Defect prediction models---classifiers that identify defect-prone software modules---have configurable parameters that control their characteristics (e.g., the number of trees in a random forest). Recent studies show that these classifiers underperform when default settings are used. In this paper, we study the impact of automated parameter optimization on defect prediction models. Through a case study of 18 datasets, we find that automated parameter optimization: (1) improves AUC performance by up to 40 percentage points; (2) yields classifiers that are at least as stable as those trained using default settings; (3) substantially shifts the importance ranking of variables, with as few as 28% of the top-ranked variables in optimized classifiers also being top-ranked in non-optimized classifiers; (4) yields optimized settings for 17 of the 20 most sensitive parameters that transfer among datasets without a statistically significant drop in performance; and (5) adds less than 30 minutes of additional computation to 12 of the 26 studied classification techniques. While widely-used classification techniques like random forest and support vector machines are not optimization-sensitive, traditionally overlooked techniques like C5.0 and neural networks can actually outperform widely-used techniques after optimization is applied. This highlights the importance of exploring the parameter space when using parameter-sensitive classification techniques.