Multiple linear regression with constrained coefficients : application of the Lagrange multiplier

In this paper, we present two unfamiliar novel estimation techniques (UNET) for the constrained regression coefficients in the frame-work of a standard multiple linear regression model. Estimation of a linear regression problem with constraints on the regression coefficients are firstly derived by minimising a formulated goal function that minimises the total sum of the squared errors, plus the sum of the linear constraints multiplied by a Lagrangian. We also show that the solution to the system of equations can be obtained without differentiating the goal function, rather expressed interms of the known matrices. This is achieved by employing properties of a blocked linear system. The UNET is justified by a numerical simulated system of linear equations in 3-dimensions. The UNET yields estimates that are comparable to those generated by the Schur complement principle.