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I am using a probit model, and margins says that my marginal effect is greater than 1. Can that be correct?

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

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 Coefficient 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
Delta-method
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 . graph export mfx_size1.png (file mfx_size1.png written in PNG format)
graph

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 Coefficient 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
Delta-method
dy/dx std. err. z P>|z| [95% conf. interval]
gear_ratio 1.37969 .2867636 4.81 0.000 .8176437 1.941736
. predict pgear, p . sort gear_ratio . twoway scatter pgear gear . graph export mfx_size2.png (file mfx_size2.png written in PNG format)
graph

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 one-unit 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.

graph

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.

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