# variance-covariance matrix of coefficients in quantile regression

I am using quantile regression to estimate the effect of a categorical variable AG_SEP2 on a response outcome ScoreGSA according to the distribution of this outcome variable, while adjusting on other variables.

quantreg <- rq(ScoreGSA ~ AG_SEP2 + Sexe + Jumeau + ZS_POIDS_NAIS_Cat + CMU + CSPTOT, method="fn", c(0.2,0.5,0.8), data=SepGSA)
a <- summary(quantreg, se="boot", R=500, cov=TRUE)


AG_SEP2 is a 4-class categorical variable, so I get the following output:

    Coefficients:
tau= 0.2      tau= 0.5      tau= 0.8
(Intercept)           4.800000e+01  5.400000e+01  5.800000e+01
AG_SEP2AG3234_SEP1    9.907373e-11  1.000000e+00  1.000000e+00
AG_SEP2AG2431_SEP0   -2.000000e+00 -2.000000e+00 -1.000000e+00
AG_SEP2AG2431_SEP1   -7.000000e+00 -5.000000e+00 -2.000000e+00
...


Now, I would like to test whether two coefficients within a class of AG_SEP2 variable are different or not using Wald test. For example, within the class "AG2431_SEP1", does -7 at tau=0.2 is different from -2 at tau=0.8 ? To do so, I use the following formula: W = (beta1 - beta2) / (var(beta1) + var(beta2) - 2*cov(beta1,beta2))

My question is: I can get the covariance between coefficients for a given tau with the following code: