This paper develops a multivariate local likelihood framework for estimating covariate-dependent copulas of arbitrary dimension \(n \ge 2\) with multivariate covariates \(\textbf{Y} \in \mathbb {R}^s\) . The proposed estimator generalizes existing bivariate approaches by locally maximizing the conditional copula log-likelihood with respect to a smooth calibration function linked to the copula parameter, thereby ensuring valid parameterization and capturing complex, heterogeneous dependence structures. Asymptotic theory is developed, including closed-form expressions for the bias, variance, and asymptotic normality of the estimator. Data-driven bandwidth and copula-family selection procedures based on cross-validation are proposed. Simulation studies demonstrate strong finite-sample performance of the estimator. An application to NHANES 2017–2018 glucose and glycohemoglobin data illustrates the method’s interpretability and empirical value. Overall, this work provides the first comprehensive asymptotic and computational treatment of high-dimensional conditional copulas within the local likelihood framework.
