Testing the Information Matrix Equality with Robust Estimators

When ^ is equivariant, IF(X,Y; ^; F_θ) = σIF(u; ^5Φ). Hence V = σ ⁴C
with C a partitioned matrix with blocks

C11 = 2 Ek [vech( XX ' )(vech( XX ' )) ' ]

+(Bιι - 2)Ek[vech(XX')]Ek[vech(XX')]',

C22 = B 22 Ek ( XX ' ),

C33 ⁼ B33 ^,

C13  =  B13 Ek [vech( XX ' )] = C 31,

C12  =  C21 = 0, C 23 = C3 2 = 0,

and all Bij as in Section 3. Again, V does not depend on the choice of the
estimator of the location parameter β . Replacing EK with sample averages
yields an estimate C of C. For a given θ = (β' , ^) ', let Ui = (Yi — X^,i∣3')/σ
and

1 n / (U2 — 1)vech(XiX') ∖    / N^1 ∖

N = - Σ      ( u 3 — 3 u i ) Xi      =   N^2   ∙

n i=1 ∖    U⁴ — 5 U 2 + 2    /    ∖ N₃ /

Then T takes the form

(ʌ » ʌ 1 1 ʌ        ʌ » ʌ nn ʌ        ʌ , ʌ OO ʌ          ʌ , ʌ to ʌ ∖

N₁ C¹¹TV₁ + N₂C²²IN₂ + N3C³³N₃ + 2N₁C¹³N3∖ ,

where Ci^j is the (i,j)-th block of (C⁺.

If θ is the ML estimator and the first column of X is a vector of ones, then
the first element of C11 is zero, B13 =0andB33 = 24. Let p = k(k + 1)/2,
with k =dim(/), and

L = 0p-1×1 Ip-1 ,

Li = L vech( XiX'),
1ⁿ

L = - ∑ Li∙
n
i=1

Then the IM test statistic with ML estimator can be written as (Hall, -987)
^{n                 n}                          -1 ⁿ

T = 2∑(U2 — 1)Li{∑(Li — L)(Li — L)T ∑(U2 — 1)Li
i=1              i=1                         i=1

nn   n    n

+6ΣU3Xi'(∑XiXiy ΣU3Xi + 2₄∑(Ui — 3)²∙
i=1        i=1            i=1             i=1

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