Structural equation models (SEMs) have been widely used in behavioural, educational, medical and socio-psychological research for exploring and confirming relations among observed and latent variables. In the existing SEMs, the unknown coefficients in the measurement and structural equations are assumed to be constant with respect to time. This assumption does not always hold, as the relation among the observed and latent variables varies with time for some situations. In this paper, we propose nonlinear dynamical structural equation models to cope with these situations, and explore the nonlinear dynamic of the relation between the variables involved. A local maximum likelihood-based estimation procedure is proposed. We investigate a bootstrap resampling-based test for the hypothesis that the coefficient is constant with respect to time, as well as confidence bands for the unknown coefficients. Intensive simulation studies are conducted to show the empirical performance of the proposed estimation procedure, hypothesis test statistic and confidence band. Finally, a real example in relation to the stock market of Hong Kong is presented to demonstrate the proposed methodologies.
