Estimation of SEM models is different from that of multiple regressions. Instead of minimizing the discrepancies between the fitted and observed values of the response variable [i.e., Σ()], SEM estimation procedures minimize the residuals that are differences between the sample variances/covariances and the variances/covariances estimated from the model.
Let use to denote the population covariance matrix of observed variables y and x; can be expressed as a function of free parameters in a hypothesized model (Appendix 1.A). The basic hypothesis in SEM is:
where is the model implied variance/covariance matrix; that is, the variance/covariance matrix implied by the population parameters for the hypothesized model. The purpose of model estimation or model fit is to find a set of model parameters to produce so that [ − ] can be minimized. The discrepancy between ...