Algorithm for Identifying Estimates of Parameters in Two-Factor Non-Elementary Linear Regressions using Ordinary Least Squares

Abstract

This article is devoted to the problem of estimating the parameters in two-factor non-elementary linear regressions using ordinary least squares. Such models are constructed using the binary min operation. Previously, to estimate non-elementary linear regression, an algorithm was developed that allowed us to obtain a «good» solution, but did not guarantee the optimality of the sum of the squared model errors. In this article, we mathematically prove in which area it is necessary to search for the optimal value of the loss function. A three-step algorithm for estimating non-elementary linear regressions has been developed. At its first step, the values of the loss function are determined at points where it is not differentiable. In the second step, the points are ordered in ascending order, after which a local minimum of the loss function is searched for in each resulting interval. In the third step, the global minimum of the loss function is selected. The developed algorithm guarantees optimal estimates of the parameters of non-elementary linear regressions based on the sum of squared errors. Computational experiments were conducted on two randomly generated samples. In the first case, the loss function has one local minimum, in the second – two. In both cases, the new algorithm provided a global minimum of the loss function, which is why the approximation quality of the two constructed non-elementary linear regressions turned out to be slightly higher than the quality of the models using the known algorithm. The algorithm developed in the article forms the foundation for constructing more complex structural specifications of regression models.