I was wondering how to express log-deviation in DYNARE (for log-linearized model):
a) x_t=log(X_t) - log(X) or
b) x_t=X_t - X,
where X_t is a variable, X is its steady state value, and x_t is log-deviation.
It seems that a) case is correct (and I use it) but as I understood DYNARE MANUAL Version 126.96.36.199 (see the middle of page 25 or very end of this post) advocates for b) case.
If I suppose to use a) case to log-linearize the model then what does DYNARE MANUAL (i.e. ** yh_t=y_t-ys**) imply?
P.S. DYNARE MANUAL Version 188.8.131.52 (see the middle of page 25)
First order approximation
y_t = ys + A yh_t-1 + B u_t
where ys is the steady state value of y and ** yh_t=y_t-ys**.
normally, one doesn’t express deviations in Dynare. This is an outcome of the approximation procedure.
Dynare, for order=1, always performs linear approximation (there is an option to perform automatically log-linear approximation for all variables in estimation, but in general, one wants only the log-approximation for some variables).
- If yo have a nonlinear equation of the form
Dynare will provide a linear approximation of the solutioni in the form
- If you want a log-linear approximation, you need to operate a change of variable in the original model and write
lx_t = log(X_t). Then Dynare will provide a linear of the model written in log
This is equivalent to the log-linear approximation of the original model.
3. If you have already performed the log-linearization of the model by hand, then you have a linear model written in the log (-deviation) of the variables and you should enter it as a linear model.
Hope it helps
Thank you for a detailed and quick reply.