Using the Gaussian copula modelling device from a perspective of latent variables, we consider the parsimonious modelling of generic types of data with the temporal dependence including those sequential observations being binary, categorical, and more. Different from the existing approaches, we directly investigate the covariance modelling in which a new kind of constraint is naturally developed to enable the identifiability of the latent Gaussian copula model. Then we develop a new latent variable framework concerning parsimoniously modelling the marginal means, variances, and covariances of the latent Gaussian variables with appealing practical interpretations. Our new framework broadly extends the joint mean-covariance modelling approaches based on the modified Cholesky decompositions to the regression analysis with broad types of response variables. Theoretical properties are then established for the resulting estimators of the parsimonious model. We demonstrate the performance of the proposed approaches with simulations and the real data analysis.