Ratios or log of ratios

[hewei2004]

Hi,

Suppose I want to estimate a stochastic dynamic system with order 3. And I want to stationarize variables, say Price/Consumption ratio, and both Price and Consumption grow at the same rate, thus P/C is in fact a stationary variable.

Suppose P/C follow the following expression:
P/C=E_t(Mt+1*Dt+1)

I wonder if it is better to write in dynare++, the above equations in terms of log(P/C) or just P/C
A.P/C=Mt+1*Dt+1
B.log(P/C)=log(Mt+1)+log(Dt+1 )
C.exp(log(P/C))=E_t(Mt+1*Dt+1)

My question is the following, please tell me, for each of A, B and C, if the expression is correct, AND, other comments on the advantage/disadvantage of such expression

[jpfeifer]

1 and 3 are correct, because they are equal to the original equation. 2 is wrong because it ignores Jensen's Inequality when pulling the log inside the expectations operator. However, in 3 the exp and the log cancel and are redundant. I guess the alternative you were looking for is an approximation in logs:
exp(P)/exp(C)=exp(Mt+1)*exp(Dt+1)
so that all variables are redefined as their logs and everything will be in percentage deviations from steady state. This may sometimes deliver a better approximation. But the main advantage is the better interpretability of the results as percentage deviations from steady state.

Finally, note that if your model is growing, you can only work with
PdivC=E_t(Mt+1*Dt+1)
i.e. the price consumption ratio must be treated as a single variable PdivC, because its components are non-stationary and cannot be determined separately.