Testing the Information Matrix Equality with Robust Estimators

where expectations are with respect to F^. From (21),

Now, expanding the integrands around β_n = 0, σ_n = 1, and γ_n = 0 gives

ψ ^Zz—^βn∖ = ψ(Z) + ψ/(Z) (-βn - Z(σn - 1)) + o(βn, σn - 1)    (24)

∖ σ_n J

and

+^o(^βn^{, σ}n

= Eφ [ψ(Z)] - βnEφ [β'(Z)] - (σn

+Yn    zψ ( Z ) d Φ( Z )+ O ( βn,σn

Jo

Since Eφ [ψ(Z)] = Eφ [Zψ'(Z)] = 0, it follows that β_n = γ_nΣ₂ + o(γ_n), with

_{ς =} ∕o∞ ^{zψ (z})^dφ(^z)

² E [Ψ(Z)]

For σ_n it holds that

Since (24) and (25) also hold with p_c replacing ψ, (26) becomes

^bc = ^eΦ ^[pc^{(z)] - β}n^eΦ [^p_c^(z)] ^{- (σ}n

+Yn / Zpc(Z)dΦ(Z) + O(βn,σn-
Jo

Now Eφ [p_c(Z)] = b_c, Eφ [p'_c(Z)] = 0 and J∞ zp'_c(z)dΦ(z) = ɪE_φ [Zp_c(Z)],
so we get

^σn = ¹ + ɪ- ^{+ o ( γ}n ).

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