Expected value sign

[dchung]

I am a very first user of Dynare. As I see an example (p.18) in the User Guide, the first equation (Euler equation of consumption) in the model block does not have an expected value sign. I wonder why it is so?

[jpfeifer]

Hi, the reason is Dynare's timing convention. Everything that has a time index of (+1) like c(+1) in the Euler equation is actually E_t(c(+1)), i.e. Dynare automatically adds the conditional expectation to all leads of variables.

[dchung]

This message is deleted.

[dchung]

Hi, the reason is Dynare's timing convention. Everything that has a time index of (+1) like c(+1) in the Euler equation is actually E_t(c(+1)), i.e. Dynare automatically adds the conditional expectation to all leads of variables.

Thanks for a quick reply. But I am still not convinced. Expected value of a function of variables is not the same as the function of the expected value of variables, ie, E [ f( c(+1), r(+1) ) ] is not the same as f( E[c(+1)], E[r(+1)] ). The Euler equation is in the former form, and the expression in the Dynare is in the latter form, if that is what you say.

By the way, would you please let me know where in the Dynare documentation, User guide or Manual, I can find info on this matter?

[jpfeifer]

Sorry for being imprecise. I am saying that in the context of a nonlinear model Dynare treats expressions that involve something like f( c(+1), r(+1) ) as E_t[ f( c(+1), r(+1) ) ]. Essentially, you can think of an conditional expectation around both sides of every equation. However, as is standard in perturbation techniques (see Schmitt-Grohé/Uribe (2004)http://www.columbia.edu/~mu2166/2nd_order.htm) the function f (and the policy functions) are approximated with a Taylor polynomial inside of the expected value. Hence, up to first order E [ f( c(+1), r(+1) ) ] is the same as f( E[c(+1)], E[r(+1)] ) because the system is linear and there is no Jensen's Inequality effect. For higher order approximations, see the linked article.

If you search the forum, you will find several answers regarding this issue. Unfortunately, I am not aware of any official documentation regarding this issue. This convention is rather taken for granted.