Testing for Smooth Structural Change in Cointegrated Systems

2026-08-04

AI Summary

This study investigates whether the long-run equilibrium relationships and short-run adjustment mechanisms in a cointegrated system undergo smooth structural changes following a known intervention date. To this end, the authors develop a time-varying vector error correction model (VECM), identifying the cointegration rank using the pre-intervention subsample and allowing both the cointegrating vectors β(δ) and adjustment coefficients α(δ) to evolve smoothly over rescaled time post-intervention. Estimation is carried out via a kernel-weighted local reduced-rank approach. The work is the first to simultaneously disentangle and model smooth transitions in both equilibrium relations and adjustment speeds within a cointegrated VAR framework, introducing corresponding tests for parameter consistency and smoothness. An empirical application to Brent crude oil prices, EUR/USD exchange rates, and gold indices during the Russia–Ukraine conflict reveals smoothly evolving cointegration rather than abrupt shifts or constancy, thereby validating the proposed methodology.

Abstract

This paper develops an econometric framework for analysing smooth structural change in cointegrated systems following a known intervention time. We consider a vector error-correction model in which the cointegration rank and the pre-intervention cointegrating structure are identified from a stable pre-intervention subsample. After the intervention, both the adjustment coefficients and the cointegrating vectors are allowed to evolve smoothly as functions of rescaled time, which are estimated using kernel-weighted local reduced-rank methods. The analysis is formulated directly in a cointegrated VAR/VECM system, which preserves the treatment of long-run relations and short-run error-correction dynamics. By working with the decomposition $Π(δ)=α(δ)β(δ)'$, the method separates changes in the equilibrium relation from those in the speed of adjustment. We also provide two tests for the parameter consistency and the post-intervention parameter smoothness respectively. An empirical application to energy market, foreign-exchange, and gold-market index around the 24 February 2022 Russia's invasion of Ukraine illustrates how the proposed approach distinguishes between a discrete regime shift and smooth post-intervention evolution. The results suggest that cointegrating relation among the price of Brent crude oil, the spot exchange rate (USD/EUR), and the Credit Suisse NASDAQ Gold Price Index has smoothly changed after the outbreak of war, instead of a constant long-run conintegration system in the pre-intervention period.

Problem

• smooth structural change
• cointegrated systems
• intervention time
• cointegration rank
• equilibrium relation

Innovation

• smooth structural change
• cointegrated systems
• vector error-correction model
• local reduced-rank estimation
• time-varying cointegration