I applied quantile regression on some data and did it for tau = 0.25, 0.5, 0.75. After i got the estimates of each model, i did some cross validation on my hold out data. When i used the estimates for my hold out data i calculated in what quantile each of my independent variables lie, 25%, 50% or 75%. For instance if my x-values (independent variable) looks as follows,
x -> 1 2 3 4 5 6 7 8 9 11 10 12 13 14 15
I calculated my quantiles using the R function cut as cut(x,4) and got the following quantile ranges (0.986; 4.49], (4.49; 8], (8; 11.15], (11.15; 15]. I then divide the x vector into the correct quantile range and depending on what range it falls determine what quantile regression estimates i use, either 0.25, 0.5 or 0.75. So for the x-values 1 to 4 i will use the estimates of the 0.25, 5 to 8 i use the estimates of 0.5 and the rest i used 0.75.
I do get very accurate results, i.t.o. forecast accuracy, but can someone please tell if this is the correct way to do my quantile cross validation.
Best Answer
You have to understand that in quantile regression, the quantile of interests is not for the predictor variables but the response variable so it is pointless to do a cross-validation using the quantile of your predictors. Here is your model for $\tau^{th}$ quantile: $$Q_\tau (Y|X=x) = X\beta(\tau)$$ In classical regression, you evaluate the model using a loss function. It's the same for the quantile regression, you need a loss function to evaluate how good your model is. In the case of quantile regression, your loss function is: $$\rho_\tau(u)=u(\tau - I(u<0)).$$