How general is managerial human capital?: Evidence from the Retention of Managers after M&As

How general is managerial human capital? Market-based views on managerial compensations explain high CEO compensation in the U.S. as resulting from the competition to employ scarce managerial talent (e.g., Gabaix and Landier, 2008; Terviö, 2008). However, the validity of these theories critically depends on the answer to this question. This question is also related to the sensitivity of managerial turnover to poor performance by a firm (e.g., Eisfeldt and Kuhnen, 2013). Thus, it is one of the most important empirical questions in the literature on corporate governance.

The empirical studies on the retention of managers after M&As may provide some evidence to assist in answering this question. Although several theories of takeovers presume that top managers in a target firm must be replaced after a takeover ⁽¹⁾ , there is increasing evidence that the retention of a management group is important to the new firm ⁽²⁾ . This evidence suggests that there are important skills held by executives of a target firm that a new management group in an acquiring company cannot easily replace.

Interestingly, the existing literature finds that the tenure of a CEO does not have any significant effect on his/her probability of retention after takeover (e.g., Buchholtz et al., 2003; Wulf and Singh, 2011). Given that experience in a firm is assumed to result in the development of firm-specific skills, the lack of such an effect places doubt on the hypothesis that firm-specific skills are required to manage a newly merged firm after a merger and acquisition (M&A).

It appears that managers’ tenures may have a negative impact on their retention rate after an M&A. If it is expected that managers with long tenures will have difficulty adapting to the new environment - on the basis that the smaller number of firms in which their experience is gained reduces their opportunity to improve their general human capital - a newly merged firm may not want to employ a manager with a long tenure (e.g., Buchholtz et al., 2003). Therefore, this could offset the positive effect of tenure on the retention rate.

This paper examines how the tenure of managers influences the retention rate of a management group after M&As in Japanese companies during the period 1990–2006. It attempts to separate the two different hypotheses regarding the effect of tenure on separation probability. We show that negative and positive effects of tenure on the managers’ separation rate coexist in the case of Japanese M&As.

The main challenge to achieving our goal is developing a way to cope with the limitations of the data. Ideally, we need a fairly rich dataset, which includes the random assignment of target managers and their history afterward, including not only their career paths, but also wage payments and obtained skills. Unfortunately, it is not possible to obtain such data for Japanese firms. These data restrictions force us to develop a new model and a new method that can extract meaningful information from the restricted dataset. We develop a general equilibrium model that is designed to analyze the separation of target-firm managers after M&As. It is shown that our structural model can be approximately estimated by a stratified Cox proportional hazards model. In addition, we propose a novel method to correct for selection biases arising from unobserved heterogeneity of managers.

Equipped with the constructed theoretical model and the new estimation method, we conduct survival analyses of Japanese titled directors, who are considered to be the top executives in Japanese companies ⁽³⁾ . Because the Japanese promotion system is known to encourage investment in firm-specific human capital, we expect that the advantages and disadvantages of long tenure can be more accurately measured and investigated in Japanese companies than is the case for other countries.

We consider two sources of managerial human capital in this paper: experience as employees and managerial experience, which is measured by experience as a board member. Our empirical analyses show that an increase in a manager's tenure as an employee in a target company increases both the probability of being appointed to the board in a merged firm and the subsequent separation probability after this appointment. However, a longer tenure as a board member within the target firm does not have any significant impacts on either appointment probability or separation probability. Through the lens of our theory, we can interpret these findings as follows: 1) Japanese firms after M&As value both the target firm-specific human capital and the general human capital of managers; 2) experience as an employee increases firm-specific skills, but at the expense of the accumulation of general human capital; and 3) Managerial experiences are likely to be more general than those as an employee.

There have been many attempts to understand the retention of managers after M&As ⁽⁴⁾ . This paper makes two main contributions to the literature. First, to the best of our knowledge, ours is the first paper that constructs a structural model of managers’ retention after M&As. Second, we propose a novel method to correct for selection biases arising from the unobserved heterogeneity of managers, which does not require a random sample from the population. These two contributions are discussed in detail below.

We develop a structural model to disentangle the several mechanisms underlying the causal effects of tenure on the separation of target managers after M&As given several data limitations. However, in contrast to the standard structural estimation approach of, for instance, Rust (1987) and Hotz and Miller (1993), we do not attempt to estimate all structural parameters. Instead, we focus on particular parameters of interest, namely the coefficients of the effects of tenure on the separation rates. Our model is designed to clarify the conditions under which we can obtain economically meaningful interpretations from these parameters. Given the data limitations that we face, this strategy allows us to utilize weaker parametric assumptions for our identification, resulting in greater transparency of the source of variations for the identification ⁽⁵⁾ .

