Testing for One-Factor Models versus Stochastic Volatility Models

Zn,m,r

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 ^{- X}j/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) L_n,_m,_r =
o_p (1) and F_m,_r = Op (1).

We can expand E_n,_m,_r as

E ™ njn,m,r l— L ( n-ɪ)rj m n ⅛ i =1	^n = 11 1 {∖Xj∕n-Xi∕n∖<ξn}n (X(j + 1)/n   Xj/n) _ ( f ) I                   n— ɪ d                                          g ∖i⅛∣El ∖           j=1=1 1 {∖Xj∕n-Xi∕n∖<ξn}                       )
— 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∕n∖<ξn}     j/n          s      j∕n         sj     ɪ,s I                         ^n—ɪ1                                             I ∖                      2~=i=i 1 {∖χj∕n-Xi∕n∖<ξn} __Z >z p n,m,r

24