Compare OLS and quantile regression coefficients

[May Fang]

Hello, I have 2 questions on testing whether the coefficients across OLS and quantile regressions are equal. Below are the models and codes I used.

The model I use is simultaneous quantile regressions like this: Y = X1 + X2 + X3 + control (1)
And the code I use is: sqreg Y X1 X2 X3 control, q(.25 .5 .75)

I used test [q25=q50=q75]: X1 to test whether the coefficients are equal across quantiles.

I was asked to cluster by permno, so I also test the model at p25 p50 p75 separately
And the code I use is:
qreg2 Y X1 X2 X3 control, q(.25) cluster(permno) (2)
qreg2 Y X1 X2 X3 control, q(.5) cluster(permno) (2)
qreg2 Y X1 X2 X3 control, q(.75) cluster(permno) (2)

Now comes my 2 question:

(1) I try to test whether the coefficients on X1 (also for X2 and X3) are equal between OLS and model 1 quantiles q25 q50 q75
To do this, I tried to store OLS regression as ols, use suest, and then test [q25]X1=[q50]X1=[q75]X1=[ols]X1, but it gave me an error message.

(2) I tried to test whether the coefficients on X1 are equal or not across the quantiles in regression 2
again suest does not allow me to do the job.

Could you please help me on this? Your inputs are greatly appreciated!

[Joro Kolev]

I do not think that what you want is possible with -suest-. -suest- is intended for use with subsambles on the same regression equation, or possibly different regression equations, but the same estimation method. In fact I think -suest- only handles OLS properly, that is it handles only -regress-.

To do what you want to do, you can follow the procedure that -sqreg- does: bootstrap. You generate a bootstrap sample you estimate all the coefficients you are interested in, you store them, then you repeats this B number of times and you have a set of B observations where each observation is an estimate from each of your methods in a given bootstrap sample. Then you can do your testing from this.

But this is not easy, requires a lot of work.

[FernandoRios]

Hi May,
As Joro suggest, -suest- doesnt work well with qreg because it mostly uses regression "scores" (or residuals for OLS), to correct standard errors when estimating models jointly.
I would suggest, however, that doing the bootstrap version is quite easy. Here a simple example you may want to use for your purposes:

 webuse womenwk
** Create wrapper program for all models of interest. Make sure you add the eclass  at the end of the program

capture program drop ols_vs_qreg
program ols_vs_qreg, eclass
** estimate OLS
reg wage age education married children
matrix bols=e(b)
** estimate qreg 10 50 90 (or other of interest)
qreg wage age education married children, q(10)
matrix bq10=e(b)
qreg wage age education married children, q(50)
matrix bq50=e(b)
qreg wage age education married children, q(90)
matrix bq90=e(b)
** name equations across matrix
matrix coleq bols = ols
matrix coleq bq10= q10
matrix coleq bq50= q50
matrix coleq bq90= q90
** Put all b together
matrix b = bols, bq10 , bq50 , bq90
** "post" them
ereturn post b
end
** Make sure you use the sample you actually need
reg wage age education married children
keep if e(sample)

** bootstrap

bootstrap: ols_vs_qreg
 ** Output

Bootstrap replications (50)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 
..................................................    50

Bootstrap results                                        Number of obs = 1,343
                                                         Replications  =    50

------------------------------------------------------------------------------
             |   Observed   Bootstrap                         Normal-based
             | coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
ols          |
         age |   .1514818   .0197969     7.65   0.000     .1126805     .190283
   education |   .8750694   .0562646    15.55   0.000     .7647928    .9853459
     married |  -.5395024   .3333384    -1.62   0.106    -1.192834    .1138288
    children |  -.6862982   .1097916    -6.25   0.000    -.9014858   -.4711106
       _cons |   7.934369   .9855672     8.05   0.000     6.002693    9.866046
-------------+----------------------------------------------------------------
q10          |
         age |   .1362803   .0322593     4.22   0.000     .0730531    .1995074
   education |   .8601218    .090122     9.54   0.000     .6834859    1.036758
     married |  -1.231153    .543483    -2.27   0.023     -2.29636   -.1659458
    children |  -1.042808   .1492278    -6.99   0.000    -1.335289   -.7503267
       _cons |   3.123572    1.78768     1.75   0.081    -.3802155     6.62736
-------------+----------------------------------------------------------------
q50          |
         age |   .1653391    .021658     7.63   0.000     .1228902    .2077881
   education |   .8940777   .0757136    11.81   0.000     .7456818    1.042474
     married |  -.5971263   .4336936    -1.38   0.169     -1.44715    .2528975
    children |  -.6206084   .1337395    -4.64   0.000    -.8827329   -.3584839
       _cons |   6.906066    1.27135     5.43   0.000     4.414266    9.397867
-------------+----------------------------------------------------------------
q90          |
         age |   .1537102   .0265448     5.79   0.000     .1016832    .2057371
   education |   .9077313   .1045996     8.68   0.000     .7027199    1.112743
     married |   .3190477    .498328     0.64   0.522    -.6576572    1.295753
    children |  -.5730253   .1777933    -3.22   0.001    -.9214937   -.2245569
       _cons |   13.28619    1.57443     8.44   0.000     10.20036    16.37202
------------------------------------------------------------------------------

