# Logistic regression with an log transformed variable, how to determine economic significance

on 2012-07-29T08:59:10-04:00
I am using a logistic regression model with continuous independent variables and two log transformed size variables (total assets and total deposits).

My question is how to interpret the results and measure the economic impact? The normal steps in interpretation I take is as follows: I firstly run stata test on summary statistics (sum), then a logistic regression (logit) and than the marginal effects for the economic significance (mfx).

These results can be explained by sentences as "a one standard deviation increase in $X$ (an independent variable) increases the probability of participation by $X\%$ I will multiply the $dy/dx$ of the marginal effects by the SD to get the % impact.

So far so good, but how do I interpreted the results with the two ln(size) variables? To clarify my problem I will show you a part of the results:

sum stat:

obs mean stdD min max
lnTCD| 2755 -1.469624 2.992985 -15.01948 4.827369
lnTA | 2767 -1.131045 2.796627 -16.51857 6.731258

logit res:

Coeff stERR Z P>Z 95%
lnTCD| 1.088141 .1424768 0.65 0.519 .8418451 1.406494
lnTA | .7880959 .105272 -1.78 0.075 .6065653 1.023954

Margin eff:

dy/dx stERR Z P>Z 95% X
lnTCD| .0104326 .01619 0.64 0.519 -.021308 .042174 -1.09399
lnTA | -.0294112 .0165 -1.78 0.075 -.061749 .002926 -.7886

In a normal situation I would multiply lnTA of the marginal effects (-0.0294112) by the Standard deviation of the summary statistic (2.796627) which results in -8.22%

Although this holds for the other non log variables, intuitively this does not sound correct for these (ln) variables.

The interpretation you want to put on the covariate change makes sense in a simple model with one covariate. If you have two or more there could be some interaction and the best you can say in general is that x% is the magnitude of change when you change variable U by one standard deviation with the others held fixed at a particular value. At other places in the covariate space the magnitud of change could be different. If a variable is changed you can still talk about this amount of change on the log scale. but if you want to make the claim on the original scale you would have to figure out how the change on th log scale translates to a change on the original scale.

Name:

With log variables, your interpretation of the mfx $dy/dx$ becomes:
$\frac{dy}{d\log x}$ which is equivalent to $\frac{dy}{dx/x}$ so the interpretation implies that 1% variation in $X$ causes a $\frac{dy}{d\log x}$ variation on $y$
$\frac{dy}{d\log x}\Delta(\log x) = -0.029*2.79= -0.08$
a 2.79% variation (or standard deviation)of $X$ causes the probability to decrease in $8\%$ .