Structural Equation Models in a Redundancy Analysis Framework With Covariates

Abstract

A recent method to specify and fit structural equation modeling in the Redundancy Analysis framework based on so-called Extended Redundancy Analysis (ERA) has been proposed in the literature. In this approach, the relationships between the observed exogenous variables and the observed endogenous variables are moderated by the presence of unobservable composites, estimated as linear combinations of exogenous variables. However, in the presence of direct effects linking exogenous and endogenous variables, or concomitant indicators, the composite scores are estimated by ignoring the presence of the specified direct effects.

To fit structural equation models, we propose a new specification and estimation method, called Generalized Redundancy Analysis (GRA), allowing us to specify and fit a variety of relationships among composites, endogenous variables, and external covariates. The proposed methodology extends the ERA method, using a more suitable specification and estimation algorithm, by allowing for covariates that affect endogenous indicators indirectly through the composites and/or directly. To illustrate the advantages of GRA over ERA we propose a simulation study of small samples. Moreover, we propose an application aimed at estimating the impact of formal human capital on the initial earnings of graduates of an Italian university, utilizing a structural model consistent with well-established economic theory.

ACKNOWLEDGMENTS

We thank the anonymous referees for providing us with constructive comments and suggestions.

Notes

Looking at Table 1, we observe for GRA a remarkable decrease of standard errors when the sample size increases from 200 to 400. Trying to explain this phenomenon, this may depend on the features of the GRA algorithm. Specifically, for sufficiently large samples, simulation results (also extended to the case of n = 800) exhibit an inverse relationship between ratios of congruence values (mean bias) and ratios of congruence's variability (across simulations) among couples of models of sizes n and 2n. In other words, the rate of reduction of standard errors (at 2n with respect to n) increases when the rate of reduction of biases increases. In the simulation study, the peak of this bias reduction is observed passing from n = 200 to n = 400; here the variability of the congruence coefficients shows the highest reduction. Decomposing such overall effect into the components of the congruence coefficient's vector, this bias reduction occurs principally for the components’ weights, whereas the components’ loadings exhibit lower bias reduction (but higher reduction of standard errors); this reflects that GRA, as ALS algorithms in general, may be prone to the problem of local minima, especially for loading parameters. Globally, however, the magnitude of the (mean) congruence coefficient continues to increase as n increases, essentially due to the strength of bias reduction observed for components’ weights.

The SAS IML macro is available upon request from the first author.