WHO WINS THE TUG OF WAR? A COMPARATIVE STUDY OF THE INFLUENCE OF BOARD POWER AND CEO POWER ON CEO-TMT PAY GAP

How to cite this paper: Wang, Z., & Li, Z. (2021). Who wins the tug of war? A comparative study of the influence of board power and CEO power on CEO-TMT pay gap [Special issue]. Corporate Ownership & Control, 19(1), 241–256. https://doi.org/10.22495/cocv19i1siart3 Copyright © 2021 The Authors This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). https://creativecommons.org/licenses/by/


INTRODUCTION
Management and finance scholars have shown increasing interest in the phenomenon of pay differentials between a CEO and other top executives of the firm, i.e., CEO-TMT pay gap (Henderson & Fredrickson, 2001 If stronger board control results in stronger board power which negatively influences CEO compensation (Boyd, 1994;Chhaochharia & Grinstein, 2009), how does stronger board power influence the CEO-TMT pay gap? Incumbent theoretical debate still does not reach a consensus. On the one hand, tournament theory suggests that a stronger board should not suppress larger CEO-TMT pay gap which motivates a CEO to do a better job (Lambert et al., 1993) and consequently benefits the firm (Burns, Minnick, & Starks, 2017); on the other hand, agency theory proposes that stronger board should discourage larger CEO-TMT pay gap which can be considered as a result of CEO rent-seeking (Bebchuk et al., 2011). Consequently, we are uncertain about the relationship between board power and the CEO-TMT pay gap.
Moreover, if stronger CEO power facilitates higher CEO compensation, does a powerful CEO capitalize his or her power to increase the CEO-TMT pay gap? In this case, agency theory predicts that a larger CEO-TMT pay gap serves the self-interest of the CEO and therefore stronger CEO power should lead to a larger pay gap; tournament theory suggests that a larger CEO-TMT pay gap is beneficial because of its motivational effect (Lambert et al., 1993). Thus, both agency theory and tournament theory lead to the same prediction that stronger CEO power should have a positive relationship with the CEO-TMT pay gap.
In sum, although these research questions regarding the antecedents of the CEO-TMT pay gap are important, prior studies voice seemingly contradicted theoretical arguments and provide inconsistent findings (Carpenter & Sanders, 2004;Eriksson, 1999;Henderson & Fredrickson, 2001). Our understanding of the antecedents of the CEO-TMT pay gap remains still limited.
In this study, we attempt to advance the research of the antecedents of the CEO-TMT pay gap by directly comparing different theoretical predictions regarding the impacts of board power and CEO power on the CEO-TMT pay gap. We seek to answer the following research question: How do board power, CEO power, and board-CEO power imbalance influence CEO-TMT pay gap? In turn, we empirically test the contrasting predictions regarding the relationships among board power, CEO power, board-CEO power imbalance, and the CEO-TMT pay gap. Particularly, we conduct dynamic panel analyses with a GMM estimator (Roodman, 2009) Lambert et al., 1993). Theoretically, our findings suggest that agency theory provides better predictions for the general relationships among board power, CEO power, and the CEO-TMT pay gap. Empirically, we contribute to corporate governance literature by exploring new aggregated proxies for board power and CEO power which reflect the multidimensional features of board-CEO relationships.
The rest of the paper is organized in the following way. In Section 2, we review relevant literature and develop hypotheses. In Section 3, we discuss the source of data, the sample of the study, definitions of the variables, and the regression model used in data analyses. Section 4 presents the results of the empirical analyses. Section 5 provides discussions of the findings. We summarize the findings, discuss the limitation and future research directions, and highlight the contributions in Section 6. analyzing the structural relationships among board members and the CEO which influence board power, agency theorists identify power-strengthening factors as the independence of board chair (Boyd, 1994), the number of inside directors (Zorn et al., 2017), the presence of lead independent director ( (Zajac & Westphal, 1996) and impacts CEO compensation (Hallock, 1997;Wong, Gygax, & Wang, 2015;Zhang, 2021). Meanwhile, corporate governance scholars find that anti-takeover-related provisions such as the classified board influence directors and CEO compensation (Bereskin & Cicero, 2013;Faleye, 2007). In summary, to examine the potential determinants of CEO compensation, prior research mainly focuses on how the structural relationship between a board and the CEO influences board power which determines CEO compensation. Further, agency theorists highlight the monitoring effects of the board of directors and document a negative relationship between board power and CEO compensation in general (Boyd, 1994;Chhaochharia & Grinstein, 2009).

