The curse of dimensionality is a challenge that researchers often face when dealing with large Vector Autoregressions (VARs). Different approaches have been proposed in the literature to address this issue. In this paper, we propose a new method based on the idea of compressed regression. In particular, we introduce two novel nonlinear compressed VARs to forecast the occupancy rate of hotels that compete within a narrow geographical area. We make the models more flexible through the introduction of neural networks, and compare their performance against several competing models. The empirical results show that the new compressed VARs outperform all other models, and their accuracy is preserved across nearly all forecast horizons from 1 to 36 months.