Let us describe the intuition on how to identify some mechanisms from the coefficients of the estimated effect of tenure on turnover in conditions where the data are constrained. In particular, we are concerned about two data problems: the lack of data about skills and the lack of data about compensation. First, because we cannot observe skills, we must differentiate two different hypotheses regarding the length of tenure and the separation probability of managers. Our identification strategy is built on the recognition that, although target-specific human capital is important as an input for current production, employment of managers with the ability to learn in a new environment is an investment in future production. More concretely, on the one hand, because managers in an acquiring firm do not have much knowledge of the target firm's organization, the acquirer initially regards the specific skills held by the target firm managers as highly valuable in managing and restructuring the target firm. On the other hand, because target managers accumulate knowledge of the new firm over time, the difference between the amount of knowledge accumulated by a manager who learns quickly and one who is a slow learner becomes larger as time goes by ⁽⁶⁾ . It is shown that the timing of separation reflects the relative importance of each type of human capital.

This seemingly intuitive argument requires nontrivial theoretical consideration of learning capability. Although the argument in the previous paragraph implicitly assumes that a newly merged firm is willing to employ managers with high learning capability, this may not be true. Because the ability to learn in a new environment is a general ability, the market competition can raise the compensation of able managers, which may make the firm reluctant to employ them. Therefore, the validity of the above argument depends on the structure of the external labor market for managers. Because we developed a general equilibrium model, we can discuss how the frictions in the managerial external labor market influence the coefficients of the effects of tenure on the separation probability.

In addition, the coefficients of the effects of tenure might be influenced by the second data problem: the lack of compensation data for Japanese managers. It is not obligatory for publicly traded companies to provide information on the compensation of managers in Japan. As suggested in the literature (e.g., Wulf, 2004, Hartzell et al., 2004, and Bargeron et al., 2009), this might cause several biases if managers in target firms can negotiate the compensation and severance pay in the newly merged firm during the process of an M&A. Our theoretical model shows that if it is impossible to make a complete contract on treatment of the new managers in the new firm during the M&A and there are some explicit or implicit transfers after the new firm starts as a result of ex post bargaining, then the separation probability does not depend either on the compensation or on the severance pay of managers. That is, the model provides a theoretical condition under which we can extract meaningful information from the coefficients of the impact of tenure on the separation rate even when compensation data are not available.

Next, we explain the second main contribution to the literature. In fact, this second contribution can be considered the more important contribution because it has a broader relevance than the post-M&A managerial retention literature. Note that because managers in a target company are not randomly assigned, the coefficient of tenure can be contaminated by another effect. If a talented person is promoted to a management position faster than is usually the case, the short tenures observed in our data may simply indicate that a manager has a high unobserved ability.

In order to deal with these selection biases, we propose a novel method to extract information about the unobserved heterogeneity of workers from the data on managers. Intuitively, if a talented person is promoted to a management position faster, the length of tenure prior to becoming a manager must contain useful information about their unobserved ability. We utilize this information to identify and control for unobserved ability to estimate the causal effects of tenure.

Compared with the standard sample selection models that have appeared in econometrics textbooks (e.g., Wooldridge, 2002; Cameron and Trivedi, 2005), our approach has three advantages. First, because we utilize the timing of selection in a selected sample to correct for selection bias, we do not need to access a random sample from the population. We are not aware of any other papers that provide a tractable method to correct a selection bias inherent to personnel data using only the selected sample. Second, because the timing of observations is generally different from the timing of promotions, we can find several exclusive variables that are required to identify parameters and obtain reliable estimates. Finally, although a standard two-step estimator makes distributional assumptions to correct selection biases, we rely on an extreme value theory to obtain robust results against misspecifications of distributional assumptions. Gabaix and Landier (2008) argue that an extreme value theory can provide a nice approximation for the upper tail of a large class of continuous distributions. We apply their argument and derive an explicit solution to the conditional expectation of unobserved ability under a selected sample, which is used to determine unbiased estimates of parameters under a selected sample.

As discussed before, this methodology is applicable to a broader literature than that of managerial retention after M&As. Although it is recognized that the characteristics of CEOs change the strategies of firms (e.g., Bertrand and Schoar, 2003), researchers face difficulties in examining the causal role of CEO characteristics in a firm because publicly available data on top managers typically involve selected samples. We propose a method to deal with this problem. Moreover, our methodology can be applied to data on internal promotions, when researchers are not able to access data on employees’ whole careers. We hope that this novel approach can help researchers struggling with data limitations to evaluate the role of leaders.

In addition to these two main contributions to the literature, our results contribute to several streams of literature, including those on managerial compensation and Japanese M&As. We clarify the contribution of our results to these fields in Section 8.