LITERATURE REVIEW AND HYPOTHESES
Although it is the prerogative of boards of directors to set CEO compensation, a CEO can rely on his or her ownership power and expert power (Finkelstein, 1992;Adams, Almeida, & Ferreira, 2005) to strengthen his or her structural power and bargain with the board for higher compensation. Prior research investigates a variety of factors that strengthen CEO power. Particularly, a CEO who is the founder possesses ownership power (Finkelstein, 1992); and a CEO with longer tenure accrues expert power (DeBoskey, Luo, & Zhou, 2019). Meanwhile, board structure related to CEO duality, CEO lone inside director, and lead independent director also influences CEO structural power (Song & Wan, 2019;Adams et al., 2005;Krause et al., 2017). Moreover, anti-takeover provisions, especially, the golden parachutes, weaken a board's power to fire the protected CEO (Singh & Harianto, 1989

Board power and CEO pay gap
If stronger board control strengthens the power of the board of directors against a CEO and is negatively associated with CEO compensation (Boyd, 1994), do powerful directors further influence the pay gap between the CEO and other TMT members? If so, does strong board power makes the CEO-TMT pay gap narrowed or enlarged? Because boards of directors bear the fiduciary duty to serve the interest of shareholders (Black, 2001) and set the compensation of CEO and other TMT members (Lorsch & MacIver, 1989;Hermanson et al., 2012), whether directors facilitate or discourage larger CEO-TMT pay gap should be influenced by the evaluation of whether CEO-TMT pay gap serves the interest of the firm or the self-interest of the CEO. Consequently, before investigating the impact of board power on the CEO-TMT pay gap, it is important to establish a premise about the implication of the CEO-TMT pay gap. In other words, we would need to know whether a larger CEO-TMT pay gap benefits the firm before we logically predict directors' impact on the phenomenon.
However, the implication of the CEO-TMT pay gap is subjected to theoretical debates. On the one hand, tournament theory argues that the CEO-TMT pay gap can act as an incentive impetus to elicit efforts from the top executives (Fisher, Sprinkle, & Walker, 2008;Henderson & Fredrickson, 2001;Lazear & Rosen, 1981). Firms pay managers differently based on their hierarchical positions and distinctive responsibilities (Lambert et al., 1993 & Wade, 1993). The theoretical and empirical divergences drive us to make contrasting predictions of the relationship between board power and the CEO pay gap.
Among the four potential scenarios that influence the CEO-TMT pay gap which we summarize in Table 1, high CEO compensation and low TMT pay would result in a larger CEO-TMT pay gap; and either low CEO compensation or high TMT pay would reduce the CEO-TMT pay gap. Because boards of directors directly set CEO compensation and influence TMT pay (Hermanson et al., 2012), a larger CEO-TMT pay gap would exist when stronger board power increases CEO compensation and facilitates low TMT compensation. Alternatively, directors facilitate a smaller CEO-TMT pay gap when stronger board power either negatively influences CEO compensation or positively impacts TMT compensation. Taking the agency theory perspective which suggests a negative effect of a larger CEO-TMT pay gap (Bebchuk et al., 2011), we expect that directors with stronger power against the CEO are more likely to serve the interest of shareholders and negatively impact the CEO-TMT pay gap. Consistent with the agency theory arguments that excessive CEO compensation is a sign of CEO rent-seeking (Bebchuk et al., 2011), prior research documents a negative relationship between board control and CEO compensation in general (Boyd, 1994;Chhaochharia & Grinstein, 2009). Accordingly, stronger board power may be associated with a smaller CEO-TMT pay gap when a board either directly reduces CEO compensation or increases TMT pay.
H1a: Board power is negatively associated with the CEO-TMT pay gap.
From the tournament theory's perspective which argues for a positive impact of the larger CEO-TMT pay gap on firm performance, we would expect that directors may facilitate a larger CEO-TMT pay gap as an incentive mechanism to motivate a CEO. Because CEO-TMT pay gap influences the behavior of a CEO (Lee, Cho, Arthurs, & Lee, 2019; Kini & Williams, 2012) and larger CEO-TMT pay gap may motivate a CEO to do a better job (Henderson & Fredrickson, 2001), especially when monitoring is costly (Connelly, Tihanyi, Crook, & Ganloff, 2014), larger CEO-TMT pay gap may benefit the firm. Consistent with this logic, Uygur (2019) shows that the CEO-toworker pay inequality has a positive connection with firm performance, especially for a more capable CEO. Therefore, it is possible that directors with stronger power may incentivize a CEO with higher compensation and facilitate lower TMT compensation.
H1b: Board power is positively associated with the CEO-TMT pay gap. Unlike the diverging predictions regarding the connection between board power and the CEO-TMT pay gap, different theoretical perspectives lead to similar conclusions regarding the impact of CEO power on the CEO-TMT pay gap. Specifically, managerial power theory and agency theory suggest that a CEO capitalizes on his or her strong power to bargain with the board for higher compensation (Lambert et al., 1993). According to the agency theory premise that a CEO pursues self-interest, it is also logical to argue that the CEO usually lacks the motivation to raise the compensation for other TMT members out of self-interest. Indeed, prior agency theory research provides evidence that CEOs tend to increase their own compensations, but this increase is not observed for the next highest-paid executive (Malmendier & Tate, 2009). In sum, managerial power theory and agency theory both suggest that stronger CEO power tends to be associated with an enlarged pay gap between a CEO and other TMT members.

