Standard deviation to percentage changes interpretation

[jacarandas]

Hi,

I know this question has been brought up in an earlier post I read. But, there are some things I am still unclear about and are always coming back to haunt me. I am just not really good with rescaling and units.

I understand that the responses of variables are interpreted in the units they have entered the model. For example, if I have a one standard deviation negative shock in the federal funds rate (where the standard deviation is equivalent to 0.07), resulting in a 0.0489 response in real GDP growth. My federal funds rate have entered the model in % as it is, and real GDP entered the model as a log first differenced term (which amounts to growth).

To convert this to percentage terms, am I right to say that a 1% decline in federal funds rate results in a 0.69% change in the GDP growth rate (0.0489/0.07)?

And, if I want to get a 1% decline in federal funds rate resulting in a X% change in GDP, do I have to cumulate my GDP growth responses before rescaling them to %?

All my variables have entered the model differently in order to ensure stationarity in the system. Some are in log levels and some are in log first differences. Do I always interpret the responses in the units they entered the system? For example, in the above case of the GDP growth rate, it is a little difficult to interpret a change in the GDP growth rate, and what that really mean, as compared to a % change in GDP?

Also, what happens if my variables are transformed with either log levels, or log first differences, and demeaned and standardised as well? How can I interpret my variables?

Would appreciate any help on this.

Thanks!

[TomDoan]

Hi,

I know this question has been brought up in an earlier post I read. But, there are some things I am still unclear about and are always coming back to haunt me. I am just not really good with rescaling and units.

I understand that the responses of variables are interpreted in the units they have entered the model. For example, if I have a one standard deviation negative shock in the federal funds rate (where the standard deviation is equivalent to 0.07), resulting in a 0.0489 response in real GDP growth. My federal funds rate have entered the model in % as it is, and real GDP entered the model as a log first differenced term (which amounts to growth).

To convert this to percentage terms, am I right to say that a 1% decline in federal funds rate results in a 0.69% change in the GDP growth rate (0.0489/0.07)?

Correct.

And, if I want to get a 1% decline in federal funds rate resulting in a X% change in GDP, do I have to cumulate my GDP growth responses before rescaling them to %?

Because of linearity, the order doesn't matter. You can either rescale then accumulate, or accumulate then rescale.

All my variables have entered the model differently in order to ensure stationarity in the system. Some are in log levels and some are in log first differences. Do I always interpret the responses in the units they entered the system? For example, in the above case of the GDP growth rate, it is a little difficult to interpret a change in the GDP growth rate, and what that really mean, as compared to a % change in GDP?

If GDP is in growth rates, then the response of (log) GDP has to be accumulated from the responses of its growth rate.

Also, what happens if my variables are transformed with either log levels, or log first differences, and demeaned and standardised as well? How can I interpret my variables?

Taking the means out has no effect. The rescaling has to be reversed out; if x*=x/lambda, then the response of x is the response of x* times lambda.

[jacarandas]

jacarandas wrote:Also, what happens if my variables are transformed with either log levels, or log first differences, and demeaned and standardised as well? How can I interpret my variables?

Taking the means out has no effect. The rescaling has to be reversed out; if x*=x/lambda, then the response of x is the response of x* times lambda.

Thanks for clarifying. I thought I'll clarify one more thing.

My Y variable is not standardised or demeaned, but only transformed. But, my X variables are transformed, demeaned and standardised. Xst=(X-mean)/stddev
So, if I do not rescale the variables, my variables are interpreted as:

One Cholesky SD factor shock results in a Z SD response in my X variables (since they are standardised variables)?

What if I do want to rescale the variables to be interpreted in percentage terms? Is this what I do?
1. Transform each of my X responses such that Actual X response = (X response from model *stddev)+mean? If the mean has no effect, do I still have to add the mean back?
2. Step 1 ensures that my Y and X are now in the same terms - not standardised or demeaned, just transformed.
3. To get my percentage interpretation, I have to divide the Actual X responses by the std deviation of the structural shock of Y? This would give me a 1% change in Y resulting in Z% change in X variables?
4. And, if my actual X responses are in growth rate terms, I will have to accumulate the responses in Step 3 to get my interpretation in log level terms.

