1. Introduction

2. Literature Review

3. Methodology

3.1. Data and Sample

Data are drawn from the audited financial statements of 11 deposit money banks between 2011 and 2022. The base period, 2011, is adopted to control the effect of the new regulatory environment occasioned by the adoption of the International Financial Reporting Standards (IFRS) in 2011 and its attendant increase in audit fees. The sample focuses on interest-bearing banks (deposit money banks) and is based on banks with complete information between the sample periods. The criteria for selecting the sample size are reported in Table I.

Table 1. Sample selection.

Table 1. Sample selection.

	Number
Total population	26
(-) Privately owned banks expunged	(12)
(-) Banks reporting in currency other than Nigerian currency (Naira)	(1)
(-) Banks that are taken over during the period under study	(2)
Study sample	11

Source: CBN 2024.

3.2. Model Specification and Analytical Technique

3.2.1. First Step

In the first step, the firm’s efficiency which comprises the observable firm characteristics and the unobservable managerial ability, is determined. Different models exist to analyse the efficiency scores based on a mix of input and output variables. Table II shows the input and output variables in some selected empirical literature (Ben Abdesslam et al. 2022; Curi and Lozano-Vivas 2020; Garcia-Meca and Garcia-Sanchez 2018). However, the literature (e.g., Jenkins and Anderson, 2003) has established the need to justify the input and output variables in determining efficiency scores by applying the DEA. Prior studies have shown that the DEA’s ability to discriminate among firms is dependent on the selection of the right number of input and output variables (Meng et al. 2008; Nataraja and Johnson 2011). Simar and Wilson (2008) argue that using a large number of input and output variables vis-à-vis the number of decision-making units can bias the efficiency frontier upward, resulting in more efficient units. For instance, using the model of Garcia-Meca and Garcia-Sanchez (2018) in our studies does not discriminate effectively among the DMUs (see Table IV); this may be attributable to a large number of inputs and output variables compared to the DMUs.

Therefore, the study conducts principal component analysis-data envelopment analysis (PCA-DEA) to determine the appropriate numbers of input and output variables. The extraction of factors from the transformation of six input and five output variables is done using the Kaiser oblique promax rotation for data collected from 11 banks between 2011 and 2022. Using the PCA, the first factor accounts for about 90% of the data’s variance, justifying its use as an uncorrelated representation of the reduced input(output) factors. The results show principal components for eigenvalues more significant than one and factor loadings greater than 0.40. This test is also supplemented by two other tests: the best-summated rating scale (Millar 2011, Stata command: optifact) and the clustering of variables around latent components (Hardouin 2019, Stata command: clv). Table III presents the diagnostic tests for determining the optimal number of variables that can discriminate efficiency levels among the DMUs. Though the results of the PCA suggest three output variables, two variables are selected based on the best-summated rating scale and the grouping of variables around latent elements. This aligns with the recommendations of prior studies (e.g., Cooper et al. 2001; Cooper et al. 2004) that adding the input and output variables multiplied by three should not be greater than the number of DMUs. The results of the adopted mix of input and output variables (Panel B of Table IV) show an excellent discriminating power. Furthermore, there are significant correlations among the input and output variables, justifying their inclusion in the model (Table V). Therefore, the study adopts two input and two output variables. The input variables comprise property, plant, and equipment (PPE), and staff cost (LABOUR), while the output variables comprise deposits (DEP) and net interest income (NETINC). Therefore, the model for the banks’ efficiency analysis is as follows:

$$MAX\theta ={\displaystyle \frac{\mu 1DEP+\mu 2NETINC}{⩗1PPE+⩗2LABOUR+}}$$

(1)

Regarding DEA orientation, the input-based model is preferred to the output-based model because the DMUs may not operate optimally. In terms of the DEA return to scale assumption, the variable return to scale (VRS) is adopted because of its good finite sample properties, most productive scale size, and its ability to converge quickly (Banker and Park 2021; Kneip et al. 2016).

