I noticed there are similar inquiries on the collinearity when there is imperfect risk sharing between countries. But I still don’t fully get it. Suppose 2 LOE with both home and foreign bonds available to both home and foreign households. Then there are four Euler equations derived as follows.

(1) MU = beta *(1+i)*MU(+1);
(2) e*MU= beta

*(1+istar)*MU(+1)*e(+1);

(3) MUstar= beta

*(1+istar)/*MUstar(+1);

(4) MUstar/e = beta *(1+i)*MUstar(+1)/e(+1);

MU is marginal utility of consumption. star refers to the foreign country.

I understand (1)(4) are redundant in the steady state, as well as (2)(3). UIP derived from (1)(2) will be redundant to the UIP derived from (3)(4) in the first order approximation. But I don’t think there would be collinearity in the orders of approximation above one.

So I am confused why dynare reports 2 collinearities in (1)-(4) when I set a second order approximation, even though it can generate steady state. Further, it works if I modify (2)(4) to

(5)e/mu = beta *(1+istar-0.0007*b)/ mu(+1)*e(+1);

(6)1/mustar/e = beta *(1+i-0.0007*bstar)/ mustar(+1)/e(+1);

Cheers!