.2. Monthly Data with 250 km Smoothing Radius and Optimal Bandwidth

#### 4.3. Robustness Exercises

## 5. Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GISTEMP | NASA Goddard Institute for Space Studies Surface Temperature Analysis |

North Hem. | Northern Hemisphere |

South Hem. | Southern Hemisphere |

GPH | Geweke and Porter-Hudak log-periodogram regression |

AG | Andrews and Guggenberger bias reduced log-periodogram estimator |

LW | Local Whittle estimator |

FELW | Feasible Exact Local Whittle estimator |

Std. Dev. | Standard deviation |

Prob. Density Est. | Probability density estimate |

No. of Grids | Number of grids |

## Appendix A

**Figure A1.**Boxplots for autocorrelation functions of individual temperature series. On each box, the central mark indicates the median, and the bottom and top edges of the box indicate the 1st and 3rd quartiles, respectively. The whiskers extend to approximately 99.3% coverage if the data are normally distributed. Also shown in a black line, the autocorrelation function for the temperature at the grid near London (${51}^{\circ}$,$-{1}^{\circ}$).

**Figure A3.**Standard deviations for long memory estimates of monthly individual grid observations of temperature anomalies. We use the optimal bandwidth and the 1200 km smoothing radius dataset.

**Figure A4.**Standard deviations for long memory estimates of monthly individual grid observations of temperature anomalies. We use the optimal bandwidth and the 250 km smoothing radius dataset.

**Figure A5.**Histograms and probability density estimates of long memory estimates for monthly individual grid observations of temperature anomalies. We use the optimal bandwidth, the 1200 km smoothing radius dataset, and the GPH estimator.

**Figure A6.**Histograms and probability density estimates of long memory estimates for monthly individual grid observations of temperature anomalies. We use the optimal bandwidth, the 1200 km smoothing radius dataset, and the LW estimator.

**Figure A7.**Histograms and probability density estimates of long memory estimates for monthly individual grid observations of temperature anomalies. We use the optimal bandwidth, the 1200 km smoothing radius dataset, and the FELW estimator.

**Figure A8.**Histograms and probability density estimates of long memory estimates for monthly individual grid observations of temperature anomalies. We use the optimal bandwidth, the 250 km smoothing radius dataset, and the AG estimator.

**Figure A9.**Histograms and probability density estimates of long memory estimates for monthly individual grid observations of temperature anomalies. We use the optimal bandwidth, the 250 km smoothing radius dataset, and the LW estimator.

**Figure A10.**Histograms and probability density estimates of long memory estimates for monthly individual grid observations of temperature anomalies. We use the optimal bandwidth, the 250 km smoothing radius dataset, and the FELW estimator.

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**Figure 1.**Author’s own plot with data from Goddard Institute for Space Studies Surface Temperature Analysis (GISTEMP). Smoothing radius of 1200 km.

**Figure 2.**Temperature anomalies for monthly regional temperatures presented by GISTEMP. For illustrative purposes, the temperature reported near London (${51}^{\circ}$,$-{1}^{\circ}$) is also shown.

**Figure 3.**Sample autocorrelation functions for monthly regional temperatures presented by GISTEMP. The autocorrelation function for the temperature reported near London (${51}^{\circ}$,$-{1}^{\circ}$) is also shown. The shaded area covers from the first to the third quartile of the autocorrelation functions of the individual grid series.

**Figure 4.**Autocorrelation functions for individual AR(1) processes with different autoregressive coefficient. Also shown, the autocorrelation function of the aggregated process.

**Figure 5.**Long memory estimates for monthly individual grid observations of temperature anomalies. We use the optimal bandwidth and the 1200 km smoothing radius dataset. GPH: Log-periodogram regression, AG: Andrews and Guggenberger (2003) estimator, LW: Local Whittle estimator, and FELW: Feasible exact local Whittle.

**Figure 6.**Histograms and probability density estimates of long memory estimates for monthly individual grid observations of temperature anomalies. We use the optimal bandwidth, the 1200 km smoothing radius dataset, and the Andrews and Guggenberger (2003) (AG) estimator.

**Figure 7.**Long memory estimates for monthly individual grid observations of temperature anomalies. We use the optimal bandwidth and the 250 km smoothing radius dataset.

**Figure 8.**Histograms and probability density estimates of long memory estimates for monthly individual grid observations of temperature anomalies. We use the optimal bandwidth, the 250 km smoothing radius dataset, and the log-periodogram regression (GPH) estimator.

