Audit Quality Inputs and Financial Statement Conformity to Benford’s Law

Abstract

We examine whether audit quality inputs are related to the conformity of financial statements to Benford’s law. We find that overall financial statement conformity increases with audit fees, nonaudit fees, and audit report lag, and decreases with audit firm tenure. We also find that these audit quality inputs are more strongly associated with income statement conformity than with cash flow statement conformity. Our findings document the role that auditing plays in enhancing the conformity of financial statements to Benford’s law.

Keywords

audit quality inputs audit fees audit report lag nonaudit service fees audit firm tenure Benford’s law

Introduction

We examine the relation between financial statement conformity (FSC) to Benford’s law and audit quality. Our study is motivated by the recent findings of Amiram et al. (2015) and Nigrini (2015) that, although accounting data tend to conform to Benford’s law on average, the degree of conformity varies significantly across firms. Amiram et al. (2015) and Amiram et al. (2018) suggest that significant deviations from Benford’s law may indicate accounting irregularities, such as fraud or error. However, Nigrini (2015) suggests that deviations may also arise due to nonfraud reasons, such as data replacement or standardization. Collectively, prior studies suggest that a number of human-related factors influence the conformity of accounting data to Benford’s law.

In this study, we propose that audit quality is one such factor. As an independent third party, auditors provide reasonable assurance that the financial statements are free of material errors. Auditors are involved directly in the firm’s financial reporting process that generates the accounting information reported in the financial statements. Therefore, we propose that the conformity of financial statements to Benford’s law is associated with variables that measure the level of auditors’ inputs into the financial reporting process.

We focus on four measures of audit quality—audit fees, nonaudit fees, audit report lag, and audit firm tenure—and examine their relations with FSC to Benford’s law. We use the mean absolute deviation and the Kolmogorov–Smirnov distance of the financial statements’ first digit distribution from the Benford distribution as measures of FSC (Amiram et al., 2015). These measures are based on an empirical phenomenon, named after Benford (1938), who showed that the leading digits in empirical data sets tend to follow the logarithmic distribution (or Benford distribution). This empirical phenomenon suggests that data sets that are free from errors and biases will conform more closely to the Benford distribution, while data sets that are falsified or plagued with errors will deviate significantly from the Benford distribution.

Our empirical tests reveal the following interesting findings. First, we find that FSC to Benford’s law increases with audit fees, nonaudit fees, and audit report lag, suggesting that, on average, FSC increases with auditor effort. For audit firm tenure, we find that FSC is greater for short tenure than for medium and long tenure. Second, we find that all four audit quality inputs are more strongly associated with the conformity of the income statement than with the conformity of the statement of cash flows. This result is consistent with expectations, because the accruals component of earnings is more susceptible to manipulation and estimation error than is the cash flow component, which is more easily verifiable. Therefore, in the absence of an audit, the numbers in the income statement are likely to deviate more from the Benford distribution than the numbers in the cash flow statement.

Our study makes at least two important contributions. First, our study extends research on Benford’s law by presenting preliminary evidence on different determinants of a firm’s FSC to (or deviation from) the Benford distribution. From a theoretical perspective, we have little understanding of the factors that explain why the financial statements conform to (deviate from) the expected Benford distribution. Our findings indicate that auditing influences the number generation process of financial statements and, therefore, firms’ FSC to Benford’s law. The results are both empirically and practically important and have implications for future research on Benford’s law.

Second, we contribute to the audit literature, by providing large-sample evidence on the relationships between various audit quality inputs and overall FSC to Benford’s law. We find that overall FSC increases with audit fees, nonaudit fees, and audit report lag and decreases with audit firm tenure. In addition, we show that auditing has greater influence on the conformity of the income statement than on the conformity of the cash flow statement. Together, these findings are meaningful additions to the literature on the relation between auditing and a mathematical property of financial statements.

The rest of the study is organized as follows. Section “Literature Review” discusses the relevant literature, section “Sample and Research Design” describes the empirical measures, sample and research design, section “Empirical Results” reports and interprets the empirical results, and section “Conclusions” concludes the study and discusses its limitations and implications for future research.

Literature Review

Newcomb (1881) was the first to document that the first digits of numbers in empirical data sets do not occur with equal frequency. Later, Benford (1938) conceptualized this phenomenon as “The Law of Anomalous Numbers.” Based on data from numerous disciplines, ranging from economics to physics, he found that the digit 1 appears as the leading digit more than 30% of the time, and the frequency of occurrence of the leading digit decreases monotonically from 1 to 9. This early evidence suggests that the leading digits in empirical data sets that arise naturally tend to follow the logarithmic rather than the uniform distribution.

Under Benford’s law, the probability distribution of the first digit, d, in a data set is as follows (see Appendix A for more details):

P (d) = Lo g_{10} (1 + \frac{1}{d}), with d \in {1, \dots, 9} .