The paper is organized as follows. Section 2 presents a general equilibrium model of the separation of target managers after M&As. It shows how target firm-specific human capital and general human capital can influence the separation decision after M&As. Section 3 links the length of tenure and each type of human capital and provides a number of testable conditions that distinguish several hypotheses on managerial human capital based on the evidence of the effect of tenure on separation probability. Section 4 introduces our empirical model and discusses how to correct a selection bias using our selected dataset. Section 5 describes our Japanese dataset and Section 6 presents our estimation model and our results, interpreted through the lens of our theory. Section 7 discusses whether the moral hazard of managers could influence the interpretation of our results. Section 8 relates our results to several streams of literature that cover managerial compensation and Japanese M&As. Section 9 concludes the paper.

In this section, we construct a general equilibrium model that is designed to examine the separation of target managers after M&As using the following procedure. First, we discuss the assumptions on the production functions, which describe how target firm-specific human capital and general human capital influence the production process. Second, using the production functions, we model the separation of a manager from a target firm after an M&A by taking the outside values of managers as given. Third, we model an external managerial labor market, which allows us to determine the outside values of the managers. Finally, we show how our structural model can be approximately represented by the proportional hazards model and discuss how target firm-specific human capital and general human capital influence the parameters of the proportional hazards model.

In this section, we explain how we utilize the timing of separation to obtain information to distinguish two hypotheses on managerial human capital in a firm (i.e., an increase in target firm specific human capital and a reduction in the general ability to learn in a new environment) from the coefficients of the effect of tenure on separation probability. For this purpose, we propose testable hypotheses to interpret the relationship between tenure and separation probability.

Taking a first-order approximation, it is shown that: (5) Πnh=α˜ThT+α˜GhG+α˜X′XV, {\Pi ^n}\left( {\bf{h}} \right) = {\tilde \alpha _T}{h^T} + {\tilde \alpha _G}{h^G} + \tilde \alpha _X^{\prime}{{\bf{X}}_V}, (6) Δπh=α¯ThT−α¯GhG+α¯X′XV, \Delta \pi \left( {\bf{h}} \right) = {\bar \alpha _T}{h^T} - {\bar \alpha _G}{h^G} + \bar \alpha _X^{\prime}{{\bf{X}}_V}, where α˜T=fTn′hT1−β1−q^≥0 {\tilde \alpha _T} = {{f_{Tn}^{\prime}\left( {{h^T}} \right)} \over {1 - \beta \left( {1 - \hat q} \right)}} \ge 0 , α˜G=fGn′hG−fGl′hG+F′hG≥0 {\tilde \alpha _G} = f_{Gn}^{\prime}\left( {{h^G}} \right) - f_{Gl}^{\prime}\left( {{h^G}} \right) + F'\left( {{h^G}} \right) \ge 0 , α˜X=∂vnxV∂XV1−β1−q^ {\tilde \alpha _X} = {{{{\partial {v^n}\left( {{{\bf{x}}_V}} \right)} \over {\partial {{\bf{X}}_V}}}} \over {1 - \beta \left( {1 - \hat q} \right)}} , α¯T=fTl′hT−fTn′hT≥0 {\bar \alpha _T} = f_{Tl}^{\prime}\left( {{h^T}} \right) - f_{Tn}^{\prime}\left( {{h^T}} \right) \ge 0 , α¯G=fGn′hG−fGl′hG≥0 {\bar \alpha _G} = f_{Gn}^{\prime}\left( {{h^G}} \right) - f_{Gl}^{\prime}\left( {{h^G}} \right) \ge 0 , and α¯X=∂vlXV∂XV−∂vnXV∂XV {\bar \alpha _X} = {{\partial {v^l}\left( {{{\bf{X}}_V}} \right)} \over {\partial {{\bf{X}}_V}}} - {{\partial {v^n}\left( {{{\bf{X}}_V}} \right)} \over {\partial {{\bf{X}}_V}}} .

Let τ_b and τ_e denote the manager's tenure as a board member and as an employee in the target firm, respectively. Following the tradition of labor economics, we assume that the target firm's firm-specific human capital is an increasing function of these tenures. hT=η0+ηbτb+ηeτe {h^T} = {\eta _0} + {\eta _b}{\tau _b} + {\eta _e}{\tau _e}

We assume that η_x ≥ 0 for x = b or e, which means that increases in both types of tenures assist in the accumulation of firm-specific human capital ⁽⁸⁾ .

