This study delves into the application of Multivariate ARCH (Autoregressive Conditional Heteroskedasticity) models to the analysis of time series data. Time series data, characterized by temporal dependencies, are abundant in various fields, including finance, economics, and environmental sciences. The primary objective of this research is to develop and implement Multivariate ARCH models to effectively capture and model the time-varying volatility and correlation structure within multivariate time series datasets. The study begins by providing a comprehensive overview of the foundational concepts of ARCH models and their extension into the multivariate context. We discuss the theoretical underpinnings and statistical properties of these models, offering a solid framework for their application. Through empirical analysis and model estimation, this research demonstrates the advantages of Multivariate ARCH models in capturing volatility clustering and the dynamic relationships between variables in multivariate time series data. Real-world applications are explored, highlighting the relevance and utility of these models in financial risk management, portfolio optimization,