Application of Benford’s law to detect signs of under-invoicing in companies in the restaurant sector during the COVID-19 pandemic

The COVID-19 pandemic is characterised by its worldwide impact and the fact that it had harmful repercussions on public health and most national economies (Strange, 2020). In Portugal, government measures to contain the pandemic were implemented, and, as a result, the economy deteriorated rapidly due to a sharp decline in employment, income, and consumption (Mamede et al., 2020).

The use of statistical methods in identifying and analysing earnings manipulation in finance and accounting is increasingly necessary nowadays, and the less complex, most popular, and still highly effective tests are the ones based on Benford’s law (Slijepčevic & Blašković, 2014). Swanson et al. (2003) state that Benford’s law is a practice most commonly used in the business sector, specifically in accounting. Several authors in the literature have analysed and examined the applicability of this law in specific areas, such as accounting, focusing on identifying the handling and alteration of results (Carslaw, 1988; Durtschi et al., 2004; Slijepčevic & Blašković, 2014; Thomas, 1989).

Carslaw (1988) applied Benford’s law to accounting data from New Zealand companies with positive results. The author analysed the second digit of the profit category, detecting an excess of zeros in the second digit and, conversely, a scarcity of the digit 9. Carslaw (1988) concluded that companies’ profits tended to be rounded up to higher values than expected to present a better image among stakeholders.

Like Carslaw (1988), Thomas (1989) discovered a similar pattern in the profits of American companies. Companies showing negative results had more nines in the second digit position and fewer zeros than expected. Niskanen and Keloharju (2000) observed that Finnish companies tended to adjust the second digit further to the left in the net profit category, as they observed an excess in the number 9, to make earnings appear sporadically larger. Nigrini (2005) used Benford’s law to detect earnings manipulation at Enron (between 2001 and 2002), finding that most of the digits were above the reference points of the respective law, suggesting earnings manipulation in Enron’s revenues and earnings per share.

Several ongoing scientific studies have analysed and applied Benford’s law in accounting. Many studies use anomalies in the frequency of digits as evidence of the intentional actions (e.g., by managers) to manipulate the results, errors, or fraud of a given organisation. Aybars and Ataunal (2016) analysed the financial reports of companies listed on the Istanbul Stock Exchange between 2005 and 2015, detecting signs of manipulation due to various deviations in the total assets and net profit data. Like Aybars and Ataunal (2016), Gong et al. (2022) and Pupokusumo et al. (2022) used Benford’s law on a set of companies to capture traces of fraud in financial data disclosed by companies in China and Brazil, respectively. Dharni (2020) proved the effectiveness of Benford’s law in detecting shell companies, as he found a considerable distinction in the conformity of the distribution of Benford’s law between real and fictitious companies in his study.

Nigrini and Mittermaier (1997) pointed out the logic of using Benford’s law to help discover unusual patterns in economic/financial activities since it is very likely that an individual making fraudulent entries will use the same or identical numbers. However, Benford’s law can never prove or disprove the presence of manipulated data; it only indicates that a given set of data does or does not follow the law, and if the analysed data do not closely follow the Benford curve, they should be considered for closer examination and scrutiny (Collins, 2017). According to Durtschi et al. (2004), Benford’s law helps to detect suspicious data in financial statements so that these data can be analysed in detail and additional tests can be carried out, as Benford’s analysis alone is not an infallible method of detecting fraud.

The following section describes the methodology used in this study.

This study is centred on exploratory quantitative research, based on a prior literature review and formulated hypotheses to be tested later using statistical techniques. The VN of a group of Portuguese companies belonging to Economic Activity Code (CAE) 56 was analysed during the COVID-19 pandemic period, specifically between 2019 and 2021, with the aim to detect signs of under-invoicing through the application of Benford’s law.

The main objective is to detect signs of anomalies, i.e., whether companies adjusted their turnover in 2020 to maximise the support they received from the State in 2021. To test the research global hypothesis, a significance level α of 5% (α=0.05) was established to test H0, measuring the risk of rejecting a true hypothesis. If H0 is rejected, it means that the VN values do not conform to Benford’s law, i.e., suggesting potential evidence of data set manipulation. If H0 is not rejected, the VN values follow the expected values according to Benford’s law, from which it can be concluded that the data analysed conform to the distribution of the law, and therefore, no evidence of under-invoicing.

The data were obtained from the Iberian Balance Sheet Analysis System (SABI) database. The data from Portuguese restaurant companies, CAE 56, were collected from 2019 to 2021 and processed and analysed in Microsoft Excel and SAS. The main purpose of choosing the period evaluated was to obtain data that would allow us to know the variations in companies’ turnover before and during the COVID-19 pandemic.