On the other hand, as h^G is a general skill, labor economists typically assume that h^G increases not only as a result of experience in the target firm, i.e., as a result of both τ_b and τ_e, but also as a result of the duration of other experiences, τ₀. Therefore, we have: hG=ω0+ωbτb+ωeτe+μτo+χ {h^G} = {\omega _0} + {\omega _b}{\tau _b} + {\omega _e}{\tau _e} + \mu {\tau _o} + \chi where ω_x ≥ 0(x = b,e) are the parameters that capture the importance of tenure for learning and μ ≥ 0 is the parameter that captures the importance of other experiences ⁽⁹⁾ . This model generalizes the standard assumption, in that it allows for different experiences to vary in their effectiveness in accumulating general human capital. Estimating τ_o by τ_o = a − τ_b − τ_e, where a is the manager's age when the M&A takes place, we can rewrite h^G as a function of tenures and age: hG=ω0+ωb−μτb+ωe−μτe+μa+χ. {h^G} = {\omega _0} + \left( {{\omega _b} - \mu } \right){\tau _b} + \left( {{\omega _e} - \mu } \right){\tau _e} + \mu a + \chi .

The parameters ω_x − μ (x = b,e) capture the relative productivity of tenure for the accumulation of general human capital compared with other experiences. Buchholtz et al. (2003) argue that longer tenure may hamper the accumulation of general skills that are needed for adaptation to a new environment. This implies that ω_x − μ < 0. Although this might be a reasonable assumption, we also allow for the possibility that ω_x − μ > 0: i.e., that experience in a target firm can assist in improving general human capital. Several studies on spin-off effects suggest that previous experience in incumbent firms can be an important source of experience in establishing new firms (e.g., Klepper, 2001). It would be possible to apply a similar reasoning in this context. This possibility is captured by ω_x − μ > 0.

The following proportional hazards model can be derived: qt=Bt:XZe−λΔπhIt=0+Πnh, Πnh=α˜ThT+α˜GhG+α˜X′XV Δπh=α¯ThT−α¯GhG+α¯X′XV hT=ηbτb+ηeτe hG=ωb−μτb+ωe−μτe+μa+χ. \matrix{ {\;\;\;\;\;\;\;\;\;{q_t} = B\left( {t:{{\bf{X}}_Z}} \right){e^{ - \lambda \left[ {\Delta \pi \left( {\bf{h}} \right)I\left( {t = 0} \right) + {\Pi ^n}\left( {\bf{h}} \right)} \right]}},} \hfill \cr {\;\,{\Pi ^n}\left( {\bf{h}} \right) = {{\tilde \alpha }_T}{h^T} + {{\tilde \alpha }_G}{h^G} + \tilde \alpha _X^{\prime}{{\bf{X}}_V}} \hfill \cr {\;\Delta \pi \left( {\bf{h}} \right) = {{\bar \alpha }_T}{h^T} - {{\bar \alpha }_G}{h^G} + \bar \alpha _X^{\prime}{{\bf{X}}_V}} \hfill \cr {\,\;\;\;\;\;\;\;{h^T} = {\eta _b}{\tau _b} + {\eta _e}{\tau _e}} \hfill \cr {\;\;\;\;\;\;\;\,{h^G} = \left( {{\omega _b} - \mu } \right){\tau _b} + \left( {{\omega _e} - \mu } \right){\tau _e} + \mu a + \chi .} \hfill \cr }

Using this derived hazard function, the theoretically predicted effects of tenure on separation probability are summarized as follows: (7) d−λΠnhdτx=−λα˜Tηx+α˜Gωx−μ, {{d\left( { - \lambda {\Pi ^n}\left( {\bf{h}} \right)} \right)} \over {d{\tau _x}}} = - \lambda \left[ {{{\tilde \alpha }_T}{\eta _x} + {{\tilde \alpha }_G}\left( {{\omega _x} - \mu } \right)} \right], (8) d−λΔπhdτx=−λα¯Tηx−α¯Gωx−μ, {{d\left( { - \lambda \Delta \pi \left( {\bf{h}} \right)} \right)} \over {d{\tau _x}}} = - \lambda \left[ {{{\bar \alpha }_T}{\eta _x} - {{\bar \alpha }_G}\left( {{\omega _x} - \mu } \right)} \right], (9) ∂−λΔπh+Πnh∂τx=−λα˜T+α¯Tηx+α˜G−α¯G(ωx−μ, {{\partial \left[ { - \lambda \left( {\Delta \pi \left( {\bf{h}} \right) + {\Pi ^n}\left( {\bf{h}} \right)} \right)} \right]} \over {\partial {\tau _x}}} = - \lambda \left[ {\left( {{{\tilde \alpha }_T} + {{\bar \alpha }_T}} \right){\eta _x} + \left( {{{\tilde \alpha }_G} - {{\bar \alpha }_G}} \right)({\omega _x} - \mu } \right], where x = b or e.

