I'm trying to determine whether Beta estimates at different quantiles obtained using quantile regression (quantreg package in R) are statistically different from each other. In reading the work of Roger Koenker ("Quanile Regression" 2005. pg 75-76) and Hao & Naiman ("Quantile Regression"2007. pg 47-51). This seems to be a relatively simple matter of constructing a Wald Statistic as follows.
$$
W= \frac{(\hat\beta_j^{(p)}-\hat\beta_j^{(q)})^2}{\hat\sigma^2_{(\hat\beta_j^{(p)}-\hat\beta_j^{(q)})}}
$$
Where $\hat\beta_j^{(p)}$ and $\hat\beta_j^{(q)}$ are the beta estimates for covariate $j$ at quantile $p$ and $q$, and $\sigma^2_{(\hat\beta_j^{(p)}-\hat\beta_j^{(q)})}$ is an estimate of the variance in those estimates as given by:
$$
\text{Var}\left(\hat\beta_j^{(p)}\right) + \text{Var}\left(\hat\beta_j^{(q)}\right) - 2\text{Cov} \left(\hat\beta_j^{(p)}, \hat\beta_j^{(q)}\right).
$$
When estimates of from two taus ($p$ and $q$) are compared this is straight forward enough. But what about when more than two taus are compared?
So for taus $p$, $q$, and $r$, do i need to compare $p$ vs $q$, $p$ vs $r$, and $q$ vs $r$ ? This problem escalates quickly when even more taus of interested are compared (in my case 19 taus = 171 unique combinations of $p$ and $q$).
My overall goal is to test whether for $n$ taus (quantiles) $$H_0 : \hat\beta_j^{(i)}=\hat\beta_j^{(i+1)}\ldots=\hat\beta_j^{(n)}$$ $$H_1 : \hat\beta_j^{(i)}\neq\hat\beta_j^{(i+1)}\ldots\neq\hat\beta_j^{(n)}$$
