Combining quantile regression with binning

Question

I'm trying to employ a framework where I uncover the marginal effects of the quantiles of one continuous variable on another continuous variable - something analogous to the Quantile-on-quantile (QQR) approach used in Sims and Zhou (2015) and others. Unfortunately, I fear my coding skills are insufficient to code up this approach myself, and packages to implement the QQR approach don't seem to exist yet on many of the major statistical programs.

Instead, to overcome this limitation, I'm thinking of an alternative technique where I:

1. Take "bins" of my independent variable around certain quantiles of that variable.
2. Create interaction terms which take the value = X(i) if X(i) is within the corresponding bin, and 0 otherwise.
3. Regress the interaction terms on the dependent variable in standard quantile regressions.
4. Repeat steps 1-3 with alternative plausible bin specifications.
5. Average over the results.

The intuition of what's seemingly achieved here seems to be fairly similar to the intuition of the original QQR approach, and I'm hopeful that steps 4-5 would remove some of the dependence of the results on binning assumptions.

Is the above technique likely to yield results that are useful, or should I give this up and start reading about splines?

Thanks in advance.

References:

Sim, N. and Zhou, H., 2015. Oil prices, US stock return, and the dependence between their quantiles. Journal of Banking & Finance, 55, pp.1-8.

Answer 1

If your predictor is continuous and your interest is in modeling quantiles of the outcome variable, then you should be able to do quantile regression with appropriately flexible modeling of the continuous predictor. The paper you cite (based on a quick look) evidently uses local linear regression, like loess modeling. Regression splines for predictors are useful approaches, as are generalized additive models.

In R, for example, the rq() function of the quantreg package, with restricted cubic spline modeling of the predictor via the standard ns() function or the rcs() function of the rms package, could do what you need. Thereafter, you can get outcome quantile estimates based on any desired quantile or binning of your predictor.

The point is not to throw away continuous information, particularly early in the modeling process. See this page for extensive discussion.

Answer 2

Quantile regression estimates conditional quantiles. You would need to bin the conditional distribution, so you would need to solve a much harder problem first of finding the full conditional distribution. If you knew the conditional distribution, you wouldn't need to do any binning, but just calculate quantities directly. So no, this is not a good idea.