Investigating equations (7), (8), and (9) provides the following propositions that we wish to reject in our empirical study. The first proposition below shows the set of conditions under which a newly merged firm does not value any specific skills of employees in the target firm and/or any ability to learn in a new environment, and under which a market friction does not depend on the general human capital.

The intuition behind equation (10) in Proposition 5 is explained as follows. If target firm-specific human capital does not increase productivity, fTn′hT=fTl′hT=0 f_{Tn}^{\prime}\left( {{h^T}} \right) = f_{Tl}^{\prime}\left( {{h^T}} \right) = 0 , an increase in tenure influences the separation probability only because it can change the learning capability of managers. If ω_x ≥ μ, where x = b or e, because an increase in tenure assists in improving general human capital relative to other work experience, it increases the productivity of the manager in the newly merged firm and, therefore, lowers the separation probability at all times. However, because learning takes time, the benefits from a general skill that improves the learning capability are not initially realized. Hence, the initial products of tenure, net of future products, are negative and the coefficient of the interaction with the t = 0 dummy must be positive. However, if ω_x ≤ μ, the opposite mechanism occurs. Because an increase in tenure reduces the experiences that improve general skills in this case, it simply lowers the productivity of the manager in a merged company and, therefore, increases the separation probability at all times. However, because the initial productivity of the manager after the M&A is not subject to this negative effect, the initial products of tenure, net of future products, are positive. Therefore, the coefficient of the interaction with the t = 0 dummy must be negative. Hence, for both cases, equation (10) must be satisfied.

Similarly, we can explain the intuition behind equations (11) (12) and (13) in Proposition 5 as follows. If there are no benefits from learning capability fGl′hG=fGn′hG f_{Gl}^{\prime}\left( {{h^G}} \right) = f_{Gn}^{\prime}\left( {{h^G}} \right) , an improvement in general human capital does not cause any productivity difference between the initial period and the later period. Because the human capital specific to the target firm possessed by the target-firm manager is more important before the managers in the acquiring firm learn the same skills, it would be more highly valued at t = 0. Therefore, the coefficient of the interaction with the t = 0 dummy must be negative. Equation (11) shows this condition.

If the cost of hiring a new manager does not depend on the general human capital of the manager, F′hG=0 F'\left( {{h^G}} \right) = 0 , as a result of market competition, there are no benefits from employing a manager with high general human capital. Therefore, as an increase in tenure raises target firm-specific human capital, it certainly lowers the initial separation probability. Equation (12) shows this condition.

If there are no benefits from learning capability fGl′hG=fGn′hG f_{Gl}^{\prime}\left( {{h^G}} \right) = f_{Gn}^{\prime}\left( {{h^G}} \right) , and the cost of hiring a new manager does not depend on the general human capital of the manager, F′hG=0 F'\left( {{h^G}} \right) = 0 , an increase in tenure changes the separation probability only because it can raise the human capital specific to the target firm. In addition to this, for the reason explained above, the coefficient of the interaction of tenure with the t = 0 dummy must be negative. Equation (13) summarizes these conditions.

Note that Proposition 5 requires that equations (10) and (11) are jointly satisfied by the coefficients for both the tenure as a board member, x = b, and the tenure as an employee, x = e. Because an acquiring firm does not care about how a target manager obtains a skill, if the skills are not profitable within the merged firm, the coefficients of tenure as a board member and as an employee will be influenced in a similar way. In other words, if the derived conditions are not satisfied for either x = b or x = e, we can reject the hypotheses in Proposition 5.

On the other hand, the difference between the coefficients for the two types of tenure, x = b and x = e, provides us with information about the differences in skills obtained as a result of experience gained as a board member or as an employee. The following proposition summarizes the hypotheses about specific human capital and experience that we wish to reject.

Equation (14) looks very similar to equation (10). The only difference is that Proposition 5 requires that equation (10) must be jointly satisfied for both x = e and x = b, whereas Proposition 6 states that we can separately use equation (14) for x = e and x = b. The intuition behind equation (14) is the same as that behind equation (10). However, by separately applying the same logic to the coefficients for the tenure as a board member and the tenure as an employee, we can determine differences in skills obtained from experience as a board member compared with experience as an employee.

Finally, the following proposition summarizes the hypotheses about general human capital and experience that we wish to reject.

