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Note: This FAQ is for Stata 10 and older versions of Stata. In Stata 11, the margins command replaced mfx.

I am using mfx after an estimation that has an offset. How does mfx take that into account?

Title   Marginal effects after estimations with offsets
Author May Boggess, StataCorp
Date April 2004; minor revisions October 2005

The command mfx evaluates at the mean value of the offset. Let’s see how this works in an example:

 . sysuse auto, clear
 (1978 Automobile Data)
  
 . probit foreign weight mpg, offset(turn) nolog
 
 Probit estimates                                  Number of obs   =         74
                                                   Wald chi2(2)    =     440.82
 Log likelihood = -130.53661                       Prob > chi2     =     0.0000
 
 ------------------------------------------------------------------------------
      foreign |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
 -------------+----------------------------------------------------------------
       weight |   -.008765   .0005981   -14.65   0.000    -.0099373   -.0075928
          mpg |  -.1921153   .0527769    -3.64   0.000    -.2955561   -.0886746
        _cons |  -10.73888   2.721338    -3.95   0.000     -16.0726   -5.405153
         turn |   (offset)
 ------------------------------------------------------------------------------
 Note: 8 failures and 0 successes completely determined.
 
 . mfx, predict(p) 
 
 Marginal effects after probit
       y  = Pr(foreign) (predict, p)
          =  .04973232
 ------------------------------------------------------------------------------
 variable |      dy/dx    Std. Err.     z    P>|z|  [    95% C.I.   ]      X
 ---------+--------------------------------------------------------------------
   weight |  -.0009001      .00031   -2.89   0.004  -.001511  -.00029   3019.46
      mpg |  -.0197293      .00802   -2.46   0.014   -.03544 -.004019   21.2973
     turn |  (offset)                                                   39.6486
 ------------------------------------------------------------------------------
 
 . summarize turn
 
     Variable |       Obs        Mean    Std. Dev.       Min        Max
 -------------+--------------------------------------------------------
         turn |        74    39.64865    4.399354         31         51
 
 . replace turn=r(mean)
 turn was int now float
 (74 real changes made)
 
 . quietly summarize weight
 
 . quietly replace weight=r(mean)
 
 . quietly summarize mpg
 
 . quietly replace mpg=r(mean)
 
 . predict y, p

 . list y in 1
 
      +----------+
      |        y |
      |----------|
   1. | .0497323 |
      +----------+

In the above example, we demonstrated that the value y (above the table in the mfx output) is the value you obtain when you predict at the average values of the covariates and the average value of the offset. In the next example, we will go further and calculate the marginal effects of two dichotomous variables by hand:

 . clear

 . set obs 50
 obs was 0, now 50
 
 . set seed 85642
 
 . generate y=uniform()>0.5
 
 . generate x1=uniform()>0.5
 
 . generate x2=uniform()>0.5
 
 . generate off=uniform()*30+50
 
 . probit y x1 x2, offset(off) nolog
 
 Probit estimates                                  Number of obs   =         50
                                                   Wald chi2(2)    =      58.13
 Log likelihood = -286.69439                       Prob > chi2     =     0.0000
 
 ------------------------------------------------------------------------------
            y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
 -------------+----------------------------------------------------------------
           x1 |   3.081539   .4533974     6.80   0.000     2.192896    3.970181
           x2 |  -1.103905   .4571051    -2.41   0.016    -1.999815   -.2079954
        _cons |  -68.40088   .4041975  -169.23   0.000    -69.19309   -67.60867
          off |   (offset)
 ------------------------------------------------------------------------------
 Note: 10 failures and 3 successes completely determined.

 . mfx, predict(p) 
 
 Marginal effects after probit
       y  = Pr(y) (predict, p)
          =  .08459144
 ------------------------------------------------------------------------------
 variable |      dy/dx    Std. Err.     z    P>|z|  [    95% C.I.   ]      X
 ---------+--------------------------------------------------------------------
       x1*|   .6777208      .11014    6.15   0.000   .461848  .893594        .4
       x2*|  -.1832137      .09128   -2.01   0.045  -.362116 -.004311       .52
      off |  (offset)                                                   66.3675
 ------------------------------------------------------------------------------
 (*) dy/dx is for discrete change of dummy variable from 0 to 1
 
 . summarize off
 
     Variable |       Obs        Mean    Std. Dev.       Min        Max
 -------------+--------------------------------------------------------
          off |        50    66.36746    8.437674   50.19214   79.75008
 
 . replace off=r(mean)
 (50 real changes made)
 
 . quietly {
 . summarize x1
 . local mean1=r(mean)
 . summarize x2
 . local mean2=r(mean)
 . replace x1=`mean1'
 . replace x2=`mean2'
 . predict p, p
 . noisily display _n "y = " p _n  
 
 y = .08459145
 
 . replace x1=0
 . predict p0,p
 . replace x1=1
 . predict p1, p
 . noisily display  _n "marginal effect x1 = " p1-p0  _n 
 
 marginal effect x1 = .67772082
 
 . drop p1 p0
 . replace x1=`mean1'
 . replace x2=0
 . predict p0,p
 . replace x2=1
 . predict p1, p
 . noisily display  _n "marginal effect x2 = " p1-p0  
 
 marginal effect x2 = -.18321373
 . }

The above example gives you the idea of what to do if you want to evaluate marginal effects at a value of the offset that is not the mean. You can replace the offset by the value you want before running mfx, as follows:

