Testing for One-Factor Models versus Stochastic Volatility Models

4.

—n∏

^

”                          S/                         ~

An,r

^{l( n-1})^rj                               „

(X(  (X(j+1)/n ^- Xj/n)^{2 -    σ2(}Xs^)dS

j=1                           ^j°

^--------------------------------------------s/---------------------------
Bn,r

1    ^{L (} n- ¹) r^j

+^—n  Σ ^σ2(Xi/n) - √n J_j ^σ2(Xs)dS.

×------—----------s/-------------------S

(18)

-/y~-
C n,r

The proof of the statement is based on the four steps below.

Step 1: An,r ^-→ ^mn (o^, 2 ʃɪ ^σ4(a) LX^a)^da) .

Step 2: B_n,_r -→ MN (θ, 2 ʃɪ σ ⁴( a ) L_χ (1 ,a )d a ) .

Step 3: Let < A_n,B_n >_r define the discretized quadratic covariation process.

plim n→∞

< An, Bn >r — 2 /∞ σ⁴(a) L^χ()2 da = 0.

∞      Lχ (1 ,a )

Step 4: C_n,_r = o_P(1).

Proof of Step 1: First note that using Ito’s formula

_ 1 _v     Vj =^{1 1} {∖^χ₃∕n-^χi∕n∖<ξn}ⁿ (X^{(j + 1)/n   Xj/n})      2_(X ʌ

An^,r = √n         ----------Vn-1 1--^{σ (}Xi/n)

ⁿ     i=1   ∖            ²=3=1 ¹ {∖χ_j∙∕n-χ_i∕n∖<ξn}

^{l (} n-¹)^rj

= 4= ∑
√n ÷-'

i=1

^

∑ⁿ=1¹¹ {∖χj_7n-χ^_n∖<ξn}²ⁿʃj/«+)/ (X^s   Xj/n) ^σ(X^{s)dw1 ,s}

∑n-1 d

j =^{1 1} {∖χ_j∙∕n-χ_i∕n∖<ξn}

Gn,r

4.

^{l (} "_y1)^rj / V j=-; ɪ {∖χ,∕n-χ,∕n∖<_tn}² n S_i^l'N^/n ( Xs - Xj/n ) μ ( Xs )d s

/ V   I                   ^vn-1₁

i =1   ∖                   ²=i=1 ¹ {∖χ_j∙∕n-χ_i∕n∖<ξn}

S/

H n,r

20