Testing for One-Factor Models versus Stochastic Volatility Models

Thus,

n Σ j =1 LX (¹ ,Xj/n ) ( g ( ^fj/n ) g ( Xj/n ))
supi LX (1 ,Xi/n)

+ oP (1)

L ( n- 1) r^j

¹ Σ

n

i=1

∑ⁿ-2{lχ_jz^-χ_iiil_iε,}M√jΛ+∑l(gl^fs2⊂^g.(Xs2)^dd

V^n-11,.            . .

j=1=1 ¹ {∣x₃∙∕_n-X_i∕_n∣<ξ_n}

n Σ j=1 LX ⁽¹ ,Xj/n ) ( g ( ^fj/n )   g ( Xj/n ))

inf i LX (1 ,Xi/n )

(29)

Note that the numerator in the lower and upper bounds of the inequality in (29) approaches
zero if and only if LX (1, a) -Lf(1,a) = 0 (almost surely) for all a ∈ A, with A having non-zero
Lebesgue measure, or in the multidimensional case, if Xs and fs have the same occupation
density, which is indeed ruled out under the alternative hypothesis. Therefore, (1 /√m)Z_n,_m,r
consists of the sum of two nondegenerate random variables which do not cancel out each
other. Thus, Z_n,_m,_r diverges at rate √m with probability approaching one.

Therefore, the statement follows.

A.4 Proof of Corollary 1

It follows directly from Theorem 1, part (i)c.

A.5 Proof of Proposition 1

(a) From equation (12), it follows that, for r1 <r2 < ... < rJ,

m,r1

d(^s )

m,r2

. ∕T(^s )
dm,rJ

d

-→

MN

0,

2 2 for¹ σ⁴(Xs)ds

2 for¹ σ⁴(Xs)ds

.

.
.

∖ 2 for¹ σ⁴(Xs)ds

2f₀r¹ σ⁴(Xs)ds ... 2f₀r¹ σ⁴(Xs)ds ∖ ʌ
2f₀r² σ⁴(Xs)ds ... 2f₀r² σ⁴(Xs)ds

.                       ..                          .

.                                            ^..

2 for² σ ⁴( Xs )d s ... 2 fo^rJ σ ⁴( Xs )d s ))