Determinant-based multiple correlation coefficient in multiplicative distortion models

This paper explores a determinant-based multiple correlation coefficient in multiplicative distortion measurement error models, where an unobservable covariate vector is distorted by a confounding variable-related multiplicative function. As a novel metric for gauging correlation among continuous variables, it proves valuable when specifying a dependent variable and covariates is difficult, especially for equally important variables. We first examine triple-variable scenarios and then extend to p-dimensional vectors under normal and general distributions. Two calibration methods are proposed to generate unobservable variable proxies. We present comprehensive asymptotic results for calibrated estimators and test statistics, with theoretical findings highlighting their asymptotic efficiency. Monte Carlo simulations and a real dataset validate the proposed estimators and test statistics.