Interpreting impulse response with log percentage as dependent variable and first difference of share as independent one

[Marius Fourne]

Hello everyone,

I estimate a VAR model where the dependent variable is log-transformed percentage (0-100) and the independent variable is first difference of a share (0-1) whereas the independent variable is not log-transformed.

I have seen similar questions here, but not specifically log-transformed percentage and first differences of share.

Can you help me to interpret the response of the dependent variable to the shock caused by the independent variable if the impulse respone would be, eg. 0.6 after 5 years?

Best
Marius

[Fei Wang]

The interpretation would be: a one percentage point increase in x is associated with a 0.6 percent increase in y.

[Marius Fourne]

Thank you, Fei.
Why is it not given by approximately multiplying 0.6 by 100 as usual in log-level settings? Is it simply because y is already given as percentage?

[Fei Wang]

No, the reason is that one percentage point increase in x is essentially a 0.01 unit increase in x, given that x is related to fraction (0-1) rather than percentage (0-100). So multiplying 0.6 by 100 is offset by dividing one unit increase in x by 100.

[Fei Wang]

BTW, it seems a little awkward to log-transform a percentage. Without log, the coefficient would be more naturally interpreted as percentage points change in y in response to percentage points change in x.

[Marius Fourne]

Thanks for clarifying! I think it's just a trade-off between interpreting the results more naturally (by using percentage points change) and emphasizing the size of the effect (by using percentage change) given that the percentage point change is relatively small compared to the percentage change.

[Marius Fourne]

Fei Wang So your interpretation also holds when y is not log-transformed percentage, but instead, e.g. log(GDP)? Then the percantage change is not given by exactly 100*(e^0,6 -1 ) but instead also just by e^0,6 -1?

[Fei Wang]

Strictly speaking, the percentage change would be 100*(exp(0.006)-1) = 0.6.