Testing CO₂ Emissions Data During Covid-19 Pandemic Using Benford’s Law

I. Introduction

On 30^th January 2020, WHO declared novel coronavirus 2019 (Covid-19) as a public health emergency of international concern (Andrews et al., 2020). Following the Covid-19 outbreak, China declared a lockdown on 23^rd January 2020, and later all countries moved to lockdown in different phases (Andrews et al., 2020; Charumathi & Mangaiyarkarasi, 2022; Koh, 2020).

The lockdowns forced people to stay at home, restricted movements in and out of countries, companies to halt production, and schools to close, which led to a halt in economic activities and, in turn, a reduction in CO₂ emissions from different sectors around the world.

Studies, such as Adhikari et al. (2021), Charumathi and Mangaiyarkarasi (2022), Le et al. (2020), Ray et al. (2022), Saadat et al. (2020), and Weir et al. (2021), have explored different countries globally and found that, though there was a huge reduction in CO₂ emissions during Covid-19, emissions increased post lockdown and restrictions. This may lead to the manipulation of daily data during the restrictions and post-Covid-19 to show a sustainable emissions report. Studies, such as Auffhammer and Carson (2008), Cole et al. (2020), and Coracioni and Danescu (2020), have applied Benford’s Law to check the authenticity of the emissions reduction data reported on the CDM project, greenhouse gas and CO₂ emissions. In line with these studies, we test whether the first, second, and first two digits of the daily CO₂ emissions data conform to Benford’s Law by taking the published global data during 2020 and 2021. In doing so, we add to the literature by examining whether the pandemic-induced government restrictions influenced the reporting of the emissions data. Our empirical investigation revealed that the CO₂ emissions data were under the category of “conformity” in 2020 and “non-conformity” in 2021. Thus, the pandemic-induced government restrictions influenced the reporting of the emissions data.

II. Method

Benford’s Law is a popular mathematical tool for identifying the patterns and anomalies of numbers. It is a concept to check the digit frequencies of the natural numbers, introduced by US astronomer Simon Newcomb (1881). After 57 years, Frank Benford (1938) identified a pattern, which follows logarithmic frequency distribution on the position of digits. This tool has been successfully applied to detect accounting and financial fraud, and tax evasion by Carslaw (1988), and Nigrini (1999). Benford’s Law can be checked through the tests of goodness of fit, such as chi-square, Kolmogorov-Smirnov test (KS), Joenssen’s JP-square, Freedman-Watson U-square, Chebyshev distance, and Z-statistics. Nigrini (2012) introduced Mean Absolute Deviation (MAD) as a method to check the conformity of data with Benford’s Law. This method is not influenced by the size of the sample compared to other methods Sadaf (2017).

Based on Benford’s law, the naturally occurring number’s first digit will be 1 or 2 compared to 8 or 9. The general logarithmic pattern of Benford’s law is

$$\scriptsize{ \begin{align} P\ (D1 = \ d1) &= \log\left( 1 + \frac{1}{d1} \right)for\ d1\ \in \ \left\{ 1,\ldots,9 \right\} \end{align} \tag{1} }$$

$$\small{ \begin{align} P\ (D2 = \ d2) &= \ \sum 9d1 \\ &= 1\log\left( 1 + \frac{1}{d1d2} \right)for\ d2\ \in \ \left\{ 1,\ldots,9 \right\} \end{align} \tag{2} }$$

$$\scriptsize{ \begin{align} P\ (D1D2 = \ d1d2)\ = \ log(1 + \frac{1}{d1\ d2})\ for\ d1\ d2\ \in \ \{ 1,\ldots,9\} \end{align} \tag{3} }$$

where $D1,\ D2\ldots D9$ denote the expected and $d1,\ d2\ldots d9$ denote the actual first digit, second digit, …, ninth digit of the naturally occurring numbers, respectively.