Table 2. Input and output variables in prior empirical studies.

Table 2. Input and output variables in prior empirical studies.

SN	Studies	Input variables	Output variables
1.	Garcia-Meca and Garcia-Sanchez (2018)	Property, plant, and equipment; Intangible assets; personnel cost; interest expenses; and operating rental expenses	Deposits, Loans, Investments, and Interest income.
2.	Curi and Lozano-Vivas (2020)	Personnel expenses, interest expenses, physical capital.	Loans; Other earning assets
3.	Ben Abdesslem et al. (2022)	Personnel expenses; property, plant, and equipment; deposit; short term funding	Loans; Other earning assets

Table 3. Results of diagnostic tests on the optimal number of input and output variables.

Table 3. Results of diagnostic tests on the optimal number of input and output variables.

SN	Techniques	Input variables	Output variables
1.	Principal component analysis	Property, plant, and equipment; Labour cost	Deposits; Net interest income; Loan
2.	Best summated rating scale (Stata command-optifact)	Property, plant, and equipment; Labour cost	Deposits; Net interest income
3.	Clustering of variables around latent components (Stata command- clv)	Property, plant, and equipment; Labour cost	Deposits; Net interest income

Table 4. DEA score description.

Table 4. DEA score description.

	Number of efficient banks based on Garcia-Meca and Garcia-Sanchez (2018)		Number of efficient banks based on PCA-DEA
	Panel A		Panel B
Year	Absolute	Relative (%)	Absolute	Relative (%)
2011	8	67	3	25
2012	6	50	2	17
2013	9	75	2	17
2014	8	67	2	17
2015	7	58	3	25
2016	8	67	2	17
2017	8	67	4	33
2018	9	75	3	25
2019	10	83	2	17
2020	9	75	2	17
2021	8	67	2	17
2022	10	83	3	25

Table 5. Descriptive statistics and correlation matrix of selected input and output variables.

Table 5. Descriptive statistics and correlation matrix of selected input and output variables.

	PPE	LABOUR	DEP	NETINC
PPE	1.000
LABOUR	0.883*	1.000
DEP	0.888*	0.892*	1.000
NETINC	0.884*	0.893*	0.944*	1.000
Descriptive Statistics:
Mean	17.671	17.229	20.921	18.236
Std. Dev	0.827	0.718	0.992	0.904
Min	15.869	15.691	18.809	16.281
Max	19.496	18.674	23.144	19.754

3.2.2. Second Step

In the first step, the efficiency scores generated (Equation 1) are attributable to the whole firm. The second step comprises the estimation of managerial ability by regressing firm efficiency scores generated in the first stage on bank-specific characteristics, such as size, market share, free cash flow, age, and number of subsidiaries, which are outside the discretionary control of the managers (Banker et al. 2019). We follow prior studies (e.g., Chen et al. 2023; Garcia-Meca and Garcia-Sanchez 2018) to extract the managerial ability score as the residual of the following model:

$$DEASC={\beta }_{1}SIZ+{\beta }_{2}MkT+{\beta }_{3}FLOW+{\beta }_{4}AGE+{\beta }_{5}SUB+{\beta }_{7}YEAR+e_{\mathrm{it}}$$

(2)

DEASC represents efficiency scores in the first stage; SIZ represents banks’ total assets; MKT represents market share; FLOW represents free cash flow; AGE represents the number of years the bank has existed; SUB represents the number of subsidiaries; and YEAR represents year dummies.

While many studies (e.g., Demerjian et al. 2012) apply the DEA-Tobit regression model to extract the managerial ability score, this study adopts the bias-corrected two-stage bootstrapped truncated regression model of Simar and Wilson (2007). DEA efficiency scores range between 0 and 1 and are not a product of data censoring (Banker et al. 2019), which makes the Tobit regression inefficient (Simar and Wilson 2011). Studies have shown that the first stage of the two-way model is fraught with serial correlation and measurement errors. The bias-corrected two-stage bootstrapped truncated regression model removes this distortion and bias in the second-stage regression occasioned by high correlation among the mix of input and output variables that generated the efficiency scores (Simar and Wilson, 2007, 2011, 2015). We adopt the bias-corrected two-stage bootstrapped truncated regression technique (Badumenko and Tauchman 2019, Stata command: simarwilson).

