On change-point detection in volatile series using GARCH models

Abstract

Abstract. We study a Cumulative Sum (CUSUM)-type test to detect a change in the unconditional variance of GARCH models. We show that, under the null hypothesis (no change), the CUSUM test statistic converges to the supremum of a standard Brownian bridge. Using Monte Carlo simulation, we demonstrate that the asymptotic power of the test is almost 1 and compare the test result with existing results in the literature. Finally, the test procedure is applied to real-world situation namely stock market returns where we are able to detect a change in the unconditional variance at a very early stage of the financial crisis in comparison to other previous analyses of the same dataset.Key words: GARCH model ; Change-point ; Squared cusum test ; Brownian bridge ; Weak convergence. AMS 2010 Mathematics Subject Classification : 37M10, 62F03, 62F05.Résumé. Nousétudions un test de type CUSUM pour la détection de rupture dans la variance inconditionnelle des modèles GARCH. Nous montrons que sous l'hypothèse nulle, notre statistique de test converge vers le supremum d'un pont Brownien standard. Utilisant des simulations de type Monte Carlo, nous démontrons que la puissance asymptotique du test est presqu'égaleà 1 et comparons le résultat du test avec les résultats existants dans la littérature. Enfin, un exemple d'application sur les données réelles des rendements du marché boursier nous a permis de détecter une rupture dans la variance inconditionnelleà un stade très précoce de la crise financière par rapportà d'autres analyses précédentes du même ensemble de données.

Change-point detection: application of Cusum method to real life data

“…In this section we present how the Cusum test statistic is constructed and used. Katchekpele et al (2017) observe a stochastic phenomenon (z 1 , z 2 , • • • , z n ) which is known to be generated by the square of a GARCH process and study a Cusumtype test to detect a change in the unconditional variance of GARCH models. Their proposed test statistic is…”

“…where q denotes the largest integer smaller than or equal to (ln(n)) 2 as in Lee et al (2003). As already mentioned, in this work, we are going to present an application of the version presented by Katchekpele et al (2017). The algorithm is given below:…”

“…Shi (2020) has proposed a survey of change-point techniques for time series data among which the Cusum procedure introduced by Brown et al (1975). Following him, numerous works devoted to this procedure, among which Katchekpele et al (2017) who studied a cumulative sum-type test to detect a change in the unconditional variance of GARCH models. In this work, we are going to explain their version and apply it to real data.…”