interpreting coefficients in level-log model

[Dana Shills]

Hello,

I am having difficulty interpreting the coefficients in the following level-log model correctly. I am estimating the following regression:

profit = a + b₁Lcontb + b₃size + b₃age + e
where profit is firm profits (%)
size is firm size proxied by total assets($)
age if firm age (years)
Lcontb is natural log of the contributions ($) paid by the firm to politicians

Suppose b₁=0.5, would the interpretation be that keeping size and age fixed, a 1% increase in $ contributions results in a (b₁/100) =0.005 percentage point increase in profit? Am I right about the percentage point increase?
Also how do I interpret this in terms of a standard deviation increase? A 1 SD increase in $ contributions results in ?? increase in profits?

Thank you for your help.

[Clyde Schechter]

So let's just go through the algebra.

If contributions increase by 1%, then they are multiplied by 1.01. So the term from Lcontb in the regression model increases by ln(1.01) = .00995 (to 5 decimal places). The increase in Lcontb is then multplied by 0.5 to yield an increase in expected profit of 0.5*.00995 = 0.00498. So it is the actual value of profit that increases by 0.00498. This is an additive absolute increase, not a percentage or percentage point change. If you want to round that to an absolute increase of 0.005 that would be reasonable. (I only carried so many decimal places because I wanted to demonstrate that any rule of thumb you have learned about logs and percentages is just an approximation. For a small enough value of b₁ and, more important, a small enough change in contributions, as here, the approximation is very good.

To answer your question about a 1SD increase in contributions, you would have to a) specify just how much 1 SD of contributions is, and b) the baseline value of contributions to which that increase is added. Then you would just go through the algebra. The key relationship is ln(x*y) = ln(x) + ln(y).

Evidently I don't know what the actual value of 1 SD contributions is. My guess is that nobody else looking at your work will either. So reporting the change in expected profits associated with a 1SD increase in contributions will probably just get you blank stares. So why would you want to even calculate this figure? Are you looking for a way to obfuscate your findings? I didn't think so.

[Dana Shills]

Thank you Clyde. I am confused though because doesn't the unit of the Profits variable matter? Shouldn't it be that if contributions increase by 1%, the profits if reported in $ increases by 0.00498 but if profits are reported in %, the interpretation would be a 1% increase in contributions would increase profits by 0.00498%.

I usually see economic effects reported in terms of a SD change and that's why I was curious on how the interpretation would change. The SD and mean are reported in the summary statistics table so the reader has this information. If a 1 SD of contributions is say 15, and the mean is 0.9, then we would say an increase of 1 SD of contributions results in an increase in expected profit of 0.5 * ln(15)?

[Clyde Schechter]

Shouldn't it be that if contributions increase by 1%, the profits if reported in $ increases by 0.00498 but if profits are reported in %, the interpretation would be a 1% increase in contributions would increase profits by 0.00498%.

So, if the profits variable is itself a percentage, then the increase is 0.00498 percentage points (not percent).

The SD and mean are reported in the summary statistics table so the reader has this information. If a 1 SD of contributions is say 15, and the mean is 0.9, then we would say an increase of 1 SD of contributions results in an increase in expected profit of 0.5 * ln(15)?

No. An increase in contributions of 15 does not correspond to a change of ln(15) in the lncontributions variable. There is no particular relationship between ln(baseline + 15) and ln(15). If the logarithm of contributions went up by 15, then sure, but that's very different from the contributions going up by 15. But that would correspond to the actual contributions being multiplied by a factor of 15--which would be quite phenomenal, wouldn't it? Does that happen in real life?

If the contributions increase by 15, then lncontributions increase by ln((baseline+15)/baseline). That number can only be evaluated after you stipulate what the baseline is. You would then multiply that logarithm by 0.5 to get the associated expected increase in profits (in percentage points).

[Dana Shills]

This is very clear. Thank you very much!

[Alex Mai]

No. An increase in contributions of 15 does not correspond to a change of ln(15) in the lncontributions variable. There is no particular relationship between ln(baseline + 15) and ln(15). If the logarithm of contributions went up by 15, then sure, but that's very different from the contributions going up by 15. But that would correspond to the actual contributions being multiplied by a factor of 15.

