Extended Least Squares Making Evident Nonlinear Relationships between Variables: Portfolios of Financial Assets

Abstract

This research work extends the least squares criterion. The regression models which have been treated so far in the literature do not study multilinear relationships between variables. Such relationships are of a nonlinear nature. They take place whenever two or more than two univariate variables are the components of a multiple variable of order 2 or an order greater than 2. A multiple variable of order 2 is not a bivariate variable, and a multiple variable of an order greater than 2 is not a multivariate variable. A multiple variable allows for the construction of a tensor. The α-norm of this tensor gives rise to an aggregate measure of a multilinear nature. In particular, given a multiple variable of order 2, four regression lines can be estimated in the same subset of a two-dimensional linear space over R. How these four regression lines give rise to an aggregate measure of a multilinear nature is shown by this paper. In this research work, such a measure is an estimate concerning the expected return on a portfolio of financial assets. The metric notion of α-product is used to summarize the sampling units which are observed.