Robust Econometrics

2.1 M -estimators

To achieve more flexibility in accommodating requirements on robustness,
Huber (1964) proposed the M -estimator by considering a general extremum
estimator based on ρ(z, θ)dF (z), thus minimizing ρ(z, θ)dFn(z) in finite
samples. Providing that the first derivative ψ(z,θ) = ∂ρ(z,θ)/∂θ exists, an
M -estimator can be also defined by an implicit equation ψ(z, θ)dFn(z) = 0.

This extremely general definition is usually adopted to a specific estima-
tion problem such as location, scale, or regression estimation. In a univariate
location model, F (z) can be parametrized as F (z - θ) and hence one limits
ρ(z, θ) and ψ(z, θ) to ρ(z - θ) and ψ(z - θ). In the case of scale estimation,
F(z) = F(z/θ) and consequently ρ(z,θ) = ρ(z/θ) and ψ(z,θ) = ψ(z/θ).
In linear regression, z = (x, y) and a zero-mean error term ε = y - x^>θ.
Analogously to the location case, one can then consider ρ(z, θ) = ρ(y - x^>θ)
and ψ(z, θ) = ψ(y - x^>θ)x, or more generally, ρ(z, θ) = ρ(y - x^>θ, x) and
ψ(z, θ) = ψ(y - x^>θ, x) (GM -estimators). Generally, we can express ρ(z, θ)
as ρ{η(z,θ)}, ψ{η(z,θ)}, where η(z,θ) ~ F.

Some well-known choices of univariate objective functions ρ and ψ are
given in Table 1; functions ρ(t) are usually assumed to be non-constant, non-
negative, even, and continuously increasing in |t|. This documents flexibility
of the concept of M -estimators, which include LS and quantile regression as
special cases.

On the other hand, many of the ρ and ψ functions in Table 1 depend

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