The sample used consisted of 19,437 companies in the restaurant sector, the majority of which were micro-sized companies, a fact explained by the fact that this is the most prevalent type of company size in Portugal. It should be emphasised that, for a company to be included in the sample for this study, the following criteria were required: –

The company’s headquarters and location must be in Portugal.

Firstly, we analysed the conformity of the data by calculating the frequency of the values found in the first digit, the second digit, and the first two digits, comparing them with the expected frequency of Benford’s law.

The first digit and second digit tests are often used as high-quality preliminary and secondary tests, respectively (Drake & Nigrini, 2000). According to Nigrini and Mittermaier (1997), the first digit test is not intended to identify abnormal duplications or irregularities but only to check whether or not a data set follows the Benford distribution. The second digit test, on the other hand, signals rounding in values with an excess of the digits 0 and 5 than expected, compared to Benford’s law (Drake & Nigrini, 2000). The first two digits test analyses and captures peaks (excess theoretical digits) in which the actual proportion, Po, is much higher than the expected proportion, Pe (Drake & Nigrini, 2000; Nigrini & Mittermaier, 1997). This test is essential for analysing a large data set and identifying duplicates and data errors in the event of poor compliance with Benford’s law (Nigrini, 2011).

For each test, statistical tests known as the Z-statistic and the chi-square test (X²) adopting Nigrini’s model (2011) will be applied to analyse the conformity of the data.

The main purpose of the Z-statistic is to determine the statistical significance of the deviation between Po and Pe, signalling the digits that appear with greater or lesser regularity in a given position (calculation of the upper and lower limits in which the Z-statistic is equal to 1.96 with a significance level of 5%). Any digit that appears with greater or lesser regularity in a given position means that the calculation of the Z-statistic has been projected above or below the limit of Benford’s law line (Nigrini, 2011). The Z-statistic is calculated as follows (Nigrini, 2011): 1Z=|Po−Pe|−12NPe(1−Pe)NZ = {{|Po - Pe| - {1 \over {2N}}} \over {\sqrt {{{Pe(1 - Pe)} \over N}} }} where: –

Pe is the expected proportion for a given digit, according to the probability stipulated by Benford’s law.

The main purpose of the X² test is to compare the relationship between Po and Pe by adding up the deviations in the position of a given digit in a sample. Like the Z-statistic, the X² test is also sensitive to the sample size (Nigrini, 2011).

The chi-square statistic for the X² test is calculated as follows (Nigrini, 2011): 2X2=∑i=1K(Po−Pe)2Pe{X^2} = \mathop \sum \limits_{i = 1}^K {{{{(Po - Pe)}^2}} \over {Pe}} where: –

Po is the proportion observed for a given digit.

With a significance level of 5%, the first and second digits are K=9 and K=10 (9 and 10 possible digits, respectively); thus, the number of degrees of freedom is eight and nine (k−1) for the X² test on the first and second digits. For the test on the first two digits, the degree of freedom is 89 (Nigrini, 2011). The critical values for the X² test are 15.51, 16.92, and 112.02, for the first digit, second digit, and first two digits tests, respectively (Nigrini, 2011). The chisquare statistic of the X² test is obtained by adding up all the parcels concerning the digits in a given sample. The null hypothesis is rejected when the value of X² exceeds the critical value defined for X² distribution.

The next sections present the main results and discussion.

This section presents and analyses the results obtained from the data set mentioned in the previous section to verify that the sample is in accordance with Benford’s law. Effectively, the sample presents a global reduction of 79% in turnover, on average, making them susceptible to under-invoicing and allowing them to benefit from the State’s incentives previously mentioned. Three groups of companies are considered, taking into account the size, “Micro-enterprises”, and the drop in turnover with “Variation 25% to 35%” and “Variation 35% to 75%”.

Table 4 shows the minimum (Min), maximum (Max), average (x¯$\bar x$), standard deviation (s), and number of observations (n) for the total and groups of companies for the following values: VN, total assets (TA), and operating subsidy (SE) for 2020.

