Log-linearization

[Sigitas]

Hi,

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 4.0.3.1 (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?


Cheers,


Sigitas


P.S. DYNARE MANUAL Version 4.0.3.1 (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.

[MichelJuillard]

Dear Sigitas,

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).
1. If yo have a nonlinear equation of the form

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F(X_t)=0

,
Dynare will provide a linear approximation of the solutioni in the form

X_t = X_bar + g_x (X_{t-1}-X_bar)+...

2. If you want a log-linear approximation, you need to operate a change of variable in the original model and write

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F(exp(lx_t))=0

where

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lx_t = log(X_t)

. Then Dynare will provide a linear of the model written in log

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x_t = x_bar + lg_lx* (lx_{t-1}-lx_bar)+...

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

Michel

[Sigitas]

Thank you for a detailed and quick reply.

Sigitas