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  • #1

    What does Stata do with margins when using weighted least squares?

    I was surprised to find that, with weight (or aweight), margins does not seem to compute the average partial (marginal) effect using the WLS estimates. And it's not obvious to me what it is doing. I don't think it can be correct. If it's basing the APEs on the weighted equation, or somehow using a weighed average, this is incorrect when using weights for heteroskedasticity. I can obtain the correct APEs by centering by hand and it's often very different from what Stata reports. The "by hand" APEs are, in the few examples I've tried, much closer to the APEs produced using OLS. The following uses the data set 401kpart, which can be found here: https://www3.nd.edu/~rwilliam/statafiles/

    Below is my code and the Stata output, which implements the WLS estimator suggested in Chapter 8 of my introductory econometrics book. The APE of mrate using OLS is 0.1597905, the same as when I use centering (not shown). My calculation using WLS is 0.1591003 but the Stata margins command returns 0.1059637. I can't figure out where the last number comes from and I'm convinced it can't have much justification.

    As a general comment, the formula for the APEs is free of the method of estimation. In the equation

    prate = b0 + b1*mrate + b2*age + b3*mrate^2 + b4*mrate*age + u

    the APE of mrate is b1 + 2*b3*mrate_bar + b4*age_bar. The averages are just the usual sample averages of mrate and age across the sample. Then plug in the estimates of the bj coefficients. The only difference should be that OLS is used in one case and WLS in the others to obtain the bj^. When this is done, the APEs are very close. This also gives the same answer as centering. The one I can't explain is what is reported by margins after using weights.

    Final comment: This weighting here has nothing to do with a nonrandom sampling scheme, where one could debate how to properly compute the APEs.


    Code:
    . qui sum mrate

. scalar mrate_bar = r(mean)

. qui sum age

. scalar age_bar = r(mean)

. reg prate c.mrate c.age c.mrate#c.mrate c.mrate#c.age, vce(r)

Linear regression                               Number of obs     =      4,075
                                                F(4, 4070)        =     176.40
                                                Prob > F          =     0.0000
                                                R-squared         =     0.1314
                                                Root MSE          =     .17482

---------------------------------------------------------------------------------
                |               Robust
          prate | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
----------------+----------------------------------------------------------------
          mrate |   .2441939   .0198724    12.29   0.000     .2052332    .2831546
            age |   .0053887   .0004146    13.00   0.000     .0045759    .0062015
                |
c.mrate#c.mrate |  -.0513863   .0104353    -4.92   0.000    -.0718452   -.0309275
                |
  c.mrate#c.age |  -.0044911   .0004759    -9.44   0.000    -.0054241   -.0035581
                |
          _cons |   .7239956   .0076064    95.18   0.000     .7090829    .7389084
---------------------------------------------------------------------------------

. di _b[mrate] + 2*_b[c.mrate#c.mrate]*mrate_bar + _b[c.mrate#c.age]*age_bar
.15979052

. margins, dydx(mrate)

Average marginal effects                                 Number of obs = 4,075
Model VCE: Robust

Expression: Linear prediction, predict()
dy/dx wrt:  mrate

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       mrate |   .1597905   .0101427    15.75   0.000     .1399053    .1796757
------------------------------------------------------------------------------

. predict uh, resid

. gen luhsq = log(uh^2)

. reg luhsq c.mrate c.age c.mrate#c.mrate c.mrate#c.age

      Source |       SS           df       MS      Number of obs   =     4,075
-------------+----------------------------------   F(4, 4070)      =    129.65
       Model |  2078.54764         4   519.63691   Prob > F        =    0.0000
    Residual |  16312.7746     4,070  4.00805272   R-squared       =    0.1130
-------------+----------------------------------   Adj R-squared   =    0.1121
       Total |  18391.3222     4,074  4.51431572   Root MSE        =     2.002

---------------------------------------------------------------------------------
          luhsq | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
----------------+----------------------------------------------------------------
          mrate |  -1.886746   .2149019    -8.78   0.000    -2.308071   -1.465421
            age |  -.0551837   .0055427    -9.96   0.000    -.0660505   -.0443168
                |
c.mrate#c.mrate |  -.0586135   .1333664    -0.44   0.660    -.3200846    .2028576
                |
  c.mrate#c.age |   .0576433   .0076463     7.54   0.000     .0426524    .0726342
                |
          _cons |  -3.727866   .0693438   -53.76   0.000    -3.863817   -3.591914
---------------------------------------------------------------------------------

. predict gh
(option xb assumed; fitted values)