Suppose that X_V can be decomposed into the characteristics of the target-firm manager and those of the firm to which the manager belongs. To analyze the determinants of the retention rate of target managers in a newly merged firm, we use the following stratified Cox proportional hazards model, which summarizes our theory: qfjt=B^t:XZ,fe−λΔπfjIt=0+Πfj Πfj=α˜bτb,fj+α˜eτe,fj+α˜D′XD,f,j+α˜F′XF,f+α˜χχfjΔπfj=α¯bτb,fj+α¯eτbf,j+α¯D′XD,f,j+α¯F′XF,f+α¯χχfj, \matrix{ {\;\;{q_{fjt}} = \hat B\left( {t:{{\bf{X}}_{Z,f}}} \right){e^{ - \lambda \left[ {\Delta {\pi _{fj}}I\left( {t = 0} \right) + {\Pi _{fj}}} \right]}}} \hfill \cr {\;\;{\Pi _{fj}} = {{\tilde \alpha }_b}{\tau _{b,fj}} + {{\tilde \alpha }_e}{\tau _{e,fj}} + \tilde \alpha _D^{\prime}{{\bf{X}}_{D,f,j}} + \tilde \alpha _F^{\prime}{{\bf{X}}_{F,f}} + {{\tilde \alpha }_\chi }{\chi _{fj}}} \hfill \cr {\Delta {\pi _{fj}} = {{\bar \alpha }_b}{\tau _{b,fj}} + {{\bar \alpha }_e}{\tau _{bf,j}} + \bar \alpha _D^{\prime}{{\bf{X}}_{D,f,j}} + \bar \alpha _F^{\prime}{{\bf{X}}_{F,f}} + {{\bar \alpha }_\chi }{\chi _{fj}},} \hfill \cr } where τ_b,fj is the j th person's tenure as a board member at firm f, τ_e,fj is the j th person's tenure as an employee at firm f, the vector X_D,f,j contains any other target-firm manager characteristics of the j th person in firm f, including the j th person's age when firm f was taken over, and the vector X_F,f contains any characteristics of the fth firm that can influence the productivity of managers. Finally, the vector X_z,f contains any firm-level variables that can influence baseline hazard and B^t:XZ,f \hat B\left( {t:{{\bf{X}}_{Z,f}}} \right) contains Bt:XZ,f B\left( {t:{{\bf{X}}_{Z,f}}} \right) and some constant terms.

This equation depends on the individual unobserved heterogeneity of a manager j at firm f, χ_fj. This unobserved heterogeneity may be the result of unobserved capabilities, or it may be the relationship with the founding family, main bank, or parent company. We do not know the cause of this heterogeneity, but we assume that the parameter χ_fj can summarize the effect of these heterogeneities. Without controlling for χ_fj, our estimates might be biased. As explained in footnote 3, we focus on titled directors in Japanese companies. However, talented workers may be appointed to titled director positions with less experience than less talented workers. Hence, χ_fj and τ_b,fj can be correlated.

In order to deal with this bias, we estimate unobserved abilities from our data and control for them in our survival analysis. The following argument explains how to estimate χ_fj.

Suppose that a person is appointed to a management position (titled director) in Japan if and only if: (15) h^fjt^≥H′Nhf,t^,h^fjt^≡1−ωhfjTt^+ωhfjGt^ \matrix{ {{{\hat h}_{fj}}\left( {\hat t} \right) \ge {\bf{H'}}{{\bf{N}}_h}\left( {f,\hat t} \right),} \hfill \cr {{{\hat h}_{fj}}\left( {\hat t} \right) \equiv \left( {1 - \omega } \right)h_{fj}^T\left( {\hat t} \right) + \omega h_{fj}^G\left( {\hat t} \right)} \hfill \cr } where hfjTt^ h_{fj}^T\left( {\hat t} \right) and hfjGt^ h_{fj}^G\left( {\hat t} \right) are the target firm-specific human capital and the general human capital of a person j at the fth firm in calendar year t^ \hat t , ω is the weight on the general human capital and H′Nhf,t^ {\bf{H'}}{{\bf{N}}_h}\left( {f,\hat t} \right) is the level of human capital required for appointment to a managerial position, where H is a parameter vector and a vector Nhf,t^ {{\bf{N}}_h}\left( {f,\hat t} \right) summarizes the conditions that influence the required level of human capital of firm f in calendar year t^ \hat t .