 . sysuse auto, clear
 (1978 Automobile Data)
 
 . replace weight=weight/1000
 weight was int now float
 (74 real changes made)
 
 . probit foreign weight length, offset(turn) nolog
 
 Probit estimates                                  Number of obs   =         74
                                                   Wald chi2(2)    =     511.38
 Log likelihood = -129.03886                       Prob > chi2     =     0.0000
 
 ------------------------------------------------------------------------------
      foreign |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
 -------------+----------------------------------------------------------------
       weight |  -3.277497   .8475066    -3.87   0.000    -4.938579   -1.616414
       length |  -.1330356   .0294001    -4.52   0.000    -.1906588   -.0754124
        _cons |  -5.978285   3.276507    -1.82   0.068    -12.40012    .4435497
         turn |   (offset)
 ------------------------------------------------------------------------------
 Note: 1 failure and 0 successes completely determined.
 
 . mfx, predict(p) 
 
 Marginal effects after probit
       y  = Pr(foreign) (predict, p)
          =  .10979816
 ------------------------------------------------------------------------------
 variable |      dy/dx    Std. Err.     z    P>|z|  [    95% C.I.   ]      X
 ---------+--------------------------------------------------------------------
   weight |  -.6154742      .19077   -3.23   0.001  -.989375 -.241573   3.01946
   length |  -.0249825      .00869   -2.87   0.004  -.042023 -.007942   187.932
     turn |  (offset)                                                   39.6486
 ------------------------------------------------------------------------------
 
 . replace turn=41
 (70 real changes made)
 
 . mfx, predict(p) 
 
 Marginal effects after probit
       y  = Pr(foreign) (predict, p)
          =  .54924357
 ------------------------------------------------------------------------------
 variable |      dy/dx    Std. Err.     z    P>|z|  [    95% C.I.   ]      X
 ---------+--------------------------------------------------------------------
   weight |  -1.297558      .34523   -3.76   0.000   -1.9742 -.620918   3.01946
   length |  -.0526687      .01159   -4.54   0.000  -.075384 -.029954   187.932
     turn |  (offset)                                                        41
 ------------------------------------------------------------------------------

If you have used exposure() rather than offset(), then mfx will evaluate at the exp(average of ln(exposure)). For example,

 . webuse airline, clear
  
 . poisson injur XYZ, exposure(n) nolog
 
 Poisson regression                                Number of obs   =          9
                                                   LR chi2(1)      =       1.77
                                                   Prob > chi2     =     0.1836
 Log likelihood = -23.027177                       Pseudo R2       =     0.0370
 
 ------------------------------------------------------------------------------
     injuries |      Coef.   Std. Err.      z    p>|z|     [95% Conf. Interval]
 -------------+----------------------------------------------------------------
     XYZowned |   .3808084   .2780192     1.37   0.171    -.1640993    .9257161
        _cons |   4.061204    .147442    27.54   0.000     3.772223    4.350185
            n | (exposure)
 ------------------------------------------------------------------------------
 
 . mfx, predict(n) 
 
 Marginal effects after poisson
       y  = predicted number of events (predict, n)
          =  6.4864947
 ------------------------------------------------------------------------------
 variable |      dy/dx    Std. Err.     z    p>|z|  [    95% C.I.   ]      X
---------+--------------------------------------------------------------------
 XYZowned*|   2.647899     2.14322    1.24   0.217  -1.55274  6.84854   .333333
    ln(n) |  (offset)                                                  -2.31842
 ------------------------------------------------------------------------------
 (*) dy/dx is for discrete change of dummy variable from 0 to 1
 
 . generate offset=ln(n)
 
 . summarize offset
 
     Variable |       Obs        Mean    Std. Dev.       Min        Max
 -------------+--------------------------------------------------------
       offset |         9   -2.318418    .5339422   -2.98975  -1.571179
 
 . replace n=exp(r(mean))
 (9 real changes made)
 
 . replace XYZ=0
 (3 real changes made)
 
 . predict n0, n
 
 . replace XYZ=1
 (9 real changes made)
 
 . predict n1, n
 
 . display _n "marginal effect XYZ = " n1-n0
 
 marginal effect XYZ = 2.6478992

Some multiple-equation estimators allow more than one offset. Let's look at an example using hetprob. First, we will create some data and run the model:

 . clear
  
 . set obs 1000
 obs was 0, now 1000
 
 . set seed 1234567
 
 . generate x = 1-2*uniform()
 
 . generate xhet=uniform()
 
 . generate sigma=exp(1.5*xhet)
 
 . generate p=normal((0.3+2*x)/sigma)
 
 . generate y = cond(uniform()<=p,1,0)
 
