Quantile regression with bootstrapping confidence intervals
Quantile Regression
USAGE: [p,stats]=quantreg(x,y,tau[,order,nboot]);
INPUTS:
x,y: data that is fitted. (x and y should be columns)
Note: that if x is a matrix with several columns then multiple
linear regression is used and the "order" argument is not used.
tau: quantile used in regression.
order: polynomial order. (default=1)
nboot: number of bootstrap surrogates used in statistical inference.(default=200)
stats is a structure with the following fields:
.pse: standard error on p. (not independent)
.pboot: the bootstrapped polynomial coefficients.
.yfitci: 95% confidence interval on polyval(p,x)
Note: uses bootstrap on residuals for statistical inference. (see help bootstrp)
check also: http://www.econ.uiuc.edu/~roger/research/intro/rq.pdf
EXAMPLE:
x=(1:1000)';
y=randn(size(x)).*(1+x/300)+(x/300).^2;
[p,stats]=quantreg(x,y,.9,2);
plot(x,y,x,polyval(p,x),x,stats.yfitci,'k:')
legend('data','2nd order 90th percentile fit','95% confidence interval','location','best')
For references on the method check e.g. and refs therein:
http://www.econ.uiuc.edu/~roger/research/rq/QRJEP.pdf
Copyright (C) 2008, Aslak Grinsted
1.3  implemented suggested change from Simeon Yurek in a FEX comment 

1.2  Fixed another small bug. 

1.1  Fixed a few issues with input parameter parsing. 
Christian Stano (view profile)
Hi Aslak, I am inexperienced using quantile regression and was wondering if you or anyone could break down what the different numbers in the p statistic mean.
Yao RuiQi (view profile)
hi,Aslak.I have some questions to ask you.Now I want to use this code to estimate the parameters of model.For example.x1=a*x2+b*x3+c*x4+d*x2.^2+e*x3.^2+f*x4.^2...
Xiaoyang Dong (view profile)
Correction: Not coefficient estimates. I mean standard errors.
Xiaoyang Dong (view profile)
Hi Aslak, thanks for your contribution. I'm wondering why when I switch the order of [x1,x2] to [x2,x1], the coefficients estimates lose the rotation invariance property?
Zhongfang He (view profile)
Hi Aslak, first thank you very much for sharing the code.
A quick question is about the way the standard errors of the parameter estimates are bootstrapped in your code. For many crosssectional data, it is not unreasonable to assume independence btw data points where it is fine to use the bootstrap method in your code. However, if one is dealing with time series data, there is usually serial correlation btw data points, a more suitable bootstrap method should be used, e.g. the stationary bootstrap of Politis&Romano(1994). I would suggest that the independence assumption is explicit in the code description and perhaps have another code specifically for correlated data points. Best regards, Zhongfang
Aslak Grinsted (view profile)
@lisa: they are the polynomial coefficients. They can be interpreted similar to the ouput from polyval
Lisa (view profile)
Hi, very helpful code, thank you.
A quick question: p outputs several values depending on the order, what do they represent?
thanks!
Aslak Grinsted (view profile)
@SimeonYurek: thank you for the suggestion. It has been implemented.
wei (view profile)
hi. just a quick question. How do we calculate the goodness of fit for quantile regression? thanks
Simeon Yurek (view profile)
Very nice code for Koenker and Hallock (2001). Thanks for posting.
One question: in your statement of the function rho (line 85), when r >= 0 (all positive residuals above x*p), does the function reduce to abs(r), and thus is not weighted by tau? It's true that [r  0.*r/tau] = r.*tau, but is the tau lost (being multiplied by zero)? Could you have stated instead:
rho=@(r)sum(abs(r).*abs(tau(r<0)))
and this way weight both over and under residuals? The stats for the bootstrap are slightly less robust but not by much. Please let me know if I'm off.
Antony: help fminsearch. 'MaxFunEvals' and 'MaxIter' can be defined as options.
Antony (view profile)
Hi, first of all, thank you very much for the code.
When I run the code I get the following message:
"Exiting: Maximum number of function evaluations has been exceeded
 increase MaxFunEvals option.
Current function value: 3183.464509 "
I get this message a number of times with just the current function value changing.
While it is not an error message and I still obtain the estimations I need, I was wondering if this had any influence on the validity of my results.
Thank you in advance for anyone who can help me out here.
Wolfgang Schwanghart (view profile)
Excellent. Well written help and code! Runs as advertised!
Satis (view profile)
If X and y have some missing observations, then
[p,stats]=quantreg(X,y,.9); does not work. I got error message.
Could you please suggest me how to resolve this problem??
Aslak Grinsted (view profile)
@mohammad: The code does do MLR. Here's an example with multiple predictors:
X=randn(100,4);
y=X*[4 2 4 1]'+randn(100,1)*.2;
[p,stats]=quantreg(X,y,.9);
Niki (view profile)
In fact I wanted to run this Mfile for multi column X, but I got error, so it is not implement for MLR? if it is , could you please put an example? because when I do not mention Order , I get error when i mention it is not MLR
Aslak Grinsted (view profile)
Thanks for the comment, and suggestions for improvement. I've just uploaded a new version (it should be online shortly).
AS (view profile)
One unexpected thing with this code. Suppose I have a Y vector and want to regress it on one explanatory variable, but also include a constant in my regression. Then my X matrix has 2 columns, the first column is just ones, the second is the explanatory variable. It should still be possible to plot this along with the results from the regression.
Currently, however, if you run:
quantreg([ones(length(x),1) x], y,.5)
You get an error because it tries to plot it, but order has been set (line 44) to [], so things get messed up.
Additionally, the plots that do get produced look odd because the default is to draw lines between all of the points, which usually isn't what you want. For example, this doesn't look like what it should:
x=randn(1000,1);
y=1+5*x+randn(1000,1);
quantreg(x,y,0.5)
Finally, there's a problem with the error checking of inputs (lines 4147) because it's all one big ifelse statement. If you only put in three inputs, then line 44 runs and order gets set to [], but then the program exits the ifelse statement and so Nboot never gets set correctly. You should split these up into separate if statements because if there are only 3 inputs then you need to set both order and Nboot.