# Log-log specified regression coefficients don't agree with level-level specification

I've got a bunch of data on charities and I'm doing a study on the effectiveness of fundraising in the sector. I've got two regressions. The first is

$$\text{revenue} = \beta_0 + \beta_1\text{fundraising} + e$$

and the second is

$$\log(\text{revenue}) = \beta_0 + \beta_1 \log (\text{fundraising}) + e$$

where $e$ is an error term, revenue is the revenue of a charity in dollars, and fundraising is the fundraising expenditure of a charity in dollars. There are a bunch of other variables too, but the question is just about this, and the problem only gets worse when I add in more variables.

The first specification gives me $\beta_1=1.4$. A way to interpret this is that on average, an increase of $1 in fundraising is associated with \$1.40 in increased revenue. This seems to also be like saying that fundraising has a 40% rate of return.

The second gives me $\beta_1 = 0.36$. This is like saying that a 1% increase in fundraising dollars leads to a 0.36% increase in revenue. But this points to a negative rate of return for fundraising.

Aren't these contradictory results? How is this possible? Any help is appreciated!

• In the second case, a 1% change in the log of fundraising dollars gives a 36% increase in the log of revenue? - I suggest visual inspection of scatterplots for both the non-log and log data. Jul 25, 2018 at 0:33
• @JamesPhillips, no, a 1 unit increase in the log of fundraising leads to a 36% increase in the log of revenue. It turns out that this can be interpreted the way that I showed, to some degree of approximation -- see here stats.stackexchange.com/questions/106278/…
– crf
Jul 25, 2018 at 0:35
• To some degree of approximation, then, you are correct. Jul 25, 2018 at 11:15

The results are not contradictory, but they are telling you different things. One might be a better model than the other, or both might be inappropriate

As an illustration, consider just two points: a large charity which undertakes serious fund raising and a small charity established as a foundation by an individual so with minimal fundraising:

You might perhaps see numbers like:

                   Revenue (R)     Fundraising (F)
Large charity      1,410,000        1,000,001
Small charity         10,000                1


Try and put a straight line through these and you might get $$R \approx 9998.6 + 1.4 F$$ but taking logs might give you $$\log_e(R) \approx 9.2103 + 0.3582 \log_e(F)$$ equivalent to $$R \approx 10000\, F^{0.3582}$$ which gives $\beta_1$s close to yours (the base of the logarithm just affects the $9.21$). Really the $1.4$ is telling you is a number close to the revenue to fundraising ratio of the larger charity, while the $0.36$ is telling you the smaller charity has disproportionately smaller fundraising to its revenues

Clearly you should not be drawing lines or curves involving just two points, but this toy example demonstrates that the results you state are not immediately contradictory

More likely than this, your other variables in your model may also be correlated with size, and so may be seriously affecting the coefficients associated with fundraising. See what happens if you drop all the other variables