Mathematicians and statisticians have provided mathematical proofs for the number generation process underlying Benford’s law (Boyle, 1994; Hill, 1995; Pinkham, 1961; Raimi, 1969, 1976; Ross, 2017). The theoretical underpinnings of Benford’s law can be explained by the following two mathematical facts. The first fact is that the mantissa (the fractional part) of the common logarithm (base 10) of a positive number will determine that number’s leading digit.¹ Formally stated, a positive number X will start with digit d if the mantissa of Log₁₀ (X) is a number between Log₁₀ (d) and Log₁₀ (d+1). The second fact is that if the distribution of the data set is smooth, symmetric, and spread over several orders of magnitude in the log scale, then the probability of d being the leading digit of a positive number X is equal to the interval between Log₁₀ (d) and Log₁₀ (d+1) (Fewster, 2009).²

Economists find that the distribution of first digits in certain economic data sets tends to conform to the theoretical Benford distribution (Judge & Schechter, 2009; Ley, 1996; Marchi & Hamilton, 2006; Michalski & Stoltz, 2013; Varian, 1972). These studies suggest that a data set that generally follows Benford’s law will diverge significantly if it contains errors or anomalies. Varian (1972) further highlights that whereas conformity to Benford’s law does not mean that data are completely free from error, nonconformity to the law should raise a red flag for further investigation.

Early research in accounting has also relied on Benford’s law to detect irregularities in accounting numbers. Carslaw (1988) finds that the occurrence of the second digit in 220 New Zealand firms’ earnings figures deviates from the expected distribution under Benford’s law. Thomas (1989) follows the same approach and also finds that U.S. firms’ earnings deviate from the expected Benford distribution. Nigrini (1995) applies Benford’s law to U.S. tax return information to detect tax evasions and hypothesizes that falsified or fabricated tax numbers will result in a first digit distribution that deviates from the expected Benford distribution. Durtschi et al. (2004), Nigrini and Miller (2009), and Nigrini and Mittermaier (1997) develop analytical tools based on Benford’s law to help auditors detect anomalies in large accounting data sets.

Using the mean absolute deviation and the Kolmogorov–Smirnov distance of the actual distribution from the Benford distribution to measure conformity, Amiram et al. (2015) find that, in aggregate, the distributions of the first digits of all the line items in firms’ income statements, balance sheets, and statements of cash flows for the entire population of US firms closely follow Benford’s law. They also document differential levels of conformity of individual firms’ income statements, balance sheets, and statements of cash flows to Benford’s law.

In this study, we use the mean absolute deviation and the Kolmogorov–Smirnov distance of the financial statements’ first digit distribution from the Benford distribution as measures of FSC. We use audit fees, audit report lag, nonaudit fees, and audit firm tenure as measures of auditors’ inputs, and propose that the conformity of financial statements to Benford’s law is associated with measures of auditors’ inputs for the following reasons. For accounting data, which generally conform to Benford’s law, nonconformity suggests that artificial factors may influence the underlying data-generation process. For this reason, we expect that the conformity to Benford’s law is associated with variables that measure the level of human inputs in the financial reporting process. Because auditors are involved directly in the firm’s financial reporting process, which generates accounting information that is reported in the financial statements, we expect that the auditor’s inputs are related to the conformity of financial statements to Benford’s law.

Sample and Research Design

We begin our sample selection with data from Compustat to identify the accounting line items that appear in a firm’s balance sheet, income statement, and statement of cash flows.³ Then, for each firm-year, we extract the first digits of each of the identified line items.⁴ We exclude line items that have missing values or are equal to 0 from the computation. Following Amiram et al. (2015), we remove any firm-year with less than 100 nonmissing financial statement line items to compute the FSC measures.⁵

We assess the conformity of a firm’s leading digits’ distribution to the Benford distribution based on the following measures: (−100) times the mean absolute deviation (MAD) and (−100) times the Kolmogorov–Smirnov distance (KS). MAD is computed as (−100) times the average absolute distance between a firm’s first digit distribution and the Benford distribution. KS is computed as (−100) times the maximum absolute distance between the firm’s first digit cumulative distribution and the Benford cumulative distribution.⁶ We conduct our analysis using both measures to ensure the robustness of our findings. Larger values of the measures indicate higher FSC to Benford’s law. We provide the details of the calculation of these measures in Appendix A.

Next, we use Audit Analytics and Compustat to obtain the data for the following observable audit quality measures: audit fees (AUD_FEES), nonaudit fees (NON_AUD_FEES), audit report lag (AUD_REPORT_LAG), and audit firm tenure (TENURE). Following prior studies, we log transform a firm’s audit fees and nonaudit fees due to the skewness in fees data. We measure audit report lag as the length of time (in days) between the fiscal year end date and the audit report date, that is, the date that the auditor finalizes and completes the audit report. Finally, we measure audit tenure as the number of continuous years the audit firm has served the same client. Following Davis et al. (2009), we control for nonlinearity in the relationship between audit firm tenure and FSC by including both tenure (TENURE) and tenure squared (TENURE2) in the model. The primary advantage of this approach is that it allows tenure to have a gradual impact on a firm’s overall FSC to Benford’s law.

We also measure audit firm tenure using indicator variables for long, medium, and short tenure, following the approach of Johnson et al. (2002) and Davis et al. (2009). We define long tenure (LONGTENURE) as audit firms that continuously serve the same client for at least 15 years and short tenure (SHORTTENURE) as audit firms that serve the same client for less than 4 years. The primary advantage of this approach is that it facilitates a more intuitive interpretation of the empirical results.

We estimate the following ordinary least squares (OLS) regression to examine the relations between the audit quality inputs and FSC:

FS C_{i, t} = α + β_{1} \times AQ_Input + \sum Controls + IndFE + YearFE + ε_{i, t},

(1)

where FSC_i,t is firm i’s measure of FSC to Benford’s law (MAD or KS) in year t, AQ_Input is one of the four audit quality inputs discussed above, ∑Controls is a vector of control variables, IndFE and YearFE are industry (based on two-digit Standard Industrial Classification [SIC] code) and year fixed effects. β₁ denotes the association between the audit quality input and FSC.

