Hi,

Coming back to this issue, I think your proposed method is indeed the right one. I tried (a lot!) to express variables in terms of the growth-rates instead of levels, but unavoidably levels show up somewhere. An additional complication is the value of the levels (e.g. Z(t)) and the growth rates (dZ(t)) at the steady-state. The only way to make both consistent with a value of zero at steady-state is to express the exogenous process in terms of growth rates of de-meaned TFP. In that case, both levels and growth rates are zero at steady state and the system can be solved.

In my model I have four TFP series - 2 countries x 2 sectors (tradables/ non-tradables) and I assume a single stochastic trend across. That means that the share of tradables TFP across-countries is stationary at the steady-state as before but also the share between tradables and non-tradables TFP within country (adjusted for the labour share in each sector, if that is different). In terms of de-meaned TFP, these ratios are all unity.

Applying the above and your recommendations below, the system can be solved when set as follows:

```
parameters
dZT_ss_1 = 1;
dZT_ss_2 = dZT_ss_1 ;
dZN_ss_1 = 1 ;
dZN_ss_2 = dZN_ss_1 ;
shNT_ss_1 = 1; % share of N to T, within country ;
shNT_ss_2 = shNT_ss_1 ;
shT12_ss = 1; % share of T, across countries
end;
```

[code]

model;

% Stochastic processes

dZT1 = -0.004*ECT(-1) + eT1(-@{q}) ;*

dZT2 = 0.004ECT(-1) + eT2(-@{q}) ;

dZN1 = -0.001*ECT(-1) + eN1(-@{q}) ;*

dZN2 = 0.001ECT(-1) + eN2(-@{q}) ;

exp(shT12) = exp(ZT1)/exp(ZT2);

exp(shNT1) = (exp(ZN1)^(1/(1-alphaN_1)))/(exp(ZT1)^(1/(1-alphaT_1)));

exp(shNT2) = (exp(ZN2)^(1/(1-alphaN_2)))/(exp(ZT2)^(1/(1-alphaT_2)));

ECT = ZT1 - ZT2 - ZN1 - ZN2;

dZT1 = ZT1 - ZT1(-1);

dZT2 = ZT2 - ZT2(-1);

dZN1 = ZN1 - ZN1(-1);

dZN2 = ZN2 - ZN2(-1);

end; [/code]

ZT (ZN) stands for tradables (non-tradables) TFP and dZT - dZN for the growth rates. Any “sh…” variable stands for share, e.g. shT12 is share of tradables TFP across the two countries (country 1 and country 2) and shNT1 is the share of ZN/ZT within country 1 - adjusted for the fact that the labour share is not the same across sectors: 1-alphaT =/ 1-alphaN. ECT stands for " Error Correction Term" of the VECM.

As you can see, this formulation is consistent with both Z(j) = dZ(j) = 0 at steady-state. This of course means that dZ(j) is not only de-meaned from the constant in the short-run part of the VECM but as well as the constant in the ECT. In my understanding, I have to estimate the VECM in these terms, i.e. de-meaned dZ(j) and no constants in neither the ECT or the SR part. Of course this does not mean that the economy is growing at a zero-rate at the steady-state, but it is simply a data trick.

Note that when running this system I get three unit-roots in the list of roots that Dynare provides. Given that the model has four unit root processes and one cointegrating relation, this is what I should expect, right? My question is: is it ok to look at the results produced by Dynare, knowing the presence of the unit roots? I know that the outcomes concern the de-trended variables as they are expressed in the model, but still are these results valid?

One more word of caution: the stability of the system depends heavily on the so called loading coefficients of the VECM, i.e. those that I set to 0.04 and 0.01 in the current example. this I guess is to be expected, given that these parameters control the speed of convergence to the LR equilibrium and hence the “jumping” or not to an explosive path. When estimating these parameters, should I then restrict them “heavily” to ensure that the algorithm does not search into explosive regions of the likelihood (not sure if this kind of language is correct…).

Thanks a lot for your input, it helped a lot. Any further advice and thoughts on the above would be very much appreciated.

Best,

Kyriacos