3.2.3. Model specification

To test the effect of managerial ability on audit pricing, the following regression model is adopted:

$$LNFEES={\beta }_{1}MGR+{\beta }_{2}JOINT+{\beta }_{3}SPEC+{\beta }_{4}LOSS+e$$

(3)

Audit pricing proxied by audit fees is denoted as LNFEES, while MGR represents managerial ability. Three control variables are introduced into the model in line with the literature. These variables are joint audit (JOINT), specialist auditor (SPEC), and financial risk (LOSS). The agency and resource dependence theories support the inclusion of joint audits (JOINT). Agency theory seeks to mitigate agency conflict by including multiple audit firms as a governance mechanism to enhance oversight and reduce collusion. The resource dependency theory provides for the diversification of risks as a result of the deployment of many resources from more than one audit firm, resulting in competitive audit pricing. The effect of a joint audit on audit fees can be either positive or negative. A joint audit is measured as the engagement of two audit firms for an audit engagement (Lesage et al. 2017).

The inclusion of audit firm industry specialisation (SPEC) in the model is justified based on the agency and differentiation strategy theories. The two theories argue that agents may employ specialist auditors to signal their stewardship to the management, and this decision can have implications on audit fees (Choi et al. 2008; Ferguson et al. 2003; Huang et al. 2007; Mayhew and Wilkins 2003). Audit firm industry specialisation is measured as an audit firm having a market share greater than 30 per cent in the respective industry (Habib and Bhuiyan 2011). Finally, the agency and risk-based audit approach facilitates the inclusion of financial risk (LOSS) into the model. Agency theory argues that the conflict between the principal and the agent may reflect high-risk exposure, necessitating a risk-based audit approach. This approach involves extensive audit work and additional effort with repercussions on audit fees. Financial risk is measured as a binary variable where ‘1’ signifies a negative net income and 0 otherwise (Asthana and Boone 2012; Zaman et al. 2011). We expect all the control variables to exert positive effects on audit fees.

We adopt the regression spline method to analyse the data. Using a regression spline model would reveal the plausibility of breakpoints that may be present in the data. Regression splines comprise a family of methods for estimating variables that change slopes at different points (Edwards and Parry 2018; Marsh and Cormier 2001).

3.2.4. Data Analysis Strategy

The data analysis process follows a 5-stage process. In the first stage, we graphically investigate the functional association between managerial ability and audit pricing (Ronkko et al. 2022). We also investigate the occurrence of switch points in the data using the closed-test approach, which automatically detects the positioning of switch points in equally spaced centiles of the variable’s distribution. In the second stage, descriptive statistics and correlation matrix are computed to gain initial insights into the data. We also conduct diagnostic tests in the third stage to ensure the model fits the data. In the fourth stage, we run spline regression, which comprises seven shape-restricted regression splines (Meyer 2008; Wang and Ghosh 2012; Wang and Yan 2021) with the restricted cubic spline (RCS) as the baseline estimator. The remaining six estimators are natural cubic spline (N-spline), basis spline (B-spline), integrated spline (I-spline), integrated basis spline (integrated B-spline), integrated natural cubic spline (integrated N-spline) and monotone spline (M-spline). The shape-restricted regression splines are preferred to kernel smoothers and smoothing splines because they are adaptively flexible, have optimal convergence, prevent over-fitting, and are robust to knot selection with capacity for smoothing functionalities (Meyer 2008). Additional analyses are conducted in the fifth stage. All studies are performed using Stata 18 with a 5% significance level.