From the definition of h^fjt^ {\hat h_{fj}}\left( {\hat t} \right) , we can find the parameters ϕ_o, ϕ_b, ϕ_e, and ϕ_o to describe the following human capital accumulation function: (16) h^fjt^=ϕ0+ϕbτb,fjt^+ϕeτe,fj+ϕoτo,fj+ωχfj, {\hat h_{fj}}\left( {\hat t} \right) = {\phi _0} + {\phi _b}{\tau _{b,fj}}\left( {\hat t} \right) + {\phi _e}{\tau _{e,fj}} + {\phi _o}{\tau _{o,fj}} + \omega {\chi _{fj}}, where τb,fjt^ {\tau _{b,fj}}\left( {\hat t} \right) is the tenure as a board member of person j at firm f in calendar year t^ \hat t and τ_o,fj is the j th person's length of experience before joining firm f. Combining equations (15) and (16), when an employee j is appointed to an executive position, the following equality must be satisfied: (17) τb,fjt^j⋆=H˜′Nhf,t^j⋆−ϕ˜0−ϕ˜eτe,fj−ϕ˜oτo,fj−χ˜fj, {\tau _{b,fj}}\left( {\hat t_j^\star} \right) = {{\bf{\tilde H}}^{\prime}}{{\bf{N}}_h}\left( {f,\hat t_j^\star} \right) - {\tilde \phi _0} - {\tilde \phi _e}{\tau _{e,fj}} - {\tilde \phi _o}{\tau _{o,fj}} - {\tilde \chi _{fj}}, where H˜=Hϕb {\bf{\tilde H}} = {{\bf{H}} \over {{\phi _b}}} , ϕ˜0=ϕ0ϕb {\tilde \phi _0} = {{{\phi _0}} \over {{\phi _b}}} , ϕ˜e=ϕeϕb {\tilde \phi _e} = {{{\phi _e}} \over {{\phi _b}}} , ϕ˜o=ϕoϕb {\tilde \phi _o} = {{{\phi _o}} \over {{\phi _b}}} , χ˜fj=ωχfjϕb {\tilde \chi _{fj}} = {{\omega {\chi _{fj}}} \over {{\phi _b}}} , and t^j⋆ \hat t_j^\star is the calendar year when employee j is appointed to an executive officer position. This equation implies that, after controlling for several observable variables, τb,fjt^j⋆ {\tau _{b,fj}}\left( {\hat t_j^\star} \right) is negatively correlated with χ˜fj {\tilde \chi _{fj}} . This means that τb,fjt^j⋆ {\tau _{b,fj}}\left( {\hat t_j^\star} \right) potentially contains useful information about χ˜fj {\tilde \chi _{fj}} .

Let us define a function R such that: Rf,j,t^≡H˜′Nhf,t^−ϕ˜0−ϕ˜eτe,fj−ϕ˜oτo,fj−τb,fjt^. R\left( {f,j,\hat t} \right) \equiv {\bf{\tilde H}}'{{\bf{N}}_h}\left( {f,\hat t} \right) - {\tilde \phi _0} - {\tilde \phi _e}{\tau _{e,fj}} - {\tilde \phi _o}{\tau _{o,fj}} - {\tau _{b,fj}}\left( {\hat t} \right).

Given the estimates of H˜ {\bf{\tilde H}} , ϕ˜0 {\tilde \phi _0} , ϕ˜e {\tilde \phi _e} , and ϕ˜o {\tilde \phi _o} , we can estimate χ˜j {\tilde \chi _j} from the following: χ˜fj=Rf,j,t^j⋆. {\tilde \chi _{fj}} = R\left( {f,j,\hat t_j^\star} \right).

Hence, we seek to obtain unbiased estimators of H˜ {\bf{\tilde H}} , ϕ˜0 {\tilde \phi _0} , ϕ˜e {\tilde \phi _e} , and ϕ˜o {\tilde \phi _o} . However, this is not easy because we can observe only a selected sample.

We define the deviation of the unobserved ability from the firm-level average in year t^ \hat t for the observed sample, εfjt^ {\varepsilon _{fj}}\left( {\hat t} \right) , as follows: εfjt^≡χ˜fj−Eχ˜fj|Ib=1,f,t^=Rf,j,t^j⋆−Eχ˜fj|Ib=1,f,t^, {\varepsilon _{fj}}\left( {\hat t} \right) \equiv {\tilde \chi _{fj}} - E\left[ {{{\tilde \chi }_{fj}}|{I_b} = 1,f,\hat t} \right] = R\left( {f,j,\hat t_j^\star} \right) - E\left[ {{{\tilde \chi }_{fj}}|{I_b} = 1,f,\hat t} \right], where I_b is the dummy variable that equals one if and only if a person is a titled director and is included in our sample and, therefore, Eχ˜fj|Ib=1,f,t^ E\left[ {{{\tilde \chi }_{fj}}|{I_b} = 1,f,\hat t} \right] is the conditional expectation of χ˜fj {\tilde \chi _{fj}} , given all relevant information at the f th firm in year t^ \hat t for the observed sample. Then, equation (17) can be rewritten as follows: (18) τb,fjt^j⋆=H˜′Nhf,t^j⋆−ϕ˜0−ϕ˜eτe,fj−ϕ˜oτo,fj−Eχ˜fj|Ib=1,f,t^−εfjt^,∀t^. {\tau _{b,fj}}\left( {\hat t_j^\star} \right) = {\bf{\tilde H}}'{{\bf{N}}_h}\left( {f,\hat t_j^\star} \right) - {\tilde \phi _0} - {\tilde \phi _e}{\tau _{e,fj}} - {\tilde \phi _o}{\tau _{o,fj}} - E\left[ {{{\tilde \chi }_{fj}}|{I_b} = 1,f,\hat t} \right] - {\varepsilon _{fj}}\left( {\hat t} \right),\forall \hat t.

