logit linear marginal effects positive but elasticity negative

[Will Hauser]

I used Stata 14 to estimate a logit model with continuous and dichotomous predictor variables. The continuous predictors x1, x2, and x3 are scores generated from a factor analysis. X1, the key variable of interest, ranges from about -2 to 1.2 with a mean of -.13.

logit y c.x1 c.x2 c.x3 i.x4 i.x5 , or

The resulting odds ratio for x1 is 1.45 indicating that unit increases in x1 cause a 45% increase in the odds of a positive outcome. When I estimate the linear margins I get a positive effect on the probability of a positive response which is consistent with the odds ratio exceeding 1.

margins, dydx(x1)

The unusual thing is that when I estimate either elasticities or semi-elasticities using the eyex or dyex I get a negative marginal effect. The negative effect emerges whenever x is specified in logarithmic terms (i.e. either the eyex or dyex specification but not the dydx and eydx specifications).


To see what's going on I evaluated the effect of x1 at different levels of x1.

margins, eyex(x1) at(x1=(-2(.2)1.2)

Whenever x1 is less than zero the elasticity is negative. This, again, is true for the margins specified with x1 in log terms but not for either of the dydx or eydx specifications.

For example, at x1=-2 a 1% increase in x1 corresponds to a .32% decrease in the probability of y but at x1=1 a 1% increase in x1 corresponds to a .08% increase in y. What's going on here and how do I reconcile this with the odds ratio which shows a positive effect? Even with x1 set to the mean (-.13) the elasticity is still negative (eyex=-.014).

I would really like to write up the effect of x1 using the dyex or eyex marginal effects which correspond to percentage changes in x because unit changes in x1 are not exactly intuitive since it is the result of factor analysis.

Thanks for any insight you can provide.

[Clyde Schechter]

eyex is defined as d(ln y)/d(ln x). If x is negative, this is undefined. Elasticities really only make sense for positives. In fact, percentage changes do not really make sense for variables that can be positive or negative.

You seem to imply that x1 has no natural metric scale because it comes from a factor analysis. This might be one situation where it makes sense to standardize (i.e. change x1 to standard deviation units).

[Will Hauser]

Clyde,
You raise an excellent point - the variable does take on negative values and that is a problem for an elasticity. This makes me wonder what the margins command is doing to generate the point estimate since the log of a negative number is undefined. You would think I would get an error message of some sort. Anyhow, this is a case of user error, I should have thought this through a bit more before posting.