CEO power and CEO-TMT pay gap
Moreover, tournament-theory-based research also supports the positive relationship between CEO power and the CEO-TMT pay gap. Theoretically, the larger CEO-TMT pay gap is consistent with the logic of tournament theory which proposes incentive structure based on organizational hierarchies (Lambert et al., 2001). When a CEO possesses stronger power over other TMT members, tournament theory predicts a larger CEO-TMT pay gap (Henderson & Fredrickson, 2001). Empirically, prior studies provide evidence that the CEO-TMT pay gap is positively related to the number of TMT members (Conyon et al., 2001;Lin et al., 2013). This phenomenon suggests that a powerful CEO who controls more subordinates receives higher pay than his or her TMT peers. Therefore, we posit H2: CEO power is positively associated with the CEO-TMT pay gap.

Board-CEO power imbalance and the CEO pay gap
We further explore the impact of the power imbalance between a board of directors and a CEO because the CEO and the board influence each other with power (Westphal & Zajac, 1995). Since power is a relative concept that reflects one actor's influence over another in a social relation (Emerson, 1962), board power and CEO power become interdependent in the negotiation process of executive compensation. In turn, we examine the differential impact of board power and CEO power on the CEO-TMT pay gap in different scenarios 1 .
1 Board power and CEO power, although relating to each other, are two different concepts reflecting unique roles and functions of the board and the CEO, respectively. Even though prior research finds that board power and CEO power exerts the opposite impact on CEO compensation (Chhaochharia & Grinstein, 2009; van Essen et al., 2015), board power and CEO power may not always be inversely related because board power and CEO power has both common and disparate bases and dimensions (French & Raven, 1959;Finkelstein, 1992). In the context of CEO compensation and CEO-TMT pay gap, factors unique to either board or CEO make different dimensions of board power and CEO power (Westphal & Zajac, 1995;Adams et al., 2005;Finkelstein, 1992) salient. For example, while strong board power due to board structure logically result in weak structural power of CEO over the board, a CEO may resort to ownership power and expert power (Finkelstein, 1992) to compensate for his or her weak structural power over the board. Meanwhile, in the context of determining CEO-TMT pay gap, we argue that the structural power of board exerts a much stronger impact on the CEO than the potential ownership and expert power of directors.
The scenario of strong CEO power and weak board power should be related to the larger CEO-TMT pay gap. In other words, a CEO can rely on his or her strong power to bargain with the board which has weak power and obtain higher compensation. Meanwhile, the CEO does not have the incentive to raise the compensation of his or her TMT peers. As a result, a larger CEO-Board power imbalance should be related to the larger CEO-TMT pay gap.
However, when a board has stronger power and the CEO has weaker power at the same time, it is less clear whether the CEO-TMT pay gap will be larger. From the agency theory's perspective, as H1a predicts, stronger board power would be related to a smaller CEO-TMT pay gap. Meanwhile, a CEO with a weaker power is unlikely able to overcome the stronger power of the board to pursue a larger CEO-TMT pay gap. From the tournament theory's perspective, as H1b posits, stronger board power should be positively related to the CEO-TMT pay gap. In this case, because the CEO would nevertheless welcome a larger CEO-TMT pay gap, the weak CEO power over the board becomes moot. As a result, a larger board-CEO power imbalance should be related to the larger CEO-TMT pay gap.
H3a: Board-CEO power imbalance is negatively associated with the CEO-TMT pay gap.
H3b: Board-CEO power imbalance is positively associated with the CEO-TMT pay gap.

