Could you have a look the attached historical and smoothed variables plot , there is no measurement error in the model. Does it look weird as for i_obs and h_obs ? Generally why does it happen?

One reason why this may happen is that you have a deterministic relationship between two variables in the model that are being fed in as observables. This would generate the observed patterns, even in the absence of stochastic singularity (i.e. even if you have #shocks>=#observables).

@kipfilet What you describe is stochastic singularity. Having as many shocks as observables is only a necessary condition, but not sufficient. If there is a linear combination only involving observables the forecast error variance matrix will be singular.

In my model, I have 6 observables, 5 structural shocks and 2 measurement error shocks, one measurement error is attached to output (more shocks than observables). I construct the data output as Y=C+I+G (not Y=C+I+G+NX) which is consistent with my model.

If I have this output measurement error, every observable matches perfectly in “Historical and smoothed variables” plot. However, if there is no output measurement error, the government spending variable does not match perfectly well (the red line and black line have some gap). Graphs are attached, please have a look.

Is this plot reliable if the red line and black line have some gap? If this is a problem, must I add output measurement error?

Is this case calling stochastic singularity, even though shocks=observables?

Why this problem can happen? I do not think I have linearity since there is no two shocks simultaneously appear in one equation.

Dear Catherine, this is the prototypical stochastic singularity example. A necessary condition is to have as many shocks as observables. But this is not sufficient. In addition, there must not be a perfect linear relationship between the observables. If you have

and you observe all variables on the right, they are going to imply a particular value for Y at every point in time. If you now observe Y at the same time without measurement error, you have exactly this issue. For this reason, you need measurement error.