Because there exists little theoretical guidance on the determinants of a firm’s FSC to Benford’s law, we control for a firm’s fundamental characteristics and economic performance. Economic intuition suggests that a firm’s fundamental characteristics and performance could influence the likelihood of occurrence of anomalies in the financial statements and, hence, the conformity of its financial statements to Benford’s law.

We control for the following fundamental firm characteristics: firm size (SIZE), firm age (AGE), market-to-book ratio (MTB), leverage (LEV), return on assets (ROA), sales growth (SALE_GROWTH), cash flow from operations (CFO), total accruals (ACCRUALS), and whether the firm reports a loss (LOSS). We also control for the following variables that potentially relate to the complexity of a firm’s financial reporting environment and tendency to manipulate accounting data: number of business segments (NUMSEGS), presence of foreign operations (FOREIGN), current ratio (CURRENT), quick ratio (QUICK), earnings volatility (NIBE_VOL), and probability of financial distress (Z_SCORE). We also control for whether the firm engages in acquisition activity (ACQUIRE), in debt financing (DEBTFIN) and in stock financing (STOCKFIN), whether the firm has extraordinary items or discontinued operations (EX_DISC), and whether the firm belongs to a high-litigation risk industry (LITIGATION). We provide detailed variable definitions in Appendix B.

In all the analyses, we control for industry fixed effects using two-digit SIC industry code. To the extent that the firm’s first digit distributions are industry-specific, including industry fixed effects in the model helps to control for the correlation between the FSC measures and industry characteristics. We also include year fixed effects to control for changes in macroeconomic conditions and regulations over time. We use all available data from Compustat and Audit Analytics and restrict the analysis to observations with nonmissing data for all of the previously discussed variables. We winsorize all continuous variables at the top and bottom 1 percentile each year to mitigate the effect of extreme values, and compute standard errors clustered at the firm level.

Empirical Results

Descriptive Statistics

Table 1, Panel A presents descriptive statistics for the relevant variables used in the primary analyses. The means (medians) of the two FSC measures, MAD and KS, are –2.89 (–2.81) and –8.27 (–7.63), respectively. Recall that these variables are defined such that higher values indicate higher conformity to Benford’s law. The distributions of both measures are comparable to those reported in Amiram et al. (2015). The means (medians) of the log of audit fees (AUD_FEES) and the log of nonaudit service fees (NON_AUD_FEES) are 13.58 (13.61) and 12.01 (12.03), respectively. The mean (median) of audit report lag (AUD_REPORT_LAG) is approximately 62 (60) days, which reflects the average length of an audit engagement. The average audit firm tenure (TENURE) is 10.1 years, and approximately 20% of client firms have auditors with either long tenure or short tenure. The distributions of the audit quality input measures and the control variables are consistent with those reported in prior studies.

Table 1.

Descriptive Statistics and Correlations.

Panel A. Descriptive Statistics of Variables Used in the Main Test.
Variables	N	M	SD	Q1	Median	Q3
MAD	27,445	−2.892	0.860	−3.435	−2.806	−2.270
KS	27,445	−8.267	3.531	−10.291	−7.626	−5.648
AUD_FEES	27,427	13.578	1.318	12.602	13.608	14.463
NON_AUD_FEES	24,619	12.005	1.786	10.820	12.029	13.212
AUD_REPORT_LAG	27,445	62.066	19.867	53.000	60.000	73.000
TENURE	27,442	10.116	8.367	4.000	8.000	13.000
LONGTENURE	27,445	0.211	0.408	0.000	0.000	0.000
SHORTTENURE	27,445	0.196	0.397	0.000	0.000	0.000
SIZE	27,445	6.293	2.106	4.735	6.229	7.775
BTM	27,445	0.667	0.727	0.279	0.484	0.786
LEV	27,445	0.500	0.221	0.324	0.510	0.671
ROA	27,445	−0.011	0.175	−0.032	0.030	0.072
SALE_GROWTH	27,445	0.124	0.361	−0.034	0.071	0.204
ACCRUALS	27,445	−0.074	0.126	−0.103	−0.054	−0.020
CFO	27,445	0.055	0.145	0.025	0.076	0.127
LOSS	27,445	0.331	0.471	0.000	0.000	1.000
EX_DISC	27,445	0.016	0.125	0.000	0.000	0.000
ACQUIRE	27,445	0.026	0.158	0.000	0.000	0.000
DEBTFIN	27,445	0.550	0.497	0.000	1.000	1.000
STOCKFIN	27,445	0.813	0.390	1.000	1.000	1.000
CURRENT	26,889	2.602	2.214	1.254	1.908	3.101
QUICK	26,609	2.121	2.005	0.959	1.456	2.462
Z_SCORE	26,889	−1.406	1.572	−2.538	−1.414	−0.435
NIBE_VOL	27,445	0.094	0.130	0.021	0.049	0.113
AGE	27,445	22.278	16.387	10.000	16.000	30.000
NUMSEGS	27,359	1.619	0.991	1.000	1.000	2.000
FOREIGN	27,445	0.350	0.477	0.000	0.000	1.000
LITIGATION	27,445	0.284	0.451	0.000	0.000	1.000

