Design and Estimation of Quadratic Term Structure Models *

Abstract

We consider the design and estimation of quadratic term structure models. We start with a list of stylized facts on interest rates and interest rate derivatives, classified into three layers: (1) general statistical properties, (2) forecasting relations, and (3) conditional dynamics. We then investigate the implications of each layer of property on model design and strive to establish a mapping between evidence and model structures. We calibrate a two-factor model that approximates these three layers of properties well, and illustrate how the model can be applied to pricing interest rate derivatives. JEL Classification Codes: G12, G13, E43.Keywords: quadratic model; term structure; positive interest rates; humps; expectation hypothesis; GMM; caps and floors.Term structure modeling has enjoyed rapid growth during the last decade. Among the vast number of different models, the affine class stands out as the most popular class due to its analytical tractability. Duffie and Kan (1996)'s characterization of the complete class has spurred a stream of studies on its empirical applications and model design. Examples include general econometric estimations by Chen and Scott (1993), Duffie and Singleton (1997), Singleton (2000b), andSingleton (1999), applications to the predictability of interest rates by Frachot and Lesne (1994), Roberds and Whiteman (1999), Backus, Foresi, Mozumdar, and Wu (2001), Duffee (1999), and Dai and Singleton (2000a), and currency pricing by Backus, Foresi, and Telmer (2001). While these applications claim success in one or two dimensions, inherent tension exists when one tries to apply the model to a wider range of properties.Even more troublesome, however, is a seemingly irreconcilable tension between delivering a relatively good empirical performance and excluding negative interest rates. Indeed, all of the relatively "successful" model designs within the affine framework, in terms of empirical performance, imply positive probabilities of negative interest rates. Examples include Backus, Foresi, Mozumdar, and Wu (2001), Backus, Foresi, and Telmer (2001), Dai and Singleton (2000b), Dai and Singleton (2000a), Duffee (1999), Singleton (1997), andSingleton (1999). Leippold and Wu (1999a) identify and characterize an alternative class, the quadratic class of term structure models, where bond yields and forward rates are quadratic functions of the state vector. Their property analysis indicates that the quadratic class is comparable to the affine class for analytical tractability but is more flexible for model design. In particular, positive interest rates can be guaranteed with little loss of generality or flexibility. In this paper, we consider the model design and estimation problem within the quadratic framework.Although examples of quadratic models have appeared in the literature since the late eighties, 1 empirical applications have at best been sporadic. The most systematic empirical study, and hence the most germane to our work, is by Ahn, Dittmar, and Gallant (2001).1 Early example...

Exchange Rate Dynamics and US Dollar-Denominated Sovereign Bond Prices in Emerging Markets

“…They find the quadratic term structure models outperform the affine term structure models, including the CIR and Vasicek models. Similar results are found by Leippold and Wu (2003), and Li and Zhao (2006). Aït-Sahalia (1996) shows that there is evidence of nonlinearities in the drift function of the interest rate term structure using a nonparametric approach.…”

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“…This result has already been shown in the continuous time by Leippold & Wu (2003). The model is finally:…”

A Note on the Relation Between Principal Components and Dynamic Factors in Affine Term Structure Models

“…Longstaff (1989) and Beaglehole and Tenney (1991) were pioneers in the exploration of quadratic term structure models. In these models the term structure of interest rates is parameterized as a quadratic function of the state vector (for a complete theoretical characterization, see Leippold and Wu (2002); for practical applications, see Leippold and Wu (2003). One of its advantages is the capability to naturally generate positive interest rates, in…”