 . generate off1=uniform()+0.5
 
 . generate off2=uniform()+0.6
 
 . hetprob y x, het(xhet, offset(off2))  offset(off1) nolog
 
 Heteroskedastic probit model                    Number of obs     =       1000
                                                 Zero outcomes     =        452
                                                 Nonzero outcomes  =        548
 
                                                 Wald chi2(1)      =      79.18
 Log likelihood = -576.5979                      Prob > chi2       =     0.0000
 
 ------------------------------------------------------------------------------
            y |      Coef.   Std. Err.      z    p>|z|     [95% Conf. Interval]
 -------------+----------------------------------------------------------------
 y            |
            x |   6.658445   .7482918     8.90   0.000      5.19182     8.12507
        _cons |  -.3720305   .2473793    -1.50   0.133     -.856885    .1128241
         off1 |   (offset)
 -------------+----------------------------------------------------------------
 lnsigma2     |
         xhet |   1.724798   .2680221     6.44   0.000     1.199484    2.250112
         off2 |   (offset)
 ------------------------------------------------------------------------------
 Likelihood-ratio test of lnsigma2=0: chi2(1) =   101.70   Prob > chi2 = 0.0000

Now we will run mfx with the predict(xb) option, which is the linear predictor from the first equation:

 . mfx, predict(xb) 
 
 Marginal effects after hetprob
       y  = Linear prediction (predict, xb)
          =  .76705778
 ------------------------------------------------------------------------------
 variable |      dy/dx    Std. Err.     z    p>|z|  [    95% C.I.   ]      X
 ---------+--------------------------------------------------------------------
        x |   6.658445      .74829    8.90   0.000   5.19182  8.12507   .020114
     xhet |          0           0       .       .         0        0   .502716
     off1 |  (offset1)                                                  1.00516
     off2 |  (offset2)                                                  1.09709
 ------------------------------------------------------------------------------
 
 . summarize x if e(sample)
 
     Variable |       Obs        Mean    Std. Dev.       Min        Max
 -------------+--------------------------------------------------------
            x |      1000    .0201138    .5846345  -.9999619   .9977288
 
 . scalar meanx=r(mean)
 
 . summarize off1  if e(sample)
 
     Variable |       Obs        Mean    Std. Dev.       Min        Max
 -------------+--------------------------------------------------------
         off1 |      1000    1.005162    .2875233   .5014016   1.497421
 
 . scalar meanoff1=r(mean)
 
 . display _n "pred xb = " meanx*_b[x]+_b[_cons] + meanoff1
 
 pred xb = .76705774

Let’s finish with a more complicated prediction option. By differentiating the formula for the probability of success, we can verify that mfx is correctly calculating the marginal effect. The formula for the probability of success is in [R] hetprob:

                  meanx*_b[x]+_b[_cons] + meanoff1
  p = normal( ------------------------------------------  )
              exp(meanxhet*_b[lnsigma2:xhet] + meanoff2)

where normal() is the cumulative distribution function for the standard normal distribution. The derivative is

 dp                  _b[x] 
 -- = ------------------------------------------   
 dx   exp(meanxhet*_b[lnsigma2:xhet] + meanoff2)

                           meanx*_b[x]+_b[_cons] + meanoff1
        *   normalden( ------------------------------------------  )
                       exp(meanxhet*_b[lnsigma2:xhet] + meanoff2)

where normalden() is the probability density function for the standard normal distribution. Let’s use these formulas to check mfx:

 . mfx, predict(p)  
 
 Marginal effects after hetprob
       y  = Pr(y) (predict, p)
          =  .54284206
 ------------------------------------------------------------------------------
 variable |      dy/dx    Std. Err.     z    p>|z|  [    95% C.I.   ]      X
 ---------+--------------------------------------------------------------------
        x |   .3704576      .03237   11.44   0.000   .307015    .4339   .020114
     xhet |  -.0736092      .02423   -3.04   0.002  -.121095 -.026124   .502716
     off1 |  (offset1)                                                  1.00516
     off2 |  (offset2)                                                  1.09709
 ------------------------------------------------------------------------------
 
 . summarize xhet if e(sample) 
 
     Variable |       Obs        Mean    Std. Dev.       Min        Max
 -------------+--------------------------------------------------------
         xhet |      1000    .5027165    .2938006   .0004071   .9983656
 
 . scalar meanxhet=r(mean)
 
 . summarize off2  if e(sample) 
 
     Variable |       Obs        Mean    Std. Dev.       Min        Max
 -------------+--------------------------------------------------------
         off2 |      1000    1.097091    .2822951   .6011194   1.597888
 
 . scalar meanoff2=r(mean)
 
 . scalar predy=normal((meanx*_b[x]+_b[_cons] + 
 > meanoff1)/exp(_b[lnsigma2:xhet]*meanxhet +meanoff2))
 
 . display _n "pred y = " predy 
 
 pred y = .54284206
 
 . scalar dpdx=_b[x]*normalden((meanx*_b[x]+_b[_cons] + 
 > meanoff1)/exp(_b[lnsigma2:xhet]*meanx
 > het  +meanoff2))/exp(_b[lnsigma2:xhet]*meanxhet +meanoff2)
 
 . display _n "dpdx = " dpdx 
 
 dpdx = .37045757
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