Note that Eεfjt^|Ib=1=0 E\left[ {{\varepsilon _{fj}}\left( {\hat t} \right)|{I_b} = 1} \right] = 0 by construction. Note also that the deviation of the unobserved ability from the conditional expectation of χ˜fj {\tilde \chi _{fj}} at the fth firm in year t^ \hat t for the observed sample, εfjt^ {\varepsilon _{fj}}\left( {\hat t} \right) , is uncorrelated with firm-level variables. Hence, we expect to satisfy an orthogonality condition that Eεfjt^Nf|Ib=1=0 E\left[ {{\varepsilon _{fj}}\left( {\hat t} \right){{\bf{N}}_f}|{I_b} = 1} \right] = 0 for the vector of firm-level variables N_f for any t^ \hat t .

Therefore, if we know a functional form of Eχ˜fj|Ib=1,f,t^ E\left[ {{{\tilde \chi }_{fj}}|{I_b} = 1,f,\hat t} \right] , we can obtain unbiased estimates of H˜ {\bf{\tilde H}} , ϕ˜0 {\tilde \phi _0} , ϕ˜e {\tilde \phi _e} , and ϕ˜o {\tilde \phi _o} from a generalized method of moments (GMM) estimation, using firm-level variables from the sample of titled directors as its instruments.

In order to implement this idea, we need to estimate Eχ˜fj|Ib=1,f,t^ E\left[ {{{\tilde \chi }_{fj}}|{I_b} = 1,f,\hat t} \right] . Because we use the sample of titled directors at target firms in the year when an M&A occurs, equation (15) implies that I_b = 1 if and only if h^fjt^fm≥H′Nhf,t^fm {\hat h_{fj}}\left( {\hat t_f^m} \right) \ge {\bf{H'}}{{\bf{N}}_h}\left( {f,\hat t_f^m} \right) , where t^fm \hat t_f^m is the calendar year in which the M&A occurs for target firm f. Using equation (16) and a function Rf,j,t^ R\left( {f,j,\hat t} \right) , h^fjt^fm≥H′Nhf,t^fm {\hat h_{fj}}\left( {\hat t_f^m} \right) \ge {\bf{H'}}{{\bf{N}}_h}\left( {f,\hat t_f^m} \right) can be rewritten as a condition for χ˜fj {\tilde \chi _{fj}} as follows: Ib=1 iff χ˜fj≥Rf,j,t^fm, {I_b} = 1\;{\rm{iff}}\;{\tilde \chi _{fj}} \ge R\left( {f,j,\hat t_f^m} \right), for any f and j. This means that the observed sample is considered to be drawn from an upper-tailed distribution of χ˜fj {\tilde \chi _{fj}} and that: Eχ˜fj|Ib=1,f,t^fm=Eχ˜fj|χ˜fj≥Rf,j,t^fm,f,t^fm. E\left[ {{{\tilde \chi }_{fj}}|{I_b} = 1,f,\hat t_f^m} \right] = E\left[ {{{\tilde \chi }_{fj}}|{{\tilde \chi }_{fj}} \ge R\left( {f,j,\hat t_f^m} \right),f,\hat t_f^m} \right].

We need to estimate Eχ˜fj|χ˜fj≥Rf,j,t^fm,f,t^fm E\left[ {{{\tilde \chi }_{fj}}|{{\tilde \chi }_{fj}} \ge R\left( {f,j,\hat t_f^m} \right),f,\hat t_f^m} \right] to obtain unbiased estimators of the parameters. The standard two-stage estimator in Heckman (1979) uses a random sample from the population to estimate selection equations, the parameters of which are used to construct an inverse Mills ratio to estimate Eχ˜fj|χ˜fj≥Rf,j,t^fm,f,t^fm E\left[ {{{\tilde \chi }_{fj}}|{{\tilde \chi }_{fj}} \ge R\left( {f,j,\hat t_f^m} \right),f,\hat t_f^m} \right] , by assuming that χ˜fj {\tilde \chi _{fj}} is drawn from a normal distribution. Typically, exclusive variables are needed for the estimation of selection equations to obtain reliable estimates.