Interpreting log base 10 transformed coefficients?

[Jun Park]

Hello! I'm having a hard time following one of the interpretation of a paper that I am reading and was wondering if anyone might be able to help me please? (It is a top tiered journal from my field, so I assume it's correct, but there are time when everyone makes mistakes )

The regression in the paper isn't specified, but the dependent variable is a dummy variable so I assume it's probably a logit model. The dependent variable is either 1 if a country has fair elections and 0 if not. The explanatory variable is the log(10) of GDP per capita.

The coefficient of the model is 0.692. So the interpretation in the paper says, the probability that a country with unfair elections transitioning to a fair election system is 0.69 higher if the country's per capita income is $10,000 per year than if it is 1,000 per year.

Now the question is, can we just use the raw coefficient 0.692 and interpret as the probability?.. Don't we need to transform it?
Also, for Log(10) values, is it correct that to interpret the changes in x values as ten-folds?

Thank you!

[Clyde Schechter]

The regression in the paper isn't specified, but the dependent variable is a dummy variable so I assume it's probably a logit model.

Well, a "top tier" journal that publishes an article based on a regression that doesn't even say what kind of regression it is isn't really doing its job, is it? Yes, makes everyone mistakes--this is a good example of a serious deficiency. Moreover, I would never assume that an article is correct just because it appears in a top-tier journal.

Moreover, it's important for the question you're asking, because it makes a huge difference whether this is a linear probability model, or a logistic regression, or a probit regression. Without knowing that, you really can't interpret the coefficient at all. So, just by this fact alone (assuming you haven't overlooked the type of regression in your reading of the article), the paper is not "correct."

[Hemanshu Kumar]

If you read carefully, I am sure that in the paper, the variable for per capita income is in '000s. So, yes, the interpretation of the coefficient is quite exact: going from a country with per capita income of 1 (000) to 10 (000), will increase the probability of a transition to fair election systems by exactly the coefficient value of 0.692, which is to say, 69.2 percentage points.

The above exposition of course assumes the linear probability model, i.e. OLS. But essentially the same thing would hold for others models as long as you've been told that 0.692 is the marginal effect. I am sure a more careful reading will reveal what model they have used.

Edit: even if the per capita income variable is not in '000s, the interpretation is the same. beta is the difference in probability for any ten-fold GDP gap, whether 1->10, or 10->100 or 1000->10000.

[Ken Chui]

Assuming this is a linear probability model, I found that interpretation a bit absurd. Who would consider "If I can increase GDP for 10 fold, then..."?

How about more moderate and plausible change, like, 5% or even 1%?

Given y = b0 + 0.692(log(x)), let's say x increases by 5% (1.05x), and the new probability (y') is then:

y' = b0 + 0.692(log(1.05x))

y' - y = 0.692(log(1.05x)) - 0.692(log(x))
y' - y = 0.692 * [log(1.05x) - log(x)]
y' - y = 0.692 * (log(1.05x/x))
y' - y = 0.692 * log(1.05)

. display 0.629 * log10(1.05)
.01332807

Given a 5% increase in GDP, the average probability of fair election increase by 1.33 percent points. It's the same model, but sounds more realistic.

[Hemanshu Kumar]

Ken Chui I am assuming the authors are making cross-country comparisons, where GDP differences of that order are not hard to come by. If they were making over-time comparisons, a ten-fold increase would be unrealistic in the short or medium-run, but more relevant if the time horizon were several decades.

[Ken Chui]

Ken Chui I am assuming the authors are making cross-country comparisons, where GDP differences of that order are not hard to come by. If they were making over-time comparisons, a ten-fold increase would be unrealistic in the short or medium-run, but more relevant if the time horizon were several decades.

Thanks for the comment, I agree with your point.

In any case, with #4, I wished to point out that it does not have to be "10-fold" all the time, the regression can be applied to any fold by substituting in the right factor into the regression model.

[Clyde Schechter]

The above exposition of course assumes the linear probability model, i.e. OLS. But essentially the same thing would hold for others models as long as you've been told that 0.692 is the marginal effect.

But O.P. doesn't say that it's a linear probability model, nor that the 0.692 number is a marginal effect. Those are very strong assumptions. If this is a logistic regression model and the 0.692 is the actual regression coefficient, the meaning is altogether different and would correspond not to any fixed probability difference but rather to an odds ratio of almost exactly 2 associated with a 10-fold difference in GDP. And if the 0.692 is the coefficient in a probit regression, it's something different yet again.

Either O.P. has missed key points in reading the article, or the article is seriously deficient and the results uninterpretable.

[Ken Chui]

Would be nice if the link to the article is provided.

[Jun Park]

Hemanshu Kumar

Thank you for your comment. Is there a reason why you thought the paper should be in 000's?

[Jun Park]

[QUOTE=Clyde Schechter;n1687514]
But O.P. doesn't say that it's a linear probability model, nor that the 0.692 number is a marginal effect. Those are very strong assumptions. If this is a logistic regression model and the 0.692 is the actual regression coefficient, the meaning is altogether different and would correspond not to any fixed probability difference but rather to an odds ratio of almost exactly 2 associated with a 10-fold difference in GDP. And if the 0.692 is the coefficient in a probit regression, it's something different yet again.

Could you explain what you mean by almost exactly 2 associated with a 10-fold difference in GDP?

[Clyde Schechter]

If 0.692 is the coefficient in a logistic regression, then the odds ratio is exp(0.692), which, in turn is 1.997, so almost exactly 2.

[Ken Chui]

Could you explain what you mean by almost exactly 2 associated with a 10-fold difference in GDP?

If it is a logistic regression and the 0.692 is in the form of logit, the odds ratio will be exp(0.692) which is 1.9977. That is nearly doubling of the odds.

Edit: Crossed #11

[Hemanshu Kumar]

Hemanshu Kumar

Thank you for your comment. Is there a reason why you thought the paper should be in 000's?

That's just a very common scaling for per capita GDP in the political science and also the economics literature.

I echo the request in #8 to provide a link to the article.

[Jun Park]

Yeah I will try to provide a link when available. But just looking at the paper again, it does say "linear probability model". Sorry everyone.. I was only looking for the term logit or profit.

[Jun Park]

[ATTACH=CONFIG]n1687499[/ATTACH]



If you read carefully, I am sure that in the paper, the variable for per capita income is in '000s. So, yes, the interpretation of the coefficient is quite exact: going from a country with per capita income of 1 (000) to 10 (000), will increase the probability of a transition to fair election systems by exactly the coefficient value of 0.692, which is to say, 69.2 percentage points.

The above exposition of course assumes the linear probability model, i.e. OLS. But essentially the same thing would hold for others models as long as you've been told that 0.692 is the marginal effect. I am sure a more careful reading will reveal what model they have used.

Edit: even if the per capita income variable is not in '000s, the interpretation is the same. beta is the difference in probability for any ten-fold GDP gap, whether 1->10, or 10->100 or 1000->10000.

Now this makes a lot of sense. I was confused by the term "probability", because I first thought it meant percentages. But I guess it can also mean percentage points, which I believe is the correct interpretation.