Panel B: Pearson Pairwise Correlations Among Selected Variables.
Variables	(1)	(2)	(3)	(4)	(5)	(6)
MAD (1)	1.000
KS (2)	0.729***	1.000
AUD_FEES (3)	0.287***	0.218***	1.000
NON_AUD_FEES (4)	0.236***	0.181***	0.602***	1.000
AUD_REPORT_LAG (5)	−0.068***	−0.053***	−0.139***	−0.252***	1.000
TENURE (6)	0.102***	0.076***	0.328***	0.263***	−0.140***	1.000

Note. Table 1, Panel A reports descriptive statistics of all relevant variables used in the main test. Panel B reports Pearson pairwise correlations between the primary dependent variables, MAD and KS, and the primary independent variables, AUD_FEES, NON_AUD_FEES, AUDIT_REPORT_LAG, and TENURE. Appendix A describes the computation of MAD and KS and Appendix B provides variable definitions.

***

, **, and * denote statistical significance at the 1%, 5% and 10% level, respectively.

Panel B shows the pairwise correlations between relevant variables used in the analysis. The correlation table shows that the two conformity measures, MAD and KS, are significantly positively correlated, with a Pearson correlation coefficient of 0.73. The correlations between the conformity measures and each of the observable audit quality inputs are also statistically significant.

Empirical Results and Discussion

Table 2, columns 1 and 2 report the regression results relating audit fees to FSC. We estimate the regression separately for each conformity measure—MAD (column 1) and KS (column 2)- and control for relevant variables as discussed in the previous section. In both columns, we find a significantly positive association between audit fees and FSC (column 1: β₁ = 0.101, t = 8.12; column 2: β₁ = 0.327, t = 6.25), which indicates greater FSC to Benford’s law when audit fees are higher. Higher audit fees can be interpreted as the compensation for the auditor’s higher level of effort and resources expended in auditing a client. Alternatively, higher audit fees can be interpreted as a premium for hiring auditors with higher expertise. Therefore, these results indicate that auditor effort, measured as audit fees, is positively associated with FSC.

Table 2, columns 3 and 4 report the regression results relating audit report lag to FSC. The model specifications are similar to those reported in columns 1 and 2. In both columns, we find a significantly positive association between audit report lag and FSC (column 3: β₂ = 0.001, t = 4.09; column 4: β₂ = 0.004, t = 2.77), which indicates that FSC increases with audit report lag. Audit report lag reflects the length of the audit engagement, with longer lag interpreted as more labor hours spent by the auditor on the audit engagement, that is, greater audit effort. These results indicate that when auditors devote more effort, the client’s financial statements are more likely to conform to Benford’s law. They reinforce the results in columns 1 and 2, where auditor effort is measured as audit fees.

Table 2, columns 5 and 6 report the regression results relating nonaudit service fees to FSC. The model specifications are similar to those in columns 1 to 4, except that we add an additional control for audit fees to disentangle the relations between audit fees and nonaudit service fees and FSC. In both columns, we find a significantly positive association between nonaudit service fees and FSC (column 5: β₃ = 0.015, t = 3.01; column 6: β₃ = 0.059, t = 2.83), which indicates that the provision of nonaudit services increases FSC. Prior research suggests that the joint provision of audit and nonaudit services may create synergies, which improve the auditor’s client-specific knowledge and auditor competence. Therefore, the results indicate that greater client-specific knowledge, as reflected by nonaudit service fees, is positively associated with FSC.

Table 2.

The Relation Between Audit Fees, Audit Report Lag, Nonaudit Service Fees, and Financial Statement Conformity.

