Specification Analysis of Affine Term Structure Models

• Author: Kenneth J. Singleton,Qiang Dai
• DOI: http://doi.org/10.1111/0022-1082.00278
• Published date: 01 October 2000
• Date: 01 October 2000

THE JOURNAL OF FINANCE * VOL. LV, NO. 5 * OCT. 2000

Specification Analysis of Affine

Term Structure Models

QIANG DAI and KENNETH J. SINGLETON*

ABSTRACT

This paper explores the structural differences and relative goodness-of-fits of af-

fine term structure models (ATSMs). Within the family of ATSMs

there is a trade-

off between flexibility in modeling the conditional correlations and volatilities of

the risk factors. This trade-off is formalized by our classification of N-factor affine

family into N + 1 non-nested subfamilies of models. Specializing to three-factor

ATSMs, our analysis suggests, based on theoretical considerations and empirical

evidence, that some subfamilies of ATSMs are better suited than others to explain-

ing historical interest rate behavior.

IN SPECIFYING A DYNAMIC TERM STRUCTURE MODEL-one that describes the co-

movement over time of short- and long-term bond yields-researchers are

inevitably confronted with trade-offs between the richness of econometric

representations of the state variables and the computational burdens of pric-

ing and estimation. It is perhaps not surprising then that virtually all of the

empirical implementations of multifactor term structure models that use

time series data on long- and short-term bond yields simultaneously have

focused on special cases of "affine" term structure models (ATSMs). An ATSM

accommodates time-varying means and volatilities of the state variables

through affine specifications of the risk-neutral drift and volatility coeffi-

cients. At the same time, ATSMs yield essentially closed-form expressions

for zero-coupon-bond prices (Duffie and Kan (1996)), which greatly facili-

tates pricing and econometric implementation.

The focus on ATSMs extends back at least to the pathbreaking studies by

Vasicek (1977) and Cox, Ingersoll, and Ross (1985), who presumed that the

instantaneous short rate r (t) was an affine function of an N-dimensional

state vector Y(t), r(t) = 50 + 8YY(t), and that Y(t) followed Gaussian and

square-root diffusions, respectively. More recently, researchers have ex-

plored formulations of ATSMs that extend the one-factor Markov represen-

* Dai is from New York

University. Singleton is from Stanford University. We thank Darrell

Duffie for extensive discussions, Ron Gallant and Jun Liu for helpful comments, the editor and

the referees for valuable inputs, and the Financial Research Initiative of the Graduate School

of Business at Stanford University for financial support. We are also grateful for comments

from seminar participants at the NBER Asset Pricing Group meeting, the Western Finance

Association Annual Meetings, Duke University, London Business School, New York

University,

Northwestern University, University of Chicago, University of Michigan, University of Wash-

ington at St. Louis, Columbia University, Carnegie Mellon University, and CIRANO.

1943

1944 The Journal of Finance

tation of the short-rate, dr(t) = (0 - r(t))dt + \/vdB(t), by introducing a

stochastic long-run mean 0(t) and a volatility v(t) of r(t) that are affine

functions of (r(t), 0(t),v(t)) (e.g., Chen (1996), Balduzzi et al. (1966)).1 These

and related ATSMs underpin extensive literatures on the pricing of bonds

and interest-rate derivatives and also underlie many of the pricing systems

used by the financial industry. Yet, in spite of their central importance in

the term structure literature, the structural differences and relative empir-

ical goodness-of-fits of ATSMs remain largely unexplored.

This paper characterizes, both formally and intuitively, the differences and

similarities among affine specifications of term structure models and as-

sesses their strengths and weaknesses as empirical models of interest rate

behavior. We begin our specification analysis by developing a comprehensive

classification of ATSMs with the following convenient features: (i) whether a

specification of an affine model leads to well-defined bond prices-a prop-

erty that we will refer to as admissibility (Section I)-is easily verified; (ii)

all admissible N-factor ATSMs are uniquely classified into N + 1 non-nested

subfamilies; and (iii) for each of the N + 1 subfamilies, there exists a max-

imal model that nests econometrically all other models within this subfam-

ily. With this classification scheme in place, we answer the following questions.

[Q1] Given an ATSM (e.g., one of the popular specifications in the litera-

ture), is it "maximally flexible," and, if not, what are the overidentifying

restrictions that it imposes on yield curve dynamics?

