Paper: arXiv 2604.19290

Authors: Robert Flassig, Emrah Gülay, Daniel Guterding

Abstract

The Nelson-Siegel-Svensson (NSS) interest rate curve model yields a separable nonlinear least-squares problem whose inner linear block is often ill-conditioned because the basis functions become nearly collinear. We analyze this instability via an exact orthogonal reparametrization of the design matrix. A thin QR decomposition produces orthogonal linear parameters for which, conditional on the nonlinear parameters, the Fisher information matrix is diagonal. We also derive a finite-horizon analytical orthogonalization: on $[0,T]$, the $4\times 4$ continuous Gram matrix has closed-form entries involving exponentials, logarithms, and the exponential integral $E_1$, yielding an explicit horizon-dependent orthogonal NSS basis. Together with Jacobian-rank and profile-likelihood arguments, this representation clarifies the degenerate manifold $λ_1=λ_2$, where the Svensson extension loses two degrees of freedom. Orthogonalization leaves the least-squares fit and uncertainty of the original linear parameters unchanged, but isolates the conditioning structure. When the decay parameters are estimated jointly, the full first-order covariance in orthogonal coordinates admits an explicit Schur-complement form. The approach also yields a scalar identifiability diagnostic through the QR element $R_{44}$ and separates model reduction from numerical instability. Synthetic experiments confirm that orthogonal parametrization eliminates correlations among the linear parameters and keeps their conditional uncertainty uniform. A daily U.S. Treasury study on a reduced fixed 9-tenor grid from 1981 to 2026 shows smoother orthogonal parameter series than classical NSS parameters while the moving QR basis remains nearly constant.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical derivation of orthogonal reparametrization using QR decomposition, Fisher information matrices, and profile likelihood analysis, while simultaneously validating these techniques with a 45-year U.S. Treasury historical dataset. It successfully bridges theoretical structural identifiability with practical, actionable improvements for interest rate curve calibration.

Research Flowchart

  flowchart TD
    Goal[Research Goal: Fix NSS Model Numerical Instability]
    Inputs[Data: NSS Nonlinear Least-Squares Problem]
    Method[Method: Thin QR Decomposition & Analytical Orthogonalization]
    Process[Process: Compute Gram Matrix & Schur-complement Covariance]
    Diagnostics[Diagnostics: Profile-likelihood & R44 Identifiability Metric]
    Results[Outcomes: Stable Parameters & Degeneracy Clarification]

    Goal --> Inputs
    Inputs --> Method
    Method --> Process
    Process --> Diagnostics
    Diagnostics --> Results