	(3)	(4)	(3)	(4)	(5)	(6)
Variables	MAD	KS	MAD	KS	MAD	KS
AUD_FEES (β₁)	0.101*** (8.12)	0.327*** (6.25)			0.095*** (7.11)	0.303*** (5.30)
AUD_REPORT_LAG (β₂)			0.001*** (4.09)	0.004*** (2.77)
NON_AUD_FEES (β₃)					0.015*** (3.01)	0.059*** (2.83)
SIZE	0.031*** (3.76)	0.066* (1.91)	0.089*** (18.23)	0.254*** (12.40)	0.021** (2.43)	0.031 (0.83)
BTM	0.026*** (2.93)	0.069* (1.78)	0.017* (1.91)	0.042 (1.08)	0.028*** (2.87)	0.084** (1.96)
LEV	−0.015 (−0.13)	0.102 (0.21)	−0.025 (−0.22)	0.069 (0.14)	−0.062 (−0.53)	−0.024 (−0.05)
ROA	0.694*** (5.45)	2.693*** (4.87)	0.704*** (5.54)	2.748*** (4.98)	0.680*** (4.97)	2.417*** (4.05)
SALE_GROWTH	−0.003 (−0.18)	−0.007 (−0.11)	−0.015 (−0.96)	−0.049 (−0.73)	−0.013 (−0.79)	−0.023 (−0.32)
ACCRUALS	−0.399*** (−3.19)	−1.606*** (−2.83)	−0.428*** (−3.42)	−1.728*** (−3.06)	−0.330** (−2.44)	−1.296** (−2.14)
CFO	0.015 (0.11)	0.099 (0.17)	−0.018 (−0.13)	−0.039 (−0.07)	0.073 (0.51)	0.541 (0.87)
LOSS	−0.022 (−1.40)	0.028 (0.41)	−0.014 (−0.86)	0.056 (0.83)	−0.027 (−1.61)	−0.007 (−0.09)
AGE	−0.000 (−0.78)	−0.001 (−0.24)	−0.000 (−0.33)	0.000 (0.06)	−0.000 (−0.89)	−0.001 (−0.44)
EX_DISC	−0.028 (−0.68)	0.030 (0.16)	−0.028 (−0.68)	0.033 (0.18)	−0.015 (−0.36)	0.092 (0.50)
ACQUIRE	0.047 (1.39)	−0.015 (−0.11)	0.052 (1.53)	0.001 (0.01)	0.035 (1.02)	−0.033 (−0.22)
DEBTFIN	0.119*** (9.10)	0.399*** (7.17)	0.115*** (8.77)	0.387*** (6.93)	0.123*** (8.97)	0.417*** (7.23)
STOCKFIN	0.049*** (3.05)	0.147** (2.10)	0.055*** (3.36)	0.166** (2.34)	0.052*** (2.98)	0.156** (2.09)
CURRENT	−0.003 (−0.23)	−0.090 (−1.45)	−0.005 (−0.34)	−0.095 (−1.52)	0.002 (0.16)	−0.097 (−1.41)
QUICK	−0.024 (−1.57)	0.007 (0.10)	−0.025 (−1.61)	0.003 (0.04)	−0.029* (−1.72)	0.013 (0.17)
Z_SCORE	0.028 (1.52)	0.060 (0.72)	0.031* (1.69)	0.071 (0.85)	0.037* (1.90)	0.088 (1.01)
NIBE_VOL	−0.087 (−1.52)	−0.266 (−1.11)	−0.049 (−0.86)	−0.159 (−0.66)	−0.088 (−1.42)	−0.220 (−0.86)
NUMSEGS	0.022*** (2.95)	0.091*** (2.85)	0.025*** (3.45)	0.103*** (3.22)	0.019** (2.53)	0.078** (2.37)
FOREIGN	0.112*** (7.30)	0.288*** (4.50)	0.139*** (9.01)	0.378*** (5.93)	0.111*** (6.93)	0.259*** (3.86)
LITIGATION	−0.080*** (−3.09)	−0.256** (−2.31)	−0.071*** (−2.72)	−0.225** (−2.02)	−0.075*** (−2.74)	−0.245** (−2.08)
Observations	26,527	26,527	26,544	26,544	23,819	23,819
R ²	.135	.084	.132	.082	.133	.084
Industry FE	Yes	Yes	Yes	Yes	Yes	Yes
Year FE	Yes	Yes	Yes	Yes	Yes	Yes

Note. Table 2 reports estimation results of the relations between overall FSC (measured by MAD and KS) and audit fees (AUD_FEES), audit report lag (AUD_REPORT_LAG), and nonaudit service fees (NON_AUD_FEES). Appendix A describes the computation of MAD and KS and Appendix B provides variable definitions. The t-values reported in the parentheses are computed based on robust standard errors clustered at the firm level. FSC = financial statement conformity; MAD = mean absolute deviation; KS = Kolmogorov–Smirnov distance; FE = fixed effects.

***

, **, and * denote statistical significance at the 1%, 5% and 10% level, respectively.

Table 3 presents the regression results relating audit firm tenure to FSC. We examine this relation using two approaches: (a) including the square of audit firm tenure to account for nonlinearity in the relationship, and (b) using indicator variables for short and long tenure⁷ to allow the relation to differ across short, medium, and long audit firm tenure in a piecewise linear fashion. Columns 1 and 2 report the regression results under the first approach, with TENURE and TENURE2 as the primary variables of interest. When the conformity measure is MAD (column 1), the coefficient on TENURE is negative and statistically significant at the 5% level (column 1: β₁ = -0.006, t = -2.55), while the coefficient on TENURE2 is positive and statistically significant at the 5% level (column 1: β₂ = 0.0002, t = 2.31). The results in column 2, using KS as the conformity measure, are consistent, although the coefficients are only weakly significant (TENURE: column 2: β₁ = -0.019, t = -1.94; β₂ = 0.0005, t = 1.69). Overall, these results indicate that the relationship between FSC and audit firm tenure follows a U-shaped pattern. Specifically, FSC is high in the early years of tenure.

Table 3.

The Relation Between Audit Firm Tenure and Financial Statement Conformity.

