Description: Cross-domain Kalman-filter application — estimating the Cox-Ingersoll-Ross (CIR) interest-rate term structure via Kalman filter: affine term structure, CIR dynamics, state-space form, quasi-maximum-likelihood estimation
My Notion Note ID: K2E-B-Z-1
Created: 2021-07-01
Updated: 2026-06-04
License: Reuse welcome — please credit Yu Zhang and link back to yuzhang.io

Cross-domain note: Kalman filter applied to finance (interest-rate term structure) rather than SLAM. The shared machinery is the Kalman filter framework itself (see Gaussian Filters); the "state" being estimated is latent interest-rate factors rather than robot pose.

1. Summary

Title: Multi-Factor Cox-Ingersoll-Ross Models of the Term Structure: Estimates and Tests from a Kalman Filter Model Authors: Ren-Raw Chen, Louis Scott Paper: Journal of Real Estate Finance and Economics, 27(2), 143–172 (2003)

The term structure of interest rates is driven by a few unobservable latent factors (short rate, etc.). Bond yields at different maturities are noisy linear observations of these factors. This is precisely a hidden-state estimation problem — the Kalman filter is a natural fit.

The paper shows how to set up the CIR (Cox-Ingersoll-Ross) multi-factor term-structure model in state-space form, apply an approximate nonlinear Kalman filter (the process noise covariance $Q_t$ is state-dependent in CIR, making it technically nonlinear), and estimate model parameters via quasi-maximum-likelihood (QML) — maximizing the likelihood implied by the KF's Gaussian innovations even though the true CIR transition density is non-central chi-square.

SLAM analogy: latent factors ↔ robot state; observed yields ↔ sensor measurements; CIR transition ↔ motion model; yield equation ↔ observation model.

2. Key Contributions

State-space formulation of affine term structure: shows affine bond-pricing functions map directly to linear observation equations, making KF applicable
Gaussian approximation of CIR dynamics: the non-central chi-square transition is approximated by Gaussian (mean = conditional expectation, variance = conditional variance from CIR), enabling standard KF
Quasi-MLE via KF innovations: standard estimation procedure for latent-factor models when exact likelihood is unavailable
Multi-factor extension: $j$ independent CIR factors, with diagonal state-transition and covariance matrices

3. Method

3.1 Affine Term Structure

Instantaneous short rate $r$ follows an SDE:

dr = \mu(r,t)\,dt + \sigma(r,t)\,dW \tag{1}

$\mu$ : deterministic drift; $\sigma\,dW$ : diffusion; $W$ : Brownian motion. Zero-coupon bond price in an affine model:

P(t,T) = A(\tau)\exp(-B(\tau)X), \quad \tau = T - t \tag{2}

$X$ : latent state factors; $A(\tau), B(\tau)$ : maturity-dependent functions; zero-coupon yield:

R(t,T) = -\frac{\ln P(t,T)}{\tau} = \frac{B(\tau)X - \ln A(\tau)}{\tau} \tag{3}

The yield is linear in $X$ — enabling a linear measurement equation for the KF.

3.2 CIR Dynamics

Cox-Ingersoll-Ross (1985) — square-root affine diffusion:

dr = k(\theta - r)\,dt + \sigma\sqrt{r}\,dW \tag{4}

$k$ : mean-reversion speed; $\theta$ : long-run mean; $\sigma\sqrt{r}$ : square-root volatility (keeps $r \geq 0$ ; volatility proportional to $\sqrt{r}$ ).

Under risk-neutral / arbitrage-free measure (subtract market price of risk $\lambda r$ from drift):

dr = \bigl(k(\theta-r) - \lambda r\bigr)\,dt + \sigma\sqrt{r}\,dW \tag{5}

Closed-form bond price $P(t,T) = A(t,T)e^{-B(t,T)r}$ with $\gamma = \sqrt{(k+\lambda)^2 + 2\sigma^2}$ .

3.3 State-Space Form

Transition (prediction) — Gaussian approximation of the CIR conditional density:

X_t = \Phi(\Psi)\,X_{t-1} + c(\Psi) + \eta_t \tag{6}

$\Phi = \text{diag}(e^{-k_j\Delta t})$ (diagonal for $j$ independent factors); $c_j = \theta_j(1 - e^{-k_j\Delta t})$ ; $\eta_t \sim \mathcal{N}(0, Q_t)$ with $Q_t$ diagonal (factor-specific CIR conditional variance $\xi_j$ ).

Measurement — $n$ observed yields:

R_t = Z(\Psi)\,X_t + d(\Psi) + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, H) \tag{7}

$R_t \in \mathbb{R}^n$ : yields at $n$ maturities; $Z \in \mathbb{R}^{n\times j}$ : loading matrix ( $Z_{ij} = B^{(j)}(\tau_i)/\tau_i$ , row $i$ = maturity $\tau_i$ , column $j$ = factor $j$ ); $d \in \mathbb{R}^n$ : constant ( $d_i = -\ln A(\tau_i)/\tau_i$ ); $H$ : diagonal measurement noise. Hyperparameter vector $\Psi = (\theta_j, k_j, \sigma_j, \lambda_j, h_i)$ .

