Least Absolute Deviation function

Question

I'm trying to estimate LAD regression, but it gives my the message: "false convergence (8)". What does it mean and why nlminb estimators are equal to lm estimators?

Sample generation step

dgp=function(){
  x=c(sample(0:9,10),sample(0:9,10));
  b0=2;
  b1=-6;
  eps=rbinom(20,1,0.05)*rnorm(20,0,1)+rbinom(20,1,0.95)*rnorm(20,0,1);
  eps=eps/sd(eps);
  y=b0+b1*x+eps;
  return(data.frame(y=y,x=x))
}
z=dgp()

Estimation step

LAD=function(...){
  z=(...);
  y=z[[1]]
  x=z[[2]]
  LADf=function(par) {(sum(y-par[1]-par[2]*x)^2)}

  outLS=lm(y~x);
  b0=as.numeric(outLS$coefficients[1]);b0
  b1=as.numeric(outLS$coefficients[2]);b1

  out=nlminb(c(b0,b1),LADf)
  return(list(out$par,out$message))
}
LAD(z)

Answer 1

Your LAD function:

LADf=function(par) {(sum(y-par[1]-par[2]*x)^2)}

Looks exactly like least squares for me. Hence what you do is minimizing sum of squares, not abs of deviance. You need something like

LADf <- function(par) { sum(abs(y - par[1] - par[2]*x)) }

Note that this function is not differentiable, so you have to use an optimizer that can handle that (Nelder-Mead or SANN for instance).

LAD estimator is also equivalent to median regression so you can do this with quantile regression package instead.