Hi Everybody,
I recently saw a paper in which a model was log-linearized around its steady-state before being solved. I wonder if one can get the same results by doing a (first or higher order) perturbation directly on the model in Dynare without log-linearizing it first. I reckon what I want to ask is why one would need to log-linearize a model? Can it not be directly solved using perturbation in Dynare?
Obviously, log-linearization is equivalent to first-order perturbation on variables after taking log. Or, the result of linearization is equivalent to that of first-order perturbation, though based on different math foundation.
Moreover, there’s an abundant literature comparing log-linearization and linearization that you can Google.
You also asked
if one can get the same results by doing a (first or higher order) perturbation directly on the model in Dynare without log-linearizing it first
The answer is yes, with one extra step. The endogenous variables in log-linearized models will have the interpretation as “percent deviation from steady state”, while those from a “directly linearized” model will have the interpretation as “deviation from steady state”. So by itself you will not get the same results. But you can obtain them by defining additional auxiliary endogenous variables which will then have the interpretation as percent deviation from steady state e.g. if Y_{t} is the level of output then define some new variable \hat{Y}_{t} \equiv \frac{Y_{t}-\bar{Y}}{Y_{t}} \approx \log\left(\frac{Y_{t}}{\bar{Y}}\right) which will then have the desired interpretation.
So if you write down the nonlinear model from that paper supplemented by auxiliary variables as above, calibrate at the same parameter values, and ask Dynare to solve/simulate, you should be able to easily reproduce IRF’s from that original paper.