If the contributions increase by 15, then lncontributions increase by ln((baseline+15)/baseline). That number can only be evaluated after you stipulate what the baseline is. You would then multiply that logarithm by 0.5 to get the associated expected increase in profits (in percentage points).

Dear Clyde,

May I follow up this post with a question on how to interpret the impact of one-standard-deviation increase in a logged variable? Can I simply multiply the standard deviation of a logged explanatory variable with its coefficient? For instance, let's say, the SD of logged GDP is 0.2 and its coefficient is 1.5, then one-standard-deviation increase in logged GDP (20% increase in GDP) increases life expectancy by 0.3 years. Is this interpretation correct?

Moreover, I have seen that some authors interpret regression coefficients by changes in standard deviation for panel data. Do you think if this is reasonable? My concern is that, for panel data, the standard deviation of a given variable is calculated across all individuals, and thus each specific individual's within-standard deviation across time may be very different from the within-standard deviation of other individuals. As a result, I am not sure if a general impact of one-standard-deviation increase can tell much (given the large across-individual difference in SD).

Thank you very much.

[Clyde Schechter]

So, yes, if you have linear model y = b0 + 1.5 log GDP + other variable + error, and SD of log GDP is 0.2, then a difference of 1SD in log GDP is associated with an expected increase of 1.5*0.2 = 0.3 in y.

Moreover, I have seen that some authors interpret regression coefficients by changes in standard deviation for panel data. Do you think if this is reasonable?

No, I don't. More generally, I think there are very few situations, panel data or otherwise, in which the use of standardized variables is appropriate. In panel data I think it is even worse for precisely the reason you give.

[Alex Mai]

No, I don't. More generally, I think there are very few situations, panel data or otherwise, in which the use of standardized variables is appropriate. In panel data I think it is even worse for precisely the reason you give.

Dear Clyde,

Thank you very much! So do you think if it is okay to interpret the coefficient by using mean within-individual standard deviation as the reference?

For example, if the mean within-country standard deviation of logged GDP (across all countries) is 0.1, then the mean impact of logged GDP can be proxied by multiplying 0.1 with the coefficient of logged GDP (1.5), say, 0.1*1.5=0.15.

I guess this approach takes into consideration, to some extent, the potential great difference in within-individual standard deviation between different individuals. But I have not seen papers using this approach.

[Clyde Schechter]

I think that approaching this based on standard deviations is just obscurantist. The purpose of statistical analysis is to make complicated things understandable. Nobody really understands standard deviations. If you know that the standard deviation of log GDP is 0.1 and you think that it is reasonable to talk about the expected difference in y associated with a log GDP difference of 0.1, then why not just say "the expected difference in y associated with a log GDP difference of 0.1 is whatever it is?" What do you gain by calling it a standard deviation? All you can do is set people wondering: standard deviation within what group? What is the distribution anyway--is it normal? Does it matter? So just say what it is and forget that the fact that you chose that particular log GDP increment happens to be the standard deviation across countries.

Using the standard deviation within countries as your metric would make things even more difficult because every country will have its own standard deviation, and nobody will have even the remotest idea what is going on!

Less gobbledygook, more plain-spoken English (or German, or whatever)!

[Alex Mai]

I think that approaching this based on standard deviations is just obscurantist. The purpose of statistical analysis is to make complicated things understandable. Nobody really understands standard deviations. If you know that the standard deviation of log GDP is 0.1 and you think that it is reasonable to talk about the expected difference in y associated with a log GDP difference of 0.1, then why not just say "the expected difference in y associated with a log GDP difference of 0.1 is whatever it is?" What do you gain by calling it a standard deviation? All you can do is set people wondering: standard deviation within what group? What is the distribution anyway--is it normal? Does it matter? So just say what it is and forget that the fact that you chose that particular log GDP increment happens to be the standard deviation across countries.

Using the standard deviation within countries as your metric would make things even more difficult because every country will have its own standard deviation, and nobody will have even the remotest idea what is going on!

Less gobbledygook, more plain-spoken English (or German, or whatever)!

Thank you very much once more. In the field of economics, people are really into using standard deviation, a kind of norm or even "correctness". People have often been expected to do so. But I have no idea about the reason for this obsession.