**Table 1.**Long memory estimates for monthly regional temperature anomalies using the optimal bandwidth. For the London grid, we use the 1200 km smoothing radius dataset. * Note that all series have equal sample sizes so that the estimates share the same standard deviations.

Series | GPH | AG | LW | FELW | Sample Size |
---|---|---|---|---|---|

Global | 0.6332 | 0.7475 | 0.6682 | 0.6304 | 1688 |

North Hem. | 0.5429 | 0.6247 | 0.5765 | 0.5584 | 1688 |

South Hem. | 0.6407 | 0.6904 | 0.6430 | 0.6159 | 1688 |

London (${51}^{\circ}$,$-{1}^{\circ}$) | 0.2346 | 0.2388 | 0.2288 | 0.2354 | 1688 |

Std. Dev. * | (0.0328) | (0.0492) | (0.0256) | (0.0256) |

**Table 2.**Regional long memory averages for monthly temperature anomalies. We use the optimal bandwidth and the 1200 km smoothing radius dataset. The bar on top denotes averages. Standard deviations given the average sample sizes are given below between parentheses.

Series | GPH | AG | LW | FELW | $\overline{\mathbf{Sample}\phantom{\rule{4.pt}{0ex}}\mathbf{Size}}$ | No. of Grids |
---|---|---|---|---|---|---|

$\overline{\mathrm{Global}}$ | 0.3427 | 0.2951 | 0.3540 | 0.3738 | 1463 | 15,981 |

(0.0347) | (0.0521) | (0.0271) | (0.0271) | |||

$\overline{\mathrm{North}\phantom{\rule{4.pt}{0ex}}\mathrm{Hem}.}$ | 0.3236 | 0.3082 | 0.3389 | 0.3539 | 1560 | 8100 |

(0.0338) | (0.0508) | (0.0264) | (0.0264) | |||

$\overline{\mathrm{South}\phantom{\rule{4.pt}{0ex}}\mathrm{Hem}.}$ | 0.3624 | 0.2817 | 0.3695 | 0.3942 | 1364 | 7881 |

(0.0357) | (0.0536) | (0.0279) | (0.0279) | |||

$\overline{\mathrm{Tropics}}$ | 0.4855 | 0.4103 | 0.5019 | 0.5216 | 1642 | 5397 |

(0.0332) | (0.0497) | (0.0259) | (0.0259) | |||

$\overline{\mathrm{Arctic}}$ | 0.2164 | 0.2588 | 0.2404 | 0.2525 | 1286 | 2160 |

(0.0366) | (0.0549) | (0.0285) | (0.0285) | |||

$\overline{\mathrm{Antarctic}}$ | 0.1174 | 0.0874 | 0.1084 | 0.1401 | 681 | 2090 |

(0.0471) | (0.0707) | (0.0368) | (0.0368) |

**Table 3.**Regional long memory averages for monthly temperature anomalies. We use the optimal bandwidth and the 250 km smoothing radius dataset. The bar on top denotes averages. Standard deviations given the average sample sizes are given below between parentheses.

Series | GPH | AG | LW | FELW | $\overline{\mathbf{Sample}\phantom{\rule{4.pt}{0ex}}\mathbf{Size}}$ | No. of Grids |
---|---|---|---|---|---|---|

$\overline{\mathrm{Global}}$ | 0.2279 | 0.2406 | 0.2277 | 0.2626 | 895 | 6185 |

(0.0423) | (0.0634) | (0.0330) | (0.0330) | |||

$\overline{\mathrm{North}\phantom{\rule{4.pt}{0ex}}\mathrm{Hem}.}$ | 0.2179 | 0.2286 | 0.2315 | 0.2596 | 1106 | 3927 |

(0.0389) | (0.0583) | (0.0303) | (0.0303) | |||

$\overline{\mathrm{South}\phantom{\rule{4.pt}{0ex}}\mathrm{Hem}.}$ | 0.2452 | 0.2615 | 0.2209 | 0.2678 | 528 | 2258 |

(0.0522) | (0.0783) | (0.0407) | (0.0407) | |||

$\overline{\mathrm{Tropics}}$ | 0.3308 | 0.3406 | 0.3315 | 0.3786 | 732 | 1685 |

(0.0458) | (0.0687) | (0.0357) | (0.0357) | |||

$\overline{\mathrm{Arctic}}$ | 0.2009 | 0.2259 | 0.2248 | 0.2678 | 845 | 797 |

(0.0432) | (0.0649) | (0.0337) | (0.0337) | |||

$\overline{\mathrm{Antarctic}}$ | 0.1680 | 0.2005 | 0.1205 | 0.1633 | 284 | 1070 |

(0.0669) | (0.1003) | (0.0521) | (0.0521) |

**Table 4.**Regional long memory averages for yearly temperature anomalies. We use the optimal bandwidth and the 250 km smoothing radius dataset. The bar on top denotes averages. Standard deviations given the average sample sizes are given below between parentheses.

