Variable selection methods have gained enormous attention in statistics, as many fields require reducing prediction error while identifying important variables simultaneously to build accurate and interpretable models. Although variable selection methods have been studied extensively in the literature, these methods generally assume that the error terms of the models are independently and symmetrically distributed. However, these assumptions may not be valid in many real-world problems where errors can exhibit dependency and asymmetry, potentially leading to bias or inefficiency. In this study, we will combine the regression model with dependent error by using penalty-based variable selection methods to perform estimation and variable selection simultaneously when innovations have skewed and/or heavy-tailed skew distributions. Through simulation studies and a real data application, we show that in the presence of outliers and/or skew distributed error terms, our approach significantly improves the precision of variable selection and model performance compared to traditional methods.