Testing for One-Factor Models versus Stochastic Volatility Models



1 I<’n-1>rj (∑. =:; 1 {⅛∕n-,Y,∕n<M" U/n (σ2(ɪ.) - σ2(Xi/n)) d«

n-1

j = 1 1 { Xj∙∕n~X≈∕nn}

~^∕""~

Dn,r

Now, given Lemma 1, Dn,r = oa.s. (1), provided that n1 /2+εξn 0, as n → ∞. It is immediate
to see that
Hn,r is of a smaller order of probability than Gn,r.

Let < Gn >r denote the discretized quadratic variation process of Fn,r. By a similar argument
as in Bandi and Phillips (2003, pp.271-272),
we have that

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 < An,Bn >r,


< An, Bn >


L(n-1’rj [(n-1’rj


i =1     j =1


'1 { iin-X∕, <<n} J/N /n ( ' - χ∕n) σ(χ. )dW 2 '
n=1 1 { χ3∕n-χi∕n n}


L(n-1’rj L(n-1’rj

= 2 n ∑  ∑

i =1     j =1


1 { χj∕ni∕n n}σ 4( Xj/n + 0a-s-(1))
V
n-1 m,          ,   ,

2=1=1 1 { Xj∕n-Xi∕n n}


(19)


= 2

1 {Xu-Xan}σ 4( XU )d U
Jθ1 1 {Xu-Xan} d U


d a + 0a.s. (1)


[∞ J ,1 u a <ξ. σ4l∏lχ(: l∏<u

J-oo -J-OO J-∞ 1 {u-an}LX(1 ,u)du


Lx(r, a)da + oa.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,


< An, Bn >

21



More intriguing information

1. A COMPARATIVE STUDY OF ALTERNATIVE ECONOMETRIC PACKAGES: AN APPLICATION TO ITALIAN DEPOSIT INTEREST RATES
2. On the Desirability of Taxing Charitable Contributions
3. Expectations, money, and the forecasting of inflation
4. The name is absent
5. LABOR POLICY AND THE OVER-ALL ECONOMY
6. Are combination forecasts of S&P 500 volatility statistically superior?
7. THE WELFARE EFFECTS OF CONSUMING A CANCER PREVENTION DIET
8. A Consistent Nonparametric Test for Causality in Quantile
9. Towards a Strategy for Improving Agricultural Inputs Markets in Africa
10. Quality Enhancement for E-Learning Courses: The Role of Student Feedback