At first, I want to say that I searched during long time on Internet and on several R books in order to clarify this question, but although there are some insights in some materials that I found in internet I need to be sure. I am not a coder, then to take a look to the function to see how it was coded is impossible for me.

I want to test for weak exogeneity in R. Lets consider the situation described under this link

Finding significance levels for cointegrating coefficients in cajorls

where the code is the following

library(vars) library(urca) finland # lets imagine that there are just 2 time series GDP in the first column and FDI in the second column sjf.vecm <- ca.jo(finland, ecdet = "none", type = "eigen", K = 2) # lets imagine that there is cointegration and, of course, one long run equation.

In the documentation of alrtest, this function has 3 arguments z, A and r which are a ca.jo object, a matrix containing the restrictions and the rank of the cointegration space (r=1, in this case), then my code should look like this

test_on_alpha <- alrtest(z = sjf.vecm, A = A1,r = 1)

My questions are:

How should I build the matrix A1 to test for weak exogeneity of GDP?

How should I build the matrix A1 to test for weak exogeneity of FDI?

In addition, Lets consider now 3 variables in the database finland (GDP, FDI and SAVING by this order) and r=2 (the second equation is between FDI and SAVING. Then the alrtest would be

test_on_alpha <- alrtest(z = sjf.vecm, A = A1,r = 2)

This leads to my last question:

How the inclusion of the second equation modifies the settings of the matrix A1?

I read this lecture https://sites.ualberta.ca/~sfossati/e50 ... /Lec12.pdf

but still is not clear for me.

Very much thank you in advance

Lastly, If someone knows about a book or link with a clear explanation about this, (s)he can go ahead and just provide the link or the title of the book. I really understand that we are all busy and time is very scarce resource.