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 $H_0\colon \beta^{QR}_{0.05}=\beta^{QR}_{0.50}=\beta^{QR}_{0.95}$?
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.
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