# Coefficient standard error of zero in quantile regression

I'm currently experimenting with quantile regression of a strongly right skewed outcome variable y on a 3-category exposure x (values 1,2,3). I wanted to model the .2, .5, and .8 quantile, using the interior point algorithm for estimation and bootstrapping (Markov chain marginal bootstrap method, n=1000) for confidence intervals. The software (SAS v9.2) gave no warning about convergence problems or the like.

For the latter two, the results appear reasonable, however, for the .2 quantile, I sometimes get standard errors of 0 for the intercept and both of the two exposure coefficients (dummy coded, 2 vs. 1 and 3 vs. 1).

Group sizes for subgroups defined by x are: 227 (x=1), 423 (x=2), and 90 (x=3). Like I said, y is strongly right skewed, with a range of 0-14 (mostly integer values, but decimal values also exist in the data). Proportion of subjects with y=0 is 31% for x=1, 18% for x=2, and 10% for x=3. I speculated that this might be the reason for the 0 standard errors, but I neither have enough experience with this particular method, nor enough mathematics background to really understand the algorithms.

Therefore my questions: Is it normal to get stanard error of 0 with this method? And what does it imply for the interpretation of these results? Should I drop the analysis of the .2 quantile altogether? Does the validity of the other two quantile regressions (for .5 and .8) remain unaffected by this?

Below is the SAS code I used, and the results of an exemplary analysis. I repeated the analysis 10 times and in 5 cases, standard error were 0 like in the example below, and in the remaining 5, they were between .01 and .10. Results for the .5 and .8 quantile were pretty stable.

PROC QUANTREG DATA=... ALGORITHM=INTERIOR(TOLERANCE=1e-4) CI=RESAMPLING(nrep=1000) ORDER=DATA; CLASS x; MODEL y = x / QUANTILE = .2 .5 .8; RUN;

I'd be highly grateful for any comments!

Quantile Parameter      x value   Estimate    StdErr       LowerCL      UpperCL

0.2      Intercept                0.0000      0.0000       0.0000       0.0000
0.2      x              3         1.0000      0.0000       1.0000       1.0000
0.2      x              2         1.0000      0.0000       1.0000       1.0000
0.5      Intercept                1.0000      0.0805       0.8419       1.1581
0.5      x              3         3.0000      0.4168       2.1817       3.8183
0.5      x              2         1.0000      0.0828       0.8375       1.1625
0.8      Intercept                3.0000      0.2288       2.5508       3.4492
0.8      x              3         4.2019      0.5462       3.1296       5.2742
0.8      x              2         1.0000      0.2444       0.5203       1.4797


SE's of zero are obviously highly suspicious. Usually, and I'm quite confident in this attribution in the present case, they arise from summary.rq when the response is really discrete and the fitted model decides that zero is a good point estimate of a covariate effect. In summary.rq the algorithm feels very proud of the fact that multiple values of the response are all predicted well by the fitted model. Of course pride goeth before a fall, and so forth, so in these cases I find it useful to consider perturbations of the data, both response and covariates, which usually produces something more sensible. The quantreg function ?dither can be used for this purpose.