Some Modifications to Calculate Regression Coefficients in Multiple Linear Regression

In a multiple linear regression model, there are instances where one has to update the regression parameters. In such models as new data become available, by adding one row to the design matrix, the least-squares estimates for the parameters must be updated to reflect the impact of the new data. We will modify two existing methods of calculating regression coefficients in multiple linear regression to make the computations more efficient. By resorting to an initial solution, we first employ the Sherman-Morrison formula to update the inverse of the transpose of the design matrix multiplied by the design matrix. We then modify the calculation of the product of the transpose of design matrix and the design matrix by the Cholesky decomposition method to solve the system. Finally, we compare these two modifications by several appropriate examples.