**4.3 Methodology and Model**

Imposing the linear econometric assumption of innovation effect on the CSR might not be accurate especially with the dynamism of the CSR conception. Aiming to define a pragmatic shape, we use in this study a semi-parametric model. Through


**Table 1.**

*The dependent and independent variables description.*


**Table 2.**

*The controls description.*

this method, we relax constraints for the innovation effect while maintaining the linearity assumption for the controls. Hence, our model is as follows:

$$\text{CSR}\_{i,t} = \alpha + f\left(i \text{in} \textit{row} \, t\_{i,t}\right) + \beta\_k \textit{controls}\_{i,t} + \varepsilon\_{i,t} \tag{1}$$

CSR refers to the CSR variables (ESG, ENV, SOC, GOV) defined previously. Innovation is measured by the Ln\_PA + 1 and controls matrix includes all the controls variables presented in **Table 2**. Finally, *i t*, ε is the estimation error term. This model has two main parts. The first is α and βj controls i,t which presents the parametric linear part while the second is *f innovation* ( ) namely a function of the CSR-innovation link. This non-parametric presentation can be estimated using several smoothing methods. In our investigation, we use the penalized splines. According to Keele [95], the splines methods provide the best mean squared error fit. Besides, the smoothing splines are designed to avert the overfitting. The splines estimation does not pre-specify ad hoc cut-off points. Hence, it minimizes the objective function to have the most pragmatic estimations.

$$\min \left\{ \frac{1}{n} \sum\_{i=1}^{n} \left( \text{CSR} - f \left( i \text{innovation} \right) - \alpha - \beta\_k \text{controls} \right)^2 + \lambda f \right\} \tag{2}$$

The n index refers to the number of observations and J presents the roughness of the objective function. This function optimum depends on the minimization of residuals squared and the maximum possible smoothing of the innovation function. The ë term is the key to this tradeoff. There are diver types of splines smoothing such as the Quadratic, the Cubic or the Natural splines. In our study, we use the penalized splines since it has fewer parameters and empirically leads to similar results. Hence, the minimization equation presented in Eq. (2) become.

( ( ) ) ( ) ( ) <sup>2</sup> 1 <sup>1</sup> min *n j i CSR f innovation controls f innovation d innovation <sup>n</sup>* αβ λ = <sup>−</sup> −− +∫ ′ <sup>∑</sup> ′ (3)

The *f* ′′ is the second derivative of the function *f* . Thus, the roughness of the innovation function is captured by this new expression. Finally, we compare the explanatory power of our semi-parametric regressions with the linear regression

using the likelihood ratio test (LR test) since our estimations apply penalized iteratively reweighted least squares.

$$LR = -2\left(\log likelihood\_{\text{restrated}} - \log likelihood\_{\text{unrestrated}}\right) \tag{4}$$

Follows an approximate χ2 distribution, the null hypothesis of this test supposes the equality between likelihoods. The degree of freedom is determined through the difference between the numbers of parameters of each model. Put differently, if the semi-parametric regression has a higher number of parameters then the linear regression is not appropriate.
