IRF as percent from baseline

[lali62]

I have a doubt related to expressing IRFs as "percent from baseline" as seen in the following link in Fig 3:
https://www.federalreserve.gov/econres/ ... 190904.htm

Some of the variables in the model are in logs (and linearly detrended) and some are in logs.
The impulse variable is also in log and similarly the response variable.
I assume the responses could be seen as in percentage change, i.e 1sd shock to log x generates a response of .01 in log y means a 1% change in y.
Now if standard devn. of the impulse variable log x is a, then if i want to see the shock in b units, can I multiply the responses by b/a and assume them to be 100 * (b/a) change in y

How do I interpret this "percent from baseline" in the responses.

[TomDoan]

The standard way to do the log transformations for VAR's is 100*log(..) which makes their responses in percentages. I assume that's what they did.

[lali62]

So instead of log x (impulse variable) and log y (response variable). I would now enter 100 * log x and 100 * log y in the model, so that responses will be in percentages to a 1SD shock in 100 * log x.

Also in order to show a response to a specific size of shock a in x (i.e defined in levels, shock of 150 in the article). So I would basically multiply my responses by
shock_size_rescale_factor = (100 * log(a))/std devn of 100 * log(x). Is this correct.

Also if i want to see the confidence bands to this rescaled shock of size a. Would I multiply the standard errors by shock_size_rescale_factor.
So rescaled standard error=1.96 * shock_size_rescale_factor * std_err
and then upper confidence band = rescaled responses + rescaled standard error
lower confidence band = rescaled responses - rescaled standard error

[TomDoan]

If you want a specific size shock, why don't you put that in, rather than trying to rescale the responses afterwards?

[lali62]

Hi Tom,
The shock size is defined in terms of the original levels of the impulse series x. But we enter the variable in log(x) (or 100 * log(x) as you have suggested) in the VAR model.
If I am entering 100 * log(x) in the VAR model, and if I want to see the response of 100 * log(y) to the shock of a units in x, is it ok to
multiply the responses by (100 * log(a))/std devn of 100 * log(x). Also could the same be done for the standard errors as mentioned above.

Thanks

[TomDoan]

You seem to be overinterpreting a label on a graph. All that IRF's are is the (linear) difference between the behavior of a system with and without a shock. Because of the linearity of the system, the actual baseline (initial conditions) have no effect on that difference. If the target variables are in 100*log(..) then that linear difference can be interpreted as percentage differences. That's all they are doing.

If the variable that you're shocking is in 100*log(..) (theirs isn't), then shocks are in percentage terms as well. If one standard deviation is 2.5, then a one-standard deviation shock is a 2.5% shock. You cannot do IRF's to a level shock in a variable that is in the model in logs---you would have to specify the base level so you could convert it to percentages.

[lali62]

Hi Tom,
Thanks for explaining the above.
I am a bit confused related to the interpretation and would apprecitae your help on this.

Lets say, I remove a constant trend from policy uncertainty index and then take 100 * logs.Lets call this transformed variable as pu_tf and orginal variable in levels as pu.
For US Industrial Production, I remove a linear trend and then take 100 * logs.Lets call this transformed variable as ip_tf and orginal variable in levels as ip.

Now a 1SD shock to pu_tf gives a response of 0.8 at month 12 for ip_tf. Since pu_tf is in 100 * logs,
is it fair to say that 1 sd shock in pu_tf, is equal to a 1% shock in pu which causes a 0.8% decline in ip at month 12.

Also lets say if I want to see the response to a specific shock of pu, lets say of size "a" units, then I will multiply the response by 100 * log (a)/std devn (pu).
Should the std deviation be for pu or pu_tf.

Thanks

[TomDoan]

First of all, are you saying that you're removing the trend, then taking logs? How would that even work?

Now a 1SD shock to pu_tf gives a response of 0.8 at month 12 for ip_tf. Since pu_tf is in 100 * logs,
is it fair to say that 1 sd shock in pu_tf, is equal to a 1% shock in pu which causes a 0.8% decline in ip at month 12.

A 1 s.d. shock in pu_tf is a 1 s.d. shock in pu_tf; it does not correspond to any specific size shock in the underlying level, 1% or otherwise.

Also lets say if I want to see the response to a specific shock of pu, lets say of size "a" units, then I will multiply the response by 100 * log (a)/std devn (pu).
Should the std deviation be for pu or pu_tf.

If you want to see the response to a specific shock in pu, you can't be analyzing the model in detrended logs. Period. If the level of PU is 10, then a shock of 1.0 is a 10% shock, while if the level of PU is 1000, it's a .1% shock, and the response of the system to those is likely to be very, very different.That's why most macro series are modeled in logs---because percent changes are more meaningful than absolute changes.