Nonparametric estimation of the observed time series trend-cycle.
In this context, we focused on the analysis and development of non-parametric techniques for the trend-cycle estimation based on the Reproducing Kernel Hilbert Space (RKHS) methodology. Following this approach, a common probabilistic representation of different trend-cycle estimators has been derived. In particular, the properties of the estimator based on the Henderson filter, commonly applied in the Census - X11 method and its variants, have been analyzed and, in particular, the study has focused on the properties of the corresponding asymmetric filters applied to the most recent observations, which they are of crucial interest for current economic analysis. These filters are shown to reduce revisions in the final estimates respect to those commonly applied in the literature, and have good properties for rapid detection of true turning points.
Latent growth models for longitudinal data.
The latent growth models, as specified in the context of a Structural Equation Model (SEM) for the study of longitudinal data, have been used in educational studies to analyse student achievements over time of each student in order to determine those factors that most often affect the success of a formative program. A generalization of these models in the wider context of generalized linear models with latent variables (GLLVM) has been proposed for the treatment of multivariate mixed and longitudinal data. Non-linear growth models developed in the context of the SEM have been the subject of study in several publications with particular interest in the analysis of the properties of Autoregressive Latent Trajectory (ALT) models, recently proposed in the literature.
Statistical inference in generalized linear models with latent variables.
Latent variable models are widely applied in all those fields of research in which the variables of interest are not directly observable, and therefore one or more factors are necessary to reduce the complexity of the data. In these contexts, integration problems in the likelihood function computation arise. First, we have studied the properties of different techniques developed in the literature to solve this problem. Then, the research interest has focused on an extension of the Laplace method, known as Fully Exponential Laplace Approximation (FELA), that provides an improvement of the accuracy of the estimates, but without increasing the computational complexity of the estimation respect to the classical Laplace approximation.
We also focused on the adaptive Gaussian quadrature that represents one of the most often applied methods in the current literature. In particular, a theoretical study of the statistical properties of maximum likelihood estimators based on this approximation have been done. In particular, we have formally proved the consistency and of asymptotic normality properties of these estimators.
In order to overcome the limitations in the methods generally applied in the literature (Laplace method and adaptive grid), our recent research has focused on the development of a new technique of integral approximation. This method allows to approximate multidimensional integrals through a sum of integrals of smaller dimension, generally one- or two-dimensional. The approach represents a good compromise compared to the techniques discussed in the literature. In fact, it provides more accurate estimates than the Laplace method, but it is not affected by the computational complexity of the adaptive Gaussian quadrature.
