A consistent test of independence and goodness-of-fit in linear regression models

We propose a new approach to simultaneously test the assumptions of independence and goodness-of-fit for a multiple linear regression model (Formula presented.) say H ₀, vs. H ₁: H ₀ is false. Our approach is based on the difference between the empirical distribution function of (Formula presented.) and a consistent estimator (Formula presented.) of (Formula presented.) where (Formula presented.) satisfies H ₀ (even if (Formula presented.) doesn’t) and (Formula presented.) iff H ₀ holds. The p-value of the test is based on the resampling distribution from (Formula presented.) The new test is consistent, i.e., its power tends to 1 as the sample size increases, even when (Formula presented.) On the contrary, the consistency of existing tests is proven under special cases of H ₁, but not all cases under H ₁. Moreover, our simulation study suggests that existing tests e.g., the test in Sen and Sen (2014) and the quantile regression test can have powers (Formula presented.) for large sample sizes.