## 2 Materials and Methods
### Discriminant Analysis  
Discriminant analysis (DA) is a statistical technique employed to classify entities into at least two distinct groups based on predictor variables. It achieves this by constructing a discriminant function that maximises group separation. DA has been widely applied in various disciplines, including finance, where it aids in predicting corporate financial health [54]. In assessing corporate financial health, the models created by DA serve to differentiate between financially robust and distressed companies by analysing financial ratios and other pertinent indicators. Mathematically, DA aims to determine a linear combination of independent variables \(X_{1},X_{2},\ldots,X_{p}\), which forms the discriminant score expressed as  
\[D=\beta_{0}+\beta_{1}X_{1}+\beta_{2}X_{2}+\cdots+\beta_{p}X_{p} \tag{3}\]  
where D is the discriminant score, \(p\) is the number of the predictor variables, \(X_{1},X_{2},\ldots,X_{p}\) are the predictor variables, and \(\beta_{0},\beta_{1},\ldots,\beta_{p}\) are the coefficients to be estimated; their estimations are often denoted by \(\mathrm{b}_{0},\mathrm{b}_{1},\ldots,\mathrm{b}_{p}\). The estimation of the coefficients relies most commonly on maximising the ratio of between-group variance (\(S_{B}\)) to within-group variance (\(S_{W}\)), defined as the objective function  
\[\frac{w^{T}S_{B}w}{w^{T}S_{W}w} \tag{4}\]  
Here, \(w\) represents the vector of discriminant coefficients. The matrices \(S_{B}\) and \(S_{W}\) are computed as:  
\[S_{B}=\sum_{g=1}^{G}n_{g}\big{(}\mu_{g}-\mu\big{)}\big{(}\mu_{g}-\mu\big{)}^{T} \tag{5}\]  
\[S_{W}=\sum_{g=1}^{G}\sum_{i=1}^{n_{g}}\big{(}X_{gi}-\mu_{g}\big{)}\big{(}X_{gi }-\mu_{g}\big{)}^{T} \tag{6}\]  
where \(G\) is the number of groups, \(n_{g}\) is the sample size in group \(g\), \(\mu_{g}\) is the mean vector of group \(g\), \(\mu\) is the overall mean vector, and \(X_{gi}\) represents the observations in group \(g\). The optimal solution for \(w\) is obtained by solving the generalised eigenvalue problem  
\[S_{B}w=\lambda S_{W}w \tag{7}\]  
where \(\lambda\) represents the eigenvalue, indicating the separation achieved by the discriminant function [55; 56]. The discriminant function is validated using metrics such as Wilks' Lambda, which tests the significance of the discriminant power and classification accuracy through cross-validation. Assumptions such as multivariate normality and homogeneity of covariance matrices are essential for the validity of DA. In practice, normality assumption is often violated; for example, financial data often exhibit non-normality. While deviations from these assumptions may impact the results, DA remains robust under moderate violations [54,56] and DA can still proceed with caution. Despite several limitations of the discriminant analysis and its stringent assumptions and the increasing preference for more advanced classification techniques in financial distress prediction research, we relied on maintaining the original method of model creation for the following reasons:  
* Employing a different, although sophisticated, machine learning technique would require constructing entirely new models instead of re-estimating existing ones. Developing such a new model would allow for the comparison of predictive performance between the models but would render it impossible to compare the re-estimated models with their pre-crisis versions regarding the coefficients of the variables. * Discriminant analysis is frequently selected by researchers due to its interpretability and transparency. It provides a clear insight into the model, enabling the interpretation of each financial ratio's contribution through its estimated coefficients. This level of interpretability is not achievable with advanced machine learning models, which only provide output predictions without exploring the underlying relationships, even though they offer a high prediction accuracy. Thus, they are very suitable for the predictive goal of model development but cannot be recommended for the descriptive goal.