This paper characterizes, interprets, and tests the over-identifying restrictions imposed in affine term structure models. The authors show that affine models can be categorized based on the different over-identifying restrictions they impose on the drift parameter δ and the parameters of the diffusion matrices. They demonstrate that these models are equivalent to models with a terraced drift structure, where one of the state variables is the stochastic long-run mean of the short rate. The paper also computes simulated method-of-moments estimates of a three-factor affine term structure model and tests the over-identifying restrictions on the joint distribution of long- and short-term interest rates implied by existing affine models. The authors find that allowing for correlated factors is key to simultaneously describing the short and long ends of the yield curve. This finding is interpreted in terms of the properties of the risk factors underlying term structure movements. The paper also discusses the identification of affine models, showing that the number of primary factors and the structure of the diffusion coefficients play a crucial role in determining the model's ability to describe the term structure. The authors conclude that the specification of affine models must account for the joint distribution of long- and short-term bond yields and that the assumptions about the structure of the diffusion coefficients are essential for the model's validity.This paper characterizes, interprets, and tests the over-identifying restrictions imposed in affine term structure models. The authors show that affine models can be categorized based on the different over-identifying restrictions they impose on the drift parameter δ and the parameters of the diffusion matrices. They demonstrate that these models are equivalent to models with a terraced drift structure, where one of the state variables is the stochastic long-run mean of the short rate. The paper also computes simulated method-of-moments estimates of a three-factor affine term structure model and tests the over-identifying restrictions on the joint distribution of long- and short-term interest rates implied by existing affine models. The authors find that allowing for correlated factors is key to simultaneously describing the short and long ends of the yield curve. This finding is interpreted in terms of the properties of the risk factors underlying term structure movements. The paper also discusses the identification of affine models, showing that the number of primary factors and the structure of the diffusion coefficients play a crucial role in determining the model's ability to describe the term structure. The authors conclude that the specification of affine models must account for the joint distribution of long- and short-term bond yields and that the assumptions about the structure of the diffusion coefficients are essential for the model's validity.