# Collinearity Problems in Identifying the Real Exchange Rate

Hi,

I’ve started working on models where international trade of assets is subject to imperfections. That is, some models assume complete financial markets, which gives rise to the Backus-Smith consumption-risk sharing relation

``q=UC_star/UC``

where q - real exchange rate, UC_star - foreign marginal utility of consumption and UC - domestic. But it typically fails to explain negative correlation between the domestic consumption and the real exchange rate and it leads to a very stable (low volatility) and rapidly mean-reverting real exchange rate (whereas in reality real exchange rates of floating currencies are extremely persistent).

So one of the ways of enhancing the model is to introduce portfolio adjustment costs. In that case, the expected change in the real exchange rate is defined from an augmented UIP condition expressed in real terms:

``1+dDELTA(-1)=(r_star(-1)/r(-1))*(q/q(-1));``

where r is the domestic real interest rate:

```1=beta*r*(UC(+1)/UC); ```

dDELTA is the slope of portfolio adjustment costs:

```dDELTA=phi*(OMEGA(+1)-STEADY_STATE(OMEGA)); ```

OMEGA is the portfolio position and r_star is the foreign real interest rate given exogenously:

```log(r_star)=(1-rho_star)*log(1/beta)+rho_star*log(r_star(-1))+sigma_star*u_star; ```

The problem is that I keep getting collinearity warnings of all the equations above when I use the augmented UIP, but not the Backus-Smith relation and I don’t understand why. The rest of the model runs smoothly without any problems and the impulse response functions seem smooth and sensible regardless of which version I use.

However, some of the results don’t make sense and in some cases the endo_simul leads to a very large gap between the theoretical mean (steady state) and the sample mean (from the simulated series over say 10000 observations).

Any idea why I might be getting such results? Many thanks.

Ok, so by toying with this set of equations I discovered that dynare has a problem with the expected real exchange rate. That is, in the above post I expressed the UIP in backward looking terms in order to avoid trivial indeterminacy, so technically it is the expected change in the real exchange rate that is creating computational problems.

If one were to simply replace the expected part with the steady state value, the collinearity disappears. In a way, it forces agents to always make a prediction error - real exchange rates are always in equilibrium except when there is a real interest rate differential, in which case they deviate from the equilibrium.

But this is not satisfactory and it doesn’t really solve the main problem, which is to predict real exchange rate volatility and persistence more accurately, because in the latter scenario they become highly unresponsive to portfolio adjustment costs.