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Say we fitted quantile regression models with a set of quantiles between 0 and 1 and a linear regression (i.e. mean regression) to a same dataset with the same set of covariates. In terms of the estimate for regression coefficient for a specific covariate, say x, we will get nine estimates from the quantile regression corresponding to each of the nine quantiles and only one estimate from the mean regression. If we plot those estimates in the same plot, with y-axis represents the values of the estimate and x-aixs for the quantiles, we should get a horizontal line for the estimate from mean regression and a zigged line connecting the estimates from quantile regression.

My question is if these two lines must cross at some point (i.e. some quantile) in every case or is it possible for them not to cross with each other for all possible quantiles?

Is the conclusion consistent in the setting of linear quantile mixed model and linear mixed model?

edit @05/13: sample plot inserted the quantile regression (solid) doesn't not cross with mean regression line (dotted)

Thanks.

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  • $\begingroup$ A special case of this situation is when there is a single constant covariate. If I understand your plot correctly, in this case your question seems to amount to "is it possible for the mean of a dataset to be less than its 10th percentile or greater than its 90th percentile?" Is this interpretation correct? If not, then some clarifying edits to this question would be welcome. $\endgroup$ – whuber May 9 '14 at 21:38
  • $\begingroup$ I'm not sure what you mean by "a zigged line" when talking about the nine quantile lines. $\endgroup$ – Glen_b May 10 '14 at 8:51
  • $\begingroup$ Oh, I mean that contrast to a horizontal line standing for a constant from the mean regression, the estimates from quantile regression are not always the same for different quantiles, so if we connect them, it won't be a straight line. $\endgroup$ – askming May 11 '14 at 21:12
  • $\begingroup$ @whuber: thank you for your comments. In my question the selection of the quantiles is just a example, so it can be any values between 0 and 1 not limited to be in the interval 0.1 and 0.9. $\endgroup$ – askming May 11 '14 at 21:14
  • $\begingroup$ Thanks. I would still like to understand your question, so my query stands with a suitable modification: are you asking whether it is possible for the mean of a dataset to be less than its $q^\text{th}$ percentile for a given $q$? $\endgroup$ – whuber May 12 '14 at 13:05

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