Software development effort estimation is one of the most critical aspect in software development process, as the success or failure of the entire project depends on the accuracy of estimations. Researchers are still conducting studies on agile effort estimation. The aim of this research is to develop a story point based agile effort estimation model using LASSO and Elastic Net regression techniques. The experimental work is applied to the agile story point approach using 21 software projects collected from six firms. The two algorithms are trained using their default parameters and tuned grid search with 5-fold cross-validation to get an enhanced model. The experiment result shows LASSO regression achieved better predictive performance PRED (8%) and PRED (25%) results of 100.0, MMRE of 0.0491, MMER of 0.0551, MdMRE of 0.0593, MdMER of 0.063, and MSE of 0.0007. The results are also compared with other related literature.