Abstract: Compound events pose significant threats to society and ecosystems, making their analysis crucial under climate change. Global climate models, the primary tools for studying future climates, require downscaling to bridge their coarse resolution to local scales. This study evaluates the performance of the two main downscaling approaches -statistical and dynamical- in reproducing compound dry-hot events (co-occurring high temperatures and low precipitation), as represented by the standardised dry and hot index (SDHI). We compare three statistical downscaling (SD) methods-generalised linear models, a posteriori random forests, and convolutional neural networks-against three EURO-CORDEX regional climate models (RCMs), over mainland Spain and the Balearic Islands. Although all the models considered in this work (both statistical and dynamical) provide good results for downscaling precipitation and temperature and are capable of capturing standard multivariate metrics (such as the Spearman correlation between both variables), their performance declines when it comes to the reproduction of compound extremes like dry-hot events. For this particular aspect, neither of the two approaches (statistical and dynamical) consistently outperforms the other. In particular, while SD methods outperform RCMs in reproducing the observed temporal variability of compound dry-hot events, RCMs are better at simulating these events' intensity, likely due to their foundation in physical processes, which enhances inter-variable consistency. Based on the different limitations of both statistical and dynamical models found for properly capturing the tails (dry and hot) of the multivariate distribution, we conclude that more advanced model development is needed for accurate analysis of compound events at the local scales needed for most practical applications.

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