After that, you can test coefficients across models as you would do with sqreg.
HTH
F

[Joao Santos Silva]

Dear FernandoRios,

Your solution is excellent, but I guess May Fang will want to do the bootstrap while accounting for the clustering, right?

Best wishes,

Joao

[FernandoRios]

Thanks Joao!
That is true! (I just saw she was doing that with qreg2, not qreg.
So, May, in that case you can add the cluster option in bootstrap. Although those results will be different from the ones produced by Joao's -qreg2-, because they will be based on the empirical, not asymptotic properties for quantile regressions.
Something like this:

. bootstrap, cluster(county): ols_vs_qreg
(running ols_vs_qreg on estimation sample)

warning: ols_vs_qreg does not set e(sample), so no observations will be excluded from the resampling
         because of missing values or other reasons. To exclude observations, press Break, save the
         data, drop any observations that are to be excluded, and rerun bootstrap.

Bootstrap replications (50)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 
..................................................    50

Bootstrap results                                        Number of obs = 1,343
                                                         Replications  =    50

                                 (Replications based on 10 clusters in county)
------------------------------------------------------------------------------
             |   Observed   Bootstrap                         Normal-based
             | coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
ols          |
         age |   .1514818   .0164449     9.21   0.000     .1192505    .1837131
   education |   .8750694   .0550743    15.89   0.000     .7671256    .9830131
     married |  -.5395024    .306771    -1.76   0.079    -1.140763    .0617577
    children |  -.6862982   .0887096    -7.74   0.000    -.8601658   -.5124306
       _cons |   7.934369   1.028707     7.71   0.000     5.918141    9.950598
-------------+----------------------------------------------------------------
q10          |
         age |   .1362803   .0232826     5.85   0.000     .0906473    .1819133
   education |   .8601218   .0656075    13.11   0.000     .7315335    .9887101
     married |  -1.231153   .5198543    -2.37   0.018    -2.250049   -.2122572
    children |  -1.042808   .1780629    -5.86   0.000    -1.391805   -.6938109
       _cons |   3.123572   1.287632     2.43   0.015     .5998602    5.647285
-------------+----------------------------------------------------------------
q50          |
         age |   .1653391   .0244855     6.75   0.000     .1173484    .2133299
   education |   .8940777   .0873886    10.23   0.000     .7227991    1.065356
     married |  -.5971263   .3937529    -1.52   0.129    -1.368868    .1746151
    children |  -.6206084   .1416982    -4.38   0.000    -.8983318    -.342885
       _cons |   6.906066   1.607285     4.30   0.000     3.755845    10.05629
-------------+----------------------------------------------------------------
q90          |
         age |   .1537102   .0231695     6.63   0.000     .1082987    .1991216
   education |   .9077313   .0903143    10.05   0.000     .7307185    1.084744
     married |   .3190477   .5291344     0.60   0.547    -.7180367    1.356132
    children |  -.5730253   .1314047    -4.36   0.000    -.8305738   -.3154768
       _cons |   13.28619   1.298303    10.23   0.000     10.74156    15.83082
------------------------------------------------------------------------------

** compared to 


. qreg2 wage age education married children, q(.90) cluster(county)

.9 Quantile regression
R-squared = .27387666
Number of obs = 1343
Objective function = .94910521

Standard errors adjusted for 10 clusters in county
------------------------------------------------------------------------------
        wage | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
         age |   .1537102   .0247034     6.22   0.000     .1052923     .202128
   education |   .9077313   .1018411     8.91   0.000     .7081264    1.107336
     married |   .3190477   .6381099     0.50   0.617    -.9316248     1.56972
    children |  -.5730253   .1229804    -4.66   0.000    -.8140625   -.3319881
       _cons |   13.28619   1.264325    10.51   0.000     10.80816    15.76422
------------------------------------------------------------------------------

HTH

[May Fang]

I do not think that what you want is possible with -suest-. -suest- is intended for use with subsambles on the same regression equation, or possibly different regression equations, but the same estimation method. In fact I think -suest- only handles OLS properly, that is it handles only -regress-.

