<p>This paper examines whether the parametric regression model is correctly specified for both source and target data and whether the regression pattern in the source domain aligns with that of the target domain. This evaluation is a critical prerequisite for applying model-based transfer learning methods under covariate shift assumptions. Traditional regression model checks and two-sample regression tests are insufficient to address this issue. To overcome these limitations, the authors propose a novel adaptive-to-regression test statistic that is asymptotically distribution-free. Under the null hypothesis, the test follows a chi-square weak limit, preserving the significance level and enabling critical value determination without resampling techniques. Additionally, the authors systematically analyze the test’s power performance, highlighting its sensitivity to different sub-local alternatives that deviate from the null hypothesis. Numerical studies, including simulations, assess finite-sample performance, and a real-world data example is provided for illustration.</p>