A Methodological Framework for Converting ARCH and GARCH Financial Time Series Models to a Kink Regression Model

Abstract

Autoregressive Conditional Heteroskedasticity (ARCH) models and their generalized form, GARCH, have been widely used for several decades as an important tool for modeling the volatility of financial time series. However, applying them to data with structural changes leads to biased estimates and unreliable results. This study aims to present a comprehensive theoretical framework for the mathematical transformation from these two models to the kink regression model developed by Hansen (2017), through a detailed mathematical derivation that demonstrates how the transition occurs mathematically. The proposed theoretical framework operates in several stages, beginning with reformulating the conditional mean function to accommodate breakpoints, followed by addressing conditional heteroskedasticity through two alternative approaches, then defining the unknown inflection points using a grid search, and concluding with a test of the significance of the transformation using the Wald statistic. The framework was applied to monthly data from the Iraq Stock Exchange spanning from February 2015 to December 2025, with a sample size of 131 observations, The T-statistic () was found to be (=47.638), indicating the presence of two inflection points with significant economic implications: the first point occurred at an exchange rate of 1,211 dinars per U.S. dollar (prior to the devaluation of the dinar against the dollar in December 2020), and the second point was at an oil price of $61.75 per barrel (i.e., close to the approximate break-even price for the Iraqi budget). This study offers a new contribution to the field of financial econometrics by establishing a mathematical and methodological approach that links modern time-series models with the threshold regression framework.