This paper provides two groups of conditions of model consistency in least-absolute-deviation mediation models. Under model consistency, we establish the asymptotic theory of the difference estimator and the product estimator, and show that the two estimators are not only numerically nonequivalent but asymptotically nonequivalent, which is dramatically different from the situation in the least squares mediation analysis where these two estimators are numerically equivalent. In all three possible scenarios of model parameters, both the asymptotic theories and simulation studies show that the product estimator is more efficient than the difference estimator.
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