Specification Analysis of Affine Term Structure Models
Source: Dai, Q. & Singleton, K. J. (2000). Journal of Finance 55(5), 1943–1978.
TL;DR
Characterizes, interprets, and tests the over-identifying restrictions imposed by affine term-structure models (ATSMs) — the workhorse class in which the short rate r(t) = δ'Y(t) is a linear combination of latent affine-diffusion state variables Y, and bond yields are affine functions of Y. The paper builds a complete classification of N-factor affine models indexed by how many state variables drive the conditional variance, derives the admissibility (existence/identification) conditions for each sub-family, and estimates a three-factor model on swap rates. The central empirical finding: allowing the factors to be correlated is essential for jointly fitting the short and long ends of the curve; restrictions imposing independent diffusions are strongly rejected.
What it models
The joint dynamics of the term structure — i.e., the joint conditional distribution of bond yields across maturities — within the affine class where drifts and instantaneous (co)variances of the state vector are affine in Y. It unifies and compares the two strands: "AY" models (r is a linear combination of latent factors interpreted as level/slope/curvature) and "Ar" models (r expressed via its own lag plus a stochastic long-run mean and stochastic volatility), showing each Ar model is equivalent to an AY model with a terraced drift.
Specification
State vector Y follows an affine diffusion: dY = K(Θ − Y)dt + Σ √S(t) dW, with the diagonal of S(t) affine in Y, so both drift and instantaneous covariance are affine.
Short rate r(t) = δ'Y(t); zero-coupon bond prices are exponential-affine, P = exp(A(τ) + B(τ)'Y), with A and B solving Riccati ODEs (per Duffie–Kan 1996).
Models are classified by the number m of state variables that drive the conditional variance (the canonical A_m(N) taxonomy), with each sub-family given a maximal, identified canonical form.
Admissibility requires conditions ensuring positive instantaneous variances and existence of solutions to the SDEs (e.g., eigenvalues of K positive for stationarity), which constrain how factor correlations and stochastic volatility can coexist.
Estimation
Simulated method of moments / EMM (Duffie–Singleton 1996; Gallant–Tauchen) applied to a three-factor affine model fit simultaneously to six-month, two-year, and ten-year swap rates — in contrast to prior work that estimated multi-factor short-rate models using the short rate alone.
What it captures
A built-in tension: the more factors allowed to drive volatility (larger m), the more restricted the admissible correlations among factors become, and vice versa.
The equivalence result shows that the factor typically labeled "stochastic volatility" in Ar models is in fact well proxied by the slope of the swap curve, and the second factor by a long-term rate — so assuming independent diffusions implicitly (and counter-factually) assumes r is conditionally uncorrelated with the level and slope of the curve.
A three-factor AY model with correlated factors adequately describes swap-rate dynamics; the independent-diffusion restriction is strongly rejected.
Use & extensions
The canonical reference framework for term-structure modeling, fixed-income derivative pricing, interest-rate risk management, and macro-finance yield-curve research; nests Vasicek- and CIR-style models as special cases and motivated subsequent quadratic and non-affine extensions.
Limitations
Affine structure, chosen for tractability, trades off correlation flexibility against stochastic volatility; the most flexible affine model examined still could not be improved within this class, motivating non-affine alternatives.
Latent factors are hard to interpret; estimation is sensitive to identification/normalization choices and to existence conditions on the diffusion coefficients.
Key references
Dai, Q. & Singleton, K. (2000) — Specification Analysis of Affine Term Structure Models — Journal of Finance (NBER WP 6128, 1997)
Duffie, D. & Kan, R. (1996) — A Yield-Factor Model of Interest Rates — Mathematical Finance
Cox, J., Ingersoll, J. & Ross, S. (1985) — A Theory of the Term Structure of Interest Rates — Econometrica
Vasicek, O. (1977) — An Equilibrium Characterization of the Term Structure — Journal of Financial Economics
Provenance: verified/generated from the paper's full text.
