# What is the interpretation of (panel data) Quantile Regression?

I estimated a (panel data) quantile regression model using qregpd in Stata 13. It is not clear what is the interpretation of the estimate. Let's say that I choose the 50th quantile, and I find that the estimate of the main parameter of interest is -0.5.

What is the interpretation of the estimate -0.5? Let's say that -0.5 is the estimated parameter for the effect of random variable x on random variable y. Let's say that x increases by 1, so that y will decrease by (-0.5 * 1) = -0.5. Is this effect true only for the guy at the median, or for all individuals?

• Can you put your optimization into a mle framework? For example, the lmm can fit into a normal log likelihood. What is the equivalent log likelihood for quantile regression? I know that the median can correspond to a laplace distribution (i.e. you are still modelling $E [y_i|x_i]$ but the errors are not from a normal distribution) – probabilityislogic Sep 8 '16 at 9:09

## 1 Answer

You can’t really say much about the guy at the median or how individuals will respons to changes in X. What you may say is that the median of a population with X equal to x is estimated to be 0.5 lower than the median of a population with X equal to (x+1), all other things being equal.

See this answer for a thorough discussion of how to interpret the result: https://stats.stackexchange.com/a/159928

• The OP is about a "mixed effects" quantile regression, not a standard cross-sectional one. Does that impact the way how you correctly interpret the effect? Btw. would you mind to change y to x in your answer? – Michael M Sep 7 '16 at 15:43
• Based on my understanding of standard panel data methods and cross sectional quantile regression I really can’t see how the interpretation of the coefficient for an explanatory variable would be any different in a panel data setting. Only the “all other things being equal” - condition now also involve “group(s)”. Hopefully someone will correct me if this wrong – SteinarV Sep 8 '16 at 7:54
• It will probably be different in that we are not talking about the median of a population with equal $X$-values but also in the same group specified by the random factor, which seems to be quite mind-twisting if we have e.g. person as random factor. – Michael M Sep 8 '16 at 7:59