Outout gap concept in RBC

[kerkkiat]

Hello every one,

This question is not about Dynare issue but the concept of output gap in RBC theory. I am working with the paper"Bayesian estimation of an Open economy DSGE model(Adolfson et al. 2007)". Thay said that the output gap is measured as the deviation form the trend value of output in the economy.
For the further information,
/////// Non-stationary production function/////////
Y(t)=z(t)^(1-alpha)*e(t)*K(t)^alpha*H(t)^(1-alpha)-z(t)*Fixed cost

where
H(t) is working hour that is detrended by its linear trend to eliminate population growth(this is stationary variable)
e(t) is a covariance stationary technology shock
K(t) is a capital service stock(this is non-stationary variable growing with z(t-1))
Y(t) is a real GDP(this is non-stationary variable growing with z(t))
z(t) is a permanent technology shock (this is non-stationary variable)

/////// Stationary production function/////////
y(t)=mu_z(t)^(-alpha)*e(t)*k(t)^alpha*H(t)^(1-alpha)-Fixed cost
where
y(t)=Y(t)/z(t) , k(t)=K(t)/z(t-1) , mu_z(t)=z(t)/z(t-1) ,

According to this work, can I summarize the y_hat(t)=ln(y(t))-ln(y_ss)=y_gap ?
And can I summarize the z(t) = output trend = potential output?

Thank you so much for your answer,
Kerkkiat Pommin

[jpfeifer]

After detrending with permanent TFP, the steady state will be the long-run trend/potential output. Thus y in steady state, y_ss, is the potential output and the output gap is
log(y)-log(y_ss).

[kerkkiat]

Thank you so much

[econ86]

Dear all,

my question is related to the previous one. We know that the historical estimates of output gap correspond to the smoothed estimates of y_hat. Now I want to plot historical estimates of potential output. Where can I find them?

Best,

[jpfeifer]

Depending on whether you use ML or Bayesian estimation (or the calib_smoother) they will be stored in fields of oo_ that start with "Smoothed", e.g. oo_.SmoothedVariables

[econ86]

Ok, but which variable corresponds to potential output, taking into account:

/////// Stationary production function/////////
y(t)=mu_z(t)^(-alpha)*e(t)*k(t)^alpha*H(t)^(1-alpha)-Fixed cost
where
y(t)=Y(t)/z(t) , k(t)=K(t)/z(t-1) , mu_z(t)=z(t)/z(t-1) , where mu_z(t) is is the stochastic growth rate of the
unit root technology shock

I know that y_hat(t)=ln(y(t))-ln(y_ss) is output gap.

As I said, I want to plot historical estimates of potential output. Which variable corresponds to potential output?

[jpfeifer]

That depends on the concept you want to use. If the output gap is defined as

y_hat(t)=ln(y(t))-ln(y_ss)

i.e. the difference between actual output and the steady state, then y_ss is the potential output.

[econ86]

In the model I use all real variables are scaled with the trend level of technology, i.e. z_t. In the model I have the following variable mu_z(t)=z(t)/z(t-1) which is the stochastic growth rate of the
unit root technology shock. The smoothed estimates of y_hat=ln(y_t)-ln(y_ss), where y=Y/z, corresponds to the trend output gap. But I want to plot historical estimates of potential level of output - trend output. Which variable should I use?