# Model specification for quantile regression

I am new to Quantile Regression and have a couple of questions. First, assume I want to study the relation between a bond index and various financial variables such as an equity market index over the period 1995-2015, would it be necessary to use crisis dummies? I have seen both ways used in the literature i.e. estimating a QR with and without dummies.

Secondly, what is the situation with specification tests for example relating to endogeneity etc? How do I know my results are unbiased? I can't find much information regarding these issues and would greatly appreciate any advice.

• Quantile regression (QR) is more of an estimation technique than a model. Consider a model $y=\beta_0+X\beta+\varepsilon$, where I explicitly separate the intercept $\beta_0$ from the slopes $\beta$. You can estimate $\beta$ (the slopes) using OLS, QR, maximum likelihood and what not. The most obvious difference between OLS and QR is in the intercept; under OLS $\hat\beta_0^{OLS}$ is an estimate of the conditional mean of $y$ when all $X$s are zeros; under QR $\hat\beta_0^{QR}$ is an estimate of a particular quantile of $y$ conditionally on all $X$s being zero. – Richard Hardy Aug 24 '17 at 7:50
• But whether you are using OLS or QR or something else, I would specify the same model -- a model that reflects my knowledge of the object being modelled. E.g. if you think it makes sense to include the dummies in the OLS regression, then include them in the QR, too. Regarding endogeneity, have you tried searching the literature? I suppose it should be a known issue/concern so there should be some material available. – Richard Hardy Aug 24 '17 at 7:52
• Hello @RichardHardy. So would you suggest estimating the equation using OLS first and conducting all the tests for specification on that and then simply carrying over the best model into a QR estimation? – Vladmir Putin Aug 24 '17 at 10:34
• No, why bother with OLS if you have decided to use QR. The point was, formulate the model sensibly just as if you were to estimate it with OLS. But then at the estimation stage use QR rather than OLS. I.e. distinguish the model building stage from the estimation stage. Your first question is about building a sensible model, not how to estimate it (since you have already chosen to do it with QR). Also, I suggest to separate the two questions into two posts. They are quite independent of each other and answering one does not depend on answering another. – Richard Hardy Aug 24 '17 at 11:42

Quantile regression (QR) is more of an estimation technique than a model. Consider a model $$y=\beta_0+X\beta+\varepsilon,$$ where I explicitly separate the intercept $\beta_0$ from the slopes $\beta$. You can estimate $\beta$ (the slopes) using OLS, QR, maximum likelihood and what not.
The most obvious difference between OLS and QR is in the intercept; under OLS $\hat\beta^{OLS}$ is an estimate of the conditional mean of $y$ when all $X$s are zeros; under QR $\hat\beta^{QR}(\tau)$ is an estimate of a particular quantile $\tau$ of $y$ conditionally on all $X$s being zero.