Identity test for regression models

Abstract

It is considered in this paper the adjustment of H regression equations in the case of juxtaposition of r polynomial submodels of k degree. The points of intersection of the submodels are supposed to be known. Appropriate restrictions are imposed in such a way that the polynomial submodels are concordant in the points of intersection. The linear model for the h^th equation is Y_h = X_h β_h+ε_h , h = 1,2,..., H, where Y_h is an n_h × 1 vector of observations, X_h is an n_h × p matrix of known constants, β_h is an p×1 vector of unknown parameters and ε_h is an n_h×1 vector of errors that is distributed NID (ε_h:ϕ, σ² I). In the parameters estimation, the Least Square Method was used. A statistic test was derived for the hypothesis that H regression models in the case of juxtapositon of r polynomial submodels of k degree were identical. The hypothesis in consideration is: H₀:β₁ = β₂ =…= β_H (H models are identical) vs. H_α:β_i ≠ β_j for at least one i ≠ j (the H models are not all identical). This method is applied to a set of H = two regression equations in the case of juxtaposition of r two polynomial submodels of first degree.