Deciding which predictors to use plays an integral role in deriving statistical models in a wide range of applications. Motivated by the challenges of predicting events across a telecommunications network, we propose a semi-automated, joint model-fitting and predictor selection procedure for linear regression models. Our approach can model and account for serial correlation in the regression residuals, produces sparse and interpretable models and can be used to jointly select models for a group of related responses. This is achieved through fitting linear models under constraints on the number of non-zero coefficients using a generalisation of a recently developed Mixed Integer Quadratic Optimisation approach. The resultant models from our approach achieve better predictive performance on the motivating telecommunications data than methods currently used by industry.