Quantile regression with linprog in Matlab

Question

I am trying to implement the quantile regression process with a simple setup in Matlab. This page contains a description of the quantile regression as a linear program, and displays the appropriate matrices and vectors. I've tried to implement it in Matlab, but I do not get the correct last element of the bhat vector. It should be around 1 but I get a very low value (<1e-10). Using another algorithm I have, I get a value of 1.0675. Where did I go wrong? I'm guessing A, b or f are wrong.

I have tried playing with optimset, but I don't think that is the problem. I think I've made a conversion mistake when going from math to code, I just can't see where.

 % set seed
 rng(1);
 % set parameters
 n=30;
 tau=0.5;
 % create regressor and regressand
 x=rand(n,1);
 y=x+rand(n,1)/10;
 % number of regressors (1)
 m=size(x,2);
 % vektors and matrices for linprog
 f=[tau*ones(n,1);(1-tau)*ones(n,1);zeros(m,1)]; 

 A=[eye(n),-eye(n),x;
   -eye(n),eye(n),-x;
   -eye(n),zeros(n),zeros(n,m);
   zeros(n),-eye(n),zeros(n,m)];
 b=[y;
   y
   zeros(n,1);
   zeros(n,1)];
 % get solution bhat=[u,v,beta] and exitflag (1=succes)
 [bhat,~,exflag]=linprog(f',A,b);

Answer 1

I solved my problem by using the formulation (in the pdf) above the one I tried to implement in the question. I've put it in a Matlab-function if you're interested in the code.

function [ bhat ] = qregressMatlab( y, x, tau )
%   bhat are the estimates
%   y is a vector of outcomes
%   x is a matrix with columns of explanatory variables
%   tau is a scalar for choosing the conditional quantile to be estimated

n=size(x,1);
m=size(x,2);
% vectors and matrices for linprog
f=[tau*ones(n,1);(1-tau)*ones(n,1);zeros(m,1)];
Aeq=[eye(n),-eye(n),x];
beq=y;
lb=[zeros(n,1);zeros(n,1);-inf*ones(m,1)];
ub=inf*ones(m+2*n,1);

% Solve the linear programme
[bhat,~,~]=linprog(f,[],[],Aeq,beq,lb,ub);

% Pick out betas from (u,v,beta)-vector.
bhat=bhat(end-m+1:end);

end