To do what you want to do, you can follow the procedure that -sqreg- does: bootstrap. You generate a bootstrap sample you estimate all the coefficients you are interested in, you store them, then you repeats this B number of times and you have a set of B observations where each observation is an estimate from each of your methods in a given bootstrap sample. Then you can do your testing from this.

But this is not easy, requires a lot of work.

Joro,

Thank you for your comment. Indeed, almost every time I have successfully used suest, it was for "regress".

[May Fang]

Thanks Joao!
That is true! (I just saw she was doing that with qreg2, not qreg.
So, May, in that case you can add the cluster option in bootstrap. Although those results will be different from the ones produced by Joao's -qreg2-, because they will be based on the empirical, not asymptotic properties for quantile regressions.
Something like this:

. bootstrap, cluster(county): ols_vs_qreg
(running ols_vs_qreg on estimation sample)

warning: ols_vs_qreg does not set e(sample), so no observations will be excluded from the resampling
because of missing values or other reasons. To exclude observations, press Break, save the
data, drop any observations that are to be excluded, and rerun bootstrap.

Bootstrap replications (50)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5
.................................................. 50

Bootstrap results Number of obs = 1,343
Replications = 50

(Replications based on 10 clusters in county)
------------------------------------------------------------------------------
| Observed Bootstrap Normal-based
| coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
ols |
age | .1514818 .0164449 9.21 0.000 .1192505 .1837131
education | .8750694 .0550743 15.89 0.000 .7671256 .9830131
married | -.5395024 .306771 -1.76 0.079 -1.140763 .0617577
children | -.6862982 .0887096 -7.74 0.000 -.8601658 -.5124306
_cons | 7.934369 1.028707 7.71 0.000 5.918141 9.950598
-------------+----------------------------------------------------------------
q10 |
age | .1362803 .0232826 5.85 0.000 .0906473 .1819133
education | .8601218 .0656075 13.11 0.000 .7315335 .9887101
married | -1.231153 .5198543 -2.37 0.018 -2.250049 -.2122572
children | -1.042808 .1780629 -5.86 0.000 -1.391805 -.6938109
_cons | 3.123572 1.287632 2.43 0.015 .5998602 5.647285
-------------+----------------------------------------------------------------
q50 |
age | .1653391 .0244855 6.75 0.000 .1173484 .2133299
education | .8940777 .0873886 10.23 0.000 .7227991 1.065356
married | -.5971263 .3937529 -1.52 0.129 -1.368868 .1746151
children | -.6206084 .1416982 -4.38 0.000 -.8983318 -.342885
_cons | 6.906066 1.607285 4.30 0.000 3.755845 10.05629
-------------+----------------------------------------------------------------
q90 |
age | .1537102 .0231695 6.63 0.000 .1082987 .1991216
education | .9077313 .0903143 10.05 0.000 .7307185 1.084744
married | .3190477 .5291344 0.60 0.547 -.7180367 1.356132
children | -.5730253 .1314047 -4.36 0.000 -.8305738 -.3154768
_cons | 13.28619 1.298303 10.23 0.000 10.74156 15.83082
------------------------------------------------------------------------------

** compared to


. qreg2 wage age education married children, q(.90) cluster(county)

.9 Quantile regression
R-squared = .27387666
Number of obs = 1343
Objective function = .94910521

Standard errors adjusted for 10 clusters in county
------------------------------------------------------------------------------
wage | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
age | .1537102 .0247034 6.22 0.000 .1052923 .202128
education | .9077313 .1018411 8.91 0.000 .7081264 1.107336
married | .3190477 .6381099 0.50 0.617 -.9316248 1.56972
children | -.5730253 .1229804 -4.66 0.000 -.8140625 -.3319881
_cons | 13.28619 1.264325 10.51 0.000 10.80816 15.76422
------------------------------------------------------------------------------

HTH

Fernando,

Really appreciate your inputs! I am going to test it and get back here.