Replacing Iterative Constrained Likelihood in Hybrid Structural Models with a Spectrally Initialized One-Step Estimator

Iterative constrained likelihood estimation can become computationally expensive in large structured models, as it requires repeated evaluations of the objective, gradient, and curvature matrix. We propose an alternative strategy that combines a computationally inexpensive spectral initializer with a single constrained Fisher-scoring correction. We specialize this strategy to hybrid factor and composite structural equation models, using a spectral estimator as an exactly feasible and root-$n$ consistent initializer. Under standard smoothness and local identification conditions, the resulting estimator is asymptotically equivalent to restricted maximum likelihood and therefore attains the same first-order efficiency. A final blockwise retraction restores the nonlinear composite normalization constraints exactly without affecting the first-order asymptotic distribution. Numerical experiments support the predicted first-order equivalence and show that the one-step estimator achieves nearly identical statistical accuracy to fully iterated restricted maximum likelihood while substantially reducing computation time, with the computational advantage becoming increasingly relevant as model dimension grows.