. gen hh = exp(gh)

. reg prate c.mrate c.age c.mrate#c.mrate c.mrate#c.age [w = 1/hh], vce(r)
(analytic weights assumed)
(sum of wgt is 684,708.534450744)

Linear regression                               Number of obs     =      4,075
                                                F(4, 4070)        =     158.57
                                                Prob > F          =     0.0000
                                                R-squared         =     0.1562
                                                Root MSE          =     .14373

---------------------------------------------------------------------------------
                |               Robust
          prate | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
----------------+----------------------------------------------------------------
          mrate |   .2386863   .0190309    12.54   0.000     .2013753    .2759974
            age |   .0039402   .0004194     9.39   0.000     .0031178    .0047625
                |
c.mrate#c.mrate |  -.0611719    .010162    -6.02   0.000    -.0810951   -.0412488
                |
  c.mrate#c.age |  -.0027945    .000486    -5.75   0.000    -.0037474   -.0018417
                |
          _cons |   .7368633   .0072664   101.41   0.000     .7226171    .7511094
---------------------------------------------------------------------------------

. di _b[mrate] + 2*_b[c.mrate#c.mrate]*mrate_bar + _b[c.mrate#c.age]*age_bar
.15910026

. margins, dydx(mrate)

Average marginal effects                                 Number of obs = 4,075
Model VCE: Robust

Expression: Linear prediction, predict()
dy/dx wrt:  mrate

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       mrate |   .1059637   .0050276    21.08   0.000     .0961069    .1158206
------------------------------------------------------------------------------

. gen mrate_dm = mrate - mrate_bar

. gen age_dm = age - age_bar

. reg prate c.mrate c.age c.mrate_dm#c.mrate_dm c.mrate_dm#c.age_dm [w = 1/hh], vce(r)
(analytic weights assumed)
(sum of wgt is 684,708.534450744)

Linear regression                               Number of obs     =      4,075
                                                F(4, 4070)        =     158.57
                                                Prob > F          =     0.0000
                                                R-squared         =     0.1562
                                                Root MSE          =     .14373

---------------------------------------------------------------------------------------
                      |               Robust
                prate | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
----------------------+----------------------------------------------------------------
                mrate |   .1591003    .009931    16.02   0.000     .1396301    .1785704
                  age |   .0026449    .000268     9.87   0.000     .0021195    .0031703
                      |
c.mrate_dm#c.mrate_dm |  -.0611719    .010162    -6.02   0.000    -.0810951   -.0412488
                      |
  c.mrate_dm#c.age_dm |  -.0027945    .000486    -5.75   0.000    -.0037474   -.0018417
                      |
                _cons |   .7606102   .0054198   140.34   0.000     .7499845    .7712359
---------------------------------------------------------------------------------------
    Tags: None
  • #2
    Hi Jeff,
    Thank you for pointing this out. I don't think I put enough emphasis on marginal effects with other functional forms and WLS on my own class.
    That being said.
    I think what you are observing is that, after applying weights to the sample, margins is using the same weights to estimate the marginal effects. And they can differ alot.

    Code:
    . sum mrate age

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
       mrate |      4,075     .463519    .4187388          0          2
         age |      4,075    8.186503    9.257011          1         71

. sum mrate age [w=1/hh]
(analytic weights assumed)

    Variable |     Obs      Weight        Mean   Std. Dev.       Min        Max
-------------+-----------------------------------------------------------------
       mrate |   4,075  684708.534    .8452094   .5874248          0          2
         age |   4,075  684708.534    10.49067   10.87822          1         71
    Now, a simple work around with margins could be requesting to not use weights

    Code:
    . margins, dydx(mrate) noweights

Average marginal effects                        Number of obs     =      4,075
Model VCE    : Robust

Expression   : Linear prediction, predict()
dy/dx w.r.t. : mrate

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       mrate |   .1591003    .009931    16.02   0.000     .1396301    .1785704
------------------------------------------------------------------------------
    Which I think provides you with what you were expecting.

    Now, I think the right way to do this when the data already had weights is use them explicitly on margins

    Code:
    margins [pw=orig_weight], dydx(*)
    Best wishes
    Fernando
    Last edited by FernandoRios; 12 May 2022, 10:40.

    Comment

    • #3
      Thanks Fernando! I didn't know about the noweights option. I the case of heteroskedasticity, I can't even interpret what using weighted averages means. I know what it means for survey sampling, but not for heteroskedasticity. And I agree with your comment about combining survey weights and heteroskedasticity.

      Comment

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