Standard deviation to percentage changes interpretation

[TomDoan]

The selection of the sign-restricted shocks has to be done on orthogonalized shocks so don't try to change that. You could normalize once you have a shock chosen, but I would suggest getting the sign restrictions working first.

[istiak]

Hi Tom
Thanks for your suggestion. After the sign restrictions are imposed, how could I normalize all the responses from a 1% increase in federal funds rate in my code? Thank you so much.
KI

[TomDoan]

Replace

ewise goodresp(accept)(i,j)=(ik=%xt(impulses,i)*i1),ik(j)

with

compute [vector] r1=%xt(impulses,1)*i1
compute i1=i1/r1(6)
ewise goodresp(accept)(i,j)=(ik=%xt(impulses,i)*i1),ik(j)

(Assuming that 6 is the slot for the interest rate). That computes the impact responses and rescales i1 so that the impact response of the 6th variable is 1.

[Andy_T]

At the risk of beating the dead horse and apologizing for bringing variations of this up again....

When estimating a VAR in which some variables are entered in logs and some are not, I have problems when trying to convert or rescale the responses such that they can be interpreted in their original units (i.e. not logged).

For example, many fiscal VARs work with variables in logs but want to interpret the results as multipliers and not in relative percentage terms. Say we have a VAR in log(G), log(Y), fin, ffr, where G is government spending, Y is real output, fin is some financial openness index and r is the federal funds rate.

The impulse responses (in a Cholesky-identified VAR) when the fiscal shock is normalized to one percent by dividing each response by the standard deviation (impact shock) of the fiscal shock would be interpreted as

the %change of Y in response to a 1% change in G;
the change (in whatever units) of fin in response to a 1% change in G;
the %change in the ffr in response to a 1% change in G.

I hope this is correct.

Now, I want to get

1. the unit change of Y in response to a unit change in G
2. the unit change of fin in response to a unit change in G.

I was thinking about this in terms of the parameter matrices of a regular regression log(Y) = A*log(G) + B*fin + error

For (1.) I would have to multiply the response of Y by Y/G because d log(Y)/d log(G) = A, which in this case would be interpreted as an elasticity (d refers to the differential operator). This can be expressed as (dY*G)/(dG*Y) = A. So to get unit changes (dY/dG = Y/G*A) I multiply by Y/G? Is this line of reasoning correct?

In case (2.) would this mean that I have to divide (the response) by G only because d fin/d log(G) = (G* d fin)/dG = B and thus (d fin/dG) = B/G? I tried this but I am not really convinced by what I get...

Y/G would have to be a scalar and thus I think most people take the sample average of Y/G (which I understand has its own problems but I would like to ignore that for now).

I don't know if this reasoning can be applied one for one to VARs and thus would greatly appreciate any feedback!

Thanks!

[TomDoan]

For (1.) I would have to multiply the response of Y by Y/G because d log(Y)/d log(G) = A, which in this case would be interpreted as an elasticity (d refers to the differential operator). This can be expressed as (dY*G)/(dG*Y) = A. So to get unit changes (dY/dG = Y/G*A) I multiply by Y/G? Is this line of reasoning correct?

Yes.

In case (2.) would this mean that I have to divide (the response) by G only because d fin/d log(G) = (G* d fin)/dG = B and thus (d fin/dG) = B/G? I tried this but I am not really convinced by what I get...

Y/G would have to be a scalar and thus I think most people take the sample average of Y/G (which I understand has its own problems but I would like to ignore that for now).

If you want to switch the levels, I wouldn't think that using averages would make sense—if you're doing that, why wouldn't you stick with percentages? In the U.S. we often hear about $100x10^9 dollars in stimulus predicted to produce something like .3% change in GDP, which is a mix of level and percentages. However, that's based upon the level of fiscal policy now as people can relate more to a hard dollar amount on fiscal policy than to %, but a % change in GDP rather than a dollar amount.

[Andy_T]

The reason for not sticking with percentages is to make the impulse responses directly interpretable as fiscal "multipliers", i.e. one dollar of G changes Y by x dollars. It is indeed very common in the literature to evaluate Y/G at the sample average.

In any case, would you say that for a specification in which the responding variable (fin) is in levels and the shock variable (log G) is in logs, would division of the impulse response of the level variable (fin) by the shock variable (not in logs, so G) make the impulse response interpretable as "fin responding by x units in response to a unit shock (if normalized of course) to G" ?

Thanks!

[TomDoan]

The reason for not sticking with percentages is to make the impulse responses directly interpretable as fiscal "multipliers", i.e. one dollar of G changes Y by x dollars. It is indeed very common in the literature to evaluate Y/G at the sample average.

If Y/G is relatively constant, then that would certainly be OK. If not, it would tend to be misleading.

In any case, would you say that for a specification in which the responding variable (fin) is in levels and the shock variable (log G) is in logs, would division of the impulse response of the level variable (fin) by the shock variable (not in logs, so G) make the impulse response interpretable as "fin responding by x units in response to a unit shock (if normalized of course) to G" ?

If it's (effectively) a unit shock in log G, then yes. If it's not, then you would have to adjust for the impact on G, that is, divide by ((impact on G) x G).

[lali62]

Hi Tom,
I have a doubt related to another variation of the above. Taking the first example for the question sake, lets say if the federal funds rate entered in log-difference and the real GDP entered in q/q saar form. Then
what do I need to do, to get the following:
1% decline in federal funds rate (level) resulting in a X% change in GDP growth rate (q/q saar, i.e same units as it was entered)

[TomDoan]

Are you sure about doing FFR in log differences? That would treat 1.75% to 1.5% the same as 17.5% to 15%. FFR (and other interest rates) are almost always done in annualized yields.

[lali62]

Are you sure about doing FFR in log differences? That would treat 1.75% to 1.5% the same as 17.5% to 15%. FFR (and other interest rates) are almost always done in annualized yields.

Hi Tom,
This was just for an example sake. Just wanted to know if the shock variable is in log-diff terms, then in that case, how to get the following
1.)1% decline in shock var result in y% decline in response var (same units as entered).

[TomDoan]

If both x and y are in 100*log differences (which is the best way to handle those), then the response to a -1.0 shock in x will be the corresponding percentage change in y.