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How do I test if the quantile regression slopes are equal for different quantiles?

E.g. I run a quantile regression at 5% quantile, 50% quantile (median) and 95% quantile and obtain the slope estimates $(\hat\beta^{QR}_{0.05},\hat\beta^{QR}_{0.50},\hat\beta^{QR}_{0.95})$. It is highly unlikely all three numbers will be exactly the same even if the true slopes are equal. So how do I test $H_0\colon \beta^{QR}_{0.05}=\beta^{QR}_{0.50}=\beta^{QR}_{0.95}$?

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The quantile regression estimators for different quantiles are asymptotically distributed as a multivariate normal random vector with a certain mean vector and a certain covariance matrix, the expressions of which are available in the literature; see Hao & Naiman (2007) Chapter 4 (short and to the point), Davino et al. (2013) Chapter 3 or Koenker (2005) Chapters 3.3 and 4, among other. Hence, testing the hypothesis that quantile regression slopes are equal for two or more different quantiles is a relatively straightforward task. As noted in Hao & Naiman (2007) Chapter 4, the covariance matrix can be tedious to calculate analytically; simple bootstrap of the original data (not of regression residuals) is an easier alternative; Hao & Naiman (2007) Chapter 4 give a detailed account of it.

References

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  • $\begingroup$ This is an extremely interesting question, and some kind of bootstrap was what first came to mind. However, by bootstrapping the original data, aren’t we saying that the predictors are random variables? For a designed experiment, that bothers me. $\endgroup$
    – Dave
    Mar 6, 2021 at 16:38
  • $\begingroup$ @Dave, thanks! Hmm, I only have limited experience with bootstrap and I do not see immediately why bootstrapping original data could imply predictors are random variables. Could you expand a bit? (Or maybe there is a thread discussing this?) $\endgroup$ Mar 6, 2021 at 17:58
  • $\begingroup$ Bootstrap is mimicking draws from the original population of interest, using the next-best option (empirical distribution) since we can’t go back to the original population. Consider a designed experiment with 5 covid+ men, 5 covid- men, 5 covid+ women, and 5 covid- women. If we bootstrap everything, we could (and sometimes would) wind up with 6 covid+ women, which is not how the experiment would go if we were to repeat it. In that regard, the 5/5/5/5 ratio is not a random variable. We always have a balanced design. $\endgroup$
    – Dave
    Mar 6, 2021 at 18:11

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