Robust Econometrics

P^{( t} )

___________^{ψ ( t} )___________

Least squares

t²

21

Least absolute deviation

∖t∖

sign(t)

Quantile estimation

{τ — I(x < 0) }x

τ — I(x < 0)

Huber: for ∖t∖ ≤ c

t²

2t

for c < ∖t∖

^c∖^t∖

c sign(t)

Hampel: for ∖t∖ ≤ a

t²

2t

for a < ∖t∖ ≤ b

a∖t∖

a sign(t)

for b < ∖t∖ ≤ c

ɪt — -∖t² sign(t)
c-b     c-b

^{a ( c —} ∖^t∖ ) /^{( c — b} )

for c < ∖t∖

a∖t∖

0

Biweight (Tukey)

— (c² — t²)³1(∖t∖ ≤ c)/6

t(c² — t²)²1(∖t∖ ≤ c)

Sine (Andrews)

—ccos(x/c)I(∖t∖ ≤ πc)

sin(x/c)I(∖t∖ ≤ πc)