Interpret the impulse response when define shocks in terms of variances of the residual of the equation

Question

I’m trying to interpret the meaning of the shocks when they are written in terms of standard errors. I have constructed a multi-country Global Projections Model similar to IMF's model here. Suppose the equations for the endogenous variables are:

Y_GAP = output gap in % terms = log(real GDP) - 100* log(potential GDP)

There is a separate equation that defines how output gap relates to its lagged values and other endogenous variables with a residual term. So it is

Y_GAP = linear combination of other variables + ERROR_Y

This is equation 13 of page 12 of the paper.

In Dynare when we compute the IRFs, we define the standard errors of the residual of the equation.

If I set the standard error on ERROR_Y to be 0.1, would that mean that we are imposing a shock of 10 percent i.e. the output gap rises by 10 percent? What if I want the output gap to fall by 10 percent, how would I reverse the direction of shock so that it’s reflected in the impulse response functions? I was thinking that since the standard errors measure the variability around the mean, we can’t obviously have negative standard errors to reflect the negative output gap.

Similarly, if there is another equation that defines:

inflation_rate = some_variables + ERROR_INF

This is eq 14.

Again, if I set the standard error on this error term to be 0.1 (i,e, se(ERROR_INF)), would that mean that we letting inflation rate to rise 10 percent ? What if I want the inflation rate to fall by 10 percent, how would I reverse the direction of shock?

Answer 1

According to the linked article, the first equation is describing the output gap $y$, defined as the difference between actual logGDP $Y:=\log X$ and potential logGDP $\bar{Y}:=\log\bar{X}$, so $y = \log X - \log\bar X = \log\frac{X}{\bar X}$.

So if $y = \mu$ initially, we are left after the shock with:

$\mu+\epsilon=\log\frac{X}{\bar X}+\epsilon = \log\frac{X}{\bar X}+\log e^\epsilon$

$ = \log [\frac{X}{\bar X} e^\epsilon]$

So if $\epsilon=0.1$, if we're looking at things in terms of GDP ratios, the change is not 10 percent but $100\times e^{0.1}$ percent.

If we're looking at the output gap directly, which is defined in log-space, we can no longer talk about a percent increase/decrease, but rather just add/subtract.

(I'm assuming natural logarithms, but now that I think of it maybe the econometric community is using base 10, keep that in mind).

It's similar for Eq 14.