Score-driven time-varying parameter models with spline-based densities

Overview

This study develops a score-driven time-varying parameter framework that employs spline-based density specification to avoid restrictive parametric assumptions about the error distribution. The methodology generates a score function as a natural cubic spline, enabling representation of flexible distributional forms including asymmetric and leptokurtic densities. The approach generalizes beyond the Gaussian case while maintaining computational tractability for parameter updating.

Methods and approach

The proposed method constructs time-varying parameters using score-driven dynamics where the score function is derived from a spline-based density rather than a pre-specified parametric family. Natural cubic splines provide flexibility in density shape while maintaining the functional form needed for score-based updates. Static parameters are estimated via maximum likelihood, with formal asymptotic theory developed for the resulting estimators. Two model specifications are examined: location models where time-varying parameters affect the conditional mean, and log-scale models where parameters influence conditional volatility.

Key Findings

Empirical validation demonstrates competitive performance against established benchmark models. Application to U.S. monthly CPI inflation filtering via the location model yields effective mean estimation. The scale model applied to daily S&P 500 stock returns across the full cross-section produces volatility estimates comparable to or exceeding those from existing methods. Both applications illustrate the practical utility of distributional flexibility without sacrificing model performance or estimation efficiency.

Implications

The framework addresses limitations of conventional score-driven models that rely on restrictive distributional assumptions. By enabling outlier-robust updating through flexible density specification, the method extends applicability to financial and economic time series exhibiting non-Gaussian features. The approach maintains the computational advantages of score-driven methodology while reducing specification risk inherent in parametric choices. The demonstrated performance on inflation filtering and volatility estimation suggests utility for applied work in macroeconomic and financial contexts where distributional assumptions substantially influence parameter evolution.