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.

翻译：纳尔逊-西格尔-斯文森利率曲线模型可化为一个可分离非线性最小二乘问题，其内部线性块常因基函数近似共线性而呈现病态条件。我们通过设计矩阵的精确正交重参数化来分析这种不稳定性。薄QR分解可产生正交线性参数，使得在给定非线性参数条件下，Fisher信息矩阵呈对角形式。我们还推导了一个有限时域的解析正交化方法：在区间$[0,T]$上，$4\times 4$连续Gram矩阵的元素呈闭式形式，涉及指数函数、对数函数及指数积分$E_1$，从而得到显式的时域依赖正交NSS基。结合雅可比秩与轮廓似然论证，该表示阐明了退化流形$λ_1=λ_2$（即斯文森扩展在此失去两个自由度）的性质。正交化不改变原始线性参数的最小二乘拟合结果与不确定性，但可分离其条件作用结构。当衰减参数联合估计时，正交坐标下的完整一阶协方差可表示为显式的Schur补形式。该方法还可通过QR分解元素$R_{44}$提供标量可辨识性诊断，并将模型降阶与数值不稳定性相分离。合成实验证实，正交参数化消除了线性参数间的相关性，并使其条件不确定性保持均匀。基于1981年至2026年间简化固定9个期限点网格的美国国债日度数据研究表明，正交参数序列比经典NSS参数序列更加平滑，且移动QR基近似保持恒定。