Testing for One-Factor Models versus Stochastic Volatility Models

plim n→∞

< ^Gn >r

- 2 ∞ σ⁴ ( a )

-∞

L X ( r, a )²
Lχ (1, a)

Thus, by the same argument as in the proof of Theorem 3 in Bandi and Phillips (2003), the
statement in Step 1 follows.

Proof of Step 2: It follows from Theorem 1 in Barndorff-Nielsen and Shephard (2004a).

Proof of Step 3: The discretized covariation process < A_n,B_n >_r,

< ^An, ^Bn >

L(n-1’rj [(n-1’rj

i =1     j =1

'¹ { ∣ .χ_iin-X∕,∣ <<n} ^J/N /n ( ' - ^χ∕n) σ(^χ. )^dW ² '
∑ⁿ=^{1 1} { ∣ ^χ₃∕n-^χ_i∕n ∣ <ξn}

L(n-1’rj L(n-1’rj

= 2 n ∑  ∑

i =1     j =1

¹ { ∣ χ_j∕n-χ_i∕n ∣ <ξn}^{σ 4( Xj/n + 0a}-^s-⁽¹⁾⁾
V^n-1 m,          ,   ,

²=1=1 ¹ {∣ Xj∕n-Xi∕n ∣ <ξn}

(19)

= 2

¹ {∖Xu-Xa∖<ξn}^{σ 4( X}U ^{)d U}
Jθ^{1 1} {∖Xu-Xa∖<ξn} ^{d U}

d a + 0a.s. (1)

[∞ J ∣^,∞ 1 u a <ξ. σ⁴l∏lχ(∣: l∏<∖u

J-oo -J-OO J-∞ ¹ {∖u-a∖<ξn}LX⁽¹ ,^u)du

Lx(r, a)da + o_a.s. (1),

where the 2 (instead of 4) on right hand side of (19) comes from Lemma 5.3 in Jacod and
Protter (1998). Along the lines of Bandi and Phillips (2001, 2003), by the change of variable

u-a

^ξn

= z,