Data and sample
The sample for our study included all publicly traded firms listed in the S&P 500 composite index in the year 2009. These firms were tracked for the study period, regardless of whether they stayed on the S&P 500 list. To identify changes over time, we collected data covering an eight-year window from 2006 and 2013. This period allowed us to track a firm's executive compensation pattern over years. To accurately reflect pay differentials between a CEO and the TMT, we excluded observations where CEO tenure is less than one year. This helped us avoid artificial low CEO-TMT pay gap ratios caused by the fact that CEOs receive lower compensations when they work only for part of the year (Bebchuk et al., 2011) 2 . We used Execucomp to gather individual executive information for the five highestpaid executives (including the CEO), which was widely used in prior studies (Bloom & Michel, 2004;Bebchuk et al., 2011). We analyzed firm 10-K filings to supplement missing executive information in Execucomp. Compustat provides firm-level financial data. In our main analyses, we eliminated financial and utility firms to follow a common practice in prior research of firm governance choices and executive compensation designs (Ridge, Aime, & White, 2015) 3 .
As a result, it's possible that CEO power and board power reaches a balanced status, rendering the offset of both power when the board and CEO have different attitudes towards CEO-TMT pay gap. In turn, it is important to simultaneously examine the differential impact of board power and CEO power on CEO-TMT pay gap. 2 Bebchuk et al. (2011) point out that a CEO receives a smaller amount of compensation if the CEO does not hold the position for the entire year. As a result, if we were to include observations where CEO tenure is less than one year, we would derive at a CEO-TMT pay gap ratio that is downward biased. 3 Ridge et al. (2015) argue that financial and utility firms are highly regulated by governmental agencies. Consequently, these firms need to satisfy unique

Dependent variables
The CEO-TMT pay gap is a ratio variable, calculated as the ratio of CEO compensation over the average compensation of the four highest highest-paid non-CEO managers (

Independent variables
We employ a composite measure of board power to represent a board's capability of controlling the CEO. Specifically, we take the standardized value of board power in the main tests, where: (1) Following Zajac and Westphal (1996), we encode an independent board chair as one if the chairperson is not the CEO, and zero otherwise. In other words, the independent board chair is a reverse coding of CEO duality which represents the case that the CEO is also the chairperson. CEO duality gives the CEO increased power over the board to exert his or her own will and pursue his or her own interests (Daily & Johnson, 1997). In turn, the separation of the board chair and CEO weakens the power of the CEO and strengthens board control (Boyd, 1994;Zajac & Westphal, 1996).
The variable of multiple insider directors takes the value of one if the CEO is not the only inside director on the board, and zero otherwise. Traditional agency theory studies argue that CEO power increases when more insiders, who are employees or managers of the firm, serve as directors (Boyd, 1994;Morse et al., 2011). In other words, more inside directors should be negatively related to board power (Boyd, 1994). However, current research finds that recent regulatory changes appear to boost an unexpected effect that CEOs appear to be more powerful when no other firm employees serve on the board (Zorn et al., data-reporting requirements which make them less comparable to firms less regulated. 4 The Execucomp database records the five highest paid executives including the CEO. The exact titles of the recorded non-CEO executives may vary from firm to firm. A cursory search of the ExecuComp database shows that the typical titles of the highest paid non-CEO executives also include Executive Vice President, Chief Audit Executive, Chief Legal Officer, etc. 2017). We agree with the assessment that non-CEO insider directors not only may share the decisionmaking process with the CEO (Adams et al., 2005) but also can strengthen the monitoring capability of independent directors by mitigating the information asymmetry between the board and the CEO (Zorn et al., 2017).
Lead independent director takes the value of one if a board designates a lead independent director (Krause et al., 2017), and zero otherwise. A lead independent director helps balance the strong power of a CEO who is also the chairperson (Krause et al., 2017) and may facilitate the removal of a poorly performed CEO (Lamoreauxa, Litov, & Mauler, 2019). Accordingly, a board strengthens its power over the CEO with a lead independent director.
Super board independence is one when a board is consisted of at least 50% of independent directors; otherwise, super board independence is zero. Chhaochharia and Grinstein (2009) show that stronger board independence is negatively associated with CEO compensation. This evidence supports the argument that a board enhances its power over the CEO with more independent directors (Cannella et al., 2009).
Board interlock takes the value of one if a member of the compensation committee also serves as a director of at least one of other companies, and zero otherwise. Zajac and Westphal (1996) suggest that the board interlocks influence board power. Regarding the effect of board power on CEO compensation, we expect that the interlocking status of a director who is a member of the compensation committee allows the interlocked director to relate CEO compensation to those of peer firms (Hallock, 1997 Effectively a reverse coding of independent board chair, CEO duality equals one if a CEO also serves as the chairperson, and zero otherwise. When a CEO serves as the chairperson of the board (CEO duality), the power of the CEO over the board is enhanced (Daily & Johnson, 1997). Meanwhile, a CEO/chairperson usually exerts more influence on the nomination process of new directors (Westphal & Zajac, 1995)  Founder is a dummy variable that takes the value of one if a CEO is also the founder of the firm, and zero otherwise. Prior research argues and provides evidence that the founder status strengthens the power of a CEO over directors with respect to decision-making and influences CEO compensation ( We define board-CEO power imbalance as the difference between the standardized value of board power and CEO power:

