Endogeneity concerns

As in most empirical corporate finance studies, endogeneity concerns play a role when examining the relation between board size and stock return volatility. To the best of our knowledge, no generally accepted fully exogenous approach, such as a natural experiment, exists for board size studies. Nakano and Nguyen (2012), who focus on Japanese firms, use the percentage of a firm’s free float as an instrument for board size. However, when we do the same with our U.S. sample, the first-stage test statistics reveal free float to be a weak instrument. Besides including industry fixed effects in our models to control for unobserved heterogeneity, we rely on two approaches that have been used in the literature to mitigate these concerns.

3.4.1 Two-stage least squares estimation

First, we conduct two-stage least squares (2SLS) estimations that address both simultaneity and omitted variable concerns. We follow a number of recent studies that use industry-level measures of their potentially endogenous variable of interest as the instrument.¹⁵ A firm’s competitive environment, such as levels of information asymmetry, will likely influence its choice of board structure, including the size of the board (e.g., Boone et al., 2007; Lehn et al., 2009). Therefore, it stands to reason that the median industry-level board size will be highly correlated with the firm’s board size. At the same time, it is highly unlikely that the industry median board structure will affect any particular firm’sidiosyncratic volatility. In other words, industry board size should only affect firm-level idiosyncratic volatility through its effect on the firm’s board size, which satisfies the exclusion restriction.

We begin by identifying the industry median of board size for each year and use it as the instrument for the firm-level board size in the first-stage regression. For the estimations by

15See, for example, Lin et al. (2011), Lin et al. (2011), Kim et al. (2014), Yang and Zhao (2014), Liu et al. (2015), and Tran and Turkiela (2020).

firm type, the industry median board size is also interacted with our firm type dummies in the first stage to provide instruments for the respective interaction terms in the second stage. The predicted values are then used in the second stage to re-estimate our main models. Likewise, we also conduct 2SLS estimations for the board category models. In those cases, we instrument each board category by their industry mean of the particular year. Since our categories are defined as dummy variables, the instruments represent the fraction of firms in the same industry that have boards in the respective category.

Panel A of Table 3.7 illustrates two examples of the first-stage results for the cases without firm type differentiation and for high growth firms. The model for complex firms is estimated accordingly. The highly significant first-stage F-statistics of the excluded instruments and the highly significant coefficient estimates of the industry median (mean) board size variables and interaction terms indicate that we do not face a weak instrument problem. This also holds for all other untabulated first-stage regressions and provides support for the relevance of our instrument choice.

Ln(Industry Board Size)×High Growth 0.0550 0.6670***

(0.4269) (0.0000)

F-Test of Excl. Instruments (p-value) 0.0000 0.0000 0.0000 Panel B: Board Size Counter (Second Stage)

All Complex High Growth

Instr. Ln(Board Size)×Firm Type 8.3215 -27.1374***

(0.1827) (0.0002)

Firm Type -16.9169 60.6003***

(Continued)

Table 3.7 Continued

Panel C: Board Size Categories (Second Stage)

All Complex High Growth

Instr. Medium Board×Firm Type 3.4385 -23.0713***

(0.2565) (0.0000)

Instr. Large Board -20.4158*** -22.8937*** -18.9520***

(0.0000) (0.0000) (0.0000)

Instr. Large Board×Firm Type 3.8791 9.1848*

(0.3944) (0.0771)

This table presents the regression results for the 2SLS estimations using the industry median (mean) as instruments for our firm-level board size counter (category) variables. Panel A presents exemplary first-stage results across all firms and for high growth firms in which board size is measured as a counter variable. Panel B illustrates the second-stage results for the models including the board size counter variable and Panel C for those measuring board size in categories. The second-stage models are the same as in previous analyses, except the board variables and the interaction terms are based on their predicted values. The (omitted) controls are the same as in previous analyses. Variable definitions can be found in Table B.1 in Appendix B. All models include industry and year fixed effects, as well as a constant term. Thep-values are based on standard errors clustered at the firm-level and are reported in parentheses. *, **, and *** indicate significance levels of 10%, 5%, and 1%, respectively.

Panel B and C of Table 3.7 contain the results based on the board size counter and categorical variables, respectively. The results clearly support our earlier findings in that board size negatively affects volatility and that this effect is especially strong in high-growth firms, where medium boards lead to a significantly stronger reduction in volatility than in low-growth firms, while large boards do not. In an unreported analysis, we also conduct 2SLS regressions similar to the ones presented by Wang (2012), in which we use the second and third lags of our board variables as instruments. The results are largely the same. However, it is hard to argue that the lagged values of board size really meet the exclusion condition.

3.4.2 Dynamic panel GMM estimation

Second, we follow the methodology suggested by Wintoki et al. (2012) and re-estimate our models using the dynamic panel GMM estimator developed by Arellano and Bond (1991), Arellano and

Bover (1995), and Blundell and Bond (1998) that addresses both unobserved heterogeneity and simultaneity. This approach considers that the relation between board size and volatility could be dynamic, that is, board size could not only affect volatility but could also be a consequence of past volatility. The estimation procedure comprises a “stacked” system of equations that uses past values of the endogenous variables as instruments.

Similar to Wintoki et al. (2012), we address concerns with respect to the serial correlation of the transient errors and only include every other year in this analysis. We also assume all independent variables to be endogenous, except Firm Age and the fixed effects. We then augment our models by including the first lagged value of stock return volatility as an additional independent variable. Second and third lagged values of the explanatory variables—including the board size variables—are then utilized as instruments in the estimation procedure.

To verify the validity of the approach, we provide the standard test statistics. Throughout the models presented in Table 3.8, the null hypothesis of the Hansen test of overidentification cannot be rejected at the ten percent level, which supports the validity of the instruments. Furthermore, the test statistics show that second-order serial correlation is not an issue.

The results presented in Table 3.8 reveal similar patterns as our baseline analyses. Across all firms, board size is still negatively associated with volatility. While complex firms do not differ in this regard from non-complex ones, the effect is again particularly strong in high-growth firms.

In Model (3) of Panel A, the interaction term fully absorbs the negative effect. The fact that the coefficient on the non-interacted board size becomes insignificant in this model is one notable difference to earlier analyses. Furthermore, the results in Panel B provide additional support for our conjecture that in high-growth firms the reduction in volatility sets in at comparatively smaller board sizes already. Surprisingly, and different to the results in our main analysis, we find a positive coefficient on the interaction term between complex firms and medium boards. Overall, the tests in this section mitigate endogeneity concerns and corroborate our earlier findings.