ARDL Bounds Test

[bobreednz]

BACKGROUND: Given two variables y and x, for which one suspects there is a LR relationship such that y = beta*x, the specification for the ARDL bounds test looks like this (ignoring deterministic regressors and lagged differenced regressors):

D.yt = b1*yt-1 + b2*xt-1 + b3*D.xt + et .

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QUESTION: It is my understanding that the ARDL specification above requires that xt is not Granger caused by yt. Specifically, it requires that a1=a3=0 in the specification below* (again, ignoring deterministic regressors and lagged differenced regressors):

D.xt = a1*yt-1 + a2*xt-1 + a3*D.yt + ut.

How can I test this?

The only way I can think of to do this would be to estimate a VEC model, test that D.xt was weakly exogenous (which equates to a1=0), then estimate a VAR in differences with no error correction term (i.e., imposing the restriction of weak exogeneity), and check that causation is unidirectional, from x->y, and not from y->x. Only after I had established these facts could I then run the ARDL bounds test. Is that correct?

* I get this from Assumption 3 on page 293 of PSS and pages 406-407 in Enders (2010)