Control variables
We first included individual-level variables that may influence the CEO-TMT pay gap as controls. Specifically, we included CEO pay and TMT pay dispersion. CEO pay was measured as the logarithm of the total compensation received by the CEO. We used the Gini coefficient to calculate TMT pay dispersion (Bloom & Michel, 2002; Messersmith, Patel, Lepak, & Gould-Williams, 2011). The Gini coefficient ranges from 0 to 1, where 1 indicates higher levels of dispersion and less equality in compensation among executive members. The Gini coefficient was calculated for each executive team for each year by using the following formula: where, is individual executive pay on the executive team j in decreasing order of size, ̅ is the mean pay on team j, and n is the number of executives on team j. We next controlled several firm-level variables that may influence the CEO-TMT pay gap. We considered the governance environment of a firm by incorporating the E-index, an indicator of managerial entrenchment and an aggregate measure of firms' status related to a poison pill, classified board, golden parachutes, supermajority requirement, limit to amend a bylaw, and limit to amend corporate charter (Bebchuk, Cohen, & Ferrell, 2009). Specifically, we control for E-index no classified board (i.e., the E-index value minus the value of classified board), E-index no golden parachutes (i.e., the E-index value minus the value of golden parachutes), and E-index no classified board and golden parachutes (i.e., the E-index value minus the values of classified board and golden parachutes) in our main tests of H1, H2, and H3, respectively. , we controlled firm size (i.e., the natural logarithm of a firm's sales), leverage (i.e., the ratio of debt to total assets), current ratio (i.e., the ratio of current assets divided by current liabilities), R&D intensity (i.e., the ratio of annual R&D expenditures over sales), and capital investment (i.e., the ratio of annual capital equipment expenditures over sales). We also control unrelated product diversification entropy, which was measured as ∑ ln , where was the percentage of total sales a firm received from its ith two-digit SIC segment (Fredrickson et al., 2010;Ridge et al., 2015). We further control for market-tobook ratio (MTB) which is measured as the ratio of the firm's market value over equity book value divided by 1,000.
Moreover, we controlled for three industry industry-level factors that may influence the CEO-TMT pay gap: complexity, munificence, and dynamism. To account for the inequalities among competitors, we measured complexity as the sum of squares of market shares of all firms in each industry (Connelly, Haynes, Tihanyi, Gamache, & Devers, 2016). Munificence refers to the capacity of an industry to support sustained growth and is the regression of industry sales over time divided by the mean of industry sales (using a 5-year window with the focal year as the last year in the series) (Bergh, 1998). Dynamism captures the level of instability or turbulence present in an industry (Lepak, Takeuchi, & Snell, 2003). We define dynamism as the standard error of the prior regression divided by mean industry sales. We calculated these factors at the two-digit SIC code level.
Lastly, we employed a set of year dummy variables to control year year-fixed effects in all models. All dependent variables were one year forwarded (t+1). The focal year's data (t) were used for all other variables in the regression models.

