Coefficients at a different scale

[Ihab Man]

Dear all.
Please, I am using logit model in my study and then Marginal effects . The Coefficient (Marginal effect test) on one of my independent variables is very low. Please, how can I interpret the coefficient at a different scale? Thanks a lot

[Clyde Schechter]

You are more likely to get a timely and helpful response if you show the exact code used for and output of both the -logit- and -margins- commands.

That said, whereas it is fairly simple to change scale of a variable when interpreting linear models, or models with a log link, this is not the case with the logistic link. There is a good chance that the only way you can see what the effect of a change in scale in a logistic model is, is to actually rerun the model with the rescaled variable.

[Ihab Man]

Dear Clyde Schechter
Thanks a lot for your answer.
Please, for more explanation: the variable is firm size (measured by firm total assets in million ). Usually, in our filed we transform this variable into its log. I understand what you mean , but please I would like to write here my commands and then I have two following questions.
1-) logit DepV X1 X2 X3 X4 i.Year, vce(cluster FirmID)
2-) margins, dydx(X4) atmeans. X4 is the logfirmsize.
Well, my quastions are:
A-) You said that in this kind of model, I can rescaled variable (X4) and then rerun the models (Logit and Marginal) (May be you will change your answer after my command).Please, my question is how can I rescaled this kind of variable (X4) in STATA (Command or steps) ?
B-)Please, this operation will change the coefficients or significance on other variables (X1,X2,and X3) ? Thanks a lot .

[Clyde Schechter]

You still are not providing enough information for a specific response. What kind of rescaling do you want to do? Change from assets in million to assets in dollars? Log transform? Remove a log transform (I can't tell if you are using it already log-transformed or not from your explanation.) Something else?

What you will find is that for a proportional scaling there will be no change in statistical significance of anything. If you rescale by a proportion (e.g. millions of dollars to thousands of dollars) there will be a corresponding proportional change in the opposite direction of the coefficient or of the marginal effect (at least approximately). But the change in odds ratio will be to raise the original odds ratio to the power of the down-scaling factor. And the results for other variables do not change at all with a proportional rescaling.

For applying or removing a log-transformation, then things are quite unpredictable both for the rescaled variable and for other variables in the model.

I really think that the only way you can be sure of what will happen if what you have in mind is something other than a proportional rescaling is to try it and see. Be aware, however, that the application of these transformations is a very material change in the model, and you may well change a model that was well fit to the data and a good description of the data generating process into a model that is just badly specified (or vice versa). Some graphical exploration is advised.

Added: If you proportionally rescale the millions of dollars, to some other unit, dollars, or billions of dollars, whatever, and then apply a log transformation, as compared to just using log original millions of dollars, all you will change is the constant term in the regression.

[Ihab Man]

Dear Clyde Schechter
Thank you so much for these information.
The thing is, my knowledge was not enough for rescaling types. I thought that the proportion (e.g. millions of dollars to thousands of dollars) is not included in the rescaling types ( I thought that there are another type of .rescaling such as multiply etc.. and I asked this question and told you what is the variable to give me what is the kind of rescaling for this kind of variable. When you talked about proportion that could be one of rescaling types make sense for me now. Well, I will do the rescaling on this type, but I could not understand your final note ( added one ) . Please, if will go to the original variable and change from assets in million to assets in other things and then apply a log transformation, the coefficient here will change ? if not, how I can change the the coefficient by using proportion and log transformation? Thanks a lot

[Clyde Schechter]

A fundamental property of logarithms is: log(a*x) = log(a) + log(x). Think of x as asset size in dollars, and a as a change of units. So if we're changing from millions of dollars to thousands of dollars, a = 1000. Then log(a*x) = log(1000*x) = log(1000) + log(x), so the new variable is nothing but log(x) with a constant, log(1000), which is 3 if you are using base 10 logs, or about 6.9 if you are using natural logs) added on. That will change only the constant term in the regression.

Another property of logarithms is that a*log(x) = log(x^a). So, while this is a proportional change in log(x), and it produces a corresponding reverse change in the coefficient of log x, it is not a proportional change in x. If you were to multiply the log of firm size (assets) by, let's say 0.01, you will increase the coefficient by a factor of 100. But what that is tantamount to saying is that the relationship of size to DepV is actually dependent more directly on the value of x^0.01 than it is on x. Those are drastically different kinds of relationship and you need to be sure that this doesn't completely destroy the validity of the model. (Or, if the original was a badly fit model, it might make it fit well.)

Again, the best advice I can offer is try it and see.

I guess the still unasked and unaswered question is why you want to make the coefficient of log firm size bigger. First of all, there was presumably some reason for using log assets in the first place. If that reason is valid, then changing it to something else is just creating some kind of illusion by using a distorted model. If there was no real reason for doing that in the beginning, well, fine. But then the first thing you should be doing is looking at the fit of the model to the data. If the model is a good model of the data, then if you don't like the coefficient you got, well, that's just too bad. If the model is not such a good fit of the data and you can improve it by some transformation of the log assets variable, all well and good. But you should not be doing it to bend a particular coefficient to your will.

[Ihab Man]

Dear Clyde Schechter
Thanks a lot . Much appreciated.
I was working on your advice. I tried to change the units of original variable in different ways , but nothing changed in the coefficient as you have mentioned, and of course I expect that based on your words. I also did the squared values of log assets but the coefficient decreases. However, regarding to the unasked and unanswered question , because the coefficient its seems very low in marginal effect in one unit increase in the logarithm asset .

[Clyde Schechter]

However, regarding to the unasked and unanswered question , because the coefficient its seems very low in marginal effect in one unit increase in the logarithm asset .

Let me rephrase the question. Why does it bother you that the marginal effect is low? If the model is a good fit to the data and makes sense conceptually, then the data are telling you that this variable doesn't have much of an effect on the outcome. It is what it is. Why do you want to reject what the data tell you?

[Ihab Man]

Dear Clyde Schechter
Thanks a lot.
The idea behind that is my advisor words. He suggest to interpret the coefficient at different scale. That is why I am seeking for a solution.

[Clyde Schechter]

Then I think you should ask your advisor to explain his or her reasoning, and clarify exactly what he or she has in mind.

[Ihab Man]

Dear Clyde Schechter
Yes, once I finished some tasks, I will talk with him about that. if his explanation needs some stata commands, I will return to this post. Thanks a lot for your help.