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Suppose I have an estimator which includes an indicator function in the objective function, then the objective function is not smooth. But if I want to approximate the behavior of this estimator in small samples I would need to have a certain number of higher order derivatives such that I can apply the usual approximation methods like Edgeworth, van Mises or Taylor expansions. I'm particularly interested in the Edgeworth expansion.

Would it be possible to smooth out somehow the objective function with a kernel such that I can apply the Edgeworth expansion? I thought of replacing the indicator with a symmetric bound kernel but I'm not sure of whether this would work. Any help would be appreciated!

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up vote 5 down vote accepted

I don't know what estimator you are considering but what you propose has certainly been done before.

Horowitz (1998) investigates whether the bootstrap can be used for asymptotic refinements of median regression. He faces the same problem as you given that the objective function has an embedded indicator $$\widehat{\beta} = min_{\beta} \frac{1}{n} \sum^{n}_{i=1} [q-1(u_i <0)]u_i$$ where the indicator is one for negative residuals. The problem is that when $u_i = 0$ we are at the kink of the check function $\rho (u_i) \equiv [q-1(u_i <0)]u_i$. Horowitz (1998) "smooths" this objective function by changing $\rho (u_i)$ to $$\rho^{S}(u_i) \equiv \left[2K\left(\frac{u_i}{h}\right) -1 \right]u_i$$ where $K(\cdot)$ is symmetric and bounded with $K \in [-1,1]$ with a differentiable function that satisfies $K(\nu)=0$ if $\nu \leq -1$ and $K(\nu) = 1$ if $\nu \geq 1$. In this context $K$ is similar to the integral of a kernel function but not a Kernel itself. Horowitz (1998) then applies the bootstrap for asymptotic refinements - which is the only difference to what you are planning to do but the reasoning is similar. For this purpose he needed the smooth objective function.

Other papers have replaced the indicator with kernels like Kaplan and Sun (2012) or Whang (2006). If I remember correctly, the Kaplan and Sun (2012) paper use a kernel smoother and then apply an Edgeworth expansion for asymptotic refinements but I don't have the details ready just now. I post the references below if you have further interest in this issue.

References

  • Horowitz, J.L. (1998) "Bootstrap Methods for Median Regression Models", Econometrica, Vol. 66(6), pp. 1327-1351 [link]
  • Kaplan, D.M. and Sun, Y. (2012) "Smoothed Estimating Equations for Instrumental Variables Quantile Regression", UC San Diego Working Paper [link]
  • Whang, Y.-J. (2006) "Smoothed empirical likelihood methods for quantile regression models", Econometric Theory, Vol. 22(2), pp. 173-205 [link]
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The application I had in mind is somewhat different but this looks promising. Thanks! –  user41384 Mar 5 at 22:41

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