I've run across several papers which have described as "Granger causality" the test on the lagged value of x in the ARDL equation
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While superficially similar (they both test lags in a bivariate dynamic regression), they are, in fact, not similar at all in the way they restrict the dynamic properties of the series. If the bivariate VAR for x and y has both represented as univariate autoregressions, with contemporaneously correlated innovations, the lagged x in the ARDL will have a non-zero coefficient, even though x (by definition) fails to Granger cause y. And, if the coefficient on the lag is zero in the ARDL, it's almost a certainty that x actually
does Granger cause y if the coefficient on current x is non-zero. The test thus has no implications one way or the other for the causal relationship. Instead, it's the test of a particular coefficient in an ARDL model.
One other difference is that the test on the lag (or lags, there is nothing about this that is restricted to single lags) in the ARDL form isn't sensitive to the unit root behavior of the series. This is because the equation can be rewritten into the form
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The test on the lag (only) is a test on a coefficient in differenced form. By the results of Sims, Stock and Watson(1990), “Inference in Linear Time Series Models with Some Unit Roots,” Econometrica, 58, 113–144 the test has standard asymptotics even if x and y have unit roots. If you test both the current and lag jointly, this is no longer the case---you still have a restriction on a coefficient in levels, so the test statistic has non-standard asymptotics in the presence of unit roots.