Transcriptome-wide association studies based on genetically predicted gene expression have the potential to identify novel regions associated with various complex traits. It has been shown that incorporating expression quantitative trait loci (eQTLs) corresponding to multiple tissue types can improve power for association studies involving complex etiology. In this article we propose a new multivariate response linear regression model and method for predicting gene expression in multiple tissues simultaneously. Unlike existing methods for multitissue joint eQTL mapping, our approach incorporates tissue-tissue expression correlation which allows us to more efficiently handle missing expression measurements and to more accurately predict gene expression using a weighted summation of eQTL genotypes. We show through simulation studies that our approach performs better than the existing methods in many scenarios. We use our method to estimate eQTL weights for 29 tissues collected by GTEx, and show that our approach significantly improves expression prediction accuracy compared to competitors. Using our eQTL weights, we perform a multitissue-based S-MultiXcan (PLoS Genet. 15 (2019) e1007889) transcriptome-wide association study and show that our method leads to more discoveries in novel regions and more discoveries overall than the existing methods. Estimated eQTL weights and code for implementing the method are available for download online at github.com/ajmolstad/MTeQTLResults.