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  • $\begingroup$ Thanks rolando2. You said "In order to test for this latter difference, one certainly needs to know just what that first-order interaction coefficient is". I totally agree. But there are three first-order effects: AxB, AxD, BxD. The first two make sense in theory, but BxD does not. In econometrics, will omitting BxD cause problem for identifying the effect of AxBxD? Thanks again! $\endgroup$ Commented Jan 19, 2015 at 6:57
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    $\begingroup$ You'll be relying heavily on your theory that BxD is exactly 0. I would be uncomfortable when my conclusion depends so strongly on theory and not on my observations, but others may feel differently. In any case, you should be explicit about this when writing up your results. $\endgroup$ Commented Jan 19, 2015 at 8:26
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    $\begingroup$ There's no term for $D$ alone either. It'd be a good idea to write down/plot out the model equations & make sure you're all right with the implications. For example, when $A=0$, the value of $D$ makes no difference at all to $Y$. Is zero a special value for $A$? $\endgroup$ Commented Jan 19, 2015 at 16:03