Estimation strategies
We conducted panel data analyses with the system generalized method of moments (GMM) procedure (Arellano & Bover, 1995;Blundell & Bond, 1998) to test our hypotheses while resolving the potential problems of endogeneity, heteroskedasticity, and autocorrelation. Studying the determinants of top executive pay disparity encounters many empirical challenges. For example, firms and top executives are heterogeneous in nature and have many unique features difficult to measure (Sanchez-Marin & Baixauli-Soler, 2015). Unobserved factors that affect the dependent variable are potentially endogenous to the independent variable. Thus, the independent variables, perhaps correlated with the past or current error terms, are not strictly exogenous (Patel, Li, del Carmen Triana, & Park, 2018). More specifically, the data used in this study contain observations of cross-sectional units (i.e., firms) over multiple time periods. The error items are often correlated across years within firm i (rather than randomly distributed), giving rise to the concern of heteroskedasticity and autocorrelation.
The GMM approach has been increasingly used to explore top executive pay disparity (Connelly et Baixauli-Soler, 2015). The system GMM model was designed especially for situations with a large number of cross-sectional observations and few time periods (small T and large N panels: Roodman, 2009) and has been widely used by empirical researchers due to its efficiency of estimation in the presence of heteroskedasticity and endogeneity (Baum, Schaffer, & Stillman, 2003;Roodman, 2009).
In general, we tested the hypotheses by estimating the GMM estimators in the following functional form of our models: (5) where, Y represents the dependent variable, i denotes the firm, and t denotes the year. X is the vector of variables including key independent variables and control variables, and represents estimated parameters. The model includes an individual effect, , to control for unobservable heterogeneity, so that the error term is , where is a random error. Specifically, we employed the xtabond2 command in Stata with the two-step estimation option and the robust standard error option for the system GMM estimators. The two-step GMM approach allows us to estimate the error terms by regressing the dependent variable against the independent variable and the endogenous control variables as well as all exogenous instrumental variables as the first step. We treated CEO pay, TMT pay dispersion, and E-index related controls (i.e., E-index no classified board, E-index no golden parachutes, E-index no classified board and golden parachutes, and E-index) as the endogenous control variables that may have an impact on the endogenous independent variables (i.e., board power, CEO power, and board-CEO power imbalance) and used all other control variables as exogeneous variables. Then, at the second step, the residuals generated from the first step were used to calculate the error terms and derive at the GMM estimators (Baum et al., 2003;Connelly et al., 2016;Roodman, 2009). In addition to the two-step system GMM approach, we followed prior studies of top executive pay disparity (Ridge et al., 2015) by adopting the robust standard error option, so that we provided more efficiency and robust estimates than other methods such as the generalized least square equation with the fixed effect and first difference GMM (Baum et al., 2003). We also included a lagged dependent variable in the regression equation to address the dynamic nature of the dependent variable and mitigate the concerns caused by autocorrelation. Table 2 shows descriptive statistics and pairwise correlations of the variables that we employ in testing our hypotheses. CEO-TMT pay gap does not have a statistically strong correlation with board power, offering no support for H1. Consistent with H2, CEO-TMT pay gap is positively correlated with CEO power. Meanwhile, board-CEO power imbalance is negatively correlated with CEO-TMT pay gap, a result consistent with H3a. Table 3 shows the main test results of Hypothese 1-3. The dependent variable for Models 1-3 is the CEO-TMT pay gap. Model 1 supplies evidence that supports H1a because the coefficient of board power is negative and significant ( = -0.214, p < 0.05). Shown in Model 2, CEO power has a marginal positive effect ( = 0.141, p < 0.10) on CEO-TMT pay gap. We thus find evidence consistent with H2. In Model 3, board-CEO power imbalance has a significant negative relationship with the CEO-TMT pay gap ( = -0.114, p < 0.05), providing support for H3a.