	(3)	(4)	(5)	(6)
Variables	MAD	KS	MAD	KS
TENURE (β₁)	−0.006** (−2.55)	−0.019* (−1.94)
TENURE2 (β₂)	0.0002** (2.31)	0.001* (1.69)
LONGTENURE (β₃)			−0.003 (−0.19)	−0.047 (−0.64)
SHORTTENURE (β₄)			0.043*** (3.00)	0.111* (1.76)
SIZE	0.086*** (17.87)	0.244*** (12.08)	0.086*** (17.91)	0.244*** (12.11)
BTM	0.021** (2.41)	0.054 (1.40)	0.021** (2.42)	0.054 (1.40)
LEV	−0.012 (−0.11)	0.108 (0.22)	−0.011 (−0.10)	0.110 (0.22)
ROA	0.699*** (5.49)	2.736*** (4.93)	0.699*** (5.50)	2.734*** (4.94)
SALE_GROWTH	−0.015 (−0.91)	−0.047 (−0.71)	−0.014 (−0.88)	−0.046 (−0.69)
ACCRUALS	−0.427*** (−3.41)	−1.730*** (−3.05)	−0.430*** (−3.44)	−1.736*** (−3.07)
CFO	−0.020 (−0.15)	−0.048 (−0.08)	−0.024 (−0.18)	−0.060 (−0.10)
LOSS	−0.011 (−0.66)	0.065 (0.97)	−0.010 (−0.65)	0.065 (0.96)
AGE	−0.000 (−0.45)	−0.000 (−0.03)	−0.000 (−0.30)	0.001 (0.22)
EX_DISC	−0.022 (−0.54)	0.048 (0.26)	−0.022 (−0.54)	0.049 (0.27)
ACQUIRE	0.053 (1.56)	0.004 (0.03)	0.054 (1.57)	0.005 (0.04)
DEBTFIN	0.117*** (8.84)	0.390*** (6.98)	0.117*** (8.88)	0.391*** (7.01)
STOCKFIN	0.053*** (3.23)	0.159** (2.26)	0.052*** (3.18)	0.157** (2.22)
CURRENT	−0.004 (−0.29)	−0.092 (−1.48)	−0.005 (−0.32)	−0.094 (−1.50)
QUICK	−0.027* (−1.71)	−0.002 (−0.03)	−0.026* (−1.68)	0.000 (0.00)
Z_SCORE	0.030* (1.65)	0.068 (0.81)	0.030 (1.63)	0.067 (0.81)
NIBE_VOL	−0.052 (−0.91)	−0.169 (−0.71)	−0.049 (−0.86)	−0.159 (−0.67)
NUMSEGS	0.026*** (3.52)	0.104*** (3.26)	0.026*** (3.50)	0.104*** (3.25)
FOREIGN	0.140*** (9.07)	0.381*** (5.98)	0.141*** (9.09)	0.382*** (6.00)
LITIGATION	−0.072*** (−2.74)	−0.227** (−2.03)	−0.073*** (−2.78)	−0.231** (−2.06)
Observations	26,541	26,541	26,544	26,544
R ²	.131	.082	.131	.082
Industry FE	Yes	Yes	Yes	Yes
Year FE	Yes	Yes	Yes	Yes

Note. Table 3 reports estimation results of the relations between overall FSC (measured by MAD and KS) and audit firm tenure (TENURE). Appendix A describes the computation of MAD and KS and Appendix B provides variable definitions. The t-values reported in the parentheses are computed based on robust standard errors clustered at the firm level. FSC = financial statement conformity; MAD = mean absolute deviation; KS = Kolmogorov–Smirnov distance; FE = fixed effects.

***

, **, and * denote statistical significance at the 1%, 5% and 10% level, respectively.

Columns 3 and 4 report the estimation results for the second approach, with SHORTTENURE and LONGTENURE as the main test variables.⁸ The coefficient on SHORTTENURE is significantly positive (Column 3: β₄ = 0.043, t = 3.00; Column 4: β₄ = 0.111, t = 1.76), indicating that short tenure is associated with higher FSC than medium tenure. The coefficient on LONGTENURE is not significantly different from zero, indicating no reliable difference in FSC between long tenure and medium tenure. These results are partially consistent with the results in columns 1 and 2. Prior research suggests that an extended auditor–client relationship is a tradeoff between improvement in auditor competence and reduction in auditor independence. Overall, the results indicate that greater auditor independence, as reflected in short audit tenure, is positively associated with FSC.

The coefficients on the control variables provide interesting insights into the determinants of the FSC measures. The overall results indicate that FSC increases with firm size (SIZE), number of segments (NUMSEGS), presence of foreign operations (FOREIGN), growth (BTM) and profitability (ROA), and decreases with total accruals (ACCRUALS).

Additional Analyses

In this section, we perform additional analyses to gain further insight into the relations between each of the audit quality inputs and FSC to Benford’s law.

Association between audit quality inputs and the conformity of individual financial statements to Benford’s law

Amiram et al. (2015) and Nigrini (2015) find that the numbers in the income statement diverge more from the Benford distribution than the numbers in the statement of cash flows, which suggests that the income statement may be more susceptible to errors and anomalies than the statement of cash flows. This finding is consistent with most studies in the accounting literature, which document that the accruals component of earnings is subject to greater manipulation and estimation errors than the cash flow component. If the line items in the income statement conform less to Benford’s law than the line items in the statement of cash flows and audit quality increases conformity to Benford’s law, as indicated by the preceding analyses, then it is likely that audit quality inputs are more strongly associated with conformity of the income statement than with conformity of the statement of cash flows.

We compute the conformity measures (MAD and KS) separately for the line items in the income statement and in the statement of cash flows, and denote the income statement conformity measures as MAD_IS and KS_IS and the corresponding statement of cash flows measures as MAD_CF and KS_CF. We then re-estimate Equation 1 using each of these statement-specific conformity measures as the dependent variable.

Table 4 presents the estimation results, with a separate panel for each audit quality input. Across all four panels, the coefficients for the primary variables in the income statement columns are qualitatively similar to those in the main tests, whereas the coefficients in the columns for the statement of cash flows are insignificant. Consistent with the view that the income statement is more prone to errors, these results indicate that the audit quality inputs have a greater influence on the conformity of the income statement to Benford’s law than on the conformity of the statement of cash flows.

Table 4.

The Relations Between Audit Quality Inputs and Conformity of the Income Statement and the Statement of Cash Flows.

