Estimation in Financial Models - RiskLab

disadvantage of ~G n and G n motivates us to consider martingale estimating
functions generated by higher order conditional moments for
example by the conditional variance:
H n () =
nX
2
h(X (i,1) ; ) Xi , F (X (i,1) ; ) , (X(i,1) ; )
i=1
; (3.59)
with given by (3.51) and F given by (3.47). In analogy to (3.50) we obtain
the optimal estimating function within the class (3.59), in the sense of
Godambe and Heyde [33]. This optimal function takes the form
H n() =
nX
i=1
_(X (i,1) ; )
(X (i,1) ; )
Xi , F (X (i,1) ; ) 2
, (X(i,1) ; )
; (3.60)
where is assumed to be dierentiable in and is the fourth conditional
cumulant
(X (i,1) ; ) =E
Xi , F (X (i,1) ; ) 4
jX(i,1)
, (X (i,1) ; ) 2 ;
where i =1;:::;n.
Combining the functions G n and H n leads to further martingale estimating
functions that may have better properties. Following the optimal way of
combining G n and H n described in Heyde [35], the function Kn is obtained:
" nX _()() , F () ()
Kn() =
(X
() () , 2 i , F ())
()
i=1
F
+ _ ()() , ()() _ # (Xi , F ()) 2 , () ; (3.61)
() () , 2 ()
where for abbreviation the rst argument of all functions on the right hand
side, that is X (i,1) , has been left out and where denotes the third conditional
central moment
(X (i,1) ; ) =E
h
(Xi , F (X (i,1) ; )) 3 jX (i,1)
i
; i =1;:::;n:
For K n, as for G n , it can be shown that an estimator obtained from the estimating
equation K n() =0exists, is consistent and asymptotically normal.
Under some regularity conditions (see Bibby [5], pp. 4{6) we have
Theorem 5 An estimator ^ n exists for every n, which on a set C n solves
the equation
K n (^ n )=0;
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