Title  Marginal effects of probabilities greater than 1  
Authors 
May Boggess, StataCorp Kristin MacDonald, StataCorp 

Date  April 2004; updated July 2009 
The marginal effect of an independent variable is the derivative (that is, the slope) of the prediction function, which, by default, is the probability of success following probit. By default, margins evaluates this derivative for each observation and reports the average of the marginal effects. We can specify the point at which we want the marginal effect to be evaluated by using the at() option.
The important thing to remember is the slope of a function can be greater than one, even if the values of the function are all between 0 and 1.
Here are some examples:
. sysuse auto, clear (1978 Automobile Data) . probit foreign mpg Iteration 0: log likelihood = 45.03321 Iteration 1: log likelihood = 39.264978 Iteration 2: log likelihood = 39.258972 Iteration 3: log likelihood = 39.258972 Probit regression Number of obs = 74 LR chi2(1) = 11.55 Prob > chi2 = 0.0007 Log likelihood = 39.258972 Pseudo R2 = 0.1282  foreign  Coef. Std. Err. z P>z [95% Conf. Interval] + mpg  .0960601 .0301523 3.19 0.001 .0369627 .1551575 _cons  2.635268 .6841462 3.85 0.000 3.97617 1.294366  . margins, dydx(mpg) at(mpg=25) Conditional marginal effects Number of obs = 74 Model VCE : OIM Expression : Pr(foreign), predict() dy/dx w.r.t. : mpg at : mpg = 25   Deltamethod  dy/dx Std. Err. z P>z [95% Conf. Interval] + mpg  .0372895 .0125636 2.97 0.003 .0126653 .0619137  . predict pmpg, p . sort mpg . twoway scatter pmpg mpg
The graph is not steep at mpg=25, and that shows the result we see from margins, which is that the marginal effect is small. By comparison
. sysuse auto, clear (1978 Automobile Data) . probit foreign gear_ratio Iteration 0: log likelihood = 45.03321 Iteration 1: log likelihood = 22.664339 Iteration 2: log likelihood = 21.653347 Iteration 3: log likelihood = 21.641904 Iteration 4: log likelihood = 21.641897 Iteration 5: log likelihood = 21.641897 Probit regression Number of obs = 74 LR chi2(1) = 46.78 Prob > chi2 = 0.0000 Log likelihood = 21.641897 Pseudo R2 = 0.5194  foreign  Coef. Std. Err. z P>z [95% Conf. Interval] + gear_ratio  3.45954 .7132767 4.85 0.000 2.061543 4.857537 _cons  11.44249 2.30258 4.97 0.000 15.95546 6.929517  . margins, dydx(gear_ratio) at(gear_ratio=3.3) Conditional marginal effects Number of obs = 74 Model VCE : OIM Expression : Pr(foreign), predict() dy/dx w.r.t. : gear_ratio at : gear_ratio = 3.3   Deltamethod  dy/dx Std. Err. z P>z [95% Conf. Interval] + gear_ratio  1.37969 .2867635 4.81 0.000 .8176435 1.941736 
Here we see the graph is quite steep at gear_ratio=3.3, so the marginal effect is large.
Many people expect the marginal effect to be less than one because we learn in calculus class that the derivative is the approximate change in y for a oneunit change in x. Because y is between 0 and 1, the change in y obviously cannot be greater than 1!
The issue comes from the word approximately. Remember the derivative at a point is the slope of the tangent line of the curve at that point.
Let’s draw the tangent line, at the point gear_ratio=3.3, on the graph produced by our last example.
Now we see that the change in the y value on the line, between gear_ratio=3 and gear_ratio=4, is greater than 1 (because at gear_ratio=4, the line has y value greater than 1).
The approximation of a curve by a tangent line is good close to the point where the tangent is drawn, but if the slope of the curve is changing quickly, this approximation is not very good further away from the point.