Functional approximation and estimation for latent variable systems

Summary Many models have been suggested in statistics and in econometrics for describing the behavior of latent random financial variables suspected to be behind volatility. Despite all the research done, this area of study remains an excellent ground for experiments on the design and analysis of stochastic systems aimed at modellingnon-linear and non-stationary processes. While it is a common strategy to rely on assumptions about the probability distributions and functions involved in representing volatility dynamics, we contribute to a shift of the emphasis toward a more generally model-free perspective, also seen as non-parametric in statistical inference terms. In pursuing this direction, we attempt to address the use of methodologies so far not exploited very much in this context. Thus, our focus is on functional approximation and semi-parametric estimation, by means of methods such as adaptive wavelet basis representations and related computational sytems. In our applications, the function dictionaries described and the algorithms implemented help in identifying and estimating the behavior of the Nikkei financial stock index, thus improving the feature detection power. From the methodological standpoint side, signal decomposition techniques such as independent component and multire solution analyses are shown to be effective for adaptive learning when used in a combined fashion.