Based on Benford’s Law (see Appendix 1), digit 1 appears 30% as the first digit in a number, and digit 9 appears only 5% as the first digit. This study applied MAD to check the conformity of the numbers to Benford’s law. MAD can be calculated by

$$MAD\ = \ \sum Ki = 1|AP - EP|\ /\ K\tag{4}$$

Here, $AP$ and $EP$ are actual and expected proportions.^[1]

III. Data and Results

This study used the world’s daily CO₂ emissions data for various sectors, viz. power, residential, industry, domestic aviation, international aviation, and ground transport from January 2020 to December 2021. The data is taken from the website https://carbonmonitor.org/. The group of countries cover in our study are Brazil, Russia, India and China (BRIC), the US, EU27 countries and UK, and the rest of the world (ROW).

First, we analysed whether the CO₂ emissions data follow the requirements of Benford’s Law considering the following assumptions:

• Data should be from natural events or naturally occurring numbers (not predefined numbers like invoice number, serial number, etc.) (see Nigrini, 2012).

• Data set or observations should be fairly large; the sample should be between 50 to 100 (see Tošić & Vičič, 2021).

• The mean of the data should be greater than the median (or the data should be right-skewed); the data should not be symmetric (see Tošić & Vičič, 2021).

Table 1 summarises the descriptive statistics of the global CO₂ emissions for 2020 and 2021. The global CO₂ emissions, CO₂ emissions by BRIC countries, sector-wise global CO₂ emissions (except ground transport), and sector-wise CO₂ emissions by BRIC countries (except international aviation in 2021) have skewness greater than 1 and, hence, these data are right-skewed (i.e. mean is greater than the median). The number of observations is greater than 100, and the daily emissions data comes under the category of naturally-occurring numbers. Hence, these data follow the basic requirements of Benford’s Law and are fit for the test.

Table 2 explains the test results of the first digit, second digit, and first two-digit frequency of the global CO₂ emissions data (only right-skewed) from January 2020 to December 2021. For 2020, the first digit frequency of CO₂ emissions has a MAD value of 0.0138, which ranges from 0.015 to 0.012 and comes under “marginally acceptable conformity”. The second digit frequency has a MAD value of 0.0050, which falls under the category of “close conformity” with the range of 0.000 to 0.008, and finally, the first two digits fall under the category of “marginally acceptable conformity” with a MAD value of 0.0020. Likewise, for 2021, the MAD value of the first digit is 0.0199 (“nonconformity”), the second digit is 0.0071 (“close conformity”), and the MAD value of the first two digits is 0.0025 (“nonconformity”). Therefore, the global CO₂ emissions data for 2021 is not conforming, and 2020 conforms with Benford’s Law.

For CO₂ emissions data of the BRIC countries, the MAD values for 2020 and 2021 for the first digit fall under “nonconformity”, the second digit under “acceptable conformity”, and the first two digits under “nonconformity” categories. For sector-wise global CO₂ emissions data, the MAD values of the first digit of all the sectors fall under the category of “nonconformity”; the second digit of the industry (2021), international aviation (2021), and residential (both 2020 and 2021) fall under the category of “nonconformity”, and the first two digits of all the sectors fall under “nonconformity”. For sector-wise emissions by BRIC countries, the first, second, and first two digits are under “nonconformity”, except for the second digit of the international aviation sector in 2020.

IV. Concluding Remarks

The reduction of CO₂ emissions is more important in this highly polluted era, and the Covid-19 restrictions led to its reduction, but it was not sustainable. Our digit analysis of the global CO₂ emissions data revealed that the emissions for 2020 conforms to Benford’s Law, whereas the emissions for 2021 do not. This may be attributed to reporting dynamics in light of the regulatory pressures to show a sustainable emission report by the countries. Our results can be used to establish whether any particular government policy during the pandemic influenced the reported data. This can be done by comparing our results with couple of years before and after the pandemic. For the sectoral analysis, we found that the second digit conforms to Benford’s law and the first and first two digits do not. Even though the calculation of daily CO₂ emissions data is difficult, reporting true and fair data is important. Normally, analysts, researchers, and regulators use Benford’s Law to identify the red flags for further investigations. Similarly, the anomalies in the global CO₂ emission data can be considered red flags, and more serious investigation can be done with the help of big data analytical tools than ever before.

Acknowledgement

Authors would also like to thank the anonymous referee and the editorial team of the journal for their valuable comments in improving this study.