4. Analysis and Discussion

4.1. Functional Relationship and Switch Point Analysis

The study uses plots to depict the functional association between managerial ability and audit pricing. The plots comprise a locally weighted scatterplot smoother (Cleveland 1979; Jacoby 2000) and the component-plus-residual plot (Chatterjee and Hadi 2015) (Stata commands: “lowess” and “cprplot,” respectively). The results of the locally weighted scatterplot smoother and the component-plus-residual plot (Figure II a & b) show a departure from linearity. We apply the closed-test approach, which automatically identifies the positioning of the switch points based on an equally spaced centile of the regression (Royston 2007) (Stata command: mvrs and uvrs). The results show the presence of three knots demarcating boundaries of four segments with markedly different slopes. Therefore, the results validate the first hypothesis (H₁), which states that breakpoints exist in the association between managerial ability and audit pricing. This result validates the threshold theory, which postulates that the association between a response variable and an explanatory variable varies depending on the breakpoints of the explanatory variable. The presence of the breakpoints results in the extension of Equation (3) as follows:

$$LNFEES={\beta }_{1}MGR1+{\beta }_{2}MGR2+{\beta }_{3}MGR3+{\beta }_{4}JOINT+{\beta }_{5}SPEC+{\beta }_{7}LOSS+e$$

(4)

The variable “managerial ability” (MGR) in Equation (3) is decomposed into three knots, demarcating four segments. The new variables are MGR1, MGR2, and MGR3, representing the first, second, and third regions of managerial ability, respectively. Following prior studies (e.g., Glover and Prawitt 2014; Lowry 2008), the first region (between the 0^th and 34^th percentiles) is labelled the low managerial ability region, and the second region (between the 35^th and 69^th percentile) is labelled the intermediate managerial ability region. The third region (70^th and 100%) is the high managerial ability region. We test the equality of matched pairs of the three regions of managerial ability by using the Wilcoxon matched-pairs signed-ranks test (Wilcoxon 1945). The null hypothesis signifies equal distribution, while the alternate hypothesis signifies unequal distribution. The results (using Stata command: signrank) show that the distributions of MGR1 and MGR 2 (z = 4.868, p = 0.000), MGR1 and MGR3 (z = 4.868, p = 0.000), and MGR2 and MGR3 (z = -6.077, p = 0.000) are not the same.

Figure 2. (a): Functional relationship between managerial ability and audit pricing (b): Functional relationship between managerial ability and audit pricing.

Figure 2. (a): Functional relationship between managerial ability and audit pricing (b): Functional relationship between managerial ability and audit pricing.

4.3. Diagnostic Tests

Table VII presents the results of the diagnostic tests. The study conducts five diagnostic tests comprising a functional form test, specification test, heteroskedasticity test, outlier test, and normality test. Ramsey’s regression specification error test (RESET) test which examines the functional form of a model is conducted, and the results show that an appropriate functional form is adopted (Wooldridge 2018). A linktest for specification error (Wooldridge 2018) is undertaken, and the results show no specification error. The Breusch-Pagan test for heteroskedasticity (Gujarati 2021) is undertaken, and the results show that the model does not have the challenge of heteroskedasticity. An outlier test using Cook's distance measure (Pardoe 2020) is also conducted; the results show the absence of any influential observation, as no distance is above the cut-off point. Fifth, the normality is conducted using the Shapiro-Wilk W normality test (Shapiro and Wilk 1965); the results show a normally distributed residual. The three regions of the managerial ability variable have variance inflation factors above a cut-off point of 10; however, Harrell (2001) argues that multicollinearity may not present a severe problem in restricted cubic splines analyses.

Table 6. Descriptive statistics and correlation matrix of the effect of managerial ability on audit pricing.

Table 6. Descriptive statistics and correlation matrix of the effect of managerial ability on audit pricing.

