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I have fitted a QR model. Then, I take the transformation of QR model.My transformed model is here,

    log(Q(Y|x))=beta0 + beta_1*x_1

How can I interpret the coefficients of X_1? Because QR model satisfies the monotony tranformation properties. Thanks in advance!

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Before moving on to the interpretation, one usually does not make any assumption about the distribution of the response variable in quantile regression. The key advantage of quantile regression is that you can look at the full conditional distribution at different quantiles and that is without making any assumption about the distribution like in linear models. In linear regression, for example, you require some distributional assumption such as normality and to meet this assumption, one applies a log transformation to the original skewed data. But you do not have this constraint here. This is why I don't understand the reason why you would apply log transformation.

If you're not convinced, suppose you are interested in the $\tau^{th}$ quantile on a log scale. Then $\beta_1$ is the increase in the $\tau^{th}$ quantile on a log scale with one unit increase in $X_1$. But without making the transformation, you do not need to think on a log scale, which is simpler.

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