*4.4. Final Structural Model Estimation and Testing*

By developing this causal model, we aimed to demonstrate that universities/institutes of higher education need to invest in and select recognized brands for developing relationships, as well as manage the corporate brand identity in the part that is more exposed to interaction with the public.

In the proposed model, the brand relationships construct was an antecedent of the corporate brand identity construct (external part), and the brand identity (external part) was an antecedent of the brand reputation construct. Corporate brand identity (external part) and reputation were latent variables. Consistent with Hair et al. (2006), Marôco (2010), and James et al. (1982), we added a parsimony fit index (PCFI) to the analysis. We selected PCFI because it represents the result of applying James et al. (1982) parsimony adjustment to the CFI:

$$\text{PCFI} = \text{CFI} \times d/db$$

where *d* is the degree of freedom for the model being evaluated, and *db* is the degree of freedom for the baseline model. Values are between [0–1], and better fits are closer to 1. Table 8 summarizes the indices of fit of the structural model:

**Table 8.** Summary of the indices of fit of the measurement model.


As expected, the χ<sup>2</sup> was higher than the one calculated with the measurement model, because a recursive structural model cannot fit better (to have a lower χ2) than the overall CFA. The difference between both <sup>χ</sup><sup>2</sup> was quite small (727.239 <sup>−</sup> 726.149 <sup>=</sup> 1.09), demonstrating that the model was strongly suggestive of adequate fit (Hair et al. 2006). The loadings, standardized residuals, and modification indices maintained approximately the same values. Regarding the standardized residuals: 2.704 between F4 and Rep2.2; 2.805 between R5 and C3; and 2.787 between R5 and C2.

The problematic items relating to the modification indices are:

$$- \text{R5 and C3} = 11.752$$

These small differences did not require further analysis, because, at this stage, the focus was on diagnosing the relationships among constructs. A good model fit alone is insufficient to support a structural theory. It is also necessary to examine the individual parameter estimates that represent each specific hypothesis (Hair et al. 2006). Table 9 summarizes the main indicators and conclusions.


**Table 9.** Structural equation model results.

Notes: S.E.—standard error; CR—Critical ratio.

Examining the paths among constructs showed that they were all statistically significant in the predicted direction. The path that represented the weight between brand relationships and external corporate brand identity was characterized by βBR.ECBI = 0.652; S.E. = 0.135; βBR.ECBI = 0.876; *p* < 0.001. This means that the regression weight for brand relationships in the prediction of external corporate brand identity was significantly different from zero at the 0.001 level (two-tailed). The path that represented the weight between external corporate brand identity and reputation was characterized by βECBI.Rep = 1.302; S.E. = 0.260; βECBI.Rep = 0.824; *p* < 0.001, meaning that the regression weight for external corporate brand identity in the prediction of reputation was significantly different from zero at the 0.001 level (two-tailed).

We analyzed the variance explained estimates for the endogenous constructs in Table 10 and found that the predictors of the physical construct explained 17.7 percent of variance. This means that the error variance of the physical dimension was approximately 82.3 percent of the variance of this dimension itself. As for the other constructs, no problems were found. We can conclude that our model supported both Hypotheses 2 and 3. Therefore, the relationships among brands (brand relationships) influenced external corporate brand identity, and later, the brand reputation.


**Table 10.** Squared correlations (R2).

Because theory has become essential in assessing the validity of a structural model, we examined an equivalent model, with the purpose of testing an alternative theory. For the previous model, we dropped the physical dimension, for comparison purposes. In line with these findings, we accepted the second and third hypotheses and concluded that the brand relationships construct influences the external part of corporate brand identity (H2) and that the brand identity influences brand reputation (H3). Therefore, the management of corporate brand identity depends on the investment and selection of strong relationships with reputed brands, to attract students and increase brand reputation.
