Asymptotic Theory and Regime-Varying Cointegration for Trend-Cycle Decomposition

Standard trend–cycle extraction methods in macroeconomics rely on assumptions of global smoothness and time-invariant dynamics that become restrictive in the presence of structural breaks and regime-dependent behavior. These limitations affect both cyclical measurement and the stability of inferred long-run relationships under cointegration. This paper develops a fuzzy clustering–based filtering framework that provides a regime-sensitive decomposition of macroeconomic time series. While fuzzy methods have been used in applied filtering, their asymptotic properties—particularly under structural breaks and cointegration—have not been formally characterized. The proposed approach assigns observations probabilistically across latent regimes, allowing for smooth transitions and mixed states. We establish √T-consistency of the cyclical component, vanishing endpoint bias, and preservation of cointegrating relationships. The filter is embedded in a regime-dependent cointegrated system (MS-F-VECM), allowing both short-run dynamics and long-run equilibria to vary across regimes. Monte Carlo simulations confirm strong finite-sample performance in terms of break detection and cointegration preservation. An application to Eurozone data (1999–2023) shows that standard measures of comovement are not invariant but reflect regime-dependent aggregation. The contribution is methodological: a unified framework for regime-dependent filtering and cointegration analysis.