Would that be right?

I actually tried the above steps. But, the responses of my variables that are entering the model as log transformed variables or as it is (as they are percents) are having some huge % changes after rescaling, while those that are log first differenced are fine. For example, I have dwelling units in my X variables, the responses are telling me that a 1% change in Y variable is giving me a 35% change in log(dwelling units). What could be wrong?

Apologies for having a lengthy question. I would really just like to clear this issue that has been bugging me for ages up. Thanks

[TomDoan]

Rather than go into the details of how the standardizing affects the interpretation of the shocks, I'm curious why you would want to standardize the variables in the first place. It doesn't change the results; it just makes interpreting them much harder. The "standard errors" in the VAR are standard errors in the residuals from the VAR, not standard errors in the raw variables.

[jacarandas]

Hi Tom,

I am actually estimating a FAVAR, and it's better to standardize the variables that goes into the estimation of the factors as they are all from very different scales. Hence, I am struggling with the rescaling. Hope you'll be able to help

Thanks!

[TomDoan]

If you are using shocks with "standard error" sizes (Cholesky or any other covariance matrix factorization), then the rescaling of the variables doesn't affect the interpretation of the "shock" (the rescaling is reversed out automatically); it only affects the interpretation of the response, which is now in terms of the rescaled data. So if you are looking at the response of y*=y/lambda, you need to scale up the response by lambda to get the response of y itself. Removing the mean (on either the shocked variable or the response variable) has no effect on the responses since an impulse response looks at the difference between the behavior of the variable with or without the shock---the mean shift is present in exactly the same way with and without the shock and so cancels out.

If you're confused about the first part of this (that the rescaling is reversed out of the shock), think about what happens in a standard regression if you rescale one of the regressors. The coefficient changes (if you multiply X by k, the coefficient is divided by k), but the effect of the variable doesn't change.

[jacarandas]

Thanks Tom! I think I understand the part on having to scale up the response now. It makes sense. But, it does mean that my rescaled responses would then be interpreted as standard deviations from the respective means in responses to one standard deviation shock?

What if I want to back out a 1 percent change in the shock resulting in X percent change in the responses? Do I just dividing the shock and the rescaled responses by the std deviation of the shock?

I have seen a few studies that have backed out percentage deviations from baseline (for which logs were taken) and percentage points deviations otherwise. I thought that is a rather intuitive way to interpret the responses. How could I actually do that, given that dividing the responses by the std deviation gives you percent changes?

Thank you so much. Much appreciated.

[TomDoan]

You can only interpret the responses as percentage changes if the data are put into the model in logs. If a series is in the model as log y, a response of .01 means 1 percent on y. If you've transformed the data from log y to (log y)/lambda, then a response of .01 means 1 x lambda percent of y.

Otherwise, they are in whatever are the units of the variable. If, for instance, you put an interest rate series in as its levels, the responses will be in points. If you transform an interest rate series y to y/lambda, a response of 1 means lambda points. If you have a series which is annualized growth rates, the responses will be in annualized growth rates. If you've transformed data from growth(y) to growth(y)/lambda then a response of 1 means a growth rate of lambda percent.

[jacarandas]

Thanks! That's so much clearer. Just a few last questions, or rather just wanting to check my understanding.

1. When you say points, I assume you are referring to percentage points?

2. I understand that the impulse responses are interpreted in the units they are in, as you have just shown me. So, if I have a one standard deviation shock in the Federal Funds Rate (equivalent say to 30 basis points), then that would be how I interpret the responses. But, if I want to convert that one standard deviation shock to a 1 percent shock, am I right to say that I will have to divide the shock and the impulse responses by the standard deviation of the shock? However, the interpretation of the impulse responses after dividing by the standard deviation of the shock will still be as before since this is just a rescalling?

So, for example, if I have a response of 0.01 on my log y after dividing throughout by standard deviation of the shock, it is still a 1*lambda percent of y right?

[TomDoan]

Thanks! That's so much clearer. Just a few last questions, or rather just wanting to check my understanding.

1. When you say points, I assume you are referring to percentage points?

Percent yield in points, yes.

