# Transformation for variables and estimation

Hello all,

I understand that when using exp(variable) in mod file for model equations, the variables are actually log(variable). So moments should be interpreted accordingly. My doubt is, since variables of the DSGE model are actually a cycle component of actual variable i.e. Y_t is already the cycle part, how does it make sense to take log of it? Sometimes this variable can be negative for actual data and taking log of a negative variable does not make sense. Also when matching the data estimation (say SMM), it is instructed to use y_obs = y_tilde−steady_state( y_tilde ) where y_obs is detrended and demeaned and y_tilde is the log(variable) where the variable is already in intensive form. So under this convention, there is somehow an equivalence between log of the cyclical component of the time series variable and log of the model variable. I dont understand how this is possible since log of the cyclical component of the time series might not be feasible for negative values.

so, any one has a clue whats going on here? Thanks in advance!

You can only conduct an exp substitution for variables that have a positive level. You cannot do it for variables that are negative or 0.

Dear Prof. Pfeifer,

Thank you very much for your reply. I still do not get it. For example, consumption can be negative if it is treated as intensive variable. So what is the rationale behind taking log of this in the model?

What is an intensive variable? Usually consumption (per capita or not) is assumed to be positive… It can be negative in deviation to a BGP or steady state. It’s this part that you may want to asociate to a cycle component.

Best,
Stéphane.

Dear Prof. Stéphane,

by intensive variable I meant a stationary/detrended variable. Is it correct to assume that all DSGE models consider a model variable as a stationary variable unless growth is explicitly modelled, right? then how come consumption variable in those models cannot be negative? say, consumption has a positive mean and fluctuates in a stationary manner around that mean but it can go to negative in some periods. so why we assume in those models that consumption cannot go negative although in the data it clearly can when taking the cycle part of the logged consumption say for example. I guess I am missing something here.

Consumption in deviation to a trend, C_t-C^{\star}_t or \log \frac{C_t}{C_t^{\star}}, can be negative if consumption is below the trend. It does not mean that consumption itself is negative. Note that when you detrend a model you divide (normalise) the trended variables by a trend (tipically population or TFP growth), If trended consumption is positive (as assumed in the models and as in the data) the detrended consumption is still always positive.

Best,
Stéphane.

So would that be the right interpretation to say that when we deal with stationary DSGE models like RBC models without growth, the variables like Y_t, C_t are level variables and not to be interpreted as deviation from a trend (yet stationary) and there exists no trend in those worlds?

The detrending of a model means that you ensure that the model admits a steady state for all endogenous variables. The resulting (normalised) variables are not centred around zero. You will get centred endogenous variables if you linearise the stationary model. Usually the user provides the (nonlinear) stationary version of the model and dynare takes care of the linearisation. Obviously the centred variables can be negative (if the variable in level is below the steady state). The uncentered variable (i.e. \hat c_t + \hat c^{\star}) can even be negative, but this is a shortcoming of local approximations: replacing a nonlinear function by a tangent comes at a cost (doesn’t mean that consumption can be negative, only approximated consumption is negative).

Best,
Stéphane

Thank you very much Prof. Stéphane. These have been utmost helpful.