Solved – How to test the equality of coefficients in a quantile regression

You can just do a Wald test on the coefficients directly or via margins:

. sysuse auto (1978 Automobile Data)  . qreg price i.rep78, quantile(0.5) nolog  Median regression                                   Number of obs =         69   Raw sum of deviations    65163 (about 5079)   Min sum of deviations    63340                    Pseudo R2     =     0.0280  ------------------------------------------------------------------------------        price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval] -------------+----------------------------------------------------------------        rep78 |           2  |        170   1745.715     0.10   0.923    -3317.467    3657.467           3  |       -185   1612.622    -0.11   0.909    -3406.584    3036.584           4  |        864   1645.876     0.52   0.601    -2424.015    4152.015           5  |        463   1697.437     0.27   0.786     -2928.02     3854.02              |        _cons |       4934   1561.415     3.16   0.002     1814.715    8053.285 ------------------------------------------------------------------------------  . test _b[5.rep78] = _b[3.rep78]    ( 1)  - 3.rep78 + 5.rep78 = 0         F(  1,    64) =    0.69             Prob > F =    0.4082  . margins rep78, pwcompare(pveffects) Warning: cannot perform check for estimable functions.  Pairwise comparisons of adjusted predictions Model VCE    : IID  Expression   : Linear prediction, predict()  -----------------------------------------------------              |            Delta-method    Unadjusted              |   Contrast   Std. Err.      z    P>|z| -------------+---------------------------------------        rep78 |      2 vs 1  |        170   1745.715     0.10   0.922      3 vs 1  |       -185   1612.622    -0.11   0.909      4 vs 1  |        864   1645.876     0.52   0.600      5 vs 1  |        463   1697.437     0.27   0.785      3 vs 2  |       -355   878.6573    -0.40   0.686      4 vs 2  |        694   938.2936     0.74   0.460      5 vs 2  |        293   1026.051     0.29   0.775      4 vs 3  |       1049   658.3504     1.59   0.111      5 vs 3  |        648   778.3381     0.83   0.405      5 vs 4  |       -401   845.0837    -0.47   0.635 ----------------------------------------------------- 

Edit:

You can do a one-sided test like this:

qreg price i.rep78, quantile(0.5) nolog local sign_diff = sign(_b[5.rep78] - _b[3.rep78]) testnl _b[5.rep78] - _b[3.rep78] = 0 display "H_0: _b[5.rep78] >= _b[3.rep78] p-value = " normal(`sign_diff'*sqrt(r(chi2))) 

or perhaps like this:

qreg price i.rep78, quantile(0.5) nolog local sign_diff = sign(_b[5.rep78] -_b[3.rep78]) test _b[5.rep78] = _b[3.rep78]  display "H_0: _b[5.rep78] >= _b[3.rep78] p-value = " 1-ttail(r(df_r),`sign_diff'*sqrt(r(F)))