Series | GPH | AG | LW | FELW | $\overline{\mathbf{Sample}\phantom{\rule{4.pt}{0ex}}\mathbf{Size}}$ | No. of Grids |
---|---|---|---|---|---|---|

$\overline{\mathrm{Global}}$ | 0.4051 | 0.4278 | 0.3458 | 0.5573 | 78 | 6006 |

(0.1116) | (0.1674) | (0.0870) | (0.0870) | |||

$\overline{\mathrm{North}\phantom{\rule{4.pt}{0ex}}\mathrm{Hem}.}$ | 0.3784 | 0.5079 | 0.3621 | 0.5218 | 94 | 3982 |

(0.1040) | (0.1560) | (0.0811) | (0.0811) | |||

$\overline{\mathrm{South}\phantom{\rule{4.pt}{0ex}}\mathrm{Hem}.}$ | 0.4542 | 0.2802 | 0.3159 | 0.6225 | 47 | 2114 |

(0.1367) | (0.2051) | (0.1066) | (0.1066) | |||

$\overline{\mathrm{Tropics}}$ | 0.4621 | 0.5088 | 0.4074 | 0.6608 | 66 | 1563 |

(0.1191) | (0.1786) | (0.0928) | (0.0928) | |||

$\overline{\mathrm{Arctic}}$ | 0.3166 | 0.4880 | 0.3083 | 0.4939 | 77 | 761 |

(0.1134) | (0.1700) | (0.0884) | (0.0884) | |||

$\overline{\mathrm{Antarctic}}$ | 0.4682 | 0.1177 | 0.2578 | 0.6449 | 24 | 1059 |

(0.1779) | (0.2668) | (0.1387) | (0.1387) |

**Table 5.**Regional long memory averages for yearly temperature anomalies. The bandwidth is given by $m={T}^{1/2}$, with T the sample size. We use the 250 km smoothing radius dataset. The bar on top denotes averages.

Series | GPH | AG | LW | FELW | $\overline{\mathbf{Sample}\phantom{\rule{4.pt}{0ex}}\mathbf{Size}}$ | No. of Grids |
---|---|---|---|---|---|---|

$\overline{\mathrm{Global}}$ | 0.5947 | 0.8443 | 0.5343 | 0.6429 | 78 | 6006 |

(0.2138) | (0.3206) | (0.1667) | (0.1667) | |||

$\overline{\mathrm{North}\phantom{\rule{4.pt}{0ex}}\mathrm{Hem}.}$ | 0.6276 | 1.0159 | 0.5650 | 0.6364 | 94 | 3982 |

(0.2028) | (0.3042) | (0.1581) | (0.1581) | |||

$\overline{\mathrm{South}\phantom{\rule{4.pt}{0ex}}\mathrm{Hem}.}$ | 0.5342 | 0.5284 | 0.4777 | 0.6549 | 47 | 2114 |

(0.2424) | (0.3636) | (0.1890) | (0.1890) | |||

$\overline{\mathrm{Tropics}}$ | 0.6718 | 0.8727 | 0.6115 | 0.6725 | 66 | 1563 |

(0.2267) | (0.3401) | (0.1768) | (0.1768) | |||

$\overline{\mathrm{Arctic}}$ | 0.5528 | 1.2723 | 0.4883 | 0.5924 | 77 | 761 |

(0.2138) | (0.3206) | (0.1667) | (0.1667) | |||

$\overline{\mathrm{Antarctic}}$ | 0.4133 | 0.4156 | 0.3766 | 0.6872 | 24 | 1059 |

(0.2868) | (0.4302) | (0.2236) | (0.2236) |

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**MDPI and ACS Style**

Vera-Valdés, J.E.
Temperature Anomalies, Long Memory, and Aggregation. *Econometrics* **2021**, *9*, 9.
https://doi.org/10.3390/econometrics9010009

**AMA Style**

Vera-Valdés JE.
Temperature Anomalies, Long Memory, and Aggregation. *Econometrics*. 2021; 9(1):9.
https://doi.org/10.3390/econometrics9010009

**Chicago/Turabian Style**

Vera-Valdés, J. Eduardo.
2021. "Temperature Anomalies, Long Memory, and Aggregation" *Econometrics* 9, no. 1: 9.
https://doi.org/10.3390/econometrics9010009