Abstract:

Multivariate modelling for time series data has evolved significantly from traditional univariate approaches, enabling the simultaneous analysis of multiple interrelated variables. Unlike univariate models, which rely solely on the historical behaviour of a single variable, multivariate models capture dynamic interdependencies and interactions among several time series, providing deeper insights into complex systems. Establishing relationships between two or more variables within a set of time series is a challenging task, as it requires accounting for lagged effects, cross-correlations, and feedback mechanisms. A multivariate time series model, particularly the Vector Autoregressive (VAR) framework, offers an adequate unrestricted approximation to the reduced form of an unknown structural simultaneous equations system. In a VAR model, all variables are treated symmetrically, without imposing exogenous or endogenous assumptions, making it highly suitable for examining interrelationships among time series. The model allows researchers to estimate a dynamic system of equations where each variable is regressed on its own lags and the lags of other variables, capturing the temporal dependencies and mutual influences within the system. In the present study, a pth order VAR model is formulated for multivariate time series data analysis, emphasizing both theoretical and practical aspects. The stationarity conditions for VAR processes are derived, ensuring that the time series do not exhibit explosive behaviour, while the autocovariance function of the system is developed to analyze temporal dependencies among variables. Further, the study employs maximum likelihood estimation (MLE) via ordinary least squares (OLS) for parameter estimation and utilizes internally Studentized residuals to estimate the error covariance structure. Additionally, the Likelihood Ratio (LR) test is implemented to determine the optimal lag length, ensuring model parsimony without sacrificing explanatory power. The proposed framework not only enhances the accuracy of forecasts but also provides a robust basis for conducting causality analysis, impulse response functions, and variance decompositions in multivariate systems. The methodologies discussed in this study are applicable across disciplines such as economics, finance, environmental science, and engineering, where multiple interdependent time series must be analysed simultaneously.

Description:

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