Granger Causality Test in PROC VARMAX with Three Variables

[tugasakhir]

Suppose there are three variables, say (x), (y), and (z), and they are modeled using a VARMA model in PROC VARMAX.

For the Granger causality test, I would like to ask about the correct way to specify and interpret the test in a multivariate setting. In many examples, Granger causality is usually explained pairwise, for example whether (x) Granger-causes (y), or whether (y) Granger-causes (x).

However, if the VARMA model includes three endogenous variables, should the Granger causality test be conducted only between two variables at a time, for example:

• (x --> y),

• (y --> x),

• (x --> z),

• (z --> x),

• (y --> z),
• (z --> y)

or should it be tested in a conditional or joint sense, for example:

• whether (x) Granger-causes (y) while controlling for (z),

• whether (x) Granger-causes (z) while controlling for (y),

• or whether (y) and (z) jointly Granger-cause (x)?

In other words, when using PROC VARMAX with three endogenous variables, should the Granger causality test be interpreted as a pairwise relationship between two variables, or as a test within the full multivariate system?

Any explanation or guidance on how to correctly conduct and interpret this Granger causality test in PROC VARMAX would be greatly appreciated.

Thank you.

[SASCom1]

In PROC VARMAX, Granger Causality test is requested using the CAUSAL statement:

SAS Help Center: CAUSAL Statement

• CAUSAL GROUP1=(variables)GROUP2=(variables);

A CAUSAL statement prints the Granger causality test **by fitting the VAR(p) model by using all variables defined in GROUP1 and GROUP2**. 

So you can specify one variable or more than one variables in the GROUP1 = ( ) and/or GROUP2 = ( ) option, but the granger causality is testing the GROUP1 variable(s) influenced only by itself and not be GROUP2 variables in the VAR(p) system that include only all the variables in GROUP1 = and GROUP2= list, regardless of what additional other variables are specified in the MODEL statement, or which variables are specified as endogenous, which are specified as exogenous. You can see the examples in the above CAUSAL statement Syntax section of documentation, as well as the example in the Getting Started section:

SAS Help Center: Causality Testing

Hope this helps.