This paper studies high-dimensional nonsparse regression when the data exhibit both strong predictor dependence and covariate shift. Many existing regression methods struggle with highly correlated predictors, while covariate shift can make components learned from source data less effective for target-domain prediction. To address these joint challenges, we develop an importance-weighted partial least squares (IWPLS) method that reweights source observations to reflect the target covariate distribution and extracts response-relevant components for coefficient estimation. The resulting estimator depends on weighted sample covariances, which generally lack the usual Wishart structure even when the target-domain covariates are Gaussian. We therefore combine a perturbation analysis of weighted PLS moments with an influence-function decomposition to establish consistency for the one- and multi-component estimators and asymptotic normality for the one-component estimator. Simulation studies and real-data analyses show that IWPLS outperforms the benchmark methods considered.

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