
Toda, H. Y. and T. Yamamoto (1995). "Statistical inference in vector autoregressions with possibly integrated processes." Journal of Econometrics 66: 225-250.



I think the above paper should also help, for testing granger-causality among non-stationary variables. 
 
JHK
> Date: Wed, 13 Aug 2008 17:01:05 -0500> From: markleeds@verizon.net> To: niheaven@hotmail.com> CC: r-sig-finance@stat.math.ethz.ch> Subject: Re: [R-SIG-Finance] Granger Causality Test> > i think your two statements deserve separate answers.> > 1)> > granger causailty is only used ( as far as i know ) in the context of > deciding on which variables in a previusly estimated VAR cause another> in the granger sense. ( defined in Lutkepohl ) . But, since it's a n > already estimated VAR, you should have already decided whether the> series under study needed to be differenced or not in order that the VAR > estimation is done on variables that are stationary.> > So, granger causality doesn't really have much to do with stationarity, > I don't think, because that should have been worked out during the> estimation fo the VAR.> > 2)> > in the 70's before cointegration was discovered, people used to > difference time series variables in order to make them stationary (> for example , if they were prices, they would make them differenced > prices ). but, it was realized that, if one did this, then the > information about levels of the> series was lost and it was always felt that there must be a better > approach so the levels info was not abruptly discarded.> > then, engle and granger figured out in the early 80's that there were > cases where two variables could be non-stationary and yet> the regression of the two on each other could still be valid > statistically ( cointegration ). So, they were able to figure out a way > to> keep the levels in the specification by rewriting the regression > relationship in the form of an error correction model. This was then > extended beyond the> bivariate case to what is called a vector-error correction model ( VECM > ) which is like a VAR but slightly different and I couldn't do the > description of a VECM justice even if i tried so I won't.> > There is a lot of econometrics literautre in this area that talks about > this in a much more eloquent way than I could.> > Some good books in order of increasing difficulty ( atleast to me ).> > Enders> Zivot ( S+Finmetrics book )> Hayashi> Lutkepohl> Hamilton> > To me, Hamilton and Lutkpohl are similar in difficulty and require a > larger time investment than the others. Enders is the most basic but it > gives nice intuition and is good for an intro. Zivot is more general in > that it covers various time series topics but also provides a brief but > nice discussion on the topics above.> > > > > > On Wed, Aug 13, 2008 at 5:23 PM, Hsiao-nan Cheung wrote:> > > Hi,> >> >> > I have some question that whether only stationary series could do > > Granger> > Causality Test. Is there any exception?> >> >> > Since I��ve found somewhere that to make a time series stationary, the> > differential series may has little economic meaning.> >> >> > Hsiao-nan Cheung> >> >> >> >> > [[alternative HTML version deleted]]> >> >> >> > ------------------------------> >> > _______________________________________________> > R-SIG-Finance@stat.math.ethz.ch mailing list> > https://stat.ethz.ch/mailman/listinfo/r-sig-finance> > -- Subscriber-posting only.> > -- If you want to post, subscribe first.> > _______________________________________________> R-SIG-Finance@stat.math.ethz.ch mailing list> https://stat.ethz.ch/mailman/listinfo/r-sig-finance> -- Subscriber-posting only.> -- If you want to post, subscribe first.
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