Main results
In all models, we performed the Arellano-Bond test for autocorrelation. Because we used a one-year lagged dependent variable in Models 1-3, we adopted the Arellano-Bond test for second-order autocorrelation to evaluate whether the lags of the dependent variable used as the instruments are endogenous (Ridge et al., 2015;Roodman, 2009). The Arellano-Bond test statistics for second-order autocorrelation in first differences -AR(2) -failed to reject the null hypothesis that no second-order autocorrelation exists in Models 1-3, respectively.
We also performed Hansen tests for the validity of the instruments. The Hansen test evaluates the validity of model specification and the exogeneity of instrumental variables (Baum et al., 2003). In Models 1-3, Hansen test statisticsreported as Hansen p-value for all GMM modelsshowed that we failed to reject the null hypothesis, indicating that the moment restrictions in our models are valid and that the instruments are exogenous.
In sum, the combined results of the tests showed that we do not have autocorrelation in the first-differenced errors and that our instruments satisfy the standard validity criterion. Our model specification enhances the validity of the results by sufficiently addressing the potential problems of autocorrelation and endogeneity.  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21   1  CEO-TMT pay

Robustness tests
In unreported analyses, we reran Model 1 without standardizing the value of board power and found result consistent with what is reported in Model 1. When we include both the one-year and two-year lagged dependent variables in Model 2, we find stronger support for H2 ( = 0.152, p < 0.05) than what is reported in Model 2.
We further carefully exam whether our main results are robust to different constructs of key independent variables. First, we modify the construct of board power by considering the argument that a larger board enhances CEO power rather than board power (Cheng, 2008;Jensen, 1993). In other words, some scholars argue that a larger board size may make it difficult to reach consensus and therefore allows the CEO to strengthen his or her power (Cheng, 2008;Jensen, 1993). We test whether this different interpretation of the impact of board size on board power changes our findings. Therefore, we take the standardized value of board power and redefine board power as: (6) where, all the components of board power are defined in the same way as in equation (1).
We reran Model 1 with the modified construct of board power. In undocumented analysis, we again found evidence that supported H1a which predicts the negative relationship between board power and CEO-TMT pay gap ( = -0.247, p < 0.05).
We further reran Model 3 with a revised construct of board-CEO power imbalance which is derived by substituting the revised construct of board power in equation (6) for the construct of board power in equation (1). In unreported analysis, we found support for H3a because the coefficient of board-CEO power imbalance is negative and significant ( = -0.114, p < 0.05).
Second, we focus on the structural power of directors and redefine board power as the following: (7) where, independent board chair, multiple insider directors, lead independent director, and super board independence are defined in the same way as in equation (1). In the meantime, we redefine CEO power as: (8) where, CEO duality, CEO lone inside director, no lead independent director, and Founder have the same definitions as in equation (2).
Lastly, we redefine board-CEO power imbalance as the logarithmic transformation of the sum of one and the ratio of the redefined board power over the redefined CEO power: ln (9) Table 4 documents the robustness test results of Hypothese 1-3. In Model 4, the coefficient of redefined board power is negative and significant ( = -0.354, p < 0.05), supporting H1a. Consistent with Model 2, the redefined CEO power has a marginally positive effect ( = 0.480, p < 0.10) on the CEO-TMT pay gap in Model 5 (Model 5 controls for a revised variable of CEO tenure which is measured by the number of years an individual had been the CEO of a given firm). We find evidence consistent with H2 again. Model 6 provides support for H3a because the redefined board-CEO power imbalance has a significant and negative coefficient ( = -0.561, p < 0.05). As a result, we show that our main results are robust to different definitions of board power, CEO power, and board-CEO power imbalance.

Supplemental analyses
We conduct supplemental tests to verify that our main results are consistent when we adopt a different construct of the CEO-TMT pay gap. Specifically, we substitute CEO pay slice for the CEO-TMT pay gap and rerun Models 1-6. Following Bebchuk et al. (2011), CEO pay slice is calculated as the ratio of CEO compensation over the aggregate compensation of the five highest-paid managers. In Table 5 which documents the abridged results of our retesting of Models 1-3, Model 7 supports H1a as board power has a negative and significant relationship with CEO pay slice ( = -0.011, p < 0.01). Model 8 shows that the coefficient of CEO power is positive and significant ( = 0.011, p < 0.01), offering support for H2. In Model 9, board-CEO power imbalance has a significant negative relationship with CEO pay slice ( = -0.007, p < 0.01). We thus find support for H3a again.
In Table 6, we provide abridged results of the retest of Models 4-6 in which we substitute the CEO pay slice for the CEO-TMT pay gap. The results offer strong support for H1a, H2, and H3a.  By showing that board power negatively influences the CEO-TMT pay gap but CEO power exerts the opposite effect, we provide evidence consistent with the agency theory argument that associates larger CEO-TMT pay gap with CEO rentseeking (Bebchuk et al., 2011). In other words, CEOs tend to use their power over the boards to improve their own compensation but not the pay of their subordinate TMT members.
We also find that the power imbalance between a board and the CEO is negatively associated with the CEO-TMT pay gap. We interpret this result as an indication that directors are sensitive to a CEO's strong power which may lead to potential rentseeking behavior exemplified by a larger CEO-TMT pay gap. Consequently, a strong board prioritizes limiting the rent-seeking opportunity of the CEO over providing tournament incentives to the CEO and his or her executive peers.