	LNFEES	MGR1	MGR2	MGR3	JOINT	SPEC	LOSS
Correlation Matrix
LNFEES	1.00
MGR1	-0.66*	1.00
MGR2	0.60*	-0.95*	1.00
MGR3	0.58*	-0.94*	0.99*	1.00
JOINT	-0.17*	-0.12	0.06	0.06	1.00
SPEC	0.46*	-0.35*	0.31*	0.30*	-0.19*	1.00
LOSS	-0.24*	0.05*	0.02*	-0.02	-0.06	-0.02	1.00
Descriptive Statistics
Observations	144	144	144	144	144	144	144
Mean	12.57	0.00	0.00	21.30	0.07	0.33	0.05
Std. Dev	0.81	0.01	0.00	0.98	0.26	0.47	0.22
Min	11.08	-0.02	-0.00	-0.00	0.00	0.00	0.00
Median	12.58	0.00	-0.00	-1.83	0.00	0.00	0.00
Max	14.33	0.02	0.02	23.42	1.00	1.00	1.00

Table 7. Diagnostic Tests.

Table 7. Diagnostic Tests.

S/N	Tests	Analytical techniques	Results
1.	Functional form	Ramsey test	F (3,134) = 2.046; p = 0.110
2.	Specification test	Linktest	t = 0.605; p = 0.546
3.	Heteroskedasticity test	Breusch-Pagan test	Chi = 0.238; p = 0.626
4.	Outlier test	Cook’s distance	No distance is above the cutoff of 1
5.	Normality test	Shapiro-Wilk W normality test	Z = 0.407; p = 0.342

4.4. Regression Results and Discussion

The study adopts shape-restricted splines to achieve its objective. Table VIII reveals the regression results of estimating Eq. (4). Panel A shows the a priori expectation for each variable. Panel B presents the results of the effect of managerial ability and audit pricing using the restricted cubic spline. Panels C, D, E, F, G and H show the results of the natural cubic splines, integrated natural cubic splines, monotone splines, integrated splines, integrated basis splines, and the basis splines respectively. The restricted cubic spline (Panel B) is adopted as the baseline result because it is a flexible model for analysing nonlinear relationships while improving the behaviour of the model by using linear functions that accommodate extreme boundaries before the first knot and after the last knot (Lusa and Ahlin 2020). In the restricted cubic spline regression analysis, we “mean-centre” the predictors to reduce the impact of multicollinearity (Cohen et al. 2013) and apply the winsorisation function at 1 and 99 per cent of the continuous predictors. The restricted cubic spline (Panel B) results show that managerial ability is significant at two breakpoints: intermediate and high regions of managerial ability. The results are interpreted in line with the hypotheses.

The second hypothesis (H₂) states that managerial ability exerts a statistically significant and positive effect on audit pricing in the low-ability region. However, the results show an insignificant negative association, invalidating the hypothesis. One of the reasons adduced for the insignificant relationship could be that managers at this level may be irresponsive to average audit pricing. The third hypothesis (H₃) predicts managerial ability exerts a statistically significant and negative effect on audit pricing in the intermediate-ability region. The results which invalidate the third hypothesis show that managerial ability exerts a significantly positive effect on audit pricing in the intermediate area of managerial ability (t = 2.57, p = 0.011). The positive effect of managerial ability on audit pricing at the intermediate region of managerial ability is consistent with the audit demand/supply theory (DeFond and Zhang 2014).

As a legitimacy/signalling measure to demonstrate effectiveness, an intermediate-ability manager may request a high-quality audit (Caramanis and Lennox 2008; Lim and Mali 2021), which aligns with the audit demand, legitimacy, and signalling theories. However, the audit supply theory infers that an audit firm will extract an audit fee premium based on specialisation, reputation, and litigation risk (Chen et al. 2019; Skinner and Srinivasan 2012). The fourth hypothesis (H₄) predicts managerial ability exerts a statistically significant and negative effect on audit pricing in the high-ability region. The results support this hypothesis (t = -2.62, p = 0.10). This result aligns with audit demand theory which suggests that high-ability managers adopt conservative estimates for reliable financial reporting systems, resulting in reduced audit risk and audit fees discount (Krishnan and Wang 2015). It also aligns with the bargaining theory, which argues that an audit client can use its bargaining ability to negotiate for audit discounts (Carcello and Nagy 2004). Furthermore, using Porter’s (1985) competitive force theory, high-ability managers with bargaining power extract audit fees discount (Huang et al. 2007) even when the firm employs specialist auditors (Fung et al. 2012). Figure III complements the baseline model results by showing that intermediate managerial ability positively correlates with audit fees. In contrast, high managerial ability exerts a negative effect on audit fees. In summary, the results show an ambiguous association between managerial ability and audit pricing, as the association exhibits positive and negative tendencies.