As shown in Table 4, in the Total group, which includes all companies under study, the maximum value for VN is €102,839,963 and that for TA is €190,580,920. In the “Variation 25% to 35%” group, it can be seen that VN has a x¯ $\bar x$ of €299,299. TA shows values between €320 and €146,861,014. The third group, “Variation 35% to 75%”, presents all the companies that showed a percentage reduction in their turnover between 35% and 75% in 2020, representing around 40% of the sample. Looking at these values of the turnover and TA, it can be seen that these are smaller companies than those in the previous division; i.e., the smaller companies recorded a greater reduction in turnover in 2020. The “Micro-enterprises” section shows all the companies with a turnover and TA of €2,000,000 or less. Thus, it can be concluded that most of the companies under study are predominantly micro-sized, and as a result, it was impossible to analyse the data set that belonged to other dimensions due to the limited number of companies available for analysis using Benford’s law.

According to Benford’s law, Table 5 shows the results obtained by applying the Z-statistic and the X² test to the first digit in the various groups of the sample under study.

By applying Benford’s law, all the groups have anomalies, except for “Variation 25% to 35%” which has a X² equal to 7.9 (below its critical limit) and a Z-statistic lower than its critical value of 1.96, in all digits from 1 to 9.

The groups “Variation 35% to 75%” and “Microenterprises” have a X² value equal to 16.9 and 53.9, respectively, rejecting H0. Thus, these groups fail to comply with Benford’s law, presenting a value of X² higher than its critical value, mainly the “Micro-enterprises” group.

The Z-statistic shows that digits 1 to 4 and 9 have a value above their critical threshold in the “All” and “Micro-enterprises” groups. The “Variation 35% to 75%” group only shows a value above its critical threshold in individual digits 2 and 4. It can be seen that the Po is higher than the Pe (excess) in individual digits 1 and 9 and, on the other hand, there is a shortage in digits 2, 3, and 4 for the “All” and “Micro-enterprises” groups. The “Variation 35% to 75%” section has a Po lower than the Pe in digits 2 and 4.

In this sense, it can be concluded that H0 is not rejected for the “Variation 25% to 35%” group, since there are no statistically significant differences between Po and Pe, at a significance level of 5%, according to Benford’s law. H0 is rejected for the remaining groups, as their data do not conform to Benford’s law, considering the values obtained by the Z-statistic and the X² test.

Compared to other studies similar to this one, there is a greater deviation between Po and Pe in the first digit test, which may suggest a potential duplication of numbers.

Next, we present the results obtained by applying the second digit test, illustrated in Table 6, for the groups analysed.

The X² test does not exceed the stipulated critical value of 16.02, for a significance level of 5%, for any of the groups analysed. There is only one violation of the critical value of Z in the “All” section for digits 0 and 9. Digit 0 is in excess, whereas digit 9 has Po<Pe (scarcity). In the other groups, the Z-statistic complies with Benford’s law, since the value of Z obtained for digits 0 to 9 does not exceed its critical limit. However, the “Micro-enterprises” section shows a Z value close to its critical limit for the individual digits 0 and 9.

For the second digit test, it can be concluded that H0 is rejected only for the “All” group since there are statistically significant differences between Po and Pe, at a significance level of 5%, according to Benford’s law, considering the values obtained by the Z-statistic. According to Benford’s law, H0 is not rejected for the remaining groups, as there are no statistically significant differences between Po and Pe at a 5% significance level.

Table 7 shows the results obtained by applying the tests to the first two digits. Table 7 only illustrates the digits with the most significant deviation between Po and Pe.

Analysing the X² test, it can be seen that all the groups have a statistic lower than their critical value, except for the “All” and “Micro-enterprises” groups. These groups are incompatible for the test of the first two digits, since the statistics obtained (120 and 131) are higher than the critical value.

If the digits are evaluated individually, there is a violation of the critical value of Z-statistic in all the groups analysed, with 14 digits in which the value obtained exceeded the threshold (1.96), for a significance level of 5%. It can be seen that individual digits 10, 11, 44, 49, 76, and 98 show a significant deviation between Po and Pe for both the “All” and “Micro-enterprises” groups. The remaining groups, “Variation 25% to 35%” and “Variation 35% to 75%”, show non-compliance with Benford’s law, according to the value obtained for the Z-statistic, in the digits 50, 57, 76, 79, and 98 and 11, 24, 44, 47, and 88, respectively. The digits 10, 11, 57, 76, 79, 88, 90, and 98 have a Po greater than Pe (excess), according to Benford’s law, while there is a scarcity in the remaining digits (24, 29, 44, 47, 49, 50, and 88), with Po<Pe.

In this sense, we conclude that H0 is rejected for all groups, since they show statistically significant differences between Po and Pe, for a significance level of 5%, according to Benford’s law, considering the values obtained by the Z-statistic and the X² test.

The results presented, especially in this last test, show signs of under-invoicing in companies in the restaurant sector in 2020, allowing them to benefit from the incentives provided by the State in 2021.