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By placing restrictions, you could ensure that the lines do not cross in the training sample, but it will in no way guarantee that crossing will not occur at the next observation you see after training the model. And if the quantile lines intersect within the training sample, it most probably means that your model is specified incorrectly: either mean or standard deviation change nonlinearly.

By placing restrictions, you could ensure that the lines do not cross in the training sample, but it will in no way guarantee that crossing will not occur at the next observation you see after training the model. And if the quantile lines intersect within the training sample, it most probably means that your model is specified incorrectly: either mean or standard deviation change nonlinearly, or you apply a wrong cost function when fitting the model.

Your model should take input $X$, multiply it by matrix(!) $\beta$ to get a matrix of forecasts $f=x\beta$ with size $n \times k$, where $n$ is number of observations and $k$ is number of estimated quantiles. I assume that quantile percentages $q$ are in increasing order. You should minimize the function

$$ L = \sum_{i=1}^n \left(\sum_{j=1}^k \max(q_j(y_i-f_{ij}), (q_j-1)(y_i-f_{ij})) + \sum_{j=1}^{k-1} \alpha \max(0, \delta - (f_{j+1}-f_{j})) \right) $$

Your model should take input $X$, multiply it by matrix(!) $\beta$ to get a matrix of forecasts $f=X\beta$ with size $n \times k$, where $n$ is number of observations and $k$ is number of estimated quantiles. I assume that quantile percentages $q$ are in increasing order. You should minimize the function

$$ L = \sum_{i=1}^n \left(\sum_{j=1}^k \max(q_j(y_i-f_{ij}), (q_j-1)(y_i-f_{ij})) + \sum_{j=1}^{k-1} \alpha \max(0, \delta - (f_{i,j+1}-f_{ij})) \right) $$

Minimizing this function by gradient descent will give you your non-crossing quantile lines (if $\alpha$ is large enough). But I still warn you that if "natural" quantile lines intersect, then ther might be problems with the functional form of your model.

Minimizing this function by gradient descent will give you your non-crossing quantile lines (if $\alpha$ is large enough). But I still warn you that if "natural" quantile lines intersect, then ther might be problems with the functional form of your model. Maybe you would prefer quantile estimates of Random Forest (like quantregForest in R), which are always consistent.