Table 8. Regression results of the association between managerial ability and audit pricing using shape-restricted regression splines.

Table 8. Regression results of the association between managerial ability and audit pricing using shape-restricted regression splines.

		Model 1	Model 2	Model 3	Model 4	Model 5	Model 6	Model 7
Panel	A	B	C	D	E	F	G	H
Var.	A priori	rcs	ns	ins	ms	is	ibs	bs
mgrA1	+/-	-18.231 (-1.09)	4.445*** (7.48)	-279.220*** (-8.58)	.006** (2.45)	1.546*** (3.33)	264.877*** (3.33)	1.085** (2.45)
mgrA2	+/-	568** (2.57)	-.357 (-1.09)	3.475* (1.97)	.005 (1.64)	-2.654*** (-2.87)	-288.820*** (-2.87)	0.580 (1.64)
mgrA3	+/-	-629** (-2.62)	0.037 (0.13)	0.639 (0.41)	-.009*** (-2.80)	.689 (0.65)	74.968 (0.65)	-.934*** (-2.80)
mgrA4	+/-				.002 (0.77)	.242 (0.23)	50.927 (0.23)	0.348 (0.77)
mgrA5	+/				-.005*** (-3.04)	-1.520 (-1.36)	-453.617 (-1.36)	-1.620*** (-3.04)
Joint	+	-.642*** (-3.46)	-.642*** (-3.46)	-.497** (-2.36)	-.341* (-1.70)	-.354* (-1.75)	-.354* (-1.75)	-.341* (-1.70)
Spec	+	.332*** (3.22)	.332*** (3.22)	.340*** (3.12)	.421*** (3.76)	.424*** (3.71)	.424*** (3.71)	.421*** (3.76)
loss	+	-.819*** (-3.99)	-.819*** (-3.99)	-.800*** (-3.71)	-.837*** (-3.71)	-.820*** (-3.58)	-.820*** (-3.58)	-.837*** (-3.71)
N		144	144	144	144	144	144	144
F Stat		34.27	34.27	29.24	18.80	18.02	18.02	18.80
Prob		0.000	0.000	0.000	0.000	0.000	0.000	0.000
Adj. R2		0.5826	0.5826	0.5423	0.4989	0.4878	0.4878	0.4989
RMSE		.52772	.52772	.55261	.57823	.58463	.58463	.57823

The restricted cubic spline is denoted as “rcs,” the natural cubic spline is denoted as “ns,” the integrated natural cubic spline is denoted as “ins,” the monotone spline is denoted as “ms,” the integrated spline is denoted as “is,” integrated basis spline is denoted as “ibs,” and basis spline is denoted as “bs.” T-statistics in parentheses; ***p<0.01, **p<0.05, *p<0.1.

Figure 3. Plot of managerial ability thresholds on audit pricing.

Figure 3. Plot of managerial ability thresholds on audit pricing.

