OLS margins, eydx(foreign) calculation by hand

[Dimitriy V. Masterov]

I am puzzled how Stata calculates the semi-elasticity in a n OLS model with a continuous outcome and a single binary regressor, y = a + b*x. The continuous case makes sense to me. The semi elasticity is (dy/dx)*(1/yhat) = b/yhat.

I thought that this would carry over to the discrete case with a finite difference instead of a derivative. In a linear model, the finite difference and the derivative should be the same, so this becomes (a + b*1 - a - b*0)/yhat = b/yhat . However, this does not seem to be the case:

. /* Continuous Case */
. sysuse auto, clear
(1978 Automobile Data)

. reg price c.mpg

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(1, 72)        =     20.26
       Model |   139449474         1   139449474   Prob > F        =    0.0000
    Residual |   495615923        72  6883554.48   R-squared       =    0.2196
-------------+----------------------------------   Adj R-squared   =    0.2087
       Total |   635065396        73  8699525.97   Root MSE        =    2623.7

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |  -238.8943   53.07669    -4.50   0.000    -344.7008   -133.0879
       _cons |   11253.06   1170.813     9.61   0.000     8919.088    13587.03
------------------------------------------------------------------------------

. margins, eydx(mpg)

Average marginal effects                        Number of obs     =         74
Model VCE    : OLS

Expression   : Linear prediction, predict()
ey/dx w.r.t. : mpg

------------------------------------------------------------------------------
             |            Delta-method
             |      ey/dx   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |  -.0421593   .0123505    -3.41   0.001    -.0667795    -.017539
------------------------------------------------------------------------------

. gen double eydx = _b[mpg]/(_b[_cons] + _b[mpg]*mpg)

. sum eydx

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
        eydx |         74   -.0421593    .0181532  -.1638066  -.0284862

. /* Binary Case */
. reg price i.foreign

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(1, 72)        =      0.17
       Model |  1507382.66         1  1507382.66   Prob > F        =    0.6802
    Residual |   633558013        72  8799416.85   R-squared       =    0.0024
-------------+----------------------------------   Adj R-squared   =   -0.0115
       Total |   635065396        73  8699525.97   Root MSE        =    2966.4

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     foreign |
    Foreign  |   312.2587   754.4488     0.41   0.680    -1191.708    1816.225
       _cons |   6072.423    411.363    14.76   0.000     5252.386     6892.46
------------------------------------------------------------------------------

. margins, eydx(foreign)

Conditional marginal effects                    Number of obs     =         74
Model VCE    : OLS

Expression   : Linear prediction, predict()
ey/dx w.r.t. : 1.foreign

------------------------------------------------------------------------------
             |            Delta-method
             |      ey/dx   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     foreign |
    Foreign  |   .0501439   .1200041     0.42   0.677    -.1890798    .2893677
------------------------------------------------------------------------------
Note: ey/dx for factor levels is the discrete change from the base level.

. gen double eydx1 = _b[1.foreign]/(_b[_cons] + _b[1.foreign]*foreign)

. gen double eydx2 = _b[1.foreign]/(_b[_cons] + _b[1.foreign]*0)

. gen double eydx3 = _b[1.foreign]/(_b[_cons] + _b[1.foreign]*1)

. gen double eydx4 = _b[1.foreign]/(_b[_cons] + _b[1.foreign]*.2972973)

. sum eydx?

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
       eydx1 |         74    .0506747    .0011573   .0489075   .0514224
       eydx2 |         74    .0514224           0   .0514224   .0514224
       eydx3 |         74    .0489075           0   .0489075   .0489075
       eydx4 |         74    .0506481           0   .0506481   .0506481

Any guidance would be appreciated.

[Dimitriy V. Masterov]

Here's the solution:

/* Continuous Case: dln(y)/dx */
sysuse auto, clear
reg price c.mpg
margins, eydx(mpg)
gen double eydx = _b[mpg]/(_b[_cons] + _b[mpg]*mpg)
sum eydx   

/* Binary Case: ln(y(x=1)) - ln(y(x=1)) */
reg price i.foreign
margins, eydx(foreign)
generate double eydx2 = log(_b[1.foreign] + _b[_cons]) - log(_b[_cons])
sum eydx2

[Laura Htd]

Here's the solution:

/* Continuous Case: dln(y)/dx */
sysuse auto, clear
reg price c.mpg
margins, eydx(mpg)
gen double eydx = _b[mpg]/(_b[_cons] + _b[mpg]*mpg)
sum eydx

/* Binary Case: ln(y(x=1)) - ln(y(x=1)) */
reg price i.foreign
margins, eydx(foreign)
generate double eydx2 = log(_b[1.foreign] + _b[_cons]) - log(_b[_cons])
sum eydx2

How would you interpret eydx for the binary case?

Also, I wonder if it would be meaningful to calculate the margins manually as
(y(x=1) - y(x=0))/ y(x=0) instead, so that the marginal effects are interpreted as the average proportional gain or loss from going from x=0 to x=1, for an individual for whom x=0. That is, if this calculation is 0.09, a subject with x=0 would gain, on average, 9% more of y if x=1 instead. Would this make sense? Thank you!

[Dimitriy V. Masterov]

. generate double eydx2 = log(_b[1.foreign] + _b[_cons]) - log(_b[_cons])

. sum eydx2

    Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
       eydx2 |         74    .0501439           0   .0501439   .0501439

My interpretation is that foreign origin is associated with a 5% price premium.

Your suggestion is correct, and what I might do if the estimated effect was large. But it will return approximately the same answer as the log approach when the delta is small:

. 
. /*  Binary Case: [y(x=1)) - y(x=0)]/y(x=0) */ 
. reg price i.foreign

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(1, 72)        =      0.17
       Model |  1507382.66         1  1507382.66   Prob > F        =    0.6802
    Residual |   633558013        72  8799416.85   R-squared       =    0.0024
-------------+----------------------------------   Adj R-squared   =   -0.0115
       Total |   635065396        73  8699525.97   Root MSE        =    2966.4

------------------------------------------------------------------------------
       price | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
     foreign |
    Foreign  |   312.2587   754.4488     0.41   0.680    -1191.708    1816.225
       _cons |   6072.423    411.363    14.76   0.000     5252.386     6892.46
------------------------------------------------------------------------------

. margins, expression((_b[1.foreign] + _b[_cons])/_b[_cons])
warning: option expression() does not contain option predict() or xb().
warning: prediction constant over observations.

Predictive margins                                          Number of obs = 74
Model VCE: OLS

Expression: (_b[1.foreign] + _b[_cons])/_b[_cons]

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
       _cons |   1.051422    .126175     8.33   0.000      .804124    1.298721
------------------------------------------------------------------------------

. generate double eydx3 = (_b[1.foreign] + _b[_cons])/_b[_cons]

. sum eydx3

    Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
       eydx3 |         74    1.051422           0   1.051422   1.051422