[Q2] Are extant models, or their maximal counterparts, sufficiently flex-

ible to describe simultaneously the historical movements in short- and

long-term bond yields?

Ideally, a specification analysis of ATSMs could begin with the specifica-

tion of an all-encompassing ATSM, and then all other ATSMs could be stud-

ied as special cases. However, for a specification to be admissible, constraints

must be imposed on the dynamic interactions among the state variables, and

these constraints turn out to preclude the existence of such an all-encompassing

model. Therefore, as a preliminary step in our specification analysis, we

characterize the family of admissible ATSMs.2 This is accomplished by clas-

sifying ATSMs into N + 1 subfamilies according to the number "m"

of the Ys

(more precisely, the number of independent linear combinations of Ys) that

determine the conditional variance matrix of Y, and then using this classi-

fication to provide a formal and intuitive characterization of (minimal known)

sufficient conditions for admissibility. Each of the N + 1 subfamilies of ad-

missible models is shown to have a maximal element, a feature we exploit in

answering Qi and Q2.

1Nonlinear models, such as those studied by Chan et al. (1992), can be extended in similar

fashion to multi-factor models, such as those of Andersen and Lund (1998), that fall outside the

affine family.

2 The problem of admissibility was not previously an issue in empirical implementations of

ATSMs, because the special structure of Gaussian and CIR-style models made verification that

bond prices are well defined relatively straightforward.

Affine Term Structure Models 1945

The usefulness of this classification scheme is illustrated by specializing to

the case of N = 3 and describing in detail the nature of the four maximal mod-

els for the three-factor family of ATSMs. This discussion highlights an impor-

tant trade-off within the family of ATSMs between the dependence of the

conditional variance of each Yi

(t) on Y(t) and the admissible structure of the

correlation matrix for Y. Gaussian models offer complete flexibility with re-

gard to the signs and magnitudes of conditional and unconditional correla-

tions among the Ys but at the "cost" of the apparently counterfactual assumption

of constant conditional variances (m = 0). At the other end of the spectrum of

volatility specifications lies (what we refer to as) the correlated square-root dif-

fusion (CSR) model that has all three state variables driving conditional vol-

atilities (m = 3). However, admissibility of models in this subfamily requires

that the conditional correlations of the state variables be zero and that their

unconditional correlations be non-negative. In between the Gaussian and CS-

Rmodels lie two subfamilies of ATSMs with time-varying conditional volatil-

ities of the state variables and unconstrained signs of (some of) their correlations.

Specializing further, we show that the Vasicek (Gaussian), BDFS, Chen,

and CIR models are classified into distinct subfamilies. Moreover, compar-

ing these models to the maximal models in their respective subfamilies, we

find that, in every case except the Gaussian models, these models impose

potentially strong overidentifying restrictions relative to the maximal model.

Thus, we answer QI by showing that there exist identified, admissible ATSMs

that allow much richer interdependencies among the factors than have here-

tofore been studied.

One notable illustration of this point is our finding that the standard as-

sumption of independent risk factors in CIR-style models (see, e.g., Chen

and Scott (1993), Pearson and Sun (1994), and Duffie and Singleton (1997))

is not necessary either for admissibility of these models or for zero-coupon-

bond prices to be known (essentially) in closed form.3 At the same time we

show that, when the correlations in these CSR models are nonzero, they

must be positive for the model to be admissible. The data on U.S. interest

rates seems to call for negative correlations among the risk factors (see Sec-

tion II.C). Because CSR models are theoretically incapable of generating

negative correlations, we conclude that they are not consistent with the his-

torical behavior of U.S. interest rates.

Given the absence of time-varying volatility in Gaussian (m = 0) models

and the impossibility of negatively correlated risk factors in CSR (m = 3)

models, in answering Q2, we focus on the two subfamilies of N = 3 models

in which the stochastic volatilities of the Ys are controlled by one (m = 1)

and two (m = 2) state variables. The maximal model for the subfamily

m = 1 nests the BDFS model, whereas the maximal model for the m = 2

subfamily nests the Chen model.

We compute simulated method of moments (SMM) estimates (Duffie and

Singleton (1993), Gallant and Tauchen (1996)) of our maximal ATSMs. Whereas

most of the empirical studies of term structure models have focused on U.S.

3 We are grateful to Jun Liu for pointing out this implication of our admissibility conditions.