The paper presents new evidence of systematic errors in real-time estimates of long-run output growth rates and, importantly, reveals a negative, nonlinear relationship between these estimates and sovereign debt spreads during the Eurozone debt crisis of the 2010s. To study the implications of these beliefs, we develop a sovereign default model in which agents infer trend growth from aggregate output and from noisy signals about the trend. The model reproduces the pattern of errors in the trend growth estimates and their negative and nonlinear relationship with spreads, unlike a comparable full-information model. Overoptimism about trend growth during booms encourages excessive borrowing, leading to persistently elevated spreads thereafter.