Panel A. Audit Quality Input Measured as Audit Fees.
Variables	Income statement		Statement of cash flows
	(1)	(2)	(3)	(4)
	MAD_IS	KS_IS	MAD_CF	KS_CF
AUD_FEES	0.285*** (11.83)	0.964*** (9.27)	0.031 (1.64)	0.074 (0.97)
Observations	26,485	26,485	26,485	26,485
R ²	.187	.116	.062	.038
Control variables	Yes	Yes	Yes	Yes
Industry FE	Yes	Yes	Yes	Yes
Year FE	Yes	Yes	Yes	Yes

Panel B. Audit Quality Input Measured as Audit Report Lag.
Variables	Income statement		Statement of cash flows
	(1)	(2)	(3)	(4)
	MAD_IS	KS_IS	MAD_CF	KS_CF
AUD_REPORT_LAG	0.003*** (5.00)	0.008*** (3.33)	0.000 (0.86)	0.002 (1.06)
Observations	26,502	26,502	26,502	26,502
R ²	.180	.111	.061	.038
Control variables	Yes	Yes	Yes	Yes
Industry FE	Yes	Yes	Yes	Yes
Year FE	Yes	Yes	Yes	Yes

Panel C. Audit Quality Input Measured as Nonaudit Fees.
Variables	Income statement		Statement of cash flows
	(1)	(2)	(3)	(4)
	MAD_IS	KS_IS	MAD_CF	KS_CF
NON_AUD_FEES	0.073*** (7.37)	0.238*** (5.78)	0.009 (1.21)	0.036 (1.11)
Observations	23,784	23,784	23,784	23,784
R ²	.181	.110	.060	.037
Control variables	Yes	Yes	Yes	Yes
Industry FE	Yes	Yes	Yes	Yes
Year FE	Yes	Yes	Yes	Yes

Panel D. Audit Quality Input Measured as Audit Firm Tenure.
Variables	Income statement		Statement of cash flows
	(1)	(2)	(3)	(4)
	MAD_IS	KS_IS	MAD_CF	KS_CF
TENURE	−0.008* (−1.87)	−0.032 (−1.61)	0.0008 (0.24)	−0.002 (−0.11)
TENURE2	0.0002* (1.73)	0.0009 (1.54)	0.000 (0.31)	0.000 (0.40)
Observations	26,499	26,499	26,499	26,499
R ²	.179	.111	.061	.038
Control variables	Yes	Yes	Yes	Yes
Industry FE	Yes	Yes	Yes	Yes
Year FE	Yes	Yes	Yes	Yes

Note. Table 4 reports estimation results of the relations between income statement conformity (measured by MAD_IS and KS_IS) and statement of cash flows conformity (measured by MAD_CF and KS_CF) and audit quality inputs. Panel A reports results for audit fees (AUD_FEES), Panel B for audit report lag (AUD_REPORT_LAG), Panel C for nonaudit service fees (NON_AUD_FEES), and Panel D for audit firm tenure (TENURE). The control variables are the same as those reported in Tables 2 and 3, but are omitted for brevity. Appendix A describes the computation of MAD and KS and Appendix B provides variable definitions. The t-values reported in the parentheses are computed based on robust standard errors clustered at the firm level. MAD = mean absolute deviation; KS = Kolmogorov–Smirnov distance; FE = fixed effects.

***

, **, and * denote statistical significance at the 1%, 5% and 10% level, respectively.

The second digit distribution test

We examine the robustness of our results to an alternative measure of FSC that is based on the distribution of the second digit in financial statement line items. We construct new measures based on the conformity of a firm’s second digit distribution to the Benford unconditional second digit distribution. Similar to the first digit test, greater deviation from the Benford second digit distribution indicates greater likelihood of error in the financial statements.⁹ In untabulated results, we find that the relations between the audit quality inputs and these alternative measures of FSC are qualitatively similar to those reported in the main tests. These results reinforce and confirm our earlier findings of a significant relation between each of the audit quality inputs and overall FSC, measured based on the first digit distribution.

Size of audit firm

We examine whether the associations between audit quality inputs and FSC to Benford’s law vary with audit firm size, by re-estimating Equation 1 separately for subsamples of client firms that are audited by Big 4 and by non-Big 4 audit firms and comparing the coefficients for the audit quality inputs across the two subsamples. In untabulated results, we do not find significant differences in the coefficients for audit report lag, nonaudit service fees, and audit firm tenure across the Big4 and the non-Big 4 subsamples. Overall, the results indicate little difference in the relation between audit quality inputs and FSC across clients audited by Big 4 and non-Big 4 audit firms.

Conclusion

Using measures of the conformity of reported financial statement information to the Benford distribution, we demonstrate that higher audit fees, nonaudit fees, and audit report lag all increase conformity, whereas the length of audit firm tenure decreases conformity. We also find that each of the audit quality inputs is more strongly associated with income statement conformity than with cash flow statement conformity. In the absence of a high-quality audit and greater auditor effort, the numbers in the income statement will be less reliable than the numbers in the cash flow statement, because the accruals component of earnings is more susceptible to manipulation and estimation errors. Therefore, finding that the audit quality inputs are more strongly associated with income statement conformity than with cash flow statement conformity is consistent with the findings of prior auditing research.

