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Lack-of-fit checking for parametric and semiparametric models is essential in reducing misspecification. The efficiency of most existing model-checking methods drops rapidly as the dimension of the covariates increases. We propose to check a model by projecting the fitted residuals along a direction that adapts to the systematic departure of the residuals from the desired pattern. Consistency of the method is proved for parametric and semiparametric regression models. A bootstrap implementation is also discussed. Simulation comparisons with several existing methods are made, suggesting that the proposed methods are more efficient than the existing methods when the dimension increases. Air pollution data from Chicago are used to illustrate the procedure.