This paper by Qiang Dai and Kenneth J. Singleton characterizes, interprets, and tests the over-identifying restrictions imposed in affine term structure models. The authors categorize these models based on the different over-identifying restrictions they impose on the parameters of the drift vector $\delta$ and the diffusion matrices. They show that every affine model can be analytically equivalent to an "Ar" model, where the drift structure is terraced with one of the state variables being the stochastic long-run mean of the short rate $r$. This equivalence allows for direct comparisons between CIR-style models and models with the state variables being the stochastic long-run mean and volatility of $r$. The authors compute simulated method-of-moments estimates of a three-factor affine term structure model and test the over-identifying restrictions on the joint distribution of long- and short-term interest rates. They find that allowing for correlated factors is crucial for simultaneously describing the short and long ends of the yield curve. The findings are interpreted in terms of the properties of the risk factors underlying term structure movements.This paper by Qiang Dai and Kenneth J. Singleton characterizes, interprets, and tests the over-identifying restrictions imposed in affine term structure models. The authors categorize these models based on the different over-identifying restrictions they impose on the parameters of the drift vector $\delta$ and the diffusion matrices. They show that every affine model can be analytically equivalent to an "Ar" model, where the drift structure is terraced with one of the state variables being the stochastic long-run mean of the short rate $r$. This equivalence allows for direct comparisons between CIR-style models and models with the state variables being the stochastic long-run mean and volatility of $r$. The authors compute simulated method-of-moments estimates of a three-factor affine term structure model and test the over-identifying restrictions on the joint distribution of long- and short-term interest rates. They find that allowing for correlated factors is crucial for simultaneously describing the short and long ends of the yield curve. The findings are interpreted in terms of the properties of the risk factors underlying term structure movements.