This paper tests parametric continuous-time models of the spot interest rate by comparing their implied parametric density to a nonparametric density estimate. The key finding is that the strong nonlinearity of the drift function is the main source of model rejection. Around the mean, the spot rate behaves like a random walk, but mean-reverts strongly when far from the mean. The volatility is higher when the rate is far from the mean. The paper develops a methodology to test the specification of the spot rate process without relying on derivative data. The test statistic is based on comparing the parametric and nonparametric density estimates. The results show that the drift function is essentially zero in the middle region (4-17%) but pulls the rate back toward this region when it moves away. The diffusion function is lower in the middle and higher at the extremes. The paper also shows that the CEV diffusion specification is not a perfect match for linear drifts, as it imposes a uniform increase in volatility, which is not supported by the data. The results suggest that a nonlinear mean-reverting drift function provides a better fit to the data. The paper concludes that the spot rate process is globally stationary, even though it is locally nonstationary on most of its support. The findings have important implications for the modeling of interest rates and the pricing of financial derivatives.This paper tests parametric continuous-time models of the spot interest rate by comparing their implied parametric density to a nonparametric density estimate. The key finding is that the strong nonlinearity of the drift function is the main source of model rejection. Around the mean, the spot rate behaves like a random walk, but mean-reverts strongly when far from the mean. The volatility is higher when the rate is far from the mean. The paper develops a methodology to test the specification of the spot rate process without relying on derivative data. The test statistic is based on comparing the parametric and nonparametric density estimates. The results show that the drift function is essentially zero in the middle region (4-17%) but pulls the rate back toward this region when it moves away. The diffusion function is lower in the middle and higher at the extremes. The paper also shows that the CEV diffusion specification is not a perfect match for linear drifts, as it imposes a uniform increase in volatility, which is not supported by the data. The results suggest that a nonlinear mean-reverting drift function provides a better fit to the data. The paper concludes that the spot rate process is globally stationary, even though it is locally nonstationary on most of its support. The findings have important implications for the modeling of interest rates and the pricing of financial derivatives.