A canonical correlation analysis of the relationship between sustainability and competitiveness

Abstract

In the present work, we apply the canonical correlation analysis (CCA) to analyze the relationship between competitiveness and environmental sustainability, studying their degree of correlation and the impact of the environmental sustainability and competitiveness indicators in this relationship. We selected seven indicators from the environmental sustainability indicators of the Environmental Performance Index, and 12 competitiveness indicators from The Global Competitiveness Index. The CCA considers 117 countries common in both databases. The method is applied to study of the intracorrelations (correlations, between the same set of indicators) and the intercorrelations (correlations between the sustainability and competitiveness sets of indicators). It is shown that the intracorrelations of the environmental and competitiveness indicators are highly correlated, indicating that the choice of the indicators adequately describes sustainability and competitiveness, separately. For the study of the intercorrelation between the sustainability and the competitiveness sets of indicators, the CCA method reduced the complex multidimensional problem to a one-dimensional situation described by one relevant canonical function. This greatly simplified the interpretation of the interrelation between competitiveness and sustainability indicators allowing a proper analysis of the different degrees of correlation between these dimensions. A discussion on the degree of correlation between these indicators is presented. The results associating competitiveness and sustainability indicators through strong intercorrelation are an indication that the conclusion of the World Economic Forum analysis “the main and very important finding is that there is no necessary trade-off between being competitive and being sustainable. On the contrary, many countries at the top of the competitiveness rankings are also the best performers in many areas of sustainability,” may be related to the specific choice of indicators. The present study suggests that the strongly correlated indicators are good candidates to be part of an adequate set of composite sustainability and competitiveness indicators. Although in this work CCA is applied to study the environmental sustainability, the method can be applied to investigate other metrics in particular the intersection of social and competitiveness toward the achievement of sustainable development.

Appendices

Appendix 1: Key terms used in this work

The following key terms are used in this paper (Hair et al. 2005; Sherry and Henson 2005):

Canonical correlation (R _c ) is the Person r relationship between the two variates in a given function and it ranges from 0 to 1. It measures the degree of the relations between the two variates. It squared value ($R_{\text{C}}^{2}$) represents the proportion of variance shared by the two variates from a given function and are also known as eigenvalues.

Canonical function relationship between two linear compositions (statistical canonical variables). Each canonical function is orthogonal to the other functions, meaning that they are statistically uncorrelated. Each canonical function has two variates (see Fig. 1), one for the set of dependent variables and other for the set of independent variables. The degree of the correlation is given by the canonical correlation coefficient (R _c). The number of function for a given canonical solution is limited to the number of the variables of the smaller set.

Standardized canonical function coefficients (Coef) are the standardized canonical coefficients (weights) used in the linear equations to combine the observed variables (predictors, independent variables) and criterion variables (dependent variables) into two respective variates. The weights are applied to the observed scores in z-scores form (standardized) to yield the structure coefficients which correlate the variables with variates.

Structure coefficients or canonical loadings (r _s) measure of bivariate correlation between an observed variable and a variate. It is the Pearson r between an observed variable (indicator in this study) a variate from a given canonical function. It may range from −1 to +1. The square value of this parameter ($r_{\text{s}}^{2}$) indicates the proportion of variance that an observed variable linearly shares with the variates.

Canonical cross-loadings (Λ) correlation of each dependent or independent observed variable with the opposite canonical statistical variable.

Redundancy index ($R_{\text{d}}$) is the amount of variance of a statistical variable (variate) explained by the other statistical variable in a given canonical function. It can be computed for the dependent statistical variables as well as for the independent statistical variable from a given canonical function. A redundancy index for the dependent variables represents the amount of variance from dependent variables explained by the independent statistical variable.

Appendix 2: Analysis of redundancy and cross loading computation

The analysis of redundancy evaluates practical significance and the amount of variance of one canonical variate (dependent or independent) by the other canonical variate. The canonical cross loading determines the relative contribution of each original variable in the canonical relationship. The cross loading method provides a direct measure of the variables relationship, since it involves directly correlating each variable with the other variates, and vice versa (Hair et al. 2005). In fact, for the above reasons cross-loadings is the most recommended among the three methods used to for interpret canonical variates: (a) standardized coefficients (canonical weights), (b) canonical loadings (structure correlations), and (c) canonical cross-loadings.

The redundancy index can be expressed as (Hair et al. 2005; Rencher 2002)

$$R_{\text{dxn}} = R_{\text{cn}}^{2} \times \bar{r}_{\text{syn}}^{2},$$

(1)

$$R_{\text{dyn}} = R_{\text{cn}}^{2} \times \bar{r}_{\text{sxn}}^{2},$$

(2)

where $R_{\text{dxn}}$ = redundancy index of independent variables of the function n, $R_{\text{dyn}}$ = redundancy index of dependent variables of the function n, $R_{\text{cn}}$ = canonical correlation coefficient of the canonical function n, $\bar{r}_{\text{sxn}}^{2}$ = mean of squared loadings of independent variables of the canonical function n, $\bar{r}_{\text{syn}}^{2}$ = mean of squared loadings of dependent variables of the canonical function n.

The cross-loadings can be computed as (Hair et al. 2005):

$$\frac{{\varLambda_{\text{xni}} }}{y} = R_{\text{cn}} \times r_{\text{sxni}},$$

(3)

$$\frac{{\varLambda_{\text{ynj}} }}{x} = R_{\text{cn}} \times r_{\text{synj}},$$

(4)

where $\frac{{\varLambda_{\text{xni}} }}{y}$ = cross loading of independent variable i in the variate X to the variate Y of function n, $\frac{{\varLambda_{\text{ynj}} }}{x}$ = cross loading of independent variable j in the variate Y to the variate X of function n, $R_{\text{cn}}$ = canonical correlation coefficient of the canonical function n, $r_{\text{sxni}}$ = loading of the independent variable i in the variate X of function n, $r_{\text{synj}}$ = loading of the independent variable j in the variate Y of function n.