How do women on corporate boards shape corporate social performance? Evidence drawn from semiparametric regression

Abstract

This study re-examines the relationship between women on corporate boards (WOCB) and corporate social performance (CSP) for a sample of companies from the Fortune 1000 ranking over the period 2004 to 2018 (ranked from 501 to 1000). To take into account the complex and non-linear relationship as well as endogeneity issues, we use a two-stage generalised additive model. This contribution is significant, as many authors have demonstrated the non-linearity of factors influencing performance, whether of a financial or social nature. Consistent with token and critical mass theories, our results shows that the effects of WOCB on CSP vary significantly depending on their numerical representation and that there are departures from linearity. Our findings provide explanations for the existing mixed empirical results, which all rely on parametric methods. As such, we suggest the use of semiparametric methods takin into account endogeneity issues to assess WOCB–CSP relationship. This study sheds some new light on that relationship, which remains a controversial issue.

Appendices

Appendix

GAM modeling

GAMs can be viewed as extensions of Generalized Linear Models, or GLMs. The classical linear regression model for a conditionally normally distributed response $y$ assumes that:

• (i) The linear predictor through which $\mu_{i} \equiv E\left( {y_{i} | x_{i} } \right)$ depends on the vector of the observations of the covariates for individual $i$ or $x_{i}$, can be written as $\eta_{i} = x_{i}^{^{\prime}} \beta$ where β represents a vector of unknown regression coefficients;

• (ii) The conditional distribution of the response variable $y_{i}$ given that covariate $x_{i}$ is normally distributed with mean $\mu_{i}$ and variance $\sigma^{2}$;

• (iii) The conditional expected response is equal to the linear predictor, or $\mu_{i} = \eta_{i}$.

GLMs extend (ii) and (iii) to more general families of distributions for $y$ and to more general relations between the expected response and the linear predictor than identity. Specifically, $y_{i}$, given covariate $x_{i}$, may now follow a probability density function as follows:

$$ f\left( {y;\theta ,{\Phi }} \right) = exp\left[ {\frac{y\theta - b\left( \theta \right)}{{a\left( {\Phi } \right)}} + c\left( {y,{\Phi }} \right)} \right] $$

(3)

where $b\left( \theta \right)$, $a\left( \Phi \right)$, and $c\left( \theta \right)$ are arbitrary functions, and, for practical modelling $a\left( \Phi \right)$ is usually set to $\Phi$$\theta$, called the “canonical parameter” of the distribution, and depends on the linear predictor, and $\Phi$ is the dispersion parameter. Eq. (3) describes the exponential family of distributions which includes a number of well-known distributions such as normal, Poisson and Gamma distributions. Finally, the linear predictor and the expected response are now related by a monotonic transformation $g\left( . \right)$, called the link function, i.e. $g\left( {\mu_{i} } \right) = \eta_{i} .$ AMs extend GLMs by allowing the determination of non-linear effects of covariates on the response variable. The linear predictor of a GAM is typically given by:

$$ g\left( {\mu_{i} } \right) = x_{i}^{^{\prime}} \beta + \mathop \sum \limits_{j} s_{j} \left( {z_{ji} } \right) $$

(4)

where β represents the vector of unknown regression coefficients for the covariates acting linearly, and $s_{j} \left( {z_{ji} } \right)$ are unknown smooth functions of the covariates $z_{ji}$. The smooth functions can be functions of a single covariate as well as of interactions between several covariates.

The smooth terms can be represented using regression splines. Specifically, the regression spline of an explanatory variable is made up of a linear combination of known basis functions, $B_{jk} \left( {z_{ji} } \right)$, and unknown regression parameters, $\delta_{jk}$, or:

$$ s_{j} \left( {z_{ji} } \right) = \mathop \sum \limits_{k = 1}^{{q_{j} }} \delta_{jk} B_{jk} \left( {z_{ji} } \right) $$

(5)

where j indicates the smooth term for the jth explanatory variable, $q_{j}$ is the number of basis functions and hence regression parameters used to represent the jh smooth term.

Recall that in order to identify (1), each smooth component is subject to a constraint such as E(s_j(z_j)) = 0. Basis functions have to be chosen in order to come up with an estimate for s_j(z_j). Common choices for representing smooth functions include natural splines and smoothing splines (Wahba, 1990). The problem with natural splines is that a spline basis can be constructed only if using knots at fixed locations throughout the range of the data. In particular, the choice of knot locations introduces some subjectivity into the model fitting process which may result in a substantial effect on the resulting smooth effect. Smoothing splines circumvent this problem by placing knots at every data point and are indeed sometimes referred to as full rank smoothers because the size of the spline basis is equal to the number of observations. However, such smoothers have as many unknown parameters as there are data and hence the difficulty is computational cost. Consequently, Wood (2003) proposes using thin plate regression splines, which are low rank smoothers, since they well approximate the behavior of a full rank thin plate spline, avoid having to choose knot locations and are reasonably computationally efficient.

GAMs are estimated using penalized maximum likelihood, typically iteratively reweighted least squares (Wood, 2006). After the basis for the function s_j(z_j) is chosen, the GAM reduces to a GLM, which makes it possible to conduct standard model building and diagnostic procedures. Model fit is estimated using either generalized cross-validation (GCV) based on the prediction mean square error, or Akaike's information criterion (AIC). Confidence intervals for parameter estimates are calculated using the posterior distribution of the model coefficients. Different models can be compared using an approximation of the likelihood ratio test for nested models.

Once the model has been estimated, it is interesting to analyze the significance of the different elements it comprises. For the parametric part of the model, this analysis is based on the usual asymptotic properties of maximum likelihood estimators. Assessing the significance of a parameter can be done using classical Student t-statistics. In turn, assessing the significance of a smooth term s_j(z_ji) proceeds differently. Firstly, the linearity of the function can be addressed as follows. Spline estimators can be shown to belong to the family of linear estimators of s_j(z_ji), i.e. estimators which can be expressed as: $\widehat{{s_{j} }}\left( {z_{ij} } \right) = A_{i} y$. The trace of A_j represents the estimated degrees of freedom (edf) of the fitted function, which is also known as the number of parameters in the function (Wood, 2006). The edf of the model is given by the sum of the degrees of freedom of the single smooth functions. Therefore, edf can indicate either the complexity of the model or that of a single smooth term. For example, if the edf of a smooth estimate is equal to 1, this means that the explanatory variable enters the model linearly.

Testing the joint nullity of the parameters δ_jk, k = 1, …, q_k, involved in the spline expansion of the smooth function s_j(z_ji), or Eq. (5), can be performed using an F-test, the degrees of freedom of which are r, the rank of the covariance matrix of estimated $\hat{\delta }_{jk} , k = 1, \ldots , q_{k} ,$ and the number of observations minus the edf corresponding to the fitted smooth function.