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
-1
1n
- ∑ξ ( Xi,Yi ; θ) ξ ( Xi ,Yi ; θ) )
i=1
For the ML estimator we have
IF( X,Y ; Θml ; K,Fθ )
=—Ee ∂sθ(χ,γ) ) sθ(χ,γ)∙
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, Fθ(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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