  As a crucial problem in statistics is to decide whether additional variables
are needed in a regression model. We propose a new multivariate test to
investigate the conditional mean independence of Y given X conditioning on some
known effect Z, i.e., E(Y|X, Z) = E(Y|Z). Assuming that E(Y|Z) and Z are
linearly related, we reformulate an equivalent notion of conditional mean
independence through transformation, which is approximated in practice. We
apply the martingale difference divergence (Shao and Zhang, 2014) to measure
conditional mean dependence, and show that the estimation error from
approximation is negligible, as it has no impact on the asymptotic distribution
of the test statistic under some regularity assumptions. The implementation of
our test is demonstrated by both simulations and a financial data example.
