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 mpgIteration 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 ------------------------------------------------------------------------------ | 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

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_ratioIteration 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 ------------------------------------------------------------------------------ | Delta-method | 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 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.

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.