2. I understand that the impulse responses are interpreted in the units they are in, as you have just shown me. So, if I have a one standard deviation shock in the Federal Funds Rate (equivalent say to 30 basis points), then that would be how I interpret the responses. But, if I want to convert that one standard deviation shock to a 1 percent shock, am I right to say that I will have to divide the shock and the impulse responses by the standard deviation of the shock? However, the interpretation of the impulse responses after dividing by the standard deviation of the shock will still be as before since this is just a rescalling?

So, for example, if I have a response of 0.01 on my log y after dividing throughout by standard deviation of the shock, it is still a 1*lambda percent of y right?

If one standard deviation is .30, then to get an own impact shock of 1.00, you would need to multiply all the responses to that shock by 1.00/.30.

[jacarandas]

An own impact shock of 1 on the federal funds rate is a 1 percent shock, or a 1 percentage point shock?

When you divide any shock by the standard deviation, are we converting the shock to a 1 percent shock in the respective variable?

For example, I perform a one std deviation shock to GDP growth( with a std deviation of 0.3). If I divide the shock by its std deviation of 0.3, does this convert a one standard deviation shock to GDP growth to a 1 percent shock to GDP growth?

Thanks for the help!

[TomDoan]

An own impact shock of 1 on the federal funds rate is a 1 percent shock, or a 1 percentage point shock?

It's a 1 in whatever you want to call the units of the federal funds rate, from (say) 2% to 3%.

When you divide any shock by the standard deviation, are we converting the shock to a 1 percent shock in the respective variable?

For example, I perform a one std deviation shock to GDP growth( with a std deviation of 0.3). If I divide the shock by its std deviation of 0.3, does this convert a one standard deviation shock to GDP growth to a 1 percent shock to GDP growth?

If what you mean is (for instance) from 2.5% to 3.5% growth rate, yes.

[jacarandas]

Thanks! Would I be right to say that by dividing a shock by its standard deviation, I am converting it to a 1 point shock. This may not necessarily be in percent form, and depends on the units of the shock variable.

For example:

- For interest rates/FFR: A one point shock, can be seen as a change from 2% to 3%, which is equivalent to a 1 percentage point change?
- For growth rate/logged first difference terms (such as GDP growth rate): A one point change, which can be seen as increasing GDP growth rate from 2% to 3%. This is again a 1 percentage point change in GDP growth rate?
- For logged variables: A one point shock in this case will be equivalent to a 1 percent shock in the variable since that is how logged variables are interpreted? This 1 percent will be equivalent to X amount of logged Y for example, and is not necessarily a 2% to 3% change.
- For factors that do not have units: A one point shock will be equivalent to an increase in the factor from 20 to 21? It won't be a percentage point shock since the factor does not have units?

Sorry, I had to edit this response and rewrite that in a clearer manner.

[TomDoan]

Your description of the response to log(x) is wrong. A response of 1 to log(x) means multiplying x by e which will be completely out of scale. It's a response of .01 to log(x) which is interpreted as a 1% change in x. If you want to work with unit shocks rather than empirically determined shocks and you have a variable that you want in logs, use 100log(x) rather than just log(x). Then a response of 1 will indeed mean a 1% change in x.

I really can't explain it more clearly than I did above. If that's still confusing, you'll have to talk with someone there.

[istiak]

Dear Tom

I am using a 6 variable sign restriction VAR model with the last variable is the federal funds rate. I am interested with finding the response of all variables from a 1% increase in federal funds rate. The attached code works well with a 1 SD increase in federal funds rate. To get the response from 1% increase in federal funds rate, I followed your post on Nov 1, 2006 about using the following formula to convert a one SD shock to a 1% point shock.

"IMPULSE(MODEL=your model,SHOCKS=%UNITV(model size,shock variable),STEPS=number of steps,PRINT)"

I used the above code in my model by replacing

"impulse(noprint,model=varmodel,decomp=%identity(6),results=impulses,steps=nstep)"

to the following:

"impulse(model=varmodel,SHOCKS=%UNITV(6x6,v1(6)),steps=nstep,print)"

But code is not working. Please help me to solve the problem.
Regards,
KI