Notwithstanding the study’s results, there are several caveats that warrant further discussion. First, this study implicitly assumes that the first digit distribution of truthful, unbiased financial statements will conform to the Benford distribution. Nigrini (2015) suggests that deviation from the Benford distribution does not always indicate irregularities, while conformity to the Benford distribution does not guarantee that the information is mostly free from error. Therefore, we acknowledge that the conformity to (or deviation from) Benford’s law may simply reflect the mathematical/distributional property of the financial statement line items and not the quality of financial reporting. Second, although our study documents that FSC to Benford’s law is associated with more intensive audit activity, there is, as yet, little theoretical guidance on how the mathematical factors that underlie Benford’s law are related to audit activities.

Footnotes

Appendix A

Following Amiram et al. (2015), we compute the following conformity measures:

Mean Absolute Deviation (MAD): (−100) times the average deviation of a firm’s first digit distribution from the Benford distribution.

MAD = (- 100) \times \frac{1}{9} \sum_{i = 1}^{9} | F_{i}^{Actual} - F_{i}^{Benford} | .

Kolmogorov–Smirnov Distance (KS): (−100) times the maximum deviation of a firm’s first digit cumulative distribution from the Benford distribution.

KS = (- 100) \times \begin{matrix} Sup \\ j = 1, \dots, 9 \end{matrix} | \sum_{i = 1}^{j} (F_{i}^{Actual} - F_{i}^{Benford}) | .

Appendix B

Table B1.

Variable Definitions.

Variable name	Definition
MAD	Mean absolute deviation, measured as (−100) times the average deviation of a firm’s first digit distribution from the Benford distribution for all financial statement line items.
MAD_IS	Mean absolute deviation for the income statement, measured as (−100) times the average deviation of a firm’s first digit distribution from the Benford distribution for all income statement line items.
MAD_CF	Mean absolute deviation for the statement of cash flows, measured as (−100) times the average deviation of a firm’s first digit distribution from the Benford distribution for all statement of cash flows line items.
KS	Kolmogorov–Smirnov distance, measured as (−100) times the maximum deviation of a firm’s first digit cumulative distribution from the Benford distribution for all financial statement line items.
KS_IS	Kolmogorov–Smirnov distance for the income statement, measured as (−100) times the maximum deviation of a firm’s first digit cumulative distribution from the Benford distribution for all income statement line items.
KS_CF	Kolmogorov–Smirnov distance for the statement of cash flows, measured as (−100) times the maximum deviation of a firm’s first digit cumulative distribution from the Benford distribution for all statement of cash flows line items.
AUD_FEES	Natural logarithm of audit fees.
NON_AUD_FEES	Natural logarithm of nonaudit fees.
AUD_REPORT_LAG	Audit report lag, measured as the number of days between the fiscal year end date and the audit report date.
TENURE	Audit firm tenure, measured as the number of continuous year the audit firm has served the same client.
TENURE2	Square of audit firm tenure.
LONGTENURE	Indicator variable that equals 1 if audit firm tenure is 15 years or more; 0 otherwise.
SHORTTENURE	Indicator variable that equals 1 if audit firm tenure is greater than 1 year but less than 4 years; 0 otherwise.
SIZE	Natural logarithm of a firm’s year-end total assets.
BTM	Book-to-market ratio, measured as the ratio of year-end book value to market value of equity.
LEV	Leverage ratio, measured as the ratio of year-end total debt to total assets
ROA	Return on asset ratio, measured as the ratio of income before extraordinary items to year-end total assets.
LOSS	Indicator variable that equals 1 if net income is negative; 0 otherwise.
AGE	Firm’s age in years, measured from the first year the company appears in the Compustat database.
NIBE_VOL	Earnings volatility, measured as the volatility of the earnings before extraordinary items over the previous 5 years.
SALE_GROWTH	Annual sales growth, measured as the sales growth percentage from the previous year.
CFO	Cash flow from operations scaled by year-end total assets.
ACCRUALS	Total accruals scaled by year-end total assets.
CURRENT	Current ratio, measured as the ratio of current assets to current liabilities.
QUICK	Quick ratio, measured as the ratio of current assets less inventory to current liabilities.
Z_SCORE	Bankruptcy score, measured as: −4.336 − 4.513 ×ROA+ 5.679 ×LEV+ 0.004 ×CURRENT (Zmijewski, 1984)
ACQUIRE	Indicator variable that equals 1 if the firm has any acquisition activities during the year; 0 otherwise.
DEBTFIN	Indicator variable that equals 1 if the firm issues debt financing during the year; 0 otherwise.
STOCKFIN	Indicator variable that equals 1 if the firm issues stock financing during the year; 0 otherwise.
NUMSEGS	Number of business segments.
FOREIGN	Indicator variable that equals 1 if the firm has foreign sales.
LITIGATION	Indicator variable that equals 1 if the firm belongs to any of the following high-litigation industries: biotechnology (SIC codes 2833–2866), computers (SIC codes 3,570–3,577 and 7,370–7374), electronics (SIC codes 3,600–3674), and retails (SIC codes 5,200–5,961; Francis et al., 1994)

Note. SIC = Standard Industrial Classification.

Acknowledgements

We thank Bharat Sarath (editor), Yuping Zhao, an anonymous reviewer, the participants at the 2019 AAA Southwest Region Meeting, and our colleagues at University of Houston for their helpful comments and suggestions.

Author’s Note

Thien Le is now affiliated with Texas A&M University-Commerce, Campbell St, Commerce, TX, USA.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Thien Le

Notes

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