theoretical mean of Welfare with second order approximations

[zhangwp]

in the model section of a dynare mod file ,i add a line
U=c-exp(n)^(1+v)/(1+v)+beta*U(+1)
when solving the model with second order approximations under stochastic shock,
i can get the theoretical first moments of U in oo_.mean.
and the theoretical mean of U in oo.mean is larger than U's steady state value.
is there someting wrong?

Anyone help me. Thanks.

[jpfeifer]

No, unless you are using Dynare 4.2.3, which had a problem with second order approximations, there is nothing wrong. At first order, you have certainty equivalence, i.e. uncertainty plays no role. At second order, there is an uncertainty correction term (stored in ghs2), which comes from the second partial derivative w.r.t. to the perturbation parameter. In Schmitt-Grohe/Uribe 2004 notation (http://www.columbia.edu/~mu2166/2nd_order.htm), this comes g_sigmasigma and h_sigmasigma.

[zhangwp]

No, unless you are using Dynare 4.2.3, which had a problem with second order approximations, there is nothing wrong. At first order, you have certainty equivalence, i.e. uncertainty plays no role. At second order, there is an uncertainty correction term (stored in ghs2), which comes from the second partial derivative w.r.t. to the perturbation parameter. In Schmitt-Grohe/Uribe 2004 notation (http://www.columbia.edu/~mu2166/2nd_order.htm), this comes g_sigmasigma and h_sigmasigma.

Thank you for your reply !
the question is :
If welfare under stochastic shock is larger than welfare under steady state. does this means the stochastic shock is a good thing ?

[jpfeifer]

It depends on the type of uncertainty. For example, if you model technology as a lognormal process, i.e.

Code: Select all

Y=z*K(-1)^alpha*L^(1-alpha)


with

Code: Select all

log(z)=rho*log(z(-1))+eps_z

where eps_z~N(0,sigma), then welfare will be higher. The reason is that technology is convex. If the world is deterministic, technology will always be z=exp(0)=1. But if there are shocks to z, the unconditional mean of z will be exp(0+1/2*sigma^2) which is larger than 0. Hence, if there are shocks, technology is higher on average due to a Jensen's Inequality effect and welfare will usually be higher too (depending on how much the agent dislikes fluctuations). However, this is not the case for all types of uncertainty.

[zhangwp]

It depends on the type of uncertainty. For example, if you model technology as a lognormal process, i.e.

Code: Select all

Y=z*K(-1)^alpha*L^(1-alpha)


with

Code: Select all

log(z)=rho*log(z(-1))+eps_z

where eps_z~N(0,sigma), then welfare will be higher. The reason is that technology is convex. If the world is deterministic, technology will always be z=exp(0)=1. But if there are shocks to z, the unconditional mean of z will be exp(0+1/2*sigma^2) which is larger than 0. Hence, if there are shocks, technology is higher on average due to a Jensen's Inequality effect and welfare will usually be higher too (depending on how much the agent dislikes fluctuations). However, this is not the case for all types of uncertainty.

Thank you very much. love you!

[ahnulxy]

No, unless you are using Dynare 4.2.3, which had a problem with second order approximations, there is nothing wrong. At first order, you have certainty equivalence, i.e. uncertainty plays no role. At second order, there is an uncertainty correction term (stored in ghs2), which comes from the second partial derivative w.r.t. to the perturbation parameter. In Schmitt-Grohe/Uribe 2004 notation (http://www.columbia.edu/~mu2166/2nd_order.htm), this comes g_sigmasigma and h_sigmasigma.

I am wondering theorectical mean has something to do with the oo_.dr.ghs2 ? It should be oo_.gamma_y{nar+3}, ie. the last element in oo_.gamma_y in 2nd approximation, which seems to be different from oo_.dr.ghs2 . The second should be the shift effect of the variance of future shocks, the first should be mean correction term. And we could get the theoretical mean by summing up the steady states and oo_.gamma_y{nar+3} not oo_.dr.ghs2.

[jpfeifer]

You are looking for the fixed point for x to
x_t=g_x*x_{t-1}+1/2*g_{xx}*x_{t-1}^2+1/2*g_{\sigma\sigma}
While the term oo_.dr.ghs2 gives you the 1/2*g_{\sigma\sigma}, i.e. the constant part of the quadratic equation above, you still need to solve that quadratic equation for x.

[ahnulxy]

You are looking for the fixed point for x to
x_t=g_x*x_{t-1}+1/2*g_{xx}*x_{t-1}^2+1/2*g_{\sigma\sigma}
While the term oo_.dr.ghs2 gives you the 1/2*g_{\sigma\sigma}, i.e. the constant part of the quadratic equation above, you still need to solve that quadratic equation for x.

Thanks very much! I need further reading on the that 2nd approximation paper.

[jpfeifer]

You might want to take a look at
Andreasen, Martin M., Jesus Fernandez-Villaverde, and Juan F. Rubio-Ramrez.
2013. "The Pruned State-Space System for Non-Linear DSGE Models: Theory and Empirical
Applications." NBER Working Papers 18983.

and

Lan, Hong, and Alexander Meyer-Gohde. 2013. "Pruning in Perturbation DSGE
Models - Guidance from Nonlinear Moving Average Approximations." SFB 649 Discussion
Papers 24.

[javiergc]

It depends on the type of uncertainty. For example, if you model technology as a lognormal process, i.e.

Code: Select all

Y=z*K(-1)^alpha*L^(1-alpha)


with

Code: Select all

log(z)=rho*log(z(-1))+eps_z

where eps_z~N(0,sigma), then welfare will be higher. The reason is that technology is convex. If the world is deterministic, technology will always be z=exp(0)=1. But if there are shocks to z, the unconditional mean of z will be exp(0+1/2*sigma^2) which is larger than 0. Hence, if there are shocks, technology is higher on average due to a Jensen's Inequality effect and welfare will usually be higher too (depending on how much the agent dislikes fluctuations). However, this is not the case for all types of uncertainty.

How would you do a welfare cost of business cycles analysis in this case then? I thought this is the way technology is usually specified (i.e. log normal), and people do perform the analysis using that specification. How can they do that? Do you have any reference on how to proceed?

[jpfeifer]

If you look at Lucas (2003): Macroeconomic Priorities you can see in equation (1) that his log-normal process for TFP accounts for the mean shift by correcting the level for volatility.