Testing for One-Factor Models versus Stochastic Volatility Models



(ii) We will prove the Theorem for the case analyzed in part (i)c; in the other cases the proof
follows straightforwardly and is therefore omitted. Under H
a , we have that

d Xt = μ ( Xt )d t + yptW Wɪ ,t

σt = g( ft )

d ft = b ( ft )d t + σ ɪ ( ft )d W2 ,t.

Pointwise in r, we can rewrite Zn,m,r as

Zn,m,r


,— L ( n-ɪ) rj

m ∑  (Sn(Xi/n) - g (fi/n))

i =1

L ( m— ɪ) rj

-m       (x(j+ɪ)/m - Xj/m)

j=i



l (n- 1)rj

∑ g (fi/n)

i =1


ʌ- l(n-ɪ)rj

m ∑  (Sn(Xi/n) - g (fi/n))

i = 1

'------------------------------->z-------------------------------'

E ʌ-.

n,m,r


([ ( m— ɪ) rj

Σ (X(j+ɪ)/m - Xj/m)
j
=1


I g ( fs )d s

Jo


4.


^>Z

Fm


m L(n- ɪ)rj                   r

+— ∑ g (fi/n) - Vm   g(fs)ds.

n    i=1                  Jo

'----------------------------------------4/-----------------------------------------'


(24)


Ln,m,r


By the same argument used in the proof of part (i)a, Step 4 and Step 2 (respectively) Ln,m,r =
op (1) and Fm,r = Op (1).

We can expand En,m,r as


E ™
njn,m,r

lL ( n-ɪ)rj
m

n ⅛

i =1

^n = 11 1 {Xj∕n-Xi∕nn}n (X(j + 1)/n   Xj/n) _ ( f )

I                   n— ɪ d                                          g iEl

          j=1=1 1 {Xj∕n-Xi∕nn}                       )

l ( n-1)rj
m

n ⅛

i =1

<-

f∑n n-ɪ 1flv    2 2n ;2 n f( j+1) /n ( Xs X X1/n )√g^(7s)d Wɪ s

j = 1 {Xj∕n-Xi∕nn}     j/n          s      jn         sj     ɪ,s

I                         ^n—ɪ1                                             I

                     2~=i=i 1 {χj∕n-Xi∕nn}

__Z
>z

p n,m,r


24




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