Interpretation of marginal effects in Logit Model with log$\times$independent variable

Question

I am totally confused by statistics and I would be glad if you could help me.

I have a difficulties to interpret marginal effects in logit model, if my independent variable is log transformed.

I will illustrate my question on the example from my data below. I run a logistic regression in stata

My dependent variable is dummy indicating whether a game is of X Genre. My independent variable is a continuous and log transformed variable (log heterogeneity)

After I run a logit regression:

logit xGenre logheterogeneity + control variables

I get the following results:

The coefficient of my independet variable is .567

STATA:

-X Genre        Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]

-log heterog|  .5655944  .1741354     3.25   0.001     .2242953    .9068935

In order to be able to interpret the results easier, I should look at the marginal effects.

I have used therefore an mfx command.

My results show following: for my independent variable I get dy/dx = .056

STATA:

-variable       dy/dx    Std. Err.     z    P>|z|  [    95% C.I.   ]      X

-log heter.  .0563382      .01688    3.34   0.001   .023259  .089417   3.51361

(*) dy/dx is for discrete change of dummy variable from 0 to 1

Now I am confused on how to interpret my results. Can I say, If my independent variable increase in 10% (log heterog.), then the probability that my Game will be of Genre X increases by 0.56%.

Answer 1

You know that in a logit:

$$Pr[y = 1 \vert x,z] = p = \frac{\exp (\alpha + \beta \cdot \ln x + \gamma z)}{1+\exp (\alpha + \beta \cdot \ln x + \gamma z )}. $$

After some tedious calculus and simplification, the partial of that with respect to $x$ becomes:

$$ \frac{\partial Pr[y=1 \vert x,z]}{\partial x} = \frac{\beta}{x} \cdot p \cdot (1-p). $$

This is (sort of) equivalent to

$$\frac{\Delta p}{\Delta x}=\frac{\beta}{x} \cdot p \cdot (1-p),$$

which can be re-written as

$$\frac{\Delta p}{100 \cdot \frac{ \Delta x}{x}}= \frac{\beta \cdot p \cdot (1-p)}{100}.$$

This is the definition of semi-elasticity, and can be interpreted as the change in probability for a 1% change in $x$.

Here's an example in Stata.* Note that I am using margins instead of the out-of-date mfx to get the average marginal effect of $x$, $\frac{1}{N}\Sigma_{i=1}^N\frac{\beta \cdot p_i \cdot (1-p_i)}{100}$:

. sysuse auto, clear
(1978 Automobile Data)

. gen ln_price = ln(price)

. logit foreign ln_price mpg weight, nolog

Logistic regression                             Number of obs     =         74
                                                LR chi2(3)        =      57.69
                                                Prob > chi2       =     0.0000
Log likelihood = -16.185932                     Pseudo R2         =     0.6406

------------------------------------------------------------------------------
     foreign |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    ln_price |   6.851215    2.11763     3.24   0.001     2.700737    11.00169
         mpg |  -.0880842   .1031317    -0.85   0.393    -.2902186    .1140503
      weight |  -.0062268   .0017269    -3.61   0.000    -.0096115   -.0028422
       _cons |  -41.32383   16.24003    -2.54   0.011    -73.15371   -9.493947
------------------------------------------------------------------------------

. margins, expression(_b[ln_price]*predict()*(1-predict())/100)

Predictive margins                              Number of obs     =         74
Model VCE    : OIM

Expression   : _b[ln_price]*predict()*(1-predict())/100

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   .0046371   .0007965     5.82   0.000      .003076    .0061982
------------------------------------------------------------------------------

This means that for a 1% increase in price, the probability that a car is foreign increases by 0.005 on a [0,1] scale. Or a 10% increase in price gives you a 0.05 increase. In this date, about 0.3 of the cars are foreign, so these are economically and statistically significant.

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Edit:

A good way to do this in Stata 10 is to install the user-written command margeff:

. margeff, dydx(ln_price) replace

Average partial effects after margeff
      y  = Pr(foreign) 

------------------------------------------------------------------------------
    variable |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    ln_price |   .4637103   .0796514     5.82   0.000     .3075964    .6198241
         mpg |  -.0059616    .006781    -0.88   0.379    -.0192522     .007329
      weight |  -.0004214   .0000417   -10.11   0.000    -.0005031   -.0003398
------------------------------------------------------------------------------

. lincom _b[ln_price]/100

 ( 1)  .01*ln_price = 0

------------------------------------------------------------------------------
    variable |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         (1) |   .0046371   .0007965     5.82   0.000      .003076    .0061982
------------------------------------------------------------------------------

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*This is actually not a great empirical example since the relationship in the data has an inverted-U shape.