HELP!! Interpreting logged dependant varaible, when its a ratio.

[Matthew Shelley]

Hi Guys,

I apologise if this question is hard to read.
I'm currently analysing a regression in which the dependant variable is in the form of a ratio (tobins q), and is also logged. The coefficient for the dependant variable I have calculated as 0.7. From what I understand in order to interpret this I need to do the exp() for this value- giving 2.01, then subtract 1 giving 1.01, then multiply by 100 giving 101%. From this I've gathered that a 0.1 unit change in my ratio will have a 10.1% change in my dependant variable (lnTobinsq). My question is what does a 10% change in a logged ratio actually mean? I'm struggling to understand the effect of this.

Thanks for the help

[Clyde Schechter]

The coefficient for the dependant variable I have calculated as 0.7. From what I understand in order to interpret this I need to do the exp() for this value- giving 2.01, then subtract 1 giving 1.01, then multiply by 100 giving 101%.

That much is fine.

From this I've gathered that a 0.1 unit change in my ratio will have a 10.1% change in my dependant variable (lnTobinsq).

But that is wrong. Here you are treating exp() as if it were a linear function, which, over the large range between 0.7 and 0.1*0.7, it is decidedly not. You have to repeat the calculation as follows. A 0.1 unit difference in the dependent variable is associated with a 0.7*0.1 = 0.07 increase in log TobinsQ. An increase of 0.07 in log TobinsQ corresponds to multiplying TobinsQ by exp(0.07) = 1.0725082. Therefore, TobinsQ goes up (multiplicatively) by about 7.25%.

My question is what does a 10% change in a logged ratio actually mean?

It means the same thing as a 10% (although, as we have seen, it is actually 7.25%) change in anything else. The fact that TobinsQ is a ratio changes nothing in this context.

[Matthew Shelley]

That much is fine.


But that is wrong. Here you are treating exp() as if it were a linear function, which, over the large range between 0.7 and 0.1*0.7, it is decidedly not. You have to repeat the calculation as follows. A 0.1 unit difference in the dependent variable is associated with a 0.7*0.1 = 0.07 increase in log TobinsQ. An increase of 0.07 in log TobinsQ corresponds to multiplying TobinsQ by exp(0.07) = 1.0725082. Therefore, TobinsQ goes up (multiplicatively) by about 7.25%.


It means the same thing as a 10% (although, as we have seen, it is actually 7.25%) change in anything else. The fact that TobinsQ is a ratio changes nothing in this context.

Hi Clyde,

Thank you so much for your help. Just to clarify, where you said "You have to repeat the calculation as follows. A 0.1 unit difference in the dependent variable is associated with a 0.7*0.1 = 0.07 increase in log TobinsQ." Did you mean-
"You have to repeat the calculation as follows. A 0.1 unit difference in the explanatory variable is associated with a 0.7*0.1 = 0.07 increase in log TobinsQ." As our dependant varaible IS log TobinsQ.

Thanks again Clyde,
Matt

[Nick Cox]

I don't have any experience of analysing data on this ratio but it's generic that ratios (of positive quantities, so that results are necessarily positive) are often best analysed on logarithmic scale. That could be done directly using Poisson regression or some other generalized linear model with a logarithmic link. Then there is some scope to use an appropriate distribution family that best matches error structure.

Economists who know related literature may be able to advise on what people do here.

[Clyde Schechter]

Re #3: Yes, you are correct, I meant to say explanatory variable. Sorry for the confusion.