Testing for One-Factor Models versus Stochastic Volatility Models

by

Thus an estimator of

σ⁴(a)

Lx(r, a)(Lχ(1 ,a) - Lχ(r, a))d
Lx (1 ,a )            a

(9)

i.e. of the quantity resulting in Theorem 1 part (i)a, is given by

∆2

∆1

LX,n(r,a) (LX,n(1 ,a) - LX,n(r,a))
σ4 ( a )-----------^k ---------------------ʌd a,

L X,n (1 ,a )

(10)

where

Σi=1 ¹ {∣¾_n-a∣<ξn}n² (^x(ⁱ+¹)/ⁿ   X^i/n)

∑n-1
i=1

1{∣x_i∕_n-a∣<ξn}

In order to implement the estimator in (10), we need to choose the interval of integration, ∆ =

(∆1, ∆2)∙ Now, if we choose ∆ too small, then we may run the risk of getting an inconsistent
estimator of the term in (9). On the other hand, if we choose ∆ too large, then for some a ∈ ∆,
LX,_n(r, a) and LX,_n(1, a) would be very close to zero, and the estimator in (10) will result in a ratio
of two terms approaching zero.

Of course, when computing (10) we can exclude all a ∈ ∆ for which, say, L_x,_n(1, a) ≤ δ_n, where
δ → 0 as n →∞∙ However, devicing a data-driven procedure for choosing δ is not an easy task.

In order to avoid this problem, we instead propose below an upper bound for the critical values of

the limiting distribution in Theorem 1, parts (i)a and (i)b.

In fact, note that almost surely,

2 ^∞ σ⁴(a)
-∞

Lχ(r, a)(Lχ(1, a) - Lχ(r, a)) d

Lx (1 ,a )            a

r

σ⁴(a)L_x(r, a)da ≡ 2

0

σ⁴(Xs)ds,

where the last equality above follows from Lemma 3 in Bandi and Phillips (2003).

Now, Barndorff-Nielsen and Shephard (2002) have shown that

^l (^n-1)".                                       „

^  (X(i +1)/n ^{- X}i/n) ^-→ ^{σ σ}4^ds,

0

(11)

¹⁰When m = n and r =1, the statistic converges to zero in probability.