DSGE models, after log-linearisation, have a state-space representation. In this representation, in most papers, the measurement/observation equation is simply stated.
I'm wondering how one deduces that equation, or if it's something ad-hoc, how does one proceed, or what guidlines should we use?
Any help would be appreciated.
First, you can also have a state-space setup for a model which is not log-linearised. It is the combination of the solution of your model (that you get by using Sims, Klein, Binder-Pesaran, or Blanchard-Kahn solution methods for rational expectation models) - the transition equation, which gives the transition of your state variables depending on shocks - and the measurement equation.
The goal of your measurement equation is then to match data into a (sub)-set of your endogenous variables. How this is done depends heavily on the way you defined your model, e.g. are your variables stationary or is there growth through for instance technological progress. But also how you treat your data and if you think that there is measurement error or not.
If for instance you have a log-linearised model without a trend then you know that all variables are zero in steady state and of course this should be matched by your data, i.e. mean zero and no trend. Now, what you can do is to make your data stationary (there is a debate how best to do that, for instance use a one-sided HP-filter) and use logs. This would lead to a one-to-one match between your observable and model-variable.
But as I said, there is much more to it. I would recommend the excellent guide by Johannes Pfeifer on how to specify observational equations. It is mainly written for usage in Dynare but I am sure it will help your understanding a lot. There is then also the topic of which observables to use but this goes beyond the question ;)
Hope it helps, but feel free to ask if it was not clear.
Hi: I can't speak for DSGE models specifically but, in more standard "rational expectations" econometrics, the measurement equation usually comes from some assumed linear relation between the dependent variable and the expectation of some other variable. For example, one might have
$y_t = \beta x^{*}_{t} + \epsilon_{t}$
where $x^{*}_{t}$ is the expectation of $x$ at time t so $E(x_t|t-1)$. Note that $x_t$ is often not observable so some formulation for how the expectation is generated must be assumed. So, for example, if one assumes the adaptive expectations hypothesis for the expectation, then this implies the following relation for $x^{*}_t$:
$x^{*}_t = x^{*}_{t-1} + \gamma(x_{t-1} - x^{*}_{t-1})$
Putting the two relations together, one ends up with an exponential smoothing relation for $y_t$ which would then be the measurement equation:
$y_t = \beta \times \gamma \sum_{j=0}^{\infty} (1-\gamma)^{j} x_{t-j-1} + \epsilon_t$
Almost all of above is taken from Harvey's "Econometric Analysis of Time Series" which I highly recommend not for DSGE models but for RE type econometrics. I assume that something similar is done in DSGE models but I'm not absolutely sure because I have no experience with them. Still, this may help some.
