  We consider structural equation modeling (SEM) with latent variables for
diffusion processes based on high-frequency data. The quasi-likelihood
estimators for parameters in the SEM are proposed. The goodness-of-fit test is
derived from the quasi-likelihood ratio. We also treat sparse estimation in the
SEM. The goodness-of-fit test for the sparse estimation in the SEM is
developed. Furthermore, the asymptotic properties of our proposed estimators
are examined.
