Testing the Information Matrix Equality with Robust Estimators

It follows that the score test statistic for testing κ = λ =0 is

which is the Jarque-Bera statistic (see (12)).

Let κ_n = k/y/n and λ_n = l∕_y∕n and F_n = F_κn ,_λn. As before, β_n and σ_n
are the solutions of (21) and (22) where expectations are now with respect
to F_κ,λ. From (21) we have, using (24),

Eφ ψ((Z) + ψ(Z)(-βn - Z(σn - 1))}{1 + ^κ6n(Z³ - 3Z)

Since EΦ [ψ(Z)] = 0 andψ is odd, it follows that

β_n = Σ3κ_n + o (β_n, σ_n - 1,κ_n,λ_n) ,                (29)

where

_ς =1 E [(Z³ - 3Z)ψ(Z)]

3   6 E [ψ(Z)]

Similarly, we have, from (22),

)   + o (β_n ,σ_n - 1,κ_n ,λ_n ) .

Now EΦ [ρc(Z)] = bc and ρc is even, so

σ_n =1+Σ4λ_n +(β_n ,σ_n - 1,κ_n ,λ_n) ,             (31)

where

1 E [(Z⁴ - 6Z2 + 3)Pc(Z)]

24 E [ZP_c(Z)]

Since

Y (1 - (σ_n - 1)) - β_n + o (κ_n, λ_n)

Y (1 - Σ4λ_n) - Σ3κ_n + o(κ_n,λ_n) ,

36

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