3.4 Kalman Filter Recursion

Prediction step:

\hat{X}_{t|t-1} = \Phi\,\hat{X}_{t-1|t-1} + c \tag{8}

P_{t|t-1} = \Phi\,P_{t-1|t-1}\,\Phi^T + Q_t \tag{9}

Measurement update (same five equations as any linear KF — see Gaussian Filters §2):

v_t = R_t - Z\hat{X}_{t|t-1} - d \quad \text{(innovation)} \tag{10}

F_t = Z\,P_{t|t-1}\,Z^T + H \quad \text{(innovation covariance)} \tag{11}

K_t = P_{t|t-1}\,Z^T\,F_t^{-1} \quad \text{(Kalman gain)} \tag{12}

\hat{X}_t = \hat{X}_{t|t-1} + K_t v_t, \qquad P_t = P_{t|t-1} - K_t Z\,P_{t|t-1} \tag{13}

3.5 Quasi-Maximum-Likelihood Estimation

The true CIR transition density is non-central chi-square — no closed-form likelihood. The KF-implied Gaussian approximation gives an innovation-based log-likelihood:

\ln L = \sum_t \left[-\frac{n}{2}\ln(2\pi) - \frac{1}{2}\ln|F_t| - \frac{1}{2}v_t^T F_t^{-1} v_t\right] \tag{14}

Maximize over $\Psi$ — quasi-MLE (quasi because the KF Gaussian is an approximation of the true non-Gaussian density). QML is consistent and asymptotically normal under mild regularity conditions.

4. Experiments & Results

The paper estimates a two-factor CIR model on U.S. Treasury yields:

Data: weekly observations, 8 maturities (3-month to 10-year), 1970–1999 (approximately 1,500 weeks)
Parameters estimated: $k_1, k_2, \theta_1, \theta_2, \sigma_1, \sigma_2, \lambda_1, \lambda_2$ (CIR dynamics) + $h_1, \ldots, h_8$ (measurement noise per maturity)

Key findings:

Two factors explain ~99% of yield variation (consistent with prior literature)
Mean-reversion speeds $k_1, k_2$ estimated on the order of 0.1–0.5 per year
Measurement noise variances $h_i$ are small but nonzero — the model doesn't fit perfectly (consistent with the "approximately linear" bond pricing)
Model-implied yield curves match observed curves well in-sample; out-of-sample fit degrades for far-horizon yields

Statistical tests: the paper runs specification tests (likelihood-ratio, parameter stability) confirming the two-factor CIR is a reasonable but not perfect description of the data. Three or more factors further improve fit at a parsimony cost.

5. Ablation & Discussion

One-factor vs multi-factor: single-factor CIR (the classic 1985 model) fits short-end rates well but fails to capture the slope dynamics of the yield curve; multi-factor models are necessary
Gaussian approximation quality: the CIR conditional variance is relatively small at typical interest-rate levels — the Gaussian approximation is adequate, though exact MCMC-based estimation gives slightly different parameter estimates
Cross-section vs time-series: the KF approach exploits both dimensions simultaneously — unlike cross-section regression (which can't identify risk premia) or pure time-series (which can't use maturity structure)
Measurement noise necessity: without $H \neq 0$ the system is over-determined (more yields than factors), so some noise allocation is mathematically required and economically justified (bid-ask spreads, reporting errors)

6. Strengths / Limitations / Future Work

Strengths:

Principled state-space framework: the KF provides optimal linear estimates of latent factors
QML is computationally cheap (no MCMC) and well-understood asymptotically
Direct analogy to SLAM: demonstrates KF universality across domains

Limitations:

Gaussian approximation of CIR dynamics introduces bias in parameter estimates at extreme interest-rate levels (near zero or very high)
Linear yield equation is only exactly valid for affine models; non-affine (stochastic volatility) models require extended/unscented KF
The CIR model imposes positivity ( $r \geq 0$ ) which the Gaussian KF does not enforce — near-zero rates can produce negative filter outputs

Extensions:

Affine jump-diffusion models (adds Poisson jumps to drift): still quasi-tractable with KF
Stochastic volatility models (Heston-type): require EKF or particle filter
The same KF framework applies to virtually any affine term-structure model (Vasicek, Hull-White, etc.)

References


Primary	Chen R.-R., Scott L. (2003). Multi-Factor Cox-Ingersoll-Ross Models of the Term Structure: Estimates and Tests from a Kalman Filter Model. J. Real Estate Finance Econ., 27(2). DOI
CIR model	Cox J.C., Ingersoll J.E., Ross S.A. (1985). A Theory of the Term Structure of Interest Rates. Econometrica, 53(2).
KF for term structure	Geyer A.L.J., Pichler S. (1999). A State-Space Approach to Estimate and Test Multifactor CIR Models. J. Financial Research.
Affine term structure	Duffie D., Kan R. (1996). A Yield-Factor Model of Interest Rates. Mathematical Finance.
KF framework	See Gaussian Filters (prediction/update, innovation, inversion lemma)

CIR Term Structure via Kalman Filter

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