# Model nonlinear and measurement error

Hi all,

Sorry this is a very basic question but I would need some help with this. I have read the guide to specify “Observation Equations for the Estimation of DSGE Models” and I’m aware of the two different ways of modeling measurement errors. My model is nonlinear and the observation equations are of the form:

QQ_obs = (q/Q_SS-1)

Hence if I would like to add a measurement error to this observable then I would have to write (choosing the exog. shock approach):

QQ_obs = (q/Q_SS-1) * eps_ME_QQ (1)

and not

QQ_obs = (q/Q_SS-1) + eps_ME_QQ (2)

In case the model is entered in a nonlinear way would (1) be the correct approach of adding the measurement error eps_ME_QQ to the observable?

Robert

That depends on the measurement error specification you want. Most people prefer additive measurement error, i.e.

Multiplicative measurement error would be

Note that the mean of eps_ME_QQ is 0 so that your formulation that multiplies by eps_ME_QQ would be strange/false as without measurement error you would always observe QQ_obs =0.

Many thanks for your reply Professor Pfeifer. Sorry to bother you with this again, I’ve read your comment and your guide again but I couldn’t figure out why the mean of eps_ME_QQ in the multiplicative case has to be zero. For example when you write “without measurement error you would always observe QQ_obs =0.”, why would be Q_obs = 0?

Sorry again for my questions.

Robert

eps_ME_QQ is an exogenous variable that has mean 0 in Dynare. If no shock occurs, it will be 0. Therefore, if no measurement error shock occurs

Many thanks, sorry I didn’t see that. Yes that would be rather strange. One last question on the measurement error topic. If I go for the additive measurement error (either explicitly modeled or using Dynare capabilities) is there a general rule of thumb how to choose the prior mean and stderr for the estimation of the ME? Or can one do it via the upper-bound and lower bound, (e.g. 10% of the data standard deviation, lower bound = 0) for the ME.

Many thanks again for all your help and I promise that was my last question on this topic.

Robert

The way Martin Uribe does it, is using a uniform prior with an upper bound equal to some fraction of the serie’s variance. I would rather go for an informative, but somewhat diffuse prior as this will usually provide fewer problems with the measurement error running to the upper or lower bound

Dear Johannes,

Sorry for the embarrassing question but I don’t know why I can’t get my head around this. In your paper “Fiscal News and Macroeconomic Volatility”, how did you work out the std. deviation of the measurement errors? I know that we can obtain the mean of the measurement errors (upper bound + lower bound)/2, where the upper bound is fixed to some fraction of the series variance. However, I’m a bit lost how to work out the correct standard deviation.

Many thanks for your help Johannes and sorry again.

Robert

Dear Robert,
you can verify at en.wikipedia.org/wiki/Uniform_distribution_(continuous) that the variance of a uniform distribution is

The standard deviation therefore is

Dear Johannes,

This wasn’t clearly my brightest moment. Many thanks again for all your help!

Robert