Orthogonal reparametrization of the Nelson-Siegel-Svensson interest rate curve model: conditioning, diagnostics, and identifiability

TL;DR

This paper introduces an orthogonal reparametrization of the Nelson-Siegel-Svensson interest rate model to improve conditioning, diagnostics, and parameter identifiability, enhancing stability and interpretability.

Contribution

It provides an explicit orthogonalization method using QR decomposition and analytical formulas, clarifying parameter degeneracies and improving model diagnostics.

Findings

Orthogonalization eliminates correlations among linear parameters.

The approach yields explicit covariance and identifiability diagnostics.

Synthetic and real data experiments confirm improved stability and interpretability.

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 $\lambda_1=\lambda_2$, 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.