Testing the Information Matrix Equality with Robust Estimators

where q is the rank of V . Remark that T can also be written as

τ = n W^My Wvwɔ ⁺ WW ,

where W is any non-singular q × q matrix. Choosing W so as to make
WVW' diagonal simplifies T, and this will be exploited below.

Note that if, say, the j-th column of D(θ) is zero, then V is independent
of the estimator θj (the j-th element of θ) that is used. When V has full
rank, an obvious estimator is

1ⁿ

- ∑ξ ( Xi,Yi ; θ) ξ ( Xi ,Yi ; θ) )

i=1

For the ML estimator we have

It may occur that (some elements of) D(θ) need to be estimated by empirical
counterparts.

3 The normal model

3.1 The IM test

For the normal model without covariates, F_θ(y) = 1 Φ(^y-β) with θ = (β, σ))
and Φ the standard normal cdf. Letting u = (Y — β)/σ, we have (White,
1994, p. 332-333)

1 / u 2 — 1

m ( Y ; θ ) = — I    u³ — 3 u

^σ    u⁴ — 5u² +2

∂             ι /    2 u         4 u² — 2

—— m ( Y ; θ ) = 3 I 3 u² — 3      5 u³ — 9 u

^{∂θ           σ}    4u³ — 10u 6u⁴ — 20u² +4

₁  02

D ( θ ) =--3   0 0   ,

σ  ₀₂

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