Testing the Information Matrix Equality with Robust Estimators

where the last element follows from

tη(u) = φ(u) [1 + 4(u⁴ - 2u² - 1)] + o(η).

See e.g. Johnson et al. (1995, p. 375). The information matrix, evaluated
at σ =1 andη =0, is

E [s (y ; β, 1, 0) s (y ; β, 1, 0) ']

u³ - u

(u² - 1)²

u⁵ - 2u³ - u

u ⁶ - 3 u 4+u 2 + 1

4

( u⁴ - 2 u 2 -1)2

16       )

0

2

7

2

(27)

Since the first two elements of s equal zero at the restricted ML estimator,
and since J ³³ = 2/3, the score test statistic equals

s=n .,

24     ^,

which is the ‘kurtosis part’ of the Jarque-Bera statistic (12).

Let Fn be F_t(pn) with pn = η-¹ = √n∣e. We then have that η₀,_n = 0,
and, from (17),

σ₀²_,n =1+2ηn + o(ηn),

since Σ1 = 4 for the ML estimator. Using (18) and (19), with Σ1 =4,it
follows that

EFn [s(Y; β0,n,σ0,n, 0)] =

0
o(ηn)
2 ηn + o ( ηn )

Hence, replacing ηn with e∣√n,

b = lim VEE_Fn [ s ( Y ; βo ,n, σ o ,n, 0)] = e
n→∞

0

0

3

2

and, using (27),

δ = bJ^-1 b = 3 e².

33