# Unit root problem (IRF)

Hi!

I have a DSGE model, which features unit root behavior. The reason for this is that some variables such as imports are influenced by relative price levels rather than by inflation rates. All impulse response functions look correct, but a shock to foreign inflation leads to irfs that do not return to the steady state. As far as I know, this is not a problem for nominal variables such as price level or exchange rates. However, real variables such as income, investment and capital also do not return to the steady state. The deviation is not that large, but it persists. Please find attached the graphs of the irfs to the foreign inflation shock. pistar1.pdf (18.4 KB)
pistar2.pdf (16.8 KB)
pistar 5.pdf (13.0 KB)
pistar3.pdf (18.1 KB)
pistar4.pdf (14.7 KB)

There is most probably a mistake in your model. Relative prices in those types of models are stationary as are inflation rates. A shock to foreign inflation should permanently move the level of the nominal exchange rate, but not affect any real variables in the long run

The problem arises, when I add capital and investment. Without investment it is perfectly fine. My assumptions are: output is shared between consumption and investment, investment depends on Tobinâ€™s q and there is variable capital utilization. However, investment takes place at the sectoral level. I did not define an aggregate capital stock as the sum of the two sectoral capital stocks. Maybe it is this part of the model, which causes the problem.

That is strange. You often get unit root problems when the two sectors are perfect substitutes and cannot be distinguished. But having the sectoral setup destroy monetary neutrality in the long-run is unusual.

According to `model_diagnostics`, the model has 4 collinear relationships, partly caused by a unit root. Indeed, the two sectors in the model are fairly similar, but there are also some differences (e.g. tradable vs. non-tradable). Now I have changed the investment equations so that they depend on sector-specific inflation instead of overall inflation. Furthermore, they react to the sector-specific rental rate of capital instead of the nominal interest rate. This improves the irfs considerably, but the problem is not entirely solved:

pistar1.pdf (10.4 KB)
pistar2.pdf (10.4 KB)
pistar3.pdf (10.3 KB)
pistar4.pdf (10.4 KB)
pistar5.pdf (8.9 KB)

This seems to be an issue only the model-builder can solve. There seems to be an economic reasoning behind what you experience. Maybe try to work with a simpler model.

The original model even contains 6 colinear relationships, but it runs. This might have to do with the similarity of both sectors. But when I specify the nominal interest rate to be determined by the second country (ie. by assuming a currency union with the interest rate determined either by a Taylor rule or an AR(1) process), the model breaks down completely.

Extending the model with investment works more or less, but there is a small deviation of irfs from the steady state in the long run. But this model version also breaks down, if the inerest rate is assumed to be determined in a currency union with a common interest rate rule.