The RATS Software Forum

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)

In order for an ARDL to make sense, yes, the x variable can't be Granger caused by the y, otherwise you can't interpret the responses of y to x as structural.

You could use a standard Granger test, but that would have a non-standard distribution in the presence of unit roots. The GRANGERBOOTSTRAP.RPF example shows how to bootstrap a p-value for it. Interestingly, I haven't seen anyone look at the behavior of the Geweke-Meese-Dent form of the Granger causality test in the presence of unit roots (that's estimate y on lagged y, current, past and future x, test leads of x), but it looks like that should avoid the non-standard distribution problem because you get the identical test statistic whether the leads are x or dx.

bobreednz wrote: 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?

It's a common misconception, but the Granger non-causality restriction does not include a3=0 in this formulation.

Thanks for this, Tom. This has been very helpful. I have a followup question. Suppose one finds evidence that y and x are cointegrated, say through an ARDL bounds test, and then one estimates the LR relationship, say using DOLS. There still remains the question of interpreting the DOLS equation, y = alpha + beta*x, as a structural equation. That is, that x affects y, but y doesn't affect x. Your reply suggests a Granger test, bootstrapping the VAR equation to get p-values. Could you also comment on the following alternative?

Estimate the ECM model below,
Dxt = a*(EC) + b1*Dxt-1 + ... + bp*Dxt-p + c1*Dyt-1 + cp*Dyt-p.
where the EC term is the EG residual. Then test Ho: a=c1=...=cp=0. This associated F-statistic should be standard since all the variables are I(0). This seems to be what Enders (2010) suggests on page 371. If we fail to reject the null, this indicates that y does not affect x, and that the LR equation y = alpha + beta*x can be interpreted as a structural equation. Would that be correct?

That would work under those circumstances.