While there are controversies in the literature on the relevance of p-values, we demonstrate the rigour, reproducibility, and scientific relevance of our results by complementing our results with the second-generation p-values, which control Type 1 error effectively (Blume et al. 2018, 2019). The results show the following second-generation p-value: mgrA1 (0.50), mgrA2 (0.00), mgrA3 (0.00), joint (0.00), spec (0.00), and loss (0.00). These p-values are similar to the traditional p-values, thus supporting the reliability of the baseline results. The results of four other estimators (monotone spline, integrated spline, integrated basis spline, and basis spline) align with the baseline results. While two estimators (natural cubic spline and integrated natural cubic spline) had different results, natural cubic splines show that managerial ability has a statistically significant and positive effect on audit pricing, and integrated natural cubic splines show that managerial ability exerts a negative effect on audit pricing. The opposing results may be due to the natural cubic spline and the integrated natural cubic spline subjective placement of knots and the forceful linearisation at the tails, which impact its flexibility at the boundaries (Perperoglou et al. 2019).

The control variables also show some interesting results. Audit firm industry specialisation (SPEC) positively correlates with audit pricing, indicating that specialist auditor charges higher audit fees (Yeboah et al. 2023). Joint audit (JOINT) negatively correlates with audit pricing, which indicates efficiency gains and economies of scale occasioned by the combined efforts of the audit firms, thereby resulting in audit fee discounts. Financial risk (LOSS) also negatively correlates with audit pricing. The results can also be justified based on the bargaining theory, where high-ability managers negotiate for audit discounts despite the higher financial risks.

4.6. Additional Analyses

4.6.1. Simultaneous Quantile Regression

For additional analyses, simultaneous quantile regression is applied, where the dependent variable is divided into five regions, namely low audit fees region (10^th quantile), fair audit fees region (25^th quantile), average audit fees region (50^th quantile), high audit fees regions (75^th quantile), and very high audit fees region (90^th quantile). The results (Table IX) show that at the low audit fees region (10^th quantile), managerial ability exhibits statistically significant positive and negative effects on audit fees respectively. The results show insignificant relationships at the fair (25^th quantile) and average (50^th quantile) levels of audit fees. However, managerial ability exerted statistically significant positive and negative effects at the high (75^th quantile) and very high (90^th quantile) levels of audit fees.

Table 9. Estimation of managerial ability on audit pricing across different quantiles using simultaneous quantile regression.

Table 9. Estimation of managerial ability on audit pricing across different quantiles using simultaneous quantile regression.

Quantile	mgrA1	mgrA2	mgrA3	joint	spec	loss
q10	7.546 (0.27)	994916*** (3.00)	-1082372*** (-3.05)	-.469 (-1.40)	.255 (1.20)	-.609*** (-2.86)
q25	-29.11 (-1.02)	345911 (0.93)	-369179 (-0.92)	-.652** (-2.91)	.360*** (3.51)	-.779*** (-5.46)
q50	-11.681 (-0.38)	541604 (1.19)	-593936 (-1.16)	-.679** (-2.43)	.412** (2.43)	-.941*** (-4.00)
q75	-25.356 (-0.96)	660681** (2.08)	-769072** (-2.21)	-.886** (-2.47)	.325*** (2.92)	-.995*** (-3.35)
q90	-.019 (-0.00)	941556*** (4.38)	-1055375*** (-4.82)	-.421 (-1.62)	.319*** (4.57)	-.846*** (-3.22)

Mgr1—mgr3 are the three knots of managerial ability. Joint represents a joint auditor, spec represents specialist auditors, and loss represents firms with negative returns on assets. *p <0.1, **p <0.05, ***p <0.01.

4.6.2. Penalised Spline

While the shape-restricted splines are regression splines, we also consider penalised splines to support the baseline results. The results show that low-region managerial ability (z = -1.09, p = 0.277) has a negative but insignificant effect on audit pricing; intermediate-region managerial ability (z = 2.57, p = 0.010) has a significantly positive effect on audit pricing; while high-region managerial ability (z = -2.62, p = 0.009) has a significantly negative effect on audit pricing. Joint audit (JOINT) and financial risk (LOSS) exerted a negative effect on audit pricing, while audit firm industry specialisation (SPEC) exerts a significantly positive effect on audit pricing. These results support the baseline results.

5. Conclusions

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