VarX2

VarX2 is a framework for multivariate time-series modeling that extends standard VARX models by incorporating second-order exogenous terms. It is designed to capture nonlinear interactions among exogenous inputs and between exogenous inputs and endogenous variables, while retaining the interpretability of linear components. In VarX2, the endogenous vector y_t is modeled as a function of past values of y, past and current values of exogenous predictors x, and additional second-order terms derived from x, such as pairwise products of exogenous components. This allows the model to represent scenarios where the impact of an exogenous variable depends on its own level or on interactions among multiple exogenous factors.

Formally, VarX2 requires specifying the lag structure for both the endogenous and exogenous sides, including a set of linear coefficients for past y and x terms and a set of nonlinear coefficients for second-order exogenous terms. The resulting specification remains linear in parameters, which facilitates estimation using ordinary least squares or regularized approaches when the parameter count is large. Regularization methods such as ridge or lasso are commonly employed to prevent overfitting in high-dimensional settings.

Estimation in VarX2 typically relies on Gaussian-orientated likelihoods or Bayesian formulations, enabling uncertainty quantification through prediction intervals and posterior distributions. Model selection often involves cross-validation or information criteria to determine the number of lags and the necessity of nonlinear exogenous terms.

VarX2 is applied in economics, finance, energy forecasting, and climate studies, where multiple time-series interact under changing external conditions. While offering improved flexibility over VARX, it requires careful attention to data availability and model complexity to ensure robust forecasting and interpretability.