Identification and Estimation of 'Maximal' Affine Term Structure Models: An Application to Stochastic Volatility

Abstract

We propose a canonical representation for affine term structure models where the state vector is comprised of the first few Taylor-series components of the yield curve and their quadratic (co-)variations. With this representation: (i) the state variables have simple physical interpretations such as level, slope and curvature, (ii) their dynamics remain affine and tractable, (iii) the model is by construction 'maximal' (i.e., it is the most general model that is econometrically identifiable), and (iv) model-insensitive estimates of the state vector process implied from the term structure are readily available. (Furthermore, this representation may be useful for identifying the state variables in a squared-Gaussian framework where typically there is no one-to-one mapping between observable yields and latent state variables). We find that the 'unrestricted' A1(3) model of Dai and Singleton (2000) estimated by 'inverting' the yield curve for the state variables generates volatility estimates that are negatively correlated with the time series of volatility estimated using a standard GARCH approach. This occurs because the 'unrestricted' A1(3) model imposes the restriction that the volatility state variable is simultaneously a linear combination of yields (i.e., it impacts the cross-section of yields), and the quadratic variation of the spot rate process (i.e., it impacts the time-series of yields). We then investigate the A1(3) model which exhibits 'unspanned stochastic volatility' (USV). This model predicts that the cross section of bond prices is independent of the volatility state variable, and hence breaks the tension between the time-series and cross-sectional features of the term structure inherent in the unrestricted model. We find that explicitly imposing the USV constraint on affine models significantly improves the volatility estimates, while maintaining a good fit cross-sectionally.