# How to get standard deviation in percentage

The version of Dynare in my computer is 4.3.2.

I write the model in normal forms and conduct the first order approximation. According to one discussion in this forum named “Moments”, the standard deviation shown in the results is in the model units. I wonder how I could convert it into the standard deviation in percentage. Is it simply std in percentage = std in units/ mean *100% ?

I know that I can do that by rewrite my model in “exp-log” forms. However, the problem is that two valuables in my model have values of zero or negative. Then when I attempt to rewrite the model using log-linearization, Dynare is not able to find the steady state and so cannot solve the model further.

Please help me either get the standard deviation in percentage from the present one or teach me how to solve the log-linearized model with the problem I mentioned.

Thanks a lot!

See Remark 19 (Variables with Negative Steady States or already in Percent) in Pfeifer (2013): “A Guide to Specifying Observation Equations for the Estimation of DSGE Models” sites.google.com/site/pfeiferecon/Pfeifer_2013_Observation_Equations.pdf

First thanks for your kind reply! The material you mentioned is really helpful. Actually I’m working on the model with the government sector imposing the distortionary taxation and issuing debt. So Remark 19 fits my case perfectly. I just want to clarify some small doubts.

1. Is it possible that I write other variable in exp() and keep government bonds and tax rates in levels and then run the model with variables in levels and in percentage mixed together?

2. Anyway I can simply write the model entirely in levels. Now I get the following moments results from simulation.

VARIABLE MEAN STD. DEV. VARIANCE SKEWNESS KURTOSIS
k 1.381102 0.045884 0.002105 0.330845 0.285354
n 0.332501 0.001285 0.000002 0.104309 0.069441
z 0.992393 0.022162 0.000491 0.151072 0.202688
c 0.278181 0.008050 0.000065 0.330593 0.267004
Y 0.520441 0.017008 0.000289 0.266787 0.224835
r_tr 0.120615 0.001565 0.000002 -0.375939 0.712287
r 0.120590 0.001724 0.000003 -0.373474 0.646802
R 1.020615 0.001565 0.000002 -0.375939 0.712287
q 0.979754 0.001203 0.000001 0.409094 0.802141
w 1.064322 0.033575 0.001127 0.209204 0.243539
w_tr 0.855896 0.026509 0.000703 0.425119 0.276798
b -1.740879 0.108770 0.011831 -0.666389 0.033384
taon 0.195887 0.005748 0.000033 -0.622081 -0.370384
taok -0.001266 0.022573 0.000510 0.048934 0.193397

Remark 19 implies that the standard deviation in percentage of tax rates, taon and taonk, should be just 0.005748 and 0.022573, respectively. But this statistics of other variables should be obtained by dividing the standard deviation here by the mean and then multiplying the result by 100%. Am I right?

1. Yes, that is possible
2. Yes, standard deviations of taxes will already be in percentages. Regarding the transformation you suggest: up to first order and for small volatilities you are correct. You are basically performing a Taylor approximation of log(x) about the steady state xbar inside of the standard deviation:

Thanks! I do the first order approximation so I think I can do that transformation.

I find another question in my model:
Basically I assume that the agent is endowed with total time 1. Do I have to regard labor as a variable already in percentage like tax rates when I try to obtain standard deviation in percentage? I have seen some Dynare codes also write labor in exp(). But in my eyes, the labor is actually a ratio.

Thanks again!

Yes, labor is a ratio. But people are tyically interested in fluctuations relative to the amount of hours worked (about 0.2 of the 1 unit endowment) instead of relative to the endowment of 1 unit.

I understand. Thanks a lot!