The connection between board power and CEO power
While extant research presents different theoretical analyses and empirical operationalizations of board power and CEO power (Finkelstein, 1992;Cannella et al., 2009), we explore beyond the structural relationship between directors and CEO to construct our proxy for board power and CEO power. To gauge the power of directors over the CEO in the context of CEO and TMT compensations, we rely on the governance mechanisms such as board composition, board interlock, and board election. Particularly, we argue that board power is positively associated with the independence of the board chair (Boyd, 1994), the number of inside directors (Adams et al., 2005;Zorn et al., 2017), the availability of lead independent director (Lamoreauxa et al., 2019), the extent of board independence (Cannella et al., 2009), the existence of board interlocks (Zajac & Westphal, 1996), the classification of board election terms (Faleye, 2007), and board size (Haynes et al., 2019).
Building upon extant theoretical interpretations and empirical constructs of CEO power (Daily & Johnson, 1997;Finkelstein, 1992), we operationalize CEO power as the combination of structural power, ownership power, expert power, and the status of CEO entrenchment. Structural power is associated with a CEO's formal organizational position (Finkelstein, 1992). Ownership power may come from a CEO's founder status (Finkelstein, 1992 However, while CEO duality and low levels of board independence tend to enhance CEO power, CEO compensation is a manifestation of CEO power rather than one of the sources. Thus, we submit that stronger CEO power over the board is associated with CEO duality, the status of a CEO as the lone inside director, the absence of a lead independent director, the founder status of a CEO (Song & Wan, 2019; Adams et al., 2005), the tenure of a CEO (Abernethy et al., 2015), and the entrenchment of a CEO (Bebchuk et al., 2014).
In sum, this study extends corporate governance literature by constructing new aggregated measures of board power and CEO power which reflect the multidimensional characteristics of board-CEO relationships.

CONCLUSION
Aiming at expanding extant literature that examines the effect of the board of directors and CEO on executive pay levels (Bebchuk et al., 2002;van Essen et al., 2015), we comparatively explore how board power and CEO power impacts CEO-TMT pay gap. We discover that board power has a negative relationship with the CEO-TMT pay gap and that CEO power is positively associated CEO-TMT pay gap.
Meanwhile, the power imbalance between board and CEO negatively connects with the CEO-TMT pay gap. These findings yield new insights that help advance the research of the antecedents of the CEO-TMT pay gap.
Although this study offers notable insights into CEO-TMT pay gap literature, there are several limitations that may limit the interpretation of the findings and offer opportunities for future research. First, the firms examined in this study are relatively large, publicly-traded firms. This sampling design is widely used in studies exploring CEO-TMT pay differential (Henderson & Fredrickson, 2001;Ridge et al., 2015); however, the results found in this study should be interpreted within the boundary conditions of the firms studied. Research using other sampling frames is needed to confirm the extent to which the results are generalizable.
Second, this study relies heavily on board composition to construct proxies for the board and CEO power. We acknowledge that board composition itself is not directly conducive to a detailed understanding of the influences of individual and social factors such as prestige (Finkelstein, 1992) on board and CEO power. Therefore, we encourage researchers to use multiple sources of data to gain further insights on board and CEO power, as well as their direct impact on the CEO-TMT pay gap.
Our study contributes to executive compensation literature by comparatively investigating the impacts of board power and CEO power on top executives' relative compensation. Our theoretical analyses and corresponding hypotheses directly compare the predictions of agency theory with those of tournament theory. Our comparative approach not only helps investigate the determinants of the CEO-TMT pay gap but also helps resolve existing debates regarding the implications of the CEO-TMT pay gap. Consistent with agency theory predictions rather than tournament theory ones, our empirical results suggest that boards of directors are conscientious about the potential negative effects of a larger CEO-TMT pay gap and therefore stronger